MMA 409 Differential Geometry pdf notes.pdf

INTRODUCTION TO
DIFFERENTIAL GEOMETRY
Joel W. Robbin
UW Madison
Dietmar A. Salamon
ETH Z¨urich
27 October 2024

ii

Preface
These are notes for the lecture course“Differential Geometry I”given by the
second author at ETH Z¨urich in the fall semester 2017. They are based on
a lecture course
1
given by the first author at the University of Wisconsin–
Madison in the fall semester 1983.
One can distinguishextrinsic differential geometryandintrinsic differ-
ential geometry. The former restricts attention to submanifolds of Euclidean
space while the latter studies manifolds equipped with a Riemannian metric.
The extrinsic theory is more accessible because we can visualize curves and
surfaces inR
3
, but some topics can best be handled with the intrinsic theory.
The definitions in Chapter 2 have been worded in such a way that it is easy
to read them either extrinsically or intrinsically and the subsequent chapters
are mostly (but not entirely) extrinsic. One can teach a self contained one
semester course in extrinsic differential geometry by starting with Chapter 2
and skipping the sections marked with an asterisk such as§2.8.
Here is a description of the content of the book, chapter by chapter.
Chapter 1 gives a brief historical introduction to differential geometry and
explains the extrinsic versus the intrinsic viewpoint of the subject.
2
This
chapter was not included in the lecture course at ETH.
The mathematical treatment of the field begins in earnest in Chapter 2,
which introduces the foundational concepts used in differential geometry
and topology. It begins by defining manifolds in the extrinsic setting as
smooth submanifolds of Euclidean space, and then moves on to tangent
spaces, submanifolds and embeddings, and vector fields and flows.
3
The
chapter includes an introduction to Lie groups in the extrinsic setting and a
proof of the Closed Subgroup Theorem. It then discusses vector bundles and
submersions and proves the Theorem of Frobenius. The last two sections
deal with the intrinsic setting and can be skipped at first reading.
1
Extrinsic Differential Geometry
2
It is shown in§1.3 how any topological atlas on a set induces a topology.
3
Our sign convention for the Lie bracket of vector fields is explained in§2.5.7.
iii

iv
Chapter 3 introduces the Levi-Civita connection as covariant derivatives
of vector fields along curves.
4
It continues with parallel transport, introduces
motions without sliding, twisting, and wobbling, and proves the Develop-
ment Theorem. It also characterizes the Levi-Civita connection in terms of
the Christoffel symbols. The last section introduces Riemannian metrics in
the intrinsic setting, establishes their existence, and characterizes the Levi-
Civita connection as the unique torsion-free Riemannian connection on the
tangent bundle.
Chapter 4 defines geodesics as critical points of the energy functional and
introduces the distance function defined in terms of the lengths of curves. It
then examines the exponential map, establishes the local existence of min-
imal geodesics, and proves the existence of geodesically convex neighbor-
hoods. A highlight of this chapter is the proof of the Hopf–Rinow Theorem
and of the equivalence of geodesic and metric completeness. The last section
shows how these concepts and results carry over to the intrinsic setting.
Chapter 5 introduces isometries and the Riemann curvature tensor and
proves the Generalized Theorema Egregium, which asserts that isometries
preserve geodesics, the covariant derivative, and the curvature.
Chapter 6 contains some answers to what can be viewed as the funda-
mental problem of differential geometry: When are two manifolds isometric?
The central tool for answering this question is the Cartan–Ambrose–Hicks
Theorem, which etablishes necessary and sufficient conditions for the exis-
tence of a (local) isometry between two Riemannian manifolds. The chapter
then moves on to examine flat spaces, symmetric spaces, and constant sec-
tional curvature manifolds. It also includes a discussion of manifolds with
nonpositive sectional curvature, proofs of the Cartan–Hadamard Theorem
and of Cartan’s Fixed Point Theorem, and as the main example a discussion
of the space of positive definite symmetric matrices equipped with a natural
Riemannian metric of nonpositive sectional curvature.
This is the point at which the ETH lecture course ended. However,
Chapter 6 contains some additional material such as a proof of the Bonnet–
Myers Theorem about manifolds with positive Ricci curvature, and it ends
with brief discussions of the scalar curvature and the Weyl tensor.
The logical progression of the book up to this point is linear in that
every chapter builds on the material of the previous one, and so no chapter
can be skipped except for the first. What can be skipped at first reading
are only the sections labelled with an asterisk that carry over the various
notions introduced in the extrinsic setting to the intrinsic setting.
4
The covariant derivative of a global vector field is deferred to§5.2.2.

v
Chapter 7 deals with various specific topics that are at the heart of the
subject but go beyond the scope of a one semester lecture course. It begins
with a section on conjugate points and the Morse Index Theorem, which
follows on naturally from Chapter 4 about geodesics. These results in turn
are used in the proof of continuity of the injectivity radius in the second
section. The third section builds on Chapter 5 on isometries and the Rie-
mann curvature tensor. It contains a proof of the Myers–Steenrod Theorem,
which asserts that the group of isometries is always a finite-dimensional Lie
group. The fourth section examines the special case of the isometry group of
a compact Lie group equipped with a bi-invariant Riemannian metric. The
last two sections are devoted to Donaldson’s differential geometric approach
to Lie algebra theory as explained in [17]. They build on all the previous
chapters and especially on the material in Chapter 6 about manifolds with
nonpositive sectional curvature. The fifth section establishes conditions un-
der which a convex function on a Hadamard manifold has a critical point.
The last section uses these results to show that the Killing form on a simple
Lie algebra is nondegenerate, to establish uniqueness up to conjugation of
maximal compact subgroups of the automorphism group of a semisimple Lie
algebra, and to prove Cartan’s theorem about the compact real form of a
semisimple complex Lie algebra.
The appendix contains brief discussions of some fundamental notions of
analysis such as maps and functions, normal forms, and Euclidean spaces,
that play a central role throughout this book.
We thank everyone who pointed out errors or typos in earlier versions of
this book. In particular, we thank Charel Antony and Samuel Trautwein for
many helpful comments. We also thank Daniel Grieser for his constructive
suggestions concerning the exposition.
28 August 2021 Joel W. Robbin and Dietmar A. Salamon

vi

Contents
1 What is Differential Geometry? 1
1.1 Cartography and Differential Geometry . . . . . . . . . . . . 1
1.2 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Topological Manifolds* . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Smooth Manifolds Defined* . . . . . . . . . . . . . . . . . . . 10
1.5 The Master Plan . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Foundations 15
2.1 Submanifolds of Euclidean Space . . . . . . . . . . . . . . . . 15
2.2 Tangent Spaces and Derivatives . . . . . . . . . . . . . . . . . 24
2.2.1 Tangent Space . . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.3 The Inverse Function Theorem . . . . . . . . . . . . . 31
2.3 Submanifolds and Embeddings . . . . . . . . . . . . . . . . . 34
2.4 Vector Fields and Flows . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.2 The Flow of a Vector Field . . . . . . . . . . . . . . . 41
2.4.3 The Lie Bracket . . . . . . . . . . . . . . . . . . . . . 46
2.5 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5.1 Definition and Examples . . . . . . . . . . . . . . . . . 52
2.5.2 The Lie Algebra of a Lie Group . . . . . . . . . . . . . 55
2.5.3 Lie Group Homomorphisms . . . . . . . . . . . . . . . 58
2.5.4 Closed Subgroups . . . . . . . . . . . . . . . . . . . . 62
2.5.5 Lie Groups and Diffeomorphisms . . . . . . . . . . . . 67
2.5.6 Smooth Maps and Algebra Homomorphisms . . . . . . 69
2.5.7 Vector Fields and Derivations . . . . . . . . . . . . . . 71
2.6 Vector Bundles and Submersions . . . . . . . . . . . . . . . . 72
2.6.1 Submersions . . . . . . . . . . . . . . . . . . . . . . . 72
2.6.2 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . 74
vii

viii CONTENTS
2.7 The Theorem of Frobenius . . . . . . . . . . . . . . . . . . . . 80
2.8 The Intrinsic Definition of a Manifold* . . . . . . . . . . . . . 87
2.8.1 Definition and Examples . . . . . . . . . . . . . . . . . 87
2.8.2 Smooth Maps and Diffeomorphisms . . . . . . . . . . 92
2.8.3 Tangent Spaces and Derivatives . . . . . . . . . . . . . 93
2.8.4 Submanifolds and Embeddings . . . . . . . . . . . . . 95
2.8.5 Tangent Bundle and Vector Fields . . . . . . . . . . . 97
2.8.6 Coordinate Notation . . . . . . . . . . . . . . . . . . . 99
2.9 Consequences of Paracompactness* . . . . . . . . . . . . . . . 101
2.9.1 Paracompactness . . . . . . . . . . . . . . . . . . . . . 101
2.9.2 Partitions of Unity . . . . . . . . . . . . . . . . . . . . 103
2.9.3 Embedding in Euclidean Space . . . . . . . . . . . . . 107
2.9.4 Leaves of a Foliation . . . . . . . . . . . . . . . . . . . 113
2.9.5 Principal Bundles . . . . . . . . . . . . . . . . . . . . 114
3 The Levi-Civita Connection 121
3.1 Second Fundamental Form . . . . . . . . . . . . . . . . . . . . 121
3.2 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . 127
3.3 Parallel Transport . . . . . . . . . . . . . . . . . . . . . . . . 129
3.4 The Frame Bundle . . . . . . . . . . . . . . . . . . . . . . . . 137
3.4.1 Frames of a Vector Space . . . . . . . . . . . . . . . . 137
3.4.2 The Frame Bundle . . . . . . . . . . . . . . . . . . . . 138
3.4.3 Horizontal Lifts . . . . . . . . . . . . . . . . . . . . . . 141
3.5 Motions and Developments . . . . . . . . . . . . . . . . . . . 146
3.5.1 Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.5.2 Sliding . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.5.3 Twisting and Wobbling . . . . . . . . . . . . . . . . . 151
3.5.4 Development . . . . . . . . . . . . . . . . . . . . . . . 154
3.6 Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . 160
3.7 Riemannian Metrics* . . . . . . . . . . . . . . . . . . . . . . . 166
3.7.1 Existence of Riemannian Metrics . . . . . . . . . . . . 166
3.7.2 Two Examples . . . . . . . . . . . . . . . . . . . . . . 169
3.7.3 The Levi-Civita Connection . . . . . . . . . . . . . . . 170
3.7.4 Basic Vector Fields in the Intrinsic Setting . . . . . . 173
4 Geodesics 175
4.1 Length and Energy . . . . . . . . . . . . . . . . . . . . . . . . 175
4.1.1 The Length and Energy Functionals . . . . . . . . . . 175
4.1.2 The Space of Paths . . . . . . . . . . . . . . . . . . . . 178
4.1.3 Characterization of Geodesics . . . . . . . . . . . . . . 180

CONTENTS ix
4.2 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.3 The Exponential Map . . . . . . . . . . . . . . . . . . . . . . 191
4.3.1 Geodesic Spray . . . . . . . . . . . . . . . . . . . . . . 191
4.3.2 The Exponential Map . . . . . . . . . . . . . . . . . . 192
4.3.3 Examples and Exercises . . . . . . . . . . . . . . . . . 195
4.4 Minimal Geodesics . . . . . . . . . . . . . . . . . . . . . . . . 197
4.4.1 Characterization of Minimal Geodesics . . . . . . . . . 197
4.4.2 Local Existence of Minimal Geodesics . . . . . . . . . 199
4.4.3 Examples and Exercises . . . . . . . . . . . . . . . . . 203
4.5 Convex Neighborhoods . . . . . . . . . . . . . . . . . . . . . . 205
4.6 Completeness and Hopf–Rinow . . . . . . . . . . . . . . . . . 209
4.7 Geodesics in the Intrinsic Setting* . . . . . . . . . . . . . . . 217
4.7.1 Intrinsic Distance . . . . . . . . . . . . . . . . . . . . . 217
4.7.2 Geodesics and the Levi-Civita Connection . . . . . . . 219
4.7.3 Examples and Exercises . . . . . . . . . . . . . . . . . 220
5 Curvature 223
5.1 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
5.2 The Riemann Curvature Tensor . . . . . . . . . . . . . . . . . 232
5.2.1 Definition and Gauß–Codazzi . . . . . . . . . . . . . . 232
5.2.2 Covariant Derivative of a Global Vector Field . . . . . 234
5.2.3 A Global Formula . . . . . . . . . . . . . . . . . . . . 237
5.2.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 239
5.2.5 Riemannian Metrics on Lie Groups . . . . . . . . . . . 241
5.3 Generalized Theorema Egregium . . . . . . . . . . . . . . . . 244
5.3.1 Pushforward . . . . . . . . . . . . . . . . . . . . . . . 244
5.3.2 Theorema Egregium . . . . . . . . . . . . . . . . . . . 245
5.3.3 Gaußian Curvature . . . . . . . . . . . . . . . . . . . . 249
5.4 Curvature in Local Coordinates* . . . . . . . . . . . . . . . . 254
6 Geometry and Topology 257
6.1 The Cartan–Ambrose–Hicks Theorem . . . . . . . . . . . . . 257
6.1.1 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . 257
6.1.2 The Global C-A-H Theorem . . . . . . . . . . . . . . . 259
6.1.3 The Local C-A-H Theorem . . . . . . . . . . . . . . . 265
6.2 Flat Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
6.3 Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 272
6.3.1 Symmetric Spaces . . . . . . . . . . . . . . . . . . . . 273
6.3.2 Covariant Derivative of the Curvature . . . . . . . . . 275
6.3.3 Examples and Exercises . . . . . . . . . . . . . . . . . 278

x CONTENTS
6.4 Constant Curvature . . . . . . . . . . . . . . . . . . . . . . . 280
6.4.1 Sectional Curvature . . . . . . . . . . . . . . . . . . . 280
6.4.2 Constant Sectional Curvature . . . . . . . . . . . . . . 281
6.4.3 Hyperbolic Space . . . . . . . . . . . . . . . . . . . . . 286
6.5 Nonpositive Sectional Curvature . . . . . . . . . . . . . . . . 293
6.5.1 The Cartan–Hadamard Theorem . . . . . . . . . . . . 293
6.5.2 Cartan’s Fixed Point Theorem . . . . . . . . . . . . . 299
6.5.3 Positive Definite Symmetric Matrices . . . . . . . . . . 302
6.6 Positive Ricci Curvature* . . . . . . . . . . . . . . . . . . . . 309
6.7 Scalar Curvature* . . . . . . . . . . . . . . . . . . . . . . . . 313
6.8 The Weyl Tensor* . . . . . . . . . . . . . . . . . . . . . . . . 319
7 Topics in Geometry 327
7.1 Conjugate Points and the Morse Index* . . . . . . . . . . . . 328
7.2 The Injectivity Radius* . . . . . . . . . . . . . . . . . . . . . 338
7.3 The Group of Isometries* . . . . . . . . . . . . . . . . . . . . 342
7.3.1 The Myers–Steenrod Theorem . . . . . . . . . . . . . 342
7.3.2 The Topology on the Space of Isometries . . . . . . . 344
7.3.3 Killing Vector Fields . . . . . . . . . . . . . . . . . . . 347
7.3.4 Proof of the Myers–Steenrod Theorem . . . . . . . . . 351
7.3.5 Examples and Exercises . . . . . . . . . . . . . . . . . 360
7.4 Isometries of Compact Lie Groups* . . . . . . . . . . . . . . . 362
7.5 Convex Functions on Hadamard Manifolds* . . . . . . . . . . 368
7.5.1 Convex Functions and The Sphere at Infinity . . . . . 368
7.5.2 Inner Products and Weighted Flags . . . . . . . . . . 377
7.5.3 Lengths of Vectors . . . . . . . . . . . . . . . . . . . . 381
7.6 Semisimple Lie Algebras* . . . . . . . . . . . . . . . . . . . . 391
7.6.1 Symmetric Inner Products . . . . . . . . . . . . . . . . 391
7.6.2 Simple Lie Algebras . . . . . . . . . . . . . . . . . . . 394
7.6.3 Semisimple Lie Algebras . . . . . . . . . . . . . . . . . 399
7.6.4 Complex Lie Algebras . . . . . . . . . . . . . . . . . . 404
A Notes 411
A.1 Maps and Functions . . . . . . . . . . . . . . . . . . . . . . . 411
A.2 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
A.3 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 414
References 417
Index 423

Chapter 1
What is Differential
Geometry?
This preparatory chapter contains a brief historical introduction to the sub-
ject of differential geometry (§1.1), explains the concept of a coordinate
chart (§1.2), discusses topological manifolds and shows how an atlas on a
set determines a topology (§1.3), introduces the notion of a smooth structure
(§1.4), and outlines the master plan for this book (§1.5).
1.1 Cartography and Differential Geometry
Carl Friedrich Gauß (1777-1855) is the father of differential geometry. He
was (among many other things) a cartographer and many terms in modern
differential geometry (chart, atlas, map, coordinate system, geodesic, etc.)
reflect these origins. He was led to hisTheorema Egregium(see 5.3.1) by
the question of whether it is possible to draw an accurate map of a portion
of our planet. Let us begin by discussing a mathematical formulation of this
problem.
Consider the two-dimensional sphereS
2
sitting in the three-dimensional
Euclidean spaceR
3
. It is cut out by the equation
x
2
+y
2
+z
2
= 1.
A map of a small regionU⊂S
2
is represented mathematically by a one-to-
one correspondence with a small region in the planez= 0. In this book we
will represent this with the notationϕ:U→ϕ(U)⊂R
2
and call such an
object achartor asystem of local coordinates(see Figure 1.1).
1

2 CHAPTER 1. WHAT IS DIFFERENTIAL GEOMETRY?U φ
Figure 1.1: A chart.
What does it mean thatϕis an “accurate” map? Ideally the user would
want to use the map to compute the length of a curve inS
2
. The length of
a curveγconnecting two pointsp, q∈S
2
is given by the formula
L(γ) =
Z
1
0
|˙γ(t)|dt, γ(0) =p, γ(1) =q,
so the user will want the chartϕto satisfyL(γ) =L(ϕ◦γ) for all curvesγ.
It is a consequence of theTheorema Egregiumthat there is no such chart.
Perhaps the user of such a map will be content to use the map to plot
the shortest path between two pointspandqinU. This path is called a
geodesicand is denoted byγpq. It satisfiesL(γpq) =dU(p, q), where
dU(p, q) = inf{L(γ)|γ(t)∈U, γ(0) =p, γ(1) =q}
so our less demanding user will be content if the chartϕsatisfies
dU(p, q) =dE(ϕ(p), ϕ(q)),
wheredE(ϕ(p), ϕ(q)) is the length of the shortest path in the plane. It is
also a consequence of theTheorema Egregiumthat there is no such chart.
Now suppose our user is content to have a map which makes it easy to
navigate close to the shortest path connecting two points. Ideally the user
would use a straight edge, magnetic compass, and protractor to do this.
S/he would draw a straight line on the map connectingpandqand steer a
course which maintains a constant angle (on the map) between the course
and meridians. Thiscanbe done by the method of stereographic projection.
This chart isconformal(which means that it preserves angles). According
to Wikipedia stereographic projection was known to the ancient Greeks
and a map using stereographic projection was constructed in the early 16th
century. Exercises 3.7.5, 3.7.12, and 6.4.22 use stereographic projection; the
latter exercise deals with thePoincar´e modelof the hyperbolic plane. The
hyperbolic plane provides a counterexample to Euclid’s Parallel Postulate.

1.1. CARTOGRAPHY AND DIFFERENTIAL GEOMETRY 3
n
p
ϕ(p)
Figure 1.2: Stereographic Projection.
Exercise 1.1.1.It is more or less obvious that for any surfaceM⊂R
3
there is a unique shortest path inMconnecting two points if they are
sufficiently close. (This will be proved in Theorem 4.5.3.) This shortest
path is called theminimal geodesicconnectingpandq. Use this fact to
prove that the minimal geodesic joining two pointspandqinS
2
is an arc
of the great circle throughpandq. (This is the intersection of the sphere
with the plane throughp,q, and the center of the sphere.) Also prove that
the minimal geodesic connecting two points in a plane is the straight line
segment connecting them.Hint:Both a great circle in a sphere and a line
in a plane are preserved by a reflection. (See also Exercise 4.2.5 below.)
Exercise 1.1.2.Stereographic projection is defined by the condition that
forp∈S
2
\nthe pointϕ(p) lies in thexy-planez= 0 and the three
pointsn= (0,0,1),p, andϕ(p) are collinear (see Figure 1.2). Using the
formula that the cosine of the angle between two unit vectors is their inner
product prove thatϕis conformal.Hint:The plane throughp,q, andn
intersects thexy-plane in a straight line and the sphere in a circle through
n. The plane throughn,p,ϕ(p), and the center of the sphere intersects the
sphere in a meridian. A proof that stereographic projection is conformal
can be found in [27, page 248]. The proof is elementary in the sense that
it doesn’t use calculus. An elementary proof can also be found online at
http://people.reed.edu/jerry/311/stereo.pdf.
Exercise 1.1.3.It may seem fairly obvious that you can’t draw an accurate
map of a portion of the earth because the sphere is curved. However, the
cylinder
C={(x, y, z)∈R
3
|x
2
+y
2
= 1}
is also curved, but the mapψ:R
2
→Cdefined byψ(s, t) = (cos(t),sin(t), s)
preserves lengths of curves, i.e.L(ψ◦γ) =L(γ) for any curveγ: [a, b]→R
2
.
Prove this.

4 CHAPTER 1. WHAT IS DIFFERENTIAL GEOMETRY?
1.1.4. The standard notationsN,N0,Z,Q,R,
Cdenote respectively the natural numbers (= positive integers), the non-
negative integers, the integers, the rational numbers, the real numbers, and
the complex numbers. We denote the identity map of a setXbyidXand the
n×nidentity matrix by1lnor simply1l. The notationV
∗
is used for the dual
of a vector spaceV, but whenKis a field such asRorCthe notationK
∗
is sometimes used for the multiplicative groupK\ {0}. The termssmooth,
infinitely differentiable, andC
∞
are all synonymous.
1.2 Coordinates
The rest of this chapter defines the category of smooth manifolds and smooth
maps between them. Before giving the precise definitions we will introduce
some terminology and give some examples.
Definition 1.2.1.Acharton a setMis a pair(ϕ, U)whereUis a subset
ofMandϕ:U→ϕ(U)is a bijection fromUto an open setϕ(U)inR
m
.
AnatlasonMis a collection
A={(ϕα, Uα)}α∈A
of charts such that the domainsUαcoverM, i.e.M=
S
α∈A
Uα.
The idea is that ifϕ(p) = (x1(p), . . . , xm(p)) forp∈U, then the func-
tionsxiform asystem of local coordinatesdefined on the subsetUofM.
ThedimensionofMshould bemsince it takesmnumbers to uniquely spec-
ify a point ofU. We will soon impose conditions on charts (ϕ, U), however
for the moment we are assuming nothing about the mapsϕ(other than that
they are bijective).
Example 1.2.2.Every open subsetU⊂R
m
has an atlas consisting of a
single chart, namely (ϕ, U) = (idU, U) where idUdenotes the identity map
ofU.
Example 1.2.3.Assume that Ω⊂R
m
is an open set, thatMis a subset
of the productR
m
×R
n
=R
m+n
, and thath: Ω→R
n
is a continuous map
whose graph isM, i.e.
graph(h) :={(x, y)∈Ω×R
n
|y=h(x)}=M.
LetU= graph(h) =Mand letϕ(x, y) =xbe the projection ofUonto Ω.
Then the pair (ϕ, U) is a chart onM. The inverse map is given by
ϕ
−1
(x) = (x, h(x))
forx∈Ω =ϕ(U). ThusMhas again an atlas consisting of a single chart.

1.2. COORDINATES 5
Example 1.2.4.Them-sphere
S
m
=
Φ
p= (x0, . . . , xm)∈R
m+1
|x
2
0+· · ·+x
2
m= 1

has an atlas consisting of the 2m+2 chartsϕi±:Ui±→D
m
whereD
m
is the
open unit disk inR
m
,Ui±={p∈S
m
| ±xi>0}, andϕi±is the projection
which discards theith coordinate. (See Example 2.1.14 below.)
Example 1.2.5.Let
A=A
T
∈R
(m+1)×(m+1)
be a symmetric matrix and define a quadratic formF:R
m+1
→Rby
F(p) :=p
T
Ap, p= (x0, . . . , xm).
After a linear change of coordinates the functionFhas the form
F(p) =x
2
0+· · ·+x
2
k
−x
2
k+1
− · · · −x
2
r.
(Herer+ 1 is the rank of the matrixA.) The setM=F
−1
(1) has an atlas
of 2m+ 2 charts by the same construction as in Example 1.2.4, in factS
m
is the special case whereA= 1lm+1, the (m+ 1)×(m+ 1) identity matrix.
(See Example 2.1.13 below for another way to construct charts.)
Figure 1.3 enumerates the familiarquadric surfacesinR
3
. Thepara-
boloidsare examples of graphs as in Example 1.2.3 with Ω =R
2
andn= 1,
and theellipsoidand the twohyperboloidsare instances of thequadric hyper-
surfacesdefined in Example 1.2.5. The sphere is an instance of the ellipsoid
(witha=b=c= 1) and the cylinder is a limit of the ellipsoid as well as of
the elliptic hyperboloid of one sheet (asa=b= 1 andc→ ∞). The pictures
were generated by computer using the parameterizations
x=acos(t) cos(s), y=bsin(t) cos(s), z=csin(s)
for the ellipsoid,
x=acos(t) cosh(s), y=bsin(t) cosh(s), z=csinh(s),
for the elliptic hyperboloid of one sheet, and
x=acos(t) sinh(s), y=bsin(t) sinh(s), z=±ccosh(s)
for the elliptic hyperboloid of two sheets. These quadric surfaces will be
often used in the sequel to illustrate important concepts.

6 CHAPTER 1. WHAT IS DIFFERENTIAL GEOMETRY?
Figure 1.3: Quadric Surfaces.
Unit Sphere x
2
+y
2
+z
2
= 1
Ellipsoid
x
2
a
2
+
y
2
b
2
+
z
2
c
2
= 1 Cylinder x
2
+y
2
= 1
Elliptic
Hyperboloid
x
2
a
2
+
y
2
b
2
−
z
2
c
2
= 1
(of one sheet)
Elliptic
Hyperboloid
x
2
a
2
+
y
2
b
2
−
z
2
c
2
=−1
(of two sheets)
Hyperbolic
Paraboloid z=
x
2
a
2
−
y
2
b
2
Elliptic
Paraboloid z=
x
2
a
2
+
y
2
b
2

1.2. COORDINATES 7
In the following two examplesKdenotes either the fieldRof real numbers
or the fieldCof complex numbers,K
∗
:={λ∈K|λ̸= 0}denotes the
corresponding multiplicative group, andVdenotes a vector space overK.
Example 1.2.6.Theprojective spaceofVis the set of lines (through
the origin) inV. In other words,
P(V) ={ℓ⊂V|ℓis a 1-dimensionalK-linear subspace}
WhenK=RandV=R
m+1
this is denoted byRP
m
and whenK=C
andV=C
m+1
this is denoted byCP
m
. For our purposes we can identify the
spacesC
m+1
andR
2m+2
but the projective spacesCP
m
andRP
2m
are very
different. The various linesℓ∈P(V) intersect in the origin, however, after
the identificationP(V) ={[v]|v∈V\ {0}}with [v] :=K
∗
v=Kv\ {0}the
elements ofP(V) become disjoint, i.e.P(V) is the set of equivalence classes
of an equivalence relation on the open setV\ {0}. Assume thatV=K
m+1
and define an atlas onP(V) as follows. For each integeri= 0,1, . . . , m
defineUi:={[v]|v= (x0, . . . , xm), xi̸= 0}and defineϕi:Ui→K
m
by
ϕi([v]) =
`
x0
xi
, . . . ,
xi−1
xi
,
xi+1
xi
. . . ,
xm
xi
´
.
This atlas consists ofm+ 1 charts.
Example 1.2.7.For each positive integerkthe set
Gk(V) :={ℓ⊂V|ℓis ak-dimensionalK-linear subspace}
is called theGrassmann manifoldofk-planes inV. ThusG1(V) =P(V).
Assume thatV=K
n
and define an atlas onGk(V) as follows. Lete1, . . . , en
be the standard basis forK
n
, i.e.eiis theith column of then×nidentity
matrix 1ln. Each partition{1,2, . . . , n}=I∪J,I={i1<· · ·< ik},
J={j1<· · ·< jn−k}of the firstnnatural numbers determines a direct
sum decompositionK
n
=V=VI⊕VJvia the formulas
VI=Kei1
+· · ·+Keik
, V J=Kej1
+· · ·+Kejn−k
.
LetUIdenote the set of allk-planesℓ∈Gk(V) which are transverse toVJ,
i.e. such thatℓ∩VJ={0}. The elements ofUIare precisely thosek-planes
of the formℓ= graph(A), whereA:VI→VJis a linear map. Define the
mapϕI:UI→K
k×(n−k)
by the formula
ϕI(ℓ) = (ars), Ae ir=
n−k
X
s=1
arsejs, r= 1, . . . , k.
Exercise:Prove that the set of all pairs (ϕI, UI) asIranges over the subsets
of{1, . . . , n}of cardinalitykform an atlas.

8 CHAPTER 1. WHAT IS DIFFERENTIAL GEOMETRY?
1.3 Topological Manifolds*
Definition 1.3.1.Atopological manifoldis a topological spaceMsuch
that each pointp∈Mhas an open neighborhoodUwhich is homeomorphic
to an open subset of a Euclidean space.
Brouwer’sInvariance of Domain Theoremasserts that, whenU⊂R
m
andV⊂R
n
are nonempty open sets andϕ:U→Vis a homeomorphism,
thenm=n. This means that ifMis a connected topological manifold
and some point ofMhas a neighborhood homeomorphic to an open subset
ofR
m
, then every point ofMhas a neighborhood homeomorphic to an open
subset of that sameR
m
. In this case we say thatMhasdimensionmor
ism-dimensional or is anm-manifold. Brouwer’s theorem is fairly difficult
(see [24, p. 126] for example) but ifϕis a diffeomorphism, then the result is
an easy consequence of the invariance of the rank in linear algebra and the
chain rule. (See equation (1.4.1) below.)
By definition, a topologicalm-manifoldMadmits an atlas where every
chart (ϕ, U) of the atlas is a homeomorphismϕ:U→ϕ(U) from an open
setU⊂Mto an open setϕ(U)⊂R
m
. The following definition and lemma
explain when a given atlas determines a topology onM.
Definition 1.3.2.LetMbe a set. Two charts(ϕ1, U1)and(ϕ2, U2)onM
are said to betopologically compatibleiffϕ1(U1∩U2)andϕ2(U1∩U2)
are open subsets ofR
m
and thetransition map
ϕ21:=ϕ2◦ϕ
−1
1
:ϕ1(U1∩U2)→ϕ2(U1∩U2)
is a homeomorphism. An atlas is said to be atopological atlasiff any two
charts in this atlas are topologically compatible.
Lemma 1.3.3.LetA={(ϕα, Uα)}α∈Abe an atlas on a setM.
(i)IfAis a topological atlas, then the collection
U:=
æ
U⊂M

ϕα(U∩Uα)is an open subset ofR
m
for everyα∈A
œ
(1.3.1)
is a topology onM, and with this topology eachUαis an open subset ofM
and eachϕαis a homeomorphism. ThusMis a topological manifold with
the topology(1.3.1).
(ii)IfMis a topological manifold and eachUαis an open set and eachϕα
is a homeomorphism, thenAis a topological atlas and the topology(1.3.1)
coincides with the topology ofM.
Proof.Exercise.

1.3. TOPOLOGICAL MANIFOLDS* 9
IfMis a topological manifold, then the collection of all charts (U, ϕ)
onMsuch thatUis open andϕis a homeomorphism is a topological atlas.
It is the uniquemaximal topological atlasin the sense that it contains
every other topological atlas as in part (ii) of Lemma 1.3.3. However, we will
often consider smaller, even finite, topological atlases, and by Lemma 1.3.3
each of these determines the topology ofM.
Exercise 1.3.4.Show that the atlas in each example in§1.2 is a topological
atlas. Conclude that each of these examples is a topological manifold.
Any subsetS⊂Xof a topological spaceXinherits a topology fromX,
called therelative topologyofS. A subsetU0⊂Sis calledrelatively
openinS(orS-open) iff there is an open setU⊂Xsuch thatU0=U∩S.
A subsetA0⊂Sis calledrelatively closed(orS-closed) iff there is a
closed setA⊂Xsuch thatA0=A∩S. The relative topology onSis the
coarsest topology such that the inclusion mapS→Xis continuous.
Exercise 1.3.5.Show that the relative topology satisfies the axioms of a
topology (i.e. arbitrary unions and finite intersections ofS-open sets areS-
open, and the empty set andSitself areS-open). Show that the complement
of anS-open set inSisS-closed and vice versa.
Exercise 1.3.6.Each of the sets defined in Examples 1.2.2, 1.2.3, 1.2.4,
and 1.2.5 is a subset of some Euclidean spaceR
k
. Show that the topology
in Exercise 1.3.4 is the relative topology inherited from the topology ofR
k
.
The topology onR
k
is of course the metric topology defined by the distance
functiond(x, y) =|x−y|.
If∼is an equivalence relation on a topological spaceX, then thequo-
tient spaceY:=X/∼:={[x]|x∈X}is the set of all equivalence classes
[x] :={x
′
∈X|x
′
∼x}. The mapπ:X→Ydefined byπ(x) = [x] will
be called theobvious projection. The quotient space inherits thequo-
tient topologyfromY. Namely, a setV⊂Yis open in this topology
iff the preimageπ
−1
(V) is open inX. This topology is the finest topol-
ogy onYsuch that projectionπ:X→Yis continuous. Since the oper-
ationV7→π
−1
(V) commutes with arbitrary unions and intersections the
quotient topology obviously satisfies the axioms of a topology.
Exercise 1.3.7.Show that the topology on the projective spaceP(V) in
Example 1.2.6 determined by the atlas is the quotient topology inherited
from the open setV\ {0}. Express the Grassmann manifoldGk(V) in
Example 1.2.7 as a quotient space and show that the topology determined
by the atlas is the quotient topology. (Recall that in both examplesV=K
n
withK=RorK=C.)

10 CHAPTER 1. WHAT IS DIFFERENTIAL GEOMETRY?
1.4 Smooth Manifolds Defined*
LetU⊂R
n
andV⊂R
m
be open sets. A mapf:U→Vis calledsmooth
iff it is infinitely differentiable, i.e. iff all its partial derivatives
∂
α
f=
∂
α1+···+αn
f
∂x
α1
1
· · ·∂x
αn
n
, α= (α1, . . . , αn)∈N
n
0,
exist and are continuous. In later chapters we will sometimes writeC
∞
(U, V)
for the set of smooth maps fromUtoV.
Definition 1.4.1.LetU⊂R
n
andV⊂R
m
be open sets. For a smooth
mapf= (f1, . . . , fm) :U→Vand a pointx∈Uthederivative offatx
is the linear mapdf(x) :R
n
→R
m
defined by
df(x)ξ:=
d
dt

t=0
f(x+tξ) = lim
t→0
f(x+tξ)−f(x)
t
, ξ∈R
n
.
This linear map is represented by theJacobian matrixoffatxwhich
will also be denoted by
df(x) :=






∂f1
∂x1
(x)· · ·
∂f1
∂xn
(x)
.
.
.
.
.
.
∂fm
∂x1
(x)· · ·
∂fm
∂xn
(x)






∈R
m×n
.
Note that we use the same notation for the Jacobian matrix and the corre-
sponding linear map fromR
n
toR
m
.
The derivative satisfies thechain rule. Namely, ifU⊂R
n
,V⊂R
m
,
andW⊂R
ℓ
are open sets andf:U→Vandg:V→Ware smooth map,
theng◦f:U→Wis smooth and
d(g◦f)(x) =dg(f(x))◦df(x) :R
n
→R
ℓ
(1.4.1)
for everyx∈U. Moreover the identity map idU:U→Uis always smooth
and its derivative at every point is the identity map ofR
n
. This implies that,
iff:U→Vis adiffeomorphism(i.e.fis bijective andfandf
−1
are both
smooth), then its derivative at every point is an invertible linear map. This
is why the Invariance of Domain Theorem (discussed after Definition 1.3.1)
is easy for diffeomorphisms: iff:U→Vis a diffeomorphism, then the
Jacobian matrixdf(x)∈R
m×n
is invertible for everyx∈Uand som=n.
The Inverse Function Theorem (see Theorem A.2.2 in Appendix A.2) is a
kind of converse.

1.4. SMOOTH MANIFOLDS DEFINED* 11
Definition 1.4.2(Smooth manifold).LetMbe a set. AchartonM
is a pair (ϕ, U) whereU⊂Mandϕis a bijection fromUto an open sub-
setϕ(U)⊂R
m
of some Euclidean space. Two charts(ϕ1, U1)and (ϕ2, U2)
are said to besmoothly compatibleiffϕ1(U1∩U2)andϕ2(U1∩U2)are
both open inR
m
and thetransition map
ϕ21=ϕ2◦ϕ
−1
1
:ϕ1(U1∩U2)→ϕ2(U1∩U2) (1.4.2)
is a diffeomorphism. Asmooth atlasonMis a collectionAof charts
onMany two of which are smoothly compatible and such that the setsU,
as(ϕ, U)ranges over the elements ofA, coverM(i.e. for everyp∈M
there is a chart(ϕ, U)∈Awithp∈U). Amaximal smooth atlasis an
atlas which contains every chart which is smoothly compatible with each of
its members. Asmooth manifoldis a pair consisting of a setMand a
maximal smooth atlasAonM.
Lemma 1.4.3.IfAis a smooth atlas, then so is the collectionAof all
charts smoothly compatible with each member ofA. The smooth atlasAis
obviously maximal. In other words, every smooth atlas extends uniquely to
a maximal smooth atlas.
Proof.Let (ϕ1, U1) and (ϕ2, U2) be charts inAand letx∈ϕ1(U1∩U2).
Choose a chart (ϕ, U)∈Asuch thatϕ
−1
1
(x)∈U. Thenϕ1(U∩U1∩U2) is
an open neighborhood ofxinR
m
and the transition maps
ϕ◦ϕ
−1
1
:ϕ1(U∩U1∩U2)→ϕ(U∩U1∩U2),
ϕ2◦ϕ
−1
:ϕ(U∩U1∩U2)→ϕ2(U∩U1∩U2)
are smooth by definition ofA. Hence so is their composition. This shows
that the mapϕ2◦ϕ
−1
1
:ϕ1(U1∩U2)→ϕ2(U1∩U2) is smooth nearx. Sincex
was chosen arbitrary, this map is smooth. Apply the same argument to its
inverse to deduce that it is a diffeomorphism. ThusAis a smooth atlas.
Definitions 1.4.2 and 1.3.2 aremutatis mutandisthe same, so every
smooth atlas on a setMisa fortioria topological atlas, i.e. every smooth
manifold is a topological manifold. (See Lemma 1.3.3.) Moreover the defi-
nitions are worded in such a way that it is obvious that every smooth map
is continuous.
Exercise 1.4.4.Show that each of the atlases from the examples in§1.2 is a
smooth atlas. (You must show that the transition maps from Exercise 1.3.4
are smooth.)

12 CHAPTER 1. WHAT IS DIFFERENTIAL GEOMETRY?
WhenAis a smooth atlas on a topological manifoldMone says thatA
is asmooth structureon the (topological) manifoldMiffA⊂B,
whereBis the maximal topological atlas onM.When no confusion can
result we generally drop the notation for the maximal smooth atlas as in the
following exercise.
Exercise 1.4.5.Define the notion of a continuous map between topological
manifolds and of a smooth map between smooth manifolds via continuity,
respectively smoothness, in local coordinates. LetM,N, andPbe smooth
manifolds andf:M→Nandg:N→Pbe smooth maps. Prove that
the identity map idMis smooth and that the compositiong◦f:M→P
is a smooth map. (This is of course an easy consequence of the chain
rule (1.4.1).)
Remark 1.4.6.It is easy to see that a topological manifold can have
many distinct smooth structures. For example,{(idR,R)}and{(ϕ,R)}
whereϕ(x) =x
3
are atlases on the real numbers which extend to distinct
smooth structures but determine the same topology. However these two
manifolds are diffeomorphic via the mapx7→x
1/3
. In the 1950’s it was
proved that there are smooth manifolds which are homeomorphic but not dif-
feomorphic and that there are topological manifolds which admit no smooth
structure. In the 1980’s it was proved in dimensionm= 4 that there are
uncountably many smooth manifolds that are all homeomorphic toR
4
but
no two of them are diffeomorphic to each other. These theorems are very
surprising and very deep.
A collection of sets and maps between them is called acategoryiff the
collection of maps contains the identity map of every set in the collection and
the composition of any two maps in the collection is also in the collection.
The sets are called theobjectsof the category and the maps are called the
morphismsof the category. An invertible morphism whose inverse is also in
the category is called anisomorphism. Some examples are the category of
allsets and maps, the category of topological spaces and continuous maps
(the isomorphisms are the homeomorphisms), the category of topological
manifolds and continuous maps between them, and the category of smooth
manifolds and smooth maps (the isomorphisms are the diffeomorphisms).
Each of the last three categories is a subcategory of the preceding one.
Often categories are enlarged by a kind of “gluing process”. For example,
the “global” category of smooth manifolds and smooth maps was constructed
from the “local” category of open sets in Euclidean space and smooth maps
between them via the device of charts and atlases. (The chain rule shows

1.5. THE MASTER PLAN 13
that this local category is in fact a category.) The point of Definition 1.3.2
is to show (via Lemma 1.3.3) that topological manifolds can be defined
in a manner analogous to the definition we gave for smooth manifolds in
Definition 1.4.2.
Other kinds of manifolds (and hence other kinds of geometry) are de-
fined by choosing other local categories, i.e. by imposing conditions on the
transition maps in Equation (1.4.2). For example, areal analytic manifold
is one where the transition maps are real analytic, acomplex manifoldis one
whose coordinate charts take values inC
n
and whose transition maps are
holomorphic diffeomorphisms, and asymplectic manifoldis one whose coor-
dinate charts take values inR
2n
and whose transition maps arecanonical
transformationsin the sense of classical mechanics. ThusCP
n
is a complex
manifold andRP
n
is a real analytic manifold.
1.5 The Master Plan
In studying differential geometry it is best to begin withextrinsic differential
geometrywhich is the study of the geometry of submanifolds of Euclidean
space as in Examples 1.2.3 and 1.2.5. This is because we can visualize
curves and surfaces inR
3
. However, there are a few topics in the later
chapters which require the more abstract Definition 1.4.2 even to say inter-
esting things about extrinsic geometry. There is a generalization to these
manifolds involving a structure called aRiemannian metric. We will call
this generalizationintrinsic differential geometry. Examples 1.2.6 and 1.2.7
fit into this more general definition so intrinsic differential geometry can be
used to study them.
Since an open set in Euclidean space is a smooth manifold the definition
of a submanifold of Euclidean space (see§2.1 below) ismutatis mutandis
the same as the definition of a submanifold of a manifold. The definitions
in Chapter 2 are worded in such a way that it is easy to read them either
extrinsically or intrinsically and the subsequent chapters are mostly (but not
entirely) extrinsic. Those sections which require intrinsic differential geom-
etry (or which translate extrinsic concepts into intrinsic ones) are marked
with a *.

14 CHAPTER 1. WHAT IS DIFFERENTIAL GEOMETRY?

Chapter 2
Foundations
This chapter introduces various fundamental concepts that are central to
the fields of differential geometry and differential topology. Both fields con-
cern the study of smooth manifolds and their diffeomorphisms. The chapter
begins with an introduction to submanifolds of Euclidean space and smooth
maps (§2.1), to tangent spaces and derivatives (§2.2), and to submanifolds
and embeddings (§2.3). In§2.4 we move on to vector fields and flows and
introduce the Lie bracket of two vector fields. Lie groups and their Lie alge-
bras, in the extrinsic setting, are the subject of§2.5, which includes a proof
of the Closed Subgroup Theorem. In§2.6 we introduce vector bundles over
a manifold as subbundles of a trivial bundle and in§2.7 we prove the theo-
rem of Frobenius. The last two sections of this chapter are concerned with
carrying over all these concepts from the extrinsic to the intrinsic setting
and can be skipped at first reading (§2.8 and§2.9).
2.1 Submanifolds of Euclidean Space
To carry out the Master Plan§1.5 we must (as was done in [50]) extend
the definition of smooth map to mapsf:X→Ybetween subsetsX⊂R
k
andY⊂R
ℓ
which are not necessarily open. In this case a mapf:X→Yis
calledsmoothiff for eachx0∈Xthere exists an open neighborhoodU⊂R
k
ofx0and a smooth mapF:U→R
ℓ
that agrees withfonU∩X. A
mapf:X→Yis called adiffeomorphismifffis bijective andfandf
−1
are smooth. When there exists a diffeomorphismf:X→YthenXandY
are calleddiffeomorphic.WhenXandYare open these definitions coin-
cide with the usage in§1.4.
15

16 CHAPTER 2. FOUNDATIONS
Exercise 2.1.1(Chain rule).LetX⊂R
k
,Y⊂R
ℓ
,Z⊂R
m
be arbitrary
subsets. Iff:X→Yandg:Y→Zare smooth maps, then so is the
compositiong◦f:X→Z. The identity map id :X→Xis smooth.
Exercise 2.1.2.LetE⊂R
k
be anm-dimensional linear subspace and
letv1, . . . , vmbe a basis ofE. Then the mapf:R
m
→Edefined by
f(x) :=
P
m
i=1
xiviis a diffeomorphism.
Definition 2.1.3.Letk, m∈N0. A subsetM⊂R
k
is called asmooth
m-dimensional submanifold ofR
k
iff every pointp∈Mhas an open
neighborhoodU⊂R
k
such thatU∩Mis diffeomorphic to an open sub-
setΩ⊂R
m
. A diffeomorphism
ϕ:U∩M→Ω
is called acoordinate chartofMand its inverse
ψ:=ϕ
−1
: Ω→U∩M
is called a(smooth) parametrizationofU∩M(see Figure 2.1).ψ
φMU M Ω
Figure 2.1: A coordinate chartϕ:U∩M→Ω.
In Definition 2.1.3 we have used the fact that the domain of a smooth
map can be an arbitrary subset of Euclidean space and need not be open.
The termm-manifold inR
k
is short form-dimensional submanifold ofR
k
.
In keeping with the Master Plan§1.5 we will sometimes saymanifoldrather
thansubmanifold ofR
k
to indicate that the context holds in both the in-
trinsic and extrinsic settings.
Lemma 2.1.4.IfM⊂R
k
is a nonempty smoothm-manifold, thenm≤k.
Proof.Fix an elementp0∈M, choose a coordinate chartϕ:U∩M→Ω
withp0∈Uand values in an open subset Ω⊂R
m
, and denote its in-
verse byψ:=ϕ
−1
: Ω→U∩M. ShrinkingU, if necessary, we may as-
sume thatϕextends to a smooth map Φ :U→R
m
. This extension satis-
fies Φ(ψ(x)) =ϕ(ψ(x)) =xand hencedΦ(ψ(x))dψ(x) = id :R
m
→R
m
for
allx∈Ω, by the chain rule. Hence the derivativedψ(x) :R
m
→R
k
is in-
jective for allx∈Ω, and hencem≤kbecause Ω is nonempty. This proves
Lemma 2.1.4.

2.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 17
Example 2.1.5.Consider the 2-sphere
M:=S
2
=
Φ
(x, y, z)∈R
3
|x
2
+y
2
+z
2
= 1

depicted in Figure 2.2 and letU⊂R
3
and Ω⊂R
2
be the open sets
U:=
Φ
(x, y, z)∈R
3
|z >0

,Ω :=
Φ
(x, y)∈R
2
|x
2
+y
2
<1

.
The mapϕ:U∩M→Ω given by
ϕ(x, y, z) := (x, y)
is bijective and its inverseψ:=ϕ
−1
: Ω→U∩Mis given by
ψ(x, y) = (x, y,
p
1−x
2
−y
2
).
Since bothϕandψare smooth, the mapϕis a coordinate chart onS
2
.
Similarly, we can use the open setsz <0,y >0,y <0,x >0,x <0 to cover
S
2
by six coordinate charts. HenceS
2
is a manifold. A similar argument
shows that the unit sphereS
m
⊂R
m+1
(see Example 2.1.14 below) is a
manifold for every integerm≥0.
Figure 2.2: The 2-sphere and the 2-torus.
Example 2.1.6.Let Ω⊂R
m
be an open set andh: Ω→R
k−m
be a smooth
map. Then the graph ofhis a smooth submanifold ofR
m
×R
k−m
=R
k
:
M:= graph(h) :={(x, y)|x∈Ω, y=h(x)}.
It can be covered by a single coordinate chartϕ:U∩M→Ω, where
U:= Ω×R
k−m
, ϕis the projection onto Ω, andψ:=ϕ
−1
: Ω→Uis given
byψ(x) = (x, h(x)) forx∈Ω.
Exercise 2.1.7(The casem= 0).Show that a subsetM⊂R
k
is a 0-
dimensional submanifold if and only ifMis discrete, i.e. for everyp∈M
there exists an open setU⊂R
k
such thatU∩M={p}.

18 CHAPTER 2. FOUNDATIONS
Exercise 2.1.8(The casem=k).Show that a subsetM⊂R
m
is an
m-dimensional submanifold if and only ifMis open.
Exercise 2.1.9(Products).IfMi⊂R
ki
is anmi-manifold fori= 1,2,
show thatM1×M2is an (m1+m2)-dimensional submanifold ofR
k1+k2
.
Prove that them-torusT
m
:= (S
1
)
m
is a smooth submanifold ofC
m
.
The next theorem characterizes smooth submanifolds of Euclidean space.
In particular condition (iii) will be useful in many cases for verifying the
manifold condition. We emphasize that the setsU0:=U∩Mthat appear
in Definition 2.1.3 are open subsets ofMwith respect to the relative topology
thatMinherits from the ambient spaceR
k
and that such relatively open
sets are also calledM-open (see§1.3).
Theorem 2.1.10(Manifolds).Letmandkbe integers with0≤m≤k.
LetM⊂R
k
be a set andp0∈M. Then the following are equivalent.
(i)There exists anM-open neighborhoodU0⊂Mofp0and a diffeomor-
phismϕ0:U0→Ω0onto an open setΩ0⊂R
m
.
(ii)There exist open setsU,Ω⊂R
k
and a diffeomorphismϕ:U→Ωsuch
thatp0∈Uand
ϕ(U∩M) = Ω∩(R
m
× {0})
(see Figure 2.3).
(iii)There exists an open setU⊂R
k
and a smooth mapf:U→R
k−m
such thatp0∈U, the derivativedf(p) :R
k
→R
k−m
is surjective for every
pointp∈U∩M, and
U∩M=f
−1
(0) ={p∈U|f(p) = 0}.
Moreover, if (i) holds, then the diffeomorphismϕ:U→Ωin (ii) can be
chosen such thatU∩M⊂U0andϕ(p) = (ϕ0(p),0)for everyp∈U∩M.p
U
M
φ
Ω
0
Figure 2.3: Submanifolds of Euclidean space.

2.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 19
Proof.First assume (ii) and denote the diffeomorphism in (ii) by
ϕ= (ϕ1, ϕ2, . . . , ϕk) :U→Ω⊂R
k
.
Then part (i) holds withU0:=U∩M, Ω0:={x∈R
m
|(x,0)∈Ω}, and
ϕ0:= (ϕ1, . . . , ϕm)|U0
:U0→Ω0,
and part (iii) holds withf:= (ϕm+1, . . . , ϕk) :U→R
k−m
. This shows that
part (ii) implies both (i) and (iii).
We prove that (i) implies (ii). Letϕ0:U0→Ω0be the coordinate chart
in part (i), letψ0:=ϕ
−1
0
: Ω0→U0be its inverse, and letx0:=ϕ0(p0)∈Ω0.
Then the derivativedψ0(x0) :R
m
→R
k
is injective by Lemma 2.1.4. Hence
there exists a matrixB∈R
k×(k−m)
such that det([dψ0(x0)B])̸= 0.Define
the mapψ: Ω0×R
k−m
→R
k
by
ψ(x, y) :=ψ0(x) +By.
Then thek×k-matrixdψ(x0,0) = [dψ0(x0)B]∈R
k×k
is nonsingular, by
choice ofB. Hence, by the Inverse Function Theorem A.2.2, there exists an
open neighborhood
e
Ω⊂Ω0×R
k−m
of (x0,0) such that
e
U:=ψ(
e
Ω)⊂R
k
is
open andψ|
e
Ω
:
e
Ω→
e
Uis a diffeomorphism. In particular, the restriction ofψ
to
e
Ω is injective. Now the set{x∈Ω0|(x,0)∈
e
Ω}is open and containsx0.
Hence the set
e
U0:=
n
ψ0(x)

x∈Ω0,(x,0)∈
e
Ω
o
=
n
p∈U0

(ϕ0(p),0)∈
e
Ω
o
⊂M
isM-open and containsp0. Hence, by the definition of the relative topology,
there exists an open setW⊂R
k
such that
e
U0=W∩M. Define
U:=
e
U∩W, Ω :=
e
Ω∩ψ
−1
(W).
ThenU∩M=
e
U0andψrestricts to a diffeomorphism from Ω toU.
Now let (x, y)∈Ω. We claim that
ψ(x, y)∈M ⇐⇒ y= 0. (2.1.1)
Ify= 0, then obviouslyψ(x, y) =ψ0(x)∈M. Conversely, let (x, y)∈Ω and
suppose thatp:=ψ(x, y)∈M. Thenp∈U∩M=
e
U∩W∩M=
e
U0⊂U0
and hence (ϕ0(p),0)∈
e
Ω, by definition of
e
U0. This implies
ψ(ϕ0(p),0) =ψ0(ϕ0(p)) =p=ψ(x, y).
Since the pairs (x, y) and (ϕ0(p),0) both belong to the set
e
Ω and the re-
striction ofψto
e
Ω is injective we obtainx=ϕ0(p) andy= 0. This
proves (2.1.1). It follows from (2.1.1) that the mapϕ:= (ψ|Ω)
−1
:U→Ω
satisfiesϕ(U∩M) ={(x, y)∈Ω|ψ(x, y)∈M}= Ω∩(R
m
× {0}). Thus we
have proved that (i) implies (ii).

20 CHAPTER 2. FOUNDATIONS
We prove that (iii) implies (ii). Letf:U→R
k−m
be as in part (iii).
Thenp0∈Uand the derivativedf(p0) :R
k
→R
k−m
is a surjective linear
map. Hence there exists a matrixA∈R
m×k
such that
det
`
A
df(p0)
´
̸= 0.
Define the mapϕ:U→R
k
by
ϕ(p) :=
`
Ap
f(p)
´
forp∈U.
Then det(dϕ(p0))̸= 0. Hence, by the Inverse Function Theorem A.2.2, there
exists an open neighborhoodU
′
⊂Uofp0such that Ω
′
:=ϕ(U
′
) is an open
subset ofR
k
and the restrictionϕ
′
:=ϕ|U
′:U
′
→Ω
′
is a diffeomorphism.
In particular, the restrictionϕ|U
′is injective. Moreover, it follows from the
assumptions onfand the definition ofϕthat
U
′
∩M=
Φ
p∈U
′

f(p) = 0

=
Φ
p∈U
′

ϕ(p)∈R
m
× {0}

and soϕ
′
(U
′
∩M) = Ω
′
∩
Γ
R
m
×{0}
∆
.Hence the diffeomorphismϕ
′
:U
′
→Ω
′
satisfies the requirements of part (ii). This proves Theorem 2.1.10.
The next corollary relates the notion of a smooth map on a smooth
submanifold as defined in the beginning of§2.1 to the standard notion of
smoothness in local coordinates used in the intrinsic setting of§2.8 below.
Corollary 2.1.11.LetM⊂R
k
be a smoothm-dimensional submanifold
and letf:M→R
ℓ
be a map. Then the following are equivalent.
(i)For everyp0∈Mthere exists an open neighborhoodU⊂R
k
ofp0and
a smooth mapF:U→R
ℓ
that agrees withfonU∩M.
(ii)IfU0⊂Mis anM-open set andϕ0:U0→Ω0is a diffeomorhism onto
an open set Ω0⊂R
m
, then the compositionf◦ϕ
−1
0
: Ω0→R
ℓ
is smooth.
Proof.Assume (ii), letp0∈M, and chooseϕ= (ϕ1, . . . , ϕk) :U→Ω⊂R
k
as in part (ii) of Theorem 2.1.10. ShrinkingU, if necessary, we may assume
that Ω = Ω0×Ω1, where Ω0⊂R
m
is an open set and Ω1⊂R
k−m
is an
open neighborhood of the origin. Then the map Ω0→R
ℓ
:x7→f◦ϕ
−1
(x,0)
is smooth by part (ii). DefineF(p) :=f◦ϕ
−1
(ϕ1(p), . . . , ϕm(p),0, . . . ,0)
forp∈U. Then the mapF:U→R
ℓ
is smooth and agrees withfonU∩M.
Thusfsatisfies (i). That (i) implies (ii) follows from Exercise 2.1.1 and this
proves Corollary 2.1.11.

2.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 21
Definition 2.1.12(Regular value).LetU⊂R
k
be an open set and
letf:U→R
ℓ
be a smooth map. An elementc∈R
ℓ
is called aregular
valueoffiff, for allp∈U, we have
f(p) =c =⇒ df(p) :R
k
→R
ℓ
is surjective.
Otherwisecis called asingular valueoff. Theorem 2.1.10 asserts that,
ifcis a regular value off, then the preimage
M:=f
−1
(c) =
Φ
p∈U

f(p) =c

is a smooth(k−ℓ)-dimensional submanifold ofR
k
.
Examples and Exercises
Example 2.1.13.LetA=A
T
∈R
k×k
be a symmetric matrix and de-
fine the functionf:R
k
→Rbyf(x) :=x
T
Ax.Thendf(x)ξ= 2x
T
Aξ
forx, ξ∈R
k
and hence the linear mapdf(x) :R
k
→Ris surjective if and
only ifAx̸= 0. Thusc= 0 is the only singular value offand hence, for
every elementc∈R\ {0}, the setM:=f
−1
(c) ={x∈R
k
|x
T
Ax=c}is a
smooth manifold of dimensionm=k−1.
Example 2.1.14(The sphere).As a special case of Example 2.1.13 con-
sider the casek=m+ 1,A= 1l, andc= 1. Thenf(x) =|x|
2
and so we
have another proof that the unit sphere
S
m
=
n
x∈R
m+1

|x|
2
= 1
o
inR
m+1
is a smoothm-manifold. (See Examples 1.2.4 and 2.1.5.)
Example 2.1.15.Define the mapf:R
3
×R
3
→Rbyf(x, y) :=|x−y|
2
.
This is another special case of Example 2.1.13 and so, for everyr >0, the
setM:={(x, y)∈R
3
×R
3
| |x−y|=r}is a smooth 5-manifold.
Example 2.1.16(The2-torus).Let 0< r <1 and definef:R
3
→Rby
f(x, y, z) := (x
2
+y
2
+r
2
−z
2
−1)
2
−4(x
2
+y
2
)(r
2
−z
2
).
This map has zero as a regular value andM:=f
−1
(0) is diffeomorphic to
the 2-torusT
2
=S
1
×S
1
. An explicit diffeomorphism is given by
(e
is
, e
it
)7→
Γ
(1 +rcos(s)) cos(t),(1 +rcos(s)) sin(t), rsin(s)
∆
.
This example corresponds to the second surface in Figure 2.2.
Exercise:Show thatf(x, y, z) = 0 if and only if (
p
x
2
+y
2
−1)
2
+z
2
=r
2
.
Verify that zero is a regular value off.

22 CHAPTER 2. FOUNDATIONS
Example 2.1.17.The set
M:=
Φ
(x
2
, y
2
, z
2
, yz, zx, xy)|x, y, z∈R, x
2
+y
2
+z
2
= 1

is a smooth 2-manifold inR
6
. To see this, define an equivalence relation on
the unit sphereS
2
⊂R
3
byp∼qiffq=±p. The quotient space (the set of
equivalence classes) is called thereal projective planeand is denoted by
RP
2
:=S
2
/{±1}.
(See Example 1.2.6.) It is equipped with the quotient topology, i.e. a sub-
setU⊂RP
2
is open, by definition, iff its preimage under the obvious projec-
tionS
2
→RP
2
is an open subset ofS
2
. Now the mapf:S
2
→R
6
defined
by
f(x, y, z) := (x
2
, y
2
, z
2
, yz, zx, xy)
descends to a homeomorphism fromRP
2
ontoM. The submanifoldMis
covered by the local smooth parameterizations
Ω→M: (x, y)7→f(x, y,
p
1−x
2
−y
2
),
Ω→M: (x, z)7→f(x,
p
1−x
2
−z
2
, z),
Ω→M: (y, z)7→f(
p
1−y
2
−z
2
, y, z),
defined on the open unit disc Ω⊂R
2
. We remark the following.
(a)Misnotthe preimage of a regular value under a smooth mapR
6
→R
4
.
(b)Misnotdiffeomorphic to a submanifold ofR
3
.
(c)The projection Σ :=
Φ
(yz, zx, xy)|x, y, z∈R, x
2
+y
2
+z
2
= 1

ofM
onto the last three coordinates is called theRoman surfaceand was dis-
covered by Jakob Steiner. The Roman surface can also be represented as
the set of solutions (ξ, η, ζ)∈R
3
of the equationη
2
ζ
2
+ζ
2
ξ
2
+ξ
2
η
2
=ξηζ.
It is not a submanifold ofR
3
.
Exercise:Prove this. Show thatMis diffeomorphic to a submanifold ofR
4
.
Show thatMis diffeomorphic toRP
2
as defined in Example 1.2.6.
Exercise 2.1.18.LetV:R
n
→Rbe a smooth function and define the
Hamiltonian functionH:R
n
×R
n
→R(kinetic plus potential energy) by
H(x, y) :=
1
2
|y|
2
+V(x).
Prove thatcis a regular value ofHif and only if it is a regular value ofV.

2.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 23
Exercise 2.1.19.Consider thegeneral linear group
GL(n,R) =
Φ
g∈R
n×n
|det(g)̸= 0

Prove that the derivative of the functionf= det :R
n×n
→Ris given by
df(g)v= det(g) trace(g
−1
v)
for allg∈GL(n,R) andv∈R
n×n
. Deduce that thespecial linear group
SL(n,R) :={g∈GL(n,R)|det(g) = 1}
is a smooth submanifold ofR
n×n
.
Example 2.1.20.Theorthogonal group
O(n) :=
n
g∈R
n×n
|g
T
g= 1l
o
is a smooth submanifold ofR
n×n
. To see this, denote by
Sn:=
n
S∈R
n×n
|S
T
=S
o
the vector space of symmetric matrices and definef:R
n×n
→Snby
f(g) :=g
T
g.
Its derivativedf(g) :R
n×n
→Snis given by
df(g)v=g
T
v+v
T
g.
This map is surjective for everyg∈O(n): ifg
T
g= 1l andS=S
T
∈Sn,
then the matrixv:=
1
2
gSsatisfies
df(g)v=
1
2
g
T
gS+
1
2
(gS)
T
g=
1
2
S+
1
2
S
T
=S.
Hence 1l is a regular value offand so O(n) is a smooth manifold. It has
the dimension
dim O(n) =n
2
−dimSn=n
2
−
n(n+ 1)
2
=
n(n−1)
2
.
Exercise 2.1.21.Prove that the set
M:=
Φ
(x, y)∈R
2
|xy= 0

is not a submanifold ofR
2
.Hint:IfU⊂R
2
is a neighborhood of the
origin andf:U→Ris a smooth map such thatU∩M=f
−1
(0), then
df(0,0) = 0.

24 CHAPTER 2. FOUNDATIONS
2.2 Tangent Spaces and Derivatives
The main reason for first discussing the extrinsic notion of embedded mani-
folds in Euclidean space as explained in the Master Plan§1.5 is that the
concept of a tangent vector is much easier to digest in the embedded case:
it is simply the derivative of a curve inM, understood as a vector in the
ambient Euclidean space in whichMis embedded.
2.2.1 Tangent Space
Definition 2.2.1(Tangent vector).LetM⊂R
k
be a smoothm-dimen-
sional manifold and fix a pointp∈M. A vectorv∈R
k
is called atangent
vectorofMatpiff there exists a smooth curveγ:R→Msuch that
γ(0) =p, ˙γ(0) =v.
The set
TpM:={˙γ(0)|γ:R→Mis smooth, γ(0) =p}
of tangent vectors ofMatpis called thetangent spaceofMatp.
Theorem 2.2.3 below shows thatTpMis a linear subspace ofR
k
. As does
any linear subspace it contains the origin; it need not actually intersectM.
Its translatep+TpMtouchesMatp; this is what you should visualize
forTpM(see Figure 2.4).p
pMT
M
p +
v
MpT
0
Figure 2.4:The tangent spaceTpMand the translated tangent spacep+TpM.
Remark 2.2.2.Letp∈Mbe as in Definition 2.2.1 and letv∈R
k
. Then
v∈TpM ⇐⇒
æ
∃ε >0∃γ: (−ε, ε)→Msuch that
γis smooth, γ(0) =p,˙γ(0) =v.
To see this suppose thatγ: (−ε, ε)→Mis a smooth curve withγ(0) =p
and ˙γ(0) =v. Defineeγ:R→Mby
eγ(t) :=γ
`
εt
√
ε
2
+t
2
´
, t∈R.
Theneγis smooth and satisfieseγ(0) =pand
˙
eγ(0) =v. Hencev∈TpM.

2.2. TANGENT SPACES AND DERIVATIVES 25
Theorem 2.2.3(Tangent spaces).LetM⊂R
k
be a smoothm-dimen-
sional manifold and fix a pointp∈M. Then the following holds.
(i)LetU0⊂Mbe anM-open set withp∈U0and letϕ0:U0→Ω0
be a diffeomorphism onto an open subsetΩ0⊂R
m
. Letx0:=ϕ0(p)and
letψ0:=ϕ
−1
0
: Ω0→U0be the inverse map. Then
TpM= im
ı
dψ0(x0) :R
m
→R
k
ȷ
.
(ii)LetU,Ω⊂R
k
be open sets andϕ:U→Ωbe a diffeomorphism such
thatp∈Uandϕ(U∩M) = Ω∩(R
m
× {0}). Then
TpM=dϕ(p)
−1
(R
m
× {0}).
(iii)LetU⊂R
k
be an open neighborhood ofpandf:U→R
k−m
be a
smooth map such that0is a regular value offandU∩M=f
−1
(0). Then
TpM= kerdf(p).
(iv)TpMis anm-dimensional linear subspace ofR
k
.
Proof.Letψ0: Ω0→U0andx0∈Ω0be as in (i) and letϕ:U→Ω be as
in (ii). We prove that
imdψ0(x0)⊂TpM⊂dϕ(p)
−1
(R
m
× {0}). (2.2.1)
To prove the first inclusion in (2.2.1), choose a constantr >0 such that
Br(x0) :={x∈R
m
| |x−x0|< r} ⊂Ω0.
Now letξ∈R
m
and chooseε >0 so small that
ε|ξ| ≤r.
Thenx0+tξ∈Ω0for allt∈Rwith|t|< ε. Defineγ: (−ε, ε)→Mby
γ(t) :=ψ0(x0+tξ) for−ε < t < ε.
Thenγis a smooth curve inMsatisfying
γ(0) =ψ0(x0) =p, ˙γ(0) =
d
dt

t=0
ψ0(x0+tξ) =dψ0(x0)ξ.
Hence it follows from Remark 2.2.2 thatdψ0(x0)ξ∈TpM, as claimed.

26 CHAPTER 2. FOUNDATIONS
To prove the second inclusion in (2.2.1) we fix a vectorv∈TpM. Then,
by definition of the tangent space, there exists a smooth curveγ:R→M
such thatγ(0) =pand ˙γ(0) =v. LetU⊂R
k
be as in (ii) and chooseε >0
so small thatγ(t)∈Ufor|t|< ε. Then
ϕ(γ(t))∈ϕ(U∩M)⊂R
m
× {0}
for|t|< εand hence
dϕ(p)v=dϕ(γ(0)) ˙γ(0) =
d
dt

t=0
ϕ(γ(t))∈R
m
× {0}.
This shows thatv∈dϕ(p)
−1
(R
m
× {0}) and thus we have proved (2.2.1).
Now the sets imdψ0(x0) anddϕ(p)
−1
(R
m
× {0}) are bothm-dimensional
linear subspaces ofR
k
. Hence it follows from (2.2.1) that these subspaces
agree and that they both agree withTpM. Thus we have proved asser-
tions (i), (ii), and (iv).
We prove (iii). Letv∈TpM. Then there is a smooth curveγ:R→M
such thatγ(0) =pand ˙γ(0) =v. Fortsufficiently small we haveγ(t)∈U,
whereU⊂R
k
is the open set in (iii), andf(γ(t)) = 0. Hence
df(p)v=df(γ(0)) ˙γ(0) =
d
dt

t=0
f(γ(t)) = 0.
This impliesTpM⊂kerdf(p). SinceTpMand the kernel ofdf(p) are both
m-dimensional linear subspaces ofR
k
, we deduce thatTpM= kerdf(p).
This proves part (iii) and Theorem 2.2.3.
Exercise 2.2.4.LetM⊂R
k
be a smoothm-dimensional manifold and
letpi∈Mbe a sequence that converges to a pointp∈M. Letτibe a
sequence of nonzero real numbers and letv∈R
k
such that
lim
i→∞
τi= 0,lim
i→0
pi−p
τi
=v.
Prove thatv∈TpM.Hint:Use part (iii) of Theorem 2.2.3.
Example 2.2.5.LetA=A
T
∈R
k×k
be a nonzero matrix as in Exam-
ple 2.1.13 and letc̸= 0. Then part (iii) of Theorem 2.2.3 asserts that the
tangent space of the manifold
M=
n
x∈R
k

x
T
Ax=c
o
at a pointx∈Mis the (k−1)-dimensional linear subspace
TxM=
n
ξ∈R
k

x
T
Aξ= 0
o
.

2.2. TANGENT SPACES AND DERIVATIVES 27
Example 2.2.6.As a special case of Example 2.2.5 withA= 1l andc= 1
we find that the tangent space of the unit sphereS
m
⊂R
m+1
at a point
x∈S
m
is the orthogonal complement ofx. i.e.
TxS
m
=x
⊥
=
Φ
ξ∈R
m+1
| ⟨x, ξ⟩= 0

.
Here⟨x, ξ⟩=
P
m
i=0
xiξidenotes the standard inner product onR
m+1
.
Exercise 2.2.7.What is the tangent space of the 5-manifold
M:=
Φ
(x, y)∈R
3
×R
3
| |x−y|=r

at a point (x, y)∈M? (See Exercise 2.1.15.)
Example 2.2.8.LetH(x, y) :=
1
2
|y|
2
+V(x) be as in Exercise 2.1.18 and
letcbe a regular value ofH. If (x, y)∈M:=H
−1
(c), then
T
(x,y)M={(ξ, η)∈R
n
×R
n
| ⟨y, η⟩+⟨∇V(x), ξ⟩= 0}.
Here∇V:= (∂V/∂x1, . . . , ∂V/∂xn) :R
n
→R
n
denotes the gradient ofV.
Exercise 2.2.9.The tangent space of SL(n,R) at the identity matrix is the
space
sl(n,R) :=T1lSL(n,R) =
Φ
ξ∈R
n×n
|trace(ξ) = 0

of traceless matrices. (Prove this, using Exercise 2.1.19.)
Example 2.2.10.The tangent space of O(n) atgis
TgO(n) =
n
v∈R
n×n
|g
T
v+v
T
g= 0
o
.
In particular, the tangent space of O(n) at the identity matrix is the space
of skew-symmetric matrices
o(n) :=T1lO(n) =
n
ξ∈R
n×n
|ξ
T
+ξ= 0
o
To see this, choose a smooth curveR→O(n) :t7→g(t). Theng(t)
T
g(t) = 1l
for allt∈Rand, differentiating this identity with respect tot, we ob-
taing(t)
T
˙g(t) + ˙g(t)
T
g(t) = 0 for everyt. Hence every matrixv∈TgO(n)
satisfies the equationg
T
v+v
T
g= 0. With this understood, the claim fol-
lows from the fact thatg
T
v+v
T
g= 0 if and only if the matrixξ:=g
−1
v
is skew-symmetric and that the space of skew-symmetric matrices inR
n×n
has dimensionn(n−1)/2.
Exercise 2.2.11.Let Ω⊂R
m
be an open set andh: Ω→R
k−m
be a
smooth map. Prove that the tangent space of the graph ofhat a point
(x, h(x)) is the graph of the derivativedh(x) :R
m
→R
k−m
:
M={(x, h(x))|x∈Ω}, T
(x,h(x))M={(ξ, dh(x)ξ)|ξ∈R
m
}.

28 CHAPTER 2. FOUNDATIONS
Exercise 2.2.12(Monge coordinates).LetMbe a smoothm-manifold
inR
k
and suppose thatp∈Mis such that the projectionTpM→R
m
× {0}
is invertible. Prove that there exists an open set Ω⊂R
m
and a smooth
maph: Ω→R
k−m
such that the graph ofhis anM-open neighborhood
ofp(see Example 2.1.6). Of course, the projectionTpM→R
m
× {0}need
not be invertible, but it must be invertible for at least one of the
Γ
k
m
∆
choices
of them-dimensional coordinate plane. Hence every point ofMhas anM-
open neighborhood which may be expressed as a graph of a function of some
of the coordinates in terms of the others as in e.g. Example 2.1.5.
2.2.2 Derivative
A key purpose behind the concept of a smooth manifold is to carry over
the notion of a smooth map and its derivatives from the realm of first year
analysis to the present geometric setting. Here is the basic definition. It ap-
peals to the notion of a smooth map between arbitrary subsets of Euclidean
spaces as introduced in the beginning of§2.1.
Definition 2.2.13(Derivative).LetM⊂R
k
be anm-dimensional smooth
manifold and let
f:M→R
ℓ
be a smooth map. Thederivativeoffat a pointp∈Mis the map
df(p) :TpM→R
ℓ
defined as follows. Given a tangent vectorv∈TpM, choose a smooth curve
γ:R→M
satisfying
γ(0) =p, ˙γ(0) =v,
and define the vectordf(p)v∈R
ℓ
by
df(p)v:=
d
dt

t=0
f(γ(t)) = lim
h→0
f(γ(h))−f(p)
h
. (2.2.2)
That the limit on the right in equation (2.2.2) exists follows from our
assumptions. We must prove, however, that the derivative is well defined,
i.e. that the right hand side of (2.2.2) depends only on the tangent vectorv
and not on the choice of the curveγused in the definition. This is the
content of the first assertion in the next theorem.

2.2. TANGENT SPACES AND DERIVATIVES 29
Theorem 2.2.14(Derivatives).LetM⊂R
k
be anm-dimensional smooth
manifold andf:M→R
ℓ
be a smooth map. Fix a pointp∈M. Then the
following holds.
(i)The right hand side of(2.2.2)is independent ofγ.
(ii)The mapdf(p) :TpM→R
ℓ
is linear.
(iii)IfN⊂R
ℓ
is a smoothn-manifold andf(M)⊂N, then
df(p)TpM⊂T
f(p)N.
(iv) (Chain Rule)LetNbe as in (iii), suppose thatf(M)⊂N, and
letg:N→R
d
be a smooth map. Then
d(g◦f)(p) =dg(f(p))◦df(p) :TpM→R
d
.
(v)Iff= id :M→M, thendf(p) = id :TpM→TpM.
Proof.We prove (i). Letv∈TpMandγ:R→Mbe as in Definition 2.2.13.
By definition there is an open neighborhoodU⊂R
k
ofpand a smooth
mapF:U→R
ℓ
such that
F(p
′
) =f(p
′
) for allp
′
∈U∩M.
LetdF(p)∈R
ℓ×k
denote the Jacobian matrix (i.e. the matrix of all first
partial derivatives) ofFatp. Then, sinceγ(t)∈U∩Mfortsufficiently
small, we have
dF(p)v=dF(γ(0)) ˙γ(0)
=
d
dt

t=0
F(γ(t))
=
d
dt

t=0
f(γ(t)).
The right hand side of this identity is independent of the choice ofFwhile
the left hand side is independent of the choice ofγ. Hence the right hand
side is also independent of the choice ofγand this proves (i). Assertion (ii)
follows immediately from the identity
df(p)v=dF(p)v
just established.

30 CHAPTER 2. FOUNDATIONS
Assertion (iii) follows directly from the definitions. Namely, ifγis as in
Definition 2.2.13, thenβ:=f◦γ:R→Nis a smooth curve inNsatisfying
β(0) =f(γ(0)) =f(p) =:q,
˙
β(0) =df(p)v=:w.
Hencew∈TqN. Assertion (iv) also follows directly from the definitions.
Ifg:N→R
d
is a smooth map andβ, q, ware as above, then
d(g◦f)(p)v=
d
dt

t=0
g(f(γ(t)))
=
d
dt

t=0
g(β(t))
=dg(q)w
=dg(f(p))df(p)v.
This proves (iv). Assertion (v) follows again directly from the definitions
and this proves Theorem 2.2.14.
Corollary 2.2.15(Diffeomorphisms).LetM⊂R
k
be a smoothm-mani-
fold andN⊂R
ℓ
be a smoothn-manifold and letf:M→Nbe a diffeomor-
phism. AssumeMandNare nonempty. Thenm=nand the deriva-
tivedf(p) :TpM→T
f(p)Nis a vector space isomorphism with inverse
df(p)
−1
=df
−1
(f(p)) :T
f(p)N→TpM
for allp∈M.
Proof.Defineg:=f
−1
:N→Mso thatg◦f= idMandf◦g= idN.
Then it follows from Theorem 2.2.14 that, forp∈Mandq:=f(p)∈N, we
havedg(q)◦df(p) = id :TpM→TpManddf(p)◦dg(q) = id :TqN→TqN.
Hencedf(p) :TpM→TqNis a vector space isomorphism and its inverse is
given bydg(q) =df(p)
−1
:TqN→TpM.Hencem=nand this proves
Corollary 2.2.15.
Exercise 2.2.16.LetM⊂R
k
be a smooth manifold and letf:M→R
ℓ
be
a smooth map. Letpi∈Mbe a sequence that converges to a pointp∈M,
letτibe a sequence of nonzero real numbers, and letv∈TpMsuch that
lim
i→∞
τi= 0, lim
i→∞
pi−p
τi
=v.
(See Exercise 2.2.4.) Prove that
lim
i→∞
f(pi)−f(p)
τi
=df(p)v.
Hint:Use the local extensionFoffin the proof of Theorem 2.2.14.

2.2. TANGENT SPACES AND DERIVATIVES 31
2.2.3 The Inverse Function Theorem
Corollary 2.2.15 is analogous to the corresponding assertion for smooth maps
between open subsets of Euclidean space. Likewise, the inverse function
theorem for manifolds is a partial converse of Corollary 2.2.15.M
φ ψ
q

V
~
RU
~
x
~
f
f
U
0 0
p V
0
0
0
00
0
0
N
R
m
m
Figure 2.5: The Inverse Function Theorem.
Theorem 2.2.17(Inverse Function Theorem).Assume thatM⊂R
k
andN⊂R
ℓ
are smoothm-manifolds andf:M→Nis a smooth map.
Letp0∈Mand suppose that the derivative
df(p0) :Tp0
M→T
f(p0)N
is a vector space isomorphism. Then there exists anM-open neighbor-
hoodU⊂Mofp0such thatV:=f(U)⊂Nis anN-open subset ofNand
the restrictionf|U:U→Vis a diffeomorphism.
Proof.Choose coordinate chartsϕ0:U0→
e
U0, defined on anM-open neigh-
borhoodU0⊂Mofp0onto an open set
e
U0⊂R
m
, andψ0:V0→
e
V0, defined
on anN-open neighborhoodV0⊂Noff(p0) onto an open set
e
V0⊂R
m
.
ShrinkingU0, if necessary, we may assume thatf(U0)⊂V0. Then the map
e
f:=ψ0◦f◦ϕ
−1
0
:
e
U0→
e
V0
(see Figure 2.5) is smooth and its derivatived
e
f(x0) :R
m
→R
m
is bijective
atx0:=ϕ0(p0). Hence the Inverse Function Theorem A.2.2 asserts that
there exists an open neighborhood
e
U⊂
e
U0ofx0such that
e
V:=
e
f(
e
U) is an
open subset of
e
V0and the restriction of
e
fto
e
Uis a diffeomorphism from
e
U
to
e
V. Hence the assertion holds withU:=ϕ
−1
0
(
e
U) andV:=ψ
−1
0
(
e
V). This
proves Theorem 2.2.17.

32 CHAPTER 2. FOUNDATIONS
Regular Values
Definition 2.2.18(Regular value).LetM⊂R
k
be a smoothm-manifold,
letN⊂R
ℓ
be a smoothn-manifold, and letf:M→Nbe a smooth map.
An elementq∈Nis called aregular valueoffiff, for everyp∈M
withf(p) =q, the derivativedf(p) :TpM→T
f(p)Nis surjective.
Theorem 2.2.19(Regular values).Letf:M→Nbe as in Defini-
tion 2.2.18 and letq∈Nbe a regular value off. Then the set
P:=f
−1
(q) ={p∈M|f(p) =q}
is a smooth submanifold ofR
k
of dimensionm−nand, for each pointp∈P,
its tangent space atpis given by
TpP= kerdf(p) ={v∈TpM|df(p)v= 0}.
Proof 1.Fix a pointp0∈Pand choose a linear mapA:R
k
→R
m−n
such
that the restriction ofAto the (m−n)-dimensional linear subspace
kerdf(p0) ={v∈Tp0
M|df(p0) = 0} ⊂Tp0
M⊂R
k
is a vector space isomorphism. Define the mapF:M→N×R
m−n
by
F(p) := (f(p), Ap)
forp∈M. The derivative ofFatp0is given bydF(p0)v= (df(p0)v, Av)
forv∈Tp0
Mand is a vector space isomorphism. Hence, by Theorem 2.2.17
there exists anM-open neighborhoodU⊂Mofp0such thatV:=F(U)
is an open subset ofN×R
m−n
in the relative topology and the restric-
tionF|U:U→Vis a diffeomorphism. Hence theP-open setU∩Pis
diffeomorphic to the open set Ω :={y∈R
m−n
|(q, y)∈V} ⊂R
m−n
by the
diffeomorphismϕ:U∩P→Ω, defined byϕ(p) :=Apforp∈U∩P, whose
inverse is the smooth mapψ: Ω→U∩Pgiven byψ(y) = (F|U)
−1
(q, y)
fory∈Ω. This shows thatPis a smooth (m−n)-manifold inR
k
.
Now letp∈Pandv∈TpP. Then there exists a smooth curveγ:R→P
such thatγ(0) =pand ˙γ(0) =v. Sincef(γ(t)) =qfor allt, we have
df(p)v=
d
dt

t=0
f(γ(t)) = 0
and sov∈kerdf(p). HenceTpP⊂kerdf(p) and equality holds because
bothTpPand kerdf(p) are (m−n)-dimensional linear subspaces ofR
k
.
This proves Theorem 2.2.19.

2.2. TANGENT SPACES AND DERIVATIVES 33
Proof 2.Here is another proof of Theorem 2.2.19 in local coordinates. Fix
a pointp0∈Pand choose a coordinate chartϕ0:U0→ϕ0(U0)⊂R
m
on
anM-open neighborhoodU0⊂Mofp0. Likewise, choose a coordinate
chartψ0:V0→ψ0(V0)⊂R
n
on anN-open neighborhoodV0⊂Nofq.
ShrinkingU0, if necessary, we may assume thatf(U0)⊂V0. Then the
pointc0:=ψ0(q) is a regular value of the map
f0:=ψ0◦f◦ϕ
−1
0
:ϕ0(U0)→R
n
.
Namely, ifx∈ϕ0(U0) satisfiesf0(x) =c0, thenp:=ϕ
−1
0
(x)∈U0∩P, so the
mapsdϕ
−1
0
(x) :R
m
→TpM,df(p) :TpM→TqN, anddψ0(q) :TqN→R
n
are all surjective, hence so is their composition, and by the chain rule this
composition is the derivativedf0(x) :R
m
→R
n
. With this understood, it
follows from Theorem 2.1.10 that the set
f
−1
0
(c0) =
Φ
x∈ϕ0(U0)|f(ϕ
−1
0
(x)) =q

=ϕ0(U0∩P)
is a manifold of of dimensionm−ncontained in the open setϕ0(U0)⊂R
m
.
Using Definition 2.1.3 and shrinkingU0further, if necessary, we may assume
that the setϕ0(U0∩P) is diffeomorphic to an open subset ofR
m−n
. Com-
posing this diffeomorphism withϕ0we find thatU0∩Pis diffeomorphic to
the same open subset ofR
m−n
. Since the setU0⊂MisM-open, there
exists an open setU⊂R
k
such thatU∩M=U0, henceU∩P=U0∩P,
and soU0∩Pis aP-open neighborhood ofp0. Thus we have proved
that every elementp0∈Phas aP-open neigborhood that is diffeomor-
phic to an open subset ofR
m−n
. ThusP⊂R
k
is a manifold of dimen-
sionm−n(Definition 2.1.3). The proof that the tangent spaces ofPare
given byTpP= kerdf(p) remains unchanged and this completes the second
proof of Theorem 2.2.19.
Definition 2.2.20.LetM⊂R
k
andN⊂R
ℓ
be smoothm-manifolds. A
smooth mapf:M→Nis called alocal diffeomorphismiff its deriva-
tivedf(p) :TpM→T
f(p)Nis a vector space isomorphism for everyp∈M.
Example 2.2.21.The inclusion of anM-open subsetU⊂MintoMand
the mapR→S
1
:t7→e
it
are examples of local diffeomorphisms.
Exercise 2.2.22.Prove that the image of a local diffeomorphism is an open
subset of the target manifold.Hint:Use the Inverse Function Theorem.
In the terminology introduced in§2.3 and§2.6.1 below, local diffeomor-
phisms are both immersions and submersions. In particular, iff:M→N
is a local diffeomorphism, then every elementq∈Nis a regular value off
and its preimagef
−1
(q) is a discrete subset ofM.

34 CHAPTER 2. FOUNDATIONS
2.3 Submanifolds and Embeddings
This section deals with subsets of a manifoldMthat are themselves mani-
folds as in Definition 2.1.3. Such subsets are called submanifolds ofM.
Definition 2.3.1(Submanifold).LetM⊂R
k
be anm-dimensional man-
ifold. A subsetP⊂Mis called asubmanifoldofMof dimensionn, iffP
itself is ann-manifold.
Definition 2.3.2(Embedding).LetM⊂R
k
be anm-dimensional mani-
fold andN⊂R
ℓ
be ann-dimensional manifold. A smooth mapf:N→M
is called animmersioniff its derivatvedf(q) :TqN→T
f(q)Mis injective
for everyq∈N. It is calledproperiff, for every compact subsetK⊂f(N),
the preimagef
−1
(K) ={q∈N|f(q)∈K}is compact. The mapfis called
anembeddingiff it is a proper injective immersion.
Remark 2.3.3.In our definition of proper maps it is important that the
compact setKis required to be contained in the image off. The literature
also contains a stronger definition ofproperwhich requires thatf
−1
(K) is
a compact subset ofNfor every compact subsetK⊂M, whether or notK
is contained in the image off. This holds if and only if the mapfis proper
in the sense of Definition 2.3.2 and has anM-closed image. (Exercise!)M
0
P
φ
p
U
(U)φ
Figure 2.6: A coordinate chart adapted to a submanifold.
Theorem 2.3.4(Submanifolds).LetM⊂R
k
be anm-dimensional man-
ifold andN⊂R
ℓ
be ann-dimensional manifold.
(i)Iff:N→Mis an embedding, thenf(N)is a submanifold ofM.
(ii)IfP⊂Mis a submanifold, then the inclusionP→Mis an embedding.
(iii)A subsetP⊂Mis a submanifold of dimensionnif and only if, for
everyp0∈P, there exists a coordinate chartϕ:U→R
m
on anM-open
neighborhoodUofp0such thatϕ(U∩P) =ϕ(U)∩(R
n
× {0})(Figure 2.6).
(iv)A subsetP⊂Mis a submanifold of dimensionnif and only if, for ev-
eryp0∈P, there exists anM-open neighborhoodU⊂Mofp0and a smooth
mapg:U→R
m−n
such that0is a regular value ofgandU∩P=g
−1
(0).
The proof is pased on the following lemma.

2.3. SUBMANIFOLDS AND EMBEDDINGS 35
Lemma 2.3.5(Embeddings).LetMandNbe as in Theorem 2.3.4,
letf:N→Mbe an embedding, letq0∈N, and define
P:=f(N), p 0:=f(q0)∈P.
Then there exists anM-open neighborhoodU⊂Mofp0, anN-open neigh-
borhoodV⊂Nofq0, an open neighborhoodW⊂R
m−n
of the origin, and
a diffeomorphismF:V×W→Usuch that, for allq∈Vand allz∈W,
F(q,0) =f(q), (2.3.1)
F(q, z)∈P ⇐⇒ z= 0. (2.3.2)
Proof.Choose any coordinate chartϕ0:U0→R
m
on anM-open neighbor-
hoodU0⊂Mofp0. Thend(ϕ0◦f)(q0) =dϕ0(f(q0))◦df(q0) :Tq0
N→R
m
is injective. Hence there is a linear mapB:R
m−n
→R
m
such that the map
Tq0
N×R
m−n
→R
m
: (w, ζ)7→d(ϕ0◦f)(q0)w+Bζ (2.3.3)
is a vector space isomorphism. Define the set
Ω :=
Φ
(q, z)∈N×R
m−n
|f(q)∈U0, ϕ0(f(q)) +Bz∈ϕ0(U0)

.
This is an open subset ofN×R
m−n
and we defineF: Ω→Mby
F(q, z) :=ϕ
−1
0
(ϕ0(f(q)) +Bz).
This map is smooth, it satisfiesF(q,0) =f(q) for allq∈f
−1
(U0), and
the derivativedF(q0,0) :Tq0
N×R
m−n
→Tp0
Mis the composition of the
map (2.3.3) withdϕ0(p0)
−1
:R
m
→Tp0
Mand so is a vector space isomor-
phism. Thus the Inverse Function Theorem 2.2.17 asserts that there is an
N-open neighborhoodV0⊂Nofq0and an open neighborhoodW0⊂R
m−n
of the origin such thatV0×W0⊂Ω, the setU0:=F(V0×W0) isM-open,
and the restriction ofFtoV0×W0is a diffeomorphism ontoU0. Thus we
have constructed a diffeomorphismF:V0×W0→U0that satisfies (2.3.1).
We claim that the restriction ofFto the productV×Wof sufficiently
small open neighborhoodsV⊂Nofq0andW⊂R
m−n
of the origin also
satisfies (2.3.2). Otherwise, there exist sequencesqi∈V0converging toq0
andzi∈W0\ {0}converging to zero such thatF(qi, zi)∈P. Hence there
exists a sequenceq
′
i
∈Nsuch thatF(qi, zi) =f(q
′
i
). This sequence converges
tof(q0). Sincefis proper we may assume, passing to a suitable subsequence
if necessary, thatq
′
i
converges to a pointq
′
0
∈N. Then
f(q
′
0) = lim
i→∞
f(q
′
i) = lim
i→∞
F(qi, zi) =f(q0).

36 CHAPTER 2. FOUNDATIONS
Sincefis injective, this impliesq
′
0
=q0. Hence (q
′
i
,0)∈V0×W0fori
sufficiently large andF(q
′
i
,0) =f(q
′
i
) =F(qi, zi). This contradicts the fact
that the mapF:V0×W0→Mis injective, and proves Lemma 2.3.5.
Proof of Theorem 2.3.4.We prove (i). Letq0∈N, denotep0:=f(q0)∈P,
and choose a diffeomorphismF:V×W→Uas in Lemma 2.3.5. Then the
setV⊂Nis diffeomorphic to an open subset ofR
n
(after schrinkingV
if necessary), the setU∩PisP-open becauseU⊂MisM-open, and we
haveU∩P={F(q,0)|q∈V}=f(V) by (2.3.1) and (2.3.2). Hence the
mapf:V→U∩Pis a diffeomorphism whose inverse is the composition
of the smooth mapsF
−1
:U∩P→V×WandV×W→V: (q, z)7→q.
Hence aP-open neighborhood ofp0is diffeomorphic to an open subset ofR
n
.
Sincep0∈Pwas chosen arbitrary, this shows thatPis ann-dimensional
submanifold ofM.
We prove (ii). The inclusionι:P→Mis obviously smooth and in-
jective (it extends to the identity map onR
k
). Moreover,TpP⊂TpMfor
everyp∈Pand the derivativedι(p) :TpP→TpMis the obvious inclusion
for everyp∈P. Thatιis proper follows immediately from the definition.
Henceιis an embedding.
We prove (iii). If a coordinate chartϕas in (iii) exists, then the setU∩P
isP-open and is diffeomorphic to an open subset ofR
n
. Sincep0∈P
was chosen arbitrary this proves thatPis ann-dimensional submanifold
ofM. Conversely, suppose thatPis ann-dimensional submanifold ofMand
letp0∈P. Choose any coordinate chartϕ0:U0→R
m
ofMdefined on an
M-open neighborhoodU0⊂Mofp0. Thenϕ0(U0∩P) is ann-dimensional
submanifold ofR
m
. Hence Theorem 2.1.10 asserts that there are open
setsV, W⊂R
m
withp0∈V⊂ϕ0(U0) and a diffeomorphismψ:V→W
such that
ϕ0(p0)∈V, ψ(V∩ϕ0(U0∩P)) =W∩(R
n
× {0}).
Now defineU:=ϕ
−1
0
(V)⊂U0. Thenp0∈U, the chartϕ0restricts to a
diffeomorphism fromUtoV, the compositionϕ:=ψ◦ϕ0|U:U→Wis a
diffeomorphism, andϕ(U∩P) =ψ(V∩ϕ0(U0∩P)) =W∩(R
n
× {0}).
We prove (iv). That the condition is sufficient follows directly from
Theorem 2.2.19. To prove that it is necessary, assume thatP⊂Mis a
submanifold of dimensionn, fix an elementp0∈P, and choose a coordi-
nate chartϕ:U→R
m
on anM-open neighborhoodU⊂Mofp0as in
part (iii). Define the mapg:U→R
m−n
byg(p) := (ϕn+1(p), . . . , ϕm(p))
forp∈U. Then 0 is a regular value ofgandg
−1
(0) =U∩P. This proves
Theorem 2.3.4.

2.3. SUBMANIFOLDS AND EMBEDDINGS 37
Exercise 2.3.6.LetM⊂R
k
be a smoothm-manifold and∅ ̸=P⊂M.
(i)IfPis ann-dimensional submanifold ofM, then 0≤n≤m.
(ii)Pis a 0-dimensional submanifold ofMif and only ifPis discrete, i.e.
everyp∈Phas anM-open neighborhoodUsuch thatU∩P={p}.
(iii)Pis anm-dimensional submanifold ofMif and only ifPisM-open.
Example 2.3.7.LetS
1
⊂R
2∼
=Cbe the unit circle and consider the map
f:S
1
→R
2
given byf(x, y) := (x, xy). This map is a proper immersion but
is not injective (the points (0,1) and (0,−1) have the same image underf).
The imagef(S
1
) is a figure 8 inR
2
and is not a submanifold (Figure 2.7).
The restriction offto the submanifoldN:=S
1
\ {(0,−1)}is an injec-
tive immersion but it is not proper. It has the same image as before and
hencef(N) is not a manifold.
Figure 2.7: A proper immersion.
Example 2.3.8.The mapf:R→R
2
given byf(t) := (t
2
, t
3
) is proper
and injective, but is not an embedding (its derivatuve att= 0 is not in-
jective). The image offis the setf(R) =C:=
Φ
(x, y)∈R
2
|x
3
=y
2

(see
Figure 2.8) and is not a submanifold. (Prove this!)
Figure 2.8: A proper injection.
Example 2.3.9.Define the mapf:R→R
2
byf(t) := (cos(t),sin(t)). This
map is an immersion, but it is neither injective nor proper. However, its
image is the unit circle inR
2
and hence is a submanifold ofR
2
. The
mapR→R
2
:t7→f(t
3
) is not an immersion and is neither injective nor
proper, but its image is still the unit circle.

38 CHAPTER 2. FOUNDATIONS
2.4 Vector Fields and Flows
This section introduces vector fields on manifolds (§2.4.1), explains the flow
of a vector field and the group of diffeomorphisms (§2.4.2), and defines the
Lie bracket of two vector fields (§2.4.3).
2.4.1 Vector Fields
Definition 2.4.1(Vector field).LetM⊂R
k
be a smoothm-manifold. A
(smooth) vector fieldonMis a smooth mapX:M→R
k
such that
X(p)∈TpM
for everyp∈M. The set of smooth vector fields onMwill be denoted by
Vect(M) :=
n
X:M→R
k
|Xis smooth, X(p)∈TpMfor allp∈M
o
.
Exercise 2.4.2.Prove that the set of smooth vector fields onMis a real
vector space.
Example 2.4.3.Denote the standard cross product onR
3
by
x×y:=


x2y3−x3y2
x3y1−x1y3
x1y2−x2y1


forx, y∈R
3
. Fix a vectorξ∈S
2
and define the mapsX, Y:S
2
→R
3
by
X(p) :=ξ×p, Y(p) := (ξ×p)×p.
These are vector fields with zeros±ξ. Their integral curves (see Defini-
tion 2.4.6 below) are illustrated in Figure 2.9.
Figure 2.9: Two vector fields on the 2-sphere.

2.4. VECTOR FIELDS AND FLOWS 39
Example 2.4.4.LetM:=R
2
. A vector field onMis then any smooth
mapX:R
2
→R
2
. As an example consider the vector field
X(x, y) := (x,−y).
This vector field has a single zero at the origin and its integral curves are
illustrated in Figure 2.10.
Figure 2.10: A hyperbolic fixed point.
Example 2.4.5.Every smooth functionf:R
m
→Rdetermines a gradient
vector field
X=∇f:=












∂f
∂x1
∂f
∂x2
.
.
.
∂f
∂xm












:R
m
→R
m
.
Definition 2.4.6(Integral curve).LetM⊂R
k
be a smoothm-manifold,
letXbe a smooth vector field onM, and letI⊂Rbe an open interval.
A continuously differentiable curveγ:I→Mis called anintegral curve
ofXiff it satisfies the equation˙γ(t) =X(γ(t))for everyt∈I. Note that
every integral curve ofXis smooth.
Theorem 2.4.7.LetM⊂R
k
be a smoothm-manifold andX∈Vect(M)
be a smooth vector field. Fix a pointp0∈M. Then the following holds.
(i)There exists an open intervalI⊂Rcontaining0and a smooth curve
γ:I→Msatisfying the equation
˙γ(t) =X(γ(t)), γ(0) =p0 (2.4.1)
for everyt∈I.
(ii)Ifγ1:I1→Mandγ2:I2→Mare two solutions of(2.4.1)on open
intervalsI1andI2containing0, thenγ1(t) =γ2(t)for everyt∈I1∩I2.

40 CHAPTER 2. FOUNDATIONS

U
0
M
0
ψ
0φ
f
X
p
0
0x
m
RΩ
Figure 2.11: Vector fields in local coordinates.
Proof.We prove (i). Letϕ0:U0→R
m
be a coordinate chart onM, defined
on anM-open neighborhoodU0⊂Mofp0. The image ofϕ0is an open set
Ω :=ϕ0(U0)⊂R
m
and we denote the inverse map byψ0:=ϕ
−1
0
: Ω→M
(see Figure 2.11). Then, by Theorem 2.2.3, the derivativedψ0(x) :R
m
→R
k
is injective and its image is the tangent spaceT
ψ0(x)Mfor everyx∈Ω.
Definef: Ω→R
m
byf(x) :=dψ0(x)
−1
X(ψ0(x)) forx∈Ω. This map
is smooth and hence, by the basic existence and uniqueness theorem for
ordinary differential equations inR
m
(see [63]), the equation
˙x(t) =f(x(t)), x(0) =x0:=ϕ0(p0), (2.4.2)
has a solutionx:I→Ω on some open intervalI⊂Rcontaining 0. Hence
the functionγ:=ψ0◦x:I→U0⊂Mis a smooth solution of (2.4.1). This
proves (i).
The local uniqueness theorem asserts that two solutionsγi:Ii→M
of (2.4.1) fori= 1,2 agree on the interval (−ε, ε)⊂I1∩I2forε >0
sufficiently small. This follows immediately from the standard uniqueness
theorem for the solutions of (2.4.2) in [63] and the fact thatx:I→Ω is a
solution of (2.4.2) if and only ifγ:=ψ0◦x:I→U0is a solution of (2.4.1).
To prove (ii) we observe that the setI:=I1∩I2is an open interval
containing zero and hence is connected. Now consider the set
A:={t∈I|γ1(t) =γ2(t)}.
This set is nonempty, because 0∈A. It is closed, relative toI, because the
mapsγ1:I→Mandγ2:I→Mare continuous. Namely, ifti∈Iis a
sequence converging tot∈I, thenγ1(ti) =γ2(ti) for everyiand, taking
the limiti→ ∞, we obtainγ1(t) =γ2(t) and hencet∈A. The setAis also
open by the local uniqueness theorem. SinceIis connected it follows that
A=I. This proves (ii) and Theorem 2.4.7.

2.4. VECTOR FIELDS AND FLOWS 41
2.4.2 The Flow of a Vector Field
Definition 2.4.8(The flow of a vector field).LetM⊂R
k
be a smooth
m-manifold andX∈Vect(M)be a smooth vector field onM. Forp0∈M
themaximal existence interval ofp0is the open interval
I(p0) :=
[
æ
I

I⊂Ris an open interval containing0
and there is a solutionγ:I→Mof(2.4.1)
œ
.
By Theorem 2.4.7 equation(2.4.1)has a solutionγ:I(p0)→M. Theflow
ofXis the mapϕ:D →Mdefined by
D:={(t, p0)|p0∈M, t∈I(p0)}
andϕ(t, p0) :=γ(t), whereγ:I(p0)→Mis the unique solution of(2.4.1).
Theorem 2.4.9.LetM⊂R
k
be a smoothm-manifold andX∈Vect(M)
be a smooth vector field onM. Letϕ:D →Mbe the flow ofX. Then the
following holds.
(i)Dis an open subset ofR×M.
(ii)The mapϕ:D →Mis smooth.
(iii)Letp0∈Mands∈I(p0). Then
I(ϕ(s, p0)) =I(p0)−s (2.4.3)
and, for everyt∈Rwiths+t∈I(p0), we have
ϕ(s+t, p0) =ϕ(t, ϕ(s, p0)). (2.4.4)
The proof is pased on the following lemma.
Lemma 2.4.10.LetM,X,D,ϕbe as in Theorem 2.4.9 and letK⊂Mbe
a compact set. Then there exists anM-open setU⊂Mand anε >0such
thatK⊂U,(−ε, ε)×U⊂ D, andϕis smooth on(−ε, ε)×U.
Proof.In the case whereM= Ω is an open subset ofR
m
this is proved
in [64, Satz 4.1.4 & Satz 4.3.1 & Satz 4.4.1]. Using local coordinates (as
in the proof of Theorem 2.4.7) we deduce that, for everyp∈M, there
exists anM-open neighborhoodUp⊂Mofpand a constantεp>0 such
that (−εp, εp)×Up⊂ Dand the restriction ofϕto (−εp, εp)×Upis smooth.
Using this observation for everyp∈K(and the axiom of choice) we obtain
anM-open coverK⊂
S
p∈K
Up. Since the setKis compact there exists a fi-
nite subcoverK⊂Up1
∪ · · · ∪UpN
=:U. Now defineε:= min{εp1
, . . . , εpN
}
to deduce that (−ε, ε)×U⊂ Dandϕis smooth on (−ε, ε)×U. This proves
Lemma 2.4.10.

42 CHAPTER 2. FOUNDATIONS
Proof of Theorem 2.4.9.We prove (iii). The mapγ:I(p0)−s→Mdefined
byγ(t) :=ϕ(s+t, p0) is a solution of the initial value problem ˙γ(t) =X(γ(t))
withγ(0) =ϕ(s, p0). HenceI(p0)−s⊂I(ϕ(s, p0)) and equation (2.4.4)
holds for everyt∈Rwiths+t∈I(p0). In particular, witht=−swe
havep0=ϕ(−s, ϕ(s, p0)). Thus we obtain equality in equation (2.4.3) by
the same argument with the pair (s.p0) replaced by (−s, ϕ(s, p0)).
We prove (i) and (ii). Let (t0, p0)∈ Dso thatp0∈Mandt0∈I(p0).
Supposet0≥0. ThenK:={ϕ(t, p0)|0≤t≤t0}is a compact subset
ofM. (It is the image of the compact interval [0, t0] under the unique
solutionγ:I(p0)→Mof (2.4.1).) Hence, by Lemma 2.4.10, there is an
M-open setU⊂Mand anε >0 such that
K⊂U, (−ε, ε)×U⊂ D,
andϕis smooth on (−ε, ε)×U. ChooseNso large thatt0/N < ε. Define
U0:=Uand, fork= 1, . . . , N, define the setsUk⊂Minductively by
Uk:={p∈U|ϕ(t0/N, p)∈Uk−1}.
These sets are open in the relative topology ofM.
We prove by induction onkthat (−ε, kt0/N+ε)×Uk⊂ Dandϕis
smooth on (−ε, kt0/N+ε)×Uk. Fork= 0 this holds by definition ofε
andU. Ifk∈ {1, . . . , N}and the assertion holds fork−1, then we have
p∈Uk=⇒p∈U, ϕ(t0/N, p)∈Uk−1
=⇒(−ε, ε)⊂I(p),(−ε,(k−1)t0/N+ε)⊂I(ϕ(t0/N, p))
=⇒(−ε, kt0/N+ε)⊂I(p).
Here the last implication follows from (2.4.3). Moreover, forp∈Ukand
t0/N−ε < t < kt0/N+ε, we have, by (2.4.4), that
ϕ(t, p) =ϕ(t−t0/N, ϕ(t0/N, p))
Sinceϕ(t0/N, p)∈Uk−1forp∈Ukthe right hand side is a smooth map
on the open set (t0/N−ε, kt0/N+ε)×Uk. SinceUk⊂U,ϕis also a
smooth map on (−ε, ε)×Ukand hence on (−ε, kt0/N+ε)×Uk. This
completes the induction. Withk=Nwe have found an open neighborhood
of (t0, p0) contained inD, namely the set (−ε, t0+ε)×UN, on whichϕis
smooth. The caset0≤0 is treated similarly. This proves (i) and (ii) and
Theorem 2.4.9.
Definition 2.4.11.A vector fieldX∈Vect(M)is calledcompleteiff, for
eachp0∈M, there exists an integral curveγ:R→MofXwithγ(0) =p0.

2.4. VECTOR FIELDS AND FLOWS 43
Lemma 2.4.12.LetM⊂R
k
be a compact manifold. Then every vector
field onMis complete.
Proof.LetX∈Vect(M). It follows from Lemma 2.4.10 withK=M
that there exists anε >0 such that (−ε, ε)⊂I(p) for allp∈M. By Theo-
rem 2.4.9 this impliesI(p) =Rfor allp∈M. HenceXis complete.
LetM⊂R
k
be a smooth manifold andX∈Vect(M). Then
Xis complete⇐⇒ I(p) =R∀p∈M ⇐⇒ D =R×M.
AssumeXis complete, letϕ:R×M→Mbe the flow ofX, and define
the mapϕ
t
:M→Mbyϕ
t
(p) :=ϕ(t, p) fort∈Randp∈M. Then
Theorem 2.4.9 asserts thatϕ
t
is smooth for everyt∈Rand that
ϕ
s+t
=ϕ
s
◦ϕ
t
, ϕ
0
= id (2.4.5)
for alls, t∈R. In particular, this implies thatϕ
t
◦ϕ
−t
=ϕ
−t
◦ϕ
t
= id.
Henceϕ
t
is bijective and (ϕ
t
)
−1
=ϕ
−t
,so eachϕ
t
is a diffeomorphism.
Exercise 2.4.13.LetM⊂R
k
be a smooth manifold. A vector fieldXonM
is said to havecompact supportiff there exists a compact subsetK⊂M
such thatX(p) = 0 for everyp∈M\K. Prove that every vector field with
compact support is complete.
We close this subsection with an important observation about incomplete
vector fields. The lemma asserts that an integral curve on a finite existence
interval must leave every compact subset ofM.
Lemma 2.4.14.LetM⊂R
k
be a smoothm-manifold, letX∈Vect(M),
letϕ:D →Mbe the flow ofX, letK⊂Mbe a compact set, and letp0∈M
be an element such that
I(p0)∩[0,∞) = [0, b),0< b <∞.
Then there exists a number0< tK< bsuch that
tK< t < b =⇒ ϕ(t, p0)∈M\K
Proof.By Lemma 2.4.10 there exists anε >0 such that (−ε, ε)⊂I(p) for
everyp∈K. Chooseεso small thatε < band define
tK:=b−ε >0.
Choose a real numbertK< t < b. ThenI(ϕ(t, p0))∩[0,∞) = [0, b−t) by
equation (2.4.3) in part (ii) of Theorem 2.4.9. Since 0< b−t < b−tK=ε,
this shows that (−ε, ε)̸⊂I(ϕ(t, p0)) and henceϕ(t, p0)/∈K. This proves
Lemma 2.4.14.

44 CHAPTER 2. FOUNDATIONS
The next corollary is an immediate consequence of Lemma 2.4.14. In
this formulation the result will be used in§4.6 and in§7.3.
Corollary 2.4.15.LetM⊂R
k
be a smoothm-manifold, letX∈Vect(M),
and letγ: (0, T)→Mbe an integral curve ofX. If there exists a compact
setK⊂Mthat contains the image ofγ, thenγextends to an integral curve
ofXon the interval(−ρ, T+ρ)for someρ >0.
Proof.Here is another more direct proof that does not rely on Lemma 2.4.10.
SinceKis compact, there exists a constantc >0 such that|X(p)| ≤cfor
allp∈K. Sinceγ(t)∈Kfor 0< t < T, this implies
|γ(t)−γ(s)|=

Z
t
s
˙γ(r)dr

≤
Z
t
s
|˙γ(r)|dr=
Z
t
s
|X(γ(r))|dr≤c(t−s)
for 0< s < t < T. Thus the limitp0:= limt↘0γ(t) exists inR
k
and, sinceK
is a closed subset ofR
k
, we havep0∈K⊂M. Defineγ0: [0, T)→Mby
γ0(t) :=
æ
p0,fort= 0,
γ(t),for 0< t < T.
We prove thatγ0is differentiable att= 0 and ˙γ0(0) =X(p0). To see this, fix
a constantε >0. Since the curve [0, T)→R
k
:t7→X(γ(t)) is continuous,
there exists a constantδ >0 such that
0< t≤δ =⇒ | X(γ(t))−X(p0)| ≤ε.
Hence, for 0< s < t≤δ, we have
|γ(t)−γ(s)−(t−s)X(p0)|=

Z
t
s
( ˙γ(r)−X(p0))dr

=

Z
t
s
(X(γ(r))−X(p0))dr

≤
Z
t
s
|X(γ(r))−X(p0)|dr
≤(t−s)ε.
Take the limits→0 to obtain

γ(t)−p0
t
−X(p0)

= lim
s→0
|γ(t)−γ(s)−(t−s)X(p0)|
t−s
≤ε
for 0< t≤δ. Thusγ0is differentiable att= 0 with ˙γ0(0) =X(p0), as
claimed. Henceγextends to an integral curveeγ: (−ρ, T)→MofXfor
someρ >0 viaeγ(t) :=ϕ(t, p0) for−ρ < t≤0 andeγ(t) :=γ(t) for 0< t < T.
Hereϕis the flow ofX. Thatγalso extends beyondt=T, follows by re-
placingγ(t) withγ(T−t) andXwith−X. This proves Corollary 2.4.15.

2.4. VECTOR FIELDS AND FLOWS 45
The Group of Diffeomorphisms
Let us denote the space of diffeomorphisms ofMby
Diff(M) :={ϕ:M→M|ϕis a diffeomorphism}.
This is a group. The group operation is composition and the neutral element
is the identity. Now equation (2.4.5) asserts that the flow of a complete
vector fieldX∈Vect(M) is a group homomorphism
R→Diff(M) :t7→ϕ
t
.
This homomorphism is smooth and is characterized by the equation
d
dt
ϕ
t
(p) =X(ϕ
t
(p)), ϕ
0
(p) =p
for allp∈Mandt∈R. We will often abbreviate this equation in the form
d
dt
ϕ
t
=X◦ϕ
t
, ϕ
0
= id. (2.4.6)
Exercise 2.4.16(Isotopy).LetM⊂R
k
be a compact manifold andI⊂R
be an open interval containing 0. Let
I×M→R
k
: (t, p)7→Xt(p)
be a smooth map such thatXt∈Vect(M) for everyt. Prove that there is
a smooth family of diffeomorphismsI×M→M: (t, p)7→ϕt(p) satisfying
d
dt
ϕt=Xt◦ϕt, ϕ 0= id (2.4.7)
for everyt∈I. Such a family of diffeomorphisms
I→Diff(M) :t7→ϕt
is called anisotopyofM. Conversely prove that every smooth isotopy
I→Diff(M) :t7→ϕtis generated (uniquely) by a smooth family of vector
fieldsI→Vect(M) :t7→Xt.

46 CHAPTER 2. FOUNDATIONS
2.4.3 The Lie Bracket
LetM⊂R
k
andN⊂R
ℓ
be smoothm-manifolds andX∈Vect(M) be
smooth vector field onM. Ifψ:N→Mis a diffeomorphism, then the
pullbackofXunderψis the vector field onNdefined by
(ψ
∗
X)(q) :=dψ(q)
−1
X(ψ(q)) (2.4.8)
forq∈N. Ifϕ:M→Nis a diffeomorphism, then thepushforwardofX
underϕis the vector field onNdefined by
(ϕ∗X)(q) :=dϕ(ϕ
−1
(q))X(ϕ
−1
(q)) (2.4.9)
forq∈N.
Lemma 2.4.17.LetM⊂R
k
,N⊂R
ℓ
, andP⊂R
n
be smoothm-dimen-
sional submanifolds, letϕ:M→Nandψ:N→Pbe diffeomorphisms,
and letX∈Vect(M)andZ∈Vect(P). Then
ϕ∗X= (ϕ
−1
)
∗
X (2.4.10)
and
(ψ◦ϕ)∗X=ψ∗ϕ∗X, (ψ◦ϕ)
∗
Z=ϕ
∗
ψ
∗
Z. (2.4.11)
Proof.Equation (2.4.10) follows from the fact that
dϕ
−1
(q) =dϕ(ϕ
−1
(q))
−1
:TqN→T
ϕ
−1
(q)M
for allq∈N(Corollary 2.2.15) and the equations in (2.4.11) follow directly
from the chain rule (Theorem 2.2.14). This proves Lemma 2.4.17.
We think of a vector field onMas a smooth map
X:M→R
k
that satisfies the conditionX(p)∈TpMfor everyp∈M. Ignoring this
condition temporarily, we can differentiateXas a map fromMtoR
k
and
its derivative atpis then a linear map
dX(p) :TpM→R
k
.
In general, this derivative will not take values in the tangent spaceTpM.
However, if we have two vector fieldsXandYonM, then the next lemma
shows that the difference of the derivative ofXin the directionYand ofY
in the directionXdoes take values in the tangent spaces ofM.

2.4. VECTOR FIELDS AND FLOWS 47
Lemma 2.4.18(Lie bracket).LetM⊂R
k
be a smoothm-manifold and
letX, Y∈Vect(M)be complete vector fields. Denote by
R→Diff(M) :t7→ϕ
t
,R→Diff(M) :t7→ψ
t
the flows ofXandY, respectively. Fix a pointp∈Mand define the smooth
mapγ:R→Mby
γ(t) :=ϕ
t
◦ψ
t
◦ϕ
−t
◦ψ
−t
(p). (2.4.12)
Then˙γ(0) = 0and
d
dt

t=0
γ
Γ√
t
∆
=
1
2
¨γ(0)
=
d
ds

s=0
((ϕ
s
)
∗
Y) (p)
=
d
dt

t=0

ψ
t
∆
∗
X
∆
(p)
=dX(p)Y(p)−dY(p)X(p) ∈TpM.
(2.4.13)
Exercise 2.4.19.Letγ:R→R
k
be aC
2
-curve and assume ˙γ(0) = 0.
Prove that
d
dt

t=0
γ(
√
t) = limt→0t
−2
Γ
γ(t)−γ(0)
∆
=
1
2
¨γ(0).
Proof of Lemma 2.4.18.Define the mapβ:R
2
→Mby
β(s, t) :=ϕ
s
◦ψ
t
◦ϕ
−s
◦ψ
−t
(p)
fors, t∈R. Thenγ(t) =β(t, t) and
∂β
∂s
(0, t) =X(p)−dψ
t
(ψ
−t
(p))X(ψ
−t
(p)), (2.4.14)
∂β
∂t
(s,0) =dϕ
s
(ϕ
−s
(p))Y(ϕ
−s
(p))−Y(p)
for alls, t∈R. Hence
˙γ(0) =
∂β
∂s
(0,0) +
∂β
∂t
(0,0) = 0.
This implies the first equality in (2.4.13) by Exercise (2.4.19). To prove
the remaining assertions, note thatβ(s,0) =β(0, t) =p, hence the second
derivatives∂
2
β/∂s
2
and∂
2
β/∂t
2
vanish ats=t= 0, and therefore
¨γ(0) = 2
∂
2
β
∂s∂t
(0,0). (2.4.16)

48 CHAPTER 2. FOUNDATIONS
Combining equations (2.4.15) and (2.4.16) we find
1
2
¨γ(0) =
∂
∂s

s=0
∂β
∂t
(s,0) =
d
ds

s=0
dϕ
s
(ϕ
−s
(p))Y(ϕ
−s
(p))
=
d
ds

s=0
((ϕ
s
)
∗
Y) (p)).
Likewise, combining equations (2.4.14) and (2.4.16) we find
1
2
¨γ(0) =
∂
∂t

t=0
∂β
∂s
(0, t) =−
d
dt

t=0
dψ
t
(ψ
−t
(p))X(ψ
−t
(p))
=
d
dt

t=0
dψ
−t
(ψ
t
(p))X(ψ
t
(p))
=
d
dt

t=0
dψ
t
(p)
−1
X(ψ
t
(p))
=
d
dt

t=0

ψ
t
∆
∗
X
∆
(p)).
In both cases the right hand side is the derivative of a smooth curve in the
tangent spaceTpMand so is itself an element ofTpM. Moreover, we have
1
2
¨γ(0) =
∂
∂s

s=0
dϕ
s
(ϕ
−s
(p))Y(ϕ
−s
(p))
=
∂
∂s

s=0
∂
∂t

t=0
ϕ
s
◦ψ
t
◦ϕ
−s
(p)
=
∂
∂t

t=0
∂
∂s

s=0
ϕ
s
◦ψ
t
◦ϕ
−s
(p)
=
∂
∂t

t=0
Γ
X(ψ
t
(p))−dψ
t
(p)X(p)
∆
=dX(p)Y(p)−
∂
∂t

t=0
∂
∂s

s=0
ψ
t
◦ϕ
s
(p)
=dX(p)Y(p)−
∂
∂s

s=0
∂
∂t

t=0
ψ
t
◦ϕ
s
(p)
=dX(p)Y(p)−
∂
∂s

s=0
Y(ϕ
s
(p))
=dX(p)Y(p)−dY(p)X(p).
This proves Lemma 2.4.18.

2.4. VECTOR FIELDS AND FLOWS 49
Definition 2.4.20(Lie bracket).LetM⊂R
k
be a smooth manifold and
letX, Y∈Vect(M)be smooth vector fields onM. TheLie bracketofX
andYis the vector field[X, Y]∈Vect(M)defined by
[X, Y](p) :=dX(p)Y(p)−dY(p)X(p). (2.4.17)
Warning:In the literature on differential geometry the Lie bracket of two
vector fields is often (but not always) defined with the opposite sign. The
rationale behind the present choice of the sign will be explained in§2.5.7.
Lemma 2.4.21.LetM⊂R
k
andN⊂R
ℓ
be smooth manifolds, letX, Y, Z
be smooth vector fields onM, and let
ϕ:N→M
be a diffeomorphism. Then
ϕ
∗
[X, Y] = [ϕ
∗
X, ϕ
∗
Y], (2.4.18)
[X, Y] + [Y, X] = 0, (2.4.19)
[X,[Y, Z]] + [Y,[Z, X]] + [Z,[X, Y]] = 0. (2.4.20)
The last equation is called theJacobi identity.
Proof.LetR→Diff(M) :t7→ψ
t
be the flow ofY. Then the map
R→Diff(N) :t7→ϕ
−1
◦ψ
t
◦ϕ
is the flow of the vector fieldϕ
∗
YonN. Hence, by Lemma 2.4.17 and
Lemma 2.4.18, we have
[ϕ
∗
X, ϕ
∗
Y] =
d
dt

t=0
Γ
ϕ
−1
◦ψ
t
◦ϕ
∆
∗
ϕ
∗
X
=
d
dt

t=0
ϕ
∗
Γ
ψ
t
∆
∗
X
=ϕ
∗
[X, Y].
This proves (2.4.18). Equation (2.4.19) is obvious. To prove (2.4.20), letϕ
t
be the flow ofX. Then by (2.4.18) and (2.4.19) and Lemma 2.4.18 we have
[[Y, Z], X] =
d
dt

t=0
(ϕ
t
)
∗
[Y, Z]
=
d
dt

t=0
[(ϕ
t
)
∗
Y,(ϕ
t
)
∗
Z]
= [[Y, X], Z] + [Y,[Z, X]]
= [Z,[X, Y]] + [Y,[Z, X]].
This proves Lemma 2.4.21.

50 CHAPTER 2. FOUNDATIONS
Definition 2.4.22.ALie algebrais a real vector spacegequipped with
a skew-symmetric bilinear mapg×g→g: (ξ, η)7→[ξ, η]that satisfies the
Jacobi identity[ξ,[η, ζ]] + [η,[ζ, ξ]] + [ζ,[ξ, η]] = 0for allξ, η, ζ∈g.
Example 2.4.23.The Vector fields on a smooth manifoldM⊂R
k
form a
Lie algebra with the Lie bracket (2.4.17). The spacegl(n,R) =R
n×n
of real
n×n-matrices is a Lie algebra with the Lie bracket
[ξ, η] :=ξη−ηξ.
It is also interesting to consider subspaces ofgl(n,R) that are invariant under
this Lie bracket. An example is the space
o(n) :=
n
ξ∈gl(n,R)

ξ
T
+ξ= 0
o
of skew-symmetricn×n-matrices. It is a nontrivial fact that every finite-
dimensional Lie algebra is isomorphic to a Lie subalgebra ofgl(n,R) for
somen. For example, the cross product defines a Lie algebra structure
onR
3
and the resulting Lie algebra is isomorphic too(3).
Remark 2.4.24.There is a linear mapR
m×m
→Vect(R
m
) :ξ7→Xξwhich
assigns to a matrixξ∈gl(m,R) the linear vector fieldXξ:R
m
→R
m
given byXξ(x) :=ξxforx∈R
m
. This map preserves the Lie bracket,
i.e. [Xξ, Xη] =X
[ξ,η], and hence is aLie algebra homomorphism.
To understand the Lie bracket geometrically, consider again the curve
γ(t) :=ϕ
t
◦ψ
t
◦ϕ
−t
◦ψ
−t
(p)
in Lemma 2.4.18, whereϕ
t
andψ
t
are the flows of the vector fieldsXandY,
respectively. Since ˙γ(0) = 0, Exercise 2.4.19 asserts that
[X, Y](p) =
1
2
¨γ(0) =
d
dt

t=0
ϕ
√
t
◦ψ
√
t
◦ϕ
−
√
t
◦ψ
−
√
t
(p). (2.4.21)
Geometrically this means that by following first the backward flow ofY
for timeε, then the backward flow ofXfor timeε, then the forward flow
ofYfor timeε, and finally the forward flow ofXfor timeε, we will not, in
general, get back to the original pointpwhere we started but approximately
obtain an“error”ε
2
[X, Y](p). An example of this (which we learned from
Donaldson) is the mathematical formulation of parking a car.

2.4. VECTOR FIELDS AND FLOWS 51
Example 2.4.25(Parking a car).The configuration space for driving a
car in the plane is the manifoldM:=C×S
1
, whereS
1
⊂Cdenotes the
unit circle. Thus a point inMis a pairp= (z, λ)∈C×Cwith|λ|= 1. The
pointz∈Crepresents the position of the car and the unit vectorλ∈S
1
represents the direction in which it is pointing. Theleft turnis represented
by a vector fieldXand theright turnby a vector fieldYonM. These
vector field are given byX(z, λ) := (λ,iλ) andY(z, λ) := (λ,−iλ).Their
Lie bracket is the vector field [X, Y](z, λ) = (−2iλ,0).This vector field
represents a sideways move of the car to the right. And a sideways move
by 2ε
2
can be achieved by following a backward right turn for timeε, then
a backward left turn for timeε, then a forward right turn for timeε, and
finally a forward left turn for timeε.
This example can be reformulated by identifyingCwithR
2
viaz=x+iy
and representing a point in the unit circle by the angleθ∈R/2πZvia
λ=e
iθ
. In this formulation the manifold isM=R
2
×R/2πZ, a point inM
is represented by a triple (x, y, θ)∈R
3
, the vector fieldsXandYare
X(x, y, θ) := (cos(θ),sin(θ),1), Y(x, y, θ) := (cos(θ),sin(θ),−1),
and their Lie bracket is [X, Y](x, y, θ) = 2(sin(θ),−cos(θ),0).
Lemma 2.4.26.LetX, Y∈Vect(M)be complete vector fields on a man-
ifoldMandϕ
t
, ψ
t
∈Diff(M)be the flows ofXandY, respectively. Then
the Lie bracket[X, Y]vanishes if and only if the flows ofXandYcommute,
i.e.ϕ
s
◦ψ
t
=ψ
t
◦ϕ
s
for alls, t∈R.
Proof.If the flows ofXandYcommute, then the Lie bracket [X, Y] vanishes
by Lemma 2.4.18. Conversely, suppose that [X, Y] = 0. Then we have
d
ds
(ϕ
s
)
∗
Y= (ϕ
s
)
∗
d
dr

r=0
(ϕ
r
)
∗
Y= (ϕ
s
)
∗
[X, Y] = 0
for everys∈Rand hence
(ϕ
s
)∗Y=Y. (2.4.22)
Fix a real numbersand define the curveγ:R→Mbyγ(t) :=ϕ
s
(ψ
t
(p))
fort∈R. Thenγ(0) =ϕ
s
(p) and
˙γ(t) =dϕ
s
(ψ
t
(p))Y(ψ
t
(p)) = ((ϕ
s
)
∗
Y) (γ(t)) =Y(γ(t))
for allt. Here the last equation follows from (2.4.22). Sinceψ
t
is the flow of
Ywe obtainγ(t) =ψ
t
(ϕ
s
(p)) for allt∈Rand this proves Lemma 2.4.26.
Exercise 2.4.27.In the situation of Lemma 2.4.26 prove that{ϕ
t
◦ψ
t
}t∈R
is the flow of the vector fieldX+Y.

52 CHAPTER 2. FOUNDATIONS
2.5 Lie Groups
Combining the concept of a group and a manifold, it is interesting to consider
groups which are also manifolds and have the property that the group op-
eration and the inverse define smooth maps. We shall only consider groups
of matrices.
2.5.1 Definition and Examples
Definition 2.5.1(Lie group).A nonempty subsetG⊂R
n×n
is called a
Lie groupiff it is a submanifold ofR
n×n
and a subgroup ofGL(n,R), i.e.
g, h∈G = ⇒ gh∈G
(whereghdenotes the product of the matricesgandh) and
g∈G = ⇒ det(g)̸= 0andg
−1
∈G.
(SinceG̸=∅it follows from these conditions that the identity matrix1lis
an element ofG.)
Example 2.5.2.The general linear group G = GL(n,R) is an open subset
ofR
n×n
and hence is a Lie group. By Exercise 2.1.19 the special linear group
SL(n,R) =
Φ
g∈GL(n,R)

det(g) = 1

is a Lie group and, by Example 2.1.20, the special orthogonal group
SO(n) :=
n
g∈GL(n,R)

g
T
g= 1l,det(g) = 1
o
is a Lie group. In fact every orthogonal matrix has determinant±1 and
so SO(n) is an open subset of O(n) (in the relative topology).
In a similar vein the group GL(n,C) :={g∈C
n×n
|det(g)̸= 0}of com-
plex matrices with nonzero (complex) determinant is an open subset ofC
n×n
and hence is a Lie group. As in the real case, the subgroups
SL(n,C) :=
Φ
g∈GL(n,C)

det(g) = 1

,
U(n) :=
Φ
g∈GL(n,C)

g
∗
g= 1l

,
SU(n) :=
Φ
g∈GL(n,C)

g
∗
g= 1l,det(g) = 1

are submanifolds of GL(n,C) and hence are Lie groups. Hereg
∗
:= ¯g
T
denotes the conjugate transpose of a complex matrix.
Exercise 2.5.3.Prove that SL(n,C), U(n), and SU(n) are Lie groups.
Prove that SO(n) is connected and that O(n) has two connected components.

2.5. LIE GROUPS 53
Exercise 2.5.4.Prove that GL(n,C) can be identified with the group
G :={Φ∈GL(2n,R)|ΦJ0=J0Φ}, J 0:=
`
0−1l
1l 0
´
.
Hint:Use the isomorphismR
n
×R
n
→C
n
: (x, y)7→x+iy. Show that a
matrix Φ∈R
2n×2n
commutes withJ0if and only if it has the form
Φ =
`
X−Y
Y X
´
, X, Y∈R
n×n
.
What is the relation between the real determinant of Φ and the complex
determinant ofX+iY?
Exercise 2.5.5.LetJ0be as in Exercise 2.5.4 and define
Sp(2n) :=
n
Ψ∈GL(2n,R)

Ψ
T
J0Ψ =J0
o
.
This is thesymplectic linear group. Prove that Sp(2n) is a Lie group.
Hint:See [49, Lemma 1.1.12].
Example 2.5.6(Unit quaternions).Thequaternionsform a four-
dimensional associative unital algebraH, equipped with a basis 1,i,j,k.
The elements ofHare vectors of the form
x=x0+ix1+jx2+kx3 x0, x1, x2, x3∈R. (2.5.1)
The product structure is the bilinear mapH×H→H: (x, y)7→xy, deter-
mined by the relations
i
2
=j
2
=k
2
=−1,ij=−ji=k,jk=−kj=i,ki=−ik=j.
This product structure is associative but not commutative. The quaternions
are equipped with an involutionH→H:x7→¯x, which assigns to a quater-
nionxof the form (2.5.1) itsconjugate¯x:=x0−ix1−jx2−kx3. This
involution satisfies the conditions
x+y= ¯x+ ¯y,xy= ¯y¯x, x¯x=|x|
2
,|xy|=|x| |y|
forx, y∈H, where|x|:=
p
x
2
0
+x
2
2
+x
2
2
+x
2
3
denotes the Euclidean norm
of the quaternion (2.5.1). Thus theunit quaternionsform a group
Sp(1) :=
Φ
x∈H

|x|= 1

with the inverse mapx7→¯x. Note that the group Sp(1) is diffeomorphic
to the 3-sphereS
3
⊂R
4
under the isomorphismH
∼
=R
4
.Warning:The
unit quaternions (a compact Lie group) are not to be confused with the
symplectic linear group in Exercise 2.5.5 (a noncompact Lie group) despite
the similarity in notation.

54 CHAPTER 2. FOUNDATIONS
Let G⊂GL(n,R) be a Lie group. Then the maps
G×G→G : (g, h)7→gh, G→G :g7→g
−1
are smooth (see [64]). Fixing an elementh∈G we find that the derivative
of the map G→G :g7→ghatg∈G is given by the linear map
TgG→TghG :bg7→bgh. (2.5.2)
Herebgandhare both matrices inR
n×n
andbghdenotes the matrix prod-
uct. In fact, ifbg∈TgG, then, since G is a manifold, there exists a smooth
curveγ:R→G withγ(0) =gand ˙γ(0) =bg. Since G is a group we obtain
a smooth curveβ:R→G given byβ(t) :=γ(t)h. It satisfiesβ(0) =ghand
sobgh=
˙
β(0)∈TghG.
The linear map (2.5.2) is obviously a vector space isomorphism whose
inverse is given by right multiplication withh
−1
. It is sometimes convenient
to define the mapRh: G→G by
Rh(g) :=gh
forg∈G (right multiplicationbyh). This is a diffeomorphism and the linear
map (2.5.2) is the derivative ofRhatg, so
dRh(g)bg=bgh forbg∈TgG.
Similarly, each elementg∈G determines a diffeomorphismLg: G→G,
given by
Lg(h) :=gh
forh∈G (left multiplicationbyg). Its derivative ath∈G is again given by
matrix multiplication, i.e. the linear mapdLg(h) :ThG→TghG is given by
dLg(h)
b
h=g
b
h for
b
h∈ThG. (2.5.3)
SinceLgis a diffeomorphism its derivativedLg(h) :ThG→TghG is again a
vector space isomorphism for everyh∈G.
Exercise 2.5.7.Prove that the map G→G :g7→g
−1
is a diffeomorphism
and that its derivative atg∈G is the vector space isomorphism
TgG→T
g
−1G :v7→ −g
−1
vg
−1
.
Hint:Use [64] or any textbook on first year analysis.

2.5. LIE GROUPS 55
2.5.2 The Lie Algebra of a Lie Group
Let
G⊂GL(n,R)
be a Lie group. Its tangent space at the identity matrix 1l∈G is called the
Lie algebraof G and will be denoted by
g= Lie(G) :=T1lG.
This terminology is justified by the fact thatgis in fact a Lie algebra, i.e.
it is invariant under the standard Lie bracket operation
[ξ, η] :=ξη−ηξ
on the spaceR
n×n
of square matrices (see Lemma 2.5.9 below). The proof
requires the notion of theexponential matrix. Forξ∈R
n×n
andt∈R
we define
exp(tξ) :=
∞
X
k=0
t
k
ξ
k
k!
. (2.5.4)
A standard result in first year analysis asserts that this series converges
absolutely (and uniformly on compactt-intervals), that the map
R→R
n×n
:t7→exp(tξ)
is smooth and satisfies the differential equation
d
dt
exp(tξ) =ξexp(tξ) = exp(tξ)ξ, (2.5.5)
and that
exp((s+t)ξ) = exp(sξ) exp(tξ),exp(0ξ) = 1l (2.5.6)
for alls, t∈R. This shows that the matrix exp(tξ) is invertible for eacht
and that the mapR→GL(n,R) :t7→exp(tξ) is a group homomorphism.
Exercise 2.5.8.Prove the following analogue of (2.4.12). Forξ, η∈g
d
dt

t=0
exp(
√
tξ) exp(
√
tη) exp(−
√
tξ) exp(−
√
tη) = [ξ, η]. (2.5.7)
In other words, the infinitesimal Lie group commutator is the matrix com-
mutator. (Compare Equations (2.5.7) and (2.4.21).)

56 CHAPTER 2. FOUNDATIONS
Lemma 2.5.9.LetG⊂GL(n,R)be a Lie group and denote byg:= Lie(G)
its Lie algebra. Then the following holds.
(i)Ifξ∈g, thenexp(tξ)∈Gfor everyt∈R.
(ii)Ifg∈Gandη∈g, thengηg
−1
∈g.
(iii)Ifξ, η∈g, then[ξ, η] =ξη−ηξ∈g.
Proof.We prove (i). For everyg∈G we have a vector space isomor-
phismg=T1lG→TgG :ξ7→ξgas in (2.5.2). Hence each elementξ∈g
determines a vector fieldXξ∈Vect(G), defined by
Xξ(g) :=ξg∈TgG, g∈G. (2.5.8)
By Theorem 2.4.7 there is an integral curveγ: (−ε, ε)→G satisfying
˙γ(t) =Xξ(γ(t)) =ξγ(t), γ(0) = 1l.
By (2.5.5), the curve (−ε, ε)→R
n×n
:t7→exp(tξ) satisfies the same initial
value problem and hence, by uniqueness, we have exp(tξ) =γ(t)∈G for
allt∈Rwith|t|< ε. Now lett∈Rand chooseN∈Nsuch that

t
N

< ε.
Then exp(
t
N
ξ)∈G and hence it follows from (2.5.6) that
exp(tξ) = exp
`
t
N
ξ
´
N
∈G.
This proves (i).
We prove (ii). Consider the smooth curveγ:R→R
n×n
defined by
γ(t) :=gexp(tη)g
−1
.
By (i) we haveγ(t)∈G for everyt∈R. Sinceγ(0) = 1l we have
gηg
−1
= ˙γ(0)∈g.
This proves (ii).
We prove (iii). Define the smooth mapη:R→R
n×n
by
η(t) := exp(tξ)ηexp(−tξ).
By (i) we have exp(tξ)∈G and, by (ii), we haveη(t)∈gfor everyt∈R.
Hence [ξ, η] = ˙η(0)∈g.This proves (iii) and Lemma 2.5.9.
By Lemma 2.5.9 the curveγ:R→G defined byγ(t) := exp(tξ)gis the
integral curve of the vector fieldXξin (2.5.8) with initial conditionγ(0) =g.
ThusXξis complete for everyξ∈g.

2.5. LIE GROUPS 57
Lemma 2.5.10.Ifξ∈gandγ:R→Gis a smooth curve satisfying
γ(s+t) =γ(s)γ(t), γ(0) = 1l, ˙γ(0) =ξ, (2.5.9)
thenγ(t) = exp(tξ)for everyt∈R.
Proof.For everyt∈Rwe have
˙γ(t) =
d
ds

s=0
γ(s+t) =
d
ds

s=0
γ(s)γ(t) = ˙γ(0)γ(t) =ξγ(t).
Henceγis the integral curve of the vector fieldXξin (2.5.8) withγ(0) = 1l.
This impliesγ(t) = exp(tξ) for everyt∈R, as claimed.
Example 2.5.11.Since the general linear group GL(n,R) is an open subset
ofR
n×n
its Lie algebra is the space of all realn×n-matrices
gl(n,R) := Lie(GL(n,R)) =R
n×n
.
The Lie algebra of the special linear group is
sl(n,R) := Lie(SL(n,R)) =
Φ
ξ∈gl(n,R)

trace(ξ) = 0

(see Exercise 2.2.9) and the Lie algebra of the special orthogonal group is
so(n) := Lie(SO(n)) =
n
ξ∈gl(n,R)

ξ
T
+ξ= 0
o
=o(n)
(see Example 2.2.10).
Exercise 2.5.12.Prove that the Lie algebras of the general linear group
overC, the special linear group overC, the unitary group, and the special
unitary group are given by
gl(n,C) := Lie(GL(n,C)) =C
n×n
,
sl(n,C) := Lie(SL(n,C)) =
Φ
ξ∈gl(n,C)

trace(ξ) = 0

,
u(n) := Lie(U(n)) =
Φ
ξ∈gl(n,R)

ξ
∗
+ξ= 0

,
su(n) := Lie(SU(n)) =
Φ
ξ∈gl(n,C)

ξ
∗
+ξ= 0,trace(ξ) = 0

.
These are vector spaces over the reals. Determine their real dimensions.
Which of these are also complex vector spaces?
Remark 2.5.13.Let G⊂GL(n,R) be a subgroup. In Theorem 2.5.27
below it is shown that G is a Lie group if and only if it is a closed subset
of GL(n,R) in the relative topology. This observation can be used in many
of the examples and exercises of the present section.

58 CHAPTER 2. FOUNDATIONS
Exercise 2.5.14.LetVbe a finite-dimensional vector space. Prove that
the vector spaceg:=V×End(V) is a Lie algebra with the Lie bracket
[(u, A),(v, B)] := (Av−Bu, AB−BA) (2.5.10)
foru, v∈VandA, B∈End(V). Find the corresponding Lie group. Find
an embedding ofginto End(R×V) as a Lie subalgebra.
Exercise 2.5.15.Let (V, ω) be a 2n-dimensional symplectic vector space,
soω:V×V→Ris a nondegenerate skew-symmetric bilinear form. The
Heisenberg algebraof (V, ω) is the Lie algebrah:=V×Rwith the Lie
bracket of two elements (v, t),(v
′
, t
′
)∈V×Rdefined by
Θ
(v, t),(v
′
, t
′
)
Λ
:=
Γ
0, ω(v, v
′
)
∆
. (2.5.11)
Find a corresponding Lie group structure on H =V×R. Embed H as a
Lie subgroup into GL(n+ 2,R) and find a formula for the exponential map.
Hint:TakeV=R
n
×R
n
andω
Γ
(x, y),(x
′
, y
′
)
∆
=⟨x, y
′
⟩ − ⟨y, x
′
⟩and
define (x, y, t)·(x
′
, y
′
, t
′
) := (x+x
′
, y+y
′
, t+t
′
+⟨x, y
′
⟩).
2.5.3 Lie Group Homomorphisms
Let G,H be Lie groups andg,hbe Lie algebras. ALie group homo-
morphismfrom G to H is a smooth mapρ: G→H that is a group
homomorphism. ALie group isomorphismis a bijective Lie group ho-
momorphism whose inverse is also a Lie group homomorphism. ALie group
automorphismis a Lie group isomorphism from a Lie group to itself. A
Lie algebra homomorphism fromgtohis a linear map Φ :g→hthat
preserves the Lie bracket. ALie algebra isomorphismis a bijective Lie
algebra homomorphism whose inverse is also a Lie algebra homomorphism.
ALie algebra automorphismis a Lie algebra isomorphism from a Lie
algebra to itself.
Lemma 2.5.16.LetGandHbe Lie groups and denote their Lie algebras
byg:= Lie(G)andh:= Lie(H). Letρ: G→Hbe a Lie group homomor-
phism and denote its derivative at1l∈Gby
˙ρ:=dρ(1l) :g→h.
Then˙ρis a Lie algebra homomorphism. Moreover,
ρ(exp(ξ)) = exp( ˙ρ(ξ)), ρ(gξg
−1
) =ρ(g) ˙ρ(ξ)ρ(g)
−1
for allξ∈gand allg∈G.

2.5. LIE GROUPS 59
Proof.The proof has three steps.
Step 1.For allξ∈gandt∈Rwe haveρ(exp(tξ)) = exp(t˙ρ(ξ)).
Fix an elementξ∈g. Then exp(tξ)∈G for everyt∈Rby Lemma 2.5.9.
Thus we can define a curveγ:R→H byγ(t) :=ρ(exp(tξ)). Sinceρis
smooth, this is a smooth curve in H and, sinceρis a group homomorphism
and the exponential map satisfies (2.5.6), our curveγsatisfies the conditions
γ(s+t) =γ(s)γ(t), γ(0) = 1l, ˙γ(0) =dρ(1l)ξ= ˙ρ(ξ).
Henceγ(t) = exp(t˙ρ(ξ)) by Lemma 2.5.10. This proves Step 1.
Step 2.For allg∈Gandη∈gwe have˙ρ(gηg
−1
) =ρ(g) ˙ρ(η)ρ(g)
−1
.
Define the smooth curveγ:R→G byγ(t) :=gexp(tη)g
−1
.It takes values
in G by Lemma 2.5.9. By Step 1 we have
ρ(γ(t)) =ρ(g)ρ(exp(tη))ρ(g)
−1
=ρ(g) exp(t˙ρ(η))ρ(g)
−1
for everyt. Sinceγ(0) = 1l and ˙γ(0) =gηg
−1
we obtain
˙ρ(gηg
−1
) =dρ(γ(0)) ˙γ(0)
=
d
dt

t=0
ρ(γ(t))
=
d
dt

t=0
ρ(g) exp(t˙ρ(η))ρ(g)
−1
=ρ(g) ˙ρ(η)ρ(g)
−1
.
This proves Step 2.
Step 3.For allξ, η∈gwe have˙ρ([ξ, η]) = [ ˙ρ(ξ),˙ρ(η)].
Define the curveη:R→gbyη(t) := exp(tξ)ηexp(−tξ) fort∈R. It takes
values in the Lie algebra of G by Lemma 2.5.9 and ˙η(0) = [ξ, η].Hence
˙ρ([ξ, η]) =
d
dt

t=0
˙ρ(exp(tξ)ηexp(−tξ))
=
d
dt

t=0
ρ(exp(tξ)) ˙ρ(η)ρ(exp(−tξ))
=
d
dt

t=0
exp (t˙ρ(ξ)) ˙ρ(η) exp (−t˙ρ(ξ))
= [ ˙ρ(ξ),˙ρ(η)].
Here the first equality follows from the fact that ˙ρis linear, the second
equality follows from Step 2 withg= exp(tξ), and the third equality follows
from Step 1. This proves Step 3 and Lemma 2.5.16.

60 CHAPTER 2. FOUNDATIONS
Exercise 2.5.17.A Lie group homomorphismρ: G→H is uniquely deter-
mined by the Lie algebra homomorphism ˙ρwhenever G is connected.Hint:
Ifρ1, ρ2: G→H are Lie group homomorphisms such that ˙ρ1= ˙ρ2, prove
that the setA:={g∈G|ρ1(g) =ρ2(g)}is both open and closed.
Exercise 2.5.18.If ˙ρ:g→his a bijective Lie algebra homomorphism, then
its inverse is also a Lie algebra homomorphism.
Exercise 2.5.19.Ifρ: G→H is a bijective Lie group homomorphism,
thenρ
−1
: H→G is smooth and henceρis a Lie group isomorphism.Hint:
Use Lemma 2.5.16 to prove that ˙ρ:g→his injective. If ˙ρis not surjective,
show thatρhas no regular value in contradiction to Sard’s theorem.
Example 2.5.20.The complex determinant defines a Lie group homomor-
phism det : U(n)→S
1
. The associated Lie algebra homomorphism is
trace =
˙
det :u(n)→iR= Lie(S
1
).
Example 2.5.21(Unit quaternions andSU(2)).The Lie group SU(2)
is diffeomorphic to the 3-sphere. Every matrix in SU(2) can be written as
g=
`
x0+ix1x2+ix3
−x2+ix3x0−ix1
´
, x
2
0+x
2
1+x
2
2+x
2
3= 1.(2.5.12)
Here thexiare real numbers. They can be interpreted as the coordinates
of a unit quaternionx=x0+ix1+jx2+kx3∈Sp(1) (see Example 2.5.6).
The reader may verify that the map Sp(1)→SU(2) :x7→gin (2.5.12) is a
Lie group isomorphism.
Exercise 2.5.22(The double cover ofSO(3)).Identify the imaginary
part ofHwithR
3
and write a vectorξ∈R
3
= Im(H) as a purely imaginary
quaternionξ=iξ1+jξ1+kξ3.Prove that ifξ∈Im(H) andx∈Sp(1),
thenxξ¯x∈Im(H). Define the mapρ: Sp(1)→SO(3) byρ(x)ξ:=xξ¯x
forx∈Sp(1) andξ∈Im(H). Prove that the linear mapρ(x) :R
3
→R
3
is
represented by the 3×3-matrix
ρ(x) =


x
2
0
+x
2
1
−x
2
2
−x
2
3
2(x1x2−x0x3) 2(x1x3+x0x2)
2(x1x2+x0x3)x
2
0
+x
2
2
−x
2
3
−x
2
1
2(x2x3−x0x1)
2(x1x3−x0x2) 2(x2x3+x0x1)x
2
0
+x
2
3
−x
2
1
−x
2
2

.
Show thatρis a Lie group homomorphism. Find a formula for the map
˙ρ:=dρ(1l) :sp(1)→so(3)
and show that it is a Lie algebra isomorphism. Forx, y∈Sp(1) prove
thatρ(x) =ρ(y) if and only ify=±x.

2.5. LIE GROUPS 61
Example 2.5.23.Letgbe a finite-dimensional Lie algebra. Then the set
Aut(g) :=



Φ :g→g

Φ is a bijective linear map and
Φ[ξ, η] = [Φξ,Φη]
for allξ, η∈g



(2.5.13)
ofLie algebra automorphismsofgis a Lie group. Its Lie algebra is the
space ofderivationsongdenoted by
Der(g) :=



δ:g→g

δis a linear map and
δ[ξ, η] = [δ ξ, η] + [ξ, δ η]
for allξ, η∈g



. (2.5.14)
Now suppose thatg= Lie(G) is the Lie algebra of a Lie group G. Then
there is a map Ad : G→Aut(g) defined by
Ad(g)η:=gηg
−1
(2.5.15)
forg∈G andη∈g. Part (ii) of Lemma 2.5.9 asserts that Ad(g) mapsg
to itself for everyg∈G. It follows directly from the definitions that the
map Ad(g) :g→gis a Lie algebra automorphism for everyg∈G and that
the map Ad : G→Aut(g) is a Lie group homomorphism. The associated
Lie algebra homomorphism is the linear map ad :g→Der(g) defined by
ad(ξ)η:= [ξ, η] (2.5.16)
forξ, η∈g. To verify the equation ad =
˙
Ad we compute
˙
Ad(ξ)η=
d
dt

t=0
Ad(exp(tξ))η=
d
dt

t=0
exp(tξ)ηexp(−tξ) = [ξ, η].
Exercise 2.5.24.Letgbe any Lie algebra. Define the map ad :g→End(g)
by (2.5.16) and prove that the endomorphism ad(ξ) :g→gis a derivation
for everyξ∈g. Prove that ad :g→Der(g) is a Lie algebra homomorphism.
Exercise 2.5.25.Letgbe any finite-dimensional Lie algebra. Prove that
the group Aut(g) in (2.5.13) is a Lie subgroup of GL(g) with the Lie al-
gebra Lie(Aut(g)) = Der(g).Hint:Show that a linear mapδ:g→gis
a derivation if and only if the linear map exp(tδ) :g→gis a Lie algebra
automorphism for everyt∈R. Use the Closed Subgroup Theorem 2.5.27.

62 CHAPTER 2. FOUNDATIONS
2.5.4 Closed Subgroups
This section deals with subgroups of a Lie group G that are also submanifolds
of G. Such subgroups are called Lie subgroups. We assume throughout
that G⊂GL(n,R) is a Lie group with the Lie algebrag:= Lie(G) =T1lG.
Definition 2.5.26(Lie subgroup).A subsetH⊂Gis called aLie sub-
group ofGiff it is both a subgroup and a smooth submanifold ofG.
A useful general criterion is the Closed Subgroup Theorem which asserts
that a subgroup H⊂G is a Lie subgroup if and only if it is a closed subset
of G. This was first proved in 1929 by John von Neumann [54] for the special
case G = GL(n,R) and then in 1930 by
´
Elie Cartan [15] in full generality.
Theorem 2.5.27(Closed Subgroup Theorem). LetHbe a subgroup
ofG. Then the following are equivalent.
(i)His a smooth submanifold (and hence a Lie subgroup) ofG.
(ii)His a closed subset ofG.
It (i) holds, then the Lie algebra ofHis the space
h:=
Φ
η∈g

exp(tη)∈Hfor allt∈R

. (2.5.17)
Proof of Theorem 2.5.27 (i)=⇒(ii) and(2.5.17).Assume that H is a Lie
subgroup of G and leth⊂gbe defined by (2.5.17). We prove thathis
the Lie algebra of H. Assume first thatη∈h. Then the curveγ:R→G
defined byγ(t) := exp(tη) fort∈Rtakes values in H and satisfiesγ(0) = 1l
and ˙γ(0) =η, and this impliesη∈T1lH = Lie(H). Conversely, ifη∈Lie(H),
then Lemma 2.5.9 asserts that exp(tη)∈H for allt∈Rand henceη∈h.
This shows thath= Lie(H).
Next we prove in three steps that H is a closed subset of G. Choose any
inner product ong, denote by|·|the associated norm, and denote byh
⊥
⊂g
the orthogonal complement ofhwith respect to this inner product.
Step 1.There exist open neighborhoodsV⊂Hof1landW⊂h
⊥
of the
origin such that the mapϕ:V×W→G, defined by
ϕ(h, ξ) :=hexp(ξ)
forh∈Vandξ∈W, is a diffeomorphism fromV×Wonto an open neigh-
borhoodU=ϕ(V×W)⊂Gof1l.
The derivative of the map H×h
⊥
→G : (h, ξ)7→hexp(ξ) at the point (1l,0)
is bijective. Hence Step 1 follows from the Inverse Function Theorem.

2.5. LIE GROUPS 63
Step 2.There exists aδ >0such that, ifξ, ξ
′
∈h
⊥
satisfy|ξ|,|ξ
′
|< δ
andexp(ξ
′
) exp(−ξ)∈H, thenξ=ξ
′
.
LetV, W, ϕbe as in Step 1, choose an open neigborhoodV
′
⊂G of 1l such
thatV
′
∩H =V, and choose a constantδ >0 such that the following holds.
(a)Ifξ∈h
⊥
satisfies|ξ|< δ, thenξ∈W.
(b)Ifξ, ξ
′
∈gsatisfy|ξ|,|ξ
′
|< δ, then exp(ξ
′
) exp(−ξ)∈V
′
.
Letξ, ξ
′
∈h
⊥
such that|ξ|,|ξ
′
|< δandh:= exp(ξ
′
) exp(−ξ)∈H. Then we
haveξ, ξ
′
∈Wby (a) andh∈V
′
∩H =Vby (b). Alsoϕ(h, ξ) =ϕ(1l, ξ
′
)
and soξ=ξ
′
, becauseϕis injective onV×W. This proves Step 2.
Step 3.Lethibe a sequence inHthat converges to an elementg∈G.
Theng∈H.
Letϕ:V×W→Ube as in Step 1 and letδ >0 be as in Step 2. Since the
sequenceh
−1
i
gconverges to 1l, there exists ani0∈Nsuch thath
−1
i
g∈Ufor
alli≥i0. Hence, for eachi≥i0, there exists a unique pair (h
′
i
, ξi)∈V×W
such thath
−1
i
g=h
′
i
exp(ξi). This sequence satisfies limi→∞ξi= 0. Hence
there exists an integeri1≥i0such that|ξi|< δfor alli≥i1. Since
hih
′
iexp(ξi) =g=hjh
′
jexp(ξj),
we also have exp(ξi) exp(−ξj) = (hih
′
i
)
−1
hjh
′
j
∈H for alli, j≥i1. By Step 3,
this impliesξi=ξjfor alli, j≥i1. Henceξi= limj→∞ξj= 0 for alli≥i1
and sog=hih
′
i
∈H. This proves Step 3.
By Step 3 the Lie subgroup H is a closed subset of G. Thus we have
proved that (i) implies (ii) and (2.5.17) in Theorem 2.5.27.
The proof of the converse implication requires three preparatory lemmas.
Lemma 2.5.28.Letξ∈gand letγ:R→Gbe a curve that is differentiable
att= 0and satisfiesγ(0) = 1land˙γ(0) =ξ. Then
exp(tξ) = lim
k→∞
γ(t/k)
k
(2.5.18)
for everyt∈R.
Proof.Fix a nonzero real numbertand defineξk:=k
Γ
γ(t/k)−1l
∆
∈R
n×n
fork∈N. Then
lim
k→∞
ξk=tlim
k→∞
γ(t/k)−γ(0)
t/k
=t˙γ(0) =tξ
and hence
exp(tξ) = lim
k→∞
`
1l +
ξk
k
´
k
= lim
k→∞
γ(t/k)
k
.
(See [64, Satz 1.5.2].) This proves Lemma 2.5.28.

64 CHAPTER 2. FOUNDATIONS
Lemma 2.5.29.LetH⊂Gbe a closed subgroup. Then the sethin(2.5.17)
is a Lie subalgebra ofg
Proof.Letξ, η∈hand define the curveγ:R→H by
γ(t) := exp(tξ) exp(tη)
fort∈R. This curve is smooth and satisfiesγ(0) = 1l and ˙γ(0) =ξ+η.
Since H is closed, it follows from Lemma 2.5.28 that
exp(t(ξ+η)) = lim
k→∞
γ(t/k)
k
∈H
for allt∈Rand soξ+η∈hby definition. Thushis a linear subspace ofg.
Now fix an elementξ∈h. Ifh∈H, then
exp(sh
−1
ξh) =h
−1
exp(sξ)h∈H
for alls∈Rand henceh
−1
ξh∈hby definition. Takeh= exp(tη) withη∈h
to obtain exp(−tη)ξexp(tη)∈hfor allt∈R. Differentiating this curve
att= 0 gives [ξ, η]∈hand this proves Lemma 2.5.29.
Lemma 2.5.30.LetH⊂Gbe a closed subgroup and leth⊂gbe the Lie
subalgebra in(2.5.17). Letξ∈g, let(ξi)i∈Nbe a sequence ing, and let(τi)i∈N
be a sequence of positive real numbers such that
exp(ξi)∈H, ξi̸= 0
for alli∈Nand
lim
i→∞
τi= 0, lim
i→∞
ξi= 0, lim
i→∞
ξi
τi
=ξ.
Thenξ∈h.
Proof.Fix a real numbert. Then, for eachi∈N, there exists a unique
integermi∈Zsuch thatmiτi≤t <(mi+ 1)τi.The sequencemisatisfies
lim
i→∞
miτi=t, lim
i→∞
miξi= lim
i→∞
miτi
ξi
τi
=tξ
and hence
exp(tξ) = lim
i→∞
exp(miξi) = lim
i→∞
exp(ξi)
mi
∈H.
Thus exp(tξ)∈H for everyt∈Rand soξ∈hby (2.5.17). This proves
Lemma 2.5.30.

2.5. LIE GROUPS 65
Proof of Theorem 2.5.27 (ii)=⇒(i).Choose any inner product ong. Let
H⊂G be a closed subgroup of G and define the seth⊂gby (2.5.17).
Thenhis a Lie subalgebra ofgby Lemma 2.5.29. Define
k:= dim(h), ℓ:= dim(g)≥k,
and choose a basisη1, . . . , ηℓofgsuch that the vectorsη1. . . . , ηkform a basis
ofhandην∈h
⊥
forν > k. Leth0∈H and define the map Θ :R
ℓ
→G by
Θ(t
1
, . . . , t
ℓ
) :=h0exp(t
1
η1+· · ·+t
k
ηk) exp(t
k+1
ηk+1+· · ·+t
ℓ
ηℓ).
Then Θ(0) =h0, Θ(R
k
× {0})⊂H, and the derivativedΘ(0) :R
ℓ
→Th0
G
is bijective. Hence the inverse function theorem asserts that Θ restricts to a
diffeomorphism from an open neighborhood Ω⊂R
ℓ
of the origin to an open
neighborhoodU:= Θ(Ω)⊂G ofh0that satisfies
Θ(0) =h0,Θ
Γ
Ω∩(R
k
× {0})
∆
⊂U∩H.
We prove the following.
Claim.There exists an open setΩ0⊂R
ℓ
such that
0∈Ω0⊂Ω,Θ
Γ
Ω0∩(R
k
× {0})
∆
=U0∩H, U 0:= Θ(Ω0).(2.5.19)
Assume, by contradiction, that such an open set Ω0does not exist. Then
there exists a sequenceti= (t
1
i
, . . . , t
ℓ
i
)∈R
ℓ
such that
lim
i→∞
ti= 0, ti∈Ω\(R
k
× {0}),Θ(ti)∈H.
Define
hi:=h0exp

k
X
ν=1
t
ν
iην
!
∈H, ξi:=
ℓ
X
ν=k+1
t
ν
iην∈h
⊥
\ {0}.
Thenhiexp(ξi) = Θ(ti)∈H and hence
lim
i→∞
ξi= 0, ξi̸= 0,exp(ξi) =h
−1
i
Θ(ti)∈H.
Passing to a subsequence, if necessary, we may assume that the sequence
ξi/|ξi|converges. Denote its limit byξ:= limi→∞ξi/|ξi|.Thenξ∈hby
Lemma 2.5.30 andξ∈h
⊥
by definition. Since|ξ|= 1, this is a contradiction.
This contradiction proves the Claim. Thus there does, after all, exist an
open set Ω0⊂R
ℓ
that satisfies (2.5.19), and the map Θ
−1
:U0→Ω0is then
a coordinate chart on G which satisfies Θ
−1
(U0∩H) = Ω0∩(R
k
× {0}).
Hence H is a submanifold of G and this proves Theorem 2.5.27.

66 CHAPTER 2. FOUNDATIONS
Exercise 2.5.31.The subgroup{exp(it)|t∈Q} ⊂S
1
is not closed.
Exercise 2.5.32.Choose a nonzero vector (ω1, . . . , ωn)∈R
n
such that at
least one of the ratiosωi/ωjis irrational. Prove that the subgroup
Sω:={(e
2πitω1
, e
2πitω2
, . . . , e
2πitωn
)|t∈R} ⊂(S
1
)
n∼
=T
n
of the torus is not closed. Similar examples exist in any Lie group that
contains a torus of dimension at least two.
Exercise 2.5.33.Let G0and G1be Lie subgroups of GL(n,R) with the
Lie algebrasg0:= Lie(G0) andg1:= Lie(G1). Prove that G := G0∩G1is a
Lie subgroup of GL(n,R) with the Lie algebrag=g0∩g1.
Exercise 2.5.34(Center).Thecenter of a groupG is the subgroup
Z(G) :=
Φ
g∈G

gh=hgfor allh∈G

. (2.5.20)
Let G⊂GL(n,R) be a Lie group. Prove that its centerZ(G) is a Lie sub-
group of G. If G is connected, prove that the Lie algebra of the centerZ(G)
is thecenter of the Lie algebrag= Lie(G), defined by
Z(g) :=
Φ
ξ∈g

[ξ, η] = 0 for allη∈g

. (2.5.21)
Hint:If G is connected, prove that an elementξ∈gsatisfies [ξ, η] = 0 for
allη∈gif and only if exp(tξ)h=hexp(tξ) for allt∈Rand allh∈G.
Exercise 2.5.35.Let G⊂GL(n,R) be a compact Lie group with the Lie
algebrag:= Lie(G) and letξ∈g. Prove that the set Tξ:=
Φ
exp(tξ)

t∈R

is a closed, connected, abelian subgroup of G and deduce that it is a Lie
subgroup of G (called thetorus generated byξ).
Exercise 2.5.36.Let G⊂GL(n,R) be a Lie group with the Lie algebrag
and letξ:R→gbe a smooth function. Prove that the differential
equation ˙γ(t) =ξ(t)γ(t),γ(0) = 1l, has a unique solutionγ:R→G.
Hint:Prove the existence of a solutionγ:R→GL(n,R) and show that
the set{t∈R|γ(t)∈G}is open and closed.
Remark 2.5.37(Malcev’s Theorem).Let G⊂GL(n,R) be a Lie group
with the Lie algebragand leth⊂gbe a Lie subalgebra. Then the set
H :=



h(1)

h: [0,1]→G is a smooth path
such thath(0) = 1l and
˙
h(t)h(t)
−1
∈hfor allt∈[0,1]



(2.5.22)
is a subgroup of G, called theintegral subgroup ofh. A theorem by
Anatolij Ivanovich Malcev [47] (see also [28, Corollary 13.4.6]) asserts that H
is a Lie subgroup of G if and only ifTη:=
Φ
exp(tη)

t∈R

⊂H for allη∈h.

2.5. LIE GROUPS 67
2.5.5 Lie Groups and Diffeomorphisms
There is a natural correspondence between Lie groups and Lie algebras on
the one hand and diffeomorphisms and vector fields on the other hand. We
summarize this correspondence in the following table.
Lie groups Diffeomorphisms
G⊂GL(n,R) Diff( M)
g= Lie(G) =T1lG Vect( M) =TidDiff(M)
exponential map flow of a vector field
t7→exp(tξ) t7→ϕ
t
= “ exp(tX)
′′
adjoint representation pushforward
ξ7→gξg
−1
X7→ϕ∗X
Lie bracket ong Lie bracket of vector fields
[ξ, η] =ξη−ηξ [X, Y] =dX·Y−dY·X
To understand the correspondence between the exponential map and the
flow of a vector field compare equation (2.4.6) with equation (2.5.5). To un-
derstand the correspondence between the adjoint representation and push-
forward observe that
ϕ∗Y=
d
dt

t=0
ϕ◦ψ
t
◦ϕ
−1
, gηg
−1
=
d
dt

t=0
gexp(tη)g
−1
,
whereψ
t
denotes the flow ofY. To understand the correspondence between
the Lie brackets recall that
[X, Y] =
d
dt

t=0
(ϕ
t
)∗Y, [ξ, η] =
d
dt

t=0
exp(tξ)ηexp(−tξ),
whereϕ
t
denotes the flow ofX. We emphasize that the analogy between
Lie groups and Diffeomorphisms only works well when the manifoldMis
compact so that every vector field onMis complete. The next exercise gives
another parallel between the Lie bracket on the Lie algebra of a Lie group
and the Lie bracket of two vector fields.
Exercise 2.5.38.Let G⊂GL(n,R) be a Lie group with Lie algebragand
letξ, η∈g. Define the smooth curveγ:R→G by
γ(t) := exp(tξ) exp(tη) exp(−tξ) exp(−tη).
Show that ˙γ(0) = 0 and
1
2
¨γ(0) = [ξ, η] (cf. Exercise 2.5.8 and Lemma 2.4.18).
Exercise 2.5.39.Let G⊂GL(n,R) be a Lie group with Lie algebragand
letξ, η∈g. Show that [ξ, η] = 0 if and only if the exponential maps com-
mute, i.e. exp(sξ) exp(tη) = exp(tη) exp(sξ) = exp(sξ+tη) for alls, t∈R.
How can this observation be deduced from Lemma 2.4.26?

68 CHAPTER 2. FOUNDATIONS
Definition 2.5.40.LetM⊂R
k
be a smooth manifold and letG⊂GL(n,R)
be a Lie group. A(smooth) group action ofGonMis a smooth map
G×M→M: (g, p)7→ϕg(p) (2.5.23)
that for each pairg, h∈Gsatisfies the condition
ϕg◦ϕh=ϕgh, ϕ 1l= id. (2.5.24)
If(2.5.23)is a smooth group action, then theinfinitesimal actionof the
Lie algebrag:= Lie(G)onMis the mapg→Vect(M) :ξ7→Xξdefined by
Xξ(p) :=
d
dt

t=0
ϕ
exp(yξ)(p) (2.5.25)
forξ∈gandp∈M.
Exercise 2.5.41.Let (2.5.23) be a smooth group action of a Lie group G
on a manifoldM. Prove that
X
g
−1
ξg=ϕ
∗
gXξ, X
[ξ,η]= [Xξ, Xη] (2.5.26)
for allg∈G and allξ, η∈g= Lie(G).
Exercise 2.5.42.Show that the maps GL(m,R)×R
m
→R
m
: (gx)7→gx,
SO(m+ 1)×S
m
→S
m
: (g, x)7→gx, andR×S
1
→S
1
: (θ, z)7→e
iθ
zare
smooth group actions. Verify the formulas in (2.5.26) in these examples.
A smooth group action of a Lie group G on a manifoldMcan the thought
of as a“Lie group homomorphism”
G→Diff(M) :g7→ϕg. (2.5.27)
While the group Diff(M) is infinite-dimensional, and so cannot cannot be a
Lie group in the formal sense, it has many properties in common with Lie
groups as explained above. For example, one can define what is meant by a
smooth path in Diff(M) and extend formally the notion of a tangent vector
(as the derivative of a path through a given element of Diff(M)) to this
setting. In particular, the tangent space of Diff(M) at the identity can then
be identified with the space of vector fieldsTidDiff(M) = Vect(M), and the
infinitesimal action in (2.5.25) is the Lie algebra homomorphism associated
to the“Lie group homomorphism”(2.5.27). In fact, we have chosen the sign
in the definition of the Lie bracket of vector fields so that the mapξ7→Xξ
is a Lie algebra homomorphism and not a Lie algebra anti-homomorphism.
This will be discussed further in§2.5.7.

2.5. LIE GROUPS 69
2.5.6 Smooth Maps and Algebra Homomorphisms
LetMbe a smooth submanifold ofR
k
. Denote byF(M) :=C
∞
(M,R)
the space of smooth real valued functionsf:M→R. ThenF(M) is a
commutative unital algebra. Eachp∈Mdetermines a unital algebra ho-
momorphismεp:F(M)→Rdefined byεp(f) =f(p) forp∈M.
Theorem 2.5.43.Every unital algebra homomorphismε:F(M)→Rhas
the formε=εpfor somep∈M.
Proof.Assume thatε:F(M)→Ris an algebra homomorphism.
Claim.For allf, g∈F(M)we haveε(g) = 0 =⇒ε(f)∈f(g
−1
(0)).
Indeed, the functionf−ε(f)·1 lies in the kernel ofεand so the func-
tionh:= (f−ε(f)·1)
2
+g
2
also lies in the kernel ofε. There must be at
least one pointp∈Mwhereh(p) = 0 for otherwise 1 =ε(h)ε(1/h) = 0.
For this pointpwe havef(p) =ε(f) andg(p) = 0, hencep∈g
−1
(0), and
thereforeε(f) =f(p)∈f(g
−1
(0)). This proves the claim.
The theorem asserts that there exists ap∈Msuch that everyf∈F(M)
satisfiesε(f) =f(p). Assume, by contradiction, that this is false. Then for
everyp∈Mthere exists a functionf∈F(M) such thatf(p)̸=ε(f). Re-
placefbyf−ε(f) to obtainf(p)̸= 0 =ε(f). Now use the axiom of choice
to obtain a family of functionsfp∈F(M), one for everyp∈M, such
thatfp(p)̸= 0 =ε(fp) for allp∈M. Then the setUp:=f
−1
p(R\ {0}) is
anM-open neighborhood ofpfor everyp∈M. Choose a sequence of com-
pact setsKn⊂Msuch thatKn⊂intM(Kn+1) for allnandM=
S
n
Kn.
Then, for eachn, there is agn∈F(M) (a finite sum of the form
P
i
f
2
pi
) such
thatε(gn) = 0 andgn(q)>0 for allq∈Kn. IfMis compact, this is already
a contradiction because a positive function cannot belong to the kernel ofε.
Otherwise, choosef∈F(M) such thatf(q)≥nfor allq∈M\Knand
alln∈N. Thenε(f)∈f(g
−1
n(0))⊂f(M\Kn)⊂[n,∞) by the claim and
soε(f)≥nfor alln. This is a contradiction and proves Theorem 2.5.43.
Now letNbe another smooth submanifold (say ofR
ℓ
) and letC
∞
(M, N)
denote the space of smooth maps fromMtoN. Ahomomorphism from
F(N) toF(M) is a (real) linear map Φ :F(N)→F(M) that satisfies
Φ(fg) = Φ(f)Φ(g),Φ(1) = 1.
Let Hom(F(N),F(M)) denote the space of homomorphisms fromF(N)
toF(M). Anautomorphismof the algebraF(M) is a bijective homo-
morphism Φ :F(M)→F(M). The automorphisms ofF(M) form a group
denoted by Aut(F(M)).

70 CHAPTER 2. FOUNDATIONS
Corollary 2.5.44.The pullback operation
C
∞
(M, N)→Hom(F(N),F(M)) :ϕ7→ϕ
∗
is bijective. In particular, the mapDiff(M)→Aut(F(M)) :ϕ7→ϕ
∗
is an
anti-isomorphism of groups.
Proof.This is an exercise with hint. Let Φ :F(N)→F(M) be a unital
algebra homomorphism. By Theorem 2.5.43 there exists a mapϕ:M→N
such thatεp◦Φ =ε
ϕ(p)for allp∈M. Prove thatf◦ϕ:M→Ris smooth
for every smooth mapf:N→Rand deduce thatϕis smooth.
Remark 2.5.45.The pullback operation isfunctorial, i.e.
(ψ◦ϕ)
∗
=ϕ
∗
◦ψ
∗
,id
∗
M= id
F(M).
forϕ∈C
∞
(M, N) andψ∈C
∞
(N, P). Here id denotes the identity map
of the space indicated in the subscript. Hence Corollary 2.5.44 may be
summarized by saying that the category of smooth manifolds and smooth
maps is anti-isomorphic to a subcategory of the category of commutative
unital algebras and unital algebra homomorphisms.
Exercise 2.5.46.IfMis compact, then there is a slightly different way to
prove Theorem 2.5.43. AnidealinF(M) is a linear subspaceJ⊂F(M)
satisfying the conditionf∈F(M), g∈J=⇒fg∈J. Amaximal
idealinF(M) is an idealJ⊊F(M) such that every idealJ
′
⊊F(M)
containingJis equal toJ. Prove that, ifMis compact andJ⊂F(M)
is an ideal with the property that for everyp∈Mthere is anf∈J
withf(p)̸= 0, thenJ=F(M). Deduce that each maximal ideal inF(M)
has the formJp:={f∈F(M)|f(p) = 0}for somep∈M.
Exercise 2.5.47.IfMis compact, give another proof of Corollary 2.5.44
as follows. The set Φ
−1
(Jp) is a maximal ideal inF(N) for eachp∈M.
Use Exercise 2.5.46 to deduce that there is a unique mapϕ:M→Nsuch
that Φ
−1
(Jp) =J
ϕ(p)for allp∈M. Show thatϕis smooth andϕ
∗
= Φ.
Exercise 2.5.48.It is a theorem of ring theory that, whenI⊂Ris an ideal
in a ringR, the quotient ringR/Iis a field if and only if the idealIis max-
imal. Show that the kernel of the ring homomorphismεp:F(M)→Rof
Theorem 2.5.43 is the idealJpof Exercise 2.5.46. Conclude thatMis com-
pact if and only if every maximal idealJinF(M) is of the formJ=Jp
for somep∈M.Hint:The functions of compact support form an ideal. It
can be shown that ifMis not compact andJis a maximal ideal contain-
ing all functions of compact support, then the quotient fieldF(M)/Jis a
non-Archimedean ordered field which properly containsR.

2.5. LIE GROUPS 71
2.5.7 Vector Fields and Derivations
AderivationofF(M) is a linear mapδ:F(M)→F(M) that satisfies
δ(fg) =δ(f)g+fδ(g).
and the derivations form a Lie algebra denoted by Der(F(M)). We may
think of Der(F(M)) as the Lie algebra of Aut(F(M)) with the Lie bracket
given by the commutator. By Theorem 2.5.43 the pullback operation
Diff(M)→Aut(F(M)) :ϕ7→ϕ
∗
(2.5.28)
can be thought of as a Lie group anti-isomorphism. Differentiating it at the
identityϕ= id gives a linear map
Vect(M)→Der(F(M)) :X7→ LX. (2.5.29)
Here the operatorLX:F(M)→F(M) is given by the derivative of a
functionfin the direction of the vector fieldX, i.e.
LXf:=df·X=
d
dt

t=0
f◦ϕ
t
,
whereϕ
t
denotes the flow ofX. Since the map (2.5.29) is the derivative
of the “Lie group” anti-homomorphism (2.5.28) we expect it to be a Lie
algebra anti-homomorphism. Indeed, one can show that
L
[X,Y]=LYLX− LXLY=−[LX,LY] (2.5.30)
forX, Y∈Vect(M). This confirms that our sign in the definition of the Lie
bracket in§2.4.3 is consistent with the standard conventions in the theory
of Lie groups. In the literature the difference between a vector field and the
associated derivationLXis sometimes neglected in the notation and many
authors writeXf:=df·X=LXf, thus thinking of a vector field on a
manifoldMas an operator on the space of functions. With this notation one
obtains the equation [X, Y]f=Y(Xf)−X(Y f) and here lies the origin for
the use of the opposite sign for the Lie bracket in many books on differential
geometry.
Exercise 2.5.49.Prove that the map (2.5.29) is bijective.Hint:Fix a
derivationδ∈Der(F(M)) and prove the following.Fact 1:IfU⊂Mis
an open set andf∈F(M) vanishes onU, thenδ(f) vanishes onU.Fact 2:
Ifp∈Mand the derivativedf(p) :TpM→Ris zero, then (δ(f))(p) = 0.
(By Fact 1, the proof of Fact 2 can be reduced to an argument in local
coordinates.)
Exercise 2.5.50.Verify the formula (2.5.30).

72 CHAPTER 2. FOUNDATIONS
2.6 Vector Bundles and Submersions
This section characterizes submersions (§2.6.1) and introduces the concept
of a smooth vector bundle in the extrinsic setting (§2.6.2).
2.6.1 Submersions
LetM⊂R
k
be a smoothm-manifold andN⊂R
ℓ
be a smoothn-manifold.
A smooth mapf:N→Mis called asubmersioniff its derivative
df(q) :TqN→T
f(q)M
is surjective for everyq∈N.q
0N
gf
UM p
0
Figure 2.12: A local right inverse of a submersion.
Lemma 2.6.1.LetM⊂R
k
be a smoothm-manifold,N⊂R
ℓ
be a smooth
n-manifold, andf:N→Mbe a smooth map. The following are equivalent.
(i)fis a submersion.
(ii)For everyq0∈Nthere is anM-open neighborhoodUofp0:=f(q0)and
a smooth mapg:U→Nsuch thatg(f(q0)) =q0andf◦g= id :U→U.
Thusfhas a local right inverse near every point inN(see Figure 2.12).
Proof.We prove that (i) implies (ii). Since the derivative
df(q0) :Tq0
N→Tp0
M
is surjective we haven≥mand
dim kerdf(q0) =n−m.
Hence there is a linear mapA:R
ℓ
→R
n−m
whose restriction to the kernel
ofdf(q0) is bijective. Now define the mapψ:N→M×R
n−m
by
ψ(q) := (f(q), A(q−q0))

2.6. VECTOR BUNDLES AND SUBMERSIONS 73
forq∈N. Thenψ(q0) = (p0,0) and its derivative
dψ(q0) :Tq0
N→Tp0
M×R
n−m
sendsw∈Tq0
Nto (df(q0)w, Aw) and is therefore bijective. Hence it follows
from the inverse function theorem for manifolds (Theorem 2.2.17) that there
exists anN-open neighborhoodV⊂Nofq0such that the set
W:=ψ(N)⊂M×R
n−m
is an open neighborhood of (p0,0) andψ|V:V→Wis a diffeomorphism.
Let
U:={p∈M|(p,0)∈W}
and define the mapg:U→Nby
g(p) :=ψ
−1
(p,0).
Thenp0∈U,gis smooth and
(p,0) =ψ(g(p)) = (f(g(p)), A(g(p)−q0)).
Hencef(g(p)) =pfor allp∈Uand
g(p0) =ψ
−1
(p0,0) =q0.
This shows that (i) implies (ii). The converse is an easy consequence of the
chain rule and is left to the reader. This proves Lemma 2.6.1
Corollary 2.6.2.The image of a submersionf:N→Mis open.
Proof.Ifp0=f(q0)∈f(N), then the neighborhoodU⊂Mofp0in
Lemma 2.6.1 (ii) is contained in the image off.
Corollary 2.6.3.IfNis a nonempty compact manifold,Mis a connected
manifold, andf:N→Mis a submersion, thenfis surjective andMis
compact.
Proof.The imagef(N) is an open subset ofMby Corollary 2.6.2, it is a
relatively closed subset ofMbecauseNis compact, and it is nonempty
becauseNis nonempty. SinceMis connected this implies thatf(N) =M.
In particular,Mis compact.
Exercise 2.6.4.Letf:N→Mbe a smooth map. Prove that the
sets{q∈N|df(q) is injective}and{q∈N|df(q) is surjective}are open (in
the relative topology ofN).

74 CHAPTER 2. FOUNDATIONS
2.6.2 Vector Bundles
LetM⊂R
k
be anm-dimensional smooth manifold.
Definition 2.6.5.A(smooth) vector bundle (overMof rankn)is a
smooth submanifoldE⊂M×R
ℓ
of dimensionm+nsuch that, for every
pointp∈M, the set
Ep:=
n
v∈R
ℓ
|(p, v)∈E
o
is ann-dimensional linear subspace ofR
ℓ
(called thefiber ofEoverp).
A vector bundleEoverMis equipped with a smooth map
π:E→M
defined byπ(p, v) :=p. This map is called thecanonical projectionofE.
IfE⊂M×R
ℓ
is a vector bundle, then a(smooth) section ofEis a
smooth maps:M→R
ℓ
such thats(p)∈Epfor everyp∈M.
A sections:M→R
ℓ
of a vector bundleEoverMdetermines a smooth
mapσ:M→Ewhich sends the pointp∈Mto the pair (p, s(p))∈E.
This map satisfiesπ◦σ= id. It is sometimes convenient to abuse notation
and eliminate the distinction betweensandσ. Thus we will sometimes use
the same lettersfor the map fromMtoR
ℓ
and the map fromMtoE.
Definition 2.6.6.LetM⊂R
k
be a smoothm-manifold. The set
T M:={(p, v)|p∈M, v∈TpM}
is called thetangent bundleofM.
The tangent bundle is a subset ofM×R
k
and, for everyp∈M, its
fiberTpMis anm-dimensional linear subspace ofR
k
by Theorem 2.2.3.
However, it is not immediately obvious from the definition thatT Mis a
submanifold ofM×R
k
. This will be proved below. The sections ofT Mare
the vector fields onM.
Exercise 2.6.7.Letf:M→Nbe a smooth map between manifolds.
Prove that the tangent mapT M→T N: (p, v)7→(f(p), df(p)v) is smooth.
Exercise 2.6.8.LetM⊂R
k
be a smoothm-manifold and letϕ:U→Ω
be a smooth coordinate chart on anM-open setU⊂Mwith values in an
open set Ω =ϕ(U)⊂R
m
. Prove that the map
e
ϕ:T U→Ω×R
m
defined
by
e
ϕ(p, v) := (ϕ(p), dϕ(p)) forp∈Uandv∈TpMis a diffeomorphism. It
is called astandard coordinate chart onT M. Deduce thatT Mis
a smooth 2m-dimensional submanifold ofM×R
k
and hence is a smooth
vector bundle overM. (See also Corollary 2.6.12 below.)

2.6. VECTOR BUNDLES AND SUBMERSIONS 75
Exercise 2.6.9.LetV⊂R
ℓ
be ann-dimensional linear subspace. The
orthogonal projectionofR
ℓ
ontoVis the matrix Π∈R
ℓ×ℓ
that satisfies
Π = Π
2
= Π
T
,im Π =V. (2.6.1)
Prove that there is a unique matrix Π∈R
ℓ×ℓ
satisfying (2.6.1). Prove that,
for every symmetric matrixS=S
T
∈R
ℓ×ℓ
, the kernel ofSis the orthogonal
complement of the image ofS. IfD∈R
ℓ×n
is any injective matrix whose
image isV, prove that det(D
T
D)̸= 0 and
Π =D(D
T
D)
−1
D
T
. (2.6.2)
Theorem 2.6.10(Vector bundles).LetM⊂R
k
be a smoothm-manifold
and letE⊂M×R
ℓ
be a subset. Assume that, for everyp∈M, the set
Ep:=
n
v∈R
ℓ
|(p, v)∈E
o
(2.6.3)
is ann-dimensional linear subspace ofR
ℓ
. LetΠ :M→R
ℓ×ℓ
be the map
that assigns to eachp∈Mthe orthogonal projection ofR
ℓ
ontoEp, i.e.
Π(p) = Π(p)
2
= Π(p)
T
,im Π(p) =Ep. (2.6.4)
Then the following are equivalent.
(i)Eis a vector bundle.
(ii)For everyp0∈Mand everyv0∈Ep0
there is a smooth maps:M→R
ℓ
such thats(p0) =v0ands(p)∈Epfor allp∈M.
(iii)The mapΠ :M→R
ℓ×ℓ
is smooth.
(iv)For everyp0∈Mthere is an open neighborhoodU⊂Mofp0and a
diffeomorphismπ
−1
(U)→U×R
n
: (p, v)7→Φ(p, v) = (p,Φp(v))such that
the mapΦp:Ep→R
n
is an isometric isomorphism for allp∈U.
(v)For everyp0∈Mthere is an open neighborhoodU⊂Mofp0and a
diffeomorphismπ
−1
(U)→U×R
n
: (p, v)7→Φ(p, v) = (p,Φp(v))such that
the mapΦp:Ep→R
n
is a vector space isomorphism for allp∈U.
Condition (i) implies that the projectionπ:E→Mis a submersion. In (ii)
the sectionscan be chosen to have compact support, i.e. there exists a
compact subsetK⊂Msuch thats(p) = 0for allp∈M\K.
Before giving the proof of Theorem 2.6.10 we explain some of its conse-
quences.

76 CHAPTER 2. FOUNDATIONS
Definition 2.6.11.The mapsΦ :π
−1
(U)→U×R
n
in Theorem 2.6.10 are
calledlocal trivializationsofE. They fit into commutative diagrams
π
−1
(U)
Φ
//
π
''NNNNNNNNNNNNN
U×R
n
pr
1
wwpppppppppppppp
U
.
Corollary 2.6.12.LetM⊂R
k
be a smoothm-manifold. ThenT Mis a
vector bundle overMand hence is a smooth2m-manifold inR
k
×R
k
.
Proof.Letϕ:U→Ω be a coordinate chart on anM-open setU⊂Mwith
values in an open subset Ω⊂R
m
. Denote its inverse byψ:=ϕ
−1
: Ω→M.
By Theorem 2.2.3 the linear mapdψ(x) :R
m
→R
k
is injective and its image
isT
ψ(x)Mfor everyx∈Ω. Hence the mapD:U→R
k×m
defined by
D(p) :=dψ(ϕ(p))∈R
k×m
is smooth and, for everyp∈U, the linear mapD(p) :R
m
→R
k
is injec-
tive and its image isTpM. Thus the function Π
T M
:M→R
k×k
defined
by (2.6.4) withEp=TpMis given by
Π
T M
(p) =D(p)
ı
D(p)
T
D(p)
ȷ
−1
D(p)
T
forp∈U.
Hence Π
T M
is smooth and soT Mis a vector bundle by Theorem 2.6.10.
LetM⊂R
k
be anm-manifold,N⊂R
ℓ
be ann-manifold,f:N→M
be a smooth map, andE⊂M×R
d
be a vector bundle. Thepullback
bundleis the vector bundlef
∗
E→Ndefined by
f
∗
E:=
n
(q, v)∈N×R
d
|v∈E
f(q)
o
and thenormal bundle ofEis the vector bundleE
⊥
→Mdefined by
E
⊥
:=
n
(p, w)∈M×R
d
| ⟨v, w⟩= 0∀v∈Ep
o
.
Corollary 2.6.13.The pullback and normal bundles are vector bundles.
Proof.Let Π = Π
E
:M→R
d×d
be the map defined by (2.6.4). This map
is smooth by Theorem 2.6.10. Moreover, the corresponding maps forf
∗
E
andE
⊥
are given by
Π
f
∗
E
= Π
E
◦f:N→R
d×d
,Π
E
⊥
= 1l−Π
E
:M→R
d×d
.
These maps are smooth and hence it follows from Theorem 2.6.10 thatf
∗
E
andE
⊥
are vector bundles.

2.6. VECTOR BUNDLES AND SUBMERSIONS 77
Proof of Theorem 2.6.10.We first assume thatEis a vector bundle and
prove thatπ:E→Mis a submersion. Letσ:M→Edenote the zero
section given byσ(p) := (p,0). Thenπ◦σ= id and hence it follows from the
chain rule that the derivativedπ(p,0) :T
(p,0)E→TpMis surjective. Now
it follows from Exercise 2.6.4 that for everyp∈Mthere is anε >0 such
that the derivativedπ(p, v) :T
(p,v)E→TpMis surjective for everyv∈Ep
with|v|< ε. Consider the mapfλ:E→Edefined by
fλ(p, v) := (p, λv).
This map is a diffeomorphism for everyλ >0. It satisfies
π=π◦fλ
and hence
dπ(p, v) =dπ(p, λv)◦dfλ(p, v) :T
(p,v)E→TpM.
Sincedfλ(p, v) is bijective anddπ(p, λv) is surjective forλ < ε/|v|it follows
thatdπ(p, v) is surjective for everyp∈Mand everyv∈Ep. Thus the
projectionπ:E→Mis a submersion for every vector bundleEoverM.
We prove that (i) implies (ii). Letp0∈Mandv0∈Ep0
. We have
already proved thatπis a submersion. Hence it follows from Lemma 2.6.1
that there exists anM-open neighborhoodU⊂Mofp0and a smooth map
σ0:U→E
such that
π◦σ0= id :U→U, σ 0(p0) = (p0, v0).
Define the maps0:U→R
ℓ
by
(p, s0(p)) :=σ0(p) forp∈U.
Thens0(p0) =v0ands0(p)∈Epfor allp∈U. Now chooseε >0 such that
{p∈M| |p−p0|< ε} ⊂U
and choose a smooth cutoff functionβ:R
k
→[0,1] such thatβ(p0) = 1
andβ(p) = 0 for|p−p0| ≥ε. Defines:M→R
ℓ
by
s(p) :=
æ
β(p)s0(p),ifp∈U,
0, ifp /∈U.
This map satisfies the requirements of (ii).

78 CHAPTER 2. FOUNDATIONS
We prove that (ii) implies (iii). Thus we assume thatEsatisfies (ii).
Choosep0∈Mand a basisv1, . . . , vnofEp0
. By (ii) there exists smooth
sectionss1, . . . , sn:M→R
ℓ
ofEsuch thatsi(p0) =vifori= 1, . . . , n. Now
there exists anM-open neighborhoodU⊂Mofp0such that the vec-
torss1(p), . . . , sn(p) are linearly independent, and hence form a basis ofEp
for everyp∈U. Hence, for everyp∈U, we have
Ep= imD(p), D(p) := [s1(p)· · ·sn(p)]∈R
ℓ×n
.
By Exercise 2.6.9, this implies Π(p) =D(p)(D(p)
T
D(p))
−1
D(p)
T
for ev-
eryp∈U. Thus everyp0∈Mhas a neighborhoodUsuch that the re-
striction of Π toUis smooth. This shows that (ii) implies (iii).
We prove that (iii) implies (iv). Fix a pointp0∈Mand choose a ba-
sisv1, . . . , vnofEp0
. Forp∈Mdefine
D(p) := [Π(p)v1· · ·Π(p)vn]∈R
ℓ×n
ThenD:M→R
ℓ×n
is a smooth map andD(p0) has rankn. Hence the set
U:={p∈M|rankD(p) =n} ⊂M
is an open neighborhood ofp0andEp= imD(p) for allp∈U. Thus
π
−1
(U) ={(p, v)∈E|p∈U} ⊂E
is an open set containingπ
−1
(p0). Define the map Φ :π
−1
(U)→U×R
n
by
Φ(p, v) :=
Γ
p,Φp(v)
∆
,Φp(v) :=
ı
D(p)
T
D(p)
ȷ
−1/2
D(p)
T
v
forp∈Uandv∈Ep. This map is bijective and its inverse is given by
Φ
−1
(p, ξ) =
Γ
p,Φ
−1
p(ξ)
∆
,Φ
−1
p(ξ) =D(p)
ı
D(p)
T
D(p)
ȷ
−1/2
ξ
forp∈Uandξ∈R
n
. Thus Φ is a diffeomorphism and|Φp(v)|=|v|for
allp∈Uand allv∈Ep. This shows that (iii) implies (iv).
That (iv) implies (v) is obvious.
We prove that (v) implies (i). ShrinkingUif necessary, we may as-
sume that there exists a coordinate chartϕ:U→Ω with values in an open
set Ω⊂R
m
. Then the composition (ϕ×id)◦Φ :π
−1
(U)→Ω×R
n
is a dif-
feomorphism. ThusE⊂R
k
×R
ℓ
is a manifold of dimensionm+nand this
proves Theorem 2.6.10.
Exercise 2.6.14.Define the notion of an isomorphism between two vector
bundlesEandFoverM. Construct a vector bundleE⊂S
1
×R
2
of rank 1
that does not admit aglobal trivialization, i.e. that is not isomorphic to the
trivial bundleS
1
×R. Such a vector bundle is called aM¨obius strip.

2.6. VECTOR BUNDLES AND SUBMERSIONS 79
The Implicit Function Theorem
Next we carry over the Implicit Function Theorem in Corollary A.2.6 to
smooth maps on vector bundles.
Theorem 2.6.15(Implicit Function Theorem).
LetM⊂R
k
be a smoothm-manifold, letN⊂R
k
be a smoothn-manifold,
letE⊂M×R
ℓ
be a smooth vector bundle of rankn, letW⊂Ebe open,
and letf:W→Nbe a smooth map. Forp∈Mdefinefp:Wp→Nby
Wp:={v∈Ep|(p, v)∈W}, f p(v) :=f(p, v).
Letp0∈Msuch that0∈Wp0
anddfp0
(0) :Ep0
→Tq0
Nis bijective,
whereq0:=f(p0,0)∈N. Then there exist a constantε >0, open neighbor-
hoodsU0⊂Mofp0andV0⊂Nofq0, and a smooth maph:U0×V0→R
ℓ
such that{(p, v)∈E|p∈U0,|v|< ε} ⊂Wand
h(p, q)∈Ep,|h(p, q)|< ε (2.6.5)
for all(p, q)∈U0×V0and
fp(v) =q ⇐⇒ v=h(p, q) (2.6.6)
for all(p, q)∈U0×V0and allv∈Epwith|v|< ε. Thush(p0, q0) = 0.
Proof.Choose a coordinate chartψ:V→R
n
on an open setV⊂Ncon-
tainingq0. Choose an open neighborhoodU⊂Mofp0such that (p,0)∈W
andf(p,0)∈Vfor allp∈U, there is a coordinate chartϕ:U→Ω⊂R
m
,
and there is a local trivialization Φ :π
−1
(U)→U×R
n
as in Theorem 2.6.10
with|Φp(v)|=|v|forp∈Uandv∈Ep. DefineBr:={ξ∈R
n
| |ξ|< r}and
chooser >0 so small that Φ
−1
(U×Br)⊂Wandf◦Φ
−1
(U×Br)⊂V.
Define the mapF: Ω×R
n
×Br→R
n
by
F(x, y, ξ) :=ψ◦f◦Φ
−1
Γ
ϕ
−1
(x), ξ
∆
−y
for (x, y)∈Ω×R
n
andξ∈Br. Letx0:=ϕ(p0) andy0:=ψ(q0). Then
we haveF(x0, y0,0) = 0 and the derivatived3F(x0, y0,0) :R
n
→R
n
ofF
with respect toξat (x0, y0,0) is bijective. Hence Corollary A.2.6 asserts
that there exist open neighborhoodsU0⊂Uofp0andV0⊂Vofq0, a con-
stant 0< ε < r, and a smooth mapg:ϕ(U0)×ψ(V0)→Bεsuch that
F(x, y, ξ) = 0 ⇐⇒ g(x, y) =ξ
for all (x, y)∈ϕ(U0)×ψ(V0) and allξ∈Bε. Thus the map
h:U0×V0→R
ℓ
, h(p, q) := Φ
−1
p
Γ
g(ϕ(p), ψ(q))
∆
,
satisfies the requirements of Theorem 2.6.15.

80 CHAPTER 2. FOUNDATIONS
2.7 The Theorem of Frobenius
LetM⊂R
k
be anm-dimensional manifold andnbe a nonnegative integer.
Asubbundle of ranknof the tangent bundleT Mis a subsetE⊂T M
that is itself a vector bundle of ranknoverM, i.e. it is a submanifold
ofT Mand the fiberEp={v∈TpM|(p, v)∈E}is ann-dimensional linear
subspace ofTpMfor everyp∈M. Note that the ranknof a subbundle
is necessarily less than or equal tom. In the literature a subbundle of the
tangent bundle is sometimes called adistributiononM. We shall, however,
not use this terminology in order to avoid confusion with the concept of a
distribution in the functional analytic setting.
Definition 2.7.1.LetM⊂R
k
be anm-dimensional manifold andE⊂T M
be a subbundle of rankn. The subbundleEis calledinvolutiveif, for any
two vector fieldsX, Y∈Vect(M), we have
X(p), Y(p)∈Ep∀p∈M =⇒ [X, Y](p)∈Ep∀p∈M.(2.7.1)
The subundleEis calledintegrableif, for everyp0∈M, there exists a
submanifoldN⊂Msuch thatp0∈NandTpN=Epfor everyp∈N.
Afoliation box forE(see Figure 2.13) is a coordinate chartϕ:U→Ω
on anM-open subsetU⊂Mwith values in an open setΩ⊂R
n
×R
m−n
such that the setΩ∩(R
n
× {y})is connected for everyy∈R
m−n
and, for
everyp∈Uand everyv∈TpM, we have
v∈Ep ⇐⇒ dϕ(p)v∈R
n
× {0}.M
U Ωφ
Figure 2.13: A foliation box.
Theorem 2.7.2(Frobenius).LetM⊂R
k
be anm-dimensional manifold
andE⊂T Mbe a subbundle of rankn. Then the following are equivalent.
(i)Eis involutive.
(ii)Eis integrable.
(iii)For everyp0∈Mthere exists a foliation boxϕ:U→Ωwithp0∈U.

2.7. THE THEOREM OF FROBENIUS 81
It is easy to show that (iii) =⇒(ii) =⇒(i) (see below). The hard part of
the theorem is to prove that (i) =⇒(iii). This requires the following lemma.
Lemma 2.7.3.LetE⊂T Mbe an involutive subbundle andX∈Vect(M)
be a complete vector field such thatX(p)∈Epfor everyp∈M. Denote by
R→Diff(M) :t7→ϕ
t
the flow ofX. Then, for allt∈Rand allp∈M, we have
dϕ
t
(p)Ep=E
ϕ
t
(p). (2.7.2)
We show first how Theorem 2.7.2 follows from Lemma 2.7.3.
Lemma 2.7.3 implies Theorem 2.7.2.We prove first that (iii) implies (ii).
Letp0∈M, choose a foliation boxϕ:U→Ω forEwithp0∈U, and define
N:= (p∈U|ϕ(p)∈R
n
× {y0}}
where (x0, y0) :=ϕ(p0)∈Ω. ThenNsatisfies the requirements of (ii).
We prove that (ii) implies (i). Choose two vector fieldsX, Y∈Vect(M)
that satisfyX(p), Y(p)∈Epfor allp∈Mand fix a pointp0∈M. Then,
by (ii), there exists a submanifoldN⊂Mcontainingp0such thatTpN=Ep
for everyp∈N. Hence the restrictionsX|NandY|Nare vector fields
onNand so is the restriction of the Lie bracket [X, Y] toN. Thus we
have [X, Y](p0)∈Tp0
N=Ep0
as claimed.
We prove that (i) implies (iii). Thus we assume thatEis an involutive
subbundle ofT Mand fix a pointp0∈M. By Theorem 2.6.10 there exist
vector fieldsX1, . . . , Xn∈Vect(M) such thatXi(p)∈Epfor alliandpand
the vectorsX1(p0), . . . , Xn(p0) form a basis ofEp0
. Using Theorem 2.6.10
again we find vector fieldsY1, . . . , Ym−n∈Vect(M) such that the vectors
X1(p0), . . . , Xn(p0), Y1(p0), . . . , Ym−n(p0)
form a basis ofTp0
M. Using cutoff functions as in the proof of Theo-
rem 2.6.10 we may assume without loss of generality that the vector fieldsXi
andYjhave compact support and hence are complete (see Exercise 2.4.13).
Denote byϕ
t
1
, . . . , ϕ
t
nthe flows of the vector fieldsX1, . . . , Xn, respectively,
and byψ
t
1
, . . . , ψ
t
m−nthe flows of the vector fieldsY1, . . . , Ym−n. Define the
map
ψ:R
n
×R
m−n
→M
by
ψ(x, y) :=ϕ
x1
1
◦ · · · ◦ϕ
xn
n◦ψ
y1
1
◦ · · · ◦ψ
ym−n
m−n
(p0).

82 CHAPTER 2. FOUNDATIONS
By Lemma 2.7.3, this map satisfies
∂ψ
∂xi
(x, y)∈E
ψ(x,y) (2.7.3)
for allx∈R
n
andy∈R
m−n
. Moreover,
∂ψ
∂xi
(0,0) =Xi(p0),
∂ψ
∂yj
(0,0) =Yj(p0),
and so the derivative
dψ(0,0) :R
n
×R
m−n
→Tp0
M
is bijective. Hence, by the Inverse Function Theorem 2.2.17, there exists an
open neighborhood Ω⊂R
n
×R
m−n
of the origin such that the set
U:=ψ(Ω)⊂M
is anM-open neighborhood ofp0andψ|Ω: Ω→Uis a diffeomorphism.
Thus the vectors∂ψ/∂xi(x, y) are linearly independent for every (x, y)∈Ω
and, by (2.7.3), form a basis ofE
ψ(x,y). Hence
ϕ:= (ψ|Ω)
−1
:U→Ω
is a foliation box and this proves Theorem 2.7.2, assuming Lemma 2.7.3.
To complete the proof of the Frobenius theorem it remains to prove
Lemma 2.7.3. This requires the following result.
Lemma 2.7.4.LetE⊂T Mbe an involutive subbundle. Ifβ:R
2
→Mis
a smooth map such that
∂β
∂s
(s,0)∈E
β(s,0),
∂β
∂t
(s, t)∈E
β(s,t), (2.7.4)
for alls, t∈R, then
∂β
∂s
(s, t)∈E
β(s,t), (2.7.5)
for alls, t∈R.
We first show how Lemma 2.7.3 follows from Lemma 2.7.4.

2.7. THE THEOREM OF FROBENIUS 83
Lemma 2.7.4 implies Lemma 2.7.3.LetX∈Vect(M) be a complete vector
field satisfyingX(p)∈Epfor everyp∈Mand letϕ
t
be the flow ofX.
Choose a pointp0∈Mand a vectorv0∈Ep0
. By Theorem 2.6.10 there is a
vector fieldY∈Vect(M) with values inEsuch thatY(p0) =v0. Moreover
this vector field may be chosen to have compact support and hence it is
complete (see Exercise 2.4.13). Thus there is a solutionγ:R→Mof the
initial value problem
˙γ(s) =Y(γ(s)), γ(0) =p0.
Defineβ:R
2
→Mby
β(s, t) :=ϕ
t
(γ(s))
fors, t∈R. Then
∂β
∂s
(s,0) = ˙γ(s) =Y(γ(s))∈E
β(s,0),
∂β
∂t
(s, t) =X(β(s, t))∈E
β(s,t)
for alls, t∈R. Hence it follows from Lemma 2.7.4 that
dϕ
t
(p0)v0=dϕ
t
(γ(0)) ˙γ(0) =
∂β
∂s
(0, t)∈E
ϕ
t
(p0)
for everyt∈R. This proves Lemma 2.7.3, assuming Lemma 2.7.4.
Proof of Lemma 2.7.4.Given any pointp0∈Mwe choose a coordinate
chartϕ:U→Ω, defined on anM-open setU⊂Mwith values in an
open set Ω⊂R
n
×R
m−n
, such thatp0∈Uanddϕ(p0)Ep0
=R
n
× {0}.
ShrinkingU, if necessary, we find that for everyp∈Uthe linear sub-
spacedϕ(p)Ep⊂R
n
×R
m−n
is the graph of a matrixA∈R
(m−n)×n
. Thus
there exists a smooth mapA: Ω→R
(m−n)×n
such that, for everyp∈U,
dϕ(p)Ep={(ξ, A(x, y)ξ)|ξ∈R
n
},(x, y) :=ϕ(p)∈Ω. (2.7.6)
For (x, y)∈Ω define the linear maps
∂A
∂x
(x, y) :R
n
→R
(m−n)×n
,
∂A
∂y
(x, y) :R
m−n
→R
(m−n)×n
by
∂A
∂x
(x, y)·ξ:=
n
X
i=1
ξi
∂A
∂xi
(x, y),
∂A
∂y
(x, y)·η:=
m−n
X
j=1
ηj
∂A
∂yj
(x, y),
forξ= (ξ1, . . . , ξn)∈R
n
andη= (η1, . . . , ηm−n)∈R
m−n
. We prove the
following.

84 CHAPTER 2. FOUNDATIONS
Claim 1.Let(x, y)∈Ω,ξ, ξ
′
∈R
n
and defineη, η
′
∈R
m−n
byη:=A(x, y)ξ
andη
′
:=A(x, y)ξ
′
. Then
`
∂A
∂x
(x, y)·ξ+
∂A
∂y
(x, y)·η
´
ξ
′
=
`
∂A
∂x
(x, y)·ξ
′
+
∂A
∂y
(x, y)·η
′
´
ξ.
The graphs of the matricesA(z) determine a subbundle
e
E⊂Ω×R
m
with
the fibers
e
Ez:=
Φ
(ξ, η)∈R
n
×R
m−n
|η=A(x, y)ξ

forz= (x, y)∈Ω. This subbundle is the image of the restriction
E|U:={(p, v)|p∈U, v∈Ep}
under the diffeomorphismT M|U→Ω×R
m
: (p, v)7→(ϕ(p), dϕ(p)v) and
hence it is involutive. Now fix two elementsξ, ξ
′
∈R
n
and define the vector
fieldsζ, ζ
′
: Ω→R
m
by
ζ(z) := (ξ, A(z)ξ), ζ
′
(z) := (ξ
′
, A(z)ξ
′
), z∈Ω.
Thenζandζ
′
are sections of
e
Eand their Lie bracket [ζ, ζ
′
] is given by
[ζ, ζ
′
](z) =
Γ
0,
Γ
dA(z)ζ
′
(z)
∆
ξ−(dA(z)ζ(z))ξ
′
∆
.
Since
e
Eis involutive the Lie bracket [ζ, ζ
′
] must take values in the graph
ofA. Hence the right hand side vanishes and this proves Claim 1.
Claim 2.LetI, J⊂Rbe open intervals and letz= (x, y) :I×J→Ωbe
a smooth map. Fix two pointss0∈Iandt0∈Jand assume that
∂y
∂s
(s0, t0) =A
Γ
x(s0, t0), y(s0, t0)
∆∂x
∂s
(s0, t0), (2.7.7)
∂y
∂t
(s, t) =A
Γ
x(s, t), y(s, t)
∆∂x
∂t
(s, t)
for alls∈Iandt∈J. Then
∂y
∂s
(s0, t) =A
Γ
x(s0, t), y(s0, t)
∆∂x
∂s
(s0, t) (2.7.9)
for allt∈J.

2.7. THE THEOREM OF FROBENIUS 85
Equation (2.7.9) holds by assumption fort=t0. Moreover, dropping the
argumentz(s0, t) =z= (x, y) for notational convenience we obtain
∂
∂t
`
∂y
∂s
−A·
∂x
∂s
´
=
∂
2
y
∂s∂t
−A
∂
2
x
∂s∂t
−
`
∂A
∂x
·
∂x
∂t
+
∂A
∂y
·
∂y
∂t
´
∂x
∂s
=
∂
2
y
∂s∂t
−A
∂
2
x
∂s∂t
−
`
∂A
∂x
·
∂x
∂t
+
∂A
∂y
·
`
A
∂x
∂t
´´
∂x
∂s
=
∂
2
y
∂s∂t
−A
∂
2
x
∂s∂t
−
`
∂A
∂x
·
∂x
∂s
+
∂A
∂y
·
`
A
∂x
∂s
´´
∂x
∂t
=
∂
2
y
∂s∂t
−A
∂
2
x
∂s∂t
−
`
∂A
∂x
·
∂x
∂s
+
∂A
∂y
·
∂y
∂s
´
∂x
∂t
+
`
∂A
∂y
·
`
∂y
∂s
−A
∂x
∂s
´´
∂x
∂t
=
`
∂A
∂y
·
`
∂y
∂s
−A
∂x
∂s
´´
∂x
∂t
.
Here the second step follows from (2.7.8), the third step follows from Claim 1,
and the last step follows by differentiating equation (2.7.8) with respect tos.
Define the curveη:J→R
m−n
by
η(t) :=
∂y
∂s
(s0, t)−A
Γ
x(s0, t), y(s0, t)
∆∂x
∂s
(s0, t).
By (2.7.7) and what we have just proved, the curveηsatisfies the linear
differential equation
˙η(t) =
`
∂A
∂y
Γ
x(s0, t), y(s0, t)
∆
·η(t)
´
∂x
∂t
(s0, t), η(t0) = 0.
Henceη(t) = 0 for allt∈J. This proves (2.7.9) and Claim 2.
Now letβ:R
2
→Mbe a smooth map satisfying (2.7.4) and fix a real
numbers0. Consider the setW:={t∈R|∂sβ(s0, t)∈E
β(s0,t)}. By going
to local coordinates, we obtain from Claim 2 thatWis open. Moreover,W
is obviously closed, andW̸=∅because 0∈Wby (2.7.4). HenceW=R.
Sinces0∈Rwas chosen arbitrarily, this proves (2.7.5) and Lemma 2.7.4.
Any subbundleE⊂T Mdetermines an equivalence relation onMvia
p0∼p1⇐⇒
there is a smooth curveγ: [0,1]→M
such thatγ(0) =p0, γ(1) =p1,˙γ(t)∈E
γ(t)∀t.
(2.7.10)
IfEis integrable, this equivalence relation is called afoliationand the
equivalence class ofp0∈Mis called theleafof the foliation throughp0.
The next example shows that the leaves do not need to be submanifolds.

86 CHAPTER 2. FOUNDATIONS
Example 2.7.5.Consider the torusM:=S
1
×S
1
⊂C
2
with the tangent
bundle
T M=
Φ
(z1, z2,iλ1z1,iλ2z2)∈C
4
| |z1|=|z2|= 1, λ1, λ2∈R

.
Letω1, ω2be real numbers and consider the subbundle
E:=
Φ
(z1, z2,itω1z1,itω2z2)∈C
4
| |z1|=|z2|= 1, t∈R

.
The leaf of this subbundle throughz= (z1, z2)∈T
2
is given by
L=
nı
e
itω1
z1, e
itω2
z2
ȷ

t∈R
o
.
It is a submanifold if and only if the quotientω1/ω2is a rational number
(orω2= 0). Otherwise each leaf is a dense subset ofT
2
.
Exercise 2.7.6.Prove that (2.7.10) defines an equivalence relation for every
subbundleE⊂T M.
Exercise 2.7.7.Each subbundleE⊂T Mof rank 1 is integrable.
Exercise 2.7.8.Consider the manifoldM=R
3
. Prove that the sub-
bundleE⊂T M=R
3
×R
3
with fiberEp=
Φ
(ξ, η, ζ)∈R
3
|ζ−yξ= 0

overp= (x, y, z)∈R
3
is not integrable and that any two points inR
3
can
be joined by a path tangent toE.
Exercise 2.7.9.Consider the manifoldM=S
3
⊂R
4
=C
2
and define
E:=
Φ
(z, ζ)∈C
2
×C
2
| |z|= 1, ζ⊥z,iζ⊥z

.
Thus the fiber
Ez⊂TzS
3
=z
⊥
is the maximal complex linear subspace ofTzS
3
. Prove thatEhas real
rank 2 and is not integrable.
Exercise 2.7.10.LetE⊂T Mbe an involutive subbundle of ranknand
letL⊂Mbe a leaf of the foliation determined byE. A subsetV⊂Lis
calledL-openiff it can be written as a union of submanifoldsNofM
with tangent spacesTpN=Epforp∈N. Prove that theL-open sets form
a topology onL(called theintrinsic topology). Prove that the obvious
inclusionι:L→Mis continuous with respect to the intrinsic topology onL.
Prove that the inclusionι:L→Mis proper if and only if the intrinsic
topology onLagrees with the relative topology inherited fromM(called
theextrinsic topology).
Remark 2.7.11.It is surprisingly difficult to prove that each closed leafL
of a foliation is a submanifold ofM. A proof due to David Epstein [19] is
sketched in§2.9.4 below.

2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 87
2.8 The Intrinsic Definition of a Manifold*
It is somewhat restrictive to only consider manifolds that are embedded in
some Euclidean space. Although we shall see that (at least) every compact
manifold admits an embedding into a Euclidean space, such an embedding is
in many cases not a natural part of the structure of a manifold. In particular,
we encounter manifolds that are described as quotient spaces and there are
manifolds that are embedded in certain infinite-dimensional Hilbert spaces.
For this reason it is convenient, at this point, to introduce a more general
intriniscdefinition of a manifold. (See Chapter 1 for an overview.) This
requires some background from point set topology that is not covered in
the first year analysis courses. We shall then see that all the definitions and
results of this chapter carry over in a natural manner to the intrinsic setting.
We begin by recalling the intrinsing definition of a smooth manifold in§1.4.
2.8.1 Definition and ExamplesM
U
α β U
βαφ
βφ
α φ
Figure 2.14: Coordinate charts and transition maps.
Definition 2.8.1(Smoothm-manifold).Letm∈N0andMbe a set.
AchartonMis a pair (ϕ, U) whereU⊂Mandϕis a bijection fromU
to an open setϕ(U)⊂R
m
. Two charts(ϕ1, U1),(ϕ2, U2)are calledcom-
patibleiffϕ1(U1∩U2)andϕ2(U1∩U2)are open and thetransition map
ϕ21=ϕ2◦ϕ
−1
1
:ϕ1(U1∩U2)→ϕ2(U1∩U2) (2.8.1)
is a diffeomorphism (see Figure 2.14). Asmooth atlasonMis a collec-
tionAof charts onMany two of which are compatible and such that the
setsU, as(ϕ, U)ranges overA, coverM(i.e. for everyp∈Mthere exists
a chart(ϕ, U)∈Awithp∈U). Amaximal smooth atlasis an atlas
which contains every chart which is compatible with each of its members.
Asmoothm-manifoldis a pair consisting of a setMand a maximal
atlasAonM.

88 CHAPTER 2. FOUNDATIONS
In Lemma 1.4.3 it was shown that, ifAis an atlas, then so is the
collectionAof all charts compatible with each member ofA. Moreover,
the atlasAis maximal, so every atlas extends uniquely to a maximal atlas.
For this reason, a manifold is usually specified by giving its underlying setM
and some atlas onM. Generally, the notation for the atlas is suppressed and
the manifold is denoted simply byM. The members of the atlas are called
coordinate chartsor simplychartsonM. By Lemma 1.3.3 a smooth
m-manifold admits a unique topology such that, for each chart (ϕ, U) of the
smooth atlas, the setU⊂Mis open and the bijection
ϕ:U→ϕ(U)
is a homeomorphism onto the open setϕ(U)⊂R
m
. This topology is called
the intrinsic topology ofMand is described in the following definition.
Definition 2.8.2.LetMbe a smoothm-manifold. Theintrinsic topology
on the setMis the topology induced by the charts, i.e. a subset
W⊂M
is open in the intrinsic topology iffϕ(U∩W)is an open subset ofR
m
for
every chart(ϕ, U)onM.
1
Remark 2.8.3.LetM⊂R
k
be smoothm-dimensional submanifold ofR
k
as in Definition 2.1.3. Then the set of all diffeomorphisms (ϕ, U∩M) as
in Definition 2.1.3 form a smooth atlas as in Definition 2.8.1. The intrin-
sic topology on the resulting smooth manifold is the same as the relative
topology defined in§1.3.
Remark 2.8.4.Atopological manifoldis a topological space such that
each point has a neighborhoodUhomeomorphic to an open subset ofR
m
.
Thus a smooth manifold (with the intrinsic topology) is a topological man-
ifold and its maximal smooth atlasAis a subset of the setA0of all
pairs (ϕ, U) whereU⊂Mis an open set andϕis a homeomorphism fromU
to an open subset ofR
m
. One says that the maximal smooth atlasAis a
smooth structureon the topological manifoldMiff the topology ofMis
the intrinsic topology of the smooth structure and every chart of the smooth
structure is a homeomorphism. As explained in§1.4 a topological manifold
can have many distinct smooth structures (see Remark 1.4.6). However, it
is a deep theorem beyond the scope of this book that there are topological
manifolds which do not admit any smooth structure.
1
At this point we do not assume that the intrinsic topology on the manifoldMis
Hausdorff or second countable. These hypotheses will be imposed after the end of the
present chapter. For explanations see the comments at the end of§2.8.1 and of§2.9.5.

2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 89
Example 2.8.5.Thecomplex projective spaceCP
n
is the set
CP
n
=
Φ
ℓ⊂C
n+1
|ℓis a 1-dimensional complex subspace

of complex lines inC
n+1
. It can be identified with the quotient space
CP
n
=
Γ
C
n+1
\ {0}
∆
/C
∗
of nonzero vectors inC
n+1
modulo the action of the multiplicative group
C
∗
=C\ {0}of nonzero complex numbers. The equivalence class of a
nonzero vectorz= (z0, . . . , zn)∈C
n+1
will be denoted by
[z] = [z0:z1:· · ·:zn] :={λz|λ∈C
∗
}
and the associated line isℓ=Cz. An atlas onCP
n
is given by the open
coverUi:={[z0:· · ·:zn]|zi̸= 0}fori= 0,1, . . . , nand the coordinate
chartsϕi:Ui→C
n
are
ϕi([z0:· · ·:zn]) :=
`
z0
zi
, . . . ,
zi−1
zi
,
zi+1
zi
, . . . ,
zn
zi
´
. (2.8.2)
Exercise:Prove that eachϕiis a homeomorphism and the transition maps
are holomorphic. Prove that the manifold topology is the quotient topology,
i.e. ifπ:C
n+1
\ {0} →CP
n
denotes the obvious projection, then a sub-
setU⊂CP
n
is open if and only ifπ
−1
(U) is an open subset ofC
n+1
\ {0}.
Example 2.8.6.Thereal projective spaceRP
n
is the set
RP
n
=
Φ
ℓ⊂R
n+1
|ℓis a 1-dimensional linear subspace

of real lines inR
n+1
. It can again be identified with the quotient space
RP
n
=
Γ
R
n+1
\ {0}
∆
/R
∗
of nonzero vectors inR
n+1
modulo the action of the multiplicative group
R
∗
=R\{0}of nonzero real numbers, and the equivalence class of a nonzero
vectorx= (x0, . . . , xn)∈R
n+1
will be denoted by
[x] = [x0:x1:· · ·:xn] :={λx|λ∈R
∗
}.
An atlas onRP
n
is given by the open cover
Ui:={[x0:· · ·:xn]|xi̸= 0}
and the coordinate chartsϕi:Ui→R
n
are again given by (2.8.2), withzj
replaced byxj. The arguments in Example 2.8.5 show that these coordinate
charts form an atlas and the manifold topology is the quotient topology. The
transition maps are real analytic diffeomorphisms.

90 CHAPTER 2. FOUNDATIONS
Example 2.8.7.Therealn-torusis the topological space
T
n
:=R
n
/Z
n
equipped with the quotient topology. Thus two vectorsx, y∈R
n
are equiv-
alent iff their differencex−y∈Z
n
is an integer vector and we denote by
π:R
n
→T
n
the obvious projection which assigns to each vectorx∈R
n
its
equivalence class
π(x) := [x] :=x+Z
n
.
Then a setU⊂T
n
is open if and only if the setπ
−1
(U) is an open subset
ofR
n
. An atlas onT
n
is given by the open cover
Uα:={[x]|x∈R
n
,|x−α|<1/2},
parametrized by vectorsα∈R
n
, and the coordinate chartsϕα:Uα→R
n
defined byϕα([x]) :=xforx∈R
n
with|x−α|<1/2.Exercise:Show
that each transition map for this atlas is a translation by an integer vector.
Example 2.8.8.Consider thecomplex Grassmannian
Gk(C
n
) :={V⊂C
n
|vis ak-dimensional complex linear subspace}.
This set can again be described as a quotient space Gk(C
n
)
∼
=Fk(C
n
)/U(k).
Here
Fk(C
n
) :=
n
D∈C
n×k
|D
∗
D= 1l
o
denotes the set of unitaryk-frames inC
n
and the group U(k) acts onFk(C
n
)
contravariantly byD7→Dgforg∈U(k). The projection
π:Fk(C
n
)→Gk(C
n
)
sends a matrixD∈ Fk(C
n
) to its imageV:=π(D) := imD. A subset
U⊂Gk(C
n
) is open if and only ifπ
−1
(U) is an open subset ofFk(C
n
). Given
ak-dimensional subspaceV⊂C
n
we can define an open setUV⊂Gk(C
n
) as
the set of allk-dimensional subspaces ofC
n
that can be represented as graphs
of linear maps fromVtoV
⊥
. This set of graphs can be identified with the
complex vector space Hom
C
(V, V
⊥
) of complex linear maps fromVtoV
⊥
and hence withC
(n−k)×k
. This leads to an atlas on Gk(C
n
) with holomorphic
transition maps and shows that Gk(C
n
) is a manifold of complex dimension
kn−k
2
.Exercise:Verify the details of this construction. Find explicit
formulas for the coordinate charts and their transition maps. Carry this
over to the real setting. Show thatCP
n
andRP
n
are special cases.

2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 91
Example 2.8.9(The real line with two zeros).A topological spaceM
is calledHausdorffiff any two points inMcan be separated by disjoint
open neighborhoods. This example shows that a manifold need not be a
Hausdorff space. Consider the quotient space
M:=R× {0,1}/≡
where [x,0]≡[x,1] forx̸= 0. An atlas onMconsists of two coordinate
chartsϕ0:U0→Randϕ1:U1→Rwhere
Ui:={[x, i]|x∈R}, ϕ i([x, i]) :=x
fori= 0,1. ThusMis a 1-manifold. But the topology onMis not
Hausdorff, because the points [0,0] and [0,1] cannot be separated by disjoint
open neighborhoods.
Example 2.8.10(A2-manifold without a countable atlas).Consider
the vector spaceX=R×R
2
with the equivalence relation
[t1, x1, y1]≡[t2, x2, y2]⇐⇒
eithery1=y2̸= 0, t1+x1y1=t2+x2y2
ory1=y2= 0, t1=t2, x1=x2.
Fory̸= 0 we have [0, x, y]≡[t, x−t/y, y], however, each point (x,0) on
thex-axis gets replaced by the uncountable setR× {(x,0)}. Our manifold
is the quotient spaceM:=X/≡. This time we do not use the quotient
topology but the topology induced by our atlas (see Definition 2.8.2). The
coordinate charts are parametrized by the reals: fort∈Rthe setUt⊂M
and the coordinate chartϕt:Ut→R
2
are given by
Ut:={[t, x, y]|x, y∈R}, ϕ t([t, x, y]) := (x, y).
A subsetU⊂Mis open, by definition, iffϕt(U∩Ut) is an open subset ofR
2
for everyt∈R. With this topology eachϕtis a homeomorphism fromUt
ontoR
2
andMadmits a countable dense subsetS:={[0, x, y]|x, y∈Q}.
However, there is no atlas onMconsisting of countably many charts. (Each
coordinate chart can contain at most countably many of the points [t,0,0].)
The functionf:M→Rgiven byf([t, x, y]) :=t+xyis smooth and each
point [t,0,0] is a critical point offwith valuet. Thusfhas no regular
value.Exercise:Show thatMis a path-connected Hausdorff space.
In Theorem 2.9.12 we will show that smooth manifolds whose topology is
Hausdorff and second countable are precisely those that can be embedded in
Euclidean space. Most authors tacitly assume that manifolds are Hausdorff
and second countable and so will we after the end of the present chapter.
However before§2.9.1 there is no need to impose these hypotheses.

92 CHAPTER 2. FOUNDATIONS
2.8.2 Smooth Maps and Diffeomorphisms
Our next goal is to carry over all the definitions from embedded manifolds
in Euclidean space to the intrinsic setting.
Definition 2.8.11(Smooth map).Let
(M,{(ϕα, Uα)}α∈A),(N,{(ψβ, Vβ)}β∈B)
be smooth manifolds. A mapf:M→Nis calledsmoothiff it is continu-
ous and the map
fβα:=ψβ◦f◦ϕ
−1
α:ϕα(Uα∩f
−1
(Vβ))→ψβ(Vβ) (2.8.3)
is smooth for everyα∈Aand everyβ∈B. It is called adiffeomorphism
iff it is bijective andfandf
−1
are smooth. The manifoldsMandNare
calleddiffeomorphiciff there exists a diffeomorphismf:M→N.
The reader may check that the notion of a smooth map is independent
of the atlas used in the definition, that compositions of smooth maps are
smooth, and that sums and products of smooth maps fromMtoRare
smooth.
Exercise 2.8.12.LetMbe a smoothm-dimensional manifold with an atlas
A={(ϕα, Uα)}
α∈A
.
Consider the quotient space
f
M:=
[
α∈A
{α} ×ϕα(Uα)
.
∼,
where
(α, x)∼(β, y)
def
⇐⇒ ϕ
−1
α(x) =ϕ
−1
β
(y).
forα, β∈A,x∈ϕα(Uα), andy∈ϕβ(Uβ). Define an atlas on
f
Mby
e
Uα:=
Φ
[α, x]

x∈ϕα(Uα)

,
e
ϕα([α, x]) :=x.
Prove that
f
Mis a smoothm-manifold and that it is diffeomorphic toM.
Exercise 2.8.13.Prove thatCP
1
is diffeomorphic toS
2
.Hint:Stereo-
graphic projection.

2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 93
2.8.3 Tangent Spaces and Derivatives
In the situation whereMis a submanifold of Euclidean space andp∈Mwe
have defined the tangent space ofMatpas the set of all derivatives ˙γ(0) of
smooth curvesγ:R→Mthat pass throughp=γ(0). We cannot do this
for manifolds in the intrinsic sense, as the derivative of a curve has yet to be
defined. In fact, the purpose of introducing a tangent space ofMis precisely
to allow us to define what we mean by the derivative of a smooth map. There
are two approaches. One is to introduce an appropriate equivalence relation
on the set of curves throughpand the other is to use local coordinates.
Definition 2.8.14.LetMbe a manifold with an atlasA={(ϕα, Uα)}α∈A
and letp∈M. Two smooth curvesγ0, γ1:R→Mwithγ0(0) =γ1(0) =p
are calledp-equivalentiff for some (and hence every)α∈Awithp∈Uα
we have
d
dt

t=0
ϕα(γ0(t)) =
d
dt

t=0
ϕα(γ1(t)).
We writeγ0
p
∼γ1iffγ0isp-equivalent toγ1and denote the equivalence class
of a smooth curveγ:R→Mwithγ(0) =pby[γ]p. Every such equivalence
class is called atangent vectorofMatp. Thetangent spaceofMatp
is the set of equivalence classes
TpM:=
Φ
[γ]p

γ:R→Mis smooth andγ(0) =p

. (2.8.4)
Definition 2.8.15.LetMbe a manifold with an atlasA={(ϕα, Uα)}α∈A
and letp∈M. TheA-tangent spaceofMatpis the quotient space
T
A
pM:=
[
p∈Uα
{α} ×R
m
.
p
∼, (2.8.5)
where the union runs over allα∈Awithp∈Uαand
(α, ξ)
p
∼(β, η) ⇐⇒ d
Γ
ϕβ◦ϕ
−1
α
∆
(x)ξ=η, x:=ϕα(p).
Each equivalence class[α, ξ]pis called atangent vectorofMatp.
In Definition 2.8.14 it is not immediately obvious that the setTpM
in (2.8.4) is a vector space. However, the quotient spaceT
A
pMin (2.8.5) is
obviously a vector space of dimensionmand there is a natural bijection
TpM→T
A
pM: [γ]p7→
ˇ
α,
d
dt

t=0
ϕα(γ(t))
˘
p
. (2.8.6)
This bijection induces a vector space structure on the setTpM. In other
words, the setTpMin (2.8.4) admits a unique vector space structure such
that the mapTpM→T
A
pMin (2.8.6) is a vector space isomorphism.

94 CHAPTER 2. FOUNDATIONS
Exercise 2.8.16.Verify the phrase“and hence every”in Definition 2.8.14
and deduce that the mapTpM→T
A
pMin (2.8.6) is well defined. Show
that it is bijective.
From now on we will use either Definition 2.8.14 or Definition 2.8.15 or
both, whichever way is most convenient, and drop the superscriptA.
Definition 2.8.17(Derivative of a smooth curve).For each smooth
curveγ:R→Mwithγ(0) =pwe define the derivative˙γ(0)∈TpMas the
equivalence class
˙γ(0) := [γ]p
∼
=
ˇ
α,
d
dt

t=0
ϕα(γ(t))
˘
p
∈TpM.
Definition 2.8.18(Derivative of a smooth map).Letf:M→N
be a smooth map between two smooth manifolds(M,{(ϕα, Uα)}α∈A)and
(N,{(ψβ, Vβ)}β∈B)and letp∈M. The derivative offatpis the map
df(p) :TpM→T
f(p)N
defined by the formula
df(p)[γ]p:= [f◦γ]
f(p) (2.8.7)
for each smooth curveγ:R→Mwithγ(0) =p. Here we use(2.8.4).
Under the isomorphism(2.8.6)this corresponds to the linear map
df(p)[α, ξ]p:= [β, dfβα(x)ξ]
f(p), x:=ϕα(p), (2.8.8)
forα∈Awithp∈Uαandβ∈Bwithf(p)∈Vβ, wherefβαis given
by(2.8.3).
Remark 2.8.19.Think ofN=R
n
as a manifold with a single coordinate
chart, namely the identity mapψβ= id :R
n
→R
n
. For everyq∈N=R
n
the tangent spaceTqNis then canonically isomorphic toR
n
via (2.8.5).
Thus for every smooth mapf:M→R
n
the derivative offatp∈Mis a
linear mapdf(p) :TpM→R
n
, and the formula (2.8.8) reads
df(p)[α, ξ]p=d(f◦ϕ
−1
α)(x)ξ, x:=ϕα(p).
This formula also applies to maps defined on some open subset ofM. In
particular, withf=ϕα:Uα→R
m
we have
dϕα(p)[α, ξ]p=ξ.
Thus the mapdϕα(p) :TpM→R
m
is the canonical vector space isomor-
phism determined byα.
With these definitions the derivative offatpis a linear map and we have
the chain rule for the composition of two smooth maps as in Theorem 2.2.14.

2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 95
2.8.4 Submanifolds and Embeddings
Definition 2.8.20(Submanifold).LetMbe a smoothm-manifold and let
n∈ {0,1, . . . , m}.A subsetN⊂Mis called ann-dimensionalsubmanifold
ofMiff, for everyp∈N, there exists a local coordinate chartϕ:U→Ω
forM, defined on an an open neighborhoodU⊂Mofpand with values in
an open setΩ⊂R
n
×R
m−n
, such thatϕ(U∩N) = Ω∩(R
n
× {0}).
By Theorem 2.1.10 anm-manifoldM⊂R
k
in the sense of Defini-
tion 2.1.3 is a submanifold ofR
k
in the sense of Definition 2.8.20. By The-
orem 2.3.4 the notion of a submanifoldN⊂Mof a manifoldM⊂R
k
in
Definition 2.3.1 agrees with the notion of a submanifold in Definition 2.8.20.
Exercise 2.8.21.LetNbe a submanifold ofM. Show that ifMis Haus-
dorff, so isN, and ifMis paracompact, so isN.
Exercise 2.8.22.LetNbe a submanifold ofMand letPbe a submanifold
ofN. Prove thatPis a submanifold ofM.Hint:Use Theorem 2.1.10.
Exercise 2.8.23.LetNbe a submanifold ofM. Prove that there exists an
open setU⊂Msuch thatN⊂UandNis closed in the relative topology
ofU.
All the theorems we have proved for embedded manifolds and their proofs
carry over almost word for word to the present setting. For example we have
the inverse function theorem, the notion of a regular value, the notions of
a submersion and of an immersion, the notion of an embedding as a proper
injective immersion, and the fact from Theorem 2.3.4 that a subsetP⊂M
is a submanifold if and only if it is the image of an embedding.
Exercise 2.8.24(Lines in Euclidean space).The tangent bundle of the
2-sphere is the 4-manifold
T S
2
=
Φ
(x, y)∈R
3
×R
3

|x|= 1,⟨x, y⟩= 0

(see Example 2.2.6). Define an equivalence relation onT S
2
by
(x, y)∼(x
′
, y
′
)
def
⇐⇒ x
′
=±x, y
′
=y
for (x, y),(x
′
, y
′
)∈T S
2
. Show that the quotient spaceT S
2
/∼can be iden-
tified with the setLof all lines inR
3
, by assigning to each pair (x, y)∈T S
2
the lineℓx,y:=
Φ
y+tx

t∈R

⊂R
3
.Show that the spaceLof lines inR
3
admits the unique structure of a smooth manifold such that the canonical
projectionT S
2
→L: (x, y)7→ℓx,yis a submersion. Show that the manifold
topology onLagrees with the quotient topology onT S
2
/∼. Show that the
mapL→RP
2
×R
3
:ℓx,y7→([x], y) is an embedding.

96 CHAPTER 2. FOUNDATIONS
Example 2.8.25(Veronese embedding).The map
CP
2
→CP
5
: [z0:z1:z2]7→[z
2
0:z
2
1:z
2
2:z1z2:z2z0:z0z1]
is an embedding. (Exercise:Prove this.) It restricts to an embedding of
the real projective plane intoRP
5
and also gives rise to embeddings ofRP
2
intoR
4
as well as to the Roman surface: an immersion ofRP
2
intoR
3
. (See
Example 2.1.17.) There are similar embeddings
CP
n
→CP
N−1
, N :=
`
n+d
d
´
,
for allnandd, defined in terms of monomials of degreedinn+ 1 variables.
These are theVeronese embeddings.
Example 2.8.26(Pl¨ucker embedding).The Grassmannian G2(R
4
) of
2-planes inR
4
is a smooth 4-manifold and can be expressed as the quotient
of the spaceF2(R
4
) of orthonormal 2-frames inR
4
by the orthogonal group
O(2). (See Example 2.8.8.) Write an orthonormal 2-frame inR
4
as a matrix
D=




x0y0
x1y1
x2y2
x3y3




, D
T
D= 1l.
Then the mapf: G2(R
4
)→RP
5
, defined by
f([D]) := [p01:p02:p03:p23:p31:p12], pij:=xiyj−xjyi,
is an embedding and its image is the quadric
X:=f(G2(R
4
)) =
Φ
p∈RP
5
|p01p23+p02p31+p03p12= 0

.
(Exercise:Prove this.) There are analogous embeddings
f: Gk(R
n
)→RP
N−1
, N :=
`
n
k
´
,
for allkandn, defined in terms of thek×k-minors of the (orthonormal)
frames. These are thePl¨ucker embeddings.

2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 97
2.8.5 Tangent Bundle and Vector Fields
LetMbe am-manifold with an atlasA={(ϕα, Uα)}
α∈A
. Thetangent
bundleofMis defined as the disjoint union of the tangent spaces, i.e.
T M:=
[
p∈M
{p} ×TpM={(p, v)|p∈M, v∈TpM}.
Denote byπ:T M→Mthe projection given byπ(p, v) :=p. Recall the
notion of a submersion as a smooth map between smooth manifolds, whose
derivative is surjective at each point.
Lemma 2.8.27.The tangent bundle ofMis a smooth2m-manifold with
coordinate charts
e
ϕα:
e
Uα:=π
−1
(Uα)→ϕα(Uα)×R
m
,
e
ϕα(p, v) := (ϕα(p), dϕα(p)v).
The projectionπ:T M→Mis a surjective submersion. IfMis second
countable and Hausdorff, so isT M.
Proof.For each pairα, β∈Athe set
e
ϕα(
e
Uα∩
e
Uβ) =ϕα(Uα∩Uβ)×R
m
is open inR
m
×R
m
and the transition map
e
ϕβα:=
e
ϕβ◦
e
ϕ
−1
α:
e
ϕα(
e
Uα∩
e
Uβ)→
e
ϕβ(
e
Uα∩
e
Uβ)
is given by
e
ϕβα(x, ξ) = (ϕβα(x), dϕβα(x)ξ)
forx∈ϕα(Uα∩Uβ) andξ∈R
m
where
ϕβα:=ϕβ◦ϕ
−1
α.
Thus the transition maps are all diffeomorphisms and so the coordinate
charts
e
ϕαdefine an atlas onT M. The topology onT Mis determined by
this atlas via Definition 2.8.2. IfMhas a countable atlas, so doesT M. The
remaining assertions are easy exercises.
Definition 2.8.28.LetMbe a smoothm-manifold. A(smooth) vector
fieldonMis a collection of tangent vectorsX(p)∈TpM, one for each
pointp∈M, such that the mapM→T M:p7→(p, X(p))is smooth. The
set of all smooth vector fields onMwill be denoted byVect(M).

98 CHAPTER 2. FOUNDATIONS
Associated to a vector field is a smooth mapM→T Mwhose composi-
tion with the projectionπ:T M→Mis the identity map onM. Strictly
speaking this map should be denoted by a symbol other thanX, for exam-
ple by
e
X. However, it is convenient at this point, and common practice,
to slightly abuse notation and denote the map fromMtoT Malso byX.
Thus a vector field can be defined as a smooth map
X:M→T M
such that
π◦X= id :M→M.
Such a map is also called asection of the tangent bundle.
Now supposeA={(ϕα, Uα)}
α∈A
is an atlas onMandX:M→T M
is a vector field onM. ThenXdetermines a collection of smooth maps
Xα:ϕα(Uα)→R
m
given by
Xα(x) :=dϕα(p)X(p), p:=ϕ
−1
α(x), (2.8.9)
forx∈ϕα(Uα). We can think of eachXαas a vector field on the open set
ϕα(Uα)⊂R
m
, representing the vector fieldXon the coordinate patchUα.
These local vector fieldsXαsatisfy the condition
Xβ(ϕβα(x)) =dϕβα(x)Xα(x) (2.8.10)
forx∈ϕα(Uα∩Uβ). This equation can also be expressed in the form
Xα|
ϕα(Uα∩Uβ)=ϕ
∗
βα
Xβ|
ϕβ(Uα∩Uβ). (2.8.11)
Conversely, any collection of smooth mapsXα:ϕα(Uα)→R
m
satisfy-
ing (2.8.10) determines a unique vectorfieldXonMvia (2.8.9). Thus we
can define the Lie bracket of two vector fieldsX, Y∈Vect(M) by
[X, Y]α(x) := [Xα, Yα](x) =dXα(x)Yα(x)−dYα(x)Xα(x) (2.8.12)
forα∈Aandx∈ϕα(Uα). It follows from equation (2.4.18) in Lemma 2.4.21
that the local vector fields
[X, Y]α:ϕα(Uα)→R
m
satisfy (2.8.11) and hence determine a unique vector field [X, Y] onMvia
[X, Y](p) :=dϕα(p)
−1
[Xα, Yα](ϕα(p)), p∈Uα. (2.8.13)

2.8. THE INTRINSIC DEFINITION OF A MANIFOLD* 99
Thus theLie bracketofXandYis defined onUαas the pullback of the Lie
bracket of the vector fieldsXαandYαunder the coordinate chartϕα. With
this understood all the results in§2.4 about vector fields and flows along with
their proofs carry over word for word to the intrinsic setting wheneverMis a
Hausdorff space. This includes the existence and uniquess result for integral
curves in Theorem 2.4.7, the concept of the flow of a vector field in Defini-
tion 2.4.8 and its properties in Theorem 2.4.9, the notion of completeness
of a vector field (that the integral curves exist for all time), and the various
properties of the Lie bracket such as the Jacobi identity (2.4.20), the formu-
las in Lemma 2.4.18, and the fact that the Lie bracket of two vector fields
vanishes if and only if the corresponding flows commute (see Lemma 2.4.26).
One can also carry over the notion of asubbundleE⊂T Mof ranknto
the intrinsic setting by the condition thatEis a smooth submanifold ofT M
and intersects each fiberTpMin ann-dimensional linear subspace
Ep:={v∈TpM|(p, v)∈E}.
Then the characterization of subbundles in Theorem 2.6.10 and the theorem
of Frobenius 2.7.2 including their proofs also carry over to the intrinsic
setting wheneverMis a Hausdorff space.
2.8.6 Coordinate Notation
Fix a coordinate chartϕα:Uα→R
m
on anm-manifoldM. The components
ofϕαare smooth real valued functions on the open subsetUαofMand it
is customary to denote them by
x
1
, . . . , x
m
:Uα→R.
The derivatives of these functions atp∈Uαare linear functionals
dx
i
(p) :TpM→R, i= 1, . . . , m. (2.8.14)
They form a basis of the dual space
T
∗
pM:= Hom(TpM,R).
(A coordinate chart onMcan in fact be characterized as anm-tuple of real
valued functions on an open subset ofMwhose derivatives are everywhere
linearly independent and which, taken together, form an injective map.)
The dual basis ofTpMwill be denoted by
∂
∂x
1
(p), . . . ,
∂
∂x
m
(p)∈TpM. (2.8.15)

100 CHAPTER 2. FOUNDATIONS
Thus
dx
i
(p)
∂
∂x
j
(p) =δ
i
j:=
æ
1,ifi=j,
0,ifi̸=j,
fori, j= 1, . . . , mand∂/∂x
i
is a vector field on the coordinate patchUα.
For eachp∈Uαit is the canonical basis ofTpMdetermined byϕα. In the
notation of (2.8.5) and Remark 2.8.19 we have
∂
∂x
i
(p) = [α, ei]p=dϕα(p)
−1
ei,
whereei= (0, . . . ,0,1,0, . . . ,0) (with 1 in theith place) denotes the stan-
dard basis vector ofR
m
. In other words, for allξ= (ξ
1
, . . . , ξ
m
)∈R
m
and
allp∈Uα, the tangent vector
v:=dϕα(p)
−1
ξ∈TpM
is given by
v= [α, ξ]p=
m
X
i=1
ξ
i
∂
∂x
i
(p). (2.8.16)
Thus the restriction of a vector fieldX∈Vect(M) toUαhas the form
X|Uα=
m
X
i=1
ξ
i
∂
∂x
i
,
whereξ
1
, . . . , ξ
m
:Uα→Rare smooth real valued functions. If the map
Xα:ϕα(Uα)→R
m
is defined by (2.8.9), then
Xα◦ϕ
−1
α= (ξ
1
, . . . , ξ
m
).
The above notation is motivated by the observation that the derivative of
a smooth functionf:M→Rin the direction of a vector fieldXon a
coordinate patchUαis given by
LXf|Uα=
m
X
i=1
ξ
i
∂f
∂x
i
.
Here the term∂f/∂x
i
is understood as first writingfas a function of
x
1
, . . . , x
m
, then taking the partial derivative, and afterwards expressing this
partial derivative again as a function ofp. Thus∂f/∂x
i
is the shorthand
notation for the function
Γ
∂
∂x
i(f◦ϕ
−1
α)
∆
◦ϕα:Uα→R.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 101
2.9 Consequences of Paracompactness*
In geometry it is often necessary to turn a construction in local coordinates
into a global geometric object. A key technical tool for such“local to global”
constructions is an existence theorem for partitions of unity.
2.9.1 Paracompactness
The existence of a countable atlas is of fundamental importance for almost
everything we will prove about manifolds. The next two remarks describe
several equivalent conditions.
Remark 2.9.1.LetMbe a smooth manifold and denote by
U⊂2
M
the topology induced by the atlas as in Definition 2.8.2. Then the following
are equivalent.
(a)Madmits a countable atlas.
(b)Misσ-compact, i.e. there is a sequence of compact subsetsKi⊂M
such thatKi⊂int(Ki+1) for everyi∈NandM=
S
∞
i=1
Ki.
(c)Every open cover ofMhas a countable subcover.
(d)Missecond countable, i.e. there is a countable collection of open sets
B⊂Usuch that every open setU∈Uis a union of open sets from the
collectionB. (Bis then called acountable basefor the topology ofM.)
That (a) =⇒(b) =⇒(c) =⇒(a) and (a) =⇒(d) follows directly from the
definitions. The proof that (d) implies (a) requires the construction of a
countable refinement and the axiom of choice. (Arefinementof an open
cover{Ui}i∈Iis an open cover{Vj}j∈Jsuch that each setVjis contained in
one of the setsUi.)
Remark 2.9.2.LetMandUbe as in Remark 2.9.1 and suppose in ad-
dition thatMis a connected Hausdorff space. Then the existence of a
countable atlas is also equivalent to each of the following conditions.
(e)Mismetrizable, i.e. there is a distance functiond:M×M→[0,∞)
such thatUis the topology induced byd.
(f)Misparacompact, i.e. every open cover ofMhas a locally finite
refinement. (An open cover{Vj}j∈Jis calledlocally finiteiff everyp∈M
has a neighborhood that intersects only finitely manyVj.)

102 CHAPTER 2. FOUNDATIONS
That (a) implies (e) follows from theUrysohn Metrization Theorem
which asserts (in its original form) that every normal second countable topo-
logical space is metrizable [51, Theorem 34.1]. A topological spaceMis
callednormaliff points are closed and, for any two disjoint closed sets
A, B⊂M, there are disjoint open setsU, V⊂Msuch thatA⊂Uand
B⊂V. It is calledregulariff points are closed and, for every closed set
A⊂Mand everyb∈M\A, there are disjoint open setsU, V⊂Msuch
thatA⊂Uandb∈V. It is calledlocally compactiff, for every open
setU⊂Mand everyp∈U, there is a compact neighborhood ofpcon-
tained inU. It is easy to show that every manifold is locally compact and
every locally compact Hausdorff space is regular.Tychonoff’s Lemma
asserts that a regular topological space with a countable base is normal [51,
Theorem 32.1]. Hence it follows from the Urysohn Metrization Theorem
that every Hausdorff manifold with a countable base is metrizable. That (e)
implies (f) follows from a more general theorem which asserts that every
metric space is paracompact (see [51, Theorem 41.4] and [62]). Conversely,
theSmirnov Metrization Theorem asserts that a paracompact Haus-
dorff space is metrizable if and only if it is locally metrizable, i.e. every
point has a metrizable neighborhood (see [51, Theorem 42.1]). Since ev-
ery manifold is locally metrizable this shows that (f) implies (e). Thus we
have (a) =⇒(e)⇐⇒(f) for every Hausdorff manifold.
The proof that (f) implies (a) does not require the Hausdorff property
but we do need the assumption thatMis connected. (A manifold with
uncountably many connected components, each of which is paracompact, is
itself paracompact but does not admit a countable atlas.) Here is a sketch.
IfMis a paracompact manifold, then there is a locally finite open cover
{Uα}α∈Aby coordinate charts. Since each setUαhas a countable dense
subset, the set{α∈A|Uα∩Uα0
̸=∅}is at most countable for eachα0∈A.
SinceMis connected we can reach each point fromUα0
through a finite
sequence of setsUα1
, . . . , Uαℓ
withUαi−1
∩Uαi
̸=∅. This implies that the
index setAis countable and henceMadmits a countable atlas.
Remark 2.9.3.ARiemann surfaceis a 1-dimensional complex manifold
(i.e. the coordinate charts take values inCand the transition maps are
holomorphic) with a Hausdorff topology. It is a deep theorem in the theory
of Riemann surfaces that every connected Riemann surface is necessarily
second countable (see [2]). Thus pathological examples of the type discussed
in Example 2.8.10 cannot be constructed with holomorphic transition maps.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 103
Exercise 2.9.4.Prove that every manifold is locally compact. Find an ex-
ample of a manifoldMand a pointp0∈Msuch that every closed neighbor-
hood ofp0is non-compact.Hint:The example is necessarly non-Hausdorff.
Exercise 2.9.5.Prove that a manifoldMadmits a countable atlas if and
only if it isσ-compact if and only if every open cover ofMhas a countable
subcover if and only if it is second countable.Hint:The topology ofR
m
is
second countable and every open subset ofR
m
isσ-compact.
Exercise 2.9.6.Prove that every submanifoldM⊂R
k
(Definition 2.1.3)
is second countable.
Exercise 2.9.7.Prove that every connected component of a manifoldMis
an open subset ofMand is path-connected.
2.9.2 Partitions of Unity
Definition 2.9.8(Partition of unity).LetMbe a smooth manifold. A
partition of unityonMis a collection of smooth functions
θα:M→[0,1], α∈A,
such that each pointp∈Mhas an open neighborhoodV⊂Mon which only
finitely manyθαdo not vanish, i.e.
#{α∈A|θα|V̸≡0}<∞, (2.9.1)
and, for everyp∈M, we have
X
α∈A
θα(p) = 1. (2.9.2)
If{Uα}
α∈A
is an open cover ofM, then a partition of unity{θα}
α∈A
(indexed
by the same setA) is calledsubordinate to the coveriff eachθαis
supported inUα, i.e.
supp(θα) :={p∈M|θα(p)̸= 0} ⊂Uα.
Theorem 2.9.9(Partitions of unity).LetMbe a smooth manifold whose
topology is paracompact and Hausdorff. Then, for every open cover ofM,
there exists a partition of unity subordinate to that cover.
The proof requires two preparatory lemmas.

104 CHAPTER 2. FOUNDATIONS
Lemma 2.9.10.LetMbe a smooth manifold with a Hausdorff topology.
Then, for every open setV⊂Mand every compact setK⊂V, there exists
a smooth functionκ:M→[0,∞)with compact support such thatκ(p)>0
for everyp∈Kandsupp(κ)⊂V.
Proof.Assume first thatK={p0}is a single point. SinceMis a mani-
fold it is locally compact. Hence there is a compact neighborhoodC⊂V
ofp0. SinceMis HausdorffCis closed and hence the setU:= int(C)
is a neighborhood ofp0whose closureU⊂Cis compact and contained
inV. ShrinkingU, if necessary, we may assume that there is a coordinate
chartϕ:U→Ω with values in some open neighborhood Ω⊂R
m
of the
origin such thatϕ(p0) = 0. (Heremis the dimension ofM.) Now choose a
smooth functionκ0: Ω→[0,∞) with compact support such thatκ0(0)>0.
Then the functionκ:M→[0,1], defined byκ|U:=κ0◦ϕandκ(p) := 0
forp∈M\Uis supported inVand satisfiesκ(p0)>0. This proves the
lemma in the case whereKis a point.
Now letKbe any compact subset ofV. Then, by the first part of
the proof, there is a collection of smooth functionsκp:M→[0,∞),
one for everyp∈K, such thatκp(p)>0 and supp(κp)⊂V. SinceK
is compact there are finitely many pointsp1, . . . , pk∈Ksuch that the
sets
Φ
p∈M|κpj
(p)>0

coverK. Hence the functionκ:=
P
j
κpj
is sup-
ported inVand is everywhere positive onK. This proves Lemma 2.9.10.
Lemma 2.9.11.LetMbe a topological space. If{Vi}
i∈I
is a locally finite
collection of open sets inM, then
[
i∈I0
Vi=
[
i∈I0
Vi
for every subsetI0⊂I.
Proof.The set
S
i∈I0
Viis obviously contained in the closure of
S
i∈I0
Vi.
To prove the converse choose a pointp0∈M\
S
i∈I0
Vi. Since the collec-
tion{Vi}i∈Iis locally finite, there exists an open neighborhoodUofp0such
that the setI1:={i∈I|Vi∩U̸=∅}is finite. Hence the set
U0:=U\
[
i∈I0∩I1
Vi
is an open neighborhood ofp0and we haveU0∩Vi=∅for everyi∈I0.
Hencep0/∈
S
i∈I0
Vi. This proves Lemma 2.9.11.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 105
Proof of Theorem 2.9.9.Let{Uα}
α∈A
be an open cover ofM. We prove in
four steps that there is a partition of unity subordinate to this cover. The
proofs of steps one and two are taken from [51, Lemma 41.6].
Step 1.There is a locally finite open cover{Vi}
i∈I
ofMsuch that, for
everyi∈I, the closureViis compact and contained in one of the setsUα.
Denote byV⊂2
M
the set of all open setsV⊂Msuch thatVis compact
andV⊂Uαfor someα∈A. SinceMis a locally compact Hausdorff
space the collectionVis an open cover ofM. (Ifp∈M, then there is an
α∈Asuch thatp∈Uα; sinceMis locally compact there is a compact
neighborhoodK⊂Uαofp; sinceMis HausdorffKis closed and thus
V:= int(K) is an open neighborhood ofpwithV⊂K⊂Uα.) SinceMis
paracompact the open coverVhas a locally finite refinement{Vi}i∈I. This
cover satisfies the requirements of Step 1.
Step 2.There is a collection of compact setsKi⊂Vi, one for eachi∈I,
such thatM=
S
i∈I
Ki.
Denote byW⊂2
M
the set of all open setsW⊂Msuch thatW⊂Vifor
somei. SinceMis a locally compact Hausdorff space, the collectionWis an
open cover ofM. SinceMis paracompactWhas a locally finite refinement
{Wj}j∈J. By the axiom of choice there is a map
J→I:j7→ij
such that
Wj⊂Vij
∀j∈J.
Since the collection{Wj}j∈Jis locally finite, we have
Ki:=
[
ij=i
Wj=
[
ij=i
Wj⊂Vi
by Lemma 2.9.11. SinceViis compact so isKi.
Step 3.There is a partition of unity subordinate to the cover{Vi}i∈I.
Choose a collection of compact setsKi⊂Vifori∈Ias in Step 2. Then,
by Lemma 2.9.10 and the axiom of choice, there is a collection of smooth
functionsκi:M→[0,∞) with compact support such that
supp(κi)⊂Vi, κ i|Ki
>0 ∀i∈I.

106 CHAPTER 2. FOUNDATIONS
Since the cover{Vi}i∈Iis locally finite the sum
κ:=
X
i∈I
κi:M→R
islocally finite(i.e. each point inMhas a neighborhood in which only
finitely many terms do not vanish) and thus defines a smooth function onM.
This function is everywhere positive, because each summand is nonnegative
and, for eachp∈M, there is ani∈Iwithp∈Kiso thatκi(p)>0. Thus
the funtionsχi:=κi/κdefine a partition of unity satisfying supp(χi)⊂Vi
for everyi∈Ias required.
Step 4.There is a partition of unity subordinate to the cover{Uα}α∈A.
Let{χi}i∈Ibe the partition of unity constructed in Step 3. By the axiom
of choice there is a mapI→A:i7→αisuch thatVi⊂Uαi
for everyi∈I.
Forα∈Adefineθα:M→[0,1] by
θα:=
X
αi=α
χi.
Here the sum runs over all indicesi∈Iwithαi=α. This sum is locally
finite and hence is a smooth function onM. Moreover, each point inMhas
an open neighborhood in which only finitely many of theθαdo not vanish.
Hence the sum of theθαis a well defined function onMand
X
α∈A
θα=
X
α∈A
X
αi=α
χi=
X
i∈I
χi≡1.
This shows that the functionsθαform a partition of unity. To prove the
inclusion supp(θα)⊂Uαwe consider the open sets
Wi:={p∈M|χi(p)>0}
fori∈I. SinceWi⊂Vithis collection is locally finite. Hence, by
Lemma 2.9.11, we have
supp(θα) =
[
αi=α
Wi=
[
αi=α
Wi=
[
αi=α
supp(χi)⊂
[
αi=α
Vi⊂Uα.
This proves Theorem 2.9.9.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 107
2.9.3 Embedding in Euclidean Space
Theorem 2.9.12.LetMbe a second countable smoothm-manifold with a
Hausdorff topology. Then there exists an embeddingf:M→R
2m+1
with a
closed image.
Proof.The proof has five steps.
Step 1.LetU⊂Mbe an open set and letK⊂Ube a compact set.
Then there exists an integerk∈N, a smooth mapf:M→R
k
, and an
open setV⊂M, such thatK⊂V⊂U, the restrictionf|V:V→R
k
is an
injective immersion, andf(p) = 0for allp∈M\U.
Choose a smooth atlasA={(ϕα, Uα)}α∈AonMsuch that, for eachα∈A,
eitherUα⊂UorUα∩K=∅. SinceMis a paracompact Hausdorff mani-
fold, Theorem 2.9.9 asserts that there exists a partition of unity{θα}α∈Asub-
ordinate to the open cover{Uα}α∈AofM. Since the sets{p∈Uα|θα(p)>0}
withUα⊂Uform an open cover ofKandKis a compact subset ofM,
there exist finitely many indicesα1, . . . , αℓ∈Asuch that
K⊂
Φ
p∈M

θα1
(p) +· · ·+θαℓ
(p)>0

=:V⊂U.
Letk:=ℓ(m+ 1) and, fori= 1, . . . , ℓ, abbreviate
ϕi:=ϕαi
, θi:=θαi
.
Define the smooth mapf:M→R
k
by
f(p) :=







θ1(p)
θ1(p)ϕ1(p)
.
.
.
θℓ(p)
θℓ(p)ϕℓ(p)







forp∈M.
Then the restrictionf|V:V→R
k
is injective. Namely, ifp0, p1∈Vsatisfy
f(p0) =f(p1),
then
I:=
Φ
i

θi(p0)>0

=
Φ
i

θi(p1)>0

̸=∅
and, fori∈I, we haveθi(p0) =θi(p1), henceϕi(p0) =ϕi(p1), and sop0=p1.
Moreover, for everyp∈Vthe derivativedf(p) :TpM→R
k
is injective, and
this proves Step 1.

108 CHAPTER 2. FOUNDATIONS
Step 2.Letf:M→R
k
be an injective immersion and letA ⊂R
(2m+1)×k
be a nonempty open set. Then there exists a matrixA∈ Asuch that the
mapAf:M→R
2m+1
is an injective immersion.
The proof of Step 2 uses the Theorem of Sard (see [1, 50]). The sets
W0:=
Φ
(p, q)∈M×M

p̸=q

,
W1:=
Φ
(p, v)∈T M

v̸= 0

are open subsets of smooth second countable Hausdorff 2m-manifolds and
the maps
F0:A ×W0→R
2m+1
, F 1:A ×W1→R
2m+1
,
defined by
F0(A, p, q) :=A(f(p)−f(q)), F 1(A, p, v) :=Adf(p)v
forA∈ A, (p, q)∈W0, and (p, v)∈W1, are smooth. Moreover, the zero
vector inR
2m+1
is a regular value ofF0becausefis injective and ofF1
becausefis an immersion. Hence it follows from the intrinsic analogue of
Theorem 2.2.19 that the sets
M0:=F
−1
0
(0) =
Φ
(A, p, q)∈ A ×W0

Af(p) =Af(q)

,
M1:=F
−1
1
(0) =
Φ
(A, p, v)∈ A ×W1

Adf(p)v= 0

are smooth manifolds of dimension
dimM0= dimM1= (2m+ 1)k−1.
SinceMis a second countable Hausdorff manifold, so areM0andM1.
Hence the Theorem of Sard asserts that the canonical projections
M0→ A: (A, p, q)7→A=:π0(A, p, q),
M1→ A: (A, p, v)7→A=:π1(A, p, v),
have a common regular valueA∈ A. Since
dimM0= dimM1<dimA,
this implies
A∈ A \(π0(M0)∪π1(M1)).
HenceAf:M→R
2m+1
is an injective immersion and this proves Step 2.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 109
IfMis compact, the result follows from Steps 1 and 2 withK=U=M.
In the noncompact case the proof requires two more steps to construct an
embedding intoR
4m+4
and a further step to reduce the dimension to 2m+ 1.
Step 3.AssumeMis not compact. Then there exists a sequence of open
setsUi⊂M, a sequence of smooth functionsρi:M→[0,1], and a sequence
of compact setsKi⊂Uisuch that
supp(ρi)⊂Ui, K i=ρ
−1
i
(1)⊂Ui, U i∩Uj=∅
for alli, j∈Nwith|i−j| ≥2andM=
S
∞
i=1
Ki.
SinceMis second countable, there exists a sequence of compact setsCi⊂M
such thatCi⊂int(Ci+1) for alli∈NandM=
S
i∈N
Ci(Remark 2.9.1).
Define the compact setsBi⊂MbyC0:=∅and
Bi:=Ci\Ci−1 fori∈N.
ThenM=
S
i∈N
Biand, for alli, j∈Nwithj≥i+ 2, we have
Bi⊂Ci⊂int(Cj−1), B j⊂Cj\int(Cj−1)
and soBi∩Bj=∅. SinceMis metrizable by Remark 2.9.2, there exists
a distance functiond:M×M→[0,∞) that induces the intrinsic topology
onM. Define
Ai:=
[
j∈N\{i−1,i,i+1}
Bj, εi:=d(Ai, Bi) = inf
p∈Ai,q∈Bi
d(p, q).
ThenAiis a closed subset ofM, because any convergent sequence inMmust
belong to a finite union of theBj. SinceAi∩Bi=∅, this impliesεi>0.
Fori∈Ndefine the setUi⊂Mby
Ui:=
Φ
p∈M

there exists aq∈Biwithd(p, q)< εi/3

.
Then{Ui}i∈Nis a sequence of open subsets ofMsuch thatBi⊂Ui⊂Ci+1
for alli∈NandUi∩Uj=∅for|i−j| ≥2. In particular, each setUihas a
compact closure.
For eachithere exists of a partition of unity subordinate to the open
coverM=Ui∪(M\Bi) and hence a smooth functionρi:M→[0,1] such
that supp(ρi)⊂Uiandρi|Bi
≡1. DefineKi:=ρ
−1
i
(1) ={p∈Ui|ρi(p) = 1}
fori∈N. ThenKiis a compact set andBi⊂Ki⊂Uifor eachi∈N.
HenceM=
S
i∈N
Kiand this proves Step 3.

110 CHAPTER 2. FOUNDATIONS
Step 4.AssumeMis not compact. Then there exists an embedding
f:M→R
4m+4
with a closed image and a pair of orthonormal vectorsx, y∈R
4m+4
such
that, for everyε >0, there exists a compact setK⊂Mwith
sup
p∈M\K
inf
s,t∈R

f(p)
|f(p)|
−sx−ty

< ε. (2.9.3)
AssumeMis not compact and letKi, Ui, ρibe as in Step 3. Then, by
Steps 1 and 2, there exists a sequence of smooth mapsgi:M→R
2m+1
such
thatgi|
M\Ui
≡0, the restrictiongi|Ki
:Ki→R
2m+1
is injective, and the
derivativedgi(p) :TpM→R
2m+1
is injective for allp∈Kiand alli∈N.
Letξ∈R
2m+1
be a unit vector and define the mapsfi:M→R
2m+1
by
fi(p) :=ρi(p)

iξ+
gi(p)
q
1 +|gi(p)|
2

 (2.9.4)
forp∈Mandi∈N. Then the restrictionfi|Ki
:Ki→R
2m+1
is injective,
the derivativedfi(p) :TpM→R
2m+1
is injective for allp∈Ki, and
supp(fi)⊂Ui, fi(Ki)⊂B1(iξ), fi(M)⊂Bi+1(0).
Define the mapsf
odd
, f
ev
:M→R
2m+1
andρ
odd
, ρ
ev
:M→Rby
ρ
odd
(p) :=
æ
ρ2i−1(p),ifi∈Nandp∈U2i−1,
0, ifp∈M\
S
i∈N
U2i−1,
f
odd
(p) :=
æ
f2i−1(p),ifi∈Nandp∈U2i−1,
0, ifp∈M\
S
i∈N
U2i−1,
ρ
ev
(p) :=
æ
ρ2i(p),ifi∈Nandp∈U2i,
0, ifp∈M\
S
i∈N
U2i,
f
ev
(p) :=
æ
f2i(p),ifi∈Nandp∈U2i,
0, ifp∈M\
S
i∈N
U2i,
and define the mapf:M→R
4m+4
by
f(p) :=
ı
ρ
odd
(p), f
odd
(p), ρ
ev
(p), f
ev
(p)
ȷ
forp∈M.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 111
We prove thatfis injective. To see this, note that
p∈K2i−1 =⇒
æ
2i−2<

f
odd
(p)

<2i,
|f
ev
(p)|<2i+ 1,
p∈K2i =⇒
æ
2i−1<|f
ev
(p)|<2i+ 1,

f
odd
(p)

<2i+ 2,
(2.9.5)
Now letp0, p1∈Msuch thatf(p0) =f(p1). Assume first thatp0∈K2i−1.
Thenρ
odd
(p1) =ρ
odd
(p0) = 1 and hencep1∈
S
j∈N
K2j−1. By (2.9.5), we
also have 2i−2<|f
odd
(p1)|=|f
odd
(p0)|<2iand hencep1∈K2i−1. This
impliesf2i−1(p1) =f
odd
(p1) =f
odd
(p0) =f2i−1(p0) and sop0=p1. Now
assumep0∈K2i. Thenρ
ev
(p1) =ρ
ev
(p0) = 1 and hencep1∈
S
j∈N
K2j.
By (2.9.5), we also have 2i−1<|f
ev
(p1)|=|f
ev
(p0)|<2i+ 1,sop1∈K2i,
which impliesf2i(p1) =f
ev
(p1) =f
ev
(p0) =f2i(p0), and so againp0=p1.
This shows thatfis injective. Thatfis an immersion follows from the fact
that the derivativedfi(p) is injective for allp∈Kiand alli∈N.
We prove thatfis proper and has a closed image. Let (pν)ν∈Nbe a
sequence inMsuch that the sequence (f(pν))ν∈NinR
4m+4
is bounded.
Choosei∈Nsuch that|f
odd
(pν)|<2iand|f
ev
(pν)|<2i+ 1 for allν∈N.
Thenpν∈
S
2i
j=1
Kjfor allν∈Nby (2.9.5). Hence (pν)ν∈Nhas a convergent
subsequence. Thusf:M→R
4m+4
is an embedding with a closed image.
Next consider the pair of orthonormal vectors
x:= (0, ξ,0,0), y:= (0,0,0, ξ)
inR
4m+4
=R×R
2m+1
×R×R
2m+1
. Let (pν)ν∈Nbe a sequence inM
that does not have a convergent subsequence and choose a sequenceiν∈N
such thatpν∈K2iν−1∪K2iνfor allν∈N. Theniνtends to infinity.
Ifpν∈K2iν−1for allν, then we have lim sup
ν→∞|f
odd
(pν)|
−1
|f
ev
(pν)| ≤1
by (2.9.5). Passing to a subsequence, still denoted by (pν)ν∈N, we may
assume that the limitλ:= limν→∞|f
odd
(fν)|
−1
|f
ev
(pν)|exists. Then
0≤λ≤1,lim
ν→∞

f
odd
(pν)

|f(pν)|
=
1
√
1 +λ
2
,lim
ν→∞
|f
ev
(pν)|
|f(pν)|
=
λ
√
1 +λ
2
,
and it follows from (2.9.4) that
lim
ν→∞
f
odd
(pν)
|f
odd
(pν)|
=ξ, lim
ν→∞
f
ev
(pν)
|f
odd
(pν)|
=λξ.
This implies
lim
ν→∞
f(pν)
|f(pν)|
=
`
0,
ξ
√
1 +λ
2
,0,
λξ
√
1 +λ
2
´
=
1
√
1 +λ
2
x+
λ
√
1 +λ
2
y.

112 CHAPTER 2. FOUNDATIONS
Similarly, ifpν∈K2iνfor allν, there exists a subsequence such that the
limitλ:= limν→∞|f
ev
(pν)|
−1
|f
odd
(pν)|exists and, by (2.9.4), this implies
lim
ν→∞
f(pν)
|f(pν)|
=
`
0,
λξ
√
1 +λ
2
,0,
ξ
√
1 +λ
2
´
=
λ
√
1 +λ
2
x+
1
√
1 +λ
2
y.
This shows that the vectorsxandysatisfy the requirements of Step 4.
Step 5.There exists an embeddingf:M→R
2m+1
with a closed image.
For compact manifolds the result was proved in Steps 1 and 2 and form= 0
the assertion is obvious, because thenMis a finite or countable set with
the discrete topology. Thus assume thatMis not compact andm≥1.
Choosef:M→R
4m+4
andx, y∈R
4m+4
as in Step 4 and define
A:=
(
A∈R
(2m+1)×(4m+4)

the vectorsAxandAy
are linearly independent
)
.
Sincem≥1, this is a nonempty open subset ofR
(2m+1)×(4m+4)
. We prove
that the mapAf:M→R
2m+1
is proper and has a closed image for ev-
eryA∈ A. To see this, fix a matrixA∈ A. Let (pν)ν∈Nbe a sequence inM
that does not have a convergent subsequence. Then by Step 4 there exists a
subsequence, still denoted by (pν)ν∈N, and real numberss, t∈Rsuch that
s
2
+t
2
= 1, lim
ν→∞
f(pν)
|f(pν)|
=sx+ty, lim
ν→∞
|f(pν)|=∞.
This implies
lim
ν→∞
Af(pν)
|f(pν)|
=sAx+tAy̸= 0
and hence limν→∞|Af(pν)|=∞. Thus the preimage of every compact sub-
set ofR
2m+1
under the mapAf:M→R
2m+1
is a compact subset ofM,
and henceAfis proper and has a closed image (Remark 2.3.3).
Now it follows from Step 2 that there exists a matrixA∈ Asuch that the
mapAf:M→R
2m+1
is an injective immersion. Hence it is an embedding
with a closed image. This proves Step 5 and Theorem 2.9.12.
TheWhitney Embedding Theorem asserts that every second count-
able Hausdorffm-manifoldMadmits an embeddingf:M→R
2m
. The
proof is based on theWhitney Trickand goes beyond the scope of this
book. The next exercise shows that Whitney’s theorem is sharp.
Remark 2.9.13.The manifoldRP
2
cannot be embedded intoR
3
. The same
is true for theKlein bottleK:=R
2
/≡where the equivalence relation is
given by [x, y]≡[x+k, ℓ−y] forx, y∈Randk, ℓ∈Z.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 113
2.9.4 Leaves of a Foliation
LetMbe anm-dimensional paracompact Hausdorff manifold andE⊂T M
be an integrable subbundle of rankn. LetL⊂Mbe aclosed leafof the
foliation determined byE. ThenLis a smoothn-dimensional submanifold
ofM. Here is a sketch of David Epstein’s proof of this fact in [19].
(a)The spaceLwith the intrinsic topology admits the structure of a mani-
fold such that the obvious inclusionι:L→Mis an injective immersion.
This is an easy exercise. For the definition of the intrinsic topology see
Exercise 2.7.10. The dimension ofLisn.
(b)Iff:X→Yis a continuous map between topological spaces such thatY
is paracompact and there is an open cover{Vj}j∈JofYsuch thatf
−1
(Vj)
is paracompact for eachj, thenXis paracompact.To see this, we may
assume that the cover{Vj}j∈Jis locally finite. Now let{Uα}α∈Abe an open
cover ofX. Then the setsUα∩f
−1
(Vj) define an open cover off
−1
(Vj).
Choose a locally finite refinement{Wij}i∈Ij
of this cover. Then the open
cover{Wij}j∈J, i∈Ij
ofMis a locally finite refinement of{Uα}α∈A.
(c)The intrinsic topology ofLis paracompact.This follows from (b) and
the fact that the intersection ofLwith every foliation box is paracompact
in the intrinsic topology.
(d)The intrinsic topology ofLis second countable.This follows from (a)
and (c) and the fact that every connected paracompact manifold is second
countable (see Remark 2.9.2).
(e)The intersection ofLwith a foliation box consists of at most countably
many connected components.This follows immediately from (d).
(f)IfLis a closed subset ofM, then the intersection ofLwith a foliation
box has only finitely many connected components.To see this, we choose
a transverse slice of the foliation atp0∈L, i.e. a connected submanifold
T⊂Mthroughp0, diffeomorphic to an open ball inR
m−n
, whose tangent
space at each pointp∈Tis a complement ofEp. By (d) we have thatT∩L
is at most countable. If this set is not finite, even after shrinkingT, there
must be a sequencepi∈(T∩L)\{p0}converging top0. Using the holonomy
of the leaf (obtained by transporting transverse slices along a curve via a
lifting argument) we find that every pointp∈T∩Lis the limit point of a
sequence in (T∩L)\ {p}. Hence the one-point set{p}has empty interior in
the relative topology ofT∩Lfor eachp∈T∩L. ThusT∩Lis a countable
union of closed subsets with empty interior. SinceT∩Ladmits the structure
of a complete metric space, this contradicts the Baire category theorem.
(g)It follows immediately from (f) thatLis a submanifold ofM.

114 CHAPTER 2. FOUNDATIONS
2.9.5 Principal Bundles
An interesting class of foliations arises from smooth Lie group actions.
Let G⊂GL(N,R) be a compact Lie group and letPbe a smoothm-
manifold whose topology is Hausdorff and second countable. Asmooth
(contravariant)G-action onPis a smooth map
P×G→P: (p, g)7→pg (2.9.6)
that satisfies the conditions
(pg)h=p(gh), p1l =p (2.9.7)
for allp∈Pand allg, h∈G. Fix any such group action. Then every
group elementg∈G determines a diffeomorphismP→P:p7→pg, whose
derivative atp∈Pis denoted byTpP→TpgP:v7→vg. Every Lie alge-
bra elementξ∈g:= Lie(G) =T1lG determines a vector fieldXξ∈Vect(P)
which assigns to eachp∈Pthe tangent vector
Xξ(p) :=pξ:=
d
dt

t=0
pexp(tξ)∈TpP. (2.9.8)
The linear mapg→Vect(P) :ξ7→Xξis called theinfinitesimal action.
It is a Lie algebra anti-homomorphism because the group action is contra-
variant. (Exercise: Prove that [Xξ, Xη] =−X
[ξ,η]forξ, η∈g.) The group
action (2.9.6) is said to be withfinite isotropyiff theisotropy subgroup
Gp:={g∈G|pg=p}
is finite for allp∈P. The isotropy subgroup Gpis a Lie subgroup of G
with Lie algebragp:= Lie(Gp) ={ξ∈g|Xξ(p) = 0}. Since G is compact,
this shows that Gpis a finite subgroup of G if and only ifgp={0}
or, equivalently, the mapg→TpP:ξ7→Xξ(p) =pξis injective. Thus, in
the finite isotropy case, the group action determines an involutive subbun-
dleE⊂T Pwith the fibersEp:=pg={Xξ(p)|ξ∈g}forp∈P. When G
is connected, the leaves of the corresponding foliation are the group or-
bitspG :={pg|g∈G}. These are the elements of theorbit space
P/G :={pG|p∈P}.
There is a natural projectionπ:P→P/G defined byπ(p) :=pG forp∈P
and the orbit spaceP/G is equipped with the quotient topology (a sub-
setU⊂P/G is open if and only ifπ
−1
(U) is an open subset ofP). The
group action is calledfreeiff Gp={1l}for allp∈P. The next theorem
shows that, in the case of a free action, the quotient space admits a unique
smooth structure such that the projectionπ:P→P/G is a submersion.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 115
Theorem 2.9.14(Principal Bundle).LetPbe a smoothm-manifold
whose topology is Hausdorff and second countable. SupposePis equipped
with a smooth contravariant action of a compact Lie groupGand assume
the group action is free. Thendim(G)≤mandB:=P/Gadmits a unique
smooth structure such that the projectionπ:P→Bis a submersion. The
intrinsic topology ofB, induced by the smooth structure, agrees with the
quotient topology, and it is Hausdorff and second countable.
Proof.For eachp∈Pthe map G→P:g7→pgis an embedding and this
impliesk:= dim(G)≤dim(P) =m. Definen:=m−k. Alocal sliceof
the group action is a smooth mapι: Ω→P, defined on an open set Ω⊂R
n
,
such that the map Ω×G→P: (x, g)7→ι(x)gis an embedding. With this
understood, we prove the assertions in five steps.
Step 1.For everyp0∈Pthere exists a local sliceι0: Ω0→P, defined on
an open neighborhoodΩ0⊂R
n
of the origin, such thatι0(0) =p0.
Choose a coordinate chartϕ:V→R
m
on an open neighborhoodV⊂P
ofp0such thatϕ(p0) = 0 andϕ(V) =R
m
. Definev1, . . . , vm∈Tp0
Pby
dϕ(p0)vi:=eifori= 1, . . . , m,
wheree1, . . . , emis the standard basis ofR
m
. Reorder the coordinates
onR
m
, if necessary, such that the vectorsv1, . . . , vnproject to a basis of the
quotient spaceTp0
P/p0g. Defineι:R
n
→Pby
ι(x1, . . . , xn) :=ϕ
−1
(x1, . . . , xn,0, . . . ,0)
and define the mapψ:R
n
×G→Pby
ψ(x, g) :=ι(x)g forx∈R
n
andg∈G.
Thenψis smooth and its derivativedψ(0,1l) :R
n
×g→Tp0
Pis given by
dψ(0,1l)(bx, ξ) =
n
X
i=1
bxivi+p0ξ
forbx= (bx1, . . . ,bxn)∈R
n
andξ∈g. Hencedψ(0,1l) is bijective and so it
follows from the Inverse Function Theorem 2.2.17 that there exist open
neighborhoods Ω⊂R
n
of 0, Ω1⊂G of 1l, andW⊂Pofp0such that the
restricted map
ψ1:=ψ|Ω×Ω1
: Ω×Ω1→W
is a diffeomorphism.

116 CHAPTER 2. FOUNDATIONS
Next we prove that there exists an open neigborhood Ω0⊂Ω of the
origin such that the restricted map
ψ0:=ψ|Ω0×G: Ω0×G→P
is injective. Suppose otherwise that no such neighborhood Ω0exists. Then
there exist sequences (xi, gi),(x
′
i
, g
′
i
)∈Ω×G such that (xi, gi)̸= (x
′
i
, g
′
i
)
andψ(xi, gi) =ψ(x
′
i
, g
′
i
) for alliand the sequences (xi)i∈Nand (x
′
i
)i∈Nin Ω
converge to the origin. Since G is compact we may assume, by passing to
a subsequence if necessary, that the sequences (gi)i∈Nand (g
′
i
)i∈Nconverge.
Denote the limits by
g:= lim
i→∞
gi∈G, g
′
:= lim
i→∞
g
′
i∈G.
Then
p0g= lim
i→∞
ι(xi)gi= lim
i→∞
ι(x
′
i)g
′
i=p0g
′
and sog=g
′
because the group action is free. Thus the sequence (g
′
i
g
−1
i
)i∈N
in G converges to 1l and hence belongs to the set Ω1forisufficiently large.
Since
ψ1(xi,1l) =ι(xi) =ι(x
′
i)g
′
ig
−1
i
=ψ1(x
′
i, g
′
ig
−1
i
)
for alli, this contradicts the injectivity ofψ1. Thus we have proved that
the mapψ0: Ω0×G→Pis injective for a suitable neighborhood Ω0⊂Ω
of the origin. That it is an immersion is a direct consequence of the formula
dψ0(x, g)(bx,bg) =
Γ
dι(x)bx+ι(x)(bgg
−1
)
∆
g=
Γ
dψ0(x,1l)(bx,bgg
−1
)
∆
g
for allx∈Ω0,bx∈R
n
,g∈G, andbg∈TgG, and the fact that the deriva-
tivedψ0(x,1l) is bijective for allx∈Ω0(even for allx∈Ω).
Thus we have proved thatψ0: Ω0×G→Pis an injective immersion.
Shrinking Ω0further, if necessary, we may assume that Ω0has a compact
closure and thatψis injective onΩ0×G. This implies thatψ0is proper.
Namely, if (xi, gi)
i∈N
is a sequence in Ω0×G and (x, g)∈Ω0×G such
thatψ0(x, g) = limi→∞ψ0(xi, gi), then there is a subsequence (xiν, giν)ν∈N
that converges to a pair (x
′
, g
′
)∈Ω0×G. This subsequence satisfies
ψ(x
′
, g
′
) = lim
ν→∞
ψ0(xiν, giν) =ψ(x, g).
Sinceψis injective onΩ0×G, this impliesx=x
′
andg=g
′
. Thus
every subsequence of (xi, gi)i∈Nhas a further subsequence that converges
to (x, g) and so the sequence (xi, gi)i∈Nitself converges to (x, g). Thus the
mapψ0: Ω0×G→Pis a proper injective immersion and this proves Step 1.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 117
Step 2.Letι: Ω→Pbe a local slice. Then the setU:=π(ι(Ω))⊂Bis
open in the quotient topology and the mapπ◦ι: Ω→Uis a homeomorphism
with respect to the quotient topology onU.
The mapψ: Ω×G→P, defined byψ(x, g) :=ι(x)gforx∈Ω andg∈G,
is an embedding. HenceW:=ψ(Ω×G) is an open G-invariant subset
ofPandψ: Ω×G→Wis a G-equivariant homeomorphism. Moreover,
for every elementp∈P, we haveπ(p)∈Uif and only if there exists
an elementx∈Ω and an elementg∈G such thatp=ι(x)g=ψ(x, g).
Thusπ
−1
(U) =ψ(Ω×G) =Wis an open subset ofP, and soUis an open
subset ofB=P/G with respect to the quotient topology. The continuity
ofπ◦ι: Ω→Ufollows directly from the definition. Moreover, if Ω
′
⊂Ω is
an open set andU
′
:=π(ι(Ω
′
)), thenπ
−1
(U
′
) =ψ(Ω
′
×G) is open by the
same argument, and soU
′
⊂Bis open with respect to the quotient topology.
Thusπ◦ι: Ω→Uis a homeomorphism and this proves Step 2.
Step 3.By Step 1 there exists a collectionια: Ωα→P,α∈A, of local
slices such that the setsUα:=π(ια(Ωα))cover the orbit spaceB=P/G.
Forα∈Adefine
ϕα:= (π◦ια)
−1
:Uα→Ωα.
ThenA={(ϕα, Uα)}α∈Ais a smooth structure onBwhich renders the
canonical projectionπ:P→Binto a submersion. Moreover, this smooth
structure is compatible with the quotient topology onB.
Forα, β∈Adefine Ωαβ:=ϕα(Uα∩Uβ) andϕβα:=ϕβ◦ϕ
−1
α: Ωαβ→Ωβα.
We must prove thatϕβαis smooth. To see this, defineψα: Ωα×G→P
byψα(x, g) :=ια(x)gforα∈A,x∈Ωα, andg∈G. Thenψαis a diffeo-
morphism onto its image andψα(Ωαβ×G) =ψβ(Ωβα×G) =π
−1
(Uα∩Uβ).
Forx∈Ωαβthe elementϕβα(x)∈Ωβαis the projection ofψ
−1
β
◦ψα(x,1l)
onto the first factor. Thusϕβαis smooth and so is its inverseϕαβ. This
shows that{(Uα, ϕα)}α∈Ais a smooth structure onB. Second,πis a sub-
mersion with respect to this smooth structure, becauseϕα◦π◦ψα(x, g) =x
for allα∈A, allx∈Ωα, and allg∈G. Third, this smooth structure is com-
patible with the quotient topology by Step 2. This proves Step 3.
Step 4.There is only one smooth structure onBwith respect to which the
projectionπ:P→Bis a submersion.
Fix any smooth structure onBfor which the projectionπ:P→Bis a
submersion. Then the dimension ofBisn= dim(P)−dim(G), and so the
smooth structure consists of bijectionsϕα:Uα→Ωαfrom subsetsUα⊂B
onto open sets Ωα⊂R
n
such that the setsUαcoverBand the transition
maps are diffeomorphisms between open subsets ofR
n
.

118 CHAPTER 2. FOUNDATIONS
We prove that the intrinsic topology onBagrees with the quotient topol-
ogy. To see this, fix a subsetU⊂B. Then the following are equivalent.
(a)Uis open with respect to the intrinsic topology onB.
(b)ϕα(U∩Uα) is open inR
n
for allα∈A.
(c)π
−1
(U∩Uα) is open inPfor allα∈A.
(d)π
−1
(U) is open inP.
(e)Uis open with respect to the quotient topology onB.
The equivalence of (a) and (b) follows from the definition of the intrinsic
topology. That (b) implies (c) follows from the three observations that the
setπ
−1
(Uα) is open inP, the mapϕα◦π:π
−1
(Uα)→Ωαis continuous,
and (ϕα◦π)
−1
(ϕα(U∩Uα)) =π
−1
(U∩Uα). That (c) implies (b) follows
from the fact that the mapϕα◦π:π
−1
(Uα)→Ωαis a submersion and hence
maps the open setπ
−1
(U∩Uα) onto an open subset of Ωα(Corollary 2.6.2).
The equivalence of (c) and (d) follows from the fact that the mapπ:P→B
is continuous andUα⊂Bis open (both with respect to the intrinsic topology
onB) and soπ
−1
(Uα) is open inPfor allα∈A. The equivalence of (d)
and (e) follows from the definition of the quotient topology onB.
Now letι: Ω→Pbe a local slice and define the setU:=π(ι(Ω))⊂B
and the mapϕ:= (π◦ι)
−1
:U→Ω. Then the composition
ϕα◦ϕ
−1
=ϕα◦π◦ι:ϕ(U∩Uα)→ϕα(U∩Uα)
is a homeomorphism between open subsets ofR
n
. Moreover,ϕα◦ϕ
−1
is the
composition of the smooth mapsι:{x∈Ω|π(ι(x))∈Uα} →π
−1
(U∩Uα),
π:π
−1
(U∩Uα)→U∩Uα, andϕα:U∩Uα→ϕα(U∩Uα). Soϕα◦ϕ
−1
is
smooth and its derivative is everywhere bijective becauseπis a submersion
and the kernel ofdπ(ι(x)) is transverse to the image ofdι(x). Thusϕα◦ϕ
−1
is a diffeomorphism by the Inverse Function Theorem and this proves Step 4.
Step 5.The quotient topology onBis a Hausdorff and second countable.
Letια: Ωα→Pforα∈Abe a collection of local slices such that the
setsUα:=π(ια(Ωα)) coverB. Then the open setsπ
−1
(Uα) form an open
cover ofPand so there is a countable subcover. ThusBis second count-
able. To prove thatBis Hausdorff, fix two distinct elementsb0, b1∈Band
choosep0, p1∈Psuch thatπ(p0) =b0andπ(p1) =b1. Thenp0G andp1G
are disjoint compact subsets ofPand hence can be separated by disjoint
open subsetsU0, U1⊂P, becausePis a Hausdorff space. Now fori= 0,1
the setVi:={p∈P|pG⊂Ui}is open (exercise) and contains the orbitpiG.
HenceW0:=π(V0) andW1:=π(V1) are disjoint open subsets ofBsuch
thatb0∈W0andb1∈W1. This proves Step 5 and Theorem 2.9.14.

2.9. CONSEQUENCES OF PARACOMPACTNESS* 119
Example 2.9.15.There are many important examples of free group actions
and principal bundles. A class of examples arises from orthonormal frame
bundles (§3.4). The complex projective spaceB=CP
n
arises from the
action of the circle G =S
1
on the unit sphereP=S
2n+1
⊂C
n+1
(Exam-
ple 2.8.5). The real projective spaceB=RP
n
arises from the action of the
finite group G =Z/2Zon the unit sphereP=S
n
⊂R
n+1
(Example 2.8.6).
The complex GrassmannianB= Gk(C
n
) arises from the action of G = U(k)
on the spaceP=Fk(C
n
) of unitaryk-frames inC
n
(Example 3.7.6). If G is
a Lie group and K⊂G is a compact subgroup, then by Theorem 2.9.14 the
homogeneous spaceG/K admits a unique smooth structure such that the
projectionπ: G→G/K is a submersion. The example SL(2,C)/SU(2) can
be identified with hyperbolic 3-space (§6.4.3), the example U(n)/O(n) can
be identified with the space of Lagrangian subspaces of a symplectic vector
space ([49, Lemma 2.3.2]), the example Sp(2n)/U(n) can be identified with
Siegel upper half space or the space of compatible linear complex structures
on a symplectic vector space (Exercise 6.5.24 and [49, Lemma 2.5.12]), and
the example G2/SO(4) can be identified with the associative Grassman-
nian ([68, Remark 8.4]).
Standing Assumption
We have seen that all the results in the present chapter carry over to the
intrinsic setting, assuming that the topology ofMis Hausdorff and paracom-
pact. In fact, in many cases it is enough to assume the Hausdorff property.
However, these results mainly deal with introducing the basic concepts such
as smooth maps, embeddings, submersions, vector fields, flows, and verifying
their elementary properties, i.e. with setting up the language for differen-
tial geometry and topology. When it comes to the substance of the subject
we shall deal with Riemannian metrics and they only exist on paracompact
Hausdorff manifolds. Another central ingredient in differential topology is
the theorem of Sard and that requires second countability. To quote Moe
Hirsch [29]: “Manifolds that are not paracompact are amusing, but they
never occur naturally and it is difficult to prove anything about them.”
Thus we will set the following convention for the remaining chapters.
We assume from now on that each intrinsic manifoldM
is Hausdorff and second countable and hence is also paracompact.
For most of this text we will in fact continue to develop the theory for
submanifolds of Euclidean space and indicate, wherever necessary, how to
extend the definitions, theorems, and proofs to the intrinsic setting.

120 CHAPTER 2. FOUNDATIONS

Chapter 3
The Levi-Civita Connection
For a submanifold of Euclidean space the inner product on the ambient space
determines an inner product on each tangent space, thefirst fundamental
form. Thesecond fundamental formis obtained by differentiating the map
which assigns to each point inM⊂R
n
the orthogonal projection onto the
tangent space (§3.1). The covariant derivative of a vector field along a curve
is the orthogonal projection of the derivative in the ambient space onto the
tangent space (§3.2). We will show how the covariant derivative gives rise
to parallel transport (§3.3), examine the frame bundle (§3.4), discuss mo-
tions without “sliding, twisting, and wobbling”, and prove the development
theorem (§3.5).
In§3.6 we will see that the covariant derivative is determined by the
Christoffel symbols in local coordinates and thus carries over to the in-
trinsic setting. The intrinsic setting of Riemannian manifolds is explained
in§3.7. The covariant derivative takes the form of a family of linear op-
erators∇: Vect(γ)→Vect(γ), one for every smooth curveγ:I→M, and
these operators are uniquely characterized by the axioms of Theorem 3.7.8.
This family of linear operators is theLevi-Civita connection.
3.1 Second Fundamental Form
LetM⊂R
n
be a smoothm-manifold. Then each tangent space ofMis an
m-dimensional real vector space and hence is isomorphic toR
m
. Thus any
two tangent spacesTpMandTqMare of course isomorphic to each other.
While there is no canonical isomorphism fromTpMtoTqMwe shall see
that every smooth curveγinMconnectingptoqinduces an isomorphism
between the tangent spaces via parallel transport of tangent vectors alongγ.
121

122 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Throughout we use the standard inner product onR
n
given by
⟨v, w⟩=v1w1+v2w2+· · ·+vnwn
forv= (v1, . . . , vn)∈R
n
andw= (w1, . . . , wn)∈R
n
. The associated Eu-
clidean norm will be denoted by
|v|=
p
⟨v, v⟩=
q
v
2
1
+v
2
2
+· · ·+v
2
n
forv= (v1, . . . , vn)∈R
n
. WhenM⊂R
n
is a smoothm-dimensional sub-
manifold, a first observation is that each tangent space ofMinherits an inner
product from the ambient spaceR
n
. The resultingfield of inner productsis
called the first fundamental form.
Definition 3.1.1.LetM⊂R
n
be a smoothm-dimensional submanifold.
Thefirst fundamental form onMis the field which assigns to eachp∈M
the bilinear map
gp:TpM×TpM→R
defined by
gp(v, w) =⟨v, w⟩ (3.1.1)
forv, w∈TpM.
A second observation is that the inner product on the ambient space also
determines an orthogonal projection ofR
n
onto the tangent spaceTpMfor
eachp∈M. This projection can be represented by the matrix Π(p)∈R
n×n
which is uniquely determined by the conditions
Π(p) = Π(p)
2
= Π(p)
T
, (3.1.2)
and
Π(p)v=v ⇐⇒ v∈TpM (3.1.3)
forp∈Mandv∈R
n
(see Exercise 2.6.9).
Lemma 3.1.2.The mapΠ :M→R
n×n
defined by(3.1.2)and(3.1.3)is
smooth.
Proof.This follows directly from Theorem 2.6.10 and Corollary 2.6.12. More
explicitly, ifU⊂Mis an open set andϕ:U→Ω is a coordinate chart onto
an open subset Ω⊂R
m
with the inverseψ:=ϕ
−1
: Ω→U, then
Π(p) =dψ(ϕ(p))
ı
dψ(ϕ(p))
T
dψ(ϕ(p))
ȷ
−1
dψ(ϕ(p))
T
forp∈Uand this proves Lemma 3.1.2.

3.1. SECOND FUNDAMENTAL FORM 123T M
M
pν( )
p
Figure 3.1: A unit normal vector field.
Example 3.1.3(Gauß map).LetM⊂R
m+1
be a submanifold of codi-
mension one. ThenT M
⊥
is a vector bundle of rank one (Corollary 2.6.13),
and so each fiberTpM
⊥
is spanned by a unit vectorν(p)∈R
m
, determined
byTpMup to a sign. By Theorem 2.6.10 eachp0∈Mhas an open neigh-
borhoodU⊂Mon which there exists a smooth map
ν:U→R
m+1
satisfying
ν(p)⊥TpM, |ν(p)|= 1 (3.1.4)
for allp∈U(see Figure 3.1). Such a mapνis called aGauß map. The
function Π :M→R
n×n
is in this case given by
Π(p) = 1l−ν(p)ν(p)
T
(3.1.5)
forp∈U.
Example 3.1.4.LetM=S
2
⊂R
3
. Thenν(p) =pand so
Π(p) = 1l−pp
T
=


1−x
2
−xy−xz
−yx1−y
2
−yz
−zx−zy1−z
2


forp= (x, y, z)∈S
2
.
Example 3.1.5(M¨obius strip).Consider the submanifold
M:=



(x, y, z)∈R
3

x= (1 +rcos(θ/2)) cos(θ),
y= (1 +rcos(θ/2)) sin(θ),
z=rsin(θ/2), r, θ∈R,|r|< ε



forε >0 sufficiently small. Show that there does not exist a global smooth
functionν:M→R
3
satisfying (3.1.4).

124 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Example 3.1.6.LetU⊂R
n
be an open set andf:U→R
n−m
be a smooth
function such that 0∈R
n−m
is a regular value offandU∩M=f
−1
(0).
ThenTpM= kerdf(p) and
Π(p) = 1l−df(p)
T
ı
df(p)df(p)
T
ȷ
−1
df(p)
for everyp∈U∩M.
Example 3.1.7.Let Ω⊂R
m
be an open set andψ: Ω→Mbe a smooth
embedding. ThenT
ψ(x)M= imdψ(x) and
Π(ψ(x)) =dψ(x)
ı
dψ(x)
T
dψ(x)
ȷ
−1
dψ(x)
T
for everyx∈Ω.
Next we differentiate the map Π :M→R
n×n
in Lemma 3.1.2. The
derivative atp∈Mtakes the form of a linear map
dΠ(p) :TpM→R
n×n
which, as usual, is defined by
dΠ(p)v:=
d
dt

t=0
Π(γ(t))∈R
n×n
forv∈TpM, whereγ:R→Mis chosen such thatγ(0) =pand ˙γ(0) =v
(see Definition 2.2.13). We emphasize that the expressiondΠ(p)vis a matrix
and can therefore be multiplied by a vector inR
n
.
Lemma 3.1.8.For allp∈Mandv, w∈TpMwe have
Γ
dΠ(p)v
∆
w=
Γ
dΠ(p)w
∆
v∈TpM
⊥
.
Proof.Choose a smooth pathγ:R→Mand a vector fieldX:R→R
n
alongγsuch that
γ(0) =p, ˙γ(0) =v, X(0) =w.
For example, we can chooseX(t) := Π(γ(t))w. Then
X(t) = Π(γ(t))X(t)
for everyt∈R. Differentiate this equation to obtain
˙
X(t) = Π(γ(t))
˙
X(t) +
Γ
dΠ(γ(t)) ˙γ(t)
∆
X(t). (3.1.6)
Hence
Γ
dΠ(γ(t)) ˙γ(t)
∆
X(t) =
Γ
1l−Π(γ(t))
∆
˙
X(t)∈T
γ(t)M
⊥
(3.1.7)
for everyt∈Rand, witht= 0, we obtain (dΠ(p)v)w∈TpM
⊥
.

3.1. SECOND FUNDAMENTAL FORM 125
Now choose a smooth map
R
2
→M: (s, t)7→γ(s, t)
satisfying
γ(0,0) =p,
∂γ
∂s
(0,0) =v,
∂γ
∂t
(0,0) =w,
(for example by doing this in local coordinates) and denote
X(s, t) :=
∂γ
∂s
(s, t)∈T
γ(s,t)M, Y (s, t) :=
∂γ
∂t
(s, t)∈T
γ(s,t)M.
Then
∂Y
∂s
=
∂
2
γ
∂s∂t
=
∂X
∂t
and hence, using (3.1.7), we obtain
`
dΠ(γ)
∂γ
∂t
´
∂γ
∂s
=
`
dΠ(γ)
∂γ
∂t
´
X
=
Γ
1l−Π(γ)
∆∂X
∂t
=
Γ
1l−Π(γ)
∆∂Y
∂s
=
`
dΠ(γ)
∂γ
∂s
´
Y
=
`
dΠ(γ)
∂γ
∂s
´
∂γ
∂t
.
Withs=t= 0 we obtain
Γ
dΠ(p)w
∆
v=
Γ
dΠ(p)v
∆
w∈TpM
⊥
and this proves Lemma 3.1.8.
Definition 3.1.9.The collection of symmetric bilinear maps
hp:TpM×TpM→TpM
⊥
,
defined by
hp(v, w) := (dΠ(p)v)w= (dΠ(p)w)v (3.1.8)
forp∈Mandv, w∈TpMis called thesecond fundamental formonM.

126 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Example 3.1.10.LetM⊂R
m+1
be anm-manifold andν:M→S
m
be
a Gauß map so thatTpM=ν(p)
⊥
for everyp∈M(see Example 3.1.3).
Then Π(p) = 1l−ν(p)ν(p)
T
and hence
hp(v, w) =−ν(p)⟨dν(p)v, w⟩
forp∈Mandv, w∈TpM.
Exercise 3.1.11.Choose a splittingR
n
=R
m
×R
n−m
and write the ele-
ments ofR
n
as tuples (x, y) = (x1, . . . , xm, y1, . . . , yn−m) LetM⊂R
n
be a
smoothm-dimensional submanifold such thatp= 0∈Mand
T0M=R
m
× {0}, T 0M
⊥
={0} ×R
n−m
.
By the implicit function theorem, there are open neighborhoods Ω⊂R
m
andV⊂R
n−m
of zero and a smooth mapf: Ω→Vsuch that
M∩(Ω×V) = graph(f) ={(x, f(x))|x∈Ω}.
Thusf(0) = 0 anddf(0) = 0. Prove that the second fundamental form
hp:TpM×TpM→TpM
⊥
is given by the second derivatives off, i.e.
hp(v, w) =

0,
m
X
i,j=1
∂
2
f
∂xi∂xj
(0)viwj


forv, w∈TpM=R
m
× {0}.
Exercise 3.1.12.LetM⊂R
n
be anm-manifold. Fix a pointp∈Mand
a unit tangent vectorv∈TpMso that|v|= 1 and define
L:={p+tv+w|t∈R, w⊥TpM}.
Letγ: (−ε, ε)→M∩Lbe a smooth curve such thatγ(0) =p, ˙γ(0) =v,
and|˙γ(t)|= 1 for allt. Prove that
¨γ(0) =hp(v, v).
Draw a picture ofMandLin the casen= 3 andm= 2.

3.2. COVARIANT DERIVATIVE 127X(t)
tγ( )
M
Figure 3.2: A vector field along a curve.
3.2 Covariant Derivative
Definition 3.2.1.LetI⊂Rbe an open interval and letγ:I→Mbe a
smooth curve. Avector field alongγis a smooth mapX:I→R
n
such
thatX(t)∈T
γ(t)Mfor everyt∈I(see Figure 3.2). The set of smooth vector
fields alongγis a real vector space and will be denoted by
Vect(γ) :=
Φ
X:I→R
n
|Xis smooth andX(t)∈T
γ(t)M∀t∈I

.
The first derivative
˙
X(t) of a vector field alongγatt∈Iwill, in general,
not be tangent toM. We may decompose it as a sum of a tangent vector
and a normal vector in the form
˙
X(t) = Π(γ(t))
˙
X(t) +
Γ
1l−Π(γ(t))
∆
˙
X(t),
where Π :M→R
n×n
is defined by (3.1.2) and (3.1.3). The tangential
component of this decomposition plays an important geometric role. It is
called the covariant derivative ofXatt.
Definition 3.2.2(Covariant derivative).LetI⊂Rbe an open inter-
val, letγ:I→Mbe a smooth curve, and letX∈Vect(γ). Thecovariant
derivative ofXis the vector field∇X∈Vect(γ), defined by
∇X(t) := Π(γ(t))
˙
X(t)∈T
γ(t)M (3.2.1)
fort∈I.
Lemma 3.2.3(Gauß–Weingarten formula).The derivative of a vector
fieldXalong a curveγis given by
˙
X(t) =∇X(t) +h
γ(t)( ˙γ(t), X(t)). (3.2.2)
Here the first summand is tangent toMand the second summand is orthog-
onal to the tangent space ofMatγ(t).
Proof.This is equation (3.1.6) in the proof of Lemma 3.1.8.

128 CHAPTER 3. THE LEVI-CIVITA CONNECTION
It follows directly from the definition that the covariant derivative along
a curveγ:I→Mis a linear operator∇: Vect(γ)→Vect(γ). The following
lemma summarizes the basic properties of this operator.
Lemma 3.2.4(Covariant derivative).The covariant derivative satisfies
the following axioms for any two open intervalsI, J⊂R.
(i)Letγ:I→Mbe a smooth curve, letλ:I→Rbe a smooth function,
and letX∈Vect(γ). Then
∇(λX) =
˙
λX+λ∇X. (3.2.3)
(ii)Letγ:I→Mbe a smooth curve, letσ:J→Ibe a smooth function
and letX∈Vect(γ). Then
∇(X◦σ) = ˙σ(∇X◦σ). (3.2.4)
(iii)Letγ:I→Mbe a smooth curve and letX, Y∈Vect(γ). Then
d
dt
⟨X, Y⟩=⟨∇X, Y⟩+⟨X,∇Y⟩. (3.2.5)
(iv)Letγ:I×J→Mbe a smooth map, denote by∇sthe covariant deriva-
tive along the curves7→γ(s, t)(withtfixed), and denote by∇tthe covariant
derivative along the curvet7→γ(s, t)(withsfixed). Then
∇s∂tγ=∇t∂sγ. (3.2.6)
Proof.Part (i) follows from the Leibniz rule
d
dt
(λX) =
˙
λX+λ
˙
Xand (ii)
follows from the chain rule
d
dt
(X◦σ) = ˙σ(
˙
X◦σ). To prove part (iii), use
the orthogonal projections Π(γ(t)) :R
n
→T
γ(t)Mto obtain
d
dt
⟨X, Y⟩=⟨
˙
X, Y⟩+⟨X,
˙
Y⟩
=⟨
˙
X,Π(γ)Y⟩+⟨Π(γ)X,
˙
Y⟩
=⟨Π(γ)
˙
X, Y⟩+⟨X,Π(γ)
˙
Y⟩
=⟨∇X, Y⟩+⟨X,∇Y⟩.
Part (iv) holds because the second derivatives commute and this proves
Lemma 3.2.4.
Part (i) in Lemma 3.2.4 asserts that the operator∇is what is called a
connection, part (iii) asserts that it is compatible with the first fundamental
form, and part (iv) asserts that it istorsion-free. Theorem 3.7.8 below as-
serts that these conditions (together with an extended chain rule) determine
the covariant derivative uniquely.

3.3. PARALLEL TRANSPORT 129
3.3 Parallel Transport
Definition 3.3.1(Parallel vector field).LetI⊂Rbe an interval and
letγ:I→Mbe a smooth curve. A vector fieldXalongγis calledparallel
iff
∇X(t) = 0
for allt∈I.
Example 3.3.2.Assumem=nso thatM⊂R
m
is an open set. Then a
vector field along a smooth curveγ:I→Mis a smooth mapX:I→R
m
.
Its covariant derivative is equal to the ordinary derivative∇X(t) =
˙
X(t)
and henceXis is parallel if and only if it is constant.
Remark 3.3.3.For everyX∈Vect(γ) and everyt∈Iwe have
∇X(t) = 0 ⇐⇒
˙
X(t)⊥T
γ(t)M.
In particular, ˙γis a vector field alongγand∇˙γ(t) = Π(γ(t))¨γ(t). Hence ˙γ
is a parallel vector field alongγif and only if ¨γ(t)⊥T
γ(t)Mfor allt∈I.
We will return to this observation in Chapter 4.
In general, a vector fieldXalong a smooth curveγ:I→Mis parallel
if and only if
˙
X(t) is orthogonal toT
γ(t)Mfor everytand, by the Gauß–
Weingarten formula (3.2.2), we have
∇X= 0 ⇐⇒
˙
X=hγ( ˙γ, X).
The next theorem shows that any given tangent vectorv0∈T
γ(t0)Mextends
uniquely to a parallel vector field alongγ.
Theorem 3.3.4(Existence and uniqueness).LetI⊂Rbe an interval
andγ:I→Mbe a smooth curve. Lett0∈Iandv0∈T
γ(t0)Mbe given.
Then there is a unique parallel vector fieldX∈Vect(γ)such thatX(t0) =v0.
Proof.Choose a basise1, . . . , emof the tangent spaceT
γ(t0)Mand let
X1, . . . , Xm∈Vect(γ)
be vector fields alongγsuch that
Xi(t0) =ei, i= 1, . . . , m.
(For example chooseXi(t) := Π(γ(t))ei.) Then the vectorsXi(t0) are lin-
early independent. Since linear independence is an open condition there is a

130 CHAPTER 3. THE LEVI-CIVITA CONNECTION
constantε >0 such that the vectorsX1(t), . . . , Xm(t)∈T
γ(t)Mare linearly
independent for everyt∈I0:= (t0−ε, t0+ε)∩I. SinceT
γ(t)Mis an
m-dimensional real vector space this implies that the vectorsXi(t) form a
basis ofT
γ(t)Mfor everyt∈I0. We express the vector∇Xi(t)∈T
γ(t)Min
this basis and denote the coefficients bya
k
i
(t) so that
∇Xi(t) =
m
X
k=1
a
k
i(t)Xk(t).
The resulting functionsa
k
i
:I0→Rare smooth. Likewise, ifX:I→R
n
is
any vector field alongγ, then there are smooth functionsξ
i
:I0→Rsuch
that
X(t) =
m
X
i=1
ξ
i
(t)Xi(t) for allt∈I0.
The derivative ofXis given by
˙
X(t) =
m
X
i=1
ı
˙
ξ
i
(t)Xi(t) +ξ
i
(t)
˙
Xi(t)
ȷ
and the covariant derivative by
∇X(t) =
m
X
i=1
ı
˙
ξ
i
(t)Xi(t) +ξ
i
(t)∇Xi(t)
ȷ
=
m
X
i=1
˙
ξ
i
(t)Xi(t) +
m
X
i=1
ξ
i
(t)
m
X
k=1
a
k
i(t)Xk(t)
=
m
X
k=1

˙
ξ
k
(t) +
m
X
i=1
a
k
i(t)ξ
i
(t)
!
Xk(t)
fort∈I0. Hence∇X(t) = 0 if and only if
˙
ξ(t) +A(t)ξ(t) = 0, A(t) :=



a
1
1
(t)· · ·a
1
m(t)
.
.
.
.
.
.
a
m
1
(t)· · ·a
m
m(t)


.
Thus we have translated the equation∇X= 0 over the intervalI0into a time
dependent linear ordinary differential equation. By a theorem in Analysis II
(see [64, Lemma 4.4.3]), this equation has a unique solution for any initial
condition at any point inI0. Thus we have proved that everyt0∈Iis

3.3. PARALLEL TRANSPORT 131
contained in an intervalI0⊂I, open in the relative topology ofI, such
that, for everyt1∈I0and everyv1∈T
γ(t1)M, there exists a unique parallel
vector fieldX:I0→R
n
alongγ|I0
satisfyingX(t1) =v1. We formulate this
condition on the intervalI0as a logical formula:
∀t1∈I0∀v1∈T
γ(t1)M∃!X∈Vect(γ|I0
)
such that∇X= 0 andX(t1) =v1.
(3.3.1)
If twoI-open intervalsI0, I1⊂Isatisfy this condition and have nonempty
intersection, then their unionI0∪I1also satisfies (3.3.1). (Prove this!) Now
define
J:=
[
{I0⊂I|I0is anI-open interval, I0satisfies (3.3.1), t0∈I0}.
This intervalJsatisfies (3.3.1). Moreover, it is nonempty and, by defi-
nition, it is open in the relative topology ofI. We prove that it is also
closed in the relative topology ofI. Thus let (ti)i∈Nbe a sequence inJ
converging to a pointt
∗
∈I. By what we have proved above, there ex-
ists a constantε >0 such that the intervalI
∗
:= (t
∗
−ε, t
∗
+ε)∩Isatis-
fies (3.3.1). Since the sequence (ti)i∈Nconverges tot
∗
, there exists ani∈N
such thatti∈I
∗
. Sinceti∈Jthere exists an intervalI0⊂I, open in the
relative topology ofI, that containst0andtiand satisfies (3.3.1). Hence
the intervalI0∪I
∗
is open in the relative topology ofI, containst0andt
∗
,
and satisfies (3.3.1). This shows thatt
∗
∈J. Thus we have proved that the
intervalJis nonempty, and open and closed in the relative topology ofI.
HenceJ=Iand this proves Theorem 3.3.4.
Definition 3.3.5(Parallel transport).LetI⊂Rbe an interval and
letγ:I→Mbe a smooth curve. Fort0, t∈Iwe define the map
Φγ(t, t0) :T
γ(t0)M→T
γ(t)M
byΦγ(t, t0)v0:=X(t)whereX∈Vect(γ)is the unique parallel vector field
alongγsatisfyingX(t0) =v0. The collection of mapsΦγ(t, t0)fort, t0∈I
is calledparallel transport alongγ.
Recall the notation
γ
∗
T M=
Φ
(s, v)|s∈I, v∈T
γ(s)M

for the pullback tangent bundle. This set is a smooth submanifold ofI×R
n
.
(See Theorem 2.6.10 and Corollary 2.6.13.) The next theorem summarizes
the properties of parallel transport. In particular, the last assertion shows
that the covariant derivative can be recovered from the parallel transport
maps.

132 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Theorem 3.3.6(Parallel transport).Letγ:I→Mbe a smooth curve
on an intervalI⊂R.
(i)The mapΦγ(t, s) :T
γ(s)M→T
γ(t)Mis linear for alls, t∈I.
(ii)For allr, s, t∈Iwe have
Φγ(t, s)◦Φγ(s, r) = Φγ(t, r),Φγ(t, t) = id.
(iii)For alls, t∈Iand allv, w∈T
γ(s)Mwe have
⟨Φγ(t, s)v,Φγ(t, s)w⟩=⟨v, w⟩.
ThusΦγ(t, s) :T
γ(s)M→T
γ(t)Mis an orthogonal transformation.
(iv)IfJ⊂Ris an interval andσ:J→Iis a smooth map, then
Φγ◦σ(t, s) = Φγ(σ(t), σ(s)).
for alls, t∈J.
(v)The map
I×γ
∗
T M→γ
∗
T M: (t,(s, v))7→(t,Φγ(t, s)v)
is smooth.
(vi)For allX∈Vect(γ)andt, t0∈Iwe have
d
dt
Φγ(t0, t)X(t) = Φγ(t0, t)∇X(t).
Proof.Assertion (i) holds because the sum of two parallel vector fields
alongγis again parallel and the product of a parallel vector field with a
constant real number is again parallel. Assertion (ii) follows directly from
the uniqueness statement in Theorem 3.3.4.
We prove (iii). Fix a numbers∈Iand two tangent vectors
v, w∈T
γ(s)M.
Define the vector fieldsX, Y∈Vect(γ) alongγby
X(t) := Φγ(t, s)v, Y(t) := Φγ(t, s)w.
These vector fields are parallel. Thus, by equation (3.2.5) in Lemma 3.2.4,
we have
d
dt
⟨X, Y⟩=⟨∇X, Y⟩+⟨X,∇Y⟩= 0.
Hence the functionI→R:t7→ ⟨X(t), Y(t)⟩is constant and this proves (iii).

3.3. PARALLEL TRANSPORT 133
We prove (iv). Fix an elements∈Jand a tangent vectorv∈T
γ(σ(s))M.
Define the vector fieldXalongγby
X(t) := Φγ(t, σ(s))v
fort∈I. ThusXis the unique parallel vector field alongγthat satisfies
X(σ(s)) =v.
Denote
eγ:=γ◦σ:J→M,
e
X:=X◦σ:I→R
n
Then
e
Xis a vector field alongeγand, by the chain rule, we have
d
dt
e
X(t) =
d
dt
X(σ(t)) = ˙σ(t)
˙
X(σ(t)).
Projecting orthogonally onto the tangent spaceT
γ(σ(t))Mwe obtain
∇
e
X(t) = ˙σ(t)∇X(σ(t)) = 0
for everyt∈J. Hence
e
Xis the unique parallel vector field alongeγthat
satisfies
e
X(s) =v. Thus
Φ
eγ(t, s)v=
e
X(t) =X(σ(t)) = Φγ(σ(t), σ(s))v.
This proves (iv).
We prove (v). Fix a pointt0∈I, choose an orthonormal basise1, . . . , em
ofT
γ(t0)M, and defineXi(t) := Φγ(t, t0)eifort∈Iandi= 1, . . . , m. Thus
Xi∈Vect(γ) is the unique parallel vector field alongγsuch thatXi(t0) =ei.
Then by (iii) we have
⟨Xi(t), Xj(t)⟩=δij
for alli, j∈ {1, . . . , m}and allt∈I. Hence the vectorsX1(t), . . . , Xm(t)
form an orthonormal basis ofT
γ(t)Mfor everyt∈I. This implies that, for
eachs∈Iand each tangent vectorv∈T
γ(s)M, we have
v=
m
X
i=1
⟨Xi(s), v⟩Xi(s).
Since each vector fieldXiis parallel it satisfiesXi(t) = Φγ(t, s)Xi(s). Hence
Φγ(t, s)v=
m
X
i=1
⟨Xi(s), v⟩Xi(t) (3.3.2)
for alls, t∈Iandv∈T
γ(s)M. This proves (v).

134 CHAPTER 3. THE LEVI-CIVITA CONNECTION
We prove (vi). LetX1, . . . , Xm∈Vect(γ) be as in the proof of (v). Thus
every vector fieldXalongγcan be written in the form
X(t) =
m
X
i=1
ξ
i
(t)Xi(t), ξ
i
(t) :=⟨Xi(t), X(t)⟩.
Since the vector fieldsXiare parallel we have
∇X(t) =
m
X
i=1
˙
ξ
i
(t)Xi(t)
for allt∈I. Hence
Φγ(t0, t)X(t) =
m
X
i=1
ξ
i
(t)Xi(t0),Φγ(t0, t)∇X(t) =
m
X
i=1
˙
ξ
i
(t)Xi(t0).
Evidently, the derivative of the first sum with respect totis equal to the
second sum. This proves (vi) and Theorem 3.3.6.
Remark 3.3.7.Fors, t∈Iwe can think of the linear map
Φγ(t, s)Π(γ(s)) :R
n
→T
γ(t)M⊂R
n
as a realn×nmatrix. The formula (3.3.2) in the proof of (v) shows that
this matrix can be expressed in the form
Φγ(t, s)Π(γ(s)) =
m
X
i=1
Xi(t)Xi(s)
T
∈R
n×n
.
The right hand side defines a smooth matrix valued function onI×Iand
this is equivalent to the assertion in (v).
Remark 3.3.8.It follows from assertions (ii) and (iii) in Theorem 3.3.6
that
Φγ(t, s)
−1
= Φγ(s, t) = Φγ(t, s)
∗
for alls, t∈I. Here the linear map Φγ(t, s)
∗
:T
γ(t)M→T
γ(s)Mis under-
stood as the adjoint operator of Φγ(t, s) :T
γ(s)M→T
γ(t)Mwith respect to
the inner products on the two subspaces ofR
n
inherited from the Euclidean
inner product on the ambient space.

3.3. PARALLEL TRANSPORT 135
The two theorems in this section carry over verbatim to any smooth
vector bundleE⊂M×R
n
over a manifold. As in the case of the tangent
bundle one can define the covariant derivative of a section ofEalongγas
the orthogonal projection of the ordinary derivative in the ambient spaceR
n
onto the fiberE
γ(t). Instead ofparallel vector fieldsone then speaks about
horizontal sectionsand one proves as in Theorem 3.3.4 that there is a unique
horizontal section alongγthrough any point in any of the fibersE
γ(t0). This
gives parallel transport maps fromE
γ(s)toE
γ(t)for any pairs, t∈Iand
Theorem 3.3.6 carries over verbatim to all vector bundlesE⊂M×R
n
. We
spell this out in more detail in the case whereE=T M
⊥
⊂M×R
n
is the
normal bundle ofM.
Letγ:I→Mbe a smooth curve. Anormal vector field alongγis
a smooth mapY:I→R
n
such thatY(t)⊥T
γ(t)Mfor everyt∈I. The set
of normal vector fields alongγwill be denoted by
Vect
⊥
(γ) :=
Φ
Y:I→R
n
|Yis smooth andY(t)⊥T
γ(t)Mfor allt∈I

.
This is again a real vector space. Thecovariant derivativeof a normal
vector fieldY∈Vect
⊥
(γ) att∈Iis defined as the orthogonal projection of
the ordinary derivative onto the orthogonal complement ofT
γ(t)Mand will
be denoted by
∇
⊥
Y(t) :=
Γ
1l−Π(γ(t))
∆
˙
Y(t). (3.3.3)
Thus the covariant derivative defines a linear operator
∇
⊥
: Vect
⊥
(γ)→Vect
⊥
(γ).
There is a version of the Gauß–Weingarten formula for the covariant deriva-
tive of a normal vector field. This is the content of the next lemma.
Lemma 3.3.9.LetM⊂R
n
be a smoothm-manifold. Forp∈Mand
u∈TpMdefine the linear maphp(u) :TpM→TpM
⊥
by
hp(u)v:=hp(u, v) =
Γ
dΠ(p)u
∆
v (3.3.4)
forv∈TpM. Then the following holds.
(i)The adjoint operatorhp(u)
∗
:TpM
⊥
→TpMis given by
hp(u)
∗
w=
Γ
dΠ(p)u
∆
w, w ∈TpM
⊥
. (3.3.5)
(ii)IfI⊂Ris an interval,γ:I→Mis a smooth curve, andY∈Vect
⊥
(γ),
then the derivative ofYsatisfies theGauß–Weingarten formula
˙
Y(t) =∇
⊥
Y(t)−h
γ(t)( ˙γ(t))
∗
Y(t). (3.3.6)

136 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Proof.Since Π(p)∈R
n×n
is a symmetric matrix for everyp∈Mso is the
matrixdΠ(p)ufor everyp∈Mand everyu∈TpM. Hence
⟨v, hp(u)
∗
w⟩=⟨hp(u)v, w⟩
=
ΩΓ
dΠ(p)u
∆
v, w

=

v,
Γ
dΠ(p)u
∆
w

for everyv∈TpMand everyw∈TpM
⊥
. This proves (i).
To prove (ii) we observe that, forY∈Vect
⊥
(γ) andt∈I, we have
Π(γ(t))Y(t) = 0.
Differentiating this identity we obtain
Π(γ(t))
˙
Y(t) +
Γ
dΠ(γ(t)) ˙γ(t)
∆
Y(t) = 0
and hence
˙
Y(t) =
˙
Y(t)−Π(γ(t))
˙
Y(t)−
Γ
dΠ(γ(t)) ˙γ(t)
∆
Y(t)
=∇
⊥
Y(t)−h
γ(t)( ˙γ(t))
∗
Y(t)
fort∈I. Here the last equation follows from (i) and the definition of∇
⊥
.
This proves Lemma 3.3.9.
Theorem 3.3.4 and its proof carry over to the normal bundleT M
⊥
.
Thus, ifγ:I→Mis a smooth curve, then for alls∈Iandw∈T
γ(s)M
⊥
there is a unique normal vector fieldY∈Vect
⊥
(γ) such that
∇
⊥
Y≡0, Y(s) =w.
This gives rise to parallel transport maps
Φ
⊥
γ(t, s) :T
γ(s)M
⊥
→T
γ(t)M
⊥
defined by
Φ
⊥
γ(t, s)w:=Y(t)
fors, t∈Iandw∈T
γ(s)M
⊥
, whereYis the unique normal vector field along
γsatisfying∇
⊥
Y≡0 andY(s) =w. These parallel transport maps satisfy
exactly the same conditions that have been spelled out in Theorem 3.3.6
for the tangent bundle and the proof carries over verbatim to the present
setting.

3.4. THE FRAME BUNDLE 137
3.4 The Frame Bundle
Each tangent space of anm-manifoldMis isomorphic to the Euclidean
spaceR
m
, however, in general there is no canonical isomorphism. The space
of all pairs consisting of a pointpin the manifoldMand an isomorphism
fromR
m
to the tangent space ofMatpis itself a smooth manifold, called
the frame bundle ofM.
3.4.1 Frames of a Vector Space
LetVbe anm-dimensional real vector space. Aframe ofVis a basis
e1, . . . , emofV. It determines a vector space isomorphisme:R
m
→Vvia
eξ:=
m
X
i=1
ξ
i
ei, ξ= (ξ
1
, . . . , ξ
m
)∈R
m
.
Conversely, each isomorphisme:R
m
→Vdetermines a basise1, . . . , em
ofVviaei=e(0, . . . ,0,1,0. . . ,0), where the coordinate 1 appears in the
ith place. The set of vector space isomorphisms fromR
m
toVwill be
denoted by
Liso(R
m
, V) :={e:R
m
→V|eis a vector space isomorphism}.
The general linear group GL(m) = GL(m,R) (of nonsingular realm×m-
matrices) acts on this space by composition on the right via
GL(m)× Liso(R
m
, V)→ Liso(R
m
, V) : (a, e)7→a
∗
e:=e◦a.
This is acontravariant group actionin that
a
∗
b
∗
e= (ba)
∗
e,1l
∗
e=e
fora, b∈GL(m) ande∈ Liso(R
m
, V). Moreover, the action isfree, i.e. for
alla∈GL(m) ande∈ Liso(R
m
, V), we have
a
∗
e=e ⇐⇒ a= 1l.
It istransitivein that for alle, e
′
∈ Liso(R
m
, V) there is a group element
a∈GL(m) such thate
′
=a
∗
e. Thus we can identify the spaceLiso(R
m
, V)
with the group GL(m) via the bijection
GL(m)→ Liso(R
m
, V) :a7→a
∗
e0
induced by a fixed elemente0∈ Liso(R
m
, V). This identification is not
canonical; it depends on the choice ofe0. The spaceLiso(R
m
, V) admits a
bijection to a group but is not itself a group.

138 CHAPTER 3. THE LEVI-CIVITA CONNECTION
3.4.2 The Frame Bundle
Definition 3.4.1(Frame bundle).LetM⊂R
n
be a smoothm-manifold.
Theframe bundleofMis the set
F(M) :={(p, e)|p∈M, e∈ F(M)p}, (3.4.1)
whereF(M)pis the space of frames of the tangent space atp, i.e.
F(M)p:=Liso(R
m
, TpM).
Define a right action ofGL(m)onF(M)by
a
∗
(p, e) := (p, a
∗
e) = (p, e◦a) (3.4.2)
fora∈GL(m)and(p, e)∈ F(M).
One can think of a framee∈ Liso(R
m
, TpM) as a linear map fromR
m
toR
n
whose image isTpMand hence as ann×m-matrix of rankm.
The basis ofTpMassociated to this frame is given by the columns of the
matrixe∈R
n×m
. Thus the frame bundleF(M) of an embedded mani-
foldM⊂R
n
is a subset of the Euclidean spaceR
n
×R
n×m
.
Lemma 3.4.2.The frame bundle
F(M)⊂R
n
×R
n×m
is a smooth manifold of dimensionm+m
2
, the group action
GL(m)× F(M)→ F(M) : (a,(p, e))7→a
∗
(p, e)
is smooth, and the projection
π:F(M)→M
defined byπ(p, e) :=pfor(p, e)∈ F(M)is a surjective submersion. The
orbits of theGL(m)-action onF(M)are the fibers of this projection, i.e.
GL(m)
∗
(p, e) =π
−1
(p)
∼
=F(M)p
for(p, e)∈ F(M), and the groupGL(m)acts freely and transitively on each
of these fibers.

3.4. THE FRAME BUNDLE 139
Proof.LetU⊂Mbe anM-open set. Amoving frameoverUis a se-
quence ofmsmooth vector fieldsE1, . . . , Em∈Vect(U) onUsuch that the
vectorsE1(p), . . . , Em(p) form a basis ofTpMfor eachp∈U. Any such
moving frame gives a bijection
U×GL(m)→ F(U) : (p, a)7→a
∗
(p, E(p)) = (p, E(p)◦a),
where
E(p) := (E1(p), . . . , Em(p))∈ F(M)p
forp∈U. This bijection (when composed with a parametrization ofU)
gives a parametrization of the open setF(U) inF(M). The assertions of
the lemma then follow from the fact that the diagram
U×GL(m)
//
pr
1
""
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
F(U)
π
¨¨~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
~
U
commutes. More precisely, suppose that there exists a coordinate chart
ϕ:U→Ω
with values in an open set Ω⊂R
m
, and denote its inverse by
ψ:=ϕ
−1
: Ω→U.
Then the open set
F(U) =π
−1
(U) ={(p, e)∈ F(M)|p∈U}= (U×R
n×m
)∩ F(M)
is parametrized by the map
Ω×GL(m)→ F(U) : (x, a)7→
Γ
ψ(x), dψ(x)◦a
∆
.
This map is amooth and so is its inverse
F(U)→Ω×GL(m) : (p, e)7→
Γ
ϕ(p), dϕ(p)◦e
∆
.
These are the desired coordinate chart onF(M). ThusF(M) is a smooth
manifold of dimensionm+m
2
. Moreover, in these coordinates the projec-
tionπ:F(U)→Uis the map Ω×GL(m)→Ω : (x, a)7→xand soπis a
submersion. The remaining assertions follow directly from the definitions
and this proves Lemma 3.4.2.

140 CHAPTER 3. THE LEVI-CIVITA CONNECTION
The frame bundleF(M) is aprincipal bundleoverMwithstruc-
ture groupGL(m). More generally, a principal bundle over a manifoldB
with structure group G is a smooth manifoldPequipped with a surjective
submersionπ:P→Band a smooth contravariant action
G×P→P: (g, p)7→pg
by a Lie group G such thatπ(pg) =π(p) for allp∈Pandg∈G and such
that the group G acts freely and transitively on the fiberPb:=π
−1
(b) for
eachb∈B. In this book we shall mostly be concerned with the frame bundle
of a manifoldMand the orthonormal frame bundle.
Definition 3.4.3.Theorthonormal frame bundle ofMis the set
O(M) :=
n
(p, e)∈R
n
×R
n×m

p∈M,ime=TpM, e
T
e= 1lm
o
.
If we denote byei:=e(0, . . . ,0,1,0, . . . ,0)(with1as theith argument) the
basis ofTpMinduced by the isomorphisme:R
m
→TpM, then we have
e
T
e= 1l⇐⇒ ⟨ei, ej⟩=δij⇐⇒
e1, . . . , emis an
orthonormal basis.
ThusO(M)is the bundle of orthonormal frames of the tangent spacesTpM
or the bundle of orthogonal isomorphismse:R
m
→TpM. It is a principal
bundle overMwith structure groupO(m).
Exercise 3.4.4.Prove thatO(M) is a submanifold ofF(M) and that the
obvious projectionπ:O(M)→Mis a submersion. Prove that the action
of GL(m) onF(M) restricts to an action of the orthogonal group O(m)
onO(M) whose orbits are the fibers
O(M)p:=
Φ
e∈R
n×m

(p, e)∈ O(M)

=
Φ
e∈ Liso(R
m
, TpM)

e
T
e= 1l

.
Hint:Ifϕ:U→Ω is a coordinate chart onMwith inverseψ: Ω→U,
then
ex:=dψ(x)(dψ(x)
T
dψ(x))
−1/2
:R
m
→T
ψ(x)M
is an orthonormal frame of the tangent spaceT
ψ(x)Mfor everyx∈Ω.

3.4. THE FRAME BUNDLE 141
3.4.3 Horizontal Lifts
We have seen in Lemma 3.4.2 that the frame bundleF(M) is a smooth
submanifold ofR
n
×R
n×m
. Next we examine the tangent space ofF(M) at
a point (p, e)∈ F(M). By Definition 2.2.1, this tangent space is given by
T
(p,e)F(M) =



( ˙γ(0),˙e(0))

R→ F(M) :t7→(γ(t), e(t))
is a smooth curve satisfying
γ(0) =pande(0) =e



.
The next lemma gives an explicit formula for this tangent space in terms of
the second fundamental formhp:TpM×TpM→TpM
⊥
in Definition 3.1.9.
Compare this formula with Lemma 4.3.1 in the next chapter.
Lemma 3.4.5.LetM⊂R
n
be a smoothm-dimensional submanifold. Then
the tangent space ofF(M)at(p, e)is given by
T
(p,e)F(M) =
(
(bp,be)

bp∈TpM,be∈R
n×m
,and
Γ
1l−Π(p)
∆
be=hp(bp)e
)
. (3.4.3)
Proof.We prove the inclusion “⊂” in (3.4.3). Let (bp,be)∈T
(p,e)F(M) and
choose a smooth curveR→ F(M) :t7→(γ(t), e(t)) such that
γ(0) =p, e(0) =e, ˙γ(0) =bp, ˙e(0) =be.
Fix a vectorξ∈R
m
and define the vector fieldX∈Vect(γ) byX(t) :=e(t)ξ
fort∈R. Then the Gauß–Weingarten formula (3.2.2) asserts that
˙e(t)ξ=
˙
X(t)
=∇X(t) +h
γ(t)( ˙γ(t), X(t))
= Π(γ(t)) ˙e(t)ξ+h
γ(t)( ˙γ(t), e(t)ξ)
for allt∈R. Taket= 0 to obtain
(1l−Π(p))beξ=hp(bp, eξ) =hp(bp)eξ
for allξ∈R
m
. This proves the inclusion “⊂” in (3.4.3). Equality holds
because both sides of the equation are (m+m
2
)-dimensional linear subspaces
ofR
n
×R
n×m
. This proves Lemma 3.4.5.
It is convenient to consider two kinds of curves inF(M), namely vertical
curves with constant projections toMand horizontal lifts of curves inM.
We denote byL(R
m
, TpM) the space of linear maps fromR
m
toTpM.

142 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Definition 3.4.6(Horizontal lift).Letγ:R→Mbe a smooth curve. A
smooth curveβ:R→ F(M)is called alift ofγiff
π◦β=γ.
Any such lift has the formβ(t) = (γ(t), e(t))withe(t)∈ Liso(R
m
, T
γ(t)M).
The associated curve of framese(t)of the tangent spacesT
γ(t)Mis called a
moving frame alongγ. A curve
β(t) = (γ(t), e(t))∈ F(M)
is calledhorizontalor ahorizontal lift ofγiff the vector field
X(t) :=e(t)ξ
alongγis parallel for everyξ∈R
m
. Thus a horizontal lift ofγhas the form
β(t) = (γ(t),Φγ(t,0)e) (3.4.4)
for somee∈ Liso(R
m
, T
γ(0)M).
Lemma 3.4.7. The tangent space ofF(M)at(p, e)∈ F(M)is the direct
sum
T
(p,e)F(M) =H
(p,e)⊕V
(p,e)
of thehorizontal space
H
(p,e):=
Φ
(v, hp(v)e)

v∈TpM

(3.4.5)
and thevertical space
V
(p,e):={0} × L(R
m
, TpM). (3.4.6)
(ii)The vertical spaceV
(p,e)at(p, e)∈ F(M)is the kernel of the linear map
dπ(p, e) :T
(p,e)F(M)→TpM.
(iii)A curveβ:R→ F(M)is horizontal if and only if it is tangent to the
horizontal spaces, i.e.
˙
β(t)∈H
β(t)for everyt∈R.
(iv)Ifβ:R→ F(M)is a horizontal curve, so isa
∗
βfor everya∈GL(m).

3.4. THE FRAME BUNDLE 143
Proof.The proof has four steps.
Step 1.Let(p, e)∈ F(M). ThenV
(p,e)= kerdπ(p, e)⊂T
(p,e)F(M).
Sinceπis a submersion, the fiberπ
−1
(p) is a submanifold ofF(M) by
Theorem 2.2.19 andT
(p,e)π
−1
(p) = kerdπ(p, e).Now let (bp,be)∈kerdπ(p, e).
Then there exists a vertical curveβ:R→ F(M) withπ◦β≡psuch that
β(0) = (p, e),
˙
β(0) = (bp,be).
Any such curve has the formβ(t) := (p, e(t)) wheree(t)∈ Liso(R
m
, TpM).
Hencebp= 0 andbe= ˙e(0)∈ L(R
m
, TpM). This shows that
kerdπ(p, e)⊂V
(p,e). (3.4.7)
Conversely, for everybe∈ L(R
m
, TpM), the curve
R→ L(R
m
, TpM) :t7→e(t) :=e+tbe
takes values in the open setLiso(R
m
, TpM) fortsufficiently small and
henceβ(t) := (p, e(t)) is a vertical curve with
˙
β(0) = (0,be). Thus
V
(p,e)⊂kerdπ(p, e)⊂T
(p,e)F(M). (3.4.8)
Combining (3.4.7) and (3.4.8) we obtain Step 1 and part (ii).
Step 2.Let(p, e)∈ F(M). ThenH
(p,e)⊂T
(p,e)F(M). Moreover, every
horizontal curveβ:R→ F(M)satisfies
˙
β(t)∈H
β(t)for allt∈R.
Fix a tangent vectorv∈TpM, letγ:R→Mbe a smooth curve satisfy-
ingγ(0) =pand ˙γ(0) =v, and letβ:R→ F(M) be the horizontal lift ofγ
withβ(0) = (p, e). Then
β(t) = (γ(t), e(t)), e(t) := Φγ(t,0)e.
Fix a vectorξ∈R
m
and consider the vector field
X(t) :=e(t)ξ= Φγ(t,0)eξ
alongγ. This vector field is parallel and hence, by the Gauß–Weingarten
formula, it satisfies
˙e(0)ξ=
˙
X(0) =h
γ(0)( ˙γ(0), X(0)) =hp(v)eξ.
Here we have used (3.3.4). Thus
(v, hp(v)e) = ( ˙γ(0),˙e(0)) =
˙
β(0)∈T
β(0)F(M) =T
(p,e)F(M)
and soH
(p,e)⊂T
(p,e)F(M). Moreover,
˙
β(0) = (v, hp(v)e)∈H
(p,e)=H
β(0)
and this proves Step 2.

144 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Step 3.We prove part (i).
We haveV
(p,e)⊂T
(p,e)F(M) by Step 1 andH
(p,e)⊂T
(p,e)F(M) by Step 2.
MoreoverH
(p,e)∩V
(p,e)={0}and soT
(p,e)F(M) =H
(p,e)⊕V
(p,e)for dimen-
sional reasons. This proves Step 3.
Step 4.We prove parts (iii) and (iv).
By Step 2 every horizontal curveβ:R→ F(M) satisfies
˙
β(t)∈H
β(t). Con-
versely, letR→ F(M) :t7→β(t) = (γ(t), e(t)) be a smooth curve satisfy-
ing
˙
β(t)∈H
β(t)for allt. Then ˙e(t) =h
γ(t)( ˙γ(t))e(t) for allt. By the Gauß–
Weingarten formula (3.2.2) this implies that the vector fieldX(t) =e(t)ξ
alongγis parallel for everyξ∈R
m
, soβis horizontal. This proves part (iii).
Part (iv) follows from (iii) and the fact that the horizontal tangent bun-
dleH⊂TF(M) is invariant under the induced action of the group GL(m)
onTF(M). This proves Lemma 3.4.7.p(M)F
π
M p
(p,e)
(M)F
=π
−1
(p)
Figure 3.3: The frame bundle.
The reason for the terminology introduced in Definition 3.4.6 is that one
draws the extremely crude picture of the frame bundle displayed in Fig-
ure 3.3. One thinks ofF(M) as “lying over”M. One would then represent
the equationγ=π◦βby the following commutative diagram:
F(M)
π
fflffl
R
β
88pppppppppppp γ
//M
;
hence the word “lift”. The vertical space is tangent to the vertical line in
Figure 3.3 while the horizontal space is transverse to the vertical space. This
crude imagery can be extremely helpful.

3.4. THE FRAME BUNDLE 145
Exercise 3.4.8.The group GL(m) acts onF(M) by diffeomorphisms. Thus
for eacha∈GL(m) the map
F(M)→ F(M) : (p, e)7→a
∗
(p, e) = (p, e◦a)
is a diffeomorphism ofF(M). The derivative of this diffeomorphism is a
diffeomorphism of the tangent bundleTF(M) and this is called the induced
action of GL(m) onTF(M). Prove that the horizontal and vertical sub-
bundles are invariant under the induced action of GL(m) onTF(M).
Exercise 3.4.9.Prove thatH
(p,e)⊂T
(p,e)O(M) and that
T
(p,e)O(M) =H
(p,e)⊕V
′
(p,e)
, V
′
(p,e)
:=V
(p,e)∩T
(p,e)O(M)
for every (p, e)∈ O(M).
The following definition introduces an important class of vector fields on
the frame bundle that will play a central role in Section 3.5. They will be
used to prove the Development Theorem 3.5.21 in§3.5.4 below.
Definition 3.4.10(Basic vector field).Every vectorξ∈R
m
determines
a vector fieldBξ∈Vect(F(M))defined by
Bξ(p, e) :=
Γ
eξ, hp(eξ)e
∆
(3.4.9)
for(p, e)∈ F(M). This vector field is horizontal, i.e.
Bξ(p, e)∈H
(p,e),
and projects toeξ, i.e.
dπ(p, e)Bξ(p, e) =eξ
for all(p, e)∈ F(M). These two conditions determine the vector fieldBξ
uniquely. It is called thebasic vector fieldcorresponding toξ.
Exercise 3.4.11. Prove that every basic vector fieldBξ∈Vect(F(M))
is tangent to the orthonormal frame bundleO(M).
(ii)LetR→ F(M) :t7→(γ(t), e(t)) be an integral curve of the vector
fieldBξanda∈GL(m). Prove thatR→ F(M) :t7→a
∗
β(t) = (γ(t), a
∗
e(t))
is an integral curve ofB
a
−1
ξ.
(iii)Prove that the vector fieldBξ∈Vect(F(M)) is complete for allξ∈R
m
if and only if the restricted vector fieldBξ|
O(M)∈Vect(O(M)) on the or-
thonormal frame bundle is complete for allξ∈R
m
.
Definition 3.4.12(Complete manifold).A smothm-manifoldM⊂R
n
is calledcompleteiff, for every smooth curveξ:R→R
m
and every
element(p0, e0)∈ F(M), there exists a smooth curveβ:R→ F(M)such
thatβ(0) = (p0, e0)and
˙
β(t) =B
ξ(t)(β(t))for allt∈R.

146 CHAPTER 3. THE LEVI-CIVITA CONNECTION
3.5 Motions and Developments
Our aim in this sections is to define motion without sliding, twisting, or
wobbling. This is the motion that results when a heavy object is rolled,
with a minimum of friction, along the floor. It is also the motion of the large
snowball a child creates as it rolls it into the bottom part of a snowman.
We shall eventually justify mathematically the physical intuition that
either of the curves of contact in such ideal rolling may be specified arbi-
trarily; the other is then determined uniquely. Thus for example the heavy
object may be rolled along an arbitrary curve on the floor; if that curve is
marked in wet ink, another curve will be traced in the object. Conversely,
if a curve is marked in wet ink on the object, the object may be rolled so as
to trace a curve on the floor. However, if both curves are prescribed, it will
be necessary to slide the object as it is being rolled, if one wants to keep the
curves in contact.
We assume throughout this section thatMandM
′
are twom-dimensio-
nal submanifolds ofR
n
. Objects onM
′
will be denoted by the same letter
as the corresponding objects onMwith primes affixed. Thus for example
Π
′
(p
′
)∈R
n×n
denotes the orthogonal projection ofR
n
onto the tangent
spaceTp
′M
′
,∇
′
denotes the covariant derivative of a vector field along a
curve inM
′
, and Φ
′
γ
′denotes parallel transport along a curve inM
′
.
3.5.1 Motion
Definition 3.5.1.Amotion ofMalongM
′
(on an intervalI⊂R)is
a triple(Ψ, γ, γ
′
)of smooth maps
Ψ :I→O(n), γ:I→M, γ
′
:I→M
′
such that
Ψ(t)T
γ(t)M=T
γ
′
(t)M
′
∀t∈I.
Note that a motion also matches normal vectors, i.e.
Ψ(t)T
γ(t)M
⊥
=T
γ
′
(t)M
′⊥
∀t∈I.
Remark 3.5.2.Associated to a motion (Ψ, γ, γ
′
) ofMalongM
′
is a family
of (affine) isometriesψt:R
n
→R
n
defined by
ψt(p) :=γ
′
(t) + Ψ(t)
Γ
p−γ(t)
∆
(3.5.1)
fort∈Iandp∈R
n
. These isometries satisfy
ψt(γ(t)) =γ
′
(t), dψ t(γ(t))T
γ(t)M=T
γ
′
(t)M
′
∀t∈I.

3.5. MOTIONS AND DEVELOPMENTS 147
Remark 3.5.3.There are three operations on motions.
Reparametrization.If (Ψ, γ, γ
′
) is a motion ofMalongM
′
on an in-
tervalI⊂Randσ:J→Iis a smooth map between intervals, then the
triple
(Ψ◦σ, γ◦σ, γ
′
◦σ)
is a motion ofMalongM
′
on the intervalJ.
Inversion.If (Ψ, γ, γ
′
) is a motion ofMalongM
′
, then
(Ψ
−1
, γ
′
, γ)
is a motion ofM
′
alongM.
Composition.If (Ψ, γ, γ
′
) is a motion ofMalongM
′
on an intervalI
and (Ψ
′
, γ
′
, γ
′′
) is a motion ofM
′
alongM
′′
on the same interval, then
(Ψ
′
Ψ, γ, γ
′′
)
is a motion ofMalongM
′′
.
We now give the three simplest examples of Φad” motions; i.e. motions
which do not satisfy the concepts we are about to define. In all three of
these examples,pis a point ofMandM
′
is the affine tangent space toM
atp:
M
′
:=p+TpM={p+v|v∈TpM}.
Example 3.5.4(Pure sliding).Take a nonzero tangent vectorv∈TpM
and let
γ(t) :=p, γ
′
(t) =p+tv,Ψ(t) := 1l.
Then ˙γ(t) = 0, ˙γ
′
(t) =v̸= 0, and so
Ψ(t) ˙γ(t)̸= ˙γ
′
(t).
(See Figure 3.4.)M
p
M’
Figure 3.4: Pure sliding.

148 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Example 3.5.5(Pure twisting).Letγandγ
′
be the constant curves
γ(t) =γ
′
(t) =p
and take Ψ(t) to be the identity onTpM
⊥
and any curve of rotations on the
tangent spaceTpM. As a concrete example withm= 2 andn= 3 one can
takeMto be the sphere of radius one centered at the point (0,1,0) andp
to be the origin:
M:=
Φ
(x, y, z)∈R
3
|x
2
+ (y−1)
2
+z
2
= 1

, p:= (0,0,0).
ThenM
′
is the (x, z)-plane andA(t) is any curve of rotations in the (x, z)-
plane, i.e. about they-axisTpM
⊥
. (See Figure 3.5.)M
p
M’
Figure 3.5: Pure twisting.
Example 3.5.6(Pure wobbling).This is the same as pure twisting except
that Ψ(t) is the identity onTpMand any curve of rotations onTpM
⊥
. As
a concrete example withm= 1 andn= 3 one can takeMto be the circle
of radius one in the (x, y)-plane centered at the point (0,1,0) andpto be
the origin:
M:=
Φ
(x, y,0)∈R
3
|x
2
+ (y−1)
2
= 1

, p:= (0,0,0).
ThenM
′
is thex-axis and Ψ(t) is any curve of rotations in the (y, z)-plane,
i.e. about the axisM
′
. (See Figure 3.6.)M
pM’
Figure 3.6: Pure wobbling.

3.5. MOTIONS AND DEVELOPMENTS 149
3.5.2 Sliding
When a train slides on the track (e.g. in the process of stopping suddenly),
there is a terrific screech. Since we usually do not hear a screech, this means
that the wheel moves along without sliding. In other words the velocity of
the point of contact in the train wheelMequals the velocity of the point
of contact in the trackM
′
. But the track is not moving; hence the point of
contact in the wheel is not moving. One may explain the paradox this way:
the train is moving forward and the wheel is rotating around the axle. The
velocity of a point on the wheel is the sum of these two velocities. When
the point is on the bottom of the wheel, the two velocities cancel.
Definition 3.5.7.A motion(Ψ, γ, γ
′
)is said to bewithout slidingiff it
satisfiesΨ(t) ˙γ(t) = ˙γ
′
(t)for everyt.
Here is the geometric picture of the no sliding condition. As explained
in Remark 3.5.2 we can view a motion as a smooth family of isometries
ψt(p) :=γ
′
(t) + Ψ(t)
Γ
p−γ(t)
∆
acting on the manifoldMwithγ(t)∈Mbeing the point of contact withM
′
.
Differentiating the curvet7→ψt(p) which describes the motion of the point
p∈Min the spaceR
n
we obtain
d
dt
ψt(p) = ˙γ
′
(t)−Ψ(t) ˙γ(t) +
˙
Ψ(t)
Γ
p−γ(t)
∆
.
Takingp=γ(t0) we find
d
dt

t=t0
ψt(γ(t0)) = ˙γ
′
(t0)−Ψ(t0) ˙γ(t0).
This expression vanishes under the no sliding condition. In general the
curvet7→ψt(γ(t0)) will be non-constant, but (when the motion is without
sliding) its velocity will vanish at the instantt=t0; i.e. at the instant when
it becomes the point of contact. In other wordsthe motion is without sliding
if and only if the point of contact is motionless.
We remark that, if the motion is without sliding, we have:

˙γ
′
(t)

=|Ψ(t) ˙γ(t)|=|˙γ(t)|
so that the curvesγandγ
′
have the same arclength:
Z
t1
t0

˙γ
′
(t)

dt=
Z
t1
t0
|˙γ(t)|dt
on any interval [t0, t1]⊂I. Hence any motion with ˙γ= 0 and ˙γ
′
̸= 0 is not
without sliding (such as the example of pure sliding above).

150 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Exercise 3.5.8.Give an example of a motion where|˙γ(t)|=|˙γ
′
(t)|for
everytbut which is not without sliding.
Example 3.5.9.We describe mathematically the motion of the train wheel.
Let the center of the wheel move right parallel to thex-axis at height one
and the wheel have radius one and make one revolution in 2πunits of time.
Then the trackM
′
is thex-axis and we take
M:=
Φ
(x, y)∈R
2
|x
2
+ (y−1)
2
= 1

.
Choose
γ(t) := (cos(t−π/2),1 + sin(t−π/2))
= (sin(t),1−cos(t)),
γ
′
(t) := (t,0),
and define Ψ(t)∈GL(2) by
Ψ(t) :=
`
cos(t) sin(t)
−sin(t) cos(t)
´
.
The reader can easily verify that this is a motion without sliding. A fixed
pointp0onM, sayp0= (0,0), sweeps out a cycloid with parametric equa-
tions
x=t−sin(t), y= 1−cos(t).
(Check that ( ˙x,˙y) = (0,0) wheny= 0; i.e. fort= 2nπ.)
Remark 3.5.10.These same formulas give a motion of a sphereMrolling
without sliding along a straight line in a planeM
′
. Namely in coordinates
(x, y, z) the sphere is given by the equation
x
2
+ (y−1)
2
+z
2
= 1,
the plane isy= 0 and the line is thex-axis. Thez-coordinate of a point is
unaffected by the motion. Note that the curveγ
′
traces out a straight line
in the planeM
′
and the curveγtraces out a great circle on the sphereM.
Exercise 3.5.11.The operations of reparametrization, inversion, and com-
position respect motion without sliding; i.e. if (Ψ, γ, γ
′
) and (Ψ
′
, γ
′
, γ
′′
)
are motions without sliding on an intervalIandσ:J→Iis a smooth
map between intervals, then the motions (Ψ◦σ, γ◦σ, γ
′
◦σ), (Ψ
−1
, γ
′
, γ),
and (Ψ
′
Ψ, γ, γ
′′
) are also without sliding.

3.5. MOTIONS AND DEVELOPMENTS 151
3.5.3 Twisting and Wobbling
A motion (Ψ, γ, γ
′
) on an intervallI⊂Rtransforms vector fields alongγ
into vector fields alongγ
′
by the formula
X
′
(t) = (ΨX)(t) := Ψ(t)X(t)∈T
γ
′
(t)M
′
fort∈IandX∈Vect(γ); soX
′
∈Vect(γ
′
).
Lemma 3.5.12.Let(Ψ, γ, γ
′
)be a motion ofMalongM
′
on an interval
I⊂R. Then the following are equivalent.
(i)The instantaneous velocity of each tangent vector is normal, i.e. fort∈I
˙
Ψ(t)T
γ(t)M⊂T
γ
′
(t)M
′⊥
.
(ii)Ψintertwines covariant differentiation, i.e. forX∈Vect(γ)
∇
′
(ΨX) = Ψ∇X.
(iii)Ψtransforms parallel vector fields alongγinto parallel vector fields
alongγ
′
, i.e. forX∈Vect(γ)
∇X= 0 = ⇒ ∇
′
(ΨX) = 0.
(iv)Ψintertwines parallel transport, i.e. fors, t∈Iandv∈T
γ(s)M
Ψ(t)Φγ(t, s)v= Φ
′
γ
′(t, s)Ψ(s)v.
A motion that satisfies these conditions is calledwithout twisting.
Proof.We prove that (i) is equivalent to (ii). A motion satisfies the equation
Ψ(t)Π(γ(t)) = Π
′
(γ
′
(t))Ψ(t)
for everyt∈I. This restates the condition that Ψ(t) maps tangent vectors
ofMto tangent vectors ofM
′
and normal vectors ofMto normal vectors
ofM
′
. Differentiating the equationX
′
(t) = Ψ(t)X(t) we obtain
˙
X
′
(t) = Ψ(t)
˙
X(t) +
˙
Ψ(t)X(t).
Applying Π
′
(γ
′
(t)) this gives
∇
′
X
′
= Ψ∇X+ Π
′
(γ
′
)
˙
ΨX.
Hence (ii) holds if and only if Π
′
(γ
′
(t))
˙
Ψ(t) = 0 for everyt∈I. Thus we
have proved that (i) is equivalent to (ii). That (ii) implies (iii) is obvious.

152 CHAPTER 3. THE LEVI-CIVITA CONNECTION
We prove that (iii) implies (iv). Lett0∈Iandv0∈T
γ(t0)M. Define the
vector fieldX∈Vect(γ) byX(t) := Φγ(t, t0)v0fort∈Iand letX
′
:= ΨX.
Then∇X= 0, hence∇
′
X
′
= 0 by (iii), and hence
X
′
(t) = Φ
′
γ
′(t, t0)X
′
(t0) = Φ
′
γ
′(t, t0)Ψ(t0)v0
for allt∈I. SinceX
′
(t) = Ψ(t)X(t) = Ψ(t)Φγ(t, t0)v0, this implies (iv).
We prove that (iv) implies (ii). LetX∈Vect(γ) andX
′
:= ΨX. By (iv)
we have
Φ
′
γ
′(t0, t)X
′
(t) = Ψ(t0)Φγ(t0, t)X(t).
Differentiating this equation with respect totatt=t0and using Theo-
rem 3.3.6, we obtain∇
′
X
′
(t0) = Ψ(t0)∇X(t0).This proves the lemma.
Lemma 3.5.13.Let(Ψ, γ, γ
′
)be a motion ofMalongM
′
on an interval
I⊂R. Then the following are equivalent.
(i)The instantaneous velocity of each normal vector is tangent, i.e. fort∈I
˙
Ψ(t)T
γ(t)M
⊥
⊂T
γ
′
(t)M
′
.
(ii)Ψintertwines normal covariant differentiation, i.e. forY∈Vect
⊥
(γ)
∇
′⊥
(ΨY) = Ψ∇
⊥
Y.
(iii)Ψtransforms parallel normal vector fields alongγinto parallel normal
vector fields alongγ
′
, i.e. forY∈Vect
⊥
(γ)
∇
⊥
Y= 0 = ⇒ ∇
′⊥
(ΨY) = 0.
(iv)Ψintertwines parallel transport of normal vector fields, i.e. fors, t∈I
andw∈T
γ(s)M
⊥
Ψ(t)Φ
⊥
γ(t, s)w= Φ
′⊥
γ
′(t, s)Ψ(s)w.
A motion that satisfies these conditions is calledwithout wobbling.
The proof that the four conditions in Lemma 3.5.13 are equivalent is
word for word analogous to the proof of Lemma 3.5.12 and will be omitted.
In summary amotion is without twisting iff tangent vectors at the point
of contact are rotating towards the normal space and it is without wobbling
iff normal vectors at the point of contact are rotating towards the tangent
space. In casem= 2 andn= 3 motion without twisting means that the
instantaneous axis of rotation is parallel to the tangent plane.

3.5. MOTIONS AND DEVELOPMENTS 153
Remark 3.5.14.The operations of reparametrization, inversion, and com-
position respect motion without twisting, respectively without wobbling; i.e.
if (Ψ, γ, γ
′
) and (Ψ
′
, γ
′
, γ
′′
) are motions without twisting, respectively with-
out wobbling, on an intervalIandσ:J→Iis a smooth map between
intervals, then the motions (Ψ◦σ, γ◦σ, γ
′
◦σ), (Ψ
−1
, γ
′
, γ), and (Ψ
′
Ψ, γ, γ
′′
)
are also without twisting, respectively without wobbling.
Remark 3.5.15.LetI⊂Rbe an interval andt0∈I. Given curves
γ:I→Mandγ
′
:I→M
′
and an orthogonal matrix Ψ0∈O(n) such that
Ψ0T
γ(t0)M=T
γ
′
(t0)M
′
there is a unique motion (Ψ, γ, γ
′
) ofMalongM
′
(with the givenγandγ
′
)
without twisting or wobblingsatisfying the initial condition:
Ψ(t0) = Ψ0.
Indeed, the path of matrices Ψ :I→O(n) is uniquely determined by the
conditions (iv) in Lemma 3.5.12 and Lemma 3.5.13. It is given by the explicit
formula
Ψ(t)v= Φ
′
γ
′(t, t0)Ψ0Φγ(t0, t)Π(γ(t))v
+ Φ
′⊥
γ
′(t, t0)Ψ0Φ
⊥
γ(t0, t)
Γ
v−Π(γ(t))v
∆ (3.5.2)
fort∈Iandv∈R
n
. We prove below a somewhat harder result where the
motion is without twisting, wobbling, or sliding. It is in this situation thatγ
andγ
′
determine one another (up to an initial condition).
Remark 3.5.16.We can now give another interpretation of parallel trans-
port. Givenγ:R→Mandv0∈T
γ(t0)MtakeM
′
to be an affine subspace of
the same dimension asM. Let (Ψ, γ, γ
′
) be a motion ofMalongM
′
without
twisting (and, if you like, without sliding or wobbling). LetX
′
∈Vect(γ
′
)
be the constant vector field alongγ
′
(so that∇
′
X
′
= 0) with value
X
′
(t) = Ψ0v0,Ψ0:= Ψ(t0).
LetX∈Vect(γ) be the corresponding vector field alongγso that
Ψ(t)X(t) = Ψ0v0
ThenX(t) = Φγ(t, t0)v0. To put it another way, imagine that M is a ball. To
define parallel transport along a given curveγroll the ball (without sliding)
along a planeM
′
keeping the curveγin contact with the planeM
′
. Letγ
′
be the curve traced out inM
′
. If a constant vector field in the planeM
′
is
drawn in wet ink along the curveγ
′
, it will mark off a (covariant) parallel
vector field alongγinM.
Exercise 3.5.17.Describe parallel transport along a great circle in a sphere.

154 CHAPTER 3. THE LEVI-CIVITA CONNECTION
3.5.4 Development
A development is an intrinsic version of motion without sliding or twisting.
Definition 3.5.18.Adevelopment ofMalongM
′
(on an intervalI)
is a triple(Φ, γ, γ
′
)whereγ:I→Mandγ
′
:I→M
′
are smooth paths
andΦis a family of orthogonal isomorphisms
Φ(t) :T
γ(t)M→T
γ
′
(t)M
′
parametrized byt∈I, such that
Φ(t) ˙γ(t) = ˙γ
′
(t) (3.5.3)
for allt∈IandΦintertwines parallel transport, i.e.
Φ(t)Φγ(t, s) = Φ
′
γ
′(t, s)Φ(s) (3.5.4)
for alls, t∈I. In particular, the familyΦof isomorphisms is smooth, i.e.
ifXis a smooth vector field alongγ, then the formulaX
′
(t) := Φ(t)X(t)
defines a smooth vector field alongγ
′
.
Lemma 3.5.19.LetI⊂Rbe an interval,γ:I→Mandγ
′
:I→M
′
be smooth curves, andΦ(t) :T
γ(t)M→T
γ
′
(t)M
′
be a family of orthogonal
isomorphisms parametrized byt∈I. Then the following are equivalent.
(i)(Φ, γ, γ
′
)is a development.
(ii)Φsatisfies(3.5.3)and
∇
′
(ΦX) = Φ∇X (3.5.5)
for allX∈Vect(γ).
(iii)There exists a motion(Ψ, γ, γ
′
)without sliding and twisting such that
Φ(t) = Ψ(t)|T
γ(t)M for allt∈I. (3.5.6)
(iv)There exists a motion(Ψ, γ, γ
′
)ofMalongM
′
without sliding, twisting,
and wobbling that satisfies(3.5.6).
Proof.That (3.5.4) is equivalent to (3.5.5) was proved in Lemma 3.5.12.
This (i) is equivalent to (ii). That (iv) implies (iii) and (iii) implies (i) is
obvious. To prove that (i) implies (iv) choose anyt0∈Iand any orthogonal
matrix Ψ0∈O(n) such that Ψ0|T
γ(t
0
)M= Φ(t0) and define Ψ(t) :R
n
→R
n
by (3.5.2). This proves Lemma 3.5.19.

3.5. MOTIONS AND DEVELOPMENTS 155
Remark 3.5.20.The operations of reparametrization, inversion, and com-
position yield developments when applied to developments; i.e. if (Φ, γ, γ
′
) is
a development ofMalongM
′
on an intervalI, (Φ
′
, γ
′
, γ
′′
) is a development
ofM
′
alongM
′′
on the same intervalI, andσ:J→Iis a smooth map of
intervals, then the triples
(Φ◦σ, γ◦σ, γ
′
◦σ),(Φ
−1
, γ
′
, γ)),(Φ
′
Φ, γ, γ
′′
)
are all developments.
Theorem 3.5.21(Development Theorem). Letp0∈Mandt0∈R,
letγ
′
:R→M
′
be a smooth curve, and let
Φ0:Tp0
M→T
γ
′
(t0)M
′
be an orthogonal isomorphism. Then the following holds.
(i)There exists a development(Φ, γ, γ
′
|I)on some open intervalI⊂R
containingt0that satisfies the initial condition
γ(t0) =p0,Φ(t0) = Φ0. (3.5.7)
(ii)Any two developments(Φ1, γ1, γ
′
|I1
)and(Φ2, γ2, γ
′
|I2
)as in (i) on two
intervalsI1andI2agree on the intersectionI1∩I2, i.e.
γ1(t) =γ2(t),Φ1(t) = Φ2(t)
for everyt∈I1∩I2.
(iii)IfMis complete, then (i) holds withI=R.
Proof.Letγ:R→Mbe any smooth curve such that
γ(t0) =p0
and, fort∈R, define the linear map
Φ(t) :T
γ(t)M→T
γ
′
(t)M
′
by
Φ(t) := Φ
′
γ
′(t, t0)Φ0Φγ(t0, t). (3.5.8)
This is an orthogonal transformation for everytand it intertwines parallel
transport. However, in general Φ(t) ˙γ(t) will not be equal to ˙γ
′
(t).

156 CHAPTER 3. THE LEVI-CIVITA CONNECTION
To construct a development that satisfies (3.5.3), we choose an orthonor-
mal framee0:R
m
→Tp0
Mand, fort∈R, definee(t) :R
m
→T
γ(t)Mby
e(t) := Φγ(t, t0)e0. (3.5.9)
We can think ofe(t) as a realn×m-matrix and the map
R→R
n×m
:t7→e(t)
is smooth. In fact, the mapt7→(γ(t), e(t)) is a smooth path in the frame
bundleF(M). Define the smooth mapξ:R→R
m
by
˙γ
′
(t) = Φ
′
γ
′(t, t0)Φ0e0ξ(t). (3.5.10)
We prove the following.
Claim:The triple(Φ, γ, γ
′
)is a development on an intervalI⊂Rif and
only if the patht7→(γ(t), e(t))satisfies the differential equation
( ˙γ(t),˙e(t)) =B
ξ(t)(γ(t), e(t)) (3.5.11)
for everyt∈I, whereB
ξ(t)∈Vect(F(M))denotes the basic vector field
associated toξ(t)∈R
m
(see equation(3.4.9)).
The triple (Φ, γ, γ
′
) is a development onIif and only if
Φ(t) ˙γ(t) = ˙γ
′
(t)
for everyt∈I. By (3.5.8) and (3.5.10) this is equivalent to the condition
Φ
′
γ
′(t, t0)Φ0Φγ(t0, t) ˙γ(t) = ˙γ
′
(t) = Φ
′
γ
′(t, t0)Φ0e0ξ(t),
hence to
Φγ(t0, t) ˙γ(t) =e0ξ(t),
and hence to
˙γ(t) = Φγ(t, t0)e0ξ(t) =e(t)ξ(t) (3.5.12)
for everyt∈I. By (3.5.9) and the Gauß–Weingarten formula, we have
˙e(t) =h
γ(t)( ˙γ(t))e(t)
for everyt∈R. Hence it follows from (3.4.9) that (3.5.12) is equivalent
to (3.5.11). This proves the claim.
Parts (i) and (ii) follow directly from the claim. Part (iii) follows from
the claim and Definition 3.4.12. This proves Theorem 3.5.21.

3.5. MOTIONS AND DEVELOPMENTS 157
Remark 3.5.22.As any two developments (Φ1, γ1, γ
′
|I1
) and (Φ2, γ2, γ
′
|I2
)
on two intervalsI1andI2that satisfy the initial condition (3.5.7) agree
onI1∩I2there is a development defined onI1∪I2. Hence there is a
uniquemaximally defined development(Φ, γ, γ
′
|I), defined on a maximal
intervalI=I(t0, p0,Φ0), associated to the initial datat0,p0, Φ0.
Denote the space of initial data by
P:=
æ
(t, p,Φ)

t∈R, p∈M,Φ :TpM→T
γ
′
(t)M
′
is an orthogonal transformation
œ
, (3.5.13)
define the setD ⊂R× Pby
D:={(t, t0, p0,Φ0)|(t0, p0,Φ0)∈ P, t∈I(t0, p0,Φ0)} (3.5.14)
and let
D → P: (t, t0, p0,Φ0)7→(t, γ(t),Φ(t)), (3.5.15)
be the map which assigns to each (t0, p0,Φ0)∈ Pand eacht∈I(t0, p0,Φ0)
the value at timetof the unique development (Φ, γ, γ
′
|I) associated to the
inital condition (t0, p0,Φ0) on the maximal time intervalI=I(t0, p0,Φ0).
Then the spacePhas a natural structure of a smooth manifold (in the
intrinsic setting), and it follows from Theorem 2.4.9 and the proof of The-
orem 3.5.21 thatDis an open subset ofR× Pand the map (3.5.15) is
smooth.
The smooth structure onPcan be understood as follows. The space
O(γ
′
) =
Φ
(t, e
′
)|(γ
′
(t), e
′
)∈ O(M
′
)

is the pullback of the orthonormal frame bundleO(M
′
)→M
′
under the
curveγ
′
:R→M
′
or, equivalently, is the orthonormal frame bundle of the
pullback tangent bundle (γ
′
)
∗
T M. ThusO(γ
′
) is a smooth submanifold
ofR×R
n×m
. The group O(m) acts diagonally onO(γ
′
)× O(M) and the
action is free. Hence the quotient (O(γ
′
)× O(M))/O(m) is a smooth man-
ifold by Theorem 2.9.14, and it can be naturally identified withPvia the
bijection [(t, e
′
),(p, e)]7→(t, p, e
′
◦e
−1
).
Remark 3.5.23.The statement of Theorem 3.5.21 is essentially symmetric
inMandM
′
as the operation of inversion carries developments to develop-
ments. Hence given
γ:R→M, p
′
0∈M
′
, t0∈R,Φ0:T
γ(t0)M→T
p
′
0
M
′
,
we may speak of the development (Φ, γ, γ
′
) corresponding toγwith initial
conditionsγ
′
(t0) =p
′
0
and Φ(t0) = Φ0.

158 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Corollary 3.5.24(Motions).Letp0∈Mandt0∈R, letγ
′
:R→M
′
be
a smooth curve, and letΨ0∈O(n)be a matrix such that
Ψ0Tp0
M=T
γ
′
(t0)M
′
.
Then the following holds.
(i)There exists a motion(Ψ, γ, γ
′
|I)without sliding, twisting and wobbling
on some open intervalI⊂Rcontainingt0that satisfies the initial condi-
tionγ(t0) =p0andΨ(t0) = Ψ0.
(ii)Any two motions as in (i) on two intervalsI1andI2agree on the
intersectionI1∩I2.
(iii)IfMis complete, then (i) holds withI=R.
Proof.Theorem 3.5.21 and Remark 3.5.15.
Corollary 3.5.25(Completeness).The following are equivalent.
(i)Mis complete, i.e. for every smooth curveξ:R→R
m
and every
element(p0, e0)∈ F(M), there exists a smooth curveβ:R→ F(M)such
that
˙
β(t) =B
ξ(t)(β(t))for allt∈Randβ(0) = (p0, e0)(Definition 3.4.12).
(ii)For every smooth curveξ:R→R
m
and every element(p0, e0)∈ O(M),
there is a smooth curveα:R→ O(M)such that˙α(t) =B
ξ(t)(α(t))for ev-
eryt∈Randα(0) = (p0, e0).
(iii)For every smooth curveγ
′
:R→R
m
, everyp0∈M, and every orthogo-
nal isomorphismΦ0:Tp0
M→R
m
there exists a development(Φ, γ, γ
′
)ofM
alongM
′
=R
m
on all ofRthat satisfiesγ(0) =p0andΦ(0) = Φ0.
Proof.We have already noted that the basic vector fields are all tangent to
the orthonormal frame bundleO(M)⊂ F(M). Now note that if a smooth
curveI→ F(M) :t7→β(t) = (γ(t), e(t)) on an intervalI⊂Rsatisfies the
differential equation
˙
β(t) =B
ξ(t)(β(t))
for allt, then so does the curve
I→ F(M) :t7→a
∗
β(t) = (γ(t), e(t)◦a)
for everya∈GL(m,R). Since any framee0:R
m
→Tp0
Mcan be car-
ried to any other (in particular an orthonormal one) by a suitable ma-
trixa∈GL(m,R), this shows that (i) is equivalent to (ii).
That (i) implies (iii) was proved in Theorem 3.5.21.

3.5. MOTIONS AND DEVELOPMENTS 159
We prove that (iii) implies (ii). Fix a smooth mapξ:R→R
m
and an
element (p0, e0)∈ O(M). Define
Φ0:=e
−1
0
:Tp0
M→R
m
and
γ
′
(t) :=
Z
t
0
ξ(s)ds∈R
m
fort∈R.
By (ii) there exists a development (Φ, γ, γ
′
) ofMalongR
m
on all ofRthat
satisfies the initial conditions
γ(0) =p0,Φ(0) = Φ0.
Then
Φ(t) = Φ0Φγ(0, t) :T
γ(t)M→R
m
,Φ(t) ˙γ(t) = ˙γ
′
(t) =ξ(t)
for allt∈Rby Definition 3.5.18. Define
e(t) := Φγ(t,0)e0= Φ(t)
−1
:R
m
→T
γ(t)M
fort∈R. Then (γ, e) :R→ F(M) is a smooth curve that satisfies the initial
condition (γ(0), e(0)) = (p0, e0) and the differential equation
˙γ(t) = Φ(t)
−1
ξ(t) =e(t)ξ(t),
˙e(t) =h
γ(t)( ˙γ(t))e(t) =h
γ(t)(e(t)ξ(t))e(t)
by the Gauß–Weingarten formula. Thus
( ˙γ(t),˙e(t)) =B
ξ(t)(γ(t), e(t))
for allt∈R. This proves Corollary 3.5.25.
It is of course easy to give an example of a manifold which is not com-
plete; e.g. if (Φ, γ, γ
′
) is any development ofMalongM
′
, thenM\ {γ(t1)}
is not complete as the given development is only defined fort̸=t1. In§4.6
we give equivalent characterizations of completeness. In particular, we will
see that any compact submanifold ofR
n
is complete.
Exercise 3.5.26.Anaffine subspaceofR
n
is a subset of the form
E=p+E=
Φ
p+v

v∈E

whereE⊂R
n
is a linear subspace andp∈R
n
. Prove that every affine
subspace ofR
n
is a complete submanifold.

160 CHAPTER 3. THE LEVI-CIVITA CONNECTION
3.6 Christoffel Symbols
The goal of this subsection is to examine the covariant derivative in local
coordinates on an embedded manifoldM⊂R
n
of dimensionm. Let
ϕ:U→Ω
be a coordinate chart, defined on anM-open subsetU⊂Mwith values in
an open set Ω⊂R
m
, and denote its inverse by
ψ:=ϕ
−1
: Ω→U⊂M.
At this point it is convenient to use superscripts for the coordinates of a
vectorx∈Ω. Thus we write
x= (x
1
, . . . , x
m
)∈Ω.
Ifp=ψ(x)∈Uis the corresponding element ofM, then the tangent space
ofMatpis the image of the linear mapdψ(x) :R
m
→R
n
(Theorem 2.2.3)
and thus two tangent vectorsv, w∈TpMcan be written in the form
v=dψ(x)ξ=
m
X
i=1
ξ
i
∂ψ
∂x
i
(x),
w=dψ(x)η=
m
X
j=1
η
j
∂ψ
∂x
j
(x)
(3.6.1)
forξ= (ξ
1
, . . . , ξ
m
)∈R
m
andη= (η
1
, . . . , η
m
)∈R
m
. Recall that the re-
striction of the inner product in the ambient spaceR
n
to the tangent space
is the first fundamental formgp:TpM×TpM→R(Definition 3.1.1). Thus
gp(v, w) =⟨v, w⟩=
m
X
i,j=1
ξ
i
gij(x)η
j
, (3.6.2)
where the functionsgij: Ω→Rare defined by
gij(x) :=
ø
∂ψ
∂x
i
(x),
∂ψ
∂x
j
(x)
Æ
forx∈Ω. (3.6.3)
In other words, the first fundamental form is in local coordinates represented
by the matrix valued functiong= (gij)
m
i,j=1
: Ω→R
m×m
.

3.6. CHRISTOFFEL SYMBOLS 161tγ( )
X(t)
c(t)
U
Ω
ψ
φ
M
ξ( )t
Figure 3.7: A vector field along a curve in local coordinates.
Now letc= (c
1
, . . . , c
m
) :I→Ω be a smooth curve in Ω, defined on an
intervalI⊂R, and consider the curve
γ=ψ◦c:I→M
(see Figure 3.7). Our goal is to describe the operatorX7→ ∇Xon the space
of vector fields alongγin local coordinates. LetX:I→R
n
be a vector
field alongγ. Then
X(t)∈T
γ(t)M=T
ψ(c(t))M= im
Γ
dψ(c(t)) :R
m
→R
n
∆
for everyt∈Iand hence there exists a unique smooth function
ξ= (ξ
1
, . . . , ξ
m
) :I→R
m
such that
X(t) =dψ(c(t))ξ(t) =
m
X
i=1
ξ
i
(t)
∂ψ
∂x
i
(c(t)). (3.6.4)
Differentiate this identity to obtain
˙
X(t) =
m
X
i=1
˙
ξ
i
(t)
∂ψ
∂x
i
(c(t)) +
m
X
i,j=1
ξ
i
(t)˙c
j
(t)
∂
2
ψ
∂x
i
∂x
j
(c(t)). (3.6.5)
We examine the projection∇X(t) = Π(γ(t))
˙
X(t) of this vector onto the tan-
gent space ofMatγ(t). The first summand on the right in (3.6.5) is already

162 CHAPTER 3. THE LEVI-CIVITA CONNECTION
tangent toM. For the second summand we simply observe that the vec-
tor Π(ψ(x))∂
2
ψ/∂x
i
∂x
j
lies in tangent spaceT
ψ(x)Mand can therefore be
expressed as a linear combination of the basis vectors∂ψ/∂x
1
, . . . , ∂ψ/∂x
m
.
The coefficients will be denoted by Γ
k
ij
(x). Thus there exist smooth func-
tions Γ
k
ij
: Ω→Rfori, j, k= 1, . . . , msuch that
Π(ψ(x))
∂
2
ψ
∂x
i
∂x
j
(x) =
m
X
k=1
Γ
k
ij(x)
∂ψ
∂x
k
(x) (3.6.6)
for allx∈Ω and alli, j∈ {1, . . . , m}. The coefficients Γ
k
ij
: Ω→Rare called
theChristoffel symbolsassociated to the coordinate chartϕ:U→Ω. To
sum up we have proved the following.
Lemma 3.6.1.Letc:I→Ωbe a smooth curve and define
γ:=ψ◦c:I→M.
Ifξ:I→R
m
is a smooth map andX∈Vect(γ)is given by(3.6.4), then
its covariant derivative at timet∈Iis given by
∇X(t) =
m
X
k=1

˙
ξ
k
(t) +
m
X
i,j=1
Γ
k
ij(c(t))ξ
i
(t)˙c
j
(t)


∂ψ
∂x
k
(c(t)), (3.6.7)
where theΓ
k
ij
are the Christoffel symbols defined by(3.6.6).
Our next goal is to understand how the Christoffel symbols are deter-
mined by the metric in local coordinates. Recall from equation (3.6.2) that
the inner products on the tangent spaces inherited from the standard Eu-
clidean inner product on the ambient spaceR
n
are in local coordinates
represented by the matrix valued function
g= (gij)
m
i,j=1: Ω→R
m×m
given by
gij:=
ø
∂ψ
∂x
i
,
∂ψ
∂x
j
Æ
R
n
. (3.6.8)
We shall see that the Christoffel symbols are completely determined by the
functionsgij: Ω→R. Here are first some elementary observations.

3.6. CHRISTOFFEL SYMBOLS 163
Remark 3.6.2.The matrixg(x)∈R
m×m
is symmetric and positive definite
for everyx∈Ω. This follows from the fact that the matrixdψ(x)∈R
n×m
has rankmand the matrixg(x) is given by
g(x) =dψ(x)
T
dψ(x)
Thusξ
T
g(x)ξ=|dψ(x)ξ|
2
>0 for allξ∈R
m
\ {0}.
Remark 3.6.3.Forx∈Ω we have det(g(x))>0 by Remark 3.6.2 and
so the matrixg(x) is invertible. Denote the entries of the inverse ma-
trixg(x)
−1
∈R
m×m
byg
kℓ
(x). They are determined by the condition
m
X
j=1
gij(x)g
jk
(x) =δ
k
i=
æ
1,ifi=k,
0,ifi̸=k.
Sinceg(x) is symmetric and positive definite, so is its inverse matrixg(x)
−1
.
In particular, we haveg
kℓ
(x) =g
ℓk
(x) for allx∈Ω and allk, ℓ∈ {1, . . . , m}.
Remark 3.6.4.Suppose thatX, Y∈Vect(γ) are vector fields along our
curveγ=ψ◦c:I→Mandξ, η:I→R
m
are defined by
X(t) =
m
X
i=1
ξ
i
(t)
∂ψ
∂x
i
(c(t)), Y(t) =
m
X
j=1
η
j
(t)
∂ψ
∂x
j
(c(t)).
Then the inner product ofX(t) andY(t) is given by
⟨X(t), Y(t)⟩=
m
X
i,j=1
ξ
i
(t)gij(c(t))η
j
(t).
Lemma 3.6.5(Christoffel symbols).LetΩ⊂R
m
be an open set and
letgij: Ω→Rfori, j= 1, . . . , mbe smooth functions such that each ma-
trix(gij(x))
m
i,j=1
is symmetric and positive definite. LetΓ
k
ij
: Ω→Rbe
smooth functions fori, j, k= 1, . . . , m. Then theΓ
k
ij
satisfy the conditions
Γ
k
ij= Γ
k
ji,
∂gij
∂x
ℓ
=
m
X
k=1
ı
gikΓ
k
jℓ
+gjkΓ
k
iℓ
ȷ
(3.6.9)
fori, j, k, ℓ= 1, . . . , mif and only if they are given by
Γ
k
ij=
m
X
ℓ=1
g
kℓ
1
2
`
∂gℓi
∂x
j
+
∂gℓj
∂x
i
−
∂gij
∂x
ℓ
´
. (3.6.10)
If theΓ
k
ij
are defined by(3.6.6)and thegijby(3.6.8), then theΓ
k
ij
sat-
isfy(3.6.9)and hence are given by(3.6.10).

164 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Proof.Suppose that the Γ
k
ij
are given by (3.6.6) and thegijby (3.6.8). Let
c:I→Ω, ξ, η:I→R
m
be smooth functions and suppose that the vector fieldsX, Yalong the curve
γ:=ψ◦c:I→M
are given by
X(t) :=
m
X
i=1
ξ
i
(t)
∂ψ
∂x
i
(c(t)), Y(t) :=
m
X
j=1
η
j
(t)
∂ψ
∂x
j
(c(t)).
Dropping the argumenttin each term, we obtain from Remark 3.6.4 and
Lemma 3.6.1 that
⟨X, Y⟩=
X
i,j
gij(c)ξ
i
η
j
,
⟨X,∇Y⟩=
X
i,k
gik(c)ξ
i

˙η
k
+
X
j,ℓ
Γ
k
jℓ
(c)η
j
˙c
ℓ

,
⟨∇X, Y⟩=
X
j,k
gkj(c)

˙
ξ
k
+
X
i,ℓ
Γ
k
iℓ
(c)ξ
i
˙c
ℓ

η
j
.
Hence it follows from equation (3.2.5) in Lemma 3.2.4 that
0 =
d
dt
⟨X, Y⟩ − ⟨X,∇Y⟩ − ⟨∇X, Y⟩
=
X
i,j

gij(c)
˙
ξ
i
η
j
+gij(c)ξ
i
˙η
j
+
X
ℓ
∂gij
∂x
ℓ
(c)ξ
i
η
j
˙c
ℓ
!
−
X
i,k
gik(c)ξ
i
˙η
k
−
X
i,j,k,ℓ
gik(c)Γ
k
jℓ
(c)ξ
i
η
j
˙c
ℓ
−
X
j,k
gkj(c)
˙
ξ
k
η
j
−
X
i,j,k,ℓ
gkj(c)Γ
k
iℓ
(c)ξ
i
η
j
˙c
ℓ
=
X
i,j,ℓ

∂gij
∂x
ℓ
(c)−
X
k
gik(c)Γ
k
jℓ
(c)−
X
k
gjk(c)Γ
k
iℓ
(c)
!
ξ
i
η
j
˙c
ℓ
.
This holds for all smooth mapsc:I→Ω andξ, η:I→R
m
, so the Γ
k
ij
satisfy
the second equation in (3.6.9). That they are symmetric iniandjis obvious.

3.6. CHRISTOFFEL SYMBOLS 165
To prove that (3.6.9) is equivalent to (3.6.10), define
Γℓij:=
m
X
k=1
gℓkΓ
k
ij. (3.6.11)
Then (3.6.9) is equivalent to
Γℓij= Γℓji,
∂gij
∂x
ℓ
= Γijℓ+ Γjiℓ. (3.6.12)
and (3.6.10) is equivalent to
Γℓij=
1
2
`
∂gℓi
∂x
j
+
∂gℓj
∂x
i
−
∂gij
∂x
ℓ
´
. (3.6.13)
If the Γℓijare given by (3.6.13), then they satisfy
Γℓij= Γℓji
and
2Γijℓ+ 2Γjiℓ=
∂gij
∂x
ℓ
+
∂giℓ
∂x
j
−
∂gjℓ
∂x
i
+
∂gji
∂x
ℓ
+
∂gjℓ
∂x
i
−
∂giℓ
∂x
j
= 2
∂gij
∂x
ℓ
for alli, j, ℓ. Conversely, if the Γℓijsatisfy (3.6.12), then
∂gij
∂x
ℓ
= Γijℓ+ Γjiℓ,
∂gℓi
∂x
j
= Γℓij+ Γiℓj= Γℓij+ Γijℓ,
∂gℓj
∂x
i
= Γℓji+ Γjℓi= Γℓij+ Γjiℓ.
Take the sum of the last two minus the first of these equations to obtain
∂gℓi
∂x
j
+
∂gℓj
∂x
i
−
∂gij
∂x
ℓ
= 2Γℓij.
Thus (3.6.12) is equivalent to (3.6.13) and so (3.6.9) is equivalent to (3.6.10).
This proves Lemma 3.6.5.

166 CHAPTER 3. THE LEVI-CIVITA CONNECTION
3.7 Riemannian Metrics*
We wish to carry over the fundamental notions of differential geometry to
the intrinsic setting. First we need an inner product on the tangent spaces
to replace the first fundamental form in Definition 3.1.1. This is the content
of Definition 3.7.1 and Lemma 3.7.4 below. Second we must introduce the
covariant derivative of a vector field along a curve. With this understood all
the definitions, theorems, and proofs in this chapter carry over in an almost
word by word fashion to the intrinsic setting.
3.7.1 Existence of Riemannian Metrics
We will always consider norms that are induced by inner products. But in
general there is no ambient space that can induce an inner product on each
tangent space. This leads to the following definition.
Definition 3.7.1.LetMbe a smoothm-manifold. ARiemannian metric
onMis a collection of inner products
TpM×TpM→R: (v, w)7→gp(v, w),
one for everyp∈M, such that the map
M→R:p7→gp(X(p), Y(p))
is smooth for every pair of vector fieldsX, Y∈Vect(M). We will also
denote the inner product by⟨v, w⟩pand drop the subscriptpif the base
point is understood from the context. A smooth manifold equipped with a
Riemannian metric is called aRiemannian manifold.
Example 3.7.2.IfM⊂R
n
is a smooth submanifold, then a Riemannian
metric onMis given by restricting the standard inner product onR
n
to
the tangent spacesTpM⊂R
n
. This is the first fundamental form of an
embedded manifold (see Definition 3.1.1).
More generally, assume thatMis a Riemannianm-manifold in the in-
trinsic sense of Definition 3.7.1 with an atlasA={(ϕα, Uα)}α∈A. Then the
Riemannian metricgdetermines a collection of smooth functions
gα= (gα,ij)
m
i,j=1:ϕα(Uα)→R
m×m
, (3.7.1)
one for eachα∈A, defined by
ξ
T
gα(x)η:=gp(v, w), ϕα(p) =x, dϕα(p)v=ξ, dϕα(p)w=η,(3.7.2)
forx∈ϕα(Uα) andξ, η∈R
m
.

3.7. RIEMANNIAN METRICS* 167
Each matrixgα(x) is symmetrix and positive definite. Note that the
tangent vectorsvandwin (3.7.2) can also be written in the form
v= [α, ξ]p, w= [α, η]p.
Choosing standard basis vectors
ξ=ei, η=ej
inR
m
we obtain
[α, ei]p=dϕα(p)
−1
ei=:
∂
∂x
i
(p)
and hence
gα,ij(x) =
ø
∂
∂x
i
(ϕ
−1
α(x)),
∂
∂x
j
(ϕ
−1
α(x))
Æ
. (3.7.3)
For different coordinate charts the mapsgαandgβare related through the
transition map
ϕβα:=ϕβ◦ϕ
−1
α:ϕα(Uα∩Uβ)→ϕβ(Uα∩Uβ)
via
gα(x) =dϕβα(x)
T
gβ(ϕβα(x))dϕβα(x) (3.7.4)
forx∈ϕα(Uα∩Uβ). Equation (3.7.4) can also be written in the shorthand
notation
gα=ϕ
∗
βα
gβ
forα, β∈A.
Exercise 3.7.3.Every collection of smooth maps
gα:ϕα(Uα)→R
m×m
with values in the set of positive definite symmetric matrices that satis-
fies (3.7.4) for allα, β∈Adetermines a global Riemannian metric via (3.7.2).
In this intrinsic setting there is no canonical metric onM(such as the
metric induced byR
n
on an embedded manifold). In fact, it is not completely
obvious that a manifold admits a Riemannian metric and this is the content
of the next lemma.

168 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Lemma 3.7.4.Every paracompact Hausdorff manifold admits a Rieman-
nian metric.
Proof.Letmbe the dimension ofMand letA={(ϕα, Uα)}α∈Abe an atlas
onM. By Theorem 2.9.9 there is a partition of unity{θα}α∈Asubordinate
to the open cover{Uα}α∈A. Now there are two equivalent ways to construct
a Riemannian metric onM.
The first method is to carry over the standard inner product onR
m
to
the tangent spacesTpMforp∈Uαvia the coordinate chartϕα, multiply
the resulting Riemannian metric onUαby the compactly supported function
θα, extend it by zero to all ofM, and then take the sum over allα. This
leads to the following formula. The inner product of two tangent vectors
v, w∈TpMis defined by
⟨v, w⟩p:=
X
p∈Uα
θα(p)⟨dϕα(p)v, dϕα(p)w⟩, (3.7.5)
where the sum runs over allα∈Awithp∈Uαand the inner product is
the standard inner product onR
m
. Since supp(θα)⊂Uαfor eachαand the
sum is locally finite we find that the function
M→R:p7→ ⟨X(p), Y(p)⟩p
is smooth for every pair of vector fieldsX, Y∈Vect(M). Moreover, the right
hand side of (3.7.5) is symmetric invandwand is positive forv=w̸= 0
because each summand is nonnegative and each summand withθα(p)>0 is
positive. Thus equation (3.7.5) defines a Riemannian metric onM.
The second method is to define the functions
gα:ϕα(Uα)→R
m×m
by
gα(x) :=
X
γ∈A
θγ(ϕ
−1
α(x))dϕγα(x)
T
dϕγα(x) (3.7.6)
forx∈ϕα(Uα) where each summand is defined onϕα(Uα∩Uγ) and is
understood to be zero forx /∈ϕα(Uα∩Uγ). We leave it to the reader to
verify that these functions are smooth and satisfy the condition (3.7.4) for
allα, β∈A. Moreover, the formulas (3.7.5) and (3.7.6) determine the same
Riemannian metric onM. (Prove this!) This proves Lemma 3.7.4.

3.7. RIEMANNIAN METRICS* 169
3.7.2 Two Examples
Example 3.7.5(Fubini–Study metric).The complex projective space
carries a natural Riemannian metric, defined as follows. IdentifyCP
n
with
the quotient of the unit sphereS
2n+1
⊂C
n+1
by the diagonal action of
the circleS
1
, i.e.CP
n
=S
2n+1
/S
1
. Then the tangent space ofCP
n
at the
equivalence class
[z] = [z0:· · ·:zn]∈CP
n
of a pointz= (z0, . . . , zn)∈S
2n+1
can be identified with the orthogonal
complement ofCzinC
n+1
. Now choose the inner product onT
[z]CP
n
to
be the one inherited from the standard inner product onC
n+1
via this
identification. The resulting metric onCP
n
is called theFubini–Study
metric.Exercise:Prove that the action of U(n+ 1) onC
n+1
induces a
transitive action of the quotient group
PSU(n+ 1) := U(n+ 1)/S
1
by isometries. Ifz∈S
2n+1
, prove that the unitary matrix
g:= 2zz
∗
−1l
descends to an isometryϕonCP
n
with fixed pointp:= [z] anddϕ(p) =−id.
Show that, in the casen= 1, the pullback of the Fubini–Study metric onCP
1
under the stereographic projection
S
2
\ {(0,0,1)} →CP
1
\ {[0 : 1]}: (x1, x2, x3)7→
ˇ
1 :
x1+ix2
1−x3
˘
is one quarter of the standard metric onS
2
.
Example 3.7.6.Think of the complex Grassmannian Gk(C
n
) ofk-planes
inC
n
as a quotient of the space
Fk(C
n
) :=
n
D∈C
n×k
|D
∗
D= 1l
o
of unitaryk-frames inC
n
by the right action of the unitary group U(k).
The spaceFk(C
n
) inherits a Riemannian metric from the ambient Euclidean
spaceC
n×k
. Show that the tangent space of Gk(C
n
) at a point Λ = imD,
withD∈ Fk(C
n
) can be identified with the space
TΛGk(C
n
) =
n
b
D∈C
n×k
|D
∗b
D= 0
o
.
Define the inner product on this tangent space to be the restriction of the
standard inner product onC
n×k
to this subspace.Exercise:Prove that
the unitary group U(n) acts on Gk(C
n
) by isometries.

170 CHAPTER 3. THE LEVI-CIVITA CONNECTION
3.7.3 The Levi-Civita Connection
A subtle point in this discussion is how to extend the notion ofcovariant
derivativeto general Riemannian manifolds. In this case the idea of project-
ing the derivative in the ambient space orthogonally onto the tangent space
has no obvious analogue. Instead we shall see how the covariant derivatives
of vector fields along curves can be characterized by several axioms and
these can be used to define the covariant derivative in the intrinsic setting.
An alternative, but somewhat less satisfactory, approach is to carry over
the formula for the covariant derivative in local coordinates to the intrinsic
setting and show that the result is independent of the choice of the coor-
dinate chart. Of course, these approaches are equivalent and lead to the
same result. We formulate them as a series of exercises. The details are
straightforward.
Assume throughout thatMis a Riemannianm-manifold with an atlas
A={(ϕα, Uα)}α∈A
and suppose that the Riemannian metric is in local coordinates given by
gα= (gα,ij)
m
i,j=1:ϕα(Uα)→R
m×m
forα∈A. These functions satisfy (3.7.4) for allα, β∈A.
Definition 3.7.7.Letf:N→Mbe a smooth map between manifolds. A
vector field alongfis a collection of tangent vectors
X(q)∈T
f(q)M,
one for eachq∈N, such that the map
N→T M:q7→(f(q), X(q))
is smooth. The space of vector fields alongfwill be denoted byVect(f).
As before we will not distinguish in notation between the collection of
tangent vectorsX(q)∈T
f(q)Mand the associated mapN→T Mand
denote them both byX. The following theorem introduces the Levi-Civita
connection as a collection of linear operators∇: Vect(γ)→Vect(γ), one for
each smooth curveγ:I→M.

3.7. RIEMANNIAN METRICS* 171
Theorem 3.7.8(Levi-Civita connection).There exists a unique collec-
tion of linear operators
∇: Vect(γ)→Vect(γ)
(called thecovariant derivative), one for every smooth curveγ:I→M
on an open intervalI⊂R, satisfying the following axioms.
(Leibniz Rule)For every smooth curveγ:I→M, every smooth function
λ:I→R, and every vector fieldX∈Vect(γ), we have
∇(λX) =
˙
λX+λ∇X. (3.7.7)
(Chain Rule)Let Ω⊂R
n
be an open set, letc:I→Ω be a smooth curve,
letγ: Ω→Mbe a smooth map, and letXbe a smooth vector field alongγ.
Denote by∇iXthe covariant derivative ofXalong the curvex
i
7→γ(x) (with
the other coordinates fixed). Then∇iXis a smooth vector field alongγand
the covariant derivative of the vector fieldX◦c∈Vect(γ◦c) is
∇(X◦c) =
n
X
j=1
˙c
j
(t)∇jX(c(t)). (3.7.8)
(Riemannian)For any two vector fieldsX, Y∈Vect(γ) we have
d
dt
⟨X, Y⟩=⟨∇X, Y⟩+⟨X,∇Y⟩. (3.7.9)
(Torsion-free)LetI, J⊂Rbe open intervals andγ:I×J→Mbe
a smooth map. Denote by∇sthe covariant derivative along the curve
s7→γ(s, t) (withtfixed) and by∇tthe covariant derivative along the curve
t7→γ(s, t) (withsfixed). Then
∇s∂tγ=∇t∂sγ. (3.7.10)
Proof.The proof is based on a reformulation of the axioms in local co-
ordinates. The (Leibnitz Rule) and (Chain Rule) axioms assert that the
covariant derivative is in local coordinates given by Christoffel symbols Γ
k
ij
as in equation (3.6.7) in Lemma 3.6.1. The (Riemannian) and (Torsion-free)
axioms assert that the Christoffel symbols satisfy the equations in (3.6.9)
and hence, by Lemma 3.6.5, are given by (3.6.10). (See also Exercise 3.7.10.)
This proves Theorem 3.7.8.

172 CHAPTER 3. THE LEVI-CIVITA CONNECTION
Exercise 3.7.9.TheChristoffel symbolsof the Riemannian metric are
the functions Γ
k
α,ij
:ϕα(Uα)→R. defined by
Γ
k
α,ij:=
m
X
ℓ=1
g
kℓ
α
1
2
`
∂gα,ℓi
∂x
j
+
∂gα,ℓj
∂x
i
−
∂gα,ij
∂x
ℓ
´
(3.7.11)
(see Lemma 3.6.5). Prove that they are related by the equation
X
k
∂ϕ
k
′
βα
∂x
k
Γ
k
α,ij=
∂
2
ϕ
k
′
βα
∂x
i
∂x
j
+
X
i
′
,j
′
ı
Γ
k
′
β,i
′
j
′◦ϕβα
ȷ
∂ϕ
i
′
βα
∂x
i
∂ϕ
j
′
βα
∂x
j
.
for allα, β∈A.
Exercise 3.7.10.Denoteψα:=ϕ
−1
α:ϕα(Uα)→M. Prove that the covari-
ant derivative of a vector field
X(t) =
m
X
i=1
ξ
i
α(t)
∂ψα
∂x
i
(cα(t))
alongγ=ψα◦cα:I→Mis given by
∇X(t) =
m
X
k=1

˙
ξ
k
α(t) +
m
X
i,j=1
Γ
k
α,ij(c(t))ξ
i
α(t)˙c
j
α(t)


∂ψα
∂x
k
(cα(t)).(3.7.12)
Prove that∇Xis independent of the choice of the coordinate chart.
Exercise 3.7.11.Let Ω⊂R
2
be open andλ: Ω→(0,∞) be a smooth
function. Letg: Ω→R
2×2
be given by
g(x) =
`
λ(x) 0
0λ(x)
´
.
Compute the Christoffel symbols Γ
k
ij
via (3.6.10).
Exercise 3.7.12.Letϕ:S
2
\ {(0,0,1)} →Cbe the stereographic projec-
tion, given by
ϕ(p) :=
`
p1
1−p3
,
p2
1−p3
´
Prove that the metricg:R
2
→R
2×2
has the formg(x) =λ(x)1l where
λ(x) :=
4
(1 +|x|
2
)
2
forx= (x
1
, x
2
)∈R
2
.

3.7. RIEMANNIAN METRICS* 173
3.7.4 Basic Vector Fields in the Intrinsic Setting
LetMbe a Riemannianm-manifold with an atlasA={(ϕα, Uα)}
α∈A
.
Then theframe bundle(3.4.1) admits the structure of a smooth manifold
with the open cover
e
Uα:=π
−1
(Uα) and coordinate charts
e
ϕα:
e
Uα→ϕα(Uα)×GL(m)
given by
e
ϕα(p, e) := (ϕα(p), dϕα(p)e).
The derivatives of the horizontal curves in Definition 3.4.6 form ahorizon-
tal subbundleH⊂TF(M) of the tangent bundle of the Frame bundle
whose fibersH
(p,e)over an element (p, e)∈ F(M) can in local coordinates
be described as follows. Let
x:=ϕα(p), a:=dϕα(p)e∈GL(m), (3.7.13)
and let (bx,ba)∈R
m
×R
m×m
. This pair has the form
(bx,ba) =d
e
ϕα(p, e)(bp,be),(bp,be)∈H
(p,e), (3.7.14)
if and only if
ba
k
ℓ
=−
m
X
i,j=1
Γ
k
α,ij(x)bx
i
a
j
ℓ
(3.7.15)
fork, ℓ= 1, . . . , m, where the functions Γ
k
α,ij
:ϕα(Uα)→Rare the Christof-
fel symbols defined by (3.7.11). Thus a tangent vector (bp,be)∈T
(p,e)F(M)
ishorizontalif and only if its coordinates (bx,ba) in (3.7.14) satisfy (3.7.15).
Hence, for every vectorξ∈R
m
, there exists a unique horizontal vector
fieldBξ∈Vect(F(M)) (thebasic vector fieldassociated toξ) such that
dπ(p, e)Bξ(p, e) =eξ
for all (p, e)∈ F(M). This vector field assigns to a pair (p, e)∈ F(M) with
the coordinates (x, a)∈R
m
×GL(m) as in (3.7.13) the horizontal tangent
vector (bp,be)∈H
(p,e)⊂T
(p,e)F(M) whose coordinates (bx,ba)∈R
m
×R
m×m
satisfy (3.7.15) andbx=aξ.
Exercise 3.7.13.Verify the equivalence of (3.7.14) and (3.7.15). Prove
that the notion of a horizontal tangent vector ofF(M) is independent of
the choice of the coordinate chart.Hint:Use Exercise 3.7.9.
Exercise 3.7.14.Examine the orthonormal frame bundleO(M) in the
intrinsic setting.
Exercise 3.7.15.Carry over the proofs of Theorem 3.3.4, Theorem 3.3.6,
and Theorem 3.5.21 to the intrinsic setting.

174 CHAPTER 3. THE LEVI-CIVITA CONNECTION

Chapter 4
Geodesics
This chapter introduces geodesics in Riemannian manifolds. It begins in§4.1
by introducing geodesics as extremals of the energy and length functionals
and characterizing them as solutions of a second order differential equation.
In§4.2 we show that minimizing the length with fixed endpoints gives rise
to an intrinsic distance functiond:M×M→Rwhich induces the topol-
ogyMinherits from the ambient spaceR
n
.§4.3 introduces the exponential
map,§4.4 shows that geodesics minimize the length on short time intervals,
§4.5 establishes the existence of geodesically convex neighborhoods, and§4.6
shows that the geodesic flow is complete if and only if (M, d) is a complete
metric space, and that in the complete case any two points are joined by a
minimal geodesic.§4.7 discusses geodesics in the intrinsic setting.
4.1 Length and Energy
This section explains the length and energy functionals on the space of paths
with fixed endpoints and introduces geodesics as their extremal points.
4.1.1 The Length and Energy Functionals
The concept of a geodesic in a manifold generalizes that of a straight line in
Euclidean space. A straight line has parametrizations of formt7→p+σ(t)v
whereσ:R→Ris a diffeomorphism andp, v∈R
n
. Different choices ofσ
yield different parametrizations of the same line. Certain parametrizations
are preferred, for example those parametrizations which are “proportional
to the arclength”, i.e. whereσ(t) =at+bfor constantsa, b∈R, so that
the tangent vector ˙σ(t)vhas constant length. The same distinctions can
175

176 CHAPTER 4. GEODESICS
be made for geodesics. Some authors use the term geodesic to include all
parametrizations of a geodesic while others restrict the term to cover only
geodesics parametrized proportional to the arclength. We follow the latter
course, referring to the more general concept as a fleparametrized geodesic”.
Thus a reparametrized geodesic need not be a geodesic.
We assume throughout thatM⊂R
n
is a smoothm-manifold.
Definition 4.1.1(Length and energy).LetI= [a, b]⊂Rbe a com-
pact interval witha < band letγ:I→Mbe a smooth curve inM. The
lengthL(γ)and theenergyE(γ)are defined by
L(γ) :=
Z
b
a
|˙γ(t)|dt, (4.1.1)
E(γ) :=
1
2
Z
b
a
|˙γ(t)|
2
dt. (4.1.2)
Avariation ofγis a family of smooth curvesγs:I→M, wheres
ranges over the reals, such that the mapR×I→M: (s, t)7→γs(t)is
smooth andγ0=γ. The variation{γs}s∈Ris said to havefixed endpoints
iffγs(a) =γ(a)andγs(b) =γ(b)for alls∈R.
Remark 4.1.2.The length of a continuous functionγ: [a, b]→R
n
can
be defined as the supremum of the numbers
P
N
i=1
|γ(ti)−γ(ti−1)|over all
partitionsa=t0< t1<· · ·< tN=bof the interval [a, b]. By a theorem in
first year analysis [64] this supremum is finite wheneverγis continuously
differentiable and is given by (4.1.1).
We shall sometimes suppress the notation for the endpointsa, b∈I.
Whenγ(a) =pandγ(b) =qwe say thatγis a curve fromptoq. One
can always composeγwith an affine reparametrizationt
′
=a+ (b−a)tto
obtain a new curveγ
′
(t) :=γ(t
′
) on the unit interval 0≤t≤1. This new
curve satisfiesL(γ
′
) =L(γ) andE(γ
′
) = (b−a)E(γ). More generally, the
lengthL(γ), but not the energyE(γ), is invariant under reparametrization.
Remark 4.1.3(Reparametrization).LetI= [a, b] andI
′
= [a
′
, b
′
] be
compact intervals. Ifγ:I→R
n
is a smooth curve andσ:I
′
→Iis a
smooth function such thatσ(a
′
) =a,σ(b
′
) =b, and ˙σ(t
′
)≥0 for allt
′
∈I
′
,
then the curvesγandγ◦σhave the same length. Namely,
L(γ◦σ) =
Z
b
′
a
′

d
dt
′
γ(σ(t
′
))

dt
′
=
Z
b
′
a
′

˙γ(σ(t
′
))

˙σ(t
′
)dt
′
=L(γ).(4.1.3)
Here second equality follows from the chain rule and the third equality
follows from the change of variables formula for the Riemann integral.

4.1. LENGTH AND ENERGY 177
Theorem 4.1.4(Characterization of geodesics).LetI= [a, b]⊂Rbe
a compact interval and letγ:I→Mbe a smooth curve. Then the following
are equivalent.
(i)γis anextremal of the energy functional, i.e. every variation{γs}s∈R
ofγwith fixed endpoints satisfies
d
ds

s=0
E(γs) = 0.
(ii)γisparametrized proportional to the arclength, i.e. the veloc-
ity|˙γ(t)| ≡c≥0is constant, and eitherγis constant, i.e.γ(t) =p=qfor
allt∈I, orc >0andγis anextremal of the length functional, i.e.
every variation{γs}s∈Rofγwith fixed endpoints satisfies
d
ds

s=0
L(γs) = 0.
(iii)The velocity vector ofγis parallel, i.e.∇˙γ(t) = 0for allt∈I.
(iv)The acceleration ofγis normal toM, i.e.¨γ(t)⊥T
γ(t)Mfor allt∈I.
(v)If(Φ, γ, γ
′
)is a development ofMalongM
′
=R
m
, thenγ
′
:I→R
m
is
a straight line parametrized proportional to the arclength, i.e.¨γ
′
≡0.
Proof.See§4.1.3.
Definition 4.1.5(Geodesic).A smooth curveγ:I→Mon an intervalI
is called ageodesiciff its restriction to each compact subinterval satisfies
the equivalent conditions of Theorem 4.1.4. Soγis a geodesic if and only if
∇˙γ(t) = 0for allt∈I. (4.1.4)
By the Gauß–Weingarten formula(3.2.2)withX= ˙γthis is equivalent to
¨γ(t) =h
γ(t)( ˙γ(t),˙γ(t))for allt∈I. (4.1.5)
Remark 4.1.6. The conditions (i) and (ii) in Theorem 4.1.4 are mean-
ingless whenIis not compact because then the curve has at most one
endpoint and the length and energy integrals may be infinite. However, the
conditions (iii), (iv), and (v) in Theorem 4.1.4 are equivalent for smooth
curves on any interval, compact or not.
(ii)The functions7→E(γs) associated to a smooth variation is always
smooth and so condition (i) in Theorem 4.1.4 is meaningful. However, more
care has to be taken in part (ii) because the functions7→L(γs) need not be
differentiable. It is differentiable ats= 0 whenever ˙γ(t)̸= 0 for allt∈I.

178 CHAPTER 4. GEODESICS
4.1.2 The Space of Paths
Fix two pointsp, q∈Mand a compact intervalI= [a, b] and denote by
Ωp,q:= Ωp,q(I) :=
Φ
γ:I→M

γis smooth andγ(a) =p, γ(b) =q

the space of smooth curves inMfromptoq, defined on the intervalI. Then
the length and energy are functionalsL, E: Ωp,q→Rand their extremal
points can be understood ascritical pointsas we now explain.
We may think of the space Ωp,qas a kind of“infinite-dimensional man-
ifold”. This is to be understood in a heuristic sense and we use these terms
here to emphasize an analogy. Of course, the space Ωp,qis not a manifold
in the strict sense of the word. To begin with it is not embedded in any
finite-dimensional Euclidean space. However, it has many features in com-
mon with manifolds. The first is that we can speak ofsmooth curves inΩp,q.
Of course Ωp,qis itself a space of curves inM. Thus a smooth curve in Ωp,q
would then be a curve of curves, namly a mapR→Ωp,q:s7→γsthat as-
signs to each real numbersa smooth curveγs:I→Msatisfyingγs(a) =p
andγs(b) =q. We shall call such a curve of curvessmoothiff the associated
mapR×I→M: (s, t)7→γs(t) is smooth. Thus smooth curves in Ωp,qare
the variations ofγwith fixed endpoints introduced in Definition 4.1.1.
Having defined what we mean by a smooth curve in Ωp,qwe can also
differentiate such a curve with respect tos. Here we can simply recall that,
sinceM⊂R
n
, we have a smooth mapR×I→R
n
and the derivative of
the curves7→γsin Ωp,qcan simply be understood as the partial derivative
of the map (s, t)7→γs(t) with respect tos. Thus, in analogy with embed-
ded manifolds, we define thetangent spaceof the space of curves Ωp,q
atγas the set of all derivatives of smooth curvesR→Ωp,q:s7→γspassing
throughγ, i.e.
TγΩp,q:=
æ
∂
∂s

s=0
γs

R→Ωp,q:s7→γsis smooth andγ0=γ
œ
.
Let us denote such a partial derivative byX(t) :=
∂
∂s

s=0
γs(t)∈T
γ(t)M.
Thus we obtain a smooth vector field alongγ. Sinceγs(a) =pandγs(b) =q
for alls, this vector field must vanish att=a, b. This suggests the formula
TγΩp,q={X∈Vect(γ)|X(a) = 0, X(b) = 0}. (4.1.6)
That every tangent vector of the path space Ωp,qatγis a vector field alongγ
vanishing at the endpoints follows from the above discussion. The converse
inclusion is the content of the next lemma.

4.1. LENGTH AND ENERGY 179
Lemma 4.1.7.Letp, q∈M,γ∈Ωp,q, andX∈Vect(γ)withX(a) = 0
andX(b) = 0. Then there exists a smooth mapR→Ωp,q:s7→γssuch that
γ0(t) =γ(t),
∂
∂s

s=0
γs(t) =X(t)for allt∈I. (4.1.7)
Proof.The proof has two steps.
Step 1.There exists smooth mapM×I→R
n
: (r, t)7→Yt(r)with compact
support such thatYt(r)∈TrMfor allt∈Iandr∈MandYa(r) =Yb(r) = 0
for allr∈M.
DefineZt(r) := Π(r)X(t) fort∈Iandr∈M. Choose an open setU⊂R
n
such thatγ(I)⊂UandU∩Mis compact (e.g. takeU:=
S
a≤t≤b
Bε(γ(t))
forε >0 sufficiently small). Now letβ:R
n
→[0,1] be a smooth cutoff
function with support in the unit ball such thatβ(0) = 1 and define the
vector fieldsYtbyYt(r) :=β(ε
−1
(r−γ(t)))Zt(r) fort∈Iandr∈M.
Step 2.We prove the lemma.
The vector fieldYt:M→T Min Step 1 is complete for eacht. Thus
there exists a unique smooth mapR×I→M: (s, t)7→γs(t) such that,
for eacht∈I, the curveR→M:s7→γs(t) is the unique solution of the
differential equation
∂
∂s
γs(t) =Yt(γs(t)) withγ0(t) =γ(t). These mapsγs
satisfies (4.1.7) by Step 1.
We can now define thederivative of the energy functionalEatγ
in the direction of a tangent vectorX∈TγΩp,qby
dE(γ)X:=
d
ds

s=0
E(γs), (4.1.8)
wheres7→γsis as in Lemma 4.1.7. Similarly, thederivative of the length
functionalLatγin the direction ofX∈TγΩp,qis defined by
dL(γ)X:=
d
ds

s=0
L(γs). (4.1.9)
To define (4.1.8) and (4.1.9) the functionss7→E(γs) ands7→L(γs) must
be differentiable ats= 0. This is true forEbut it only holds forL
when ˙γ(t)̸= 0 for allt∈I. Second, we must show that the right hand sides
of (4.1.8) and (4.1.9) depend only onXand not on the choice of{γs}s∈R.
Third, we must verify thatdE(γ) :TγΩp,q→RanddL(γ) : Ωp,q→Rare lin-
ear maps. This is an exercise in first year analysis (see also the proof of The-
orem 4.1.4). A curveγ∈Ωp,qis then an extremal point ofE(respectivelyL
when ˙γ(t)̸= 0 for allt) if and only ifdE(γ) = 0 (respectivelydL(γ) = 0).
Such a curve is also called acritical pointofE(respectivelyL).

180 CHAPTER 4. GEODESICS
4.1.3 Characterization of Geodesics
Proof of Theorem 4.1.4.The equivalence of (iii) and (iv) follows directly
from the equations∇˙γ(t) = Π(γ(t))¨γ(t) and ker(Π(γ(t))) =T
γ(t)M
⊥
.
We prove that (i) is equivalent to (iii) and (iv). LetX∈TγΩp,qand
choose a smooth curve of curvesR→Ωp,q:s7→γssatisfying (4.1.7). Then
the function (s, t)7→ |˙γs(t)|
2
is smooth and hence
dE(γ)X=
d
ds

s=0
E(γs)
=
d
ds

s=0
1
2
Z
b
a
|˙γs(t)|
2
dt
=
1
2
Z
b
a
∂
∂s

s=0
|˙γs(t)|
2
dt
=
Z
b
a
ø
˙γ(t),
∂
∂s

s=0
˙γs(t)
Æ
dt
=
Z
b
a
D
˙γ(t),
˙
X(t)
E
dt
=−
Z
b
a
⟨¨γ(t), X(t)⟩dt.
(4.1.10)
That (iii) implies (i) follows directly from this identity. To prove that (i) im-
plies (iv) we argue indirectly and assume that there exists a pointt0∈[0,1]
such that ¨γ(t0) is not orthogonal toT
γ(t0)M. Then there exists a vec-
torv0∈T
γ(t0)Msuch that⟨¨γ(t0), v0⟩>0. We may assume without loss
of generality thata < t0< b. Then there exists a constantε >0 such
thata < t0−ε < t0+ε < band
t0−ε < t < t0+ε =⇒ ⟨ ¨γ(t),Π(γ(t))v0⟩>0.
Now choose a smooth cutoff functionβ:I→[0,1] such thatβ(t) = 0 for
allt∈Iwith|t−t0| ≥εandβ(t0) = 1. DefineX∈TγΩp,qby
X(t) :=β(t)Π(γ(t))v0 fort∈I.
Then⟨¨γ(t), X(t)⟩ ≥0 for alltand⟨¨γ(t0), X(t0)⟩>0. Hence
dE(γ)X=−
Z
b
a
⟨¨γ(t), X(t)⟩dt <0
and soγdoes not satisfy (i). Thus (i) is equivalent to (iii) and (iv).

4.1. LENGTH AND ENERGY 181
We prove that (i) is equivalent to (ii). Assume first thatγsatisfies (i).
Thenγalso satisfies (iv) and hence ¨γ(t)⊥T
γ(t)Mfor allt∈I. This implies
0 =⟨¨γ(t),˙γ(t)⟩=
1
2
d
dt
|˙γ(t)|
2
.
Hence the functionI→R:t7→ |˙γ(t)|
2
is constant. Choosec≥0 such
that|˙γ(t)| ≡c. Ifc= 0, thenγ(t) is constant and soγ(t)≡p=q. Ifc >0,
then
dL(γ)X=
d
ds

s=0
Z
b
a
|˙γs(t)|dt
=
Z
b
a
∂
∂s

s=0
|˙γs(t)|dt
=
Z
b
a
|˙γ(t)|
−1
ø
˙γ(t),
∂
∂s

s=0
˙γs(t)
Æ
dt
=
1
c
Z
b
a
D
˙γ(t),
˙
X(t)
E
dt
=
1
c
dE(γ)X.
Thus, in the casec >0,γis an extremal point ofEif and only if it is an
extremal point ofL. Hence (i) is equivalent to (ii).
We prove that (iii) is equivalent to (v). Let (Φ, γ, γ
′
) be a development
ofMalongM
′
=R
m
. Then
˙γ
′
(t) = Φ(t) ˙γ(t),
d
dt
Φ(t)X(t) = Φ(t)∇X(t)
for allX∈Vect(γ) and allt∈I. TakeX= ˙γto obtain ¨γ
′
(t) = Φ(t)∇˙γ(t) for
allt∈I. Thus∇˙γ≡0 if and only if ¨γ
′
≡0. This proves Theorem 4.1.4.
Remark 4.1.3 shows that reparametrization by a nondecreasing surjective
mapσ:I
′
→Igives rise to map
Ωp,q(I)→Ωp,q(I
′
) :γ7→γ◦σ
which preserves the length functional, i.e.
L(γ◦σ) =L(γ)
for allγ∈Ωp,q(I). Thus the chain rule in infinite dimensions should assert
that ifγ◦σis an extremal (i.e. critical) point ofL, thenγis an extremal point

182 CHAPTER 4. GEODESICS
ofL. Moreover, ifσis a diffeomorphism, the mapγ7→γ◦σis bijective and
should give rise to a bijective correspondence between the extremal points
ofLon Ωp,q(I) and those on Ωp,q(I
′
). Finally, if the tangent vector field ˙γ
vanishes nowhere, thenγcan be parametrized by the arclength. This is
spelled out in more detail in the next exercise.
Exercise 4.1.8.Letγ:I= [a, b]→Mbe a smooth curve such that
˙γ(t)̸= 0
for allt∈Iand define
T:=L(γ) =
Z
b
a
|˙γ(t)|dt.
(i)Prove that there exists a unique diffeomorphismσ: [0, T]→Isuch that
σ(t
′
) =t ⇐⇒ t
′
=
Z
t
a
|˙γ(s)|ds
for allt
′
∈[0, T] and allt∈[a, b]. Prove thatγ
′
:=γ◦σ: [0, T]→Mis
parametrized by the arclength, i.e.|˙γ
′
(t
′
)|= 1 for allt
′
∈[0, T].
(ii)Prove that
dL(γ)X=−
Z
b
a
⟨
˙
V(t), X(t)⟩dt, V(t) :=|˙γ(t)|
−1
˙γ(t). (4.1.11)
Hint:See the relevant formula in the proof of Theorem 4.1.4.
(iii)Prove thatγis an extremal point ofLif and only if the curveγ
′
in
part (i) is a geodesic.
(iv)Prove thatγis an extremal point ofLif and only if there exists a
geodesicγ
′
:I
′
→Mand a diffeomorphismσ:I
′
→Isuch thatγ
′
=γ◦σ.
Next we generalize this exercise to cover the case where ˙γis allowed to
vanish. Recall from Remark 4.1.6 that the functions7→L(γs) need not be
differentiable. As an example consider the case whereγ=γ0is constant
(see also Exercise 4.4.12 below).
Exercise 4.1.9.Letγ:I→Mbe a smooth curve and letX∈TγΩp,q(I).
Choose a smooth curve of curvesR→Ωp,q(I) :s7→γsthat satisfies (4.1.7).
Prove that the one-sided derivatives of the functions7→L(γs) exist ats= 0
and satisfy the inequalities
−
Z
I

˙
X(t)

dt≤
d
ds
L(γs)

s=0
≤
Z
I

˙
X(t)

dt.
Exercise 4.1.10.Let (Φ, γ, γ
′
) be a development ofMalongM
′
. Show
thatγis a geodesic inMif and only ifγ
′
is a geodesic inM
′
.

4.2. DISTANCE 183
4.2 Distance
Assume thatM⊂R
n
is a connected smoothm-dimensional submanifold.
Two pointsp, q∈Mare of distance|p−q|apart in the ambient Euclidean
spaceR
n
. In this section we define a distance function which is more in-
timately tied toMby minimizing the length functional over the space of
curves inMwith fixed endpoints. Thus it may happen that two points
inMhave a very short distance inR
n
but can only be joined by very long
curves inM(see Figure 4.1). This leads to theintrinsic distance inM.
Throughout we denote byI= [0,1] the unit interval and, forp, q∈M, by
Ωp,q:={γ: [0,1]→M|γis smooth andγ(0) =p, γ(1) =q}(4.2.1)
the space of smooth paths on the unit interval joiningptoq. SinceMis
connected the set Ωp,qis nonempty for allp, q∈M. (Prove this!)M
p
q
p q
Figure 4.1: Curves inM.
Definition 4.2.1.Theintrinsic distancebetween two pointsp, q∈Mis
the real numberd(p, q)≥0defined by
d(p, q) := inf
γ∈Ωp,q
L(γ). (4.2.2)
The inequalityd(p, q)≥0holds because each curve has nonnegative length
and the inequalityd(p, q)<∞holds becauseΩp,q̸=∅.
Remark 4.2.2.Every smooth curveγ: [0,1]→R
n
with endpointsγ(0) =p
andγ(1) =qsatisfies the inequality
L(γ) =
Z
1
0
|˙γ(t)|dt≥

Z
1
0
˙γ(t)dt

=|p−q|.
Thusd(p, q)≥ |p−q|. Forγ(t) :=p+t(q−p) we have equality and hence
the straight lines minimize the length among all curves fromptoq.

184 CHAPTER 4. GEODESICS
Lemma 4.2.3.The functiond:M×M→[0,∞)defines a metric onM:
(i)Ifp, q∈Msatisfyd(p, q) = 0, thenp=q.
(ii)For allp, q∈Mwe haved(p, q) =d(q, p).
(iii)For allp, q, r∈Mwe haved(p, r)≤d(p, q) +d(q, r).
Proof.By Remark 4.2.2 we haved(p, q)≥ |p−q|for allp, q∈Mand this
proves part (i). Part (ii) follows from the fact that the curveeγ(t) :=γ(1−t)
has the same length asγand belongs to Ωq,pwheneverγ∈Ωp,q. To prove
part (iii) fix a constantε >0 and choose curvesγ0∈Ωp,qandγ1∈Ωq,r
such thatL(γ0)< d(p, q) +εandL(γ1)< d(q, r) +ε. By Remark 4.1.3 we
may assume without loss of generality thatγ0(1−t) =γ1(t) =qfort >0
sufficiently small. Under this assumption the curve
γ(t) :=
æ
γ0(2t), for 0≤t≤1/2,
γ1(2t−1),for 1/2< t≤1
is smooth. Moreover,γ(0) =pandγ(1) =rand soγ∈Ωp,r. Thus
d(p, r)≤L(γ) =L(γ0) +L(γ1)< d(p, q) +d(q, r) + 2ε.
Henced(p, r)< d(p, q) +d(q, r) + 2εfor everyε >0. This proves part (iii)
and Lemma 4.2.3.
Remark 4.2.4.It is natural to ask if the infimum in (4.2.2) is always
attained. This is easily seen not to be the case in general. For example,
letMresult from the Euclidean spaceR
m
by removing a pointp0. Then
the distanced(p, q) =|p−q|is equal to the length of the line segment fromp
toqand any other curve from p to q is longer. Hence ifp0is in the interior
of this line segment, the infimum is not attained. We shall prove below that
the infimum is attained wheneverMis complete.q
p
Figure 4.2: A geodesic on the 2-sphere.

4.2. DISTANCE 185
Example 4.2.5.Let
M:=S
2
=
Φ
p∈R
3
| |p|= 1

be the unit sphere inR
3
and fix two pointsp, q∈S
2
. Thend(p, q) is the
length of the shortest curve on the 2-sphere connectingpandq. Such a
curve is a segment on a great circle throughpandq(see Figure 4.2) and its
length is
d(p, q) = cos
−1
(⟨p, q⟩), (4.2.3)
where⟨p, q⟩denotes the standard inner product, and we have
0≤d(p, q)≤π.
(See Example 4.3.11 below for details.) By Lemma 4.2.3 this defines a metric
onS
2
.Exercise:Prove directly that (4.2.3) is a distance function onS
2
.
We now have two topologies on our manifoldM⊂R
n
, namely the topol-
ogy determined by the metricdin Lemma 4.2.3 and the relative topology
inherited fromR
n
. The latter is also determined by a distance function,
namely theextrinsic distance functiondefined as the restriction of the Eu-
clidean distance function onR
n
to the subsetM. We denote it by
d0:M×M→[0,∞), d 0(p, q) :=|p−q|. (4.2.4)
A natural question is if these two metricsdandd0induce the same topology
onM. In other words is a subsetU⊂Mopen with respect tod0if and only
if it is open with respect tod? Or, equivalently, does a sequencepν∈M
converge top0∈Mwith respect todif and only if it converges top0with
respect tod0? Lemma 4.2.7 answers this question in the affirmative.
Exercise 4.2.6.Prove that every translation ofR
n
and every orthogonal
transformation preserves the lengths of curves.
Lemma 4.2.7.For everyp0∈Mwe have
lim
p,q→p0
d(p, q)
|p−q|
= 1.
Lemma 4.2.8.Letp0∈Mand letϕ0:U0→Ω0be a coordinate chart onto
an open subset ofR
m
such that its derivativedϕ0(p0) :Tp0
M→R
m
is an
orthogonal transformation. Then
lim
p,q→p0
d(p, q)
|ϕ0(p)−ϕ0(q)|
= 1.

186 CHAPTER 4. GEODESICS
The proofs will be given below. The lemmas imply that the topologyM
inherits as a subset ofR
m
, the topology onMdetermined by the metricd,
and the topology onMinduced by the local coordinate systems onMare
all the same.
Corollary 4.2.9.For every subsetU⊂Mthe following are equivalent.
(i)Uis open with respect to the metricdin(4.2.2).
(ii)Uis open with respect to the metricd0in(4.2.4).
(iii)For every coordinate chartϕ0:U0→Ω0ofMonto an open sub-
setΩ0⊂R
m
the setϕ0(U0∩U)is an open subset ofR
m
.
Proof.By Remark 4.2.2 we have
|p−q| ≤d(p, q) (4.2.5)
for allp, q∈M. Thus the identity idM: (M, d)→(M, d0) is Lipschitz
continuous with Lipschitz constant one and so everyd0-open subset ofM
isd-open. Conversely, letU⊂Mbe ad-open subset ofMand letp0∈U
andε >0. Then, by Lemma 4.2.7, there exists a constantδ >0 such that
allp, q∈Mwith|p−p0|< δand|q−p0|< δsatisfy
d(p, q)≤(1 +ε)|p−q|.
SinceUisd-open, there exists a constantρ >0 such that
Bρ(p0, d)⊂U.
With
ρ0:= min
æ
δ,
ρ
1 +ε
œ
this impliesBρ0
(p0, d0)⊂U. Namely, ifp∈Msatisfies
|p−p0|< ρ0≤δ,
then
d(p, p0)≤(1 +ε)|p−p0|<(1 +ε)ρ0≤ρ
and sop∈U. ThusUisd0-open and this proves that (i) is equivalent to (ii).
That (ii) implies (iii) follows from the fact that each coordinate chartϕ0
is a homeomorphism. To prove that (iii) implies (i), we argue indirectly
and assume thatUis notd-open. Then there exists a sequencepν∈M\U
that converges to an elementp0∈U. Letϕ0:U0→Ω0be a coordinate
chart withp0∈U0. Then limν→∞|ϕ0(pν)−ϕ0(p0)|= 0 by Lemma 4.2.8.
Thusϕ0(U0∩U) is not open and soUdoes not satisfy (iii). This proves
Corollary 4.2.9.

4.2. DISTANCE 187T M
p
0
⊥
M
T M
p
0
p
0
Figure 4.3: Locally,Mis the graph off.
Proof of Lemma 4.2.7.By Remark 4.2.2 the estimate|p−q| ≤d(p, q) holds
for allp, q∈M. The lemma asserts that, for allp0∈Mand allε >0, there
exists ad0-open neighborhoodU0⊂Mofp0such that allp, q∈U0satisfy
|p−q| ≤d(p, q)≤(1 +ε)|p−q|. (4.2.6)
Letp0∈Mandε >0, and definex:R
n
→Tp0
Mandy:R
n
→Tp0
M
⊥
by
x(p) := Π(p0)(p−p0), y(p) := (1l−Π(p0)) (p−p0),
where Π(p0) :R
n
→Tp0
Mdenotes the orthogonal projection as usual.
Then the derivative of the mapx|M:M→Tp0
Matp=p0is the iden-
tity onTp0
M. Hence the Inverse Function Theorem 2.2.17 asserts that
the mapx|M:M→Tp0
Mis locally invertible nearp0. Extending this
inverse to a smooth map fromTp0
MtoR
n
and composing it with the
mapy:M→Tp0
M
⊥
, we obtain a smooth map
f:Tp0
M→Tp0
M
⊥
and an open neighborhoodW⊂R
n
ofp0such that
p∈M ⇐⇒ y(p) =f(x(p))
for allp∈W(see Figure 4.3). Moreover, by definition the mapfsatisfies
f(0) = 0∈Tp0
M
⊥
, df(0) = 0 :Tp0
M→Tp0
M
⊥
.
Hence there exists a constantδ >0 such that, for everyx∈Tp0
M, we have
|x|< δ=⇒ x+f(x)∈Wand∥df(x)∥= sup
0̸=bx∈Tp
0
M
|df(x)bx|
|bx|
< ε.

188 CHAPTER 4. GEODESICS
Define
U0:={p∈M∩W| |x(p)|< δ}.
Givenp, q∈U0letγ: [0,1]→Mbe the curve whose projection to thex-axis
is the straight line joiningx(p) tox(q), i.e.
x(γ(t)) =x(p) +t(x(q)−x(p)) =:x(t),
y(γ(t)) =f(x(γ(t))) =f(x(t)) =:y(t).
Thenγ(t)∈U0for allt∈[0,1] and
L(γ) =
Z
t
0
|˙x(t) + ˙y(t)|dt
=
Z
t
0
|˙x(t) +df(x(t)) ˙x(t)|dt
≤
Z
t
0
Γ
1 +∥df(x(t))∥
∆
|˙x(t)|dt
≤(1 +ε)
Z
t
0
|˙x(t)|dt
= (1 +ε)|x(p)−x(q)|
= (1 +ε)|Π(p0)(p−q)|
≤(1 +ε)|p−q|.
Henced(p, q)≤L(γ)≤(1 +ε)|p−q|and this proves Lemma 4.2.7.
Proof of Lemma 4.2.8.By assumption we have
|dϕ0(p0)v|=|v|
for allv∈Tp0
M. Fix a constantε >0. Then, by continuity of the derivative,
there exists ad0-open neighborhoodM0⊂Mofp0such that for allp∈M0
and allv∈TpMwe have
(1−ε)|dϕ0(p)v| ≤ |v| ≤(1 +ε)|dϕ0(p)v|.
Thus for every curveγ: [0,1]→M0we have
(1−ε)L(ϕ0◦γ))≤L(γ)≤(1 +ε)L(ϕ0◦γ).
One is tempted to take the infimum over all curvesγ: [0,1]→M0joining
two pintsp, q∈M0to obtain the inequality
(1−ε)|ϕ0(p)−ϕ0(q)| ≤d(p, q)≤(1 +ε)|ϕ0(p)−ϕ0(q)|. (4.2.7)
However, we must justify these inequalities by showing that the infimum
over all curves inM0agrees with the infimum over all curves inMjoining
the pointspandq.

4.2. DISTANCE 189
It suffices to show that the inequalities hold on a smaller heighbor-
hoodM1⊂M0ofp0. Choose such a smaller neighborhoodM1such that the
open setϕ0(M1) is a convex subset of Ω0. Then the right inequality in (4.2.7)
follows by taking the curveγ: [0,1]→M1fromγ(0) =ptoγ(1) =qsuch
thatϕ0◦γ: [0,1]→ϕ0(M1) is a straight line. To prove the left inequality
in (4.2.7) we use the fact thatM0isd-open by Lemma 4.2.7. Hence, after
shrinkingM1if necessary, there exists a constantr >0 such that
p0∈M1⊂Br(p0, d)⊂B3r(p0, d)⊂M0.
Then, forp, q∈M1we haved(p, q)≤2rwhileL(γ)≥4rfor any curveγ
fromptoqwhich leavesM0. Hence the distanced(p, q) ofp, q∈M1is the
infimum of the lengthsL(γ) over all curvesγ: [0,1]→M0that joinγ(0) =p
toγ(1) =q. This proves the left inequality in (4.2.7) and Lemma 4.2.8.
A next question one might ask is the following. Can we choose a coor-
dinate chartϕ:U→Ω onMwith values in an open set Ω⊂R
m
so that
the length of each smooth curveγ: [0,1]→Uis equal to the length of the
curvec:=ϕ◦γ: [0,1]→Ω? We examine this question by considering the
inverse mapψ:=ϕ
−1
: Ω→U. Denote the components ofxandψ(x) by
x= (x
1
, . . . , x
m
)∈Ω, ψ(x) = (ψ
1
(x), . . . , ψ
n
(x))∈U.
Given a smooth curve [0,1]→Ω :t7→c(t) = (c
1
(t), . . . , c
m
(t)) we can write
the length of the compositionγ=ψ◦c: [0,1]→Min the form
L(ψ◦c) =
Z
1
0

d
dt
ψ(c(t))

dt
=
Z
1
0
v
u
u
t
n
X
ν=1
`
d
dt
ψ
ν
(c(t))
´
2
dt
=
Z
1
0
v
u
u
t
n
X
ν=1

m
X
i=1
∂ψ
ν
∂x
i
(c(t))˙c
i
(t)
!
2
dt
=
Z
1
0
v
u
u
t
n
X
ν=1
m
X
i,j=1
∂ψ
ν
∂x
i
(c(t))
∂ψ
ν
∂x
j
(c(t))˙c
i
(t)˙c
j
(t)dt
=
Z
1
0
v
u
u
t
m
X
i,j=1
˙c
i
(t)gij(c(t))˙c
j
(t)dt.

190 CHAPTER 4. GEODESICS
Here the functionsgij: Ω→Rare defined by
gij(x) :=
n
X
ν=1
∂ψ
ν
∂x
i
(x)
∂ψ
ν
∂x
j
(x) =
ø
∂ψ
∂x
i
(x),
∂ψ
∂x
j
(x)
Æ
. (4.2.8)
Thus we have a smooth functiong= (gij) : Ω→R
m×m
with values in the
positive definite matrices given byg(x) =dψ(x)
T
dψ(x) such that
L(ψ◦c) =
Z
1
0
q
˙c(t)
T
g(c(t))˙c(t)dt (4.2.9)
for every smooth curvec: [0,1]→Ω. Thus the conditionL(ψ◦c) =L(c)
for every such curve is equivalent to
gij(x) =δij
for allx∈Ω or, equivalently,
dψ(x)
T
dψ(x) = 1l. (4.2.10)
This means thatψpreserves angles and areas. The next example shows that
forM=S
2
it is impossible to find such coordinates.Aβ
α
γ
Figure 4.4: A spherical triangle.
Example 4.2.10.Consider the manifoldM=S
2
. If there is a diffeomor-
phismψ: Ω→Ufrom an open set Ω⊂R
2
onto an open setU⊂S
2
that
satisfies (4.2.10), it has to map straight lines onto arcs of great circles and
it preserves the area. However, the areaAof a spherical triangle bounded
by three arcs on great circles satisfies the angle sum formula
α+β+γ=π+A.
(See Figure 4.4.) Hence there can be no such mapψ.

4.3. THE EXPONENTIAL MAP 191
4.3 The Exponential Map
Geodesics give rise to a flow on the tangent bundle, thegeodesic flow. It is
generated by a vector field on the tangent bundle, called thegeodesic spray.
The time-1-map of the geodesic flow gives rise to theexponential map.
4.3.1 Geodesic Spray
The tangent bundleT Mis a smooth 2m-dimensional manifold inR
n
×R
n
by Corollary 2.6.12. The next lemma characterizes the tangent bundle of
the tangent bundle. Compare this with Lemma 3.4.5.
Lemma 4.3.1.The tangent space ofT Mat(p, v)∈T Mis given by
T
(p,v)T M=
æ
(bp,bv)∈R
n
×R
n

bp∈TpMand
Γ
1l−Π(p)
∆
bv=hp(bp, v)
œ
.(4.3.1)
Proof.We prove the inclusion “⊂” in (4.3.1). Let (bp,bv)∈T
(p,v)T Mand
choose a smooth curveR→T M:t7→(γ(t), X(t)) such that
γ(0) =p, X(0) =v, ˙γ(0) =bp,
˙
X(0) =bv.
Then
˙
X=∇X+hγ( ˙γ, X) by the Gauß–Weingarten formula (3.2.2) and
hence (1l−Π(γ(t)))
˙
X(t) =h
γ(t)( ˙γ(t), X(t)) for allt∈R. Taket= 0 to
obtain (1l−Π(p))bv=hp(bp, v). This proves the inclusion “⊂” in (4.3.1).
Equality holds because both sides of the equation are 2m-dimensional linear
subspaces ofR
n
×R
n
.
By Lemma 4.3.1 a smooth mapS= (S1, S2) :T M→R
n
×R
n
is a vector
field onT Mif and only if
S1(p, v)∈TpM, (1l−Π(p))S2(p, v) =hp(S1(p, v), v)
for all (p, v)∈T M. A special case is whereS1(p, v) =v. Such vector fields
correspond to second order differential equations onM.
Definition 4.3.2(Spray).A vector fieldS∈Vect(T M)is called aspray
iff it has the formS(p, v) = (v, S2(p, v))whereS2:T M→R
n
is a smooth
map satisfying
(1l−Π(p))S2(p, v) =hp(v, v), S 2(p, λv) =λ
2
S2(p, v) (4.3.2)
for all(p, v)∈T Mandλ∈R. The vector fieldS∈Vect(T M)defined by
S(p, v) := (v, hp(v, v))∈T
(p,v)T M (4.3.3)
forp∈Mandv∈TpMis called thegeodesic spray.

192 CHAPTER 4. GEODESICS
4.3.2 The Exponential Map
Lemma 4.3.3.Letγ:I→Mbe a smooth curve on an open intervalI⊂R.
Thenγis a geodesic if and only if the curveI→T M:t7→(γ(t),˙γ(t))is
an integral curve of the geodesic spraySin(4.3.3).
Proof.A smooth curveI→T M:t7→(γ(t), X(t)) is an integral curve ofS
if and only if ˙γ(t) =X(t) and
˙
X(t) =h
γ(t)(X(t), X(t)) for allt∈I. By
equation (4.1.5), this holds if and only ifγis a geodesic and ˙γ=X.
Combining Lemma 4.3.3 with Theorem 2.4.7 we obtain the following
existence and uniqueness result for geodesics.
Lemma 4.3.4.LetM⊂R
n
be anm-dimensional submanifold.
(i)For everyp∈Mand everyv∈TpMthere is anε >0and a smooth
curveγ: (−ε, ε)→Msuch that
∇˙γ≡0, γ(0) =p, ˙γ(0) =v. (4.3.4)
(iI)Ifγ1:I1→Mandγ2:I2→Mare geodesics andt0∈I1∩I2with
γ1(t0) =γ2(t0), ˙γ1(t0) = ˙γ2(t0),
thenγ1(t) =γ2(t)for allt∈I1∩I2.
Proof.Lemma 4.3.3 and Theorem 2.4.7.
Definition 4.3.5(Exponential map).Forp∈Mandv∈TpMthe in-
terval
Ip,v:=
[
æ
I⊂R

Iis an open interval containing0and there is a
geodesicγ:I→Msatisfyingγ(0) =p,˙γ(0) =v
œ
.
is called themaximal existence intervalfor the geodesic throughpin the
directionv. Forp∈Mdefine the setVp⊂TpMby
Vp:={v∈TpM|1∈Ip,v}. (4.3.5)
Theexponential mapatpis the mapexp
p:Vp→Mthat assigns to every
tangent vectorv∈Vpthe pointexp
p(v) :=γ(1), whereγ:Ip,v→Mis the
unique geodesic satisfyingγ(0) =pand˙γ(0) =v.

4.3. THE EXPONENTIAL MAP 193p
M
Figure 4.5: The exponential map.
Lemma 4.3.6. The set
V:={(p, v)|p∈M, v∈Vp} ⊂T M
is open and the mapV→M: (p, v)7→exp
p(v)is smooth.
(ii)Ifp∈Mandv∈Vp, then
Ip,v={t∈R|tv∈Vp}
and the geodesicγ:Ip,v→Mwithγ(0) =pand˙γ(0) =vis given by
γ(t) = exp
p(tv), t∈Ip,v.
Proof.Part (i) follows directly from Lemma 4.3.3 and Theorem 2.4.9. To
prove part (ii), fix an elementp∈Mand a tangent vectorv∈Vp, and
letγ:Ip,v→Mbe the unique geodesic withγ(0) =pand ˙γ(0) =v. Fix a
nonzero real numberλand define the mapγλ:λ
−1
Ip,v→Mby
γλ(t) :=γ(λt) fort∈λ
−1
Ip,v.
Then ˙γλ(t) =λ˙γ(λt) ans ¨γλ(t) =λ
2
¨γ(λt) and hence
∇˙γλ(t) = Π(γλ(t))¨γλ(t) =λ
2
Π(γ(λt))¨γ(λt) =λ
2
∇˙γ(λt) = 0
for everyt∈λ
−1
Ip,v. This shows thatγλis a geodesic with
γλ(0) =p, ˙γλ(0) =λv.
In particular, we haveλ
−1
Ip,v⊂Ip,λv. Interchanging the roles ofvandλv
we obtainλ
−1
Ip,v=Ip,λv. Thus
λ∈Ip,v ⇐⇒ 1∈Ip,λv ⇐⇒ λv∈Vp
and
γ(λ) =γλ(1) = exp
p(λv)
forλ∈Ip,v. This proves Lemma 4.3.6.

194 CHAPTER 4. GEODESICS
Since exp
p(0) =pby definition, the derivative of the exponential map
atv= 0 is a linear map fromTpMto itself. This derivative is the identity
map as illustrated in Figure 4.5 and proved in the following corollary.
Corollary 4.3.7.The mapexp
p:Vp→Mis smooth and its derivative at
the origin isdexp
p(0) = id :TpM→TpM.
Proof.The setVpis an open subset of the linear subspaceTpM⊂R
n
,
with respect to the relative topology, and hence is a manifold. The tan-
gent space ofVpat each point isTpM. By Lemma 4.3.6 the exponential
map exp
p:Vp→Mis smooth and its derivative at the origin is given by
dexp
p(0)v=
d
dt

t=0
exp
p(tv) = ˙γ(0) =v,
whereγ:Ip,v→Mis once again the unique geodesic throughpin the
directionv. This proves Corollary 4.3.7.
Corollary 4.3.8.Letp∈Mand, forr >0, denote
Br(p) :={v∈TpM| |v|< r}.
Ifr >0is sufficiently small, thenBr(p)⊂Vp, the set
Ur(p) := exp
p(Br(p))
is an open subset ofM, and the restriction of the exponential map toBr(p)
is a diffeomorphism fromBr(p)toUr(p).
Proof.This follows directly from Corollary 4.3.7 and Theorem 2.2.17.
Definition 4.3.9(Injectivity radius).LetM⊂R
n
be a smoothm-
manifold. Theinjectivity radius ofMatpis the supremum of all real
numbersr >0such thatBr(p)⊂Vpand the restriction of the exponential
mapexp
ptoBr(p)is a diffeomorphism onto its image
Ur(p) := exp
p(Br(p)).
It will be denoted by
inj(p) := inj(p;M) := sup



r >0

Br(p)⊂Vpand
exp
p:Br(p)→Ur(p)
is a diffeomorphism



.
Theinjectivity radius ofMis the infimum of the injectivity radii ofM
atpover allp∈M. It will be denoted by
inj(M) := inf
p∈M
inj(p;M).

4.3. THE EXPONENTIAL MAP 195
4.3.3 Examples and Exercises
Example 4.3.10.The exponential map onR
m
is given by
exp
p(v) =p+v forp, v∈R
m
.
For everyp∈R
m
this map is a diffeomorphism fromTpR
m
=R
m
toR
m
and
hence the injectivity radius ofR
m
is infinity.
Example 4.3.11.The exponential map onS
m
is given by
exp
p(v) = cos(|v|)p+
sin(|v|)
|v|
v
for everyp∈S
m
and every nonzero tangent vectorv∈TpS
m
=p
⊥
. The re-
striction of this map to the open ball of radiusrinTpMis a diffeomorphism
onto its image if and only ifr≤π. Hence the injectivity radius ofS
m
at every point isπ.Exercise:Givenp∈S
m
and 0̸=v∈TpS
m
=p
⊥
,
prove that the geodesicγ:R→S
m
withγ(0) =pand ˙γ(0) =vis given
byγ(t) = cos(t|v|)p+
sin(t|v|)
|v|
vfort∈R. Show that in the case 0≤ |v| ≤π
there is no shorter curve inS
m
connectingpandq:=γ(1) and deduce that
the intrinsic distance onS
m
is given byd(p, q) = cos
−1
(⟨p, q⟩) forp, q∈S
m
(see Example 4.2.5 form= 2).
Example 4.3.12.Consider the orthogonal group O(n)⊂R
n×n
with the
standard inner product⟨v, w⟩:= trace(v
T
w) onR
n×n
. The orthogonal
projection Π(g) :R
n×n
→TgO(n) is given by
Π(g)v:=
1
2
ı
v−gv
T
g
ȷ
and the second fundamental form by
hg(v, v) =−gv
T
v.
Hence a curveγ:R→O(n) is a geodesic if and only ifγ
T
¨γ+ ˙γ
T
˙γ= 0 or,
equivalently,γ
T
˙γis constant. This shows that geodesics in O(n) have the
formγ(t) =gexp(tξ) forg∈O(n) andξ∈o(n). It follows that
exp
g(v) =gexp(g
−1
v) = exp(vg
−1
)g
forg∈O(n) andv∈TgO(n). In particular, forg= 1l the exponential
map exp
1l:o(n)→O(n) agrees with the exponential matrix.
Exercise 4.3.13.What is the injectivity radius of the 2-torusT
2
=S
1
×S
1
,
the punctured 2-planeR
2
\ {(0,0)}, and the orthogonal group O(n)?

196 CHAPTER 4. GEODESICS
Geodesics in Local Coordinates
Lemma 4.3.14.LetM⊂R
n
be anm-dimensional manifold and choose a
coordinate chartϕ:U→Ωwith inverse
ψ:=ϕ
−1
: Ω→U.
LetΓ
k
ij
: Ω→Rbe the Christoffel symbols defined by(3.6.6)and letc:I→Ω
be a smooth curve. Then the curve
γ:=ψ◦c:I→M
is a geodesic if and only ifcsatisfies the 2nd order differential equation
¨c
k
+
m
X
i,j=1
Γ
k
ij(c)˙c
i
˙c
j
= 0 (4.3.6)
fork= 1, . . . , m.
Proof.This follows immediately from the definition of geodesics and equa-
tion (3.6.7) in Lemma 3.6.1 withX= ˙γandξ= ˙c.
We remark that Lemma 4.3.14 gives rise to another proof of Lemma 4.3.4
that is based on the existence and uniqueness of solutions of second order
differential equations in local coordinates.
Exercise 4.3.15.Let Ω⊂R
m
be an open set andg= (gij) : Ω→R
m×m
be a smooth map with values in the space of positive definite symmetric
matrices. Consider the energy functional
E(c) :=
Z
1
0
L(c(t),˙c(t))dt
on the space of pathsc: [0,1]→Ω, whereL: Ω×R
m
→Ris defined by
L(x, ξ) :=
1
2
m
X
i,j=1
ξ
i
gij(x)ξ
j
. (4.3.7)
TheEuler–Lagrange equationsof this variational problem have the form
d
dt
∂L
∂ξ
k
(c(t),˙c(t)) =
∂L
∂x
k
(c(t),˙c(t)), k= 1, . . . , m.(4.3.8)
Prove that the Euler–Lagrange equations (4.3.8) are equivalent to the geo-
desic equations (4.3.6), where the Γ
k
ij
: Ω→Rare given by (3.6.10).

4.4. MINIMAL GEODESICS 197
4.4 Minimal Geodesics
Any straight line segment in Euclidean space is the shortest curve joining its
endpoints. The analogous assertion for geodesics in a manifoldMis false;
consider for example an arc which is more than half of a great circle on a
sphere. In this section we consider curves which realize the shortest distance
between their endpoints.
4.4.1 Characterization of Minimal Geodesics
Lemma 4.4.1.LetI= [a, b]be a compact interval, letγ:I→Mbe a
smooth curve, and definep:=γ(a)andq:=γ(b). Then the following are
equivalent.
(i)γis parametrized proportional to the arclength, i.e.|˙γ(t)|=cis constant,
andγminimizes the length, i.e.L(γ)≤L(γ
′
)for every smooth curveγ
′
inM
joiningpandq.
(ii)γminimizes the energy, i.e.E(γ)≤E(γ
′
)for every smooth curveγ
′
inMjoiningpandq.
Definition 4.4.2(Minimal geodesic).A smooth curveγ:I→Mon a
compact intervalI⊂Ris called aminimal geodesiciff it satisfies the
equivalent conditions of Lemma 4.4.1.
Remark 4.4.3. Condition (i) says that (the velocity|˙γ|is constant
and)L(γ) =d(p, q), i.e. thatγis a shortest curve fromptoq. It is not
precluded that there be more than one suchγ; consider for example the
case whereMis a sphere andpandqare antipodal.
(ii)Condition (ii) implies that
d
ds

s=0
E(γs) = 0
for every smooth variationR×I→M:s7→γs(t) ofγwith fixed endpoints.
Hence a minimal geodesic is a geodesic.
(iii)Finally, we remark thatL(γ) (but notE(γ)) is independent of the
parametrization ofγ. Hence, ifγis a minimal geodesic, thenL(γ)≤L(γ
′
)
for everyγ
′
(fromptoq) whereasE(γ)≤E(γ
′
) for thoseγ
′
defined on (an
interval the same length as)I.

198 CHAPTER 4. GEODESICS
Proof of Lemma 4.4.1.We prove that (i) implies (ii). Letcbe the (constant)
value of|˙γ(t)|. Then
L(γ) = (b−a)c, E(γ) =
(b−a)c
2
2
.
Then, for every smooth curveγ
′
:I→Mwithγ
′
(a) =pandγ
′
(b) =q, we
have
4E(γ)
2
=c
2
L(γ)
2
≤c
2
L(γ
′
)
2
=c
2
`Z
b
a

˙γ
′
(t)

dt
´2
≤c
2
(b−a)
Z
b
a

˙γ
′
(t)

2
dt
= 2(b−a)c
2
E(γ
′
)
= 4E(γ)E(γ
′
).
Here the fourth step follows from the Cauchy–Schwarz inequality. Now
divide by 4E(γ) to obtainE(γ)≤E(γ
′
).
We prove that (ii) implies (i). We have already shown in Remark 4.4.3
that (ii) implies thatγis a geodesic. It is easy to dispose of the case where
M is one-dimensional. In that case anyγminimizingE(γ) orL(γ) must be
monotonic onto a subarc; otherwise it could be altered so as to make the
integral smaller. Hence supposeMis of dimension at least two. Suppose,
by contradiction, thatL(γ
′
)< L(γ) for some curveγ
′
fromptoq. Since
the dimension ofMis bigger than one, we may approximateγ
′
by a curve
whose tangent vector nowhere vanishes, i.e. we may assume without loss of
generality that ˙γ
′
(t)̸= 0 for allt. Then we can reparametrizeγ
′
proportional
to arclength without changing its length, and by a further transformation
we can make its domain equal toI. Thus we may assume without loss of
generality thatγ
′
:I→Mis a smooth curve withγ
′
(a) =pandγ
′
(b) =q
such that|γ
′
(t)|=c
′
and
(b−a)c
′
=L(γ
′
)< L(γ) = (b−a)c.
This impliesc
′
< cand hence
E(γ
′
) =
(b−a)c
′2
2
<
(b−a)c
2
2
=E(γ).
This contradicts (ii) and proves Lemma 4.4.1.

4.4. MINIMAL GEODESICS 199
4.4.2 Local Existence of Minimal Geodesics
The next theorem asserts the existence of minimal geodesics joining two
points that are sufficiently close to each other. It also shows that the
setUr(p) = exp
p(Br(p)) that was introduced in Definition 4.3.9 is actually
the open ballUr(p) ={q∈M|d(p, q)< r}wheneverr≤inj(p;M).
Theorem 4.4.4(Existence of minimal geodesics).LetM⊂R
n
be a
smoothm-manifold, fix a pointp∈M, and letr >0be smaller than the
injectivity radius ofMatp. Letv∈TpMsuch that|v|< r. Then
d(p, q) =|v|, q:= exp
p(v),
and a curveγ∈Ωp,qhas minimal lengthL(γ) =|v|if and only if there is a
smooth mapβ: [0,1]→[0,1]satisfying
β(0) = 0, β(1) = 1,
˙
β≥0
such thatγ(t) = exp
p(β(t)v)for0≤t≤1.
The proof is based on the following lemma.U
p
r
Figure 4.6: The Gauß Lemma.
Lemma 4.4.5(Gauß Lemma). LetM,p,rbe as in Theorem 4.4.4,
letI⊂Rbe an open interval, and letw:I→Vpbe a smooth curve whose
norm
|w(t)|=:r
is constant. Define
α(s, t) := exp
p(sw(t))
for(s, t)∈R×Iwithsw(t)∈Vp. Then
ø
∂α
∂s
,
∂α
∂t
Æ
≡0.
Thus the geodesics throughpare orthogonal to the boundaries of the embed-
ded ballsUr(p)in Corollary 4.3.8 (see Figure 4.6).

200 CHAPTER 4. GEODESICS
Proof of Lemma 4.4.5.For everyt∈Iwe have
α(0, t) = exp
p(0) =p
and so the assertion holds fors= 0, i.e.
ø
∂α
∂s
(0, t),
∂α
∂t
(0, t)
Æ
= 0.
Moreover, each curves7→α(s, t) is a geodesic, i.e.
∇s
∂α
∂s
= Π(α)
∂
2
α
∂s
2
≡0.
By Theorem 4.1.4, the function
s7→

∂α
∂s
(s, t)

is constant for everyt, so that

∂α
∂s
(s, t)

=

∂α
∂s
(0, t)

=|w(t)|=rfor (s, t)∈R×I.
This implies
∂
∂s
ø
∂α
∂s
,
∂α
∂t
Æ
=
ø
∇s
∂α
∂s
,
∂α
∂t
Æ
+
ø
∂α
∂s
,∇s
∂α
∂t
Æ
=
ø
∂α
∂s
,Π(α)
∂
2
α
∂s∂t
Æ
=
ø
Π(α)
∂α
∂s
,
∂
2
α
∂s∂t
Æ
=
ø
∂α
∂s
,
∂
2
α
∂s∂t
Æ
=
1
2
∂
∂t

∂α
∂s

2
= 0.
Since the function⟨
∂α
∂s
,
∂α
∂t
⟩vanishes fors= 0 we obtain
ø
∂α
∂s
(s, t),
∂α
∂t
(s, t)
Æ
= 0
for allsandt. This proves Lemma 4.4.5.

4.4. MINIMAL GEODESICS 201
Proof of Theorem 4.4.4.Letr >0 be as in Corollary 4.3.8 and letv∈TpM
such that 0<|v|=:ε < r. Denoteq:= exp
p(v) and letγ∈Ωp,q. Assume
first that
γ(t)∈exp
p
Γ
Bε(p)
∆
=Uε ∀t∈[0,1].
Then there is a unique smooth function [0,1]→TpM:t7→v(t) such that
|v(t)| ≤εandγ(t) = exp
p(v(t)) for everyt. The set
I:={t∈[0,1]|γ(t)̸=p}={t∈[0,1]|v(t)̸= 0} ⊂(0,1]
is open in the relative topology of (0,1]. ThusIis a union of open intervals
in (0,1) and one half open interval containing 1. Defineβ: [0,1]→[0,1]
andw:I→TpMby
β(t) :=
|v(t)|
ε
, w(t) :=ε
v(t)
|v(t)|
.
Thenβis continuous, bothβandware smooth onI,
β(0) = 0, β(1) = 1, w(1) =v,
and
|w(t)|=ε, γ(t) = exp
p(β(t)w(t))
for allt∈I. We prove thatL(γ)≥ε. To see this letα: [0,1]×I→Mbe
the map of Lemma 4.4.5, i.e.
α(s, t) := exp
p(sw(t)).
Thenγ(t) =α(β(t), t) and hence
˙γ(t) =
˙
β(t)
∂α
∂s
(β(t), t) +
∂α
∂t
(β(t), t)
for everyt∈I. Hence it follows from Lemma 4.4.5 that
|˙γ(t)|
2
=
˙
β(t)
2

∂α
∂s
(β(t), t)

2
+

∂α
∂t
(β(t), t)

2
≥
˙
β(t)
2
ε
2
for everyt∈I. Hence
L(γ) =
Z
1
0
|˙γ(t)|dt=
Z
I
|˙γ(t)|dt≥ε
Z
I

˙
β(t)

dt≥ε
Z
I
˙
β(t)dt=ε.

202 CHAPTER 4. GEODESICS
Here the last equality follows by applying the fundamental theorem of cal-
culus to each interval inIand using the fact thatβ(0) = 0 andβ(1) = 1.
IfL(γ) =ε, we must have
∂α
∂t
(β(t), t) = 0,
˙
β(t)≥0 for all t∈I.
ThusIis a single half open interval containing 1 and on this interval the
condition
∂α
∂t
(β(t), t) = 0 implies ˙w(t) = 0. Sincew(1) =vwe havew(t) =v
for everyt∈I. Henceγ(t) = exp
p(β(t)v) for everyt∈[0,1]. It follows
thatβis smooth on the closed interval [0,1] (and not just onI). Thus we
have proved that everyγ∈Ωp,qwith values inUεhas lengthL(γ)≥εwith
equality if and only ifγis a reparametrized geodesic. But ifγ∈Ωp,qdoes not
take values only inUε, there must be aT∈(0,1) such thatγ([0, T])⊂Uε
andγ(T)∈∂Uε. ThenL(γ|
[0,T])≥ε, by what we have just proved,
andL(γ|
[T,1])>0 because the restriction ofγto [T,1] cannot be constant;
so in this case we haveL(γ)> ε. This proves Theorem 4.4.4.
The next corollary gives a partial answer to our problem of finding length
minimizing curves. It asserts that geodesics minimize the lengthlocally.
Corollary 4.4.6.LetM⊂R
n
be a smoothm-manifold, letI⊂Rbe an
open interval, and letγ:I→Mbe a geodesic. Fix a pointt0∈I. Then
there exists a constantε >0such that
t0−ε < s < t < t0+ε =⇒ L(γ|
[s,t]) =d(γ(s), γ(t)).
Proof.Sinceγis a geodesic its derivative has constant norm|˙γ(t)| ≡c(see
Theorem 4.1.4). Chooseδ >0 so small that the interval [t0−δ, t0+δ] is
contained inI. Then there is a constantr >0 such thatr≤inj(γ(t))
whenever|t−t0| ≤δ. Chooseε >0 such that
ε < δ,2εc < r.
Ift0−ε < s < t < t0+ε, then
γ(t) = exp
γ(s)((t−s) ˙γ(s))
and
|(t−s) ˙γ(s)|=|t−s|c <2εc < r≤inj(γ(s)).
Hence it follows from Theorem 4.4.4 that
L(γ|
[s,t]) =|t−s|c=d(γ(s), γ(t)).
This proves Corollary 4.4.6.

4.4. MINIMAL GEODESICS 203
4.4.3 Examples and Exercises
Exercise 4.4.7.How large can the constantεin Corollary 4.4.6 be chosen
in the caseM=S
2
? Compare this with the injectivity radius.
Remark 4.4.8.We conclude from Theorem 4.4.4 that
Sr(p) :=
Φ
q∈M

d(p, q) =r

= exp
p
ΓΦ
v∈TpM| |v|=r
Ψ∆
(4.4.1)
for 0< r <inj(p;M). The Gauß Lemma 4.4.5 shows that the geodesic
rays [0,1]→M:s7→exp
p(sv) emanating frompare the orthogonal tra-
jectories to the concentric spheresSr(p).
Exercise 4.4.9.Let
M⊂R
3
be of dimension two and suppose thatMis invariant under the (orthogonal)
reflection about some planeE⊂R
3
. Show thatEintersectsMin a geodesic.
(Hint:Otherwise there would be pointsp, q∈Mvery close to one an-
other joined by two distinct minimal geodesics.) Conclude for example that
the coordinate planes intersect the ellipsoid (x/a)
2
+ (y/b)
2
+ (z/c)
2
= 1 in
geodesics.
Exercise 4.4.10.Choose geodesic normal coordinates nearp∈Mvia
q= exp
p

m
X
i=1
x
i
(q)ei
!
,
wheree1, . . . , emis an orthonormal basis ofTpM(see Corollary 4.5.4 below).
Then we havex
i
(p) = 0 and
Br(p) ={q∈M|d(p, q)< r}=
(
q∈M

m
X
i=1

x
i
(q)

2
< r
2
)
(4.4.2)
for 0< r <inj(p;M). Hence Theorem 4.5.3 below asserts thatBr(p) is
convex forr >0 sufficiently small.
(i)Show that it can happen that a geodesic inBr(p) is not minimal.Hint:
TakeMto be the hemisphere{(x, y, z)∈R
3
|x
2
+y
2
+z
2
= 1, z >0}to-
gether with the disc{(x, y, z)∈R
3
|x
2
+y
2
≤1, z= 0}, but smooth the cor-
ners along the circlex
2
+y
2
= 1, z= 0. Takep= (0,0,1) andr=π/2.
(ii)Show that, ifr >0 is sufficiently small, then the unique geodesicγ
inBr(p) joining two pointsq, q
′
∈Br(p) is minimal and that in fact any
curveγ
′
fromqtoq
′
which is not a reparametrization ofγis strictly longer,
i.e.L(γ
′
)> L(γ) =d(q, q
′
).

204 CHAPTER 4. GEODESICS
Exercise 4.4.11.Letγ:I= [a, b]→Mbe a smooth curve with end-
pointsγ(a) =pandγ(b) =qand nowhere vanishing derivative, i.e. ˙γ(t)̸= 0
for allt∈I. Prove that the following are equivalent.
(i)The curveγis anextremal of the length functional, i.e. every
smooth mapR×I→M: (s, t)7→γs(t) withγs(a) =pandγs(b) =qfor
allssatisfies
d
ds
L(γs)

s=0
= 0.
(ii)The curveγis a reparametrized geodesic, i.e. there exists a smooth
mapσ: [a, b]→[0,1] withσ(a) = 0,σ(b) = 1, ˙σ(t)≥0 for allt∈I, and a
vectorv∈TpMsuch that
q= exp
p(v), γ(t) = exp
p(σ(t)v)
for allt∈I. (We remark that the hypothesis ˙γ(t)̸= 0 implies thatσis
actually a diffeomorphism, i.e. ˙σ(t)>0 for allt∈I.)
(iii)The curveγminimizes the length functional locally, i.e. there ex-
ists anε >0 such thatL(γ|
[s,t]) =d(γ(s), γ(t)) for every closed subinter-
val [s, t]⊂Iof lengtht−s < ε.
It is often convenient to consider curvesγwhere ˙γ(t) is allowed to vanish
for some values oft; thenγcannot (in general) be parametrized by arclength.
Such a curveγ:I→Mcan be smooth (as a map) and yet its image may
have corners (where ˙γnecessarily vanishes). Note that a curve with corners
can never minimize the distance, even locally.
Exercise 4.4.12.Show that conditions (ii) and (iii) in Exercise 4.4.11 are
equivalent, even without the assumption that ˙γis nowhere vanishing. De-
duce that, ifγ:I→Mis a shortest curve joiningptoq, i.e.L(γ) =d(p, q),
thenγis a reparametrized geodesic.
Show by example that one can have a variation{γs}s∈Rof a reparame-
trized geodesicγ0=γfor which the maps7→L(γs) is not even differentiable
ats= 0. (Hint:Takeγto be constant. See also Exercise 4.1.9.)
Show, however, that conditions (i), (ii) and (iii) in Exercise 4.4.11 remain
equivalent if the hypothesis that ˙γis nowhere vanishing is weakened to the
hypothesis that ˙γ(t)̸= 0 for all but finitely manyt∈I. Conclude that a bro-
ken geodesic is a reparametrized geodesic if and only if it minimizes arclength
locally. (Abroken geodesicis a continuous mapγ:I= [a, b]→Mfor
which there exista=t0< t1<· · ·< tn=bsuch thatγ|
[ti−1,ti]is a geodesic
fori= 1, . . . , n. It is thus a geodesic if and only if ˙γis continuous at the
break points, i.e. ˙γ(t
−
i
) = ˙γ(t
+
i
) fori= 1, . . . , n−1.)

4.5. CONVEX NEIGHBORHOODS 205
4.5 Convex Neighborhoods
A subset of an affine space is called convex iff it contains the line segment
joining any two of its points. The definition carries over to a submanifoldM
of Euclidean space (or indeed more generally to any manifoldMequipped
with a spray) once we reword the definition so as to confront the difficulty
that a geodesic joining two points might not exist nor, if it does, need it be
unique.
Definition 4.5.1(Geodesically convex set).LetM⊂R
n
be a smooth
m-dimensional manifold. A subsetU⊂Mis calledgeodesically con-
vexiff, for allp0, p1∈U, there exists a unique geodesicγ: [0,1]→Usuch
thatγ(0) =p0andγ(1) =p1.
It is not precluded in Definition 4.5.1 that there be other geodesics fromp
toqwhich leave and then re-enterU, and these may even be shorter than
the geodesic inU.
Exercise 4.5.2. Find a geodesically convex setUin a manifoldMand
pointsp0, p1∈Usuch that the unique geodesicγ: [0,1]→Uwithγ(0) =p0
andγ(1) =p1has lengthL(γ)> d(p0, p1).Hint:An interval of length
bigger thanπinS
1
.
(b)Find a setUin a manifoldMsuch that any two points inUcan be
joined by a minimal geodesic inU, butUis not geodesically convex.Hint:
A closed hemisphere inS
2
.
Theorem 4.5.3(Convex Neighborhood Theorem). LetM⊂R
n
be a
smoothm-dimensional submanifold and fix a pointp0∈M. Letϕ:U→Ω
be any coordinate chart on an open neighborhoodU⊂Mofp0with values
in an open setΩ⊂R
m
. Then the set
Ur:={p∈U| |ϕ(p)−ϕ(p0)|< r} (4.5.1)
is geodesically convex forr >0sufficiently small.
Before giving the proof of Theorem 4.5.3 we derive a useful corollary.
Corollary 4.5.4.LetM⊂R
n
be a smoothm-manifold and letp0∈M.
Then, forr >0sufficiently small, the open ball
Ur(p0) :={p∈M|d(p0, p)< r} (4.5.2)
is geodesically convex.

206 CHAPTER 4. GEODESICS
Proof.Choose an orthonormal basise1, . . . , emofTp0
Mand define
Ω :={x∈R
m
| |x|<inj(p0;M)},
U:={p∈M|d(p0, p)<inj(p0;M)}.
(4.5.3)
Define the mapψ: Ω→Uby
ψ(x) := exp
p0

m
X
i=1
x
i
ei
!
(4.5.4)
forx= (x
1
, . . . , x
m
)∈Ω. Thenψis a diffeomorphism andd(p0, ψ(x)) =|x|
for allx∈Ω by Theorem 4.4.4. Hence its inverse
ϕ:=ψ
−1
:U→Ω (4.5.5)
satisfiesϕ(p0) = 0 and|ϕ(p)|=d(p0, p) for allp∈U. Thus
Ur(p0) ={p∈U| |ϕ(p)−ϕ(p0)|< r} for 0< r <inj(p0;M)
and so Corollary 4.5.4 follows from Theorem 4.5.3.
Definition 4.5.5(Geodesically normal coordinates).The coordinate
chartϕ:U→Ωin(4.5.4)and(4.5.5)sends geodesics throughp0to straight
lines through the origin. Its componentsx
1
, . . . , x
m
:U→Rare calledgeo-
desically normal coordinatesatp0.
Proof of Theorem 4.5.3.Assume without loss of generality thatϕ(p0) = 0.
Let Γ
k
ij
: Ω→Rbe the Christoffel symbols of the coordinate chart and,
forx∈Ω, define the quadratic functionQx:R
m
→Rby
Qx(ξ) :=
m
X
k=1
ı
ξ
k
ȷ
2
−
m
X
i,j,k=1
x
k
Γ
k
ij(x)ξ
i
ξ
j
.
ShrinkingU, if necessary, we may assume that
max
i,j=1,...,m

m
X
k=1
x
k
Γ
k
ij(x)

≤
1
2m
for allx∈Ω.
Then, for allx∈Ω and allξ∈R
m
we have
Qx(ξ)≥ |ξ|
2
−
1
2m

m
X
i=1

ξ
i

!
2
≥
1
2
|ξ|
2
≥0.
HenceQxis positive definite for everyx∈Ω.

4.5. CONVEX NEIGHBORHOODS 207
Now letγ: [0,1]→Ube a geodesic and define
c(t) :=ϕ(γ(t))
for 0≤t≤1. Then, by Lemma 4.3.14,csatisfies the differential equation
¨c
k
+
X
i,j
Γ
k
ij(c)˙c
i
˙c
j
= 0.
Hence
d
2
dt
2
|c|
2
2
=
d
dt
⟨˙c, c⟩=|˙c|
2
+⟨¨c, c⟩=Qc(˙c)≥
|˙c|
2
2
≥0
and so the functiont7→ |ϕ(γ(t))|
2
is convex. Thus, ifγ(0), γ(1)∈Urfor
somer >0, it follows thatγ(t)∈Urfor allt∈[0,1].
Consider the exponential map
V={(p, v)∈T M|v∈Vp} →M: (p, v)7→exp
p(v)
in Lemma 4.3.6. Its domainVis open and the exponential map is smooth.
Since it sends the pair (p0,0)∈Vto exp
p0
(0) =p0∈U, it follows from con-
tinuity that there exist constantsε >0 andr >0 such that
p∈Ur, v∈TpM,|v|< ε =⇒ v∈Vp,exp
p(v)∈U.(4.5.6)
Moreover, we have
dexp
p0
(0) = id :Tp0
M→Tp0
M
by Corollary 4.3.7. Hence the Implicit Function Theorem 2.6.15 asserts that
the constantsε >0 andr >0 can be chosen such that (4.5.6) holds and there
exists a smooth maph:Ur×Ur→R
n
that satisfies the conditions
h(p, q)∈TpM, |h(p, q)|< ε (4.5.7)
for allp, q∈Urand
exp
p(v) =q ⇐⇒ v=h(p, q) (4.5.8)
for allp, q∈Urand allv∈TpMwith|v|< ε. In particular, we have
h(p0, p0) = 0
and exp
p(h(p, q)) =qfor allp, q∈Ur.

208 CHAPTER 4. GEODESICS
Fix two constantsε >0 andr >0 and a smooth maph:Ur×Ur→R
n
such that (4.5.6), (4.5.7), (4.5.8) are satisfied. We show that any two
pointsp, q∈Urare joined by a geodesic inUr. Letp, q∈Urand define
γ(t) := exp
p(th(p, q)) for 0≤t≤1.
This curveγ: [0,1]→Mis well defined by (4.5.6) and (4.5.7), it is a
geodesic satisfyingγ(0) =p∈Urby Lemma 4.3.6, it satisfiesγ(1) =q∈Ur
by (4.5.8), it takes values inUby (4.5.6) and (4.5.7), and soγ([0,1])⊂Ur
because the function [0,1]→R:t7→ |ϕ(γ(t))|
2
is convex.
We show that there exists at most one geodesic inUrjoiningpandq.
Letp, q∈Urand letγ: [0,1]→Urbe any geodesic such thatγ(0) =p
andγ(1) =q. Definev:= ˙γ(0)∈TpM. Thenγ(t) = exp
p(tv) for 0≤t≤1
by Lemma 4.3.6. We claim that|v|< ε. Suppose, by contradiction, that
|v| ≥ε.
Then
T:=
ε
|v|
≤1
and, for 0< t < T, we have|tv|< εand exp
p(tv) =γ(t)∈Urand so
h(p, γ(t)) =tv.
by (4.5.8). Thus
|h(p, γ(t))|=t|v|for 0< t < T.
Take the limitt↗Tto obtain
|h(p, γ(T))|=T|v|=ε
in contradiction to (4.5.7). This contradiction shows that|v|< ε. Since
exp
p(v) =γ(1) =q∈Ur
it follows from (4.5.8) thatv=h(p, q). This proves Theorem 4.5.3.
Remark 4.5.6.Theorem 4.5.3 and its proof carry over to general sprays
(see Definition 4.3.2).
Exercise 4.5.7.Consider the setUr(p) ={q∈M|d(p, q)< r}forp∈M
andr >0. Corollary 4.5.4 asserts that this set is geodesically convex forr
sufficiently small. How large can you chooserin the cases
M=S
2
, M =T
2
=S
1
×S
1
, M =R
2
, M =R
2
\ {0}.
Compare this with the injectivity radius. If the setUr(p) in these exam-
ples is geodesically convex, does it follow that every geodesic inUr(p) is
minimizing?

4.6. COMPLETENESS AND HOPF–RINOW 209
4.6 Completeness and Hopf–Rinow
For a Riemannian manifold there are different notions of completeness.
First, in§3.4 completeness was defined in terms of the completeness of time
dependent basic vector fields on the frame bundle (Definition 3.4.10). Sec-
ond, there is a distance function
d:M×M→[0,∞)
defined by equation (4.2.2) so that we can speak of completeness of the
metric space (M, d) in the sense that every Cauchy sequence converges.
Third, there is the question of whether geodesics through any point in any
direction exist for all time; if so we call a Riemannian manifold geodesically
complete. The remarkable fact is that these three rather different notions of
completeness are actually equivalent and that, in the complete case, any two
points inMcan be joined by a shortest geodesic. This is the content of the
Hopf–Rinow theorem. We will spell out the details of the proof for embedded
manifolds and leave it to the reader (as a straight forward exercise) to extend
the proof to the intrinsic setting.
Geodesic Completeness
Definition 4.6.1(Geodesically complete manifold).LetM⊂R
n
be an
m-dimensional manifold. Given a pointp∈Mwe say thatMis geodesi-
cally complete atpiff, for every tangent vectorv∈TpM, there exists a
geodesicγ:R→M(on the entire real axis) satisfyingγ(0) =pand˙γ(0) =v
(or equivalentlyVp=TpMwhereVp⊂TpMis defined by(4.3.5)). The man-
ifoldMis calledgeodesically completeiff it is geodesically complete at
every pointp∈M.
Definition 4.6.2.Let(M, d)be a metric space. A subsetA⊂Mis called
boundediff
sup
p∈A
d(p, p0)<∞
for some (and hence every) pointp0∈M.
Example 4.6.3.A manifoldM⊂R
n
can be contained in a bounded subset
ofR
n
and still not be bounded with respect to the metric (4.2.2). An
example is the 1-manifoldM=
Φ
(x, y)∈R
2
|0< x <1, y= sin(1/x)

.
Exercise 4.6.4.Let (M, d) be a metric space. Prove that every compact
subsetK⊂Mis closed and bounded. Find an example of a metric space
that contains a closed and bounded subset that is not compact.

210 CHAPTER 4. GEODESICS
Theorem 4.6.5(Completeness).LetM⊂R
n
be a connectedm-dimen-
sional manifold and letd:M×M→[0,∞)be the distance function defined
by(4.1.1),(4.2.1), and(4.2.2). Then the following are equivalent.
(i)Mis geodesically complete.
(ii)There exists a pointp∈Msuch thatMis geodesically complete atp.
(iii)Every closed and bounded subset ofMis compact.
(iv)(M, d)is a complete metric space.
(v)Mis complete, i.e. for every smooth curveξ:R→R
m
and every ele-
ment(p0, e0)∈ F(M)there exists a smooth curveβ:R→ F(M)satisfying
˙
β(t) =B
ξ(t)(β(t)), β(0) = (p0, e0). (4.6.1)
(vi)The basic vector fieldBξ∈Vect(F(M))is complete for everyξ∈R
m
.
(vii)For every smooth curveγ
′
:R→R
m
, everyp0∈M, and every or-
thogonal isomorphismΦ0:Tp0
M→R
m
there exists a development(Φ, γ, γ
′
)
ofMalongR
m
on all ofRthat satisfiesγ(0) =p0andΦ(0) = Φ0.
Proof.The proof relies on Theorem 4.6.6 below.
Global Existence of Minimal Geodesics
Theorem 4.6.6(Hopf–Rinow).LetM⊂R
n
be a connectedm-manifold
and letp∈M. AssumeMis geodesically complete atp. Then, for ev-
eryq∈M, there exists a geodesicγ: [0,1]→Msuch that
γ(0) =p, γ(1) =q, L(γ) =d(p, q).
Before giving the proof of the Hopf–Rinow Theorem we show that it
implies Theorem 4.6.5.
Theorem 4.6.6 implies Theorem 4.6.5.That (i) implies (ii) follows directly
from the definitions.
We prove that (ii) implies (iii). Thus assume thatMis geodesically
complete at the pointp0∈Mand letK⊂Mbe a closed and bounded
subset. Thenr:= sup
q∈Kd(p0, q)<∞. Hence Theorem 4.6.6 asserts that,
for everyq∈K, there exists a vectorv∈Tp0
Msuch that|v|=d(p0, q)≤r
and exp
p0
(v) =q. Thus
K⊂exp
p0
(Br(p0)),Br(p0) ={v∈Tp0
M| |v| ≤r}.
ThenB:={v∈Tp0
M| |v| ≤r,exp
p0
(v)∈K}is a closed and bounded sub-
set of the Euclidean spaceTp0
M. HenceBis compact andK= exp
p0
(B).
Since the exponential map exp
p0
:Tp0
M→Mis continuous it follows thatK
is compact. This shows that (ii) implies (iii).

4.6. COMPLETENESS AND HOPF–RINOW 211
We prove that (iii) implies (iv). Thus assume that every closed and
bounded subset ofMis compact and choose a Cauchy sequencepi∈M.
Choosei0∈Nsuch thatd(pi, pj)≤1 for alli, j∈Nwithi, j≥i0. Define
c:= max
1≤i≤i0
d(p1, pi) + 1.
Thend(p1, pi)≤d(p1, pi0
) +d(pi0
, pi)≤d(p1, pi0
) + 1≤cfor alli≥i0and
sod(p1, pi)≤cfor alli∈N. Hence the set{pi|i∈N}is bounded and so
is its closure. By (iii) this implies that the sequencepihas a convergent
subsequence. Sincepiis a Cauchy sequence, this implies thatpiconverges.
Thus we have proved that (iii) implies (iv).
We prove that (iv) implies (v). Fix a smooth curveξ:R→R
m
and
an element (p0, e0)∈ F(M). Assume, by contradiction, that there exists
a real numberT >0 such that there exists a solutionβ: [0, T)→ F(M)
of equation (4.6.1) that cannot be extended to the interval [0, T+ε) for
anyε >0. Writeβ(t) =: (γ(t), e(t)) so thatγandesatisfy the equations
˙γ(t) =e(t)ξ(t),˙e(t) =h
γ(t)( ˙γ(t))e(t), γ(0) =p0, e(0) =e0.
This impliese(t)η∈T
γ(t)Mand ˙e(t)η∈T
⊥
γ(t)
Mfor allη∈R
m
and therefore
d
dt
⟨η, e(t)
T
e(t)ζ⟩=
d
dt
⟨e(t)η, e(t)ζ⟩=⟨˙e(t)η, e(t)ζ⟩+⟨e(t)η,˙e(t)ζ⟩= 0
for allη, ζ∈R
m
and allt∈[0, T). Thus the functiont7→e(t)
T
e(t) is con-
stant, hence
e(t)
T
e(t) =e
T
0e0,∥e(t)∥= sup
0̸=η∈R
m
|e(t)η|
|η|
=∥e0∥ (4.6.2)
for 0≤t < T, hence
|˙γ(t)|=|e(t)ξ(t)| ≤ ∥e0∥ |ξ(t)| ≤ ∥e0∥sup
0≤s≤T
|ξ(s)|=:cT
and sod(γ(s), γ(t))≤L(γ|
[s,t])≤(t−s)cTfor 0≤s < t < T. Since (M, d)
is a complete metric space, this shows that the limitp1:= limt↗Tγ(t)∈M
exists. Thus the setK:=γ([0, T))∪ {p1} ⊂Mis compact and so is the set
e
K:=
n
(p, e)∈ F(M)|p∈K, e
T
e=e
T
0e0
o
⊂ F(M).
By equation (4.6.2) the curve [0, T)→R× F(M) :t7→(t, γ(t), e(t)) takes
values in the compact set [0, T]×
e
Kand is the integral curve of a vector field
on the manifoldR× F(M). Hence Corollary 2.4.15 asserts that [0, T) cannot
be the maximal existence interval of this integral curve, a contradiction. This
shows that (iv) implies (v).

212 CHAPTER 4. GEODESICS
That (v) implies (vi) follows by takingξ(t)≡ξin (v).
We prove that (vi) implies (i). Fix an elementp0∈Mand a tan-
gent vectorv0∈Tp0
M. Lete0∈ Liso(R
m
, Tp0
M) be any isomorphism and
chooseξ∈R
m
such thate0ξ=v0. By (vi) the vector fieldBξhas a unique
integral curveR→ F(M) :t7→β(t) = (γ(t), e(t)) with
β(0) = (p0, e0).
Thus
˙γ(t) =e(t)ξ, ˙e(t) =h
γ(t)(e(t)ξ)e(t),
and hence
¨γ(t) = ˙e(t)ξ=h
γ(t)(e(t)ξ)e(t)ξ=h
γ(t)( ˙γ(t),˙γ(t)).
By the Gauß–Weingarten formula, this implies∇˙γ(t) = 0 for everytand
henceγ:R→Mis a geodesic withγ(0) =p0and ˙γ(0) =e0ξ=v0. ThusM
is geodesically complete and this shows that (vi) implies (i).
The equivalence of (v) and (vii) was established in Corollary 3.5.25 and
this shows that Theorem 4.6.6 implies Theorem 4.6.5.
Proof of the Hopf–Rinow Theorem
The proof of Theorem 4.6.6 relies on the next two lemmas.
Lemma 4.6.7.LetM⊂R
n
be a connectedm-manifold andp∈M. Sup-
poseε >0is smaller than the injectivity radius ofMatpand denote
Σ1(p) :={v∈TpM| |v|= 1}, S ε(p) :=
Φ
p
′
∈M|d(p, p
′
) =ε

.
Then the mapΣ1(p)→Sε(p) :v7→exp
p(εv)is a diffeomorphism and, for
allq∈M, we have
d(p, q)> ε =⇒ d(Sε(p), q) =d(p, q)−ε.
Proof.By Theorem 4.4.4, we have
d(p,exp
p(v)) =|v|for allv∈TpMwith|v| ≤ε
and
d(p, p
′
)> ε for allp
′
∈M\
Φ
exp
p(v)|v∈TpM,|v| ≤ε

.
This shows thatSε(p) = exp
p(εΣ1(p)) and, sinceεis smaller than the injec-
tivity radius, the map
Σ1(p)→Sε(p) :v7→exp
p(εv)
is a diffeomorphism.

4.6. COMPLETENESS AND HOPF–RINOW 213
To prove the second assertion, letq∈Msuch that
r:=d(p, q)> ε.
Fix a constantδ >0 and choose a smooth curveγ: [0,1]→Msuch that
γ(0) =p, γ(1) =q, L(γ)≤r+δ.
Chooset0>0 such thatγ(t0) is the last point of the curve onSε(p), i.e.
γ(t0)∈Sε(p), γ(t)/∈Sε(p) fort0< t≤1.
Then
d(γ(t0), q)≤L(γ|
[t0,1])
=L(γ)−L(γ|
[0,t0])
≤L(γ)−ε
≤r+δ−ε.
This shows thatd(Sε(p), q)≤r+δ−εfor everyδ >0 and therefore
d(Sε(p), q)≤r−ε.
Moreover,
d(p
′
, q)≥d(p, q)−d(p, p
′
) =r−ε
for allp
′
∈Sε(p). Thus
d(Sε(p), q) =r−ε
and this proves Lemma 4.6.7.
Lemma 4.6.8(Curve Shortening Lemma).LetM⊂R
n
be anm-mani-
fold, letp∈M, and letεbe a real number such that
0< ε <inj(p;M).
Then, for allv, w∈TpM, we have
|v|=|w|=ε, d(exp
p(v),exp
p(w)) = 2ε =⇒ v+w= 0.

214 CHAPTER 4. GEODESICSw
v
Figure 4.7: Two unit tangent vectors.
Proof.We will prove that, for allv, w∈TpM, we have
lim
δ→0
d(exp
p(δv),exp
p(δw))
δ
=|v−w|. (4.6.3)
Assume this holds and suppose, by contradiction, that there exist two tan-
gent vectorsv, w∈TpMsuch that
|v|=|w|= 1, d(exp
p(εv),exp
p(εw)) = 2ε, v+w̸= 0.
Then
|v−w|<2
(see Figure 4.7). Thus by (4.6.3) there exists a constant 0< δ < εsuch that
d(exp
p(δv),exp
p(δw))<2δ.
Then
d(exp
p(εv),exp
p(εw))
≤d(exp
p(εv),exp
p(δv)) +d(exp
p(δv),exp
p(δw)) +d(exp
p(δw),exp
p(εw))
< ε−δ+ 2δ+ε−δ= 2ε
and this contradicts our assumption.
It remains to prove (4.6.3). For this we observe that
lim
δ→0
d(exp
p(δv),exp
p(δw))
δ
= lim
δ→0
d(exp
p(δv),exp
p(δw))

exp
p(δv)−exp
p(δw)

exp
p(δv)−exp
p(δw)

δ
= lim
δ→0

exp
p(δv)−exp
p(δw)

δ
= lim
δ→0

exp
p(δv)−p
δ
−
exp
p(δw)−p
δ

=|v−w|.
Here the second equality follows from Lemma 4.2.7.

4.6. COMPLETENESS AND HOPF–RINOW 215
Proof of Theorem 4.6.6.By assumptionM⊂R
n
is a connected submani-
fold, andp∈Mis given such that the exponential map exp
p:TpM→Mis
defined on the entire tangent space atp. Fix a pointq∈M\ {p}so that
0< r:=d(p, q)<∞.
Choose a constantε >0 smaller than the injectivity radius ofMatpand
smaller thanr. Then, by Lemma 4.6.7, we have
d(Sε(p), q) =r−ε.
Hence there exists a tangent vectorv∈TpMsuch that
d(exp
p(εv), q) =r−ε,|v|= 1.
Define the curveγ: [0, r]→Mby
γ(t) := exp
p(tv) for 0≤t≤r.
By Lemma 4.3.6, this is a geodesic and it satisfiesγ(0) =p. We must
prove thatγ(r) =qandL(γ) =d(p, q). Instead we will prove the follow-
ing stronger statement.
Claim.For everyt∈[0, r]we have
d(γ(t), q) =r−t.
In particular,γ(r) =qandL(γ) =r=d(p, q).
Consider the subset
I:={t∈[0, r]|d(γ(t), q) =r−t} ⊂[0, r].
This set is nonempty, becauseε∈I, it is obviously closed, and
t∈I =⇒ [0, t]⊂I. (4.6.4)
Namely, ift∈Iand 0≤s≤t, then
d(γ(s), q)≤d(γ(s), γ(t)) +d(γ(t), q)≤t−s+r−t=r−s
and
d(γ(s), q)≥d(p, q)−d(p, γ(s))≥r−s.
Henced(γ(s), q) =r−sand hences∈I. This proves (4.6.4).

216 CHAPTER 4. GEODESICS
We prove thatIis open (in the relative topology of [0, r]). Lett∈I
be given witht < r. Choose a constantε >0 smaller than the injectivity
radius ofMatγ(t) and smaller thanr−t. Then, by Lemma 4.6.7 withp
replaced byγ(t), we have
d(Sε(γ(t)), q) =r−t−ε.
Next we choosew∈T
γ(t)Msuch that
|w|= 1, d(exp
γ(t)(εw), q) =r−t−ε.
Then
d(γ(t−ε),exp
γ(t)(εw))≥d(γ(t−ε), q)−d(exp
γ(t)(εw), q)
= (r−t+ε)−(r−t−ε)
= 2ε.
The converse inequality is obvious, because both points have distanceε
toγ(t) (see Figure 4.8).γ
ε
ε
exp ( w)ε
(t)
p
q
γ
γ(t)
S ( (t))
ε
ε
r−t−
Figure 4.8: The proof of the Hopf–Rinow theorem.
Thus we have proved that
d(γ(t−ε),exp
γ(t)(εw)) = 2ε.
Since
γ(t−ε) = exp
γ(t)(−ε˙γ(t)),
it follows from Lemma 4.6.8 that
w= ˙γ(t).
Hence exp
γ(t)(sw) =γ(t+s) and this implies that
d(γ(t+ε), q) =r−t−ε.
Thust+ε∈Iand, by (4.6.4), we have [0, t+ε]∈I. Thus we have proved
thatIis open. In other words, I is a nonempty subset of [0, r] which is
both open and closed, and henceI= [0, r]. This proves the claim and
Theorem 4.6.6.

4.7. GEODESICS IN THE INTRINSIC SETTING* 217
4.7 Geodesics in the Intrinsic Setting*
This section examines the distance function on a Riemannian manifold,
shows how the results of this chapter extend to the intrinsic setting, and
discusses several examples.
4.7.1 Intrinsic Distance
LetMbe a connected smooth manifold (§2.8) equipped with a Riemannian
metric (§3.7). Then we can define the length of a curveγ: [0,1]→M
by the formula (4.1.1) and it is invariant under reparametrization as in
Remark 4.1.3. Thedistance functiond:M×M→Ris then given by
the same formula (4.2.2). We prove that it still defines a metric onMand
that this metric induces the same topology as the smooth structure.
Lemma 4.7.1.LetMbe a connected smooth Riemannian manifold and
define the functiond:M×M→[0,∞)by(4.1.1),(4.2.1), and(4.2.2).
Thendis a metric and induces the same topology as the smooth structure.
Proof.The proof has three steps.
Step 1.Fix a pointp0∈Mand letϕ:U→Ωbe a coordinate chart ofM
onto an open subsetΩ⊂R
m
such thatp0∈U. Then there exists an open
neighborhoodV⊂Uofp0and constantsδ, r >0such that
δ|ϕ(p)−ϕ(p0)| ≤d(p, p0)≤δ
−1
|ϕ(p)−ϕ(p0)| (4.7.1)
for everyp∈Vandd(p, p0)≥δrfor everyp∈M\V.
Denote the inverse of the coordinate chartϕbyψ:=ϕ
−1
: Ω→Mand
define the mapg= (gij)
m
i,j=1
: Ω→R
m×m
by
gij(x) :=
ø
∂ψ
∂x
i
(x),
∂ψ
∂x
j
(x)
Æ
ψ(x)
forx∈Ω. Then a smooth curveγ: [0,1]→Uhas the length
L(γ) =
Z
1
0
q
˙c(t)
T
g(c(t))˙c(t)dt, c(t) :=ϕ(γ(t)). (4.7.2)
Letx0:=ϕ(p0)∈Ω and chooser >0 such thatBr(x0)⊂Ω. Then there is
a constantδ∈(0,1] such that
δ|ξ| ≤
q
ξ
T
g(x)ξ≤δ
−1
|ξ| (4.7.3)
for allx∈Br(x0) andξ, η∈R
m
. DefineV:=ϕ
−1
(Br(x0))⊂U.

218 CHAPTER 4. GEODESICS
Now letp∈Vand denotex:=ϕ(p)∈Br(x0). Then, for every smooth
curveγ: [0,1]→Vwithγ(0) =p0andγ(1) =p, the curve
c:=ϕ◦γ
takes values inBr(x0) and satisfiesc(0) =x0andc(1) =x. Hence, by (4.7.2)
and (4.7.3), we have
L(γ)≥δ
Z
1
0
|˙c(t)|dt≥δ

Z
1
0
˙c(t)dt

=δ|x−x0|.
Ifγ: [0,1]→Mis a smooth curve with endpointsγ(0) =p0andγ(1) =p
whose image is not entirely contained inV, then there exists aT∈(0,1]
such thatγ(t)∈Vfor 0≤t < Tandγ(T)∈∂V, soc(t) =ϕ(γ(t))∈Br(x0)
for 0≤t < Tand|c(T)−x0|=r. Hence, by the above argument, we have
L(γ)≥δr.
This shows thatd(p0, p)≥δrforp∈M\Vandd(p0, p)≥δ|ϕ(p)−ϕ(p0)|
forp∈V. Ifp∈V,x:=ϕ(p), andc(t) :=x0+t(x−x0), thenγ:=ψ◦cis a
smooth curve inVwithγ(0) =p0andγ(1) =pand, by (4.7.2) and (4.7.3),
L(γ)≤δ
−1
Z
1
0
|˙c(t)|dt=δ
−1
|x−x0|.
This proves Step 1.
Step 2.dis a distance function.
Step 1 shows thatd(p, p0)>0 for everyp∈M\ {p0}and hencedsatisfies
condition (i) in Lemma 4.2.3. The proofs of (ii) and (iii) remain unchanged
in the intrinsic setting and this proves Step 2.
Step 3.The topology onMinduced bydagrees with the topology induced
by the smooth structure.
Assume first thatW⊂Mis open with respect to the manifold topology
and letp0∈W. Letϕ:U→Ω be a coordinate chart ofMonto an open
subset Ω⊂R
m
such thatp0∈U, and chooseV⊂Uandδ, ras in Step 1.
Thenϕ(V∩W) is an open subset of Ω containing the pointϕ(p0). Hence
there exists a constant 0< ε≤δrsuch thatB
δ
−1
ε(ϕ(p0))⊂ϕ(V∩W).
Thus by Step 1 we haved(p, p0)≥δr≥εfor allp∈M\V. Hence, ifp∈M
satisfiesd(p, p0)< ε, thenp∈V, so|ϕ(p)−ϕ(p0)|< δ
−1
d(p, p0)< δ
−1
ε
by (4.7.1), and thereforeϕ(p)∈ϕ(V∩W). ThusBε(p0;d)⊂Wand this
shows thatWis open with respect tod.

4.7. GEODESICS IN THE INTRINSIC SETTING* 219
Conversely, assume thatW⊂Mis open with respect todand choose
a coordinate chartϕ:U→Ω onto an open set Ω⊂R
m
. We must prove
thatϕ(W∩U) is an open subset of Ω. To see this, choosex0∈ϕ(W∩U)
and letp0:=ϕ
−1
(x0)∈W∩U. Now chooseV⊂Uandδ, ras in Step 1.
Chooseε >0 such thatB
δ
−1
ε(p0;d)⊂WandBε(x0)⊂ϕ(V). Letx∈R
n
such that|x−x0|< ε. Thenx∈ϕ(V) and thereforep:=ϕ
−1
(x)∈V. This
impliesd(p, p0)< δ
−1
|ϕ(p)−ϕ(p0)|=δ
−1
|x−x0|< δ
−1
ε, thusp∈W∩U,
and sox=ϕ(p)∈ϕ(W∩U). Thusϕ(W∩U) is open, and soWis open in
the manifold topology ofM. This proves Step 3 and Lemma 4.7.1.
4.7.2 Geodesics and the Levi-Civita Connection
With the covariant derivative understood (Theorem 3.7.8), we can define
geodesics onMas smooth curvesγ:I→Mthat satisfy the equation
∇˙γ= 0, as in Definition 4.1.5. Then all the above results about geodesics,
as well as their proofs, carry over almost verbatim to the intrinsic setting. In
particular, geodesics are in local coordinates described by equation (4.3.6)
(Lemma 4.3.14) and they are the critical points of the energy functional
E(γ) :=
1
2
Z
1
0
|˙γ(t)|
2
dt
on the space Ωp,qof all pathsγ: [0,1]→Mwith fixed endpointsγ(0) =p
andγ(1) =q. Here we use the fact that Lemma 4.1.7 extends to the in-
trinsic setting via the Embedding Theorem 2.9.12. So for every vector
fieldX∈Vect(γ) alongγwithX(0) = 0 andX(1) = 0 there exists a curve
of curvesR→Ωp,q:s7→γswithγ0=γand∂sγs|s=0=X. Then, by the
properties of the Levi-Civita connection, we have
dE(γ)X=
1
2
Z
1
0
∂s|∂tγs(t)|
2
dt
=
Z
1
0
⟨˙γ(t),∇tX(t)⟩dt
=−
Z
1
0
⟨∇t˙γ(t), X(t)⟩dt.
The right hand side vanishes for allXif and only if∇˙γ≡0 (Theorem 4.1.4).
With this understood, we find that, for allp∈Mandv∈TpM, there exists a
unique geodesicγ:Ip,v→Mon a maximal open intervalIp,v⊂Rcontaining
zero that satisfiesγ(0) =pand ˙γ(0) =v(Lemma 4.3.4).

220 CHAPTER 4. GEODESICS
This gives rise to a smooth exponential map
exp
p:Vp={v∈TpM|1∈Ip,v} →M
as in§4.3 which satisfies
dexp
p(0) = id :TpM→TpM
as in Corollary 4.3.7. This leads directly to the injectivity radius, the Gauß
Lemma 4.4.5, the local length minimizing property of geodesics in Theo-
rem 4.4.4, and the Convex Neighborhood Theorem 4.5.3. Also the proof of
the equivalence of metric and geodesic completeness in Theorem 4.6.5 and of
the Hopf–Rinow Theorem 4.6.6 carry over verbatim to the intrinsic setting
of general Riemannian manifolds. The only place where some care must be
taken is in the proof of the Curve Shortening Lemma 4.6.8 as is spelled out
in Exercise 4.7.2 below.
4.7.3 Examples and Exercises
Exercise 4.7.2.Choose a coordinate chartϕ:U→Ω withϕ(p0) = 0 such
that the metric in local coordinates satisfies
gij(0) =δij.
Refine the estimate (4.7.1) in the proof of Lemma 4.7.1 and show that
lim
p,q→p0
d(p, q)
|ϕ(p)−ϕ(q)|
= 1.
This is the intrinsic analogue of Lemma 4.2.8. Use this to prove that equa-
tion (4.6.3) continues to hold for all Riemannian manifolds, i.e.
lim
δ→0
d(exp
p(δv),exp
p(δw))
δ
=|v−w|
forp∈Mandv, w∈TpM. With this understood, the proof of theCurve
Shortening Lemma4.6.8 carries over verbatim to the intrinsic setting.
Exercise 4.7.3.The real projective spaceRP
n
inherits a Riemannian met-
ric fromS
n
as it is a quotient ofS
n
by an isometric involution. Prove that
each geodesic inS
n
with its standard metric descends to a geodesic inRP
n
.

4.7. GEODESICS IN THE INTRINSIC SETTING* 221
Exercise 4.7.4.Letf:S
3
→S
2
be theHopf fibrationdefined by
f(z, w) =
ı
|z|
2
− |w|
2
,2Re ¯zw,2Im ¯zw
ȷ
Prove that the image of a great circle inS
3
is a nonconstant geodesic inS
2
if and only if it is orthogonal to the fibers off, which are also great circles.
Here we identifyS
3
with the unit sphere inC
2
. (See also Exercise 2.5.22.)
Exercise 4.7.5.Prove that a nonconstant geodesicγ:R→S
2n+1
de-
scends to a nonconstant geodesic inCP
n
with the Fubini–Study metric (see
Example 3.7.5) if and only if ˙γ(t)⊥Cγ(t) for everyt∈R.
Exercise 4.7.6.Consider the manifold
Fk(R
n
) :=
n
D∈R
n×k

D
T
D= 1l
o
of orthonormalk-frames inR
n
, equipped with the Riemannian metric inher-
ited from the standard inner product⟨X, Y⟩:= trace(X
T
Y) on the space of
realn×k-matrices.
(a)Prove that
TDFk(R
n
) =
n
X∈R
n×k

D
T
X+X
T
D= 0
o
,
TDFk(R
n
)
⊥
=
n
DA

A=A
T
∈R
k×k
o
.
and that the orthogonal projection Π(D) :R
n×k
→TDFk(R
n
) is given by
Π(D)X=X−
1
2
D
Γ
D
T
X+X
T
D
∆
.
(b)Prove that the second fundamental form ofFk(R
n
) is given by
hD(X)Y=−
1
2
D
Γ
X
T
Y+Y
T
X
∆
forD∈ Fk(R
n
) andX, Y∈TDFk(R
n
).
(c)Prove that a smooth mapR→ Fk(R
n
) :t7→D(t) is a geodesic if and
only if it satisfies the differential equation
¨
D=−D
˙
D
T˙
D. (4.7.4)
Prove that the functionD
T˙
Dis constant for every geodesic inFk(R
n
). Com-
pare this with Example 4.3.12.

222 CHAPTER 4. GEODESICS
Exercise 4.7.7.Let Gk(R
n
) =Fk(R
n
)/O(k) be the real Grassmannian of
k-dimensional subspaces inR
n
, equipped with a Riemannian metric as in
Example 3.7.6. Prove that a geodesicsR→ Fk(R
n
) :t7→D(t) descends
to a nonconstant geodesic in Gk(R
n
) if and only ifD
T˙
D≡0 and
˙
D̸≡0.
Deduce that the exponential map on Gk(R
n
) is given by
exp
Λ(
b
Λ) = im
`
Dcos
`
ı
b
D
Tb
D
ȷ
1/2
´
+
b
D
ı
b
D
Tb
D
ȷ
−1/2
sin
`
ı
b
D
Tb
D
ȷ
1/2
´´
for Λ∈ Fk(R
n
) and
b
Λ∈TΛFk(R
n
)\ {0}. Here we identify the tangent
spaceTΛFk(R
n
) with the space of linear maps from Λ to Λ
⊥
, and choose
the matricesD∈ Fk(R
n
) and
b
D∈R
n×k
such that
Λ = imD, D
Tb
D= 0,
b
Λ◦D=
b
D:R
k
→Λ
⊥
= kerD
T
.
Prove that the group O(n) acts on Gk(R
n
) by isometries. Which subgroup
acts trivially?
Exercise 4.7.8.Carry over Exercises 4.7.6 and 4.7.7 to the complex Grass-
mannian Gk(C
n
). Prove that the group U(n) acts on Gk(C
n
) by isometries.
Which subgroup acts trivially?

Chapter 5
Curvature
This chapter begins by introducing the notion of an isometry (§5.1). It
shows that isometries of embedded manifolds preserve the lengths of curves
and can be characterized as diffeomorphisms whose derivatives preserve the
inner products. The chapter then moves on to the Riemann curvature tensor
and establishes its symmetry properties (§5.2). That section also includes
a discussion of the covariant derivative of a global vector field. The next
section is devoted to the generalized Gauß Theorema Egregium which as-
serts that isometries preserve geodesics, the covariant derivative, and the
Riemann curvature tensor (§5.3). The final section examines the Riemann
curvature tensor in local coordinates and shows how the definitions and re-
sults of the present chapter carry over to the intrinsic setting of Riemannian
manifolds (§5.4).
5.1 Isometries
LetMandM
′
be connected submanifolds ofR
n
. An isometry is an isomor-
phism of the intrinsic geometries ofMandM
′
. Recall the definition of the
intrinsic distance function
d:M×M→[0,∞)
in§4.2 by
d(p, q) := inf
γ∈Ωp,q
L(γ), L(γ) =
Z
1
0
|˙γ(t)|dt
forp, q∈M. Letd
′
denote the intrinisic distance function onM
′
.
223

224 CHAPTER 5. CURVATURE
Theorem 5.1.1(Isometries).Letϕ:M→M
′
be a bijective map. Then
the following are equivalent.
(i)ϕintertwines the distance functions onMandM
′
, i.e.
d
′
(ϕ(p), ϕ(q)) =d(p, q)
for allp, q∈M.
(ii)ϕis a diffeomorphism and
dϕ(p) :TpM→T
ϕ(p)M
′
is an orthogonal isomorphism for everyp∈M.
(iii)ϕis a diffeomorphism and
L(ϕ◦γ) =L(γ)
for every smooth curveγ: [a, b]→M.
The bijectionϕis called anisometryiff it satisfies these equivalent condi-
tions. In the caseM=M
′
the isometriesϕ:M→Mform a group denoted
byI(M)and called theisometry groupofM.
The proof is based on the following lemma.
Lemma 5.1.2.For everyp∈Mthere exists a constantε >0such that,
for allv, w∈TpMwith0<|w|<|v|< ε, we have
d(exp
p(w),exp
p(v)) =|v| − |w| =⇒ w=
|w|
|v|
v. (5.1.1)
Remark 5.1.3.It follows from the triangle inequality and Theorem 4.4.4
that
d(exp
p(v),exp
p(w))≥d(exp
p(v), p)−d(exp
p(w), p)
=|v| − |w|
whenever 0<|w|<|v|<inj(p). Lemma 5.1.2 asserts that equality can
only hold whenwis a positive multiple ofvor, to put it differently, that the
distance between exp
p(v) and exp
p(w) must be strictly bigger that|v| − |w|
wheneverwis not a positive multiple ofv.

5.1. ISOMETRIES 225
Proof of Lemma 5.1.2.As in Corollary 4.3.8 we denote
Bε(p) :={v∈TpM| |v|< ε},
Uε(p) :={q∈M|d(p, q)< ε}.
By Theorem 4.4.4 and the definition of the injectivity radius, the exponential
map atpis a diffeomorphism exp
p:Bε(p)→Uε(p) forε <inj(p). Choose
0< r <inj(p). Then the closure ofUr(p) is a compact subset ofM. Hence
there is a constantε >0 such thatε < randε <inj(p
′
) for everyp
′
∈Ur(p).
Sinceε < rwe have
ε <inj(p
′
)∀p
′
∈Uε(p). (5.1.2)
Thus exp
p
′:Bε(p
′
)→Uε(p
′
) is a diffeomorphism for everyp
′
∈Uε(p).
Definep1:= exp
p(w) andp2:= exp
p(v). Then, by assumption, we have
d(p1, p2) =|v|−|w|< ε.Sincep1∈Uε(p) it follows from our choice ofεthat
ε <inj(p1). Hence there is a unique tangent vectorv1∈Tp1
Msuch that
|v1|=d(p1, p2) =|v| − |w|,exp
p1
(v1) =p2.
Following first the shortest geodesic fromptop1and then the shortest
geodesic fromp1top2we obtain (after suitable reparametrization) a smooth
curveγ: [0,2]→Msuch that
γ(0) =p, γ(1) =p1, γ(2) =p2,
and
L(γ|
[0,1]) =d(p, p1) =|w|, L(γ|
[1,2]) =d(p1, p2) =|v| − |w|.
ThusL(γ) =|v|=d(p, p2). Hence, by Theorem 4.4.4, there is a smooth
functionβ: [0,2]→[0,1] satisfying
β(0) = 0, β(2) = 1,
˙
β(t)≥0, γ(t) = exp
p(β(t)v)
for everyt∈[0,2]. This implies
exp
p(w) =p1=γ(1) = exp
p(β(1)v),0≤β(1)≤1.
Sincewandβ(1)vare both elements ofBε(p) and exp
pis injective onBε(p),
this impliesw=β(1)v. Sinceβ(1)≥0 we haveβ(1) =|w|/|v|. This
proves (5.1.1) and Lemma 5.1.2.

226 CHAPTER 5. CURVATURE
Proof of Theorem 5.1.1.That (ii) implies (iii) follows from the definition of
the length of a curve. Namely
L(ϕ◦γ) =
Z
b
a

d
dt
ϕ(γ(t))

dt
=
Z
b
a
|dϕ(γ(t)) ˙γ(t)|dt
=
Z
b
a
|˙γ(t)|dt
=L(γ).
In the third equation we have used (ii). That (iii) implies (i) follows imme-
diately from the definition of the intrinsic distance functionsdandd
′
.
We prove that (i) implies (ii). Fix a pointp∈Mand chooseε >0
so small thatε <min{inj(p;M),inj(ϕ(p);M
′
)}and that the assertion of
Lemma 5.1.2 holds for the pointp
′
:=ϕ(p)∈M
′
. Then there is a unique
homeomorphism Φp:Bε(p)→Bε(ϕ(p)) such that the following diagram
commutes.
TpM ⊃ Bε(p)
exp
p
fflffl
Φp
//Bε(ϕ(p))
exp
′
ϕ(p)
fflffl
⊂ T
ϕ(p)M
′
M ⊃ Uε(p)
ϕ
//Uε(ϕ(p)) ⊂ M
′
.
Here the vertical maps are diffeomorphisms andϕ:Uε(p)→Uε(ϕ(p)) is a
homeomorphism by (i). Hence Φp:Bε(p)→Bε(ϕ(p)) is a homeomorphism.
Claim 1.The mapΦpsatisfies the equations
exp
′
ϕ(p)
(Φp(v)) =ϕ(exp
p(v)), (5.1.3)
|Φp(v)|=|v|, (5.1.4)
Φp(tv) =tΦp(v)
for everyv∈Bε(p)and everyt∈[0,1].
Equation (5.1.3) holds by definition. To prove (5.1.4) we observe that, by
Theorem 4.4.4, we have
|Φp(v)|=d
′
(ϕ(p),exp
′
ϕ(p)
(Φp(v)))
=d
′
(ϕ(p), ϕ(exp
p(v)))
=d(p,exp
p(v))
=|v|.

5.1. ISOMETRIES 227
Here the second equation follows from (5.1.3) and the third equation from (i).
Equation (5.1.5) holds fort= 0 because Φp(0) = 0 and fort= 1 it is a
tautology. Hence assume 0< t <1. Then
d
′
(exp
′
ϕ(p)
(Φp(tv)),exp
′
ϕ(p)
(Φp(v))) =d
′
(ϕ(exp
p(tv)), ϕ(exp
p(v)))
=d(exp
p(tv),exp
p(v))
=|v| − |tv|
=|Φp(v)| − |Φp(tv)|.
Here the first equation follows from (5.1.3), the second equation from (i),
the third equation from Theorem 4.4.4 and the fact that|v|<inj(p), and
the last equation follows from (5.1.4). Since 0<|Φp(tv)|<|Φp(v)|< εwe
can apply Lemma 5.1.2 and obtain
Φp(tv) =
|Φp(tv)|
|Φp(v)|
Φp(v) =tΦp(v).
This proves Claim 1.
By Claim 1, Φpextends to a bijective map Φp:TpM→T
ϕ(p)M
′
via
Φp(v) :=
1
δ
Φp(δv),
whereδ >0 is chosen so small thatδ|v|< ε. The right hand side of
this equation is independent of the choice ofδ. Hence the extension is well
defined. It is bijective because the original map Φpis a bijection fromBε(p)
toBε(ϕ(p)). The reader may verify that the extended map satisfies the
conditions (5.1.4) and (5.1.5) for allv∈TpMand allt≥0.
Claim 2.The extended mapΦp:TpM→T
ϕ(p)M
′
is linear and preserves
the inner product.
It follows from the equation (4.6.3) in the proof of Lemma 4.6.8 that
|v−w|= lim
t→0
d(exp
p(tv),exp
p(tw))
t
= lim
t→0
d
′
(ϕ(exp
p(tv)), ϕ(exp
p(tw)))
t
= lim
t→0
d
′
(exp
′
ϕ(p)
(Φp(tv)),exp
′
ϕ(p)
(Φp(tw)))
t
= lim
t→0
d
′
(exp
′
ϕ(p)
(tΦp(v)),exp
′
ϕ(p)
(tΦp(w)))
t
=|Φp(v)−Φp(w)|.

228 CHAPTER 5. CURVATURE
Here the second equation follows from (i), the third from (5.1.3), the fourth
from (5.1.4), and the last equation follows again from (4.6.3). By polariza-
tion we obtain
2⟨v, w⟩=|v|
2
+|w|
2
− |v−w|
2
=|Φp(v)|
2
+|Φp(w)|
2
− |Φp(v)−Φp(w)|
2
= 2⟨Φp(v),Φp(w)⟩.
Thus Φppreserves the inner product. Hence, for allv1, v2, w∈TpM, we
have
⟨Φp(v1+v2),Φp(w)⟩=⟨v1+v2, w⟩
=⟨v1, w⟩+⟨v2, w⟩
=⟨Φp(v1),Φp(w)⟩+⟨Φp(v2),Φp(w)⟩
=⟨Φp(v1) + Φp(v2),Φp(w)⟩.
Since Φpis surjective, this implies
Φp(v1+v2) = Φp(v1) + Φp(v2)
for allv1, v2∈TpM. Withv1=vandv2=−vwe obtain
Φp(−v) =−Φp(v)
for everyv∈TpMand by (5.1.5) this gives
Φp(tv) =tΦp(v)
for allv∈TpMandt∈R. This proves Claim 2.
Claim 3.ϕis smooth anddϕ(p) = Φp.
By (5.1.3) we have
ϕ= exp
′
ϕ(p)
◦Φp◦exp
−1
p:Uε(p)→Uε(ϕ(p)).
Since Φpis linear, this shows that the restriction ofϕto the open setUε(p)
is smooth. Moreover, for everyv∈TpMwe have
dϕ(p)v=
d
dt

t=0
ϕ(exp
p(tv)) =
d
dt

t=0
exp
′
ϕ(p)
(tΦp(v)) = Φp(v).
Here we have used equations (5.1.3) and (5.1.5) as well as Lemma 4.3.6.
This proves Claim 3 and Theorem 5.1.1.

5.1. ISOMETRIES 229
Exercise 5.1.4.Prove that every isometryψ:R
n
→R
n
is an affine map
ψ(p) =Ap+b
whereA∈O(n) andb∈R
n
. Thusψis a composition of translation and
rotation.Hint:Lete1, . . . , enbe the standard basis ofR
n
. Prove that any
two vectorsv, w∈R
n
that satisfy
|v|=|w|
and
|v−ei|=|w−ei|fori= 1, . . . , n
must be equal.
Remark 5.1.5.Ifψ:R
n
→R
n
is an isometry of the ambient Euclidean
space withψ(M) =M
′
, then certainlyϕ:=ψ|Mis an isometry fromM
ontoM
′
. On the other hand, ifMis a plane manifold
M={(0, y, z)∈R
3
|0< y < π/2}
andM
′
is the cylindrical manifold
M
′
={(x, y, z)∈R
3
|x
2
+y
2
= 1, x >0, y >0},
Then the mapϕ:M→M
′
defined by
ϕ(0, y, z) := (cos(y),sin(y), z)
is an isometry which isnotof the formϕ=ψ|M. Indeed, an isometry of the
formϕ=ψ|Mnecessarily preserves the second fundamental form (as well
as the first) in the sense that
dψ(p)hp(v, w) =h
′
ψ(p)
(dψ(p)v, dψ(p)w)
forv, w∈TpMbut in the examplehvanishes identically whileh
′
does not.
We may thus distinguish two fundamental question:
I. MandM
′
when are they extrinsically isomorphic, i.e. when is
there an ambient isometryψ:R
n
→R
n
withψ(M) =M
′
?
II. MandM
′
when are they intrinsically isomorphic, i.e. when is
there an isometryϕ:M→M
′
fromMontoM
′
?

230 CHAPTER 5. CURVATURE
As we have noted, both the first and second fundamental forms are
preserved by extrinsic isomorphisms while only the first fundamental form
need be preserved by an intrinsic isomorphism (i.e. an isometry).
A question which occurred to Gauß (who worked for a while as a cartog-
rapher) is this: Can one draw a perfectly accurate map of a portion of the
earth? (i.e. a map for which the distance between points on the map is pro-
portional to the distance between the corresponding points on the surface
of the earth). We can now pose this question as follows: Is there an isom-
etry from an open subset of a sphere to an open subset of a plane? Gauß
answered this question negatively by associating an invariant, the Gaußian
curvatureK:M→R,to a surfaceM⊂R
3
. According to hisTheorema
Egregium
K
′
◦ϕ=K
for an isometryϕ:M→M
′
. The sphere has positive curvature; the plane
has zero curvature; hence the perfectly accurate map does not exist. Our
aim is to explain these ideas.
Local Isometries
We shall need a concept slightly more general than that of “isometry”.
Definition 5.1.6(Local isometry).A smooth mapϕ:M→M
′
is called
alocal isometryiff its derivative
dϕ(p) :TpM→T
ϕ(p)M
′
is an orthogonal linear isomorphism for everyp∈M.
Remark 5.1.7.LetM⊂R
n
andM
′
⊂R
n
′
be manifolds andϕ:M→M
′
be a map. The following are equivalent.
(i)ϕis a local isometry.
(ii)For everyp∈Mthere are open neighborhoodsU⊂MandU
′
⊂M
′
such that the restriction ofϕtoUis an isometry fromUontoU
′
.
That (ii) implies (i) follows immediately from Theorem 5.1.1. On the other
hand (i) implies thatdϕ(p) is invertible so that (ii) follows from the inverse
function theorem.
Example 5.1.8.The map
R→S
1
:θ7→e
iθ
is a local isometry but not an isometry.

5.1. ISOMETRIES 231
Exercise 5.1.9.LetM⊂R
n
be a compact connected 1-manifold. Prove
thatMis diffeomorphic to the circleS
1
. Define the length of a compact
connected Riemannian 1-manifold. Prove that two compact connected 1-
manifoldsM, M
′
⊂R
n
are isometric if and only if they have the same
length.Hint:Letγ:R→Mbe a geodesic with|˙γ(t)| ≡1. Show that
γis not injective; otherwise construct an open cover ofMwithout finite
subcover. Ift0< t1withγ(t0) =γ(t1), show that ˙γ(t0) = ˙γ(t1); otherwise
show thatγ(t0+t) =γ(t1−t) for alltand find a contradiction.
The next result asserts that two local isometries that have the same value
and the same derivative at a single point must agree everywhere, provided
that the domain is connected.
Lemma 5.1.10.LetM⊂R
n
andM
′
⊂R
n
′
be smoothm-manifolds and
assume thatMis connected. Letϕ:M→M
′
andψ:M→M
′
be local
isometries and letp0∈Msuch that
ϕ(p0) =ψ(p0) =:p
′
0, dϕ(p0) =dψ(p0) :Tp0
M→T
p
′
0
M
′
.
Thenϕ(p) =ψ(p)for everyp∈M.
Proof.Define the set
M0:={p∈M|ϕ(p) =ψ(p), dϕ(p) =dψ(p)}.
This set is obviously closed. We prove thatM0is open. Letp∈M0and
chooseU⊂MandU
′
⊂M
′
as in Remark 5.1.7 (ii). Denote
Φp:=dϕ(p) =dψ(p) :TpM→Tp
′M
′
, p
′
:=ϕ(p) =ψ(p)
Then it follows from equation (5.1.3) in the proof of Theorem 5.1.1 that
there exists a constantε >0 such thatUε(p)⊂UandUε(p
′
)⊂U
′
and
q∈Uε(p) = ⇒ ϕ(q) = exp
′
p
′◦Φp◦exp
−1
p(q) =ψ(q).
HenceUε(p)⊂M0. ThusM0is open, closed, and nonempty. SinceMis
connected it follows thatM0=Mand this proves Lemma 5.1.10.
Exercise 5.1.11. If a sequence of local isometriesϕi:M→M
′
con-
verges uniformly to a local isometryϕ:M→M
′
, then it converges in the
C
∞
topology.Hint:Letp∈M. Then every sufficiently small tangent vec-
torv∈TpMsatisfies the equationdϕ(p)v= (exp
′
ϕ(p)
)
−1
(ϕ(exp
p(v))). Use
this to prove thatdϕi(p) converges todϕ(p). Deduce thatϕiconverges toϕ
uniformly with all derviatives in a neighborhood ofp.
(ii)TheC
∞
topology on the space of local isometries fromMtoM
′
agrees
with theC
0
topology.

232 CHAPTER 5. CURVATURE
5.2 The Riemann Curvature Tensor
This section defines the Riemann curvature tensor and proves the Gauß–
Codazzi formula (§5.2.1), introduces the covariant derivative of a global
vector field (§5.2.2), expresses the curvature tensor in terms of a global
formula (§5.2.3), establishes its symmetry properties (§5.2.4), and examines
the curvature for a class of Riemannian metrics on Lie groups (§5.2.5).
5.2.1 Definition and Gauß–Codazzi
LetM⊂R
n
be a smooth manifold andγ:R
2
→Mbe a smooth map.
Denote by (s, t) the coordinates onR
2
. LetZ∈Vect(γ) be a smooth vector
field alongγ, i.e.Z:R
2
→R
n
is a smooth map such thatZ(s, t)∈T
γ(s,t)M
for allsandt. Thecovariant partial derivativesofZwith respect to
the variablessandtare defined by
∇sZ:= Π(γ)
∂Z
∂s
,∇tZ:= Π(γ)
∂Z
∂t
.
In particular∂sγ=∂γ/∂sand∂tγ=∂γ/∂tare vector fields alongγand we
have∇s∂tγ− ∇t∂sγ= 0 as both terms on the left are equal to Π(γ)∂s∂tγ.
Thus ordinary partial differentiation and covariant partial differentiation
commute. The analogous formula (which results on replacing∂by∇andγ
byZ) is in general false. Instead we have the following.
Definition 5.2.1.TheRiemann curvature tensorassigns to eachp∈M
the bilinear mapRp:TpM×TpM→ L(TpM, TpM)characterized by the
equation
Rp(u, v)w=
Γ
∇s∇tZ− ∇t∇sZ
∆
(0,0) (5.2.1)
foru, v, w∈TpMwhereγ:R
2
→Mis a smooth map andZ∈Vect(γ)is a
smooth vector field alongγsuch that
γ(0,0) =p, ∂ sγ(0,0) =u, ∂ tγ(0,0) =v, Z(0,0) =w.(5.2.2)
We must prove thatRis well defined, i.e. that the right hand side of
equation (5.2.1) is independent of the choice ofγandZ. This follows from
the Gauß–Codazzi formula which we prove next. Recall that the second fun-
damental form can be viewed as a linear maphp:TpM→ L(TpM, TpM
⊥
)
and that, foru∈TpM, the linear maphp(u)∈ L(TpM, TpM
⊥
) and its
dualhp(u)
∗
∈ L(TpM
⊥
, TpM) are given by
hp(u)v=
Γ
dΠ(p)u
∆
v, h p(u)
∗
w=
Γ
dΠ(p)u
∆
w
forv∈TpMandw∈TpM
⊥
.

5.2. THE RIEMANN CURVATURE TENSOR 233
Theorem 5.2.2.The Riemann curvature tensor is well defined and given
by theGauß–Codazzi formula
Rp(u, v) =hp(u)
∗
hp(v)−hp(v)
∗
hp(u) (5.2.3)
foru, v∈TpM.
Proof.Letu, v, w∈TpMand choose a smooth mapγ:R
2
→Mand a
smooth vector fieldZalongγsuch that (5.2.2) holds. Then, by the Gauß–
Weingarten formula (3.2.2), we have
∇tZ=∂tZ−hγ(∂tγ)Z
=∂tZ−
Γ
dΠ(γ)∂tγ
∆
Z
=∂tZ−
Γ
∂t
Γ
Π◦γ
∆∆
Z.
Hence
∂s∇tZ=∂s∂tZ−∂s
ı
Γ
∂t
Γ
Π◦γ
∆∆
Z
ȷ
=∂s∂tZ−
Γ
∂s∂t
Γ
Π◦γ
∆∆
Z−
Γ
∂t
Γ
Π◦γ
∆∆
∂sZ
=∂s∂tZ−
Γ
∂s∂t
Γ
Π◦γ
∆∆
Z−
Γ
dΠ(γ)∂tγ
∆Γ
∇sZ+hγ(∂sγ)Z
∆
=∂s∂tZ−
Γ
∂s∂t
Γ
Π◦γ
∆∆
Z−hγ(∂tγ)∇sZ−hγ(∂tγ)
∗
hγ(∂sγ)Z.
Interchangingsandtand taking the difference we obtain
∂s∇tZ−∂t∇sZ=hγ(∂sγ)
∗
hγ(∂tγ)Z−hγ(∂tγ)
∗
hγ(∂sγ)Z
+hγ(∂sγ)∇tZ−hγ(∂tγ)∇sZ.
Here the first two terms on the right are tangent toMand the last two
terms on the right are orthogonal toTγM. Hence
∇s∇tZ− ∇t∇sZ= Π(γ)
Γ
∂s∇tZ−∂t∇sZ
∆
=hγ(∂sγ)
∗
hγ(∂tγ)Z−hγ(∂tγ)
∗
hγ(∂sγ)Z.
Evaluating the right hand side ats=t= 0 we find that
Γ
∇s∇tZ− ∇t∇sZ
∆
(0,0) =hp(u)
∗
hp(v)w−hp(v)
∗
hp(u)w.
This proves the Gauß–Codazzi equation and shows that the left hand side
is independent of the choice ofγandZ. This proves Theorem 5.2.2.

234 CHAPTER 5. CURVATURE
5.2.2 Covariant Derivative of a Global Vector Field
So far we have only defined the covariant derivatives of vector fields along
curves. The same method can be applied to global vector fields. This leads
to the following definition.
Definition 5.2.3(Covariant derivative).LetM⊂R
n
be anm-dimen-
sional submanifold andXbe a vector field onM. Fix a pointp∈Mand
a tangent vectorv∈TpM. Thecovariant derivative ofXatpin the
directionvis the tangent vector
∇vX(p) := Π(p)dX(p)v∈TpM,
whereΠ(p)∈R
n×n
denotes the orthogonal projection ontoTpM.
Remark 5.2.4.Letγ:I→Mbe a smooth curve on an intervalI⊂Rand
letX∈Vect(M) be a smooth vector field onM. ThenX◦γis a smooth
vector field alongγand the covariant derivative ofX◦γis related to the
covariant derivative ofXby the formula
∇(X◦γ)(t) =∇
˙γ(t)X(γ(t)). (5.2.4)
Remark 5.2.5(Gauß–Weingarten formula). Differentiating the equa-
tionX= ΠX(understood as a function fromMtoR
n
) and using the
notation∂vX(p) :=dX(p)vfor the derivative ofXatpin the directionv
we obtain theGauß–Weingarten formula for global vector fields:
∂vX(p) =∇vX(p) +hp(v)X(p). (5.2.5)
Remark 5.2.6(Levi-Civita connection).Differentiating a vector fieldY
onMcovariantly in the direction of another vector fieldXwe obtain a vector
field∇XY∈Vect(M) defined by
(∇XY)(p) :=∇
X(p)Y(p)
forp∈M. This gives rise to a family of linear operators
∇X: Vect(M)→Vect(M),
one for each vector fieldX∈Vect(M), and the assignment
Vect(M)→ L(Vect(M),Vect(M)) :X7→ ∇X (5.2.6)
is itself a linear operator. This linear operator is called theLevi-Civita
connectionon the tangent bundleT M.

5.2. THE RIEMANN CURVATURE TENSOR 235
The Levi-Civita connection (5.2.6) satisfies the conditions
∇f X(Y) =f∇XY, (5.2.7)
∇X(fY) =f∇XY+ (LXf)Y, (5.2.8)
LX⟨Y, Z⟩=⟨∇XY, Z⟩+⟨Y,∇XZ⟩, (5.2.9)
∇YX− ∇XY= [X, Y]
for allX, Y, Z∈Vect(M) and allf∈F(M), whereLXf=df◦X
and [X, Y]∈Vect(M) denotes the Lie bracket of the vector fieldsXandY.
The conditions (5.2.7) and (5.2.8) assert that the linear operator (5.2.6) is
aconnectionon the tangent bundleT M, condition (5.2.9) asserts that
the connection (5.2.6) isRiemannian(i.e. it is compatible with the first
fundamental form), and condition (5.2.10) asserts that it istorsion-free.
The next lemma shows that the Levi-Civita connection (5.2.6) is uniquely
determined by (5.2.9) and (5.2.10), and hence is the unique torsion-free
Riemannian connection on the tangent bundleT M.
Lemma 5.2.7(Uniqueness Lemma).There is a unique linear operator
Vect(M)→ L(Vect(M),Vect(M)) :X7→ ∇X
satisfying equations(5.2.9)and(5.2.10)for allX, Y, Z∈Vect(M).
Proof.Existence follows from the properties of the Levi-Civita connection.
We prove uniqueness. LetX7→DXbe any linear operator from Vect(M)
toL(Vect(M),Vect(M)) that satisfies (5.2.9) and (5.2.10). Then we have
LX⟨Y, Z⟩=⟨DXY, Z⟩+⟨Y, DXZ⟩,
LY⟨X, Z⟩=⟨DYX, Z⟩+⟨X, DYZ⟩,
−LZ⟨X, Y⟩=−⟨DZX, Y⟩ − ⟨X, DZY⟩.
Adding these three equations we find
LX⟨Y, Z⟩+LY⟨Z, X⟩ − LZ⟨X, Y⟩
= 2⟨DXY, Z⟩+⟨DYX−DXY, Z⟩
+⟨X, DYZ−DZY⟩+⟨Y, DXZ−DZX⟩
= 2⟨DXY, Z⟩+⟨[X, Y], Z⟩+⟨X,[Z, Y]⟩+⟨Y,[Z, X]⟩.
The same equation holds for the Levi-Civita connection and hence
⟨DXY, Z⟩=⟨∇XY, Z⟩.
This impliesDXY=∇XYfor allX, Y∈Vect(M).

236 CHAPTER 5. CURVATURE
Exercise 5.2.8.In the proof of Lemma 5.2.7 we did not actually use the
assumption that the operatorDX: Vect(M)→Vect(M) is linear nor that
the operatorX7→DXis linear. Prove directly that if a map
DX:L(M)→ L(M)
satisfies (5.2.9) for allY, Z∈Vect(M), thenDXis linear. Prove that every
map Vect(M)→ L(Vect(M),Vect(M)) :X7→DXthat satisfies (5.2.10) is
linear.
Exercise 5.2.9.Letϕ
t
be the flow of a complete vector fieldXonMand
letψ
t
be the flow of a complete vector fieldYonM.
(i)Prove that the formula
e
X(p, v) := (X(p), dX(p)v) defines a vector field
on the tangent bundleT M.Hint:Lemma 4.3.1 and equation (5.2.5).
(ii)Prove that the flow of
e
Xis given by
e
ϕ
t
(p, v) := (ϕ
t
(p), dϕ
t
(p)v).
(iii)Prove that the vector fieldst
−1
((ψ
t
)
∗
X−X) converge to [X, Y] in
theC
1
topology asttends to zero.Hint:EstablishC
0
convergence in
Lemma 2.4.18 and then use this result for the vector fields
e
Xand
e
Y.
Remark 5.2.10(The Levi-Civita connection in local coordinates).
Letϕ:U→Ω be a coordinate chart on an open setU⊂Mwith values in
an open set Ω⊂R
m
. In such a coordinate chart a vector fieldX∈Vect(M)
is represented by a smooth mapξ= (ξ
1
, . . . , ξ
m
) : Ω→R
m
defined by
ξ(ϕ(p)) =dϕ(p)X(p)
forp∈U. IfY∈Vect(M) is represented byη, then∇XYis represented by
the map
(∇ξη)
k
:=
m
X
i=1
∂η
k
∂x
i
ξ
i
+
m
X
i,j=1
Γ
k
ijξ
i
η
j
. (5.2.11)
Here the Γ
k
ij
: Ω→Rare the Christoffel symbols defined by
Γ
k
ij:=
m
X
ℓ=1
g
kℓ
1
2
`
∂gℓi
∂x
j
+
∂gℓj
∂x
i
−
∂gij
∂x
ℓ
´
, (5.2.12)
wheregijis the metric tensor andg
ij
is the inverse matrix so that
X
j
gijg
jk
=δ
k
i
(see Lemma 3.6.5). This formula can be used to prove the existence state-
ment in Lemma 5.2.7 and hence define the Levi-Civita connection in the
intrinsic setting.

5.2. THE RIEMANN CURVATURE TENSOR 237
5.2.3 A Global Formula
Lemma 5.2.11.ForX, Y, Z∈Vect(M)we have
R(X, Y)Z=∇X∇YZ− ∇Y∇XZ+∇
[X,Y]Z. (5.2.13)
Proof.Fix a pointp∈M. Then the right hand side of equation (5.2.13) at
premains unchanged if we multiply each of the vector fieldsX, Y, Zby a
smooth functionf:M→[0,1] that is equal to one nearp. Choosingfwith
compact support we may therefore assume that the vector fieldsXandY
are complete. Letϕ
s
denote the flow ofXandψ
t
the flow ofY. Define the
mapγ:R
2
→Mby
γ(s, t) :=ϕ
s
◦ψ
t
(p), s, t∈R.
Then
∂sγ=X(γ), ∂ tγ= (ϕ
s
∗Y)(γ).
Hence, by Remark 5.2.4 we have
∇s(Z◦γ) = (∇XZ) (γ),∇t(Z◦γ) =
Γ
∇ϕ
s
∗YZ
∆
(γ).
Using Remark 5.2.4 again we obtain
∇s∇t(Z◦γ) =∇∂sγ
Γ
∇ϕ
s
∗YZ
∆
(γ) +
Γ
∇∂sϕ
s
∗YZ
∆
(γ),
∇t∇s(Z◦γ) =
Γ
∇ϕ
s
∗Y∇XZ
∆
(γ).
Since
∂
∂s

s=0
ϕ
s
∗Y= [X, Y]
and∂sγ=X(γ), it follows that
∇s∇t(Z◦γ)(0,0) =
Γ
∇X∇YZ+∇
[X,Y]Z
∆
(p),
∇t∇s(Z◦γ)(0,0) =
Γ
∇Y∇XZ
∆
(p).
Hence
Rp(X(p), Y(p))Z(p) =
Γ
∇s∇t(Z◦γ)− ∇t∇s(Z◦γ)
∆
(0,0)
=
Γ
∇X∇YZ− ∇Y∇XZ+∇
[X,Y]Z
∆
(p).
This proves Lemma 5.2.11.

238 CHAPTER 5. CURVATURE
Remark 5.2.12.Equation (5.2.13) can be written succinctly as
[∇X,∇Y] +∇
[X,Y]=R(X, Y). (5.2.14)
This can be contrasted with the equation
[LX,LY] +L
[X,Y]= 0 (5.2.15)
for the operatorLXon the space of real valued functions onM.
Remark 5.2.13.Equation (5.2.13) can be used to define the Riemann
curvature tensor. To do this one must again prove that the right hand side
of equation (5.2.13) atpdepends only on the valuesX(p), Y(p), Z(p) of the
vector fieldsX, Y, Zat the pointp. For this it suffices to prove that the map
Vect(M)×Vect(M)×Vect(M)→Vect(M) : (X, Y, Z)7→R(X, Y)Z
is linear over the RingF(M) of smooth real valued functions onM, i.e.
R(fX, Y)Z=R(X, fY)Z=R(X, Y)fZ=fR(X, Y)Z (5.2.16)
forX, Y, Z∈Vect(M) andf∈F(M). The formula (5.2.16) follows from
the equations (5.2.7), (5.2.8), (5.2.15), and [X, fY] =f[X, Y]−(LXf)Y.It
follows from (5.2.16) that the right hand side of (5.2.13) atpdepends only
on the vectorsX(p),Y(p),Z(p). The proof requires two steps. One first
shows that ifXvanishes nearp, then the right hand side of (5.2.13) vanishes
atp(and similarly forYandZ). Just multiplyXby a smooth function
equal to zero atpand equal to one on the support ofX; thenfX=Xand
hence the vector fieldR(X, Y)Z=R(fX, Y)Z=fR(X, Y)Zvanishes atp.
Second, we choose a local frameE1, . . . , Em∈Vect(M), i.e. vector fields
that form a basis ofTpMfor eachpin some open setU⊂M. Then we may
write
X=
m
X
i=1
ξ
i
Ei, Y =
m
X
j=1
η
j
Ej, Z=
m
X
k=1
ζ
k
Ek
inU. Using the first step and theF(M)-multilinearity we obtain
R(X, Y)Z=
m
X
i,j,k=1
ξ
i
η
j
ζ
k
R(Ei, Ej)Ek
inU. IfX
′
(p) =X(p), thenξ
i
(p) =ξ
′i
(p) so ifX(p) =X
′
(p),Y(p) =Y
′
(p),
Z(p) =Z
′
(p), then (R(X, Y)Z)(p) = (R(X
′
, Y
′
)Z
′
)(p) as required.

5.2. THE RIEMANN CURVATURE TENSOR 239
5.2.4 Symmetries
Theorem 5.2.14.The Riemann curvature tensor satisfies
R(Y, X) =−R(X, Y) =R(X, Y)
∗
, (5.2.17)
R(X, Y)Z+R(Y, Z)X+R(Z, X)Y= 0, (5.2.18)
⟨R(X, Y)Z, W⟩=⟨R(Z, W)X, Y⟩ (5.2.19)
forX, Y, Z, W∈Vect(M). Equation(5.2.18)is thefirst Bianchi identity.
Proof.The first equation in (5.2.17) is obvious from the definition and the
second follows from the Gauß–Codazzi formula (5.2.3). Alternatively, choose
a smooth mapγ:R
2
→Mand two vector fieldsZ, Walongγ. Then
0 =∂s∂t⟨Z, W⟩ −∂t∂s⟨Z, W⟩
=∂s⟨∇tZ, W⟩+∂s⟨Z,∇tW⟩ −∂t⟨∇sZ, W⟩ −∂t⟨Z,∇sW⟩
=⟨∇s∇tZ, W⟩+⟨Z,∇s∇tW⟩ − ⟨∇t∇sZ, W⟩ − ⟨Z,∇t∇sW⟩
=⟨R(∂sγ, ∂tγ)Z, W⟩+⟨Z, R(∂sγ, ∂tγ)W⟩.
This proof has the advantage that it carries over to the intrinsic setting. We
prove the first Bianchi identity using (5.2.10) and (5.2.13):
R(X, Y)Z+R(Y, Z)X+R(Z, X)Y
=∇X∇YZ− ∇Y∇XZ+∇
[X,Y]Z+∇Y∇ZX− ∇Z∇YX+∇
[Y,Z]X
+∇Z∇XY− ∇X∇ZY+∇
[Z,X]Y
=∇
[Y,Z]X− ∇X[Y, Z] +∇
[Z,X]Y− ∇Y[Z, X] +∇
[X,Y]Z− ∇Z[X, Y]
= [X,[Y, Z]] + [Y,[Z, X]] + [Z,[X, Y]].
The last term vanishes by the Jacobi identity. We prove (5.2.19) by com-
bining the first Bianchi identity with (5.2.17):
⟨R(X, Y)Z, W⟩ − ⟨R(Z, W)X, Y⟩
=−⟨R(Y, Z)X, W⟩ − ⟨R(Z, X)Y, W⟩ − ⟨R(Z, W)X, Y⟩
=⟨R(Y, Z)W, X⟩+⟨R(Z, X)W, Y⟩+⟨R(W, Z)X, Y⟩
=⟨R(Y, Z)W, X⟩ − ⟨R(X, W)Z, Y⟩
=⟨R(Y, Z)W, X⟩ − ⟨R(W, X)Y, Z⟩.
Note that the first line is related to the last by a cyclic permutation. Re-
peating this argument we find
⟨R(Y, Z)W, X⟩ − ⟨R(W, X)Y, Z⟩=⟨R(Z, W)X, Y⟩ − ⟨R(X, Y)Z, W⟩.
Combining the last two identities we obtain (5.2.19). This proves Theo-
rem 5.2.14.

240 CHAPTER 5. CURVATURE
Remark 5.2.15.We may think of a vector fieldXonMas a section of
the tangent bundle. This is reflected in the alternative notation
Ω
0
(M, T M) := Vect(M).
A 1-form onMwith values in the tangent bundleis a collection of
linear mapsA(p) :TpM→TpM, one for everyp∈M, which is smooth
in the sense that for every smooth vector fieldXonMthe assignment
p7→A(p)X(p) defines again a smooth vector field onM. We denote by
Ω
1
(M, T M)
the space of smooth 1-forms onMwith values inT M. The covariant deriva-
tive of a vector fieldYis such a 1-form with values in the tangent bundle
which assigns to everyp∈Mthe linear mapTpM→TpM:v7→ ∇vY(p).
Thus we can think of the covariant derivative as a linear operator
∇: Ω
0
(M, T M)→Ω
1
(M, T M).
The equation (5.2.7) asserts that the operatorsX7→ ∇Xindeed determine
a linear operator from Ω
0
(M, T M) to Ω
1
(M, T M). Equation (5.2.8) as-
serts that this linear operator∇is aconnectionon the tangent bundle
ofM. Equation (5.2.9) asserts that∇is aRiemannian connectionand
equation (5.2.10) asserts that∇istorsion-free. Thus Lemma 5.2.7 can
be restated as asserting that theLevi-Civita connectionis the unique
torsion-free Riemannian connection on the tangent bundle.
Exercise 5.2.16.Extend the notion of a connection to a general vector bun-
dleE, both as a collection of linear operators∇X: Ω
0
(M, E)→Ω
0
(M, E),
one for every vector fieldX∈Vect(M), and as a linear operator
∇: Ω
0
(M, E)→Ω
1
(M, E)
satisfying the analogue of equation (5.2.8). Interpret this equation as a Leib-
niz rule for the product of a function onMwith a section ofE. Show that
∇
⊥
is a connection onT M
⊥
. Extend the notion of curvature to connections
on general vector bundles.
Exercise 5.2.17.Show that the field which assigns to eachp∈Mthe
multi-linear mapR
⊥
p:TpM×TpM→ L(TpM
⊥
, TpM
⊥
) characterized by
R
⊥
(∂sγ, ∂tγ)Y=∇
⊥
s∇
⊥
tY− ∇
⊥
t∇
⊥
sY
forγ:R
2
→MandY∈Vect
⊥
(γ) satisfies the equation
R
⊥
p(u, v) =hp(u)hp(v)
∗
−hp(v)hp(u)
∗
forp∈Mandu, v∈TpM.

5.2. THE RIEMANN CURVATURE TENSOR 241
5.2.5 Riemannian Metrics on Lie Groups
We begin with a calculation of the Riemann curvature tensor on a Lie sub-
group of the orthogonal group O(n) with the Riemannian metric inherited
from the standard inner product
⟨v, w⟩:= trace(v
T
w) (5.2.20)
on the ambient spacegl(n,R) =R
n×n
. This fits into the extrinsic setting
used throughout most of this book. Note that every Lie subgroup of O(n)
is a closed subset of O(n) by Theorem 2.5.27 and hence is compact.
Example 5.2.18.Let G⊂O(n) be a Lie subgroup and let
g:= Lie(G) =T1lG
be the Lie algebra of G. Consider the Riemannian metric on G induced by
the inner product (5.2.20) onR
n×n
. Then the Riemann curvature tensor
on G can be expressed in terms of the Lie bracket (see item (d) below).
(a)The mapsg7→ag,g7→ga,g7→g
−1
are isometries of G for everya∈G.
(b)Letγ:R→G be a smooth curve andX∈Vect(γ) be a smooth vector
field alongγ. Then the covariant derivative ofXis given by
γ(t)
−1
∇X(t) =
d
dt
γ(t)
−1
X(t) +
1
2
Θ
γ(t)
−1
˙γ(t), γ(t)
−1
X(t)
Λ
.(5.2.21)
(Exercise:Prove equation (5.2.21).Hint:Sinceg⊂o(n) we have the
identity trace((ξη+ηξ)ζ) = 0 for allξ, η, ζ∈g.)
(c)A smooth mapγ:R→G is a geodesic if and only if there exist matrices
g∈G andξ∈gsuch that
γ(t) =gexp(tξ). (5.2.22)
For G = O(n) we have seen this in Example 4.3.12 and in the general
case this follows from equation (5.2.21) withX= ˙γ. Hence the exponential
map exp :g→G defined by the exponential matrix (as in§2.5) agrees with
the time-1-map of the geodesic flow (as in§4.3).
(d)The Riemann curvature tensor on G is given by
g
−1
Rg(u, v)w=−
1
4
[[g
−1
u, g
−1
v], g
−1
w] (5.2.23)
forg∈G andu, v, w∈TgG. Note that the first Bianchi identity is equivalent
to the Jacobi identity. (Exercise:Prove equation (5.2.23).)

242 CHAPTER 5. CURVATURE
Definition 5.2.19(Bi-invariant Riemannian metric).LetGbe a Lie
subgroup ofGL(n,R)and letg= Lie(G) =T1lGbe its Lie algebra. A Rie-
mannian metric onGis calledbi-invariantiff it has the form
⟨v, w⟩g:=⟨vg
−1
, wg
−1
⟩ (5.2.24)
forg∈Gandv, w∈TgG, where⟨·,·⟩is an inner product on the Lie algebrag
that isinvariant under conjugation, i.e. it satisfies the equation
⟨ξ, η⟩=⟨gξg
−1
, gηg
−1
⟩. (5.2.25)
for allξ, η∈gand allg∈G.
Exercise 5.2.20.Prove that the Riemannian metric induced by (5.2.20)
on any Lie subgroup G⊂O(n) is bi-invariant.
Exercise 5.2.21.Use the Haar measure ([66, Chapter 8]) to prove that
every compact Lie group admits a bi-invariant Riemannian metric.
Exercise 5.2.22.Prove that all the assertions in Example 5.2.18 carry over
verbatim to any Lie group equipped with a bi-invariant Riemann metric.
Exercise 5.2.23(Invariant inner product).Prove that, if an inner prod-
uct on the Lie algebragof a Lie group G is invariant under conjugation,
then it satisfies the equation
⟨[ξ, η], ζ⟩=⟨ξ,[η, ζ]⟩ (5.2.26)
for allξ, η, ζ∈g. If G is connected, prove that, conversely, equation (5.2.26)
implies (5.2.25). An inner product on an arbitrary Lie algebragis called
invariantiff it satisfies equation (5.2.26).
Exercise 5.2.24(Commutant).Letgbe a finite-dimensional Lie algebra.
The linear subspace spanned by all vectors of the form [ξ, η] is called the
commutant ofgand is denoted by [g,g] := span{[ξ, η]|ξ, η∈g}. Ifg
is equipped with an invariant inner product, prove that [g,g]
⊥
=Z(g) is
the center ofg(Exercise 2.5.34) and henceg= [g,g]⊕Z(g).Prove that the
Heisenberg algebrahin Exercise 2.5.15 satisfies [h,h] =Z(h) and hence does
not admit an invariant inner product.
Example 5.2.25(Killing form).Every finite-dimensional Lie algebrag
admits a natural symmetric bilinear formκ:g×g→Rthat satisfies equa-
tion (5.2.26). It is called theKilling formand is defined by
κ(ξ, η) := trace
Γ
ad(ξ)ad(η)
∆
, ξ, η∈g, (5.2.27)
where ad :g→Der(g) is the adjoint representation in Example 2.5.23. The
Killing form may have a kernel (which always contains the center ofg), and
even if it is nondegenerate, it may be indefinite.

5.2. THE RIEMANN CURVATURE TENSOR 243
Exercise 5.2.26.Prove thatκ([ξ, η], ζ) =κ(ξ,[η, ζ]) for allξ, η, ζ∈g.
Exercise 5.2.27.Assume thatgadmits an invariant inner product. For
eachξ∈gprove that the derivation ad(ξ) is skew-adjoint with respect to this
inner product and deduce thatκ(ξ, ξ) =−trace(ad(ξ)
∗
ad(ξ)) =−|ad(ξ)|
2
.
Deduce that the Killing form is nondegenerate wheneverghas a trivial
center and admits an invariant inner product.
Example 5.2.28(Right invariant Riemannian metric). Let G be
any Lie subgroup of GL(n,R) (not necessarily contained in O(n)), and
letg:= Lie(G) =T1lG be its Lie algebra. Fix any inner product on the
Lie algebrag(not necessarily invariant under conjugation) and consider the
Riemannian metric on G defined by
⟨v, w⟩g:=⟨vg
−1
, wg
−1
⟩ (5.2.28)
forv, w∈TgG. This metric is calledright invariant.
(a)The mapg7→gais an isometry of G for everya∈G.
(b)Define the linear mapA:g→End(g) by
⟨A(ξ)η, ζ⟩=
1
2
ı
⟨ξ,[η, ζ]⟩ − ⟨η,[ζ, ξ]⟩ − ⟨ζ,[ξ, η]⟩
ȷ
(5.2.29)
forξ, η, ζ∈g. ThenAis the unique linear map that satisfies
A(ξ) +A(ξ)
∗
= 0, A(η)ξ−A(ξ)η= [ξ, η]
for allξ, η∈g, whereA(ξ)
∗
is the adjoint operator with respect to the inner
product ong. Letγ:R→G be a smooth curve andX∈Vect(γ) be a
smooth vector field alongγ. Then the covariant derivative ofXis given by
∇X=
`
d
dt
(Xγ
−1
) +A( ˙γγ
−1
)Xγ
−1
´
γ. (5.2.30)
(Exercise:Prove this. Moreover, if the inner produt ongis invariant, prove
thatA(ξ)η=−
1
2
[ξ, η] for allξ, η∈g.)
(c)A smooth curveγ:R→G is a geodesic if and only if it satisfies
d
dt
( ˙γγ
−1
) +A( ˙γγ
−1
) ˙γγ
−1
= 0. (5.2.31)
(Exercise:G is complete.)
(d)The Riemann curvature tensor on G is given by
Γ
Rg(ξg, ηg)ζg
∆
g
−1
=
ı
A([ξ, η]) + [A(ξ), A(η)]
ȷ
ζ (5.2.32)
forg∈G andξ, η, ζ∈g. (Exercise:Prove this.)

244 CHAPTER 5. CURVATURE
5.3 Generalized Theorema Egregium
We will now show that geodesics, covariant differentiation, parallel trans-
port, and the Riemann curvature tensor are all intrinsic, i.e. they are in-
tertwined by isometries. In the extrinsic setting these results are somewhat
surprising since these objects are all defined using the second fundamental
form, whereas isometries need not preserve the second fundamental form in
any sense but only the first fundamental form.
Below we shall give a formula expressing the Gaußian curvature of a
surfaceM
2
inR
3
in terms of the Riemann curvature tensor and the first
fundamental form. It follows that the Gaußian curvature is also intrinsic.
This fact was called by Gauß the “Theorema Egregium” which explains the
title of this section.
5.3.1 Pushforward
We assume throughout this section thatM⊂R
n
andM
′
⊂R
n
′
are smooth
submanifolds of the same dimensionm. As in§5.1 we denote objects onM
′
by the same letters as objects inMwith primes affixed. In particular,g
′
denotes the first fundamental form onM
′
andR
′
denotes the Riemann
curvature tensor onM
′
.
Letϕ:M→M
′
be a diffeomorphism. Usingϕwe can move objects
onMtoM
′
. For example the pushforward of a smooth curveγ:I→Mis
the curve
ϕ∗γ:=ϕ◦γ:I→M
′
,
the pushforward of a smooth functionf:M→Ris the function
ϕ∗f:=f◦ϕ
−1
:M
′
→R,
the pushforward of a vector fieldX∈Vect(γ) along a curveγ:I→Mis
the vector fieldϕ∗X∈Vect(ϕ∗γ) defined by
(ϕ∗X)(t) :=dϕ(γ(t))X(t)
fort∈I, and the pushforward of a global vector fieldX∈Vect(M) is the
vector fieldϕ∗X∈Vect(M
′
) defined by
(ϕ∗X)(ϕ(p)) :=dϕ(p)X(p)
forp∈M. Recall that the first fundamental form onMis the Riemannian
metricgdefined as the restriction of the Euclidean inner product on the

5.3. GENERALIZED THEOREMA EGREGIUM 245
ambient space to each tangent space ofM. It assigns to eachp∈Mthe
bilinear mapgp:TpM×TpM→Rgiven by
gp(u, v) =⟨u, v⟩, u, v∈TpM.
Its pushforward is the Riemannian metric which assigns to eachp
′
∈M
′
the
inner product (ϕ∗g)p
′:Tp
′M
′
×Tp
′M
′
→Rdefined by
(ϕ∗g)
ϕ(p)(dϕ(p)u, dϕ(p)v) :=gp(u, v)
forp:=ϕ
−1
(p
′
)∈Mandu, v∈TpM. The pushforward of the Riemann
curvature tensor is the tensor which assigns to eachp
′
∈M
′
the bilinear
map (ϕ∗R)p
′:Tp
′M
′
×Tp
′M
′
→ L
Γ
Tp
′M
′
, Tp
′M
′
∆
,defined by
(ϕ∗R)
ϕ(p)(dϕ(p)u, dϕ(p)v) :=dϕ(p)Rp(u, v)dϕ(p)
−1
forp:=ϕ
−1
(p
′
)∈Mandu, v∈TpM.
5.3.2 Theorema Egregium
Theorem 5.3.1(Theorema Egregium).The first fundamental form, co-
variant differentiation, geodesics, parallel transport, and the Riemann cur-
vature tensor are intrinsic. This means that for every isometryϕ:M→M
′
the following holds.
(i)ϕ∗g=g
′
.
(ii)IfX∈Vect(γ)is a vector field along a smooth curveγ:I→M, then
∇
′
(ϕ∗X) =ϕ∗∇X, (5.3.1)
and ifX, Y∈Vect(M)are global vector fields, then
∇
′
ϕ∗X
ϕ∗Y=ϕ∗(∇XY). (5.3.2)
(iii)Ifγ:I→Mis a geodesic, thenϕ◦γ:I→M
′
is a geodesic.
(iv)Ifγ:I→Mis a smooth curve, then for alls, t∈I, we have
Φ
′
ϕ◦γ
(t, s)dϕ(γ(s)) =dϕ(γ(t))Φγ(t, s). (5.3.3)
(v)ϕ∗R=R
′
.

246 CHAPTER 5. CURVATURE
Proof.Assertion (i) is simply a restatement of Theorem 5.1.1. To prove (ii)
we choose a local smooth parametrizationψ: Ω→Uof an open setU⊂M,
defined on an open set Ω⊂R
m
, so thatψ
−1
:U→Ω is a coordinate chart.
Suppose without loss of generality thatγ(t)∈Ufor allt∈Iand define
c:I→Ω andξ:I→R
m
by
γ(t) =ψ(c(t)), X(t) =
m
X
i=1
ξ
i
(t)
∂ψ
∂x
i
(c(t)).
Recall from equations (3.6.6) and (3.6.7) that
∇X(t) =
m
X
k=1

˙
ξ
k
(t) +
m
X
i,j=1
Γ
k
ij(c(t))˙c
i
(t)ξ
j
(t)


∂ψ
∂x
k
(c(t)),
where the Christoffel symbols Γ
k
ij
: Ω→Rare defined by
Π(ψ)
∂
2
ψ
∂x
i
∂x
j
=
m
X
k=1
Γ
k
ij
∂ψ
∂x
k
.
Now consider the same formula forϕ∗Xusing the parametrization
ψ
′
:=ϕ◦ψ: Ω→U
′
:=ϕ(U)⊂M
′
.
The Christoffel symbols Γ
′k
ij: Ω→Rassociated to this parametrization of
U
′
are defined by the same formula as the Γ
k
ij
withψreplaced byψ
′
. But
the metric tensor forψagrees with the metric tensor forψ
′
:
gij=
ø
∂ψ
∂x
i
,
∂ψ
∂x
j
Æ
=
ø
∂ψ
′
∂x
i
,
∂ψ
′
∂x
j
Æ
.
Hence it follows from Lemma 3.6.5 that Γ
′k
ij= Γ
k
ij
for alli, j, k. This implies
that the covariant derivative ofϕ∗Xis given by
∇
′
(ϕ∗X) =
m
X
k=1

˙
ξ
k
+
m
X
i,j=1
Γ
k
ij(c)˙c
i
ξ
j


∂ψ
′
∂x
k
(c)
=dϕ(ψ(c))
m
X
k=1

˙
ξ
k
+
m
X
i,j=1
Γ
k
ij(c)˙c
i
ξ
j


∂ψ
∂x
k
(c)
=ϕ∗∇X.
This proves (5.3.1). Equation (5.3.2) follows immediately from (5.3.1) and
Remark 5.2.4.

5.3. GENERALIZED THEOREMA EGREGIUM 247
Here is a second proof of (ii). For every vector fieldX∈Vect(M) we
define the operatorDX: Vect(M)→Vect(M) by
DXY:=ϕ
∗
(∇ϕ∗Xϕ∗Y).
Then, for allX, Y∈Vect(M), we have
DYX−DXY=ϕ
∗
(∇ϕ∗Yϕ∗X− ∇ϕ∗Xϕ∗Y) =ϕ
∗
[ϕ∗X, ϕ∗Y] = [X, Y].
Moreover, it follows from (i) that
ϕ∗LX⟨Y, Z⟩=Lϕ∗X⟨ϕ∗Y, ϕ∗Z⟩
=⟨∇ϕ∗Xϕ∗Y, ϕ∗Z⟩+⟨ϕ∗Y,∇ϕ∗Xϕ∗Z⟩
=⟨ϕ∗DXY, ϕ∗Z⟩+⟨ϕ∗Y, ϕ∗DXZ⟩
=ϕ∗
Γ
⟨DXY, Z⟩+⟨Y, DXZ⟩
∆
.
and henceLX⟨Y, Z⟩=⟨DXY, Z⟩+⟨Y, DXZ⟩for allX, Y, Z∈Vect(M).
Thus the operatorX7→DXsatisfies equations (5.2.9) and (5.2.10) and, by
Lemma 5.2.7, it follows thatDXY=∇XYfor allX, Y∈Vect(M). This
completes the second proof of (ii).
We prove (iii). Sinceϕpreserves the first fundamental form it also
preserves the energy of curves, namely
E(ϕ◦γ) =E(γ)
for every smooth mapγ: [0,1]→M. Henceγis a critical point of the energy
functional if and only ifϕ◦γis a critical point of the energy functional.
Alternatively it follows from (ii) that
∇
′
`
d
dt
ϕ◦γ
´
=∇
′
ϕ∗˙γ=ϕ∗∇˙γ
for every smooth curveγ:I→M. Ifγis a geodesic, the last term vanishes
and henceϕ◦γis a geodesic as well. As a third proof we can deduce (iii) from
the formulaϕ(exp
p(v)) = exp
ϕ(p)(dϕ(p)v) in the proof of Theorem 5.1.1.
We prove (iv). Fort0∈Iandv0∈T
γ(t0)Mdefine
X(t) := Φγ(t, t0)v0, X
′
(t) := Φ
′
ϕ◦γ
(t, t0)dϕ(γ(t0))v0.
By (ii) the vector fieldsX
′
andϕ∗Xalongϕ◦γare both parallel and they
agree att=t0. HenceX
′
(t) =ϕ∗X(t) for allt∈Iand this proves (5.3.3).

248 CHAPTER 5. CURVATURE
We prove (v). Fix a smooth mapγ:R
2
→Mand a smooth vector
fieldZalongγ, and define
γ
′
=ϕ◦γ:R
2
→M
′
, Z
′
:=ϕ∗Z∈Vect(γ
′
).
Then it follows from (ii) that
R
′
(∂sγ
′
, ∂tγ
′
)Z
′
=∇
′
s∇
′
tZ
′
− ∇
′
t∇
′
sZ
′
=ϕ∗(∇s∇tZ− ∇t∇sZ)
=dϕ(γ)R(∂sγ, ∂tγ)Z
= (ϕ∗R)(∂sγ
′
, ∂tγ
′
)Z
′
.
This proves (v) and Theorem 5.3.1.
The assertions of Theorem 5.3.1 carry over in slightly modified form to
local isometriesϕ:M→M
′
. In particular, the pushforward of a vector field
onMunderϕis only defined whenϕis a diffeomorphism while the pushfor-
ward of a vector field along a curve is defined for any smooth mapϕ. Also,
the pushforward of the Riemann curvature tensor under a local isometry is
only defined locally, and local isometries satisfy the local analogue of the
first assertion in Theorema Egregium by definition.
Corollary 5.3.2(Theorema Egregium for Local Isometries). Every
local isometryϕ:M→M
′
has the following properties.
(i)Every vector fieldXalong a smooth curveγ:I→Msatisfies(5.3.1).
(ii)Ifγ:I→Mis a geodesic, then so isϕ◦γ:I→M
′
.
(iii)Parallel transport along a smooth curveγ:I→Msatisfies(5.3.3).
(iv)The curvature tensorsRofMandR
′
ofM
′
are related by the formula
R
′
ϕ(p)
(dϕ(p)u, dϕ(p)v) =dϕ(p)Rp(u, v)dϕ(p)
−1
(5.3.4)
for allp∈Mand allu, v∈TpM.
Proof.Letp0∈M. Then, by the Inverse Function Theorem 2.2.17, there
exists an open neighborhoodU⊂Mofp0such thatU
′
:=ϕ(U) is an open
subset ofM
′
and the restrictionϕ|U:U→U
′
is a diffeomorphism. This
restriction is an isometry by Theorem 5.1.1. Hence, by Theorem 5.3.1 the
assertions (i) and (ii) hold for the restriction ofγtoI0:=γ
−1
(U) (a union
of subintervals ofI) and (iv) holds for allp∈U. Since these are local state-
ments andp0was chosen arbitrary, this proves (i), (ii), and (iv). Part (iii)
follows directly from (i) as in the proof of Theorem 5.3.1 and this proves
Corollary 5.3.2.

5.3. GENERALIZED THEOREMA EGREGIUM 249
The next corollary spells out a useful consequence of Corollary 5.3.2. For
sufficiently small tangent vectors equation (5.3.5) below already appeared
in the proof of Theorem 5.1.1 and was used in Lemma 5.1.10 and Exer-
cise 5.1.11. WhenMis not complete, recall the notationVp⊂TpMfor
the domain of the exponential map ofMat a pointp(Definition 4.3.5).
Forp
′
∈M
′
denote the domain of the exponential map byV
′
p
′⊂Tp
′M
′
.
Corollary 5.3.3.Letϕ:M→M
′
be a local isometry and letp∈M.
Thendϕ(p)Vp⊂V
′
ϕ(p)
and, for everyv∈Vp,
ϕ(exp
p(v)) = exp
′
ϕ(p)
(dϕ(p)v). (5.3.5)
Proof.Letv∈Vp⊂TpMand defineγ(t) := exp
p(tv) for 0≤t≤1.
Thenγ: [0,1]→Mis a geodesic by Lemma 4.3.6, and hence so is the
curveγ
′
:=ϕ◦γ: [0,1]→M
′
by Corollary 5.3.2. Moreover,
γ
′
(0) =ϕ(γ(0)) =ϕ(p), ˙γ
′
(0) =dϕ(γ(0)) ˙γ(0) =dϕ(p)v
by the chain rule. Hence it follows from the definition of the exponential
map (Definition 4.3.5) thatdϕ(p)v∈V
′
ϕ(p)
and
exp
′
ϕ(p)
(dϕ(p)v) =γ
′
(1) =ϕ(γ(1)) =ϕ(exp
p(v)).
This proves Corollary 5.3.3.
5.3.3 Gaußian Curvature
As a special case we shall now consider ahypersurfaceM⊂R
m+1
,i.e.
a smooth submanifold of codimension one. We assume that there exists a
smooth mapν:M→R
m+1
such that, for everyp∈M, we have
ν(p)⊥TpM, |ν(p)|= 1.
Such a map always exists locally (see Example 3.1.3). Note thatν(p) is
an element of the unit sphere inR
m+1
for everyp∈Mand hence we can
regardνas a map fromMtoS
m
, i.e.ν:M→S
m
.Such a map is called a
Gauß mapforM. Note that ifν:M→S
m
is a Gauß map, so is−ν, but
this is the only ambiguity whenMis connected. Differentiatingνatp∈M
we obtain a linear map
dν(p) :TpM→T
ν(p)S
m
=TpM
Here we use the fact thatT
ν(p)S
m
=ν(p)
⊥
and, by definition of the Gauß
mapν, the tangent space ofMatpis also equal toν(p)
⊥
. Thusdν(p) is a
linear map from the tangent space ofMatpto itself.

250 CHAPTER 5. CURVATURE
Definition 5.3.4.TheGaußian curvatureof the hypersurfaceMis the
real valued functionK:M→Rdefined by
K(p) := det
Γ
dν(p) :TpM→TpM
∆
forp∈M. (Replacingνby−νhas the effect of replacingKby(−1)
m
K;
soKis independent of the choice of the Gauß map whenmis even.)
Remark 5.3.5.Given a subsetB⊂M, the setν(B)⊂S
m
is often called
thespherical imageofB. Ifνis a diffeomorphism on a neighborhood
ofB, the change of variables formula for an integral gives
Z
ν(B)
µS=
Z
B
|K|µM.
HereµMandµSdenote the volume elements onMandS
m
, respectively.
Introducing the notation AreaM(B) :=
R
B
µMwe obtain the formula
|K(p)|= lim
B→p
AreaS(ν(B))
AreaM(B)
.
This says that the curvature atpis roughly the ratio of the (m-dimensional)
area of the spherical imageν(B) to the area ofBwhereBis a very small open
neighborhood ofpinM. The sign ofK(p) is positive when the linear map
dν(p) :TpM→TpMpreserves orientation and negative when it reverses
orientation.
Remark 5.3.6.We see that the Gaußian curvature is a natural general-
ization ofEuler’s curvaturefor a plane curve. Indeed ifM⊂R
2
is a
1-manifold andp∈M, we can choose a curveγ= (x, y) : (−ε, ε)→Msuch
thatγ(0) =pand|˙γ(s)|= 1 for everys. This curve parametrizesMby the
arclength and the unit normal vector pointing to the right with respect to
the orientation ofγisν(x, y) = ( ˙y,−˙x). This is a local Gauß map and its
derivative (¨y,−¨x) is tangent to the curve. The inner product of the latter
with the unit tangent vector ˙γ= ( ˙x,˙y) is the Gaußian curvature. Thus
K:=
dx
ds
d
2
y
ds
2
−
dy
ds
d
2
x
ds
2
=
dθ
ds
wheresis the arclength parameter andθis the angle made by the normal
(or the tangent) with some constant line. With this conventionKis positive
at a left turn and negative at a right turn.

5.3. GENERALIZED THEOREMA EGREGIUM 251
Exercise 5.3.7.The Gaußian curvature of anm-dimensional sphere of ra-
diusris constant and has the valuer
−m
(with respect to an outward pointing
Gauß map whenmis odd).
Exercise 5.3.8.Show that the Gaußian curvature of the surfacez=x
2
−y
2
is−4 at the origin.
We now restrict to the case ofsurfaces, i.e. of 2-dimensional submani-
folds ofR
3
. Figure 5.1 illustrates the difference between positive and nega-
tive Gaußian curvature in dimension two.K > 0 K < 0K = 0
Figure 5.1: Positive and negative Gaußian curvature.
Theorem 5.3.9(Gaußian curvature).LetM⊂R
3
be a surface and fix
a pointp∈M. Ifu, v∈TpMis a basis, then
K(p) =
⟨R(u, v)v, u⟩
|u|
2
|v|
2
− ⟨u, v⟩
2
. (5.3.6)
Moreover,
R(u, v)w=−K(p)⟨ν(p), u×v⟩ν(p)×w (5.3.7)
for allu, v, w∈TpM.
Proof.The orthogonal projection ofR
3
onto the tangent spaceTpM=ν(p)
⊥
is given by the 3×3-matrix
Π(p) = 1l−ν(p)ν(p)
T
.
Hence
dΠ(p)u=−ν(p)(dν(p)u)
T
−
Γ
dν(p)u
∆
ν(p)
T
.
Here the first summand is the second fundamental form, which mapsTpM
toTpM
⊥
, and the second summand is its dual, which mapsTpM
⊥
toTpM.
Thus
hp(v) =−ν(p)
Γ
dν(p)v
∆
T
:TpM→TpM
⊥
,
hp(u)
∗
=−
Γ
dν(p)u
∆
ν(p)
T
:TpM
⊥
→TpM.

252 CHAPTER 5. CURVATURE
By the Gauß–Codazzi formula this implies
Rp(u, v)w=hp(u)
∗
hp(v)w−hp(v)
∗
hp(u)w
=
Γ
dν(p)u
∆Γ
dν(p)v
∆
T
w−
Γ
dν(p)v
∆Γ
dν(p)u
∆
T
w
=⟨dν(p)v, w⟩dν(p)u− ⟨dν(p)u, w⟩dν(p)v
and hence
⟨Rp(u, v)w, z⟩=⟨dν(p)u, z⟩⟨dν(p)v, w⟩ − ⟨dν(p)u, w⟩⟨dν(p)v, z⟩.(5.3.8)
Now fix four tangent vectorsu, v, w, z∈TpMand consider the composition
R
3A
−→R
3B
−→R
3C
−→R
3
of the linear maps
Aξ:=ξ
1
ν(p) +ξ
2
u+ξ
3
v,
Bη:=
æ
dν(p)η,ifη⊥ν(p),
η, ifη∈Rν(p),
Cζ:=


⟨ζ, ν(p)⟩
⟨ζ, z⟩
⟨ζ, w⟩

.
This composition is represented by the matrix
CBA=


1 0 0
0⟨dν(p)u, z⟩ ⟨dν(p)v, z⟩
0⟨dν(p)u, w⟩ ⟨dν(p)v, w⟩

.
Hence, by (5.3.8), we have
⟨Rp(u, v)w, z⟩= det(CBA)
= det(A) det(B) det(C)
=⟨ν(p), u×v⟩K(p)⟨ν(p), z×w⟩
=−K(p)⟨ν(p), u×v⟩⟨ν(p)×w, z⟩.
This implies (5.3.7) and
⟨Rp(u, v)v, u⟩=K(p)⟨ν(p), u×v⟩
2
=K(p)|u×v|
2
=K(p)
ı
|u|
2
|v|
2
− ⟨u, v⟩
2
ȷ
.
This proves Theorem 5.3.9.

5.3. GENERALIZED THEOREMA EGREGIUM 253
Remark 5.3.10.Equation (5.3.6) implies
⟨Rp(u, v)w, z⟩=K(p)
ı
⟨u, z⟩⟨v, w⟩ − ⟨u, w⟩⟨v, z⟩
ȷ
(5.3.9)
for allp∈Mand allu, v, w, z∈TpM. This is proved in Theorem 6.4.8
below.Exercise:Deduce this formula from (5.3.7).
Corollary 5.3.11(Theorema Egregium of Gauß). The Gaußian cur-
vature is intrinsic, i.e. if
ϕ:M→M
′
is an isometry of surfaces inR
3
, then
K=K
′
◦ϕ:M→R.
Proof.Theorem 5.3.1 and Theorem 5.3.9.
Exercise 5.3.12.Form= 1 the Gaußian curvature is clearlynotintrinsic
as any two curves are locally isometric (parameterized by arclength). Show
that the curvatureK(p) is intrinsic for evenmwhile its absolute value|K(p)|
is intrinsic for oddm≥3.Hint:We still have the equation (5.3.8) which,
forz=uandv=w, can be written in the form
⟨Rp(u, v)v, u⟩= det
`
⟨dν(p)u, u⟩ ⟨dν(p)u, v⟩
⟨dν(p)v, u⟩ ⟨dν(p)v, v⟩
´
.
Thus, for every orthonormal basisv1, . . . , vmofTpM, the 2×2 minors of
the matrix
(⟨dν(p)vi, vj⟩)
i,j=1,...,m
are intrinsic. Hence everything reduces to the following assertion.
Lemma.The determinant of anm×mmatrix is an expression in its2×2
minors ifmis even; the absolute value of the determinant is an expression
in the2×2minors ifmis odd and greater than or equal to3.
The lemma is proved by induction onm. For the absolute value, note the
formula
det(A)
m
= det(det(A)1lm) = det(AB) = det(A) det(B)
for anm×m-matrixAwhereBis the transposed matrix of cofactors.

254 CHAPTER 5. CURVATURE
5.4 Curvature in Local Coordinates*
Riemann
LetM⊂R
k
be anm-dimensional manifold and letϕ=ψ
−1
:U→Ω
be a local coordinate chart on an open setU⊂Mwith values in an open
set Ω⊂R
m
. Define the vector fieldsE1, . . . , Emalongψby
Ei(x) :=
∂ψ
∂x
i
(x)∈T
ψ(x)M.
These vector fields form a basis ofT
ψ(x)Mfor everyx∈Ω and the coeffi-
cientsgij: Ω→Rof the first fundamental form aregij=⟨Ei, Ej⟩.Recall
from Lemma 3.6.5 that the Christoffel Γ
k
ij
: Ω→Rare the coefficients of
the Levi-Civita connection, defined by
∇iEj=
m
X
k=1
Γ
k
ijEk
and that they are given by the formula
Γ
k
ij:=
m
X
ℓ=1
g
kℓ
1
2
Γ
∂igjℓ+∂jgiℓ−∂ℓgij
∆
.
Define the coefficientsR
ℓ
ijk
: Ω→RandRijkℓ: Ω→Rof the Riemann cur-
vature tensor by
R(Ei, Ej)Ek=
m
X
ℓ=1
R
ℓ
ijk
Eℓ, (5.4.1)
Rijkℓ:=⟨R(Ei, Ej)Ek, Eℓ⟩=
m
X
ν=1
R
ν
ijk
gνℓ. (5.4.2)
These coefficients are given by
R
ℓ
ijk
=∂iΓ
ℓ
jk
−∂jΓ
ℓ
ik
+
m
X
ν=1
ı
Γ
ℓ
iνΓ
ν
jk
−Γ
ℓ
jνΓ
ν
ik
ȷ
. (5.4.3)
The coefficients of the Riemann curvature tensor have the symmetries
Rijkℓ=−Rjikℓ=−Rijℓk=Rkℓij (5.4.4)
and thefirst Bianchi identityhas the form
R
ℓ
ijk
+R
ℓ
jki
+R
ℓ
kij
= 0, R ijkℓ+Rjkiℓ+Rkijℓ= 0. (5.4.5)
Warning:Care must be taken with the ordering of the indices. Some
authors use the notationR
ℓ
kij
for what we callR
ℓ
ijk
andRℓkijfor what we
callRijkℓ.

5.4. CURVATURE IN LOCAL COORDINATES* 255
Exercise 5.4.1.Prove equations (5.4.3), (5.4.4), and (5.4.5). Use (5.4.3)
to give an alternative proof of Theorem 5.3.1.
Gauß
IfM⊂R
n
is a 2-manifold (not necessarily embedded inR
3
), we can use
equation (5.3.6) as the definition of the Gaußian curvatureK:M→R.
Letψ: Ω→Ube a local parametrization of an open setU⊂Mdefined on
an open set Ω⊂R
2
. Denote the coordinates inR
2
by (x, y) and define the
functionsE, F, G: Ω→Rby
E:=|∂xψ|
2
, F :=⟨∂xψ, ∂yψ⟩, G :=|∂yψ|
2
.
We abbreviateD:=EG−F
2
.Then the composition of the Gaußian curva-
tureK:M→Rwith the parametrizationψis given by
K◦ψ=
1
D
2
det


E F ∂ yF−
1
2
∂xG
F G
1
2
∂yG
1
2
∂xE ∂xF−
1
2
∂yE−
1
2
∂
2
yE+∂x∂yF−
1
2
∂
2
xG


−
1
D
2
det


E F
1
2
∂yE
F G
1
2
∂xG
1
2
∂yE
1
2
∂xG 0


=−
1
2
√
D
∂
∂x
`
∂xG−∂yF
√
D
´
−
1
2
√
D
∂
∂y
`
∂yE−∂xF
√
D
´
−
1
4D
2
det


E ∂xE ∂yE
F ∂xF ∂yF
G ∂xG ∂yG

.
This expression simplifies dramatically whenF= 0 and we get
K◦ψ=−
1
2
√
EG
`
∂
∂x
∂xG
√
EG
+
∂
∂y
∂yE
√
EG
´
. (5.4.6)
Exercise 5.4.2.Prove that the Riemannian metric
E=G=
4
(1 +x
2
+y
2
)
2
, F = 0,
onR
2
has constant constant curvatureK= 1 and the Riemannian metric
E=G=
4
(1−x
2
−y
2
)
2
, F = 0,
on the open unit disc has constant curvatureK=−1.

256 CHAPTER 5. CURVATURE

Chapter 6
Geometry and Topology
In this chapter we address what might be called the fiundamental problem
of intrinsic differential geometry”: when are two manifolds isometric? The
central tool for addressing this question is the Cartan–Ambrose–Hicks The-
orem (§6.1). In the subsequent sections we will use this result to examine
flat spaces (§6.2), symmetric spaces (§6.3), and constant sectional curvature
manifolds (§6.4). The chapter then examines manifolds of nonpositive sec-
tional curvature and includes a proof of the Cartan Fixed Point Theorem
(§6.5). The last three sections introduce the Ricci tensor and show that
complete manifolds with uniformly positive Ricci tensor are compact (§6.6)
and discuss the scalar curvature (§6.7) and the Weyl tensor (§6.8).
6.1 The Cartan–Ambrose–Hicks Theorem
The Cartan–Ambrose–Hicks Theorem answers the question (at least locally)
when two manifolds are isometric. In general the equivalent conditions given
there are probably more difficult to verify in most examples than the con-
dition that there exist an isometry. However, under additional assumptions
it has many important consequences. The section starts with some basic
observations about homotopy and simple connectivity.
6.1.1 Homotopy
Definition 6.1.1.LetMbe a manifold and letI= [a, b]be a compact
interval. A(smooth) homotopyof maps fromItoMis a smooth map
γ: [0,1]×I→M. We often writeγλ(t) =γ(λ, t)forλ∈[0,1]andt∈I
and callγa(smooth) homotopy between γ0andγ1. We say the
257

258 CHAPTER 6. GEOMETRY AND TOPOLOGY
homotopy hasfixed endpointsifγλ(a) =γ0(a)andγλ(b) =γ0(b)for all
λ∈[0,1]. (See Figure 6.1.)
We remark that a homotopy and a variation are essentially the same
thing, namely a curve of maps (curves). The difference is pedagogical. We
used the word “variation” to describe a curve of maps through a given
map; when we use this word we are going to differentiate the curve to find
a tangent vector (field) to the given map. The word “homotopy” is used
to describe a curve joining two maps; it is a global rather than a local
(infinitesimal) concept.1
γ
γ
0
M
Figure 6.1: A homotopy with fixed endpoints.
Definition 6.1.2.A manifoldMis calledsimply connectediff for any
two curvesγ0, γ1: [a, b]→Mwithγ0(a) =γ1(a)andγ0(b) =γ1(b)there ex-
ists a homotopy fromγ0toγ1with endpoints fixed. (The idea is that the
spaceΩp,qof curves fromptoqis connected.)
Remark 6.1.3.Two smooth mapsγ0, γ1: [a, b]→Mwith the same end-
points can be joined by a continuous homotopy if and only if they can be
joined by a smooth homotopy. This follows from the Weierstrass approxi-
mation theorem.
Remark 6.1.4.AssumeMis a connected smooth manifold. Then the
topological space Ωp,qof all smooth curves inMwith the endpointspandq
is connected for some pair of pointsp, q∈Mif and only if it is connected
for every pair of pointsp, q∈M. (Prove this!)
Example 6.1.5.The Euclidean spaceR
m
is simply connected; any two
curvesγ0, γ1: [a, b]→R
m
with the same endpoints can be joined by the
homotopyγλ(t) :=γ0(t) +λ(γ1(t)−γ0(t)).
Example 6.1.6.The punctured planeC\ {0}is not simply connected; two
curves of the form
γn(t) :=e
2πint
,0≤t≤1,
are not homotopic with fixed endpoints for distinct integersn.
Exercise 6.1.7.Prove that them-sphereS
m
is simply connected form̸= 1.

6.1. THE CARTAN–AMBROSE–HICKS THEOREM 259
6.1.2 The Global C-A-H Theorem
Theorem 6.1.8(Global C-A-H Theorem). LetM⊂R
n
andM
′
⊂R
n
′
be nonempty, connected, simply connected, completem-manifolds. Fix two
elementsp0∈Mandp
′
0
∈M
′
and letΦ0:Tp0
M→T
p
′
0
M
′
be an orthogonal
linear isomorphism. Then the following are equivalent.
(i)There exists an isometryϕ:M→M
′
satisfying
ϕ(p0) =p
′
0, dϕ(p0) = Φ0. (6.1.1)
(ii)If(Φ, γ, γ
′
)is a development satisfying the initial condition
γ(0) =p0, γ
′
(0) =p
′
0,Φ(0) = Φ0, (6.1.2)
then
γ(1) =p0 =⇒ γ
′
(1) =p
′
0,Φ(1) = Φ0
(iii)If(Φ0, γ0, γ
′
0
)and(Φ1, γ1, γ
′
1
)are developments satisfying(6.1.2), then
γ0(1) =γ1(1) =⇒ γ
′
0(1) =γ
′
1(1).
(iv)If(Φ, γ, γ
′
)is a development satisfying(6.1.2), thenΦ∗Rγ=R
′
γ
′.M
p
M’
0
Figure 6.2: Diagram for Example 6.1.9.
Example 6.1.9.Before giving the proof let us interpret the conditions in
caseMandM
′
are two-dimensional spheres of radiusrandr
′
respectively in
three-dimensional Euclidean spaceR
3
. Imagine that the spheres are tangent
atp0=p
′
0
. Clearly the spheres will be isometric exactly whenr=r
′
.
Condition (ii) says that if the spheres are rolled along one another with-
out sliding or twisting, then the endpointγ
′
(1) of one curve of contact
depends only on the endpointγ(1) of the other and not on the intervening
curveγ(t). This condition is violated in the caser̸=r
′
(see Figure 6.2).

260 CHAPTER 6. GEOMETRY AND TOPOLOGY
By Theorem 5.3.9 the Riemann curvature of a 2-manifold atpis deter-
mined by the Gaußian curvatureK(p); and for spheres we haveK(p) = 1/r
2
.
Exercise 6.1.10.Letγbe the closed curve which bounds an octant as
shown in the diagram for Example 6.1.9. Findγ
′
.
Exercise 6.1.11.Show that in caseMis two-dimensional, the condition
Φ(1) = Φ0in Theorem 6.1.8 may be dropped from (ii).
Lemma 6.1.12.Letϕ:M→M
′
be a local isometry and letγ:I→Mbe
a smooth curve on an intervalI. Fix an elementt0∈Iand define
p0:=γ(t0), q0:=ϕ(p0),Φ0:=dϕ(p0). (6.1.3)
Then there exists a unique development(Φ, γ, γ
′
)ofMalongM
′
on the
entire intervalIsatisfying the initial conditions
γ
′
(t0) =q0,Φ(t0) = Φ0. (6.1.4)
This development is given by
γ
′
(t) =ϕ(γ(t)),Φ(t) =dϕ(γ(t)) (6.1.5)
fort∈I.
Proof.Define
γ
′
(t) :=ϕ(γ(t)),Φ(t) :=dϕ(γ(t))
fort∈I. Then ˙γ
′
(t) = Φ(t) ˙γ(t) for allt∈Iby the chain rule, and ev-
ery vector fieldXalongγsatisfies Φ∇X=∇
′
(ΦX) by Corollary 5.3.2.
Hence (Φ, γ, γ
′
) is a development by Lemma 3.5.19. By (6.1.3) this devel-
opment satisfies the initial condition (6.1.4). Hence the assertion follows
from the uniqueness result for developments in Theorem 3.5.21. This proves
Lemma 6.1.12.
Proof of Theorem 6.1.8.We first prove a slightly different theorem. Namely,
we weaken condition (i) to assert thatϕis a local isometry (i.e. not neces-
sarily bijective), and prove that this weaker condition is equivalent to (ii),
(iii), and (iv) wheneverMis connected and simply connected andM
′
is
complete. Thus we drop the hypotheses thatMbe complete andM
′
be
connected and simply connected.
We prove that (i) implies (ii). Given a development as in (ii) we have,
by Lemma 6.1.12,
γ
′
(1) =ϕ(γ(1)) =ϕ(p0) =p
′
0,Φ(1) =dϕ(γ(1)) =dϕ(p0) = Φ0,
as required.

6.1. THE CARTAN–AMBROSE–HICKS THEOREM 261
We prove that (ii) implies (iii) whenM
′
is complete. Choose develop-
ments (Φi, γi, γ
′
i
) fori= 0,1 as in (iii). Define a curveγ: [0,1]→Mby
“composition”, i.e.
γ(t) :=
æ
γ0(2t), 0≤t≤1/2,
γ1(2−2t),1/2≤t≤1,
so thatγis continuous and piecewise smooth andγ(1) =p0. By Theo-
rem 3.5.21 there exists a development (Φ, γ, γ
′
) on the interval [0,1] satis-
fying (6.1.2) (becauseM
′
is complete). Sinceγ(1) =p0it follows from (ii)
thatγ
′
(1) =p
′
0
and Φ(1) = Φ0.By the uniqueness of developments and the
invariance under reparametrization, we have
(Φ(t), γ(t), γ
′
(t)) =
æ
(Φ0(2t), γ0(2t), γ
′
0
(2t)), 0≤t≤1/2,
(Φ1(2−2t), γ1(2−2t), γ
′
1
(2−2t)),1/2≤t≤1.
Henceγ
′
0
(1) =γ
′
(1/2) =γ
′
1
(1) as required.
We prove that (iii) implies (i) whenM
′
is complete andMis connected.
Define the mapϕ:M→M
′
as follows. Fix an elementp∈M. SinceMis
connected, there exists a smooth curveγ: [0,1]→Msuch thatγ(0) =p0
andγ(1) =p. SinceM
′
is complete, there exists a development (Φ, γ, γ
′
)
withγ
′
(0) =p
′
0
and Φ(0) = Φ0(Theorem 3.5.21). Now defineϕ(p) :=γ
′
(1).
By (iii) the endpointp
′
:=γ
′
(1) is independent of the choice of the curveγ,
and soϕis well-defined. We prove that this mapϕsatisfies the following
(a)If(Φ, γ, γ
′
)is a development satisfyingγ(0) =p0,γ
′
(0) =p
′
0
,Φ(0) = Φ0,
thenϕ(γ(t)) =γ
′
(t)for0≤t≤1.
(b)Ifp, q∈Msatisfy0< d(p, q)<inj(p;M)andd(p, q)<inj(ϕ(p);M
′
),
thend
′
(ϕ(p), ϕ(q)) =d(p, q).
Thatϕsatisfies (a) follows directly from the definition and the fact that the
triple (Φt, γt, γ
′
t) defined by Φt(s) := Φ(st), γt(s) :=γ(st), γ
′
t(s) :=γ
′
(st)
for 0≤s≤1 is a development. To prove (b), choosev∈TpMsuch that
|v|=d(p, q) exp
p(v) =q
(Theorem 4.4.4) and letγ: [0,1]→Mbe a smooth curve with
γ(0) =p0, γ(t) = exp
p((2t−1)v)
for
1
2
≤t≤1. Let (Φ, γ, γ
′
) be the unique development ofMalongM
′
satisfyingγ
′
(0) =p
′
0
and Φ(0) = Φ0(Theorem 3.5.21). Then, by (a),
γ
′
(
1
2
) =ϕ(p), γ
′
(1) =ϕ(q).

262 CHAPTER 6. GEOMETRY AND TOPOLOGY
Also, by part (ii) of Lemma 3.5.19 withX= ˙γ, the restriction ofγ
′
to the
interval [
1
2
,1] is a geodesic. Thusγ
′
(t) = exp
ϕ(p)((2t−1)v
′
) for
1
2
≤t≤1,
where the tangent vectorv
′
∈T
ϕ(p)M
′
is given byv
′
:= ˙γ
′
(
1
2
) = Φ(
1
2
)vand
hence satisfies|v
′
|=|v|=d(p, q)<inj(ϕ(p), M
′
). Thus it follows from The-
orem 4.4.4 thatd
′
(ϕ(p), ϕ(q)) =d
′
(ϕ(p),exp
′
ϕ(p)
(v
′
)) =|v
′
|=d(p, q) and this
proves (b). It follows from (b) and Theorem 5.1.1 thatϕis a local isometry.
We prove that (i) implies (iv). Given a development as in (ii) we have
γ
′
(t) =ϕ(γ(t)),Φ(t) =dϕ(γ(t))
for everyt, by Lemma 6.1.12. Hence it follows from part (iv) of Corol-
lary 5.3.2 (Theorema Egregium for local isometries) that
Φ(t)∗R
γ(t)= (ϕ∗R)
γ
′
(t)=R
′
γ
′
(t)
for alltas required.
We prove that (iv) implies (iii) whenM
′
is complete andMis simply
connected. Choose developments (Φi, γi, γ
′
i
) fori= 0,1 as in (iii). SinceM
is simply connected there exists a homotopy
[0,1]×[0,1]→M: (λ, t)7→γ(λ, t) =γλ(t)
fromγ0toγ1with endpoints fixed. By Theorem 3.5.21 there is, for eachλ,
a development (Φλ, γλ, γ
′
λ
) on the interval [0,1] with initial conditions
γ
′
λ
(0) =p
′
0,Φλ(0) = Φ0
(becauseM
′
is complete). The proof of Theorem 3.5.21 also shows that
γλ(t) and Φλ(t) depend smoothly on bothtandλ. We must prove that
γ
′
1(1) =γ
′
0(1).
To see this we will show that, for each fixedt, the curve
λ7→(Φλ(t), γλ(t), γ
′
λ
(t))
is a development; then by the definition of development we have that the
curveλ7→γ
′
λ
(1) is smooth and
∂λγ
′
λ
(1) = Φλ(1)∂λγλ(1) = 0
as required.

6.1. THE CARTAN–AMBROSE–HICKS THEOREM 263
First choose a basise1, . . . , emofTp0
Mand extend it to obtain vector
fieldsEi∈Vect(γ) along the homotopyγby imposing the conditions that
the vector fieldst7→Ei(λ, t) be parallel, i.e.
∇tEi(λ, t) = 0, E i(λ,0) =ei. (6.1.6)
Then the vectorsE1(λ, t), . . . Em(λ, t) form a basis ofT
γλ(t)Mfor allλandt.
Second, define the vector fieldsE
′
i
alongγ
′
by
E
′
i(λ, t) := Φλ(t)Ei(λ, t) (6.1.7)
so that∇
′
tE
′
i
= 0. Third, define the functionsξ
1
, . . . , ξ
m
: [0,1]
2
→Rby
∂tγ=:
m
X
i=1
ξ
i
Ei, ∂ tγ
′
=
m
X
i=1
ξ
i
E
′
i. (6.1.8)
Here the second equation follows from (6.1.7) and the fact that Φλ∂tγ=∂tγ
′
.
Now consider the vector fields
X
′
:=∂λγ
′
, Y
′
i:=∇
′
λ
E
′
i (6.1.9)
alongγ
′
. They satisfy the equations
∇
′
tX
′
=∇
′
t∂λγ
′
=∇
′
λ
∂tγ
′
=∇
′
λ

m
X
i=1
ξ
i
E
′
i
!
=
m
X
i=1
Γ
∂λξ
i
E
′
i+ξ
i
Y
′
i
∆
and
∇
′
tY
′
i=∇
′
t∇
′
λ
E
′
i− ∇
′
λ
∇
′
tE
′
i=R
′
(∂tγ
′
, ∂λγ
′
)E
′
i.
To sum up we haveX
′
(λ,0) =Y
′
i
(λ,0) = 0 and
∇
′
tX
′
=
m
X
i=1
Γ
∂λξ
i
E
′
i+ξ
i
Y
′
i
∆
,∇
′
tY
′
i=R
′
(∂tγ
′
, X
′
)E
′
i. (6.1.10)
On the other hand, the vector fields
X
′
:= Φλ∂λγ, Y
′
i:= Φλ∇λEi (6.1.11)
alongγ
′
satisfy the same equations, namely
∇
′
tX
′
= Φλ∇t∂λγ= Φλ∇λ∂tγ= Φλ∇λ

m
X
i=1
ξ
i
Ei
!
= Φλ
m
X
i=1
Γ
∂λξ
i
Ei+ξ
i
∇λEi
∆
=
m
X
i=1
Γ
∂λξ
i
E
′
i+ξ
i
Y
′
i
∆
,
∇
′
tY
′
i= Φλ
Γ
∇t∇λEi− ∇λ∇tEi
∆
= ΦλR(∂tγ, ∂λγ)Ei
=R
′
(Φλ∂tγ,Φλ∂λγ)ΦλEi=R
′
(∂tγ
′
, X
′
)E
′
i.
Here the last but one equation follows from (iv).

264 CHAPTER 6. GEOMETRY AND TOPOLOGY
Since the tuples (6.1.9) and (6.1.11) satisfy the same differential equa-
tion (6.1.10) and vanish att= 0 they must agree. Hence
∂λγ
′
= Φλ∂λγ, ∇
′
λ
E
′
i= Φλ∇λEi
fori= 1, . . . , m. This says thatλ7→(Φλ(t), γλ(t), γ
′
λ
(t)) is a development.
Fort= 1 we obtain∂λγ
′
(λ,1) = 0 as required.
Now the modified theorem (whereϕis a local isometry) is proved. The
original theorem follows immediately. Condition (iv) is symmetric inM
andM
′
. Thus, if we assume (iv), there are local isometriesϕ:M→M
′
andψ:M
′
→Msatisfyingϕ(p0) =p
′
0
,dϕ(p0) = Φ0andψ(p
′
0
) =p0,
dψ(p
′
0
) = Φ
−1
0
. But thenψ◦ϕis a local isometry withψ◦ϕ(p0) =p0and
d(ψ◦ϕ)(p0) = id. Henceψ◦ϕis the identity. Similarlyϕ◦ψis the identity
soϕis bijective (andψ=ϕ
−1
) as required. This proves Theorem 6.1.8.
Remark 6.1.13.The proof of Theorem 6.1.8 shows that the various im-
plications in theweak versionof the theorem (whereϕis only a local
isometry) require the following conditions onMandM
′
:
(i) always implies (ii), (iii), and (iv);
(ii) implies (iii) wheneverM
′
is complete;
(iii) implies (i) wheneverM
′
is complete andMis connected;
(iv) implies (iii) wheneverM
′
is complete andMis simply connected.
Remark 6.1.14.The proof that (iii) implies (i) in Theorem 6.1.8 can be
slightly shortened by using the following observation.Letϕ:M→M
′
be
a map between smooth manifolds. Assume thatϕ◦γis smooth for every
smooth curveγ: [0,1]→M. Thenϕis smooth.
Corollary 6.1.15.LetMandM
′
be nonempty, connected, simply con-
nected, complete Riemannian manifolds and letϕ:M→M
′
be a local
isometry. Thenϕis bijective and hence is an isometry.
Proof.This follows by combining the weak and strong versions of the global
C-A-H Theorem 6.1.8. Letp0∈Mand definep
′
0
:=ϕ(p0) and Φ0:=dϕ(p0).
Then the tupleM, M
′
, p0, p
′
0
,Φ0satisfies condition (i) of the weak version
of Theorem 6.1.8. Hence this tuple also satisfies condition (iv) of Theo-
rem 6.1.8. SinceMandM
′
are connected, simply connected, and complete
we may apply the strong version of Theorem 6.1.8 to obtain an isometry
ψ:M→M
′
satisfyingψ(p0) =p
′
0
anddψ(p0) = Φ0.Since every isometry
is also a local isometry andMis connected it follows from Lemma 5.1.10
thatϕ(p) =ψ(p) for allp∈M. Henceϕis an isometry, as required.

6.1. THE CARTAN–AMBROSE–HICKS THEOREM 265
Remark 6.1.16.Refining the argument in the proof of Corollary 6.1.15 one
can show that a local isometryϕ:M→M
′
must be surjective wheneverM
is complete andM
′
is connected. None of these assumptions can be removed.
(Take an isometric embedding of a disc in the plane or an embedding of
a complete spaceMinto a space with two components, one of which is
isometric toM.)
Likewise, one can show that a local isometryϕ:M→M
′
must be
injective wheneverMis complete and connected andM
′
is simply connected.
Again none of these asumptions can be removed. (Take a coveringR→S
1
,
or a covering of a disjoint union of two isometric complete simply connected
spaces onto one copy of this space, or some noninjective immersion of a disc
into the plane and choose the pullback metric on the disc.)
6.1.3 The Local C-A-H Theorem
Theorem 6.1.17(Local C-A-H Theorem). LetMandM
′
be smooth
m-manifolds, letp0∈Mandp
′
0
∈M
′
, and letΦ0:Tp0
M→T
p
′
0
M
′
be an
orthogonal linear isomorphism. Letr >0be smaller than the injectvity radii
ofMatp0and ofM
′
atp
′
0
and define
Ur:={p∈M|d(p0, p)< r}, U
′
r:=
Φ
p
′
∈M
′
|d
′
(p
′
0, p
′
)< r

.
Then the following are equivalent.
(i)There exists an isometryϕ:Ur→U
′
rsatisfying(6.1.1).
(ii)If(Φ, γ, γ
′
)is a development on an intervalI⊂Rwith0∈I, satisfying
the initial condition(6.1.2)as well asγ(I)⊂Urandγ
′
(I)⊂U
′
r, then
γ(1) =p0 =⇒ γ
′
(1) =p
′
0,Φ(1) = Φ0.
(iii)If(Φ0, γ0, γ
′
0
)and(Φ1, γ1, γ
′
1
)are developments as in (ii), then
γ0(1) =γ1(1) =⇒ γ
′
0(1) =γ
′
1(1).
(iv)Ifv∈Tp0
Mwith|v|< rand
γ(t) := exp
p0
(tv), γ
′
(t) := exp
′
p
′
0
(tΦ0v),Φ(t) := Φ
′
γ
′(t,0)Φ0Φγ(0, t),
thenΦ(t)∗R
γ(t)=R
′
γ
′
(t)
for0≤t≤1.
If these equivalent conditions are satisfied, then
ϕ(exp
p0
(v)) = exp
′
p
′
0
(Φ0v)
for allv∈Tp0
Mwith|v|< r.
The proof is based on the following lemma.

266 CHAPTER 6. GEOMETRY AND TOPOLOGY
Lemma 6.1.18.Letp∈Mandv, w∈TpM. For0≤t≤1define
γ(t) := exp(tv), X(t) :=
∂
∂λ

λ=0
exp
p
Γ
t(v+λw)
∆
∈T
γ(t)M.
Then
∇t∇tX+R(X,˙γ) ˙γ= 0, X(0) = 0,∇tX(0) =w. (6.1.12)
A vector field alongγsatisfying the first equation in(6.1.12)is called a
Jacobi field alongγ.
Proof.Define
γ(λ, t) := exp
p(t(v+λw)), X(λ, t) :=∂λγ(λ, t)
for allλandt. Sinceγ(λ,0) =pfor allλwe haveX(λ,0) = 0 and
∇tX(λ,0) =∇t∂λγ(λ,0) =∇λ∂tγ(λ,0) =
d
dλ
Γ
v+λw
∆
=w.
Moreover,∇t∂tγ= 0 and hence
∇t∇tX=∇t∇t∂λγ
=∇t∇λ∂tγ− ∇λ∇t∂tγ
=R(∂tγ, ∂λγ)∂tγ
=R(∂tγ, X)∂tγ.
This proves Lemma 6.1.18.
Proof of Theorem 6.1.17.The proofs (i) =⇒(ii) =⇒(iii) =⇒(i) =⇒(iv)
are as before; the reader might note that whenL(γ)≤rwe also have
L(γ
′
)≤rfor any development so that there are plenty of developments
withγ: [0,1]→Urandγ
′
: [0,1]→U
′
r. The proof that (iv) implies (i) is
a little different since (iv) here is somewhat weaker than (iv) of the global
theorem: the equation Φ∗R=R
′
is only assumed for certain developments.
Hence assume (iv) and defineϕ:Ur→U
′
rby
ϕ:= exp
′
p
′
0
◦Φ0◦exp
−1
p0
:Ur→U
′
r.
We must prove thatϕis an isometry. Thus we fix a pointq∈Urand a
tangent vectoru∈TqMand choosev, w∈TpMwith|v|< rsuch that
exp
p0
(v) =q, dexp
p0
(v)w=u. (6.1.13)

6.1. THE CARTAN–AMBROSE–HICKS THEOREM 267
Defineγ: [0,1]→Ur,γ
′
: [0,1]→U
′
r,X∈Vect(γ), andX
′
∈Vect(γ
′
) by
γ(t) = exp
p0
(tv), X(t) :=
∂
∂λ

λ=0
exp
p0
(t(v+λw)),
γ
′
(t) = exp
′
p
′
0
(tΦ0v), X
′
(t) :=
∂
∂λ

λ=0
exp
′
p
′
0
(t(Φ0v+λΦ0w)).
Then, by definition ofϕ, we have
γ
′
:=ϕ◦γ, dϕ(γ)X=X
′
. (6.1.14)
By Lemma 6.1.18,Xis a solution of (6.1.12) andX
′
is a solution of
∇t∇tX
′
=R
′
(∂tγ
′
, X
′
)∂tγ
′
, X
′
(λ,0) = 0,∇
′
tX
′
(λ,0) = Φ0w.(6.1.15)
Now define Φ(t) :T
γ(t)M→T
γ
′
(t)M
′
by
Φ(t) := Φ
′
γ
′(t,0)Φ0Φγ(0, t).
Then Φ intertwines covariant differentiation. Since ˙γand ˙γ
′
are parallel
vector fields with ˙γ
′
(0) = Φ0v= Φ(0) ˙γ(0), we have
Φ(t) ˙γ(t) = ˙γ
′
(t)
for everyt. Moreover, it follows from (iv) that Φ∗Rγ=R
′
γ
′. Combining this
with (6.1.12) we obtain
∇
′
t∇
′
t(ΦX) = Φ∇t∇tX=R
′
(Φ ˙γ,ΦX)Φ ˙γ=R
′
( ˙γ
′
,ΦX) ˙γ
′
.
Hence the vector field ΦXalongγ
′
also satisfies the initial value prob-
lem (6.1.15) and thus
ΦX=X
′
=dϕ(γ)X.
Here we have also used (6.1.14). Using (6.1.13) we find
γ(1) = exp
p0
(v) =q, X(1) =dexp
p0
(v)w=u,
and so
dϕ(q)u=dϕ(γ(1))X(1) =X
′
(1) = Φ(1)u.
Since Φ(1) :T
γ(1)M→T
γ
′
(1)M
′
is an orthogonal transformation this gives
|dϕ(q)u|=|Φ(1)u|=|u|.
Henceϕis an isometry as claimed. This proves Theorem 6.1.17.

268 CHAPTER 6. GEOMETRY AND TOPOLOGY
6.2 Flat Spaces
Our aim in the next few sections is to give applications of the Cartan-
Ambrose-Hicks Theorem. It is clear that the hypothesis Φ∗R=R
′
forall
developments will be difficult to verify without drastic hypotheses on the
curvature. The most drastic such hypothesis is that the curvature vanishes
identically.
Definition 6.2.1.A Riemannian manifoldMis calledflatiff the Riemann
curvature tensorRvanishes identically.
Theorem 6.2.2.LetM⊂R
n
be a smoothm-manifold.
(i)Mis flat if and only if every point has a neighborhood which is isometric
to an open subset ofR
m
, i.e. at each pointp∈Mthere exist local coordinates
x
1
, . . . , x
m
such that the coordinate vectorfieldsEi=∂/∂x
i
are orthonormal.
(ii)AssumeMis connected, simply connected, and complete. ThenMis
flat if and only if there is an isometryϕ:M→R
m
onto Euclidean space.
Proof.Assertion (i) follows immediately from Theorem 6.1.17 and (ii) fol-
lows immediately from Theorem 6.1.8.
Exercise 6.2.3.Carry over the Cartan–Ambrose–Hicks theorem and The-
orem 6.2.2 to the intrinsic setting.
Exercise 6.2.4.A one-dimensional manifold is always flat.
Exercise 6.2.5.IfM1andM2are flat, so isM=M1×M2.
Example 6.2.6.By Exercises 6.2.4 and 6.2.5 the standard torus
T
m
=
Φ
z= (z1, . . . , zm)∈C
m

|z1|=· · ·=|zm|= 1

is flat.
Exercise 6.2.7.Fora, b >0 andc≥0 defineM⊂C
3
by
M:=M(a, b, c) :=
n
(u, v, w)∈C
3

|u|=a,|v|=b, w=c
u
a
v
b
o
.
ThenMis diffeomorphic to a torus (a product of two circles) andMis flat. If
a
′
, b
′
>0 andc
′
≥0, prove that there is an isometryϕfromM
′
=M(a
′
, b
′
, c
′
)
toM=M(a, b, c) if and only if the triples (a
′
, b
′
, c
′
) and (a, b, c) are related
by a permutation.

6.2. FLAT SPACES 269
Hint:Show first that an isometryϕ:M
′
→Mthat satisfies the condi-
tionϕ(a
′
, b
′
, c
′
) = (a, b, c) must have the form
ϕ(u
′
, v
′
, w
′
) =

a
`
u
′
a
′
´
α`
v
′
b
′
´
β
, b
`
u
′
a
′
´
γ`
v
′
b
′
´
δ
, c
`
u
′
a
′
´
α+γ`
v
′
b
′
´
β+δ
!
for integersα, β, γ, δthat satisfyαδ−βγ=±1. Show that this mapϕis an
isometry if and only if
a
′2
+c
′2
=α
2
a
2
+γ
2
b
2
+ (α+γ)
2
c
2
,
c
′2
=αβa
2
+γδb
2
+ (α+γ)(β+δ)c
2
,
b
′2
+c
′2
=β
2
a
2
+δ
2
b
2
+ (β+δ)
2
c
2
.
Exercise 6.2.8(Developable manifolds).Letn=m+1 and letE(t) be
a one-parameter family of hyperplanes inR
n
. Then there exists a smooth
mapu:R→R
n
such that
E(t) =u(t)
⊥
,|u(t)|= 1, (6.2.1)
for everyt. We assume that ˙u(t)̸= 0 for everytso thatu(t) and ˙u(t) are
linearly independent. Show that
L(t) :=u(t)
⊥
∩˙u(t)
⊥
= lim
s→t
E(t)∩E(s). (6.2.2)
ThusL(t) is a linear subspace of dimensionm−1. Now letγ:R→R
n
be
a smooth curve such that
⟨˙γ(t), u(t)⟩= 0,⟨˙γ(t),˙u(t)⟩ ̸= 0 (6.2.3)
for allt. This means that ˙γ(t)∈E(t) and ˙γ(t)/∈L(t); thusE(t) is spanned
byL(t) and ˙γ(t). Fort∈Randε >0 define
L(t)ε:={v∈L(t)| |v|< ε}.
LetI⊂Rbe a bounded open interval such that the restriction ofγto the
closure ofIis injective. Prove that, forε >0 sufficiently small, the set
M0:=
[
t∈I
ı
γ(t) +L(t)ε
ȷ
is a smooth manifold of dimensionm=n−1. A manifold which arises this
way is calleddevelopable. Show that the tangent spaces ofM0are the
original subspacesE(t), i.e.
TpM0=E(t) for p∈γ(t) +L(t)ε.

270 CHAPTER 6. GEOMETRY AND TOPOLOGY
(One therefore callsM0the“envelope”of the hyperplanesγ(t) +E(t).)
Show thatM0is flat. (Hint:use Gauß–Codazzi.) If (Φ, γ, γ
′
) is a de-
velopment ofM0alongR
m
, show that the mapϕ:M0→R
m
, defined
by
ϕ(γ(t) +v) :=γ
′
(t) + Φ(t)v
forv∈L(t)ε, is an isometry onto an open setM
′
0
⊂R
m
. Thus a development
“unrolls”M0onto the Euclidean spaceR
m
. Whenn= 3 andm= 2 one
can visualizeM0as a twisted sheet of paper (see Figure 6.3).
Figure 6.3: Developable surfaces.
Remark 6.2.9.Given a codimension-1 submanifold
M⊂R
m+1
and a curveγ:R→Mwe may form theosculating developableM0toM
alongγby taking
E(t) :=T
γ(t)M.
This developable has common affine tangent spaces withMalongγas
T
γ(t)M0=E(t) =T
γ(t)M
for everyt. This gives a nice interpretation of parallel transport:M0may be
unrolled onto a hyperplane where parallel transport has an obvious meaning
and the identification of the tangent spaces thereby defines parallel transport
inM. (See Remark 3.5.16.)
Exercise 6.2.10.Each of the following is a developable surface inR
3
.
(i)Acone on a plane curveΓ⊂H, i.e.
M={tp+ (1−t)q|t >0, q∈Γ}
whereH⊂R
3
is an affine hyperplane,p∈R
3
\H, and Γ⊂His a 1-manifold.

6.2. FLAT SPACES 271
(ii)Acylinder on a plane curveΓ, i.e.
M={q+tv|q∈Γ, t∈R}
whereHand Γ are as in (i) andvis a fixed vector not parallel toH. (This
is a cone with the cone pointpat infinity.)
(iii)Thetangent developableto a space curveγ:R→R
3
, i.e.
M={γ(t) +s˙γ(t)| |t−t0|< ε,0< s < ε},
where ˙γ(t0) and ¨γ(t0) are linearly independent andε >0 is sufficiently small.
(iv)Thepaper model of a M¨obius strip(see Figure 6.3).
Figure 6.4: A circular one-sheeted hyperboloid.
Remark 6.2.11.A 2-dimensional submanifoldM⊂R
3
is called aruled
surfaceiff there is a straight line inMthrough every point. Every devel-
opable surface is ruled, however, there are ruled surfaces that are not devel-
opable. An example is the manifoldM={γ(t) +s¨γ(t)| |t−t0|< ε,|s|< ε}
whereγ:R→R
3
is a smooth curve with|˙γ| ≡1 and ¨γ(t0)̸= 0, andε >0 is
sufficiently small; this surface is not developable in general. Other examples
are theelliptic hyperboloid of one sheet
M:=
æ
(x, y, z)∈R
3

x
2
a
2
+
y
2
b
2
−
z
2
c
2
= 1
œ
(6.2.4)
depicted in Figure 6.4, thehyperbolic paraboloid
M:=
æ
(x, y, z)∈R
3

z=
x
2
a
2
−
y
2
b
2
œ
. (6.2.5)
(both with two straight lines through every point inM),Pl¨ucker’s conoid
M:=
æ
(x, y, z)∈R
3

x
2
+y
2
̸= 0, z=
2xy
x
2
+y
2
œ
, (6.2.6)

272 CHAPTER 6. GEOMETRY AND TOPOLOGY
thehelicoid
M:=
(
(x, y, z)∈R
3

x+iy
p
x
2
+y
2
=e
iαz
)
, (6.2.7)
and theM¨obius strip
M:=





cos(s)
sin(s)
0

+
t
2


cos(s/2) cos(s)
cos(s/2) sin(s)
sin(s/2)



s∈Rand
−1< t <1



.(6.2.8)
These five surfaces have negative Gaußian curvature. The M¨obius strip
in (6.2.8) is not developable, while the paper model of the M¨obius strip is.
The helicoid in (6.2.7) is aminimal surface, i.e. itsmean curvature(the
trace of the second fundamental form) vanishes. A minimal surface which
is not ruled is thecatenoid
M:={(x, y, z)∈R
3
|x
2
+y
2
=c
2
cosh (z/c)}.
(Exercise:Prove all this.)
6.3 Symmetric Spaces
In the last section we applied the Cartan-Ambrose-Hicks Theorem in the flat
case; the hypothesis Φ∗R=R
′
was easy to verify since both sides vanish. To
find more general situations where we can verify this hypothesis note that
for any development (Φ, γ, γ
′
) satisfying the initial conditions
γ(0) =p0, γ
′
(0) =p
′
0,Φ(0) = Φ0,
we have
Φ(t) = Φ
′
γ
′(t,0)Φ0Φγ(0, t)
so that the hypothesis Φ∗R=R
′
is certainly implied by the three hypotheses
Φγ(t,0)∗Rp0
=R
γ(t)
Φ
′
γ
′(t,0)∗R
′
p
′
0
=R
′
γ
′
(t)
(Φ0)∗Rp0
=R
′
p
′
0
.
The last hypothesis is a condition on the initial linear isomorphism
Φ0:Tp0
M→T
p
′
0
M
′
while the former hypotheses are conditions onMandM
′
respectively,
namely, that the Riemann curvature tensor is invariant by parallel trans-
port. It is rather amazing that this condition is equivalent to a simple
geometric condition as we now show.

6.3. SYMMETRIC SPACES 273
6.3.1 Symmetric Spaces
Definition 6.3.1.A Riemannian manifoldMis calledsymmetric about
the pointp∈Miff there exists a (necessarily unique) isometry
ϕ:M→M
satisfying
ϕ(p) =p, dϕ(p) =−id; (6.3.1)
Mis called asymmetric spaceiff it is symmetric about each of its points.
A Riemannian manifoldMis calledlocally symmetric about the point
p∈Miff, forr >0sufficiently small, there exists an isometry
ϕ:Ur(p, M)→Ur(p, M), U r(p, M) :={q∈M|d(p, q)< r},
satisfying(6.3.1);Mis called alocally symmetric spaceiff it is locally
symmetric about each of its points.
Remark 6.3.2.The proof of Theorem 6.3.4 below will show that, ifMis
locally symmetric, the isometryϕ:Ur(p, M)→Ur(p, M) withϕ(p) =p
anddϕ(p) =−id exists whenever 0< r≤inj(p).
Exercise 6.3.3.Every symmetric space is complete.Hint:Ifγ:I→Mis
a geodesic andϕ:M→Mis a symmetry about the pointγ(t0) fort0∈I,
thenϕ(γ(t0+t)) =γ(t0−t) for allt∈Rwitht0+t, t0−t∈I.
Theorem 6.3.4.LetM⊂R
n
be anm-dimensional submanifold. Then the
following are equivalent.
(i)Mis locally symmetric.
(ii)The covariant derivative∇R(defined below) vanishes identically, i.e.
(∇vR)p(v1, v2)w= 0
for allp∈Mandv, v1, v2, w∈TpM.
(iii)The curvature tensorRis invariant under parallel transport, i.e.
Φγ(t, s)∗R
γ(s)=R
γ(t) (6.3.2)
for every smooth curveγ:R→Mand alls, t∈R.
Proof.See§6.3.2.

274 CHAPTER 6. GEOMETRY AND TOPOLOGY
Corollary 6.3.5.LetMandM
′
be locally symmetric spaces and fix two
pointsp0∈Mandp
′
0
∈M
′
, and letΦ0:Tp0
M→T
p
′
0
M
′
be an orthogonal
linear isomorphism. Letr >0be less than the injectivity radius ofMatp0
and the injectivity radius ofM
′
atp
′
0
. Then the following holds.
(i)There exists an isometryϕ:Ur(p0, M)→Ur(p
′
0
, M
′
)withϕ(p0) =p
′
0
anddϕ(p0) = Φ0if and only ifΦ0intertwinesRandR
′
, i.e.
(Φ0)∗Rp0
=R
′
p
′
0
. (6.3.3)
(ii)AssumeMandM
′
are connected, simply connected, and complete.
Then there exists an isometryϕ:M→M
′
withϕ(p0) =p
′
0
anddϕ(p0) = Φ0
if and only ifΦ0satisfies(6.3.3).
Proof.In (i) and (ii) the “only if” statement follows from Theorem 5.3.1
(Theorema Egregium) with Φ0:=dϕ(p0). To prove the “if” statement, let
(Φ, γ, γ
′
) be a development satisfyingγ(0) =p0,γ
′
(0) =p
′
0
, and Φ(0) = Φ0.
SinceRandR
′
are invariant under parallel transport, by Theorem 6.3.4, it
follows from the discussion in the beginning of this section that Φ∗R=R
′
.
Hence assertion (i) follows from the local C-A-H Theorem 6.1.17 and (ii)
follows from the global C-A-H Theorem 6.1.8.
Corollary 6.3.6.A connected, simply connected, complete, locally symmet-
ric space is symmetric.
Proof.Corollary 6.3.5 (ii) withM
′
=M,p
′
0
=p0, and Φ0=−id.
Corollary 6.3.7.A connected symmetric spaceMishomogeneous; i.e.
givenp, q∈Mthere exists an isometryϕ:M→Mwithϕ(p) =q.
Proof.IfMis simply connected, the assertion follows from part (ii) of Corol-
lary 6.3.5 withM=M
′
,p0=p,p
′
0
=q, and Φ0= Φγ(1,0) :TpM→TqM,
whereγ: [0,1]→Mis a curve fromptoq. IfMis not simply connected,
we can argue as follows. There is an equivalence relation onMdefined by
p∼q:⇐⇒ ∃ isometryϕ:M→M∋ϕ(p) =q.
Letp, q∈Mand suppose thatd(p, q)<inj(p). By Theorem 4.4.4 there
is a unique shortest geodesicγ: [0,1]→Mconnectingptoq. SinceMis
symmetric there is an isometryϕ:M→Msuch thatϕ(γ(1/2)) =γ(1/2)
anddϕ(γ(1/2)) =−id. This isometry satisfiesϕ(γ(t)) =γ(1−t) and hence
ϕ(p) =q. Thusp∼qwheneverd(p, q)<inj(p). This shows that each
equivalence class is open, hence each equivalence class is also closed, and
hence there is only one equivalence class becauseMis connected. This
proves Corollary 6.3.7.

6.3. SYMMETRIC SPACES 275
6.3.2 Covariant Derivative of the Curvature
For two vector spacesV, Wand an integerk≥1 we denote byL
k
(V, W)
the vector space of all multi-linear maps fromV
k
=V× · · · ×VtoW.
ThusL
1
(V, W) =L(V, W) is the space of all linear maps fromVtoW.
Definition 6.3.8.Thecovariant derivative of the Riemann curvature
tensorassigns to everyp∈Ma linear map
(∇R)p:TpM→ L
2
(TpM,L(TpM, TpM))
such that
(∇R)(X)(X1, X2)Y=∇X
Γ
R(X1, X2)Y
∆
−R(∇XX1, X2)Y
−R(X1,∇XX2)Y−R(X1, X2)∇XY
(6.3.4)
for allX, X1, X2, Y∈Vect(M). We also use the notation
(∇vR)p:= (∇R)p(v)
forp∈Mandv∈TpMso that
(∇XR)(X1, X2)Y:= (∇R)(X)(X1, X2)Y
for allX, X1, X2, Y∈Vect(M).
Remark 6.3.9.One verifies easily that the map
Vect(M)
4
→Vect(M) : (X, X1, X2, Y)7→(∇XR)(X1, X2)Y,
defined by the right hand side of equation (6.3.4), is multi-linear over the
ring of functionsF(M). Hence it follows as in Remark 5.2.13 that∇Ris
well defined, i.e. that the right hand side of (6.3.4) atp∈Mdepends only
on the tangent vectorsX(p), X1(p), X2(p), Y(p).
Remark 6.3.10.Letγ:I→Mbe a smooth curve on an intervalI⊂R
and
X1, X2, Y∈Vect(γ)
be smooth vector fields alongγ. Then equation (6.3.4) continues to hold
withXreplaced by ˙γand each∇Xon the right hand side replaced by the
covariant derivative of the respective vector field alongγ:
(∇˙γR)(X1, X2)Y=∇(R(X1, X2)Y)−R(∇X1, X2)Y
−R(X1,∇X2)Y−R(X1, X2)∇Y.
(6.3.5)

276 CHAPTER 6. GEOMETRY AND TOPOLOGY
Theorem 6.3.11. Ifγ:R→Mis a smooth curve such thatγ(0) =p
and˙γ(0) =v, then
(∇vR)p=
d
dt

t=0
Φγ(0, t)∗R
γ(t) (6.3.6)
(ii)The covariant derivative of the Riemann curvature tensor satisfies the
second Bianchi identity
(∇XR)(Y, Z) + (∇YR)(Z, X) + (∇ZR)(X, Y) = 0. (6.3.7)
Proof.We prove (i). Letv1, v2, w∈TpMand choose parallel vector fields
X1, X2, Y∈Vect(γ) alongγsatisfying the initial conditionsX1(0) =v1,
X2(0) =v2,Y(0) =w. Thus
X1(t) = Φγ(t,0)v1, X 2(t) = Φγ(t,0)v2, Y(t) = Φγ(t,0)w.
Then the last three terms on the right vanish in equation (6.3.5) and hence
(∇vR)(v1, v2)w=∇(R(X1, X2)Y)(0)
=
d
dt

t=0
Φ
γ(0,t)R
γ(t)(X1(t), X2(t))Y(t)
=
d
dt

t=0
Φ
γ(0,t)R
γ(t)(Φγ(t,0)v1,Φγ(t,0)v2)Φγ(t,0)w
=
d
dt

t=0
Γ
Φγ(0, t)∗R
γ(t)
∆
(v1, v2)w.
Here the second equation follows from Theorem 3.3.6. This proves (i).
We prove (ii). Choose a smooth functionγ:R
3
→Mand denote by
(r, s, t) the coordinates onR
3
. IfYis a vector field alongγ, we have
(∇∂rγR)(∂sγ, ∂tγ)Y=∇r
Γ
R(∂sγ, ∂tγ)Y
∆
−R(∂sγ, ∂tγ)∇rY
−R(∇r∂sγ, ∂tγ)Y−R(∂sγ,∇r∂tγ)Y
=∇r(∇s∇tY− ∇t∇sY)−(∇s∇t− ∇t∇s)∇rY
+R(∂tγ,∇r∂sγ)Y−R(∂sγ,∇t∂rγ)Y.
Permuting the variablesr, s, tcyclically and taking the sum of the resulting
three equations we obtain
(∇∂rγR)(∂sγ, ∂tγ)Y+ (∇∂sγR)(∂tγ, ∂rγ)Y+ (∇∂tγR)(∂rγ, ∂sγ)Y
=∇r(∇s∇tY− ∇t∇sY)−(∇s∇t− ∇t∇s)∇rY
+∇s(∇t∇rY− ∇r∇tY)−(∇t∇r− ∇r∇t)∇sY
+∇t(∇r∇sY− ∇s∇rY)−(∇r∇s− ∇s∇r)∇tY.
The terms on the right cancel out. This proves Theorem 6.3.11.

6.3. SYMMETRIC SPACES 277
Proof of Theorem 6.3.4.We prove that (iii) implies (i). This follows from
the local Cartan–Ambrose–Hicks Theorem 6.1.17 with
p
′
0=p0=p,Φ0=−id :TpM→TpM.
This isomorphism satisfies
(Φ0)∗Rp=Rp.
Hence it follows from the discussion in the beginning of this section that
Φ∗R=R
′
for every development (Φ, γ, γ
′
) ofMalong itself satisfying
γ(0) =γ
′
(0) =p,Φ(0) =−id.
Hence, by the local C-A-H Theorem 6.1.17, there is an isometry
ϕ:Ur(p, M)→Ur(p, M)
satisfying
ϕ(p) =p, dϕ(p) =−id
whenever 0< r <inj(p;M).
We prove that (i) implies (ii). By Theorem 5.3.1 (Theorema Egregium),
every isometryϕ:M→M
′
preserves the Riemann curvature tensor and
covariant differentiation, and hence also the covariant derivative of the Rie-
mann curvature tensor, i.e.
ϕ∗(∇R) =∇
′
R
′
.
Applying this to the local isometryϕ:Ur(p, M)→Ur(p, M) we obtain
Γ
∇
dϕ(p)vR
∆
ϕ(p)
(dϕ(p)v1, dϕ(p)v2) =dϕ(p) (∇vR) (v1, v2)dϕ(p)
−1
for allv, v1, v2∈TpM. Since
dϕ(p) =−id
this shows that∇Rvanishes atp.
We prove that (ii) imlies (iii). If∇Rvanishes, then equation (6.3.6) in
Theorem 6.3.11 shows that the function
s7→Φγ(t, s)∗R
γ(s)= Φγ(t,0)∗Φγ(0, s)∗R
γ(s)
is constant and hence is everywhere equal toR
γ(t). This implies (6.3.2) and
completes the proof of Theorem 6.3.4.

278 CHAPTER 6. GEOMETRY AND TOPOLOGY
Covariant Derivative of the Curvature in Local Coordinates
Letϕ:U→Ω be a local coordinate chart onMwith values in an open set
Ω⊂R
m
, denote its inverse byψ:=ϕ
−1
: Ω→U,and let
Ei(x) :=
∂ψ
∂x
i
(x)∈T
ψ(x)M, x ∈Ω, i= 1, . . . , m,
be the local frame of the tangent bundle determined by this coordinate
chart. Let Γ
k
ij
: Ω→Rdenote the Christoffel symbols andR
ℓ
ijk
: Ω→Rthe
coefficients of the Riemann curvature tensor so that
∇iEj=
X
k
Γ
k
ijEk, R(Ei, Ej)Ek=
X
ℓ
R
ℓ
ijk
Eℓ.
Giveni, j, k, ℓ∈ {1, . . . , m}we can express the vector field (∇Ei
R)(Ej, Ek)Eℓ
alongψfor eachx∈Ω as a linear combination of the basis vectorsEi(x).
This gives rise to functions
∇iR
ν
jkℓ
: Ω→R
defined by
(∇Ei
R)(Ej, Ek)Eℓ=:
X
ν
∇iR
ν
jkℓ
Eν. (6.3.8)
These functions are given by
∇iR
ν
jkℓ
=∂iR
ν
jkℓ
+
X
µ
Γ
ν
iµR
µ
jkℓ
−
X
µ
Γ
µ
ij
R
ν
µkℓ
−
X
µ
Γ
µ
ik
R
ν
jµℓ
−
X
µ
Γ
µ
iℓ
R
ν
jkµ
.
(6.3.9)
The second Bianchi identity has the form
∇iR
ν
jkℓ
+∇jR
ν
kiℓ
+∇kR
ν
ijℓ
= 0. (6.3.10)
Exercise:Prove equations (6.3.9) and (6.3.10).Warning:As in§5.4,
care must be taken with the ordering of the indices. Some authors use the
notation∇iR
ν
ℓjk
for what we call∇iR
ν
jkℓ
.
6.3.3 Examples and Exercises
Example 6.3.12.Every flat manifold is locally symmetric.
Example 6.3.13.IfM1andM2are (locally) symmetric, so isM1×M2.

6.3. SYMMETRIC SPACES 279
Example 6.3.14.M=R
m
with the standard metric is a symmetric space.
Recall that the isometry groupI(R
m
) consists of all affine transformations
of the form
ϕ(x) =Ax+b, A∈O(m), b∈R
m
.
(See Exercise 5.1.4.) The isometry with fixed pointp∈R
m
anddϕ(p) =−id
is given byϕ(x) = 2p−xforx∈R
m
.
Example 6.3.15.The flat tori of Exercise 6.2.7 in the previous section are
symmetric (but not simply connected). This shows that the hypothesis of
simple connectivity cannot be dropped in part (ii) of Corollary 6.3.5.
Example 6.3.16.Below we define manifolds of constant curvature and
show that they are locally symmetric. The simplest example, after a flat
space, is the unit sphereS
m
=
Φ
x∈R
m+1
| |x|= 1

.The symmetryϕof
the sphere about a pointp∈Mis given by
ϕ(x) :=−x+ 2⟨p, x⟩p
forx∈S
m
. This extends to an orthogonal linear transformation of the
ambient space. In fact the group of isometries ofS
m
is the group O(m+ 1)
of orthogonal linear transformations ofR
m+1
(see Example 6.4.16 below).
In accordance with Corollary 6.3.7 this group acts transitively onS
m
.
Example 6.3.17.A compact two-dimensional manifold of constant neg-
ative curvature is locally symmetric (as its universal cover is symmetric)
but not homogeneous (as closed geodesics of a given period are isolated).
Hence it is not symmetric. This shows that the hypothesis thatMbe simply
connected cannot be dropped in Corollary 6.3.6.
Example 6.3.18.The real projective spaceRP
n
with the metric inherited
fromS
n
is a symmetric space and the orthogonal group O(n+1) acts on it by
isometries. The complex projective spaceCP
n
with the Fubini–Study metric
in Example 3.7.5 is a symmetric space and the unitary group U(n+ 1) acts
on it by isometries. The complex Grassmannian Gk(C
n
) in Example 3.7.6
is a symmetric space and the unitary group U(n) acts on it by isometries.
(Exercise:Prove this.)
Example 6.3.19.The simplest example of a symmetric space which is not
of constant curvature is the orthogonal group O(n) =
Φ
g∈R
n×n
|g
T
g= 1l

with the Riemannian metric (5.2.20) of Example 5.2.18. The symmetryϕ
about the pointa∈O(n) is given byϕ(g) =ag
−1
a.This discussion extends
to every Lie subgroup G⊂O(n). (Exercise:Prove this.)

280 CHAPTER 6. GEOMETRY AND TOPOLOGY
6.4 Constant Curvature
In the§5.3 we saw that the Gaußian curvature of a two-dimensional surface
is intrinsic: we gave a formula for it in terms of the Riemann curvature
tensor and the first fundamental form. We may use this formula to define the
Gaußian curvature foranytwo-dimensional manifold (even if its codimension
is greater than one). We make a slightly more general definition.
6.4.1 Sectional Curvature
Definition 6.4.1.LetM⊂R
n
be a smoothm-dimensional submanifold.
Letp∈Mand letE⊂TpMbe a2-dimensional linear subspace of the tan-
gent space. Thesectional curvatureofMat(p, E)is the number
K(p, E) =
⟨Rp(u, v)v, u⟩
|u|
2
|v|
2
− ⟨u, v⟩
2
, (6.4.1)
whereu, v∈Eare linearly independent (and hence form a basis ofE).
The right hand side of (6.4.1) remains unchanged if we multiplyuorv
by a nonzero real number or add to one of the vectors a real multiple of the
other; hence it depends only on the linear subspace spanned byuadv.
Example 6.4.2.IfM⊂R
3
is a 2-manifold, then by Theorem 5.3.9 the
sectional curvatureK(p, TpM) =K(p) is the Gaußian curvature ofMatp.
More generally, for any 2-manifoldM⊂R
n
(whether or not it has codimen-
sion one) wedefinetheGaußian curvatureofMatpby
K(p) :=K(p, TpM). (6.4.2)
Example 6.4.3.IfM⊂R
m+1
is a submanifold of codimension one and
ν:M→S
m
is a Gauß map, then the sectional curvature of a 2-dimensional
subspaceE⊂TpMspanned by two linearly independent tangent vectors
u, v∈TpMis given by
K(p, E) =
⟨u, dν(p)u⟩⟨v, dν(p)v⟩ − ⟨u, dν(p)v⟩
2
|u|
2
|v|
2
− ⟨u, v⟩
2
. (6.4.3)
This follows from equation (5.3.8) in the proof of Theorem 5.3.9 which holds
in all dimensions. In particular, whenM=S
m
, we haveν(p) =pand hence
K(p, E) = 1 for allpandE. For a sphere of radiusrwe haveν(p) =p/r
and henceK(p, E) = 1/r
2
.

6.4. CONSTANT CURVATURE 281
Example 6.4.4.Let G⊂O(n) be a Lie subgroup equipped with the Rie-
mannian metric
⟨v, w⟩:= trace(v
T
w)
forv, w∈TgG⊂R
n×n
. Then, by Example 5.2.18, the sectional curvature
of G at the identity matrix 1l is given by
K(1l, E) =
1
4
|[ξ, η]|
2
for every 2-dimensional linear subspaceE⊂g= Lie(G) =T1lG with an
orthonormal basisξ, η.
Exercise 6.4.5.LetE⊂TpMbe a 2-dimensional linear subspace, letr >0
be smaller than the injectivity radius ofMatp, and letN⊂Mbe the 2-
dimensional submanifold given by
N:= exp
p({v∈E| |v|< r}).
Show that the sectional curvatureK(p, E) ofMat (p, E) agrees with the
Gaußian curvature ofNatp.
Exercise 6.4.6.Letp∈M⊂R
n
and letE⊂TpMbe a 2-dimensional lin-
ear subspace. Forr >0 letLdenote the ball of radiusrin the (n−m+ 2)-
dimensional affine subspace ofR
n
throughpand parallel to the vector sub-
spaceE+TpM
⊥
:
L=
n
p+v+w|v∈E, w∈TpM
⊥
,|v|
2
+|w|
2
< r
2
o
.
Show that, forrsufficiently small,L∩Mis a 2-dimensional manifold with
Gaußian curvatureKL∩M(p) atpgiven byKL∩M(p) =K(p, E).
6.4.2 Constant Sectional Curvature
Definition 6.4.7.Letk∈Randm≥2be an integer. Anm-manifold
M⊂R
n
is said to haveconstant sectional curvaturekiffK(p, E) =k
for everyp∈Mand every2-dimensional linear subspaceE⊂TpM.
Theorem 6.4.8.LetM⊂R
n
be anm-manifold and fix an elementp∈M
and a real numberk. Then the following are equivalent.
(i)K(p, E) =kfor every2-dimensional linear subspaceE⊂TpM.
(ii)The Riemann curvature tensor ofMatpis given by
⟨Rp(v1, v2)v3, v4⟩=k
ı
⟨v1, v4⟩⟨v2, v3⟩ − ⟨v1, v3⟩⟨v2, v4⟩
ȷ
(6.4.4)
for allv1, v2, v3, v4∈TpM.

282 CHAPTER 6. GEOMETRY AND TOPOLOGY
Proof.That (ii) implies (i) follows directly from the definition of the sec-
tional curvature in (6.4.1) by takingv1=v4=uandv2=v3=vin (6.4.4).
Conversely, assume (i) and define the multi-linear mapQ:TpM
4
→Rby
Q(v1, v2, v3, v4) :=⟨Rp(v1, v2)v3, v4⟩ −k
ı
⟨v1, v4⟩⟨v2, v3⟩ − ⟨v1, v3⟩⟨v2, v4⟩
ȷ
.
Then, for allu, v, v1, v2, v3, v4∈TpM, the mapQsatisfies the equations
Q(v1, v2, v3, v4) +Q(v2, v1, v3, v4) = 0, (6.4.5)
Q(v1, v2, v3, v4) +Q(v2, v3, v1, v4) +Q(v3, v1, v2, v4) = 0, (6.4.6)
Q(v1, v2, v3, v4)−Q(v3, v4, v1, v2) = 0, (6.4.7)
Q(u, v, u, v) = 0. (6.4.8)
Here the first three equations follow from Theorem 5.2.14 and the last follows
from the definition ofQand the hypothesis that the sectional curvature
isK(p, E) =kfor every 2-dimensional linear subspaceE⊂TpM.
We must prove thatQvanishes. Using (6.4.7) and (6.4.8) we find
0 =Q(u, v1+v2, u, v1+v2)
=Q(u, v1, u, v2) +Q(u, v2, u, v1)
= 2Q(u, v1, u, v2)
for allu, v1, v2∈TpM. This implies
0 =Q(u1+u2, v1, u1+u2, v2)
=Q(u1, v1, u2, v2) +Q(u2, v1, u1, v2)
for allu1, u2, v1, v2∈TpM. Hence
Q(v1, v2, v3, v4) =−Q(v3, v2, v1, v4)
=Q(v2, v3, v1, v4)
=−Q(v3, v1, v2, v4)−Q(v1, v2, v3, v4).
Here the second equation follows from (6.4.5) and the last from (6.4.6). Thus
Q(v1, v2, v3, v4) =−
1
2
Q(v3, v1, v2, v4) =
1
2
Q(v1, v3, v2, v4)
for allv1, v2, v3, v4∈TpMand, repeating this argument,
Q(v1, v2, v3, v4) =
1
4
Q(v1, v2, v3, v4).
HenceQ≡0 as claimed. This proves Theorem 6.4.8.

6.4. CONSTANT CURVATURE 283
Remark 6.4.9.The symmetric groupS4on four symbols acts naturally
on the spaceL
4
(TpM,R) of multi-linear maps fromTpM
4
toR. The condi-
tions (6.4.5), (6.4.6), (6.4.7), and (6.4.8) say that the four elements
a= id + (12),
c= id + (123) + (132),
b= id−(34),
d= id + (13) + (24) + (13)(24)
of the group ring ofS4annihilateQ. This suggests an alternate proof of
Theorem 6.4.8. A representation of a finite group is completely reducible
so one can prove thatQ= 0 by showing that any vector in any irreducible
representation ofS4which is annihilated by the four elementsa, b, cand
dmust necessarily be zero. This can be checked case by case for each
irreducible representation. (The groupS4has 5 irreducible representations:
two of dimension 1, two of dimension 3, and one of dimension 2.)
IfMandM
′
are twom-dimensional manifolds with constant curvaturek,
then every orthogonal isomorphism Φ :TpM→Tp
′M
′
intertwines the Rie-
mann curvature tensors by Theorem 6.4.8. Hence by the appropriate version
(local or global) of the C-A-H Theorem we have the following corollaries.
Corollary 6.4.10.Every Riemannian manifold with constant sectional cur-
vature is locally symmetric.
Proof.Theorem 6.3.4 and Theorem 6.4.8.
Corollary 6.4.11.LetMandM
′
bem-dimensional Riemannian manifolds
with constant curvaturekand letp∈Mandp
′
∈M
′
. Ifr >0is smaller
than the injectivity radii ofMatpand ofM
′
atp
′
, then for every orthogonal
isomorphism
Φ :TpM→Tp
′M
′
there exists an isometry
ϕ:Ur(p, M)→Ur(p
′
, M
′
)
such that
ϕ(p) =p
′
, dϕ(p) = Φ.
Proof.This follows from Corollary 6.3.5 and Corollary 6.4.10. Alternatively
one can use Theorem 6.4.8 and the local C-A-H Theorem 6.1.17.

284 CHAPTER 6. GEOMETRY AND TOPOLOGY
Corollary 6.4.12.Any two connected, simply connected, complete Rieman-
nian manifolds with the same constant sectional curvature and the same
dimension are isometric.
Proof.Theorem 6.4.8 and the global C-A-H Theorem 6.1.8.
Corollary 6.4.13.LetM⊂R
n
be a connected, simply connected, complete
manifold. Then the following are equivalent.
(i)Mhas constant sectional curvature.
(ii)For every pair of pointsp, q∈Mand every orthogonal linear isomor-
phismΦ :TpM→TqMthere exists an isometryϕ:M→Msuch that
ϕ(p) =q, dϕ(p) = Φ.
Proof.That (i) implies (ii) follows immediately from Theorem 6.4.8 and the
global C-A-H Theorem 6.1.8. Conversely assume (ii). Then, for every pair
of pointsp, q∈Mand every orthogonal linear isomorphism
Φ :TpM→TqM,
it follows from Theorem 5.3.1 (Theorema Egregium) that
Φ∗Rp=Rq
and so
K(p, E) =K(q,ΦE)
for every 2-dimensional linear subspaceE⊂TpM. Since, for every pair of
pointsp, q∈Mand of 2-dimensional linear subspaces
E⊂TpM, F ⊂TqM,
we can find an orthogonal linear isomorphism Φ :TpM→TqMsuch that
ΦE=F,
this implies (i).
Corollary 6.4.13 asserts that a connected, simply connected, complete
Riemannianm-manifoldMhas constant sectional curvature if and only if
the isometry groupI(M) acts transitively on its orthonormal frame bun-
dleO(M). Note that, by Lemma 5.1.10, this group action is also free.

6.4. CONSTANT CURVATURE 285
Examples and Exercises
Example 6.4.14.Any flat Riemannian manifold has constant sectional
curvaturek= 0.
Example 6.4.15.The manifold
M=R
m
with its standard metric is, up to isometry, the unique connected, simply
connected, complete Riemannianm-manifold with constant sectional curva-
ture
k= 0.
Example 6.4.16.Form≥2 the unit sphere
M=S
m
with its standard metric is, up to isometry, the unique connected, simply
connected, complete Riemannianm-manifold with constant sectional curva-
ture
k= 1.
Hence, by Corollary 6.4.12, every connected simply connected, complete
Riemannian manifold with positive sectional curvaturek= 1 is compact.
Moreover, by Corollary 6.4.13, the isometry groupI(S
m
) is isomorphic to
the group O(m+ 1) of orthogonal linear transformations ofR
m+1
. Thus,
by Corollary 6.4.13, the orthonormal frame bundleO(S
m
) is diffeomorphic
to O(m+ 1). This follows also from the fact that, if
v1, . . . , vm
is an orthonormal basis ofTpS
m
=p
⊥
then
p, v1, . . . , vm
is an orthonormal basis ofR
m+1
.
Example 6.4.17.A product of spheres isnota space of constant sectional
curvature, but itisa symmetric space.Exercise:Prove this.
Example 6.4.18.Forn≥4 the orthogonal group O(n) is not a space of
constant sectional curvature, but it is a symmetric space and has nonnegative
sectional curvature (see Example 6.4.4).

286 CHAPTER 6. GEOMETRY AND TOPOLOGY
6.4.3 Hyperbolic Space
Fix an integerm≥2. Thehyperbolic spaceH
m
is, up to isometry, the
unique connected, simply connected, complete Riemannianm-manifold with
constant sectional curvature
k=−1.
A model forH
m
can be constructed as follows. A point inR
m+1
will be
denoted by
p= (x0, x), x 0∈R, x= (x1, . . . , xm)∈R
m
.
LetQ:R
m+1
×R
m+1
→Rdenote the symmetric bilinear form given by
Q(p, q) :=−x0y0+x1y1+· · ·+xmym (6.4.9)
forp= (x0, x), q= (y0, y)∈R
m+1
. SinceQis nondegenerate the space
H
m
:=
Φ
p= (x0, x)∈R
m+1
|Q(p, p) =−1, x0>0

is a smoothm-dimensional submanifold ofR
m+1
and the tangent space
ofH
m
atpis given by
TpH
m
=
Φ
v∈R
m+1
|Q(p, v) = 0

.
Forp= (x0, x)∈R
m+1
andv= (ξ0, ξ)∈R
m+1
we have
p∈H
m
⇐⇒ x0=
p
1 +|x|
2
,
v∈TpH
m
⇐⇒ ξ0=
⟨ξ, x⟩
p
1 +|x|
2
.
Now let us define a Riemannian metric onH
m
by
gp(v, w) :=Q(v, w)
=⟨ξ, η⟩ −ξ0η0
=⟨ξ, η⟩ −
⟨ξ, x⟩⟨η, x⟩
1 +|x|
2
(6.4.10)
forv= (ξ0, ξ)∈TpH
m
andw= (η0, η)∈TpH
m
.
Theorem 6.4.19.H
m
is a connected, simply connected, complete Rieman-
nianm-manifold with constant sectional curvaturek=−1.

6.4. CONSTANT CURVATURE 287
We remark that the manifoldH
m
does not quite fit into the extrinsic
framework of most of this book as it is not exhibited as a submanifold
of Euclidean space but rather of “pseudo-Euclidean space”: the positive
definite inner product⟨v, w⟩of the ambient spaceR
m+1
is replaced by a
nondegenerate symmetric bilinear formQ(v, w). However, all the theory
developed thus far goes through (readingQ(v, w) for⟨v, w⟩) provided we
impose the additional hypothesis (true in the exampleM=H
m
) that the
first fundamental formgp=Q|TpMis positive definite. For thenQ|TpMis
nondegenerate and we may define the orthogonal projection Π(p) ontoTpM
as before. The next lemma summarizes the basic observations; the proof is
an exercise in linear algebra.
Lemma 6.4.20.LetQbe a symmetric bilinear form on a vector spaceV
and for each subspaceEofVdefine its orthogonal complement by
E
⊥Q
:={u∈V|Q(u, v) = 0∀v∈E}.
AssumeQis nondegenerate, i.e.V
⊥Q={0}. Then, for every linear sub-
spaceE⊂V, we have
V=E⊕E
⊥Q
⇐⇒ E∩E
⊥Q
={0},
i.e.E
⊥Qis a vector space complement ofEif and only if the restriction
ofQtoEis nondegenerate.
Proof of Theorem 6.4.19.The proofs of the various properties ofH
m
are
entirely analogous to the corresponding proofs forS
m
. Thus the unit normal
field toH
m
is given by
ν(p) =p
forp∈H
m
although the “square of its length” is
Q(p, p) =−1.
Forp∈H
m
we introduce theQ-orthogonal projection Π(p) ofR
m+1
onto
the subspaceTpH
m
. It is characterized by the conditions
Π(p)
2
= Π(p),ker Π(p)⊥QimΠ(p),imΠ(p) =TpH
m
,
and is given by the explicit formula
Π(p)v=v+Q(v, p)p
forv∈R
m+1
.

288 CHAPTER 6. GEOMETRY AND TOPOLOGY
The covariant derivative of a vector fieldX∈Vect(γ) along a smooth
curveγ:R→H
m
is given by
∇X(t) = Π(γ(t))
˙
X(t)
=
˙
X(t) +Q(
˙
X(t), γ(t))γ(t)
=
˙
X(t)−Q(X(t),˙γ(t))γ(t).
The last identity follows by differentiating the equationQ(X, γ)≡0. This
can be interpreted as the hyperbolic Gauß–Weingarten formula as follows.
Forp∈H
m
andu∈TpH
m
we introduce, as before, the second fundamental
form
hp(u) :TpH
m
→(TpH
m
)
⊥Q
via
hp(u)v:=
Γ
dΠ(p)u
∆
v
and denote itsQ-adjoint by
hp(u)
∗
: (TpH
m
)
⊥Q
→TpH
m
.
For allp∈H
m
,u∈TpH
m
, andv∈R
m+1
we have
ı
dΠ(p)u
ȷ
v=
d
dt

t=0
Γ
v+Q(v, p+tu)(p+tu)
∆
=Q(v, p)u+Q(v, u)p,
where the first summand on the right is tangent toH
m
and the second
summand isQ-orthogonal toTpH
m
. Hence
hp(u)v=Q(v, u)p, h p(u)
∗
w=Q(w, p)u (6.4.11)
forv∈TpH
m
andw∈(TpH
m
)
⊥Q.
With this understood, the Gauß-Weingarten formula
˙
X=∇X+hγ( ˙γ)X
extends to the present setting. The reader may verify that the operators
∇: Vect(γ)→Vect(γ)
thus defined satisfy the axioms of Theorem 3.7.8 and hence define the Levi-
Civita connection onH
m
.

6.4. CONSTANT CURVATURE 289
Now a smooth curveγ:I→H
m
is a geodesic if and only if it satisfies
the equivalent conditions
∇˙γ≡0⇐⇒ ¨γ(t)⊥QT
γ(t)H
m
∀t∈I⇐⇒ ¨γ=Q( ˙γ,˙γ)γ.
A geodesic must satisfy the equation
d
dt
Q( ˙γ,˙γ) = 2Q(¨γ,˙γ) = 0
because ¨γis a scalar multiple ofγ, and henceQ( ˙γ,˙γ) is constant. Fix an
elementp∈H
m
and a tangent vectorv∈TpH
m
such that
Q(v, v) = 1.
Then the geodesicγ:R→H
m
withγ(0) =pand ˙γ(0) =vis given by
γ(t) = cosh(t)p+ sinh(t)v, (6.4.12)
where
cosh(t) :=
e
t
+e
−t
2
,sinh(t) :=
e
t
−e
−t
2
.
In fact we have ¨γ(t) =γ(t)⊥QT
γ(t)H
m
. It follows that the geodesics
exist for all time and henceH
m
is geodesically complete. Moreover, being
diffeomorphic to Euclidean space,H
m
is connected and simply connected.
It remains to prove thatH
m
has constant sectional curvaturek=−1.
To see this we use the Gauß–Codazzi formula in the hyperbolic setting, i.e.
Rp(u, v) =hp(u)
∗
hp(v)−hp(v)
∗
hp(u). (6.4.13)
By equation (6.4.11), this gives
⟨Rp(u, v)v, u⟩=Q(hp(u)u, hp(v)v)−Q(hp(v)u, hp(u)v)
=Q(Q(u, u)p, Q(v, v)p)−Q(Q(u, v)p, Q(u, v)p)
=−Q(u, u)Q(v, v) +Q(u, v)
2
=−gp(u, u)gp(v, v) +gp(u, v)
2
for allu, v∈TpH
m
. Hence, for everyp∈Mand every 2-dimensional linear
subspaceE⊂TpMwith a basisu, v∈Ewe have
K(p, E) =
⟨Rp(u, v)v, u⟩
gp(u, u)gp(v, v)−gp(u, v)
2
=−1.
This proves Theorem 6.4.19.

290 CHAPTER 6. GEOMETRY AND TOPOLOGY
Exercise 6.4.21.Prove that the pullback of the metric onH
m
under the
diffeomorphism
R
m
→H
m
:x7→
ıp
1 +|x|
2
, x
ȷ
is given by
|bx|
x
=
s
|bx|
2
−
⟨x, ξ⟩
2
1 +|x|
2
forx∈R
m
andbx∈R
m
=TxR
m
. Thus the metric tensor is given by
gij(x) =δij−
xixj
1 +|x|
2
(6.4.14)
forx= (x1, . . . , xm)∈R
m
.
Exercise 6.4.22.ThePoincar´e modelof hyperbolic space is the open
unit discD
m
⊂R
m
equipped with thePoincar´e metric
|by|
y
=
2|by|
1− |y|
2
fory∈D
m
andby∈R
m
=TyD
m
. Thus the metric tensor is given by
gij(y) =
4δij
ı
1− |y|
2
ȷ
2
, y∈D
m
. (6.4.15)
Prove that the diffeomorphism
D
m
→H
m
:y7→

1 +|y|
2
1− |y|
2
,
2y
1− |y|
2
!
is an isometry with the inverse
H
m
→D
m
: (x0, x)7→
x
1 +x0
.
Interpret this map as a stereographic projection from thesouth pole(−1,0).
Exercise 6.4.23.The composition of the isometries in Exercise 6.4.21 and
Exercise 6.4.22 is the diffeomorphismR
m
→D
m
:x7→ygiven by
y=
x
p
1 +|x|
2
+ 1
, x=
2y
1− |y|
2
,
p
1 +|x|
2
=
1 +|y|
2
1− |y|
2
.
Prove that this is an isometry intertwining the Riemannian metrics (6.4.14)
and (6.4.15).

6.4. CONSTANT CURVATURE 291
Exercise 6.4.24.This exercise shows that every nonconstant geodesic in
the Poincar´e modelD
m
of hyperbolic space in Exercise 6.4.22 converges to
two points on the boundaryS
m−1
=∂D
m
in forward and backward time,
and that any two distinct points on the boundary are the asymptotic limits
of a unique geodesic inD
m
up to reparametrization.
Fix an elementy∈D
m
and a tangent vectorby∈TyD
m
=R
m
of norm
one in the hyperbolic metric, i.e.
λ|by|= 1, λ:=
2
1− |y|
2
. (6.4.16)
Letγ:R→D
m
be the unique geodesic satisfyingγ(0) =yand ˙γ(0) =by.
Prove the following.
(a)The geodesicγis given by the explicit formula
γ(t) =
cosh(t)λy+ sinh(t)
Γ
λby+⟨λy, λby⟩y
∆
1 + cosh(t)
Γ
λ−1
∆
+ sinh(t)⟨λy, λby⟩
(6.4.17)
fort∈R.Hint:Use (6.4.12) and the isometries in Exercise 6.4.22.
(b)The limits
y±:= lim
t→±∞
γ(t)∈S
m−1
exist and are given by
y+=
λy+λby+⟨λy, λby⟩y
λ−1 +⟨λy, λby⟩
, y−=
λy−λby− ⟨λy, λby⟩y
λ−1− ⟨λy, λby⟩
.(6.4.18)
(c)Assumeby /∈Ry. Then there is a unique circle inR
m
throughy−andy+
that is orthogonal toS
m−1
aty±. The centerc∈R
m
and the radiusrof
this circle are given by
c=
y++y−
1 +⟨y+, y−⟩
=
Γ
λ
2
−λ− ⟨λy, λby⟩
2
∆
y− ⟨λy, λby⟩λby
λ
2
−2λ− ⟨λy, λby⟩
2
,
r
2
=
1− ⟨y+, y−⟩
1 +⟨y+, y−⟩
=
1
λ
2
−2λ− ⟨λy, λby⟩
2
.
(6.4.19)
(d)Letcandrbe as in (c). Then the geodesicγin (a) satisfies|γ(t)−c|=r
for allt.Hint:It suffices to verify this equation fort= 0.
(e)Ifby∈Ry, theny−+y+= 0 and the geodesicγtraverses a segment of
a straight line through the origin.
(f)Fix two distinct pointsy−andy+on the unit sphereS
m−1
. Then there
exists a geodesicγ:R→D
m
such that limt→±∞γ(t) =y±. Ifγ
′
:R→D
m
is any other geodesic satisfying limt→±∞γ
′
(t) =y±, then there exist real
numbersa, bsuch thata >0 andγ
′
(t) =γ(at+b) for allt∈R.

292 CHAPTER 6. GEOMETRY AND TOPOLOGY
Exercise 6.4.25.Prove that the isometry group ofH
m
is the pseudo-ortho-
gonal group
I(H
m
) = O(m,1) :=
æ
g∈GL(m+ 1)

Q(gv, gw) =Q(v, w)
for allv, w∈R
m+1
œ
.
Thus, by Corollary 6.4.13, the orthonormal frame bundleO(H
m
) is diffeo-
morphic to O(m,1).
Exercise 6.4.26.Prove that the exponential map exp
p:TpH
m
→H
m
is
given by
exp
p(v) = cosh
ıp
Q(v, v)
ȷ
p+
sinh
ı
p
Q(v, v)
ȷ
p
Q(v, v)
v (6.4.20)
forv∈TpH
m
=p
⊥Q. Prove that this map is a diffeomorphism for every
p∈H
m
. Thus any two points inH
m
are connected by a unique geodesic.
Prove that the intrinsic distance function on hyperbolic space is given by
d(p, q) = cosh
−1
(Q(p, q)) (6.4.21)
forp, q∈H
m
. Compare this with Example 4.3.11.
Exercise 6.4.27.In the casem= 2 the Poincar´e model of hyperbolic space
in Exercise 6.4.22 is the open unit discD⊂Cin the complex plane. It can
be identified with the upper half plane
H={z∈C|Im(z)>0}
via the diffeomorphism
H→D:z7→
i−z
i+z
.
Show that the pullback of the Poincar´e metric onDunder this diffeomor-
phism is the Riemannian metric onHgiven by
|bz|z=
|bz|
y
forz=x+iy∈Handbz∈TzH=C. Show that the isometries ofH(in the
identity component) have the form
ϕ(z) =
az+b
cz+d
,
`
a b
c d
´
∈SL(2,R),
and deduce that the Lie group PSL(2,R) := SL(2,R)/{±1l}is isomorphic to
the identity component of O(2,1). Prove that every nonconstant geodesic
inHtraverses either a vertical half line or a semicircle centered at a point
on the boundary∂H=R.

6.5. NONPOSITIVE SECTIONAL CURVATURE 293
6.5 Nonpositive Sectional Curvature
In the previous section we have seen that any two points in a connected,
simply connected, complete manifoldMof constant negative curvature are
joined by a unique geodesic (Exercise 6.4.26). Thus the entire manifoldM
is geodesically convex and its injectivity radius is infinity. This continues
to hold in much greater generality for manifolds with nonpositive sectional
curvature. It is convenient, at this point, to extend the discussion to Rie-
mannian manifolds in the intrinsic setting. In particular, at some point in
the proof of the main theorem of this section and in our main example, we
shall work with a Riemannian metric that does not arise (in any obvious
way) from an embedding.
Definition 6.5.1.A Riemannian manifoldMis said to havenonpos-
itive sectional curvatureiffK(p, E)≤0for everyp∈Mand every2-
dimensional linear subspaceE⊂TpMor, equivalently,⟨Rp(u, v)v, u⟩ ≤0for
allp∈Mand allu, v∈TpM. A nonempty, connected, simply connected,
complete Riemannan manifold with nonpositive sectional curvature is called
aHadamard manifold.
6.5.1 The Cartan–Hadamard Theorem
The next theorem shows that every Hadamard manifold is diffeomorphic to
Euclidean space and has infinite injectivity radius. This is in sharp contrast
to positive curvature manifolds as the exampleM=S
m
shows.
Theorem 6.5.2(Cartan–Hadamard). LetMbe a connected, simply con-
nected, complete Riemannan manifold. Then the following are equivalent.
(i)Mhas nonpositive sectional curvature.
(ii)The derivative of each exponential map is length increasing, i.e.

dexp
p(v)bv

≥ |bv|
for allp∈Mand allv,bv∈TpM.
(iii)Each exponential map isdistance increasing, i.e.
d(exp
p(v0),exp
p(v1))≥ |v0−v1|
for allp∈Mand allv0, v1∈TpM.
Moreover, if these equivalent conditions are satisfied, then the exponential
mapexp
p:TpM→Mis a diffeomorphism for everyp∈M. Thus any two
points inMare joined by a unique geodesic.
The proof makes use of the following two exercises.

294 CHAPTER 6. GEOMETRY AND TOPOLOGY
Exercise 6.5.3.Letξ: [0,∞)→R
n
be a smooth function such that
ξ(0) = 0,
˙
ξ(0)̸= 0, ξ(t)̸= 0∀t >0.
Prove that the functionf: [0,∞)→Rgiven byf(t) :=|ξ(t)|is smooth.
Hint:The functionη: [0,∞)→R
n
defined by
η(t) :=
æ
t
−1
ξ(t),fort >0,
˙
ξ(0),fort= 0,
is smooth. Show thatfis differentiable and
˙
f=|η|
−1
⟨η,
˙
ξ⟩.
Exercise 6.5.4.Letξ:R→R
n
be a smooth function such that
ξ(0) = 0,
¨
ξ(0) = 0.
Prove that there exist constantsε >0 andc >0 such that, for allt∈R,
|t|< ε =⇒ | ξ(t)|
2
|
˙
ξ(t)|
2
− ⟨ξ(t),
˙
ξ(t)⟩
2
≤c|t|
6
.
Hint:Write
ξ(t) =tv+η(t),
˙
ξ(t) =v+ ˙η(t)
withη(t) =O(t
3
) and ˙η(t) =O(t
2
). Show that the terms of order 2 and 4
cancel in the Taylor expansion att= 0.
Proof of Theorem 6.5.2.We prove that (i) implies (ii). Fix a pointp∈M
and two tangent vectorsv,bv∈TpM. Assume without loss of generality
thatbv̸= 0 and define the curveγ:R→Mand the vector fieldX∈Vect(γ)
alongγby
γ(t) := exp
p(tv), X(t) :=
∂
∂s

s=0
exp
p(t(v+sbv))∈T
γ(t)M (6.5.1)
fort∈R. Then
X(0) = 0, X(t) =dexp
p(tv)tbv,∇X(0) =bv̸= 0. (6.5.2)
To see this, define the mapβ:R
2
→Mbyβ(s, t) := exp
p(t(v+sbv)). It
satisfiesβ(0, t) =γ(t),∂sβ(0, t) =X(t),β(s,0) =p, and∂tβ(s,0) =v+sbv
for alls, t∈R. Hence∇X(0) =∇t∂sβ(0,0) =∇s∂tβ(0,0) =bv. Moreover,
the curveβ(s,·) is a geodesic for everys, and hence Lemma 6.1.18 asserts
thatX=∂sβ(0,·) is a Jacobi field alongγ, i.e.
∇∇X+R(X,˙γ) ˙γ= 0. (6.5.3)

6.5. NONPOSITIVE SECTIONAL CURVATURE 295
It follows from Exercise 6.5.3 withξ(t) := Φγ(0, t)X(t) that the function
[0,∞)→R:t7→ |X(t)|
is smooth and
d
dt

t=0
|X(t)|=|∇X(0)|=|bv|.
Moreover, fort >0, we have
d
2
dt
2
|X|=
d
dt
⟨X,∇X⟩
|X|
=
|∇X|
2
+⟨X,∇∇X⟩
|X|
−
⟨X,∇X⟩
2
|X|
3
=
|X|
2
|∇X|
2
− ⟨X,∇X⟩
2
|X|
3
+
⟨X, R( ˙γ, X) ˙γ⟩
|X|
≥0.
(6.5.4)
Here the third equality follows from the fact thatXis a Jacobi field alongγ,
and the inequality follows from the nonpositive sectional curvature condition
in (i) and from the Cauchy–Schwarz inequality. Thus the second derivative
of the function [0,∞)→R:t7→ |X(t)| −t|bv|is nonnegative; so its first
derivative is nondecreasing and it vanishes att= 0; thus
|X(t)| −t|bv| ≥0
for everyt≥0. In particular, fort= 1 we obtain

dexp
p(v)bv

=|X(1)| ≥ |bv|.
as claimed. Thus we have proved that (i) implies (ii).
We prove that (ii) implies (i). Assume, by contradiction, that (ii) holds
but there exists a pointp∈Mand a pair of vectorsv,bv∈TpMsuch that
⟨Rp(v,bv)v,bv⟩<0. (6.5.5)
Defineγ:R→MandX∈Vect(γ) by (6.5.1) so that (6.5.2) and (6.5.3)
are satisfied. ThusXis a Jacobi field with
X(0) = 0,∇X(0) =bv̸= 0.
Hence it follows from Exercise 6.5.4 withξ(t) := Φγ(0, t)X(t) that there is a
constantc >0 such that, fort >0 sufficiently small, we have the inequality
|X(t)|
2
|∇X(t)|
2
− ⟨X(t),∇X(t)⟩
2
≤ct
6
.

296 CHAPTER 6. GEOMETRY AND TOPOLOGY
Moreover, by (6.5.1) and (6.5.2), limt↘0˙γ(t) =vand limt↘0t
−1
X(t) =bv.
Hence, by (6.5.5) there exist constantsδ >0 andε >0 such that
|X(t)| ≥δt,⟨X(t), R( ˙γ(t), X(t)) ˙γ(t)⟩ ≤ −εt
2
,
fort >0 sufficiently small. By (6.5.4) this implies
d
2
dt
2
|X|=
|X|
2
|∇X|
2
− ⟨X,∇X⟩
2
|X|
3
+
⟨X, R( ˙γ, X) ˙γ⟩
|X|
≤
ct
3
δ
3
−
εt
δ
.
Integrate this inequality over an interval [0, t] withct
2
< εδ
2
to obtain
d
dt
|X(t)|<
d
dt

t=0
|X(t)|=|∇X(0)|
Integrating this inequality again gives|X(t)|< t|∇X(0)|for small positivet.
Hence it follows from (6.5.2) that

dexp
p(tv)tbv

=|X(t)|< t|∇X(0)|=t|bv|
fort >0 sufficiently small. This contradicts (ii).
We prove that (ii) implies that the exponential map exp
p:TpM→Mis
a diffeomorphism for everyp∈M. By (ii) exp
pis alocal diffeomorphism, i.e.
its derivativedexp
p(v) :TpM→T
exp
p
(v)Mis bijective for everyv∈TpM.
Hence we can define a Riemannian metric onM
′
:=TpMby pulling back
the metric onMunder the exponential map. To make this more explicit we
choose a basise1, . . . , emofTpMand define the mapψ:R
m
→Mby
ψ(x) := exp
p

m
X
i=1
x
i
ei
!
forx= (x
1
, . . . , x
m
)∈R
m
. Define the metric tensor by
gij(x) :=
ø
∂ψ
∂x
i
(x),
∂ψ
∂x
j
(x)
Æ
, i, j= 1, . . . , m.
Then (R
m
, g) is a Riemannian manifold (covered by a single coordinate
chart) andψ: (R
m
, g)→Mis a local isometry, by definition ofg. The
manifold (R
m
, g) is clearly connected and simply connected. Moreover, for
everyξ= (ξ
1
, . . . , ξ
n
)∈R
m
=T0R
m
, the curveR→R
m
:t7→tξis a
geodesic with respect tog(becauseψis a local isometry and the image of
the curve underψis a geodesic inM). Hence it follows from Theorem 4.6.5
that (R
m
, g) is complete.

6.5. NONPOSITIVE SECTIONAL CURVATURE 297
Since both (R
m
, g) andMare connected, simply connected, and com-
plete, it follows from Corollary 6.1.15 that the local isometryψis bijective.
Thus the exponential map exp
p:TpM→Mis a diffeomorphism as claimed.
It follows that any two points inMare joined by a unique geodesic.
We prove that (ii) implies (iii). Fix a pointp∈Mand two tangent
vectorsv0, v1∈TpM. Letγ: [0,1]→Mbe the geodesic with the endpoints
γ(0) = exp
p(v0), γ(1) = exp
p(v1)
and letv: [0,1]→TpMbe the unique curve satisfying exp
p(v(t)) =γ(t) for
allt. Thenv(0) =v0,v(1) =v1, and
d(exp
p(v0),exp
p(v1)) =L(γ)
=
Z
1
0

dexp
p(v(t)) ˙v(t)

dt
≥
Z
1
0
|˙v(t)|dt
≥

Z
1
0
˙v(t)dt

=|v1−v0|.
Here the third inequality follows from (ii). This shows that (ii) implies (iii).
We prove that (iii) implies (ii). Fix a pointp∈Mand a tangent
vectorv∈TpMand denoteq:= exp
p(v).By (iii) the exponential map
exp
q:TqM→Mis injective and, sinceMis complete, it is bijective (see
Theorem 4.6.6). Hence there exists a unique geodesic fromqto any other
point inMand therefore, by Theorem 4.4.4, we have
|w|=d(q,exp
q(w)) (6.5.6)
for everyw∈TqM. Now defineϕ:= exp
−1
q◦exp
p:TpM→TqM.This
map satisfiesϕ(v) = 0.Moreover, it is differentiable in a neighborhood ofv
and, by the chain rule,dϕ(v) =dexp
p(v) :TpM→TqM.Now choose
w:=ϕ(v+bv) withbv∈TpM. Then exp
q(w) = exp
q(ϕ(v+bv)) = exp
p(v+bv)
and hence it follows from (6.5.6) and part (iii) that
|ϕ(v+bv))|=|w|=d(q,exp
q(w)) =d(exp
p(v),exp
p(v+bv))≥ |bv|.
This gives

dexp
p(v)bv

=|dϕ(v)bv|= lim
t→0
|ϕ(v+tbv)|
t
≥lim
t→0
|tbv|
t
=|bv|.
Thus we have proved that (iii) implies (ii) and this proves Theorem 6.5.2.

298 CHAPTER 6. GEOMETRY AND TOPOLOGY
The next lemma establishes a useful inequality for Hadamard manifolds
that amplifies the expanding property of the exponential map.
Lemma 6.5.5.LetMbe a Hadamard manifold. Fix an elementp∈M
and two tangent vectorsv0, v1∈TpM. Then, for0< t≤T,
|v0−v1| ≤
d(exp
p(tv0),exp
p(tv1))
t
≤
d(exp
p(T v0),exp
p(T v1))
T
.(6.5.7)
Proof.The first inequality in (6.5.7) is part (iii) of Theorem 6.5.2. To prove
the second inequality, assumev0̸=v1and define
γ0(t) := exp
p(tv0), γ 1(t) := exp
p(tv1)
fort∈R. For eacht∈Rlet the curve [0,1]→M:s7→γ(s, t) be the unique
geodesic with the endpointsγ(0, t) =γ0(t) andγ(1, t) =γ1(t). Then
ρ(t) :=d(γ0(t), γ1(t)) =
Z
1
0
|∂sγ|ds=|∂sγ(s, t)|
for allsandtand hence
˙ρ(t) =
Z
1
0
⟨∂sγ,∇t∂sγ⟩
|∂sγ|
ds
=
⟨∂sγ(1, t), ∂tγ(1, t)⟩ − ⟨∂sγ(0, t), ∂tγ(0, t)⟩
ρ(t)
.
(6.5.8)
Since
d
dt
(ρ˙ρ) =ρ¨ρ+ ˙ρ
2
andγ0andγ1are geodesics, this implies
ρ(t)¨ρ(t) + ˙ρ(t)
2
=
d
dt
ı
⟨∂sγ(1, t), ∂tγ(1, t)⟩ − ⟨∂sγ(0, t), ∂tγ(0, t)⟩
ȷ
=⟨∇t∂sγ(1, t), ∂tγ(1, t)⟩ − ⟨∇t∂sγ(0, t), ∂tγ(0, t)⟩
=
Z
1
0
∂
∂s
⟨∇t∂sγ, ∂tγ⟩ds
=
Z
1
0
ı
|∇t∂sγ|
2
+⟨∇s∇t∂sγ, ∂tγ⟩
ȷ
ds
=
Z
1
0
ı
|∇t∂sγ|
2
+⟨R(∂sγ, ∂tγ)∂sγ, ∂tγ⟩
ȷ
ds
≥˙ρ(t)
2
.
Here the last step follows from (6.5.8), the Cauchy–Schwarz inequality, and
the nonpositive sectional curvature assumption. Thusρ: [0, T]→Ris a
convex function satisfyingρ(0) = 0 and henceρ(t)≤tρ(T)/Tfor 0≤t≤T.
This proves Lemma 6.5.5.

6.5. NONPOSITIVE SECTIONAL CURVATURE 299
6.5.2 Cartan’s Fixed Point Theorem
Recall from Definition 6.5.1 that a Hadamard manifold is a nonempty, con-
nected, simply connected, complete Riemannian manifold of nonpositive
sectional curvature.
Theorem 6.5.6(Cartan).LetMbe a Hadamard manifold and letGbe a
compact topological group that acts onMby isometries. Then there exists
a pointp∈Msuch thatgp=pfor everyg∈G.
The proof follows the argument given by Bill Casselmann in [16] and
requires the following two lemmas. The first lemma asserts that every com-
plete, connected, simply connected Riemannian manifold of nonpositive sec-
tional curvature is a semi-hyperbolic space in the sense of Alexandrov [3].v
m
q
0 1
pp
Figure 6.5: Alexandrov semi-hyperbolic space.
Lemma 6.5.7(Alexandrov).LetMbe a Hadamard manifold, letm∈M
andv∈TmM, and define
p0:= exp
m(−v), p 1:= exp
m(v).
Then
2d(m, q)
2
+
d(p0, p1)
2
2
≤d(p0, q)
2
+d(p1, q)
2
(6.5.9)
for everyq∈M(see Figure 6.5).

300 CHAPTER 6. GEOMETRY AND TOPOLOGY
Proof.By Theorem 6.5.2 the exponential map exp
m:TmM→Mis a
diffeomorphism. Henced(p0, p1) = 2|v|. Now letq∈M. Then there is a
unique tangent vectorw∈TmMsuch that
q= exp
m(w), d(m, q) =|w|.
Since the exponential map is expanding, by Theorem 6.5.2, we have
d(p0, q)≥ |w+v|, d(p1, q)≥ |w−v|.
Hence
d(m, q)
2
=|w|
2
=
|w+v|
2
+|w−v|
2
2
− |v|
2
≤
d(p0, q)
2
+d(p1, q)
2
2
−
d(p0, p1)
2
4
.
This proves Lemma 6.5.7.
Exercise 6.5.8.Equality holds in (6.5.9) wheneverMis flat.
The next lemma isSerre’s Uniqueness Theoremfor thecircumcentre
of a bounded set in asemi-hyperbolic space.r
Ω
Ω
Ω
p
Figure 6.6: The circumcenter of a bounded set.
Lemma 6.5.9(Serre).LetMbe a Hadamard manifold and, forp∈M
andr≥0, denote byB(p, r)⊂Mthe closed ball of radiusrcentered atp.
LetΩ⊂Mbe a nonempty bounded set and define
rΩ:= inf{r >0|there exists ap∈Msuch thatΩ⊂B(p, r)}.
Then there exists a unique pointpΩ∈Msuch thatΩ⊂B(pΩ, rΩ)(see
Figure 6.6).

6.5. NONPOSITIVE SECTIONAL CURVATURE 301
Proof.We prove existence. Choose sequencesri> rΩandpi∈Msuch that
Ω⊂B(pi, ri), lim
i→∞
ri=rΩ.
Chooseq∈Ω. Thend(q, pi)≤rifor everyi. Since the sequenceriis
bounded andMis complete, it follows thatpihas a convergent subsequence,
still denoted bypi. Its limit
pΩ:= lim
i→∞
pi
satisfies Ω⊂B(pΩ, rΩ).
We prove uniqueness. Letp0, p1∈Msuch that
Ω⊂B(p0, rΩ)∩B(p1, rΩ).
Since the exponential map exp
p:TpM→Mis a diffeomorphism (by Theo-
rem 6.5.2), there exists a unique vectorv0∈Tp0
Msuch thatp1= exp
p0
(v0).
Denote the midpoint betweenp0andp1by
m:= exp
p0
Γ
1
2
v0
∆
.
Then it follows from Lemma 6.5.7 that
d(m, q)
2
≤
d(p0, q)
2
+d(p1, q)
2
2
−
d(p0, p1)
2
4
≤r
2
Ω−
d(p0, p1)
2
4
for everyq∈Ω. Since sup
q∈Ωd(m, q)≥rΩ(by definition ofrΩ), it follows
thatd(p0, p1) = 0 and hencep0=p1. This proves Lemma 6.5.9.
Proof of Theorem 6.5.6.Letq∈Mand consider the group orbit
Ω :={gq|g∈G}.
Since G is compact, this set is bounded. LetrΩandpΩbe as in Lemma 6.5.9.
Then Ω⊂B(pΩ, rΩ).Since G acts onMby isometries, this implies
Ω =gΩ⊂B(gpΩ, rΩ)
for allg∈G. Hence it follows from the uniqueness statement in Lemma 6.5.9
thatgpΩ=pΩfor everyg∈G. This proves Theorem 6.5.6.

302 CHAPTER 6. GEOMETRY AND TOPOLOGY
6.5.3 Positive Definite Symmetric Matrices
We close this section with an example of a nonpositive sectional curvature
manifold which plays a key role in Donaldson’s approach to Lie algebra
theory [17] (see§7.5.2). Letmbe a positive integer and consider the space
P:=P(R
m
) :=
n
P∈R
m×m

P
T
=P >0
o
(6.5.10)
of positive definite symmetricm×m-matrices. (Here the notation “P >0”
means⟨x, P x⟩>0 for every nonzero vectorx∈R
m
.) ThusPis an open
subset of the vector spaceS:={S∈R
m×m
|S
T
=S}of symmetric ma-
trices and hence the tangent space ofPisTPP=Sfor everyP∈P.
However, we do not use the metric inherited from the inclusion intoSbut
define a Riemannian metric by
⟨
b
P1,
b
P2⟩P:= trace
Γ
b
P1P
−1b
P2P
−1
∆
(6.5.11)
forP∈Pand
b
P1,
b
P2∈TPP=S.
Theorem 6.5.10.The spacePwith the Riemannian metric(6.5.11)is a
connected, simply connected, complete Riemannian manifold with nonposi-
tive sectional curvature, and the distance function onPis given by
d(P, Q) =
r
trace
ı
Γ
log(P
−1/2
QP
−1/2
)
∆
2
ȷ
(6.5.12)
forP, Q∈P. Moreover,Pis a symmetric space and the groupGL(m,R)
of nonsingularm×m-matrices acts onPby isometries viaP7→gP g
T
forg∈GL(m,R).
Proof.See below.
Remark 6.5.11.LetVbe anm-dimensional vector space andH⊂S
2
V
∗
be the set of inner products onV. Define a Riemannian metric onHby
⟨
b
h1,
b
h2⟩h:= trace(S1S2), h(·, Si·) :=
b
hi, (6.5.13)
forh∈Hand
b
h1,
b
h2∈ThH=S
2
V
∗
. Then every vector space isomor-
phismα:R
m
→Vdetermines a diffeomorphismϕα:H→Pvia
ϕα(h) =P ⇐⇒ h(αξ, αη) =⟨ξ, P
−1
η⟩R
m. (6.5.14)
The derivative ofϕαathin the direction
b
h∈ThHis given by
dϕα(h)
b
h=
b
P ⇐⇒
b
P P
−1
=−α
−1
Sα, h(·, S·) :=
b
h.(6.5.15)
Thusϕαis an isometry with respect to the Riemannian metrics (6.5.13)
onHand (6.5.11) onP. Theϕαform an atlas onHwith the transition
mapsϕβα(P) :=ϕβ◦ϕ
−1
α(P) =gβαP g
T
βα
, wheregβα:=β
−1
α∈GL(m,R).

6.5. NONPOSITIVE SECTIONAL CURVATURE 303
Remark 6.5.12.The submanifold
P0:=P0(R
m
) :={P∈P|det(P) = 1} (6.5.16)
of positive definite symmetricm×m-matrices with determinant one is to-
tally geodesic (see Remark 6.5.13 below). Hence all the assertions of Theo-
rem 6.5.10 (with GL(m,R) replaced by SL(m,R)) remain valid forP0.
Remark 6.5.13.LetMbe a Riemannian manifold andL⊂Mbe a sub-
manifold. Then the following are equivalent.
(i)Ifγ:I→Mis a geodesic on an open intervalIsuch that 0∈Iand
γ(0)∈L, ˙γ(0)∈T
γ(0)L,
then there is a constantε >0 such thatγ(t)∈Lfor|t|< ε.
(ii)Ifγ:I→Lis a smooth curve on an open intervalIand Φγdenotes
parallel transport alongγinM, then
Φγ(t, s)T
γ(s)L=T
γ(t)L ∀s, t∈I.
(iii)Ifγ:I→Lis a smooth curve on an open intervalIandX∈Vect(γ)
is a vector field alongγ(with values inT M), then
X(t)∈T
γ(t)L∀t∈I =⇒ ∇ X(t)∈T
γ(t)L∀t∈I.
A submanifold that satisfies these equivalent conditions is calledtotally
geodesic.
Exercise 6.5.14.Prove the equivalence of (i), (ii), (iii) in Remark 6.5.13.
Hint:Choose suitable coordinates and translate each of the three assertions
into conditions on the Christoffel symbols.
Exercise 6.5.15.Prove thatP0is a totally geodesic submanifold ofP.
Prove thatPis diffeomorphic to the quotient GL(m,R)/O(m) via polar de-
composition and thatP0is diffeomorphic to the quotient SL(m,R)/SO(m).
Hint:Consider the map GL(m,R)→P:g7→
p
gg
T
.
Exercise 6.5.16.In the casem= 2 prove thatP0is isometric to the
hyperbolic spaceH
2
.
The proof of Theorem 6.5.10 is based on the calculation of the Levi-Civita
connection and the formulas for geodesics and the Riemann curvature tensor
in the following three lemmas.

304 CHAPTER 6. GEOMETRY AND TOPOLOGY
Lemma 6.5.17.LetI→P:t7→P(t)be a smooth path inPon an
intervalI⊂Rand letI→S:t7→S(t)be a vector field alongP. Then
the covariant derivative ofSis given by
∇S=
˙
S−
1
2
SP
−1˙
P−
1
2
˙
P P
−1
S. (6.5.17)
Proof.The formula (6.5.17) determines a family of linear operators on the
spaces of vector fields along paths that satisfy the torsion-free condition
∇s∂tP=∇t∂sP
for every smooth mapR
2
→P: (s, t)7→P(s, t) and the Leibniz rule
∇ ⟨S1, S2⟩
P
=⟨∇S1, S2⟩
P
+⟨S1,∇S2⟩
P
for any two vector fieldsS1andS2alongP. These two conditions determine
the covariant derivative uniquely (see Lemma 3.6.5 and Theorem 3.7.8).
This proves Lemma 6.5.17.
Lemma 6.5.18.The geodesics inPare given by
γ(t) =Pexp
Γ
tP
−1b
P
∆
= exp
Γ
t
b
P P
−1
∆
P
=P
1/2
exp
Γ
tP
−1/2b
P P
−1/2
∆
P
1/2
(6.5.18)
forP∈P,
b
P∈TPP=S, andt∈R. In particular,Pis complete.
Proof.The curveγ:R→Pdefined by (6.5.18) satisfies
˙γ(t) =
b
Pexp
Γ
tP
−1b
P
∆
=
b
P P
−1
γ(t).
Hence it follows from Lemma 6.5.17 that
∇˙γ(t) = ¨γ(t)−˙γ(t)γ(t)
−1
˙γ(t)
= ¨γ(t)−
b
P P
−1
˙γ(t)
= 0
for everyt∈R. Henceγis a geodesic. Since the curveγ:R→Pin (6.5.18)
satisfiesγ(0) =Pand ˙γ(0) =
b
P, this proves Lemma 6.5.18.

6.5. NONPOSITIVE SECTIONAL CURVATURE 305
Lemma 6.5.19.ForP∈P,S, T, A∈Sthe curvature tensor onPis
RP(S, T)A=−
1
4
SP
−1
T P
−1
A−
1
4
AP
−1
T P
−1
S
+
1
4
T P
−1
SP
−1
A+
1
4
AP
−1
SP
−1
T
=−
1
4
ΘΘ
SP
−1
, T P
−1
Λ
, AP
−1
Λ
P.
(6.5.19)
Proof.Choose smooth mapsP:R
2
→PandA:R
2
→Sand define
S:=∂sPandT:=∂tP. Then∂sT=∂tSandRP(S, T)A=∇s∇tA−∇t∇sA.
Moreover, by Lemma 6.5.17 we have∇sA=∂sA−
1
2
AP
−1
S−
1
2
SP
−1
A
and∇tA=∂tA−
1
2
AP
−1
T−
1
2
T P
−1
A. Hence
RP(S, T)A=∂s∇tA−
1
2
(∇tA)P
−1
S−
1
2
SP
−1
(∇tA)
−∂t∇sA+
1
2
(∇sA)P
−1
T+
1
2
T P
−1
(∇sA)
=∂s
ı
∂tA−
1
2
AP
−1
T−
1
2
T P
−1
A
ȷ
−
1
2
ı
∂tA−
1
2
AP
−1
T−
1
2
T P
−1
A
ȷ
P
−1
S
−
1
2
SP
−1
ı
∂tA−
1
2
AP
−1
T−
1
2
T P
−1
A
ȷ
−∂t
ı
∂sA−
1
2
AP
−1
S−
1
2
SP
−1
A
ȷ
+
1
2
ı
∂sA−
1
2
AP
−1
S−
1
2
SP
−1
A
ȷ
P
−1
T
+
1
2
T P
−1
ı
∂sA−
1
2
AP
−1
S−
1
2
SP
−1
A
ȷ
.
A term by term inspection shows that the partial derivatives ofA,S,T
cancel because∂sT=∂tS. Hence we are left with the drivatives ofP, so
RP(S, T)A
=
1
2
AP
−1
(∂sP)P
−1
T+
1
2
T P
−1
(∂sP)P
−1
A
+
1
4
AP
−1
T P
−1
S+
1
4
T P
−1
AP
−1
S+
1
4
SP
−1
AP
−1
T+
1
4
SP
−1
T P
−1
A
−
1
2
AP
−1
(∂tP)P
−1
S−
1
2
SP
−1
(∂tP)P
−1
A
−
1
4
AP
−1
SP
−1
T−
1
4
SP
−1
AP
−1
T−
1
4
T P
−1
AP
−1
S−
1
4
T P
−1
SP
−1
A.
Insert∂sP=S,∂tP=Tto obtain (6.5.19). This proves Lemma 6.5.19.

306 CHAPTER 6. GEOMETRY AND TOPOLOGY
Proof of Theorem 6.5.10.The manifoldPis obviously connected and sim-
ply connected as it is a convex open subset of a finite-dimensional vector
space. That the map GL(m,R)×P→P: (g, P)7→gP g
T
defines a group
action of GL(m,R) onPby isometries follows directly from the definitions.
The remaining assertions will be proved in three steps.
Step 1.The manifoldPhas nonpositive sectional curvature.
By Lemma 6.5.19 withA=Tand equation (6.5.11) we have
⟨S, RP(S, T)T⟩
P
= trace
Γ
SP
−1
(RP(S, T)T)P
−1
∆
=−
1
4
trace
Γ
SP
−1
ΘΘ
SP
−1
, T P
−1
Λ
, T P
−1
Λ∆
=
1
2
trace
Γ
SP
−1
T P
−1
SP
−1
T P
−1
∆
−
1
2
trace
Γ
SP
−1
T P
−1
T P
−1
SP
−1
∆
=
1
2
trace
Γ
X
2
∆
−
1
2
trace
ı
X
T
X
ȷ
,
whereX:=P
−1/2
SP
−1
T P
−1/2
∈R
m×m
.WriteX=: (xij)i,j=1,...,m.Then,
by the Cauchy–Schwarz inequality, we have
trace(X
2
) =
X
i,j
xijxji≤
X
i,j
x
2
ij= trace(X
T
X).
Thus⟨S, RP(S, T)T⟩
P
≤0 for allP∈PandS, T∈S. This proves Step 1.
Step 2.Pis a symmetric space.
Fix an elementA∈Pand define the mapϕ:P→Pbyϕ(P) :=AP
−1
A
forP∈P. This map is a diffeomorphism, fixes the matrixA=ϕ(A),and
its derivative atP∈Pis given bydϕ(P)
b
P=−AP
−1b
P P
−1
Afor
b
P∈TPP.
Hencedϕ(A) =−id.Moreover, (dϕ(P)
b
P)ϕ(P)
−1
=−AP
−1b
P A
−1
and so
|dϕ(P)
b
P|
2
ϕ(P)
= trace
ı
Γ
AP
−1b
P A
−1
∆
2
ȷ
= trace
ı
Γ
P
−1b
P
∆
2
ȷ
=|
b
P|
2
P
for allP∈Pand
b
P∈TPP. Thusϕis an isometry and this proves Step 2.
Step 3.The distance function onPis given by(6.5.12).
LetP, Q∈P. Then, sincePis a Hadamard manifold by Step 1 and
Lemma 6.5.18, there exists a unique matrix
b
P∈Ssuch that exp
P(
b
P) =Q.
By Lemma 6.5.18 this equation readsP
1/2
exp(P
−1/2b
P P
−1/2
)P
1/2
=Q.
ThusS:=P
−1/2b
P P
−1/2
= log(P
−1/2
QP
−1/2
) and
d(P, Q)
2
=|
b
P|
2
P= trace
Γ
b
P P
−1b
P P
−1
∆
= trace
Γ
S
2
∆
.
This proves Step 3 and Theorem 6.5.10.

6.5. NONPOSITIVE SECTIONAL CURVATURE 307
Remark 6.5.20.Theorem 6.5.10 carries over to the complex setting as
follows. ReplacePby the space
Q:=
Φ
Q∈C
m×m

Q
∗
=Q >0

(6.5.20)
of positive definite Hermitian matrices. HereQ
∗
denotes the conjugate trans-
posed matrix ofQ∈C
m×m
and the notation “Q >0” meansz
∗
Qz >0 for
every nonzero vectorz∈C
m
. ThusQis an open subset of the vector space
of Hermitianm×m-matrices. Define the Riemannian metric onQby
⟨H1, H2⟩
Q
:= Re
Γ
trace(H1Q
−1
H2Q
−1
)
∆
forQ∈QandH1, H2∈TQQ. Then all the assertions of Theorem 6.5.10
(with GL(m,R) replaced by GL(m,C)) carry over toQ. The proof is verba-
tim the same, with the transposed matric replaced by the conjugate tram-
sposed matrix and the trace replaced by the real part of the trace.
Remark 6.5.21.The setQ0:=
Φ
Q∈Q

det(Q) = 1

of positive definite
Hermitian matrices with determinant one is a totally geodesic submanifold
ofQ. Hence all the assertions of Theorem 6.5.10 (with GL(m,R) replaced
by SL(m,C)) remain valid forQ0.
Exercise 6.5.22.Show that Theorem 6.5.10 remains valid forQ, andQ0
is a totally geodesic submanifold ofQ. Prove thatQis diffeomorphic to
the quotient GL(m,C)/U(m) andQ0is diffeomorphic to SL(m,C)/SU(m).
Hint:Consider the map SL(m,C)→Q0:g7→
√
gg
∗
. Show that the pull-
back metric on SL(m,C)/SU(m) is given by the norm of the Hermitian part
of the matrixg
−1
bgforbg∈TgSL(m,C).
Exercise 6.5.23.In the casem= 2 prove thatQ0is isometric to the
hyperbolic spaceH
3
.
The spaceQ0
∼
=SL(m,C)/SU(m) (with nonpositive sectional curvature)
can be viewed as a kind of dual space of the Lie group SU(m) (with non-
negative sectional curvature). They have the same dimension and in both
cases the Riemann curvature tensor is given by the Lie bracket (see equa-
tion (5.2.23) for SU(m) and equation (6.5.19) forQ0). One can think of
the noncompact Lie group G
c
:= SL(m,C) as thecomplexificationof the
compact Lie group G := SU(m). It can be written in the form
G
c
={exp(iη)u|u∈G, η∈g}, (6.5.21)
the Lie algebra of G
c
is the complexificationg
c
=g⊕igof the Lie algebra
of G, and the quotient G
c
/G is a Hadamard manifold. These observations
carry over to all Lie subgroups G⊂SU(m). For an exposition see [20].

308 CHAPTER 6. GEOMETRY AND TOPOLOGY
Exercise 6.5.24(Siegel upper half space). The standard symplectic
formω0onR
2n
=R
n
×R
n
is given by
ω0(z, ζ) := (J0z)
T
ζ, J 0:=
`
0−1l
1l 0
´
,
forz, ζ∈R
2n
and the space ofω0-compatible linear complex structures is
the (n
2
+n)-dimensional manifold
J(R
2n
, ω0) :=
æ
J∈R
2n×2n

J
2
=−1l, JJ0+J0J
T
= 0,
ω0(ζ, Jζ)>0 for 0̸=ζ∈R
2n
œ
.(6.5.22)
Define a Riemannian metric onJ(M, ω0) by
⟨
b
J1,
b
J2⟩:= trace
Γ
b
J1
b
J2) (6.5.23)
for
b
J1,
b
J2∈TJJ(R
2n
, ω0). Show that the symplectic linear group Sp(2n)
(Exercise 2.5.5) acts on the spaceJ(R
2n
, ω0) by isometriesJ7→gJg
−1
.
IfJ∈ J0(R
2n
, ω0) andP:=−JJ0=P
T
, show thatω0(·, J·) =⟨·, P
−1
·⟩.
Show that the mapJ(R
2n
, ω0)→P0(R
2n
) :J7→ −JJ0is an Sp(2n)-
equivariant isometric embedding, whose image is a totally geodesic subman-
ifold ofP0(R
2n
). Deduce thatJ(R
2n
, ω0) is a Hadamard manifold and a
symmetric space. For everyJ∈ J(R
2n
, ω0) show that the mapJ
′
7→ −JJ
′
J
is an isometry fixingJwhose derivative atJis−id.
(ii) Siegel upper half spaceis the manifoldSn⊂C
n×n
of symmetric
complexn×n-matrices with positive definite imaginary part [71]. The sym-
plectic linear group Sp(2n) acts on this space via
g∗Z:= (AZ+B)(CZ+D)
−1
, g=
`
A B
C D
´
,
A
T
C=C
T
A, B
T
D=D
T
B, A
T
D−C
T
B= 1l,
(6.5.24)
forg∈Sp(2n) andZ∈ Sn. Show that this is a well-defined group action.
(For hints see [49, page 72].) Show that there is a unique Sp(2n)-equivariant
diffeomorphism fromSntoJ(R
2n
, ω0) that sendsi1l toJ0. Show that this
map is given by the explicit formula
J(Z) =
`
XY
−1
−Y−XY
−1
X
Y
−1
−Y
−1
X
´
∈ J(R
2n
, ω0), Z=X+iY∈ Sn.
Show that the diffeomorphismSn→ J(R
2n
, ω0) :Z7→J(Z) is an isometry
with respect to the Riemannian metric onSngiven by
|
b
Z|
2
Z= 2trace
Γ
(Y
−1b
X)
2
+ (Y
−1b
Y)
2
∆
(6.5.25)
forZ=X+iY∈ Snand
b
Z=
b
X+i
b
Y∈TZSn.

6.6. POSITIVE RICCI CURVATURE* 309
6.6 Positive Ricci Curvature*
In this section we prove that every complete connected manifoldM⊂R
n
whose Ricci curvature satisfies a uniform positive lower bound is necessarily
compact. If the sectional curvature is constant and positive, this follows
from Corollary 6.4.12 as was noted in Example 6.4.16.
Definition 6.6.1(Ricci tensor).LetM⊂R
n
be anm-dimensional sub-
manifold and fix an elementp∈M. TheRicci tensorofMatpis the
symmetric bilinear form
Ricp:TpM×TpM→R
defined by
Ricp(u, v) :=
m
X
i=1
⟨Rp(ei, u)v, ei⟩, (6.6.1)
wheree1, . . . , emis an orthonormal basis ofTpM. The Ricci tensor is inde-
pendent of the choice of this orthonormal frame and is symmetric by equa-
tions(5.2.17)and(5.2.19)in Theorem 5.2.14.
The Ricci Tensor in Local Coordinates
Letϕ:U→Ω be a local coordinate chart on an open setU⊂Mwith values
in an open set Ω⊂R
m
, denote its inverse byψ:=ϕ
−1
: Ω→U,and let
Ei(x) :=
∂ψ
∂x
i
(x)∈T
ψ(x)M, x ∈Ω, i= 1, . . . , m,
be the local frame of the tangent bundle determined by this coordinate chart.
Denote the coefficients of the first fundamental form bygij:=⟨Ei, Ej⟩and
the coefficients of the Riemann curvature tensor byR
ℓ
ijk
: Ω→Rso that
R(Ei, Ej)Ek=
X
ℓ
R
ℓ
ijk
Eℓ.
(see Section 5.4). Then
Ricij:= Ric(Ei, Ej) =
m
X
ν=1
R
ν
νij=
m
X
µ,ν=1
Rνijµg
µν
. (6.6.2)
(Exercise:Prove this.)

310 CHAPTER 6. GEOMETRY AND TOPOLOGY
The Bonnet–Myers Theorem
Theorem 6.6.2(Bonnet–Myers).LetM⊂R
n
be a complete, connected
manifold of dimensionm≥2and suppose that there exists aδ >0such that
Ricp(v, v)≥(m−1)δ|v|
2
(6.6.3)
for everyp∈Mand everyv∈TpM. Thend(p, q)≤π/
√
δfor allp, q∈M
and henceMis compact.
The proof is based on the following lemma.
Lemma 6.6.3.LetR×[0,1]→M: (s, t)7→γs(t)be a smooth map such
thatγ:=γ0: [0,1]→Mis a geodesic andγs(0) =γ(0)andγs(1) =γ(1)
for alls∈R. Define the vector fieldXalongγbyX(t) :=
∂
∂s

s=0
γs(t)
for0≤t≤1. Then
d
2
ds
2

s=0
E(γs) =−
Z
1
0
⟨∇∇X+R(X,˙γ) ˙γ, X⟩dt
=
Z
1
0
ı
|∇X|
2
− ⟨R(X,˙γ) ˙γ, X⟩
ȷ
dt.
(6.6.4)
Proof.In the proof of Theorem 4.1.4 we have seen that
d
ds
E(γs) =−
Z
1
0
⟨∇t∂tγs(t), ∂sγs(t)⟩dt
for alls∈R(see equation (4.1.10)). Differentiate this equation again with
respect tosand use the identity∇s∂t=∇t∂sto obtain
d
2
ds
2
E(γs) =−
d
ds

s=0
Z
1
0
⟨∇t∂tγs, ∂sγs⟩dt
=−
Z
1
0
⟨∇s∇t∂tγs, ∂sγs⟩dt−
Z
1
0
⟨∇t∂tγs,∇s∂sγs⟩dt
=−
Z
1
0
⟨∇t∇t∂sγs+R(∂sγs, ∂tγs)∂tγs, ∂sγs⟩dt
−
Z
1
0
⟨∇t∂tγs,∇s∂sγs⟩dt.
Now takes= 0. Then∇t∂tγ= 0 becauseγis a geodesic and hence
d
2
ds
2

s=0
E(γs) =−
Z
1
0
⟨∇t∇tX+R(X,˙γ) ˙γ, X⟩dt
This proves the first equality in (6.6.4). To prove the second equality
in (6.6.4) use integration by parts and the fact thatX(0) = 0 andX(1) = 0.
This proves Lemma 6.6.3.

6.6. POSITIVE RICCI CURVATURE* 311
Proof of Theorem 6.6.2.Letp, q∈M. By the Hopf–Rinow Theorem 4.6.6
there exists a geodesicγ: [0,1]→Msuch that
γ(0) =p, γ(1) =q, L(γ) =d(p, q).
LetX∈Vect(γ) be a vector field alongγsuch thatX(0) = 0 andX(1) = 0
and defineγs(t) := exp
γ(t)(sX(t)) fors∈Rand 0≤t≤1. Thens= 0 is
the absolute minimum of the functionR→R:s7→E(γs) by Lemma 4.4.1.
Hence
d
2
ds
2|s=0E(γs)≥0 and by Lemma 6.6.3 this implies
Z
1
0
⟨R(X,˙γ) ˙γ, X⟩dt≤
Z
1
0
|∇tX(t)|
2
dt. (6.6.5)
Now assumep̸=qand choose an orthonormal frameE1, . . . , Emalongγ
such thatE1= ˙γ/|˙γ|and∇tEi≡0 fori= 1, . . . m. Define
Xi(t) := sin(πt)Ei(t)
fori= 1, . . . mand 0≤t≤1. Then|∇tXi(t)|=πcos(πt) for alliandtand
δ(m−1)|˙γ(t)|
2
≤Ric
γ(t)( ˙γ(t),˙γ(t)) =
m
X
i=2
⟨R(Ei(t),˙γ(t)) ˙γ(t), Ei(t)⟩
for 0≤t≤1. Multiply this inequality by sin
2
(πt), integrate over the unit
interval, and use the identities|˙γ(t)|=d(p, q) and
R
1
0
sin
2
(πt)dt= 1/2 to
obtain the estimate
δ(m−1)
2
d(p, q)
2
=
Z
1
0
δ(m−1) sin
2
(πt)|˙γ(t)|
2
dt
≤
m
X
i=2
Z
1
0
⟨R(Xi(t),˙γ(t)) ˙γ(t), Xi(t)⟩dt
≤
m
X
i=2
Z
1
0
|∇tXi(t)|
2
dt
= (m−1)
Z
1
0
π
2
cos
2
(πt)dt
=
π
2
(m−1)
2
.
Here the third step uses (6.6.5). Sincem≥2 it follows from this estimate
thatd(p, q)
2
≤π
2
/δand this proves Theorem 6.6.2.
A direct consequence of Theorem 6.6.2 is that every compact manifold
with positive Ricci curvature has a compact universal cover and hence has
a finite fundamental group.

312 CHAPTER 6. GEOMETRY AND TOPOLOGY
Positive Sectional Curvature
Corollary 6.6.4.LetM⊂R
n
be a complete, connected manifold of dimen-
sionm≥2and suppose that there exists aδ >0such that
K(p, E)≥δ
for everyp∈Mand every2-dimensional linear subspaceE⊂TpM. Then
d(p, q)≤
π
√
δ
for allp, q∈Mand henceMis compact.
Proof.The conditionK≥δimplies (6.6.3) withm:= dim(M) and hence
the assertion follows from Theorem 6.6.2. This proves Corollary 6.6.4.
The example of them-sphere shows that the estimate in Corollary 6.6.4
is sharp. Namely,M:=S
m
has sectional curvatureK= 1 and diameterπ.
The paraboloidM:= (x, y, z)∈R
3
|z=x
2
+y
2
}has positive Gaußian
curvature and so positive Ricci curvature (Lemma 6.7.2) but is noncompact.
Remark 6.6.5(Sphere Theorem).TheTopological Sphere Theorem
asserts that every complete, connected, simply connected Riemannianm-
manifoldMwhose sectional curvature satisfies the estimate
1/4< K(p, E)≤1
for everyp∈Mand every 2-dimensional linear subspaceE⊂TpMmust be
homeomorphic to them-sphere. TheDifferentiable Sphere Theorem
asserts under the same assumptions thatMis diffeomorphic toS
m
.
The problem goes back to a question posed by Heinz Hopf [31, 32] in the
1920s. After intermediate results by Rauch [57] (with 1/4 replaced by 3/4)
and others, the Toplogical Sphere Theorem was proved in 1961 by Berger [6]
and Klingenberg [41]. The Differentiable Sphere Theorem was proved in
2007 by Brendle and Schoen [8, 9, 10]. They even weakened the assumption
to 0<maxEK(p, E)<4 minEK(p, E) for allp∈M, where the maximum
and minimum are taken over all 2-dimensional linear subspacesE⊂TpM.
The Topological Sphere Theorem is sharp, as the suitably scaled Fubini–
Study metric on complex projective space satisfies 1/4≤K(p, E)≤1 for
allpandE. The Differentiable Sphere Theorem is a significant improve-
ment, because in many dimensions there exist smoothm-manifolds that are
homeomorphic, but not diffeomorphic, toS
m
. These are the so-calledex-
otic spheresand many of those do not even admit metrics of positivescalar
curvature [30]. (For the definition of scalar curvature see§6.7.)

6.7. SCALAR CURVATURE* 313
6.7 Scalar Curvature*
This section introduces the scalar curvature and explains how it is related to
the Ricci tensor and the Riemann curvature tensor in dimensions 2 and 3.
The section also includes a brief discussion of several problems in differential
geometry in which the scalar curvature plays a central role.
Definition and Basic Properties
Letmbe a positive integer and letM⊂R
n
be anm-manifold. For each
elementp∈Mdenote byRp:TpM×TpM→End(TpM) the Riemann cur-
vature tensor and by Ricp:TpM×TpM→Rthe Ricci tensor ofMatp.
Definition 6.7.1(Scalar curvature).Fix an elementp∈M. Thescalar
curvatureS(p)ofMatpis the trace of the Ricci tensor and is given by
S(p) :=
m
X
i=1
Ricp(ei, ei) =
m
X
i,j=1
⟨Rp(ei, ej)ej, ei⟩, (6.7.1)
wheree1, . . . , emis an orthonormal basis ofTpM. The scalar curvature is
independent of the choice of this orthonormal frame.
Lemma 6.7.2.Assumem= 2and letK:M→Rbe the Gaußian curvature
ofMin(6.4.2). Then
S(p) = 2K(p),Ricp(u, v) =K(p)⟨u, v⟩, (6.7.2)
⟨Rp(u, v)w, z⟩=K(p)
ı
⟨u, z⟩⟨v, w⟩ − ⟨u, w⟩⟨v, z⟩
ȷ
(6.7.3)
for allp∈Mandu, v, w, z∈TpM.
Proof.By definition, the Gaußian curvature is given by
K(p) =
⟨Rp(u, v)v, u⟩
|u|
2
|v|
2
− ⟨u, v⟩
2
(6.7.4)
for every pair of linearly independent tangent vectorsu, v∈TpM. Takeu
to be a unit vector orthogonal tovto obtain Ricp(v, v) =K(p)|v|
2
for
everyv∈TpM. Since Ricp:TpM×TpM→Ris a symmetric bilinear form,
this implies the second equality in (6.7.2). With this understood the first
equality in (6.7.2) follows directly from the definition of the scalar curvature
in (6.7.1). Equation (6.7.3) follows from (6.7.4) by Theorem 6.4.8. This
proves Lemma 6.7.2.

314 CHAPTER 6. GEOMETRY AND TOPOLOGY
Lemma 6.7.3.Assumem= 3. Then
⟨Rp(u, v)w, z⟩= Ricp(v, w)⟨u, z⟩ −Ricp(u, w)⟨v, z⟩
+ Ricp(u, z)⟨v, w⟩ −Ricp(v, z)⟨u, w⟩
−
S(p)
2
ı
⟨u, z⟩⟨v, w⟩ − ⟨u, w⟩⟨v, z⟩
ȷ
(6.7.5)
for allp∈Mand allu, v, w, z∈TpM.
Proof.Choose an orthonormal basise1, e2, e3of the tangent spaceTpMand
define the endomorphism
Qp:TpM→TpM
by
Qpu:=
X
i
Rp(u, ei)ei
foru∈TpM. The right hand side of this equation is independent of the
choice of the orthonormal basis and the endomorphismQpsatisfies
trace(Qp) =S(p)
and
⟨Qpu, v⟩= Ricp(u, v)
for allu, v∈TpM. With this notation equation (6.7.5) takes the form
Rp(u, v)w= Ricp(v, w)u+⟨v, w⟩
`
Qpu−
S(p)
2
u
´
−Ricp(u, w)v− ⟨u, w⟩
`
Qpv−
S(p)
2
v
´
.
(6.7.6)
It suffices to verify equation (6.7.6) in the following three cases.
(a)u, vare linarly dependent.
(b)u, v, ware orthonormal.
(c)u, vare orthonormal andw=v.
In the case (a) both sides of equation (6.7.6) vanish. In the case (b) we have
Rp(u, v)w=⟨Rp(u, v)w, u⟩u+⟨Rp(u, v)w, v⟩v
= Ricp(v, w)u−Ricp(u, w)v,
and this is equivalent to (6.7.6).

6.7. SCALAR CURVATURE* 315
In the case (c) equaton (6.7.6) takes the form
Rp(u, v)v= Ricp(v, v)u−Ricp(u, v)v+Qpu−
S(p)
2
u. (6.7.7)
To verify this formula, choose a unit vectorwthat is orthogonal touandv.
Then it follows from the definition ofS(p) andQpthat
S(p)
2
=⟨Rp(u, v)v, u⟩+⟨Rp(w, v)v, w⟩+⟨Rp(u, w)w, u⟩
= Ricp(v, v) +⟨Rp(u, w)w, u⟩,
and
Qpu=Rp(u, v)v+Rp(u, w)w
=Rp(u, v)v+⟨Rp(u, w)w, v⟩v+⟨Rp(u, w)w, u⟩u
=Rp(u, v)v+ Ricp(u, v)v+
S(p)
2
u−Ricp(v, v)u.
This proves (6.7.7) and Lemma 6.7.3.
Scalar Curvature in Local Coordinates
Letϕ:U→Ω be a local coordinate chart on an open setU⊂Mwith values
in an open set Ω⊂R
m
, denote its inverse by
ψ:=ϕ
−1
: Ω→U,
and denote by
Ei(x) :=∂iψ(x)∈T
ψ(x)M
forx∈Ω andi= 1, . . . , mthe local frame of the tangent bundle determined
by this coordinate chart. Denote the coefficients of the first fundamental
form bygij:=⟨Ei, Ej⟩, of the Ricci tensor by
Ricij:= Ric(Ei, Ej),
and of the Riemann curvature tensor byRijkℓandR
ℓ
ijk
so that
Rijkℓ=⟨R(Ei, Ej)Ek, Eℓ⟩, R(Ei, Ej)Ek=
m
X
ℓ=1
R
ℓ
ijk
Eℓ.
Then the scalar curvature is the functionS: Ω→Rgiven by
S=
m
X
i,j=1
Ricijg
ij
=
m
X
i,j,ν=1
R
ν
νijg
ij
=
m
X
i,j,k,ℓ=1
Rijkℓg
jk
g
iℓ
. (6.7.8)
(Exercise:Prove this.)

316 CHAPTER 6. GEOMETRY AND TOPOLOGY
Positive Scalar Curvature
An important question for a compact smooth manifoldMis whether or not
it admits a Riemannian metric of positive scalar curvature. A theorem of
Lichnerowicz [46] asserts that, ifMis a compact spin manifold of dimen-
sionm= 4nthat admits a Riemannian metric of positive scalar curvature,
then a certain characteristic class of this manifold (the
b
A-genus) must van-
ish. The definitions of the terms that appear in this sentence (spin structure
and
b
A-genus) as well as in the proof, which involves the Dirac operator, the
Atiyah–Singer index theorem, and the Weitzenb¨ock formula, go beyond the
scope of the present book. For an exposition see [65, Theorem 6.30].
A nonlinear variant of Lichnerowicz’ theorem asserts that a compact
oriented smooth 4-manifold withb
+
2
−b1odd andb
+
2
>1 that admits a
Riemannian metric of positive scalar curvature has vanishing Seiberg-Witten
invariants (see [65, Proposition 7.32]).
In another direction Gromov and Lawson [22] proved that ifM1andM2
are two compact manifolds of dimensionm≥3 that admit Riemannian met-
rics of positive scalar curvature, then so does their connected sumM1#M2
(see also [65, Theorem 2.18]).
In the late 1970’s Schoen and Yau [70] proved, using minimal surfaces,
that the torusT
m
=R
m
/Z
m
does not admit a metric of positive scalar
curvature form≤7. We remark that the
b
A-genus of the torus vanishes and
so Lichnerowicz’ theorem does not apply. In [22] Gromov and Lawson refined
the techniques of Lichnerowicz to prove that, for anym, them-torus does not
admit a metric of positive scalar curvature. In fact, they proved that for any
compact spin manifoldMof dimensionmthe connected sumN:=M#T
m
does not admit a metric of positive scalar curvature. Moreover, they proved
that ifNadmits a metric of nonnegative scalar curvature, then this metric
must be flat andNmust be the standardm-torus.
Constant Scalar Curvature
An interesting class of Riemannian metrics consists of those that have con-
stant scalar curvature. In dimensionm= 2 the existence of such a metric
is the content of theuniformisation theorem. The proof involves the
solution of the Kazdan-Warner equation and goes beyond the scope of this
book. For an exposition see [65, Theorem 2.20 and Theorem D.1].
In dimension two Lemma 6.7.2 shows that the constant scalar curva-
ture condition is equivalent to constant sectional curvature. However, in
higher dimensions the constant scalar curvature condition is more general.

6.7. SCALAR CURVATURE* 317
By Corollary 6.4.12 every compact simply connectedm-manifold with con-
stant sectional curvature is diffeomorphic to them-sphere, while constant
scalar curvature metrics exist on every compact manifold. Examples are
the Fubini-Study metric on complex projective space (Example 2.8.5 and
Example 3.7.5), locally symmetric spaces (Theorem 6.3.4), and products of
Riemannian manifolds with constant scalar curvature.
Definition 6.7.4.LetMbe a Riemannian manifold with the metricg. A
Riemannian metricg
′
onMis calledconformally equivalenttogiff there
exists a smooth functionλ:M→(0,∞)such thatg
′
=λg.The set of all
such Riemannian metrics is called theconformal classofg.
Remark 6.7.5(Yamabe problem).TheYamabe problemasserts that
the conformal class of every Riemannian metric on a compactm-manifoldM
of dimensionm≥2contains a metric of constant scalar curvature.
This problem was formulated in 1960 by Hidehiko Yamabe and was even-
tually settled in the affirmative in 1984 by the combined work of Hide-
hiko Yamabe [78], Thierry Aubin [5], Neil Trudinger [75], and Richard
Schoen [69]. The proof for a compact manifoldMof dimensionm >2
relies on finding a positive functionf:M→Rand a real numbercthat
satisfy theYamabe equation
4(m−1)
m−2
∆gf+Sgf=cf
(m+2)/(m−2)
. (6.7.9)
HereSg:M→Rdenotes the scalar curvature of the Riemannian metricg
and ∆gdenotes itsLaplace–Beltrami operator. In local coordinates this
operator is given by the formula
∆g=
1
p
det(g)
X
i,j
∂
∂x
i
g
ij
p
det(g)
∂
∂x
j
. (6.7.10)
Iff:M→(0,∞) is a positive solution of (6.7.9), then the Riemannian
metricf
4/(m−2)
ghas the constant scalar curvaturec.Exercise: Prove this.
Hint:Show first that
S
u
2
g=u
−2
Sg+ 2(m−1)u
−3
∆gu−(m−1)(m−4)u
−4
|du|
2
g.(6.7.11)
Then takef:=u
m/2−1
and use the identities
|du
n
|
2
g=n
2
u
2n−2
|du|
2
g, (6.7.12)
∆gu
n
=nu
n−1
∆gu+n(n−1)u
n−2
|du|
2
g (6.7.13)
for a smooth functionu:M→(0,∞) and a real numbern >0.

318 CHAPTER 6. GEOMETRY AND TOPOLOGY
Examples of constant scalar curvature metrics arise from the Einstein
condition (see Lemma 6.7.7 below).
Definition 6.7.6(Einstein metric).A Riemannian manifoldMis called
anEinstein manifoldiff its Ricci tensor is a scalar multiple of the first
fundamental form, i.e. there exists a smooth functionλ:M→R
Ricp(u, v) =λ(p)⟨u, v⟩ (6.7.14)
for allp∈Mand allu, v∈TpM. It follows from the definitions that the
factorλin(6.7.14)is related to the scalar curvatureSby
λ=
S
m
. (6.7.15)
Lemma 6.7.2 shows that every Riemannian metric on a 2-manifold is an
Einstein metric and the factorλ=Kis the Gaußian curvature.
Lemma 6.7.7.LetMbe an Einstein manifold of dimensionm≥3. Then
the scalar curvature ofMis locally constant.
Proof.Letp∈M. Then there exists a local orthonormal frame of the
tangent bundleE1, . . . , Em∈Vect(U) in a neighborhoodUofpsuch that the
covariant derivatives∇Eiall vanish atp. (Exercise:Prove this.) Denote
the deriviative of a functionfatpin the directionEi(p) by∂if. Then
0 =
X
j,k
⟨(∇Ei
R)(Ej, Ek)Ek, Ej⟩
+
X
j,k
⟨(∇Ej
R)(Ek, Ei)Ek, Ej⟩+
X
j,k
⟨(∇Ek
R)(Ei, Ej)Ek, Ej⟩
=∂i
X
j,k
⟨R(Ej, Ek)Ek, Ej⟩
+
X
j
∂j
X
k
⟨R(Ek, Ei)Ek, Ej⟩+
X
k
∂k
X
j
⟨R(Ei, Ej)Ek, Ej⟩
=∂iS−
X
j
∂jRic(Ei, Ej)−
X
k
∂kRic(Ei, Ek)
=
m−2
m
∂iS.
Here the the first equality follows from the second Bianchi identity (6.3.7),
the second holds atpbecause∇Ei(p) = 0, the third follows from the def-
initions of Ric andS, and the last uses the identity Ric(Ei, Ej) =δijS/m,
which holds by (6.7.14) and (6.7.15). Sincem≥3 it follows that the deriva-
tive ofSvanishes everywhere, and this proves Lemma 6.7.7.

6.8. THE WEYL TENSOR* 319
Examples of Einstein metrics include all constant sectional curvature
metrics by Theorem 6.4.8, the Fubini-Study metric on complex projective
space, and all metrics with vanishing Ricci tensor. Examples of the lat-
ter are Calabi–Yau metrics on complex manifolds andG2-structures on 7-
manifolds. These are again subjects that go far beyond the scope of this
book. In general, the construction of Einstein metrics and the question of
their existence is a highly nontrivial problem in differential geonetry. The
study of this problem has a long history and there are many deep theorems
and interesting open questions about this subject.
6.8 The Weyl Tensor*
This section introduces the Weyl tensor and explains some of its basic prop-
erties. The section closes with brief discussions of locally conformally flat
metrics and self-dual four-manifolds.
Definition and Basic Properties
Letmbe a positive integer and letM⊂R
n
be anm-manifold. For each
elementp∈Mdenote byRp:TpM×TpM→End(TpM) the Riemann cur-
vature tensor and by Ricp:TpM×TpM→Rthe Ricci tensor ofMatp.
Also letS:M→Rbe the scalar curvature in Definition 6.7.1.
Definition 6.8.1(Weyl tensor).Assumem≥3. TheWeyl tensorofM
at an elementp∈Mis the bilinear map
Wp:TpM×TpM→End(TpM)
defined by
⟨Wp(u, v)w, z⟩:=⟨Rp(u, v)w, z⟩
−
1
m−2
ı
Ricp(v, w)⟨u, z⟩ −Ricp(u, w)⟨v, z⟩
ȷ
−
1
m−2
ı
Ricp(u, z)⟨v, w⟩ −Ricp(v, z)⟨u, w⟩
ȷ
+
S(p)
(m−1)(m−2)
ı
⟨u, z⟩⟨v, w⟩ − ⟨v, z⟩⟨u, w⟩
ȷ
(6.8.1)
foru, v, w, z∈TpM.
Lemma 6.7.3 shows that the Weyl tensor vanishes in dimension three.
In higher dimensions the Weyl tensor may be nonzero. The next lemma
summarizes the basic algebraic properties of the Weyl tensor.

320 CHAPTER 6. GEOMETRY AND TOPOLOGY
Lemma 6.8.2.The Weyl tensorWp:TpM×TpM→End(TpM)at an el-
ementp∈Mis a skew-symmetric bilinear map with values in the space of
skew-adjoint endomorphisms ofTpMand, for allu, v, w, z∈TpMand every
orthonormal basise1, . . . , emofTpM, it satisfies
⟨Wp(u, v)w, z⟩=⟨Wp(w, z)u, v⟩, (6.8.2)
Wp(u, v)w+Wp(v, w)u+Wp(w, u)v= 0, (6.8.3)
m
X
i=1
⟨Wp(ei, u)v, ei⟩= 0. (6.8.4)
Proof.The skew-symmetry of the Weyl tensor and equation (6.8.2) follow
directly from the definition and Theorem 5.2.14. It then follows from (6.8.2)
thatWp(u, v) is a skew-adjoint endomorphism ofTpMfor allu, v∈TpM.
The verification of the Bianchi identity (6.8.3) is a straight forward compu-
tation which we leave as an exercise. To prove (6.8.4), letu, v∈TpMand
choose an orthonormal basise1, . . . , emofTpM. Then
m
X
i=1
⟨Wp(ei, u)v, ei⟩=
m
X
i=1
⟨Rp(ei, u)v, ei⟩
−
1
m−2
m
X
i=1
ı
Ricp(u, v)⟨ei, ei⟩ −Ricp(ei, v)⟨u, ei⟩
ȷ
−
1
m−2
m
X
i=1
ı
Ricp(ei, ei)⟨u, v⟩ −Ricp(u, ei)⟨ei, v⟩
ȷ
+
S(p)
(m−1)(m−2)
m
X
i=1
ı
⟨ei, ei⟩⟨u, v⟩ − ⟨u, ei⟩⟨ei, v⟩
ȷ
= Ricp(u, v)
−
m−1
m−2
Ricp(u, v)
−
S(p)
m−2
⟨u, v⟩+
1
m−2
Ricp(u, v)
+
S(p)
m−2
⟨u, v⟩
= 0.
This proves (6.8.4) and Lemma 6.8.2.

6.8. THE WEYL TENSOR* 321
The Weyl Tensor in Local Coordinates
Letϕ:U→Ω be a local coordinate chart on an open setU⊂Mwith val-
ues in an open set Ω⊂R
m
, denote its inverse byψ:=ϕ
−1
: Ω→Uand
letEi(x) :=∂iψ(x)∈T
ψ(x)Mforx∈Ω andi= 1, . . . , mbe the be the local
frame of the tangent bundle determined by this coordinate chart. Denote
the coefficients of the first fundamental form bygij:=⟨Ei, Ej⟩, of the Ricci
tensor by Ricij:= Ric(Ei, Ej), and of the Riemann curvature tensor byR
ℓ
ijk
.
LetS=
P
i,j
Ricijg
ij
=
P
m
i,j,ν
R
ν
νij
g
ij
be the scalar curvature in local coor-
dinates. Then the cofficientsW
ℓ
ijk
: Ω→Rof the Weyl tensor are defined
byW(Ei, Ej)Ek=
P
ℓ
W
ℓ
ijk
Eℓ.and they can be expresses in the form
Wijkℓ:=⟨W(Ei, Ej)Ek, Eℓ⟩=
X
ν
W
ν
ijk
gνℓ
=
X
ν
R
ν
ijk
gνℓ+
S
(m−1)(m−2)
ı
giℓgjk−gikgjℓ
ȷ
−
1
m−2
ı
Ricjkgiℓ−Ricikgjℓ+ Riciℓgjk−Ricjℓgik
ȷ
.
(6.8.5)
Conformal Invariance
Definition 6.8.3(Locally conformally flat metric).LetMbe a Rie-
mannian manifold. The metricgonMis calledlocally conformally flat,
iff for eachp∈Mthere exists a Riemannian metric onMthat is confor-
mally equivalent tog(see Definition 6.7.4) and flat in a neighborhood ofp.
By the local C-A-H Theorem 6.1.17 a Riemannianm-manifoldMis lo-
cally conformally flat if and only if eachp∈Mhas an open neighborhoodU
that isconformally diffeomorphicto an open subset Ω⊂R
m
, i.e. there
exists a coordinate chartϕ:U→Ω and a smooth functionλ:U→(0,∞)
such that|v|=λ(p)|dϕ(p)v|R
mfor allp∈Uand allv∈TpM.
A remarkable property of the Weyl tensor is that it remains unchanged
under multiplication of the Riemannian metric by a positive function and
so is an invariant of the conformal class of the metric. This is easy to see
when the function is constant. In that case the Riemann curvature tensor
and the Ricci tensor remain unchanged, the scalar curvature gets multiplied
by the inverse, and so the Weyl tensor remains unchanged. As Remark 6.7.5
shows, the situation is more complicated when instead of multiplying the
metric by a constant, we multiply it by anonconstantpositive function. The
conformal invariance of the Weyl tensor can then be proved by a somewhat
cumbersome calculation in local coordinates.

322 CHAPTER 6. GEOMETRY AND TOPOLOGY
Remark 6.8.4.This discussion shows that the Weyl tensor vanishes for
every Riemannian metric that is locally conformally flat. In fact, it turns
out that in dimensionm≥4 the Weyl tensor ofMvanishes if and only if
the Riemannian metric onMis locally conformally flat (see [43]). More-
over, atheorem of Kuiper[33, 42] asserts that every compact, connected,
simply connected Riemannianm-manifold that is locally conformally flat is
conformally diffeomorphic to them-sphere with its constant sectional curva-
ture metric. Thus every compact, connected, simply connected Riemannian
manifold of dimensionm≥4 that is not diffeomorphic toS
m
must have a
nonvanishing Weyl tensor.
Self-Dual Four-Manifolds
The lowest dimension in which the study of the Weyl tensor is interesting
ism= 4. To explain this, it is useful to consider the notion of anoriented
manifold.
Definition 6.8.5(Orientation).Anorientationof anm-manifoldMis
a collection of orientations of the tangent spacesTpM, one for eachp∈M,
that depend continuously onp, i.e. ifE1, . . . , Emare pointwise linearly inde-
pendent vector fields in a connected open neighborhoodU⊂Mofpand the
vectorsE1(p), . . . , Em(p)form a positive basis ofTpM, then for everyq∈U
the vectorsE1(q), . . . , Em(q)form a positive basis ofTqM. In the intrinsic
language an oriented manifold is one equipped with an atlas such that all the
transition maps are orientation preserving diffeomorphisms.
Definition 6.8.6(2-Form).LetMbe a smooth manifold. A2-formonM
is a collection of skew-symmetric bilinear mapsωp:TpM×TpM→R, one
for eachp∈M, which is smooth in the sense that for every pair of smooth
vector fieldsX, YonMthe assignmentp7→ωp(X(p), Y(p))defines a smooth
function onM. The space of all2-forms onMis denoted byΩ
2
(M).
In the similar vein the Weyl tensor of a Riemannianm-manifoldMcan
be thought of as 2-form with values in the endomorphism bundle End(T M).
By the symmetry properties in Lemma 6.8.2 the Weyl tensor induces a linear
mapW: Ω
2
(M)→Ω
2
(M) via the formula
(Wω)p(u, v) :=
X
1≤i<j≤m
⟨Wp(u, v)ei, ej⟩ωp(ei, ej) (6.8.6)
forω∈Ω
2
(M),p∈M, andu, v∈TpM, wheree1, . . . , emis an orthonormal
basis ofTpM. The right hand side of equation (6.8.6) is independent of the
choice of this orthonormal basis and is a 2-form by Lemma 6.8.2.

6.8. THE WEYL TENSOR* 323
Now letMbe an oriented Riemannian 4-manifold. Then a 2-formω
onMis calledself-dualiff it satisfies the condition
ωp(e0, e1) =ωp(e2, e3) (6.8.7)
for everyp∈Mand every positive orthonormal basise0, e1, e2, e3ofTpM. It
is calledanti-self-dualiff it satisfies (6.8.7) for everyp∈Mand every neg-
ative orthonormal basise0, e1, e2, e3ofTpM. Thusωis anti-self-dual if and
only if it is self-dual for the opposite orientation. Denote the space of self-
dual 2-forms by Ω
2,+
(M) and the space of anti-self-dual 2-forms by Ω
2,−
(M).
Then there is a direct sum decomposition
Ω
2
(M) = Ω
2,+
(M)⊕Ω
2,−
(M).
Lemma 6.8.7.LetMbe an oriented Riemannian4-manifold. Then the
linear operatorW: Ω
2
(M)→Ω
2
(M)in(6.8.6)preserves the subspace of
self-dual2-forms and the subspace of anti-self-dual2-forms.
Proof.Fix any orthonormal basise0, e1, e2, e3ofTpMand abbreviate
wijkℓ:=⟨Wp(ei, ej)ek, eℓ⟩
fori, j, k, ℓ∈ {0,1,2,3}. Then by Lemma 6.8.2 we have
wijkℓ=−wjikℓ=−wijℓk=wkℓij, w ijkℓ+wjkiℓ+wkijℓ= 0,(6.8.8)
w0ij0+w1ij1+w2ij2+w3ij3= 0 (6.8.9)
for alli, j, k, ℓ. It follows from (6.8.8) and (6.8.9) that
w0102=w2331, w 0103=w2312, w 0203=w3112,
w2302=w0131, w 2303=w0112, w 3103=w0212.
(6.8.10)
Namely, by (6.8.8) each of these six identities is equivalent to an equation
of the form
P
3
i=0
wijki= 0 withj̸=kand this holds by (6.8.9). Use (6.8.8)
and (6.8.9) again to obtain
w0101−w2323−w0202+w3131
=w0220+w1221+w3223−w0110−w2112−w3113
= 0
and hence, by cyclic permutation of 1,2,3,
ε:=w0101−w2323=w0202−w3131=w0303−w1212.

324 CHAPTER 6. GEOMETRY AND TOPOLOGY
Sincew0101+w0202+w0303= 0 by (6.8.9), this implies
3ε=w1221+w2332+w1331=w1221−w0330=w0303−w1212=ε.
Thusε= 0 and so
w0101=w2323, w 0202=w3131, w 0303=w1212. (6.8.11)
Now assume thatτis a nonzero self-dual 2-form and fix an elementp∈M.
Then there exists a positive orthonormal basise0, e1, e2, e3ofTpMand a
real numberλ̸= 0 such thatτ(e0, e1) =τ(e2, e3) =λandτ(ei, ej) = 0
for all other pairsi, j. Hence (Wτ)(ei, ej) =λw01ij+λw23ijfor alliandj.
Takeλ= 1 and use the equations (6.8.8), (6.8.10), and (6.8.11) to obtain
(Wτ)(e0, e1) =w0101+w2301=w0123+w2323= (Wτ)(e2, e3),
(Wτ)(e0, e2) =w0102+w2302=w0131+w2331= (Wτ)(e3, e1),
(Wτ)(e0, e3) =w0103+w2303=w0112+w2312= (Wτ)(e1, e2).
ThusWτis self-dual. The anti-self-dual case follows by reversing the orien-
tation. This proves Lemma 6.8.7.
Lemma 6.8.8.LetMbe an oriented Riemannian4-manifold and denote
byW: Ω
2
(M)→Ω
2
(M)the linear operator determined by the Weyl tensor
via(6.8.6). Then the following are equivalent.
(i)Ifτ∈Ω
2
(M)is anti-self-dual, thenWτ= 0.
(ii)The Weyl tensorWsatisfies satisfies the equation
Wp(e0, e1) =Wp(e2, e3) (6.8.12)
for everyp∈Mand every positive orthonormal basise0, e1, e2, e3ofTpM.
Proof.We prove that (i) implies (ii). Thus assume (i) and choose a positive
orthonormal basise0, e1, e2, e3ofTpM. Letτbe the 2-form defined by
τ(e0, e1) := 1, τ(e2, e3) :=−1
andτ(ei, ej) := 0 for all other pairsi, j. Thenτis anti-self-dual and
henceWτ= 0 by (i). Thus it follows from (6.8.2) and (6.8.6) that
⟨W(e0, e1)u−W(e2, e3)u, v⟩= (Wτ)(u, v) = 0
for allu, v∈TpMand this proves (ii).

6.8. THE WEYL TENSOR* 325
Conversely, suppose that (ii) holds, choose a positive orthonormal ba-
sise0, e1, e2, e3ofTpM, and letτbe an anti-self-dual 2-form. Then
λ1:=τ(e0, e1) =−τ(e2, e3),
λ2:=τ(e0, e2) =−τ(e3, e1),
λ3:=τ(e0, e3) =−τ(e1, e2)
and hence
Wτ=λ1⟨(W(e0, e1)−W(e2, e3))·,·⟩
+λ2⟨(W(e0, e2)−W(e3, e1))·,·⟩
+λ3⟨(W(e0, e3)−W(e1, e2))·,·⟩= 0
by (ii). This proves Lemma 6.8.8.
Definition 6.8.9.An oriented Riemannian4-manifold is calledself-dual
iff its Weyl tensor satisfies the equivalent conditions of Lemma 6.8.8. It is
calledanti-self-dualiff it is self-dual for the opposite orientation.
Examples of self-dual 4-manifolds are all 4-manifolds with constant sec-
tional curvature by Theorem 6.4.8, or more generally all locally conformally
flat 4-manifolds such asS
1
×S
3
. Other examples are the complex projec-
tive plane with its Fubini–Study metric (Examples 2.8.5 and 3.7.5), and
Ricci flat K¨ahler surfaces with the opposite of the complex orientation (the
K3-surface and the Enriques surface). Compact simply connected smooth
4-manifolds with signature zero that are not diffeomorphic to the 4-sphere,
such asS
2
×S
2
or the one-point blowup of the projective plane, do not
admit any self-dual metrics by Kuiper’s theorem (Remark 6.8.4), because
every self-dual metric on such a manifold is locally conformally flat. For a
survey of these basic examples see Kalafat [34].
The study of self-dual 4-manifolds was initiated by Penrose [56] and
Atiyah–Hitchin–Singer [4] and was later taken up by many authors including
Taubes [73], LeBrun [45], and Donaldson [18].
In [18] Donaldson proposes to study the moduli space of self-dual metrics
on a compact oriented smooth 4-manifold (without boundary) modulo the
action of the group of diffeomorphisms. This is a very difficult problem. The
self-duality equation is a system of nonlinear partial differential equations
and they do give rise to a“finite-dimensional moduli space”. However, this
space may be highly singular and it is noncompact unless it is empty. It
would need to be“compactified”in a suitable way, one would need to find
a way to understand the singularities, and one would have to assign to it
some kind of“virtual fundamental class”to obtain numerical invariants by
pairing this class with suitable cohomology classes in the ambient space.

326 CHAPTER 6. GEOMETRY AND TOPOLOGY

Chapter 7
Topics in Geometry
This chapter explores various topics in differential geometry that are cen-
tral to the subject and accessible with the material covered in this book,
but go beyond the scope of a one semester lecture course. The first sec-
tion builds on the material in Chapter 4 about geodesics and can be read
directly after that chapter. It introduces conjugate points on geodesics and
proves the Morse Index Theorem. This result is then used to show that every
geodesic without conjugate points minimizes the lengthlocally (and strictly)
in the space of all curves joining the same endpoints and, conversely, that
locally minimizing geodesics have no interior conjugate points (§7.1). This
result in turn plays an essential role in the proof of the Continuity Theorem
for the injectivity radius (§7.2). The next section examines the group of
isometries of a connected Riemannian manifold and contains a proof of the
Myers–Steenrod Theorem, which asserts that the isometry group admits the
natural structure of a finite–dimensional Lie group (§7.3). The proof given
here has several parallels to the proof of the Closed Subgroup Theorem.
This section is based on the study of isometries and the Riemann curva-
ture tensor in Chapter 5 and can be read directly after that chapter. The
present chapter then deals with the specific example of the isometry group
of a compact connected Lie group equipped with a bi-invariant Riemannian
metric (§7.4). The last two sections are devoted to Donaldson’s differential
geometric approach to Lie algebra theory [17]. They build on the material in
Chapter 6 and include Donaldson’s existence theorems for critical points of
convex functions on Hadamard manifolds (§7.5) and his existence proof for
symmetric inner products on simple Lie algebras (§7.6). Corollaries include
the uniqueness of maximal compact subgroups, and Cartan’s theorem about
the compact real form of a semisimple complex Lie algebra.
327

328 CHAPTER 7. TOPICS IN GEOMETRY
7.1 Conjugate Points and the Morse Index*
This section introduces conjugate points on geodesics and contains a proof
of the Morse Index Theorem. As an application we prove that every geodesic
without conjugate points minimizes the length and the energylocally (and
strictly)among all nearby curves joining the same endpoints. Conversely,
locally minimizing geodesics have no interior conjugate points. The results
of this section will be used in§7.2 to prove the Continuity Theorem for the
injectivity radius. Assume throughout thatM⊂R
n
is a smoothm-manifold
and recall the notation Ωp,qfor the space of all smooth pathsγ: [0,1]→M
with the endpointsγ(0) =pandγ(1) =q(§4.1.2).
Conjugate Points
We have seen in Theorem 4.4.4 and Corollary 4.4.6 that geodesics minimize
the length on short time intervals. A natural question arising from this
observation is how long the time interval can be chosen on which a geodesic
minimizes the length at least locally in some neighborhood in the space of
paths. An answer to this question is closely related to the Hessian of the
energy functionalE: Ωp,q→Rin (4.1.2). It was established in Lemma 6.6.3
that the Hessian of the energy functionalEat a geodesicγ: [0,1]→Mis
the linear operatorHγ: Vect0(γ)→Vect(γ) defined by
HγX:=−∇∇X−R(X,˙γ) ˙γ. (7.1.1)
The domain of this operator is the space Vect0(γ) of all vector fieldsX
alongγthat vanish at the endoints, i.e.X(0) = 0 andX(1) = 0. Recall
that a Jacobi field along a geodesic is a solution of the equation
∇∇X+R(X,˙γ) ˙γ= 0, (7.1.2)
and so the kernel ofHγis the space of Jacobi fields alongγthat vanish at
the endpoints (Lemma 6.1.18).
Definition 7.1.1.Letγ: [a, b]→Mbe a geodesic. Aconjugate point
is a real numberτin the intervala < τ≤bsuch that there exists a non-
vanishing Jacobi fieldXalong the restrictionγ|
[a,τ]that satisfiesX(a) = 0
andX(τ) = 0. The dimension of the space of solutions of this equation is
called themultiplicityof the conjugate pointτand will be denoted by
mγ(τ) := dim
æ
X∈Vect(γ|
[a,τ])

∇∇X+R(X,˙γ) ˙γ= 0,
X(a) = 0, X(τ) = 0
œ
.(7.1.3)
Thusmγ(τ) = 0wheneverτis not a conjugate point ofγ.

7.1. CONJUGATE POINTS AND THE MORSE INDEX* 329
In§4.4 we have addressed the question when a geodesicγ: [0,1]→M
with the endpointsγ(0) =pandγ(1) =qminimizes the lengths of curves
with the same endpointsglobally, i.e. when it satisfiesL(γ) =d(p, q). A
weaker variant of this question is whether it minimizes the lengthlocally,
i.e. only among nearby curves in Ωp,q. Here the wordΩearby”can have
several meanings, depending on which topology on the space Ωp,qis used.
The relevant topologies in the present setting are theC
1
topology and the
C
∞
topology. Since our manifoldMis embedded inR
n
these topologies are
induced by the distance functions defined by
d
C
1(γ, γ
′
) := sup
0≤t≤1
|γ(t)−γ
′
(t)|+ sup
0≤t≤1
|˙γ(t)−˙γ
′
(t)|,
dC
∞(γ, γ
′
) :=
∞
X
k=0
2
−k
sup
0≤t≤1|
d
k
dt
k(γ(t)−γ
′
(t))|
1 + sup
0≤t≤1|
d
k
dt
k(γ(t)−γ
′
(t))|
(7.1.4)
forγ, γ
′
∈Ωp,q. Note that, ifγ∈Ωp,qsatisfies mint|˙γ(t)|>0, then so does
every curveγ
′
in a sufficiently smallC
1
-neighborhood ofγin Ωp,q.
Theorem 7.1.2.Letγ: [0,1]→Mbe a nonconstant geodesic with the end-
pointsγ(0) =pandγ(1) =q. Then the following holds.
(i)If there exists aC
∞
-open neigborhoodU ⊂Ωp,qofγsuch that every
curveγ
′
∈ UsatisfiesL(γ
′
)≥L(γ), thenγhas no conjugate pointsτin the
open interval0< τ <1.
(ii)Ifγhas no conjugate pointsτin the interval0< τ≤1, then there ex-
ists aC
1
-open neighborhoodU ⊂Ωp,qofγsuch that every curveγ
′
∈ U
satisfiesL(γ
′
)≥L(γ)with equality if and only if there exists a diffeomor-
phismρ: [0.1]→[0,1]such thatρ(0) = 0,ρ(1) = 1, andγ
′
=γ◦ρ.
The proof will be based on the Morse Index Theorem explained below.
Example 7.1.3.The archetypal example of a conjugate point is the end-
point of a geodesicγ: [0,1]→S
2
on the unit sphereS
2
⊂R
3
of lengthπ.
This is the endpoint of an arc around half of a great circle, and at this point
the geodesic seizes to be locally unique as there is a continuous family of
geodesics joining north and south pole (the meridians).
Example 7.1.4.The absence of conjugate points does not signify that a
geodesicγ∈Ωp,qminimizes the lengthglobally. The sinplest example is
a geodesic around the unit circleM=S
1
of lengthL(γ)> π. Similar
examples exist on the torusM=T
m
, on the cylinderM=R×S
1
, and
also on simply connected manifolds such as the sphereS
2
with a suitably
chosen Riemannian metric (so that an open subsetU⊂S
2
is isometric to a
cylinder (0,1)×S
1
, for example).

330 CHAPTER 7. TOPICS IN GEOMETRY
The Morse Index Theorem
Definition 7.1.5(Morse index).Letγ: [0,1]→Mbe a geodesic. The
Morse indexofγis defined as the maximal dimension of a linear sub-
spaceX ⊂Vect0(γ), such that
Z
1
0
ı
|∇X|
2
− ⟨R(X,˙γ) ˙γ, X⟩
ȷ
dt <0 (7.1.5)
for every nonzero elementX∈ X. It will be denoted by
µ(γ) := max
æ
dim(X)

Xis a linear subspace ofVect0(γ)
and(7.1.5)holds for allX∈ X \ {0}
œ
.(7.1.6)
This is also the number of negative eigenvalues, counted with multiplicities,
of the operatorHγin(7.1.1).
Theorem 7.1.6(Morse Index Theorem). Letγ: [0,1]→Mbe a
geodesic. Thenγhas only finitely many conjugate pointsτ, each with multi-
plicity1≤mγ(τ)≤m, and the Morse index ofγis the number of conjugate
points in the open interval0< τ <1, counted with multiplicities, i.e.
µ(γ) =
X
0<τ <1
mγ(τ). (7.1.7)
Exercise 7.1.7.Prove that geodesics on manifolds with nonpositive sec-
tional curvature have no conjugate points. So their Morse indices are zero.
Exercise 7.1.8.Find geodesics on the unit sphereM=S
2
⊂R
3
with
arbitrarily large Morse index.
The proof of Theorem 7.1.6 requires some background in functional anal-
ysis which goes beyond the scope of this book and for which the interested
reader is referred to [12]. Remark 7.1.9 below lists the main properties of
the operatorHγthat are used in the proofs of Theorems 7.1.2 and 7.1.6.
It will be convenient in some places to use theL
2
inner product
⟨X, Y⟩
L
2:=
Z
1
0
⟨X(t), Y(t)⟩dt
and the associated norm
∥X∥
L
2:=
s
Z
1
0
|X(t)|
2
dt
for vector fieldsX, Yalongγ.

7.1. CONJUGATE POINTS AND THE MORSE INDEX* 331
Remark 7.1.9(Properties ofHγ).We begin with the properties that
can be easily verified directly, without an appeal to Hilbert space theory.
(a)Each eigenvalue ofHγis a real number.
This assertion is a consequence of the fact that the operatorHγis symmetric,
i.e.⟨HγX, Y⟩
L
2=⟨X,HγY⟩
L
2for allX, Y∈Vect0(γ). (Exercise:Verify
assertion (a) by complexifying the space of vector fields alongγ.)
(b)Each eigenvalue ofHγhas multiplicity at mostm.
The eigenspace ofHγfor an eigenvalueλconsists of solutions of the equa-
tion−∇∇X−R(X,˙γ) =λXwithX(0) = 0. Every solution is uniquely
determined by∇X(0) and is an eigenfunction forλif and only ifX(1) = 0.
(c)EveryX∈Vect0(γ)satisfies thePoincar´e inequality
Z
1
0
|X(t)|
2
≤
1
π
2
Z
1
0
|∇X|
2
dt. (7.1.8)
To see this, choose a parallel orthonormal frameE1(t), . . . , Em(t)∈T
γ(t)M
of the tangent bundle alongγand writeX(t) =
P
m
i=1
ξi(t)Ei(t). Then the
inequality (7.1.8) takes the formπ
2
R
1
0
|ξ|
2
dt≤
R
1
0
|
˙
ξ|
2
dtand this can be
proved by writingξas a Fourier seriesξ(t) =
P
∞
k=1
sin(πkt)ξk. (Exercise:
Prove the estimate (7.1.8). Verify that it is sharp.)
(d)Letcγ:= sup
tsup
|v|=1⟨R(v,˙γ(t)) ˙γ(t), v⟩. Then, for allX∈Vect0(γ),
Z
1
0
ı
|∇X|
2
− ⟨R(X,˙γ) ˙γ, X⟩
ȷ
dt≥(π
2
−cγ)
Z
1
0
|X|
2
dt
This follows directly from the Poincar´e inequality in (c).
(e)Letλbe an eigenvalue ofHγ. Thenλ≥π
2
−cγ.
There exists a nonzero elementX∈Vect0(γ) such thatHγX=λX. By (d)
this impliesλ∥X∥
2
L
2=⟨X,HγX⟩
L
2≥(π
2
−cγ)∥X∥
2
L
2and soλ≥π
2
−cγ.
(f)The setσ(Hγ)of eigenvalues ofHγis a discrete subset ofR.
Letλ∈σ(Hγ) and let Πλbe the orthogonal projection onto the eigenspace
ofλ. Then the operatorHγ−λid + Πλ: Vect0(γ)→Vect(γ) is bijective,
and hence so is the operatorHγ−λ
′
id+Πλforλ
′
sufficently close toλ. Such
a numberλ
′
cannot be an eigenvalue ofHγ. This argument uses the fact
thatHγis a Fredholm operator between appropriate Sobolev completions.
(g)The smallest eigenvalue ofHγis the supremum of all real numbersa
that satisfy the inequality
Z
1
0
ı
|∇X|
2
− ⟨R(X,˙γ) ˙γ, X⟩
ȷ
dt≥a
Z
1
0
|X(t)|
2
dt (7.1.9)
for allX∈Vect0(γ).

332 CHAPTER 7. TOPICS IN GEOMETRY
To prove this, definea∈Rto be the infimum of the integrals on the left
in (7.1.9) over allX∈Vect0(γ) with∥X∥
L
2= 1. This infimum is finite
by (d). Now choose a minimizing sequenceXi∈Vect0(γ) with∥Xi∥
L
2= 1.
This sequence satisfies a uniform upper bound on∥∇Xi∥
L
2. Hence, by the
theorems of Banach–Alaoglu and Arzel`a-Ascoli, there exists a subsequence
that converges weakly in the Sobolev spaceW
1,2
and strongly in the supre-
mum norm to a weak solution of the equation−∇∇X−R(X,˙γ) ˙γ=aX
with zero boundary condition. Now one can verify by a bootstrapping ar-
gument that every weak solution is smooth. Hence (7.1.9) holds,ais an
eigenvalue ofHγ, and every eigenvalueλofHγsatisfiesλ≥a.
(h)The Morse index ofγis finite.
The operatorHγhas only finitely many negative eigenvalues by (e) and (f),
and the direct sumXγof their eigenspaces is finite-dimensional by (b). More-
over, every nonzero elementX∈ Xγsatisfies the inequality⟨X,HγX⟩
L
2<0.
One can then repeat the argument sketched in (g) for theL
2
orthogonal
complement ofXγto show that⟨X,HγX⟩
L
2≥0 for allX∈ X
⊥
γ. Hence the
dimension ofXγis the Morse index ofγ(Definition 7.1.5).
Proof of Theorem 7.1.6.Choose a parallel orthonormal frame
E1(t), . . . , Em(t)∈T
γ(t)M
and, for 0≤t≤1, define the symmetric matrixS(t) =S(t)
T
∈R
m×m
by
S(t) := (Sij(t))
m
i,j=1, S ij(t) :=⟨R(Ei(t),˙γ(t)) ˙γ(t), Ej(t)⟩,
LetI:= [0,1] and define the operatorA:C
∞
0
(I,R
m
)→C
∞
(I.R
m
) with
the domainC
∞
0
(I,R
m
) :={ξ∈C
∞
(I,R
m
)|ξ(0) =ξ(1) = 0},by
Aξ:=−
¨
ξ−Sξ.
This operator is isomorphic to the operator (7.1.1) via the isomorphism that
sendsξ∈C
∞
0
(I,R
m
) toX:=
P
i
ξiEi∈Vect0(γ). Now define a family of
operatorsAτ:C
∞
0
(I,R
m
)→C
∞
(I.R
m
) by
(Aτξ)(t) :=−
1
τ
2
¨
ξ(t)−S(τt)ξ(t)
for 0< τ≤1. ThenAτis isomorphic to the operator (7.1.1) on the in-
terval [0, τ] via the isomorphism that sendsξ∈C
∞
0
(I,R
m
) to the vector
fieldX(t) :=
P
i
ξ(τ
−1
t)Ei(t)∈T
γ(t)M, 0≤t≤τ, along the curveγ|
[0,τ].
Thusτis a conjugate point ofγif and only ifAτhas a nontrivial kernel,
and the dimension of the kernel is the multiplicitymγ(τ).

7.1. CONJUGATE POINTS AND THE MORSE INDEX* 333
By Remark 7.1.9Aτhas only positive eigenvalues for small positiveτ.
For 0< τ≤1 define the operator
˙
Aτ:C
∞
0
(I,R
m
)→C
∞
(I,R
m
) by
(
˙
Aτξ)(t) :=
d
dτ
(Aτξ)(t) =
2
τ
3
¨
ξ(t)−t
˙
S(τt)ξ(t).
Ifτis a conjugate point, we claim that every elementξ∈ker(Aτ) satisfies
Γτ(ξ) :=
Z
1
0
⟨ξ,
˙
Aτξ⟩dt=−
1
τ
3
|
˙
ξ(1)|
2
. (7.1.10)
To see this, note that
¨
ξ(t) +τ
2
S(τt)ξ(t) = 0 and
d
dt
S(τt) =τ
˙
S(τt). Hence
Γτ(ξ) =
Z
1
0
ı
2
τ
3
⟨ξ(t),
¨
ξ(t)⟩ −
t
τ
⟨ξ(t), τ
˙
S(τt)ξ(t)⟩
ȷ
dt
=
Z
1
0
ı
2
τ
3
⟨ξ(t),
¨
ξ(t)⟩+
1
τ
⟨ξ(t), S(τt)ξ(t)⟩+
2t
τ
⟨
˙
ξ(t), S(τt)ξ(t)⟩
ȷ
dt
=
Z
1
0
ı
1
τ
3
⟨ξ(t),
¨
ξ(t)⟩ −
2t
τ
3
⟨
˙
ξ(t),
¨
ξ(t)⟩
ȷ
dt
=−
1
τ
3
Z
1
0
ı
|
˙
ξ(t)|
2
+ 2t⟨
˙
ξ(t),
¨
ξ(t)⟩
ȷ
dt
=−
1
τ
3
Z
1
0
d
dt
ı
t|
˙
ξ(t)|
2
ȷ
dt
=−
1
τ
3
|
˙
ξ(1)|
2
.
This proves (7.1.10). Note also that Γτ(ξ)<0 unlessξ(t) = 0 for allt.
With this understood, we appeal to the Kato Selection Theorem [35,
Theorem II.5.4 and Theorem II.6.8]. It asserts in the case at hand that,
near each pointτ0withk:= dim(ker(Aτ0
))>0, there existkcontinuously
differentiable functionsλi: (τ0−ε, τ0+ε)→Rsuch thatλi(τ0) = 0, the
numbersλ1(τ), . . . , λk(τ) are the eigenvalues ofAτnear zero, with mul-
tiplicities accounted for by repetitions, and the derivatives
˙
λi(τ0) are the
eigenvalues of the crossing form Γτ0
: kerAτ0
→Rin (7.1.10), again with
multiplicities accounted for by repetitions. The derivatives are all negative
by (7.1.10). Hence there exists aδ >0 such thatλi(τ)>0 forτ0−δ < τ < τ0
andλi(τ)<0 forτ0< τ < τ0+δ. Hence conjugate points are isolated, and
forτ0< τ < τ0+δthe operatorAτhas preciselykmore negative eigenvalues
than forτ0−δ < τ < τ0. Hence the number of the negative eigenvalues ofA1
with multiplicities is
P
0<τ <1
dim(ker(Aτ)). This proves Theorem 7.1.6.
For more detailed explanations and other closely related applications of
the Kato Selection Theorem the reader is referred to [61, 76].

334 CHAPTER 7. TOPICS IN GEOMETRY
Locally Minimal Geodesics
Proof of Theorem 7.1.2.We prove part (i). LetX∈Vect0(M) and define
γs(t) := exp
γ(t)(sX(t))
for 0≤t≤1 and−δ < s < δ. Hereδ >0 is chosen so small thatsX(t)
is contained in the domainV
γ(t)⊂T
γ(t)Mof the exponential map for allt
and allswith|s|< δ. Then lims→0dC
∞(γ, γs) = 0 and soγs∈ Ufors
sufficiently small. Hence the function (−δ, δ)→R:s7→L(γs) has a local
minimum ats= 0, and sinceE(γ) =L(γ)
2
/2 andL(γs)
2
/2≤E(γs), so
does the functions7→E(γs). By Lemma 6.6.3 this implies
0≤
d
2
ds
2

s=0
E(γs) =
Z
1
0
ı
|∇X|
2
− ⟨R(X,˙γ) ˙γ, X⟩
ȷ
dt.
This inequality shows that the operatorHγhas no negative eigenvalues.
Hence, by Theorem 7.1.6 the geodesicγhas no conjugate pointsτin the
interval 0< τ <1. This proves (i).
We prove part (ii) in six steps. Letγ∈Ωp,qbe a nonconstant geodesic
without conjugate points.
Step 1.There exist constantsδ1>0andε >0such that
Z
1
0
ı
|∇X|
2
− ⟨R(X,˙γ
′
) ˙γ
′
, X⟩
ȷ
dt≥ε
Z
1
0
ı
|∇X(t)|
2
+|X(t)|
2
ȷ
dt(7.1.11)
for every curveγ
′
∈Ωp,qwithd
C
1(γ, γ
′
)< δ1and everyX∈Vect0(γ
′
).
By Theorem 7.1.6 the HessianHγhas no negative eigenvalues and, since 1
is not a conjugate point ofγ, also zero is not an eigenvalue ofHγ. Hence
the smallest eigenvalue ofHγis positive and so, by part (e) of Remark 7.1.9,
the estimate (7.1.9) holds for allX∈Vect0(γ) with a positive constanta. If
the constantb >0 is chosen such that⟨R(v,˙γ(t)) ˙γ(t), v⟩ ≤b|v|
2
for alltand
allv∈T
γ(t)M, we obtain the estimate (7.1.11) forγ
′
=γandX∈Vect0(γ)
withε:=a/(a+b+ 1). (Exercise:Verify this.) In a local coordinat chart
onMthe integrand on the left in (7.1.11) has the form
X
k,ℓ
ı
˙
ξ
k
+
X
i,j
Γ
k
ijξ
i
˙c
j
ȷ
gkℓ
ı
˙
ξ
ℓ
+
X
µ,ν
Γ
ℓ
µνξ
µ
˙c
ν
ȷ
−
X
i,j,k,ℓ
Rijkℓξ
i
˙c
j
˙c
k
ξ
ℓ
,
wherec(t) :=ϕ(γ
′
(t)) andξ(t) =dϕ(γ
′
(t))X(t) (and the coefficientsgij, Γ
k
ij
,
andRijkℓare functions ofc(t)). This expression depends continuously on the
curvecand its derivative ˙c. Hence there exists a constantδ1>0 such that
every curveγ
′
∈Ωp,qwithd
C
1(γ, γ
′
)< δ1and everyX∈Vect0(γ
′
) satisfies
the estimate (7.1.11) withε=a/2(a+b+ 1). This proves Step 1.

7.1. CONJUGATE POINTS AND THE MORSE INDEX* 335
Step 2.Letδ1>0be as in Step 1. Then there exists a constantδ2>0with
the following significance. IfY∈Vect0(γ)satisfies
sup
0≤t≤1
ı
|Y(t)|+|∇Y(t)|
ȷ
< δ2, (7.1.12)
then the curveγ
′
:= exp
γ(Y)∈Ωp,qsatisfiesd
C
1(γ, γ
′
)< δ1.
For 0≤t≤1 define the mapft:V
γ(t)×T
γ(t)M→T Mby
ft(v,bv) :=
ı
exp(γ(t), v), dexp(γ(t), v)
Γ
˙γ(t),bv+h
γ(t)( ˙γ(t), v)
∆
ȷ
(7.1.13)
forv∈V
γ(t)andbv∈T
γ(t)M. By Lemma 4.3.6 the map (t, v,bv)7→ft(v,bv)
fromU:=
Φ
(t, v,bv)|0≤t≤1, v∈V
γ(t),bv∈T
γ(t)M

toT Mis smooth and
satisfiesft(0,0) = (γ(t),˙γ(t)). Thus there exist constantsr1>0 andC1≥1
such that alltand allv,bv∈T
γ(t)Mwith|v|+|bv| ≤r1satisfyv∈V
γ(t)and
|p−γ(t)|+|bp−˙γ(t)| ≤C1
ı
|v|+|bv|
ȷ
,(p,bp) :=ft(v,bv).(7.1.14)
ChooseY∈Vect0(γ) such that|Y(t)|+|∇Y(t)| ≤r1for 0≤t≤1 and
defineγ
′
(t) := exp
γ(t)(Y(t)). Then it follows from the Gauß–Weingarten
formula in Lemma 3.2.3 that (γ
′
(t),˙γ
′
(t)) =ft(Y(t),∇Y(t)) for allt. Hence
the estimate (7.1.14) shows that Step 2 holds withδ2:= min{r1, δ1/C1}.
Step 3Letδ2>0be as in Step 2. Then there exists a constantδ3>0with
the following significance. Ifγ
′
∈Ωp,qsatisfies the inequalityd
C
1(γ, γ
′
)< δ3,
then there exists a unique vector fieldY∈Vect0(γ)satisfying(7.1.12)and
|Y(t)|<inj(γ(t);M), γ
′
(t) = exp
γ(t)(Y(t)) (7.1.15)
for0≤t≤1.
Defineρ:= inf0≤t≤1inj(γ(t);M)>0 and letftbe the map in (7.1.13).
Its derivative at the origin is given bydft(0,0)(η,bη) = (η,bη+h
γ(t)( ˙γ(t), η))
and so is invertible. Hence, by the implicit function theorem, there exist
constantsr2>0 andC2≥1 such that, for 0≤t≤1, the set
Qt:=
Φ
(p,bp)∈T M

|p−γ(t)|+|bp−˙γ(t)| ≤r2

is compact,ftrestricts to a diffeomorphism from a compact neighborhood
of the origin inBρ(γ(t))×T
γ(t)MtoQt, and every pair (p,bp)∈Qtsatisfies
|v|+|bv| ≤C2
ı
|p−γ(t)|+|bp−˙γ(t)|
ȷ
,(v,bv) :=f
−1
t
(p,bp).(7.1.16)
Chooseγ
′
∈Ωp,qsuch thatd
C
1(γ, γ
′
)≤r2. Then (γ
′
(t),˙γ
′
(t))∈Qtfor allt
and hence there exists a unique vector fieldY∈Vect0(γ) satisfying (7.1.15).
By (7.1.13) it also satisfies (Y(t),∇Y(t)) =f
−1
t
(γ
′
(t),˙γ
′
(t)) for allt. Hence
the estimate (7.1.16) shows that Step 3 holds withδ3:= min{r2, δ2/C2}.

336 CHAPTER 7. TOPICS IN GEOMETRY
Step 4.Letδ3be as in Step 3. Then there exists a constantδ4>0with
the following significance. Ifγ
′
∈Ωp,qsatisfiesd
C
1(γ, γ
′
)< δ4, then the
mapρ: [0,1]→[0,1]defined byρ(t) :=L(γ
′
)
−1
R
t
0
|˙γ
′
(s)|dsfor0≤t≤1is
a diffeomorphism andd
C
1(γ, γ
′
◦ρ
−1
)< δ3.
Definec:=L(γ) andC:= sup
t|¨γ(t)|. Thenc >0 becauseγis nonconstant.
We claim that Step 4 holds for every constantδ4>0 that satisfies
δ4<
c
2
,
`
32 +
12C
c
´
δ4< δ3. (7.1.17)
To see this, fix a constantδ4>0 that satisfies (7.1.17) and letγ
′
∈Ωp,qsuch
thatd
C
1(γ, γ
′
)< δ4. Since the geodesicγis parametrized by the arclength,
it satisfies|˙γ(t)|=L(γ) =c. Hence||˙γ
′
(t)| −c|< δ4for allt. Sinceδ4< c
by (7.1.17), this implies thatρis a diffeomorphism. It also implies the length
inequality|L(γ
′
)−c|< δ4and hence
1
3
<
c−δ4
c+δ4
<˙ρ(t) =
|˙γ
′
(t)|
L(γ
′
)
<
c+δ4
c−δ4
<3,|˙ρ(t)−1|<
2δ4
c−δ4
<
4δ4
c
.
Defineβ:=ρ
−1
andγ
′′
:=γ
′
◦β. Then
˙
β(ρ(t)) = 1/˙ρ(t) and hence
|
˙
β(ρ(t))−1|=
|1−˙ρ(t)|
˙ρ(t)
<
12δ4
c
.
This implies
|
˙
β(t)−1|<
12δ4
c
,|β(t)−t|<
12δ4
c
,
˙
β(t)<7.
Hence
|γ(t)−γ
′′
(t)| ≤ |γ(t)−γ(β(t))|+|γ(β(t))−γ
′
(β(t))|
≤c|t−β(t)|+d
C
1(γ, γ
′
)
<13δ4
and
|˙γ(t)−˙γ
′′
(t)| ≤ |˙γ(t)−˙γ(β(t))|+|1−
˙
β(t)||˙γ(β(t))|
+
˙
β(t)|˙γ(β(t))−˙γ
′
(β(t))|
≤C|t−β(t)|+c|1−
˙
β(t)|+
˙
β(t)d
C
1(γ, γ
′
)
<
`
12C
c
+ 12 + 7
´
δ4.
These inequalities implyd
C
1(γ, γ
′′
)<(12C/c+ 32)δ4< δ3by (7.1.17) and
this proves Step 4.

7.1. CONJUGATE POINTS AND THE MORSE INDEX* 337
Step 5.Letδ3>0be as in Step 3 and letγ
′
∈Ωp,qsuch thatd
C
1(γ, γ
′
)< δ3.
ThenE(γ
′
)≥E(γ)with equality if and only ifγ
′
=γ.
Assumeγ
′
̸=γ. By Step 3 there exists a nonzero vector fieldY∈Vect0(γ)
satisfying (7.1.12), (7.1.15), andγ
′
= exp
γ(Y). For 0≤s, t≤1 define
γs(t) := exp
γ(t)(sY(t)).
Thenγ0=γ,γ1=γ
′
, and Step 2 asserts that
d
C
1(γ, γs)< δ1 for 0≤s≤1.
Hence it follows from the argument in the proof of Lemma 6.6.3 that
d
2
ds
2
E(γs) =
Z
1
0
ı
|∇t∂sγs|
2
− ⟨R(∂sγs, ∂tγs)∂tγs, ∂sγs⟩
ȷ
dt >0
for alls. Here the equality uses the identity∇s∂sγs= 0 and the inequality
follows from Step 1 and the fact that∂sγs=dexp
γ(sY)Yis a nonzero vector
field alongγs. Thus the curve [0,1]→R:s7→E(γs) is strictly convex. Since
its derivative vanishes ats= 0, it follows thatE(γs)> E(γ) for 0< s≤1.
Sinceγ1=γ
′
, this proves Step 5.
Step 6.Letδ4be as in Step 4. Then theC
1
-open set
U:=
Φ
γ
′
∈Ωp,q

d
C
1(γ, γ
′
)< δ4

satisfies the requirements of part (ii) in Theorem 7.1.2.
Letγ
′
∈ Uand define the diffeomorphismρ: [0,1]→[0,1] as in Step 4.
Then, by Step 4 we haved
C
1(γ, γ
′
◦ρ
−1
)< δ3and hence, by Step 5 this
impliesE(γ
′
◦ρ
−1
)≥E(γ) with equality if and only ifγ
′
◦ρ
−1
=γ. Sinceγ
andγ
′
◦ρ
−1
are parametrized by the arclength, we find that
L(γ
′
) =L(γ
′
◦ρ
−1
) =
p
2E(γ
′
◦ρ
−1
)≥
p
2E(γ) =L(γ),
and that equality holds if and only ifγ
′
◦ρ
−1
=γ. Thus every curveγ
′
∈ U
that arises fromγby reparametrization has the same length asγand every
other curve inUis strictly longer. This proves Step 6 and Theorem 7.1.2.
We remark that there is a precise analogy between Theorem 7.1.2 about
geodesics (extrema of the energy functionalE) and extrema of a function
f:R
n
→R
on a finite-dimensional vector space. If the functionfhas a local minimum
at a pointx0∈R
n
, then the Hessian offatx0has no negative eigenvalues
and, conversely, ifx0is a critical point offand all the eigenvalues of the
Hessian atx0are positive, thenx0is a strict local minimum off.

338 CHAPTER 7. TOPICS IN GEOMETRY
7.2 The Injectivity Radius*
Assume throughout thatM⊂R
n
is a smoothm-manifold. In this section
we prove that the function which assigns to each pointp∈Mits injectivity
radius inj(p;M)∈(0,∞] is continuous. Recall the concept of a local diffeo-
morphism as a smooth map between manifolds of the same dimension whose
derivative at each point is a vector space isomorphism (Definition 2.2.20).
Recall also the notationVp⊂TpMfor the domain of the exponential map
atpand the notationBr(p) ={v∈TpM| |v|< r}forp∈Mandr >0.
Theorem 7.2.1.The functionM→(0,∞] :p7→inj(p;M)is continuous.
The proof of Theorem 7.2.1 is based on two lemmas.
Lemma 7.2.2.The setUr:={p∈M|r <inj(p;M)}is open for each real
numberr >0.
Proof.Letr >0 andp0∈Ur. We prove in three steps that there exists
aδ >0 such thatr+δ≤inj(p;M) for everyp∈Mwithd(p, p0)< δ.
Step 1.There exists aδ >0such thatBr+δ(p)⊂Vpfor everyp∈M
withd(p, p0)< δ.
Suppose, by contradiction, that such a constantδdoes not exist. Then
there exist sequencespi∈Mandvi∈Tpi
M\Visuch thatd(pi, p0)<1/i
and|vi|< r+ 1/i. Passing to a subsequence, if necessary, we may assume
thatviconverges to a vectorv0∈Tp0
M. Then|v0| ≤rand so (p0, v0)∈V.
Since (pi, vi)∈T M\Vconverges to (p0, v0), this contradicts the fact thatV
is open. This proves Step 1.
Step 2.There exists aδ >0such that the mapexp
p:Br+δ(p)→Mis a
local diffeomorphism for everyp∈Mwithd(p, p0)< δ.
Suppose, by contradiction, that such a constantδdoes not exist. Then
there exist sequencespi∈Mandvi∈Tpi
Msuch thatd(pi, p0)<1/iand
|vi|< r+ 1/i,and the derivativedexp
pi
(vi) :Tpi
M→T
exp
p
i
(vi)Mis not
injective. Passing to a subsequence, if necessary, we may assume thatvi
converges to a vectorv0∈Tp0
M. Then, by smoothness of the exponential
map, the derivative
dexp
p0
(v0) :Tp0
M→T
exp
p
0
(v0)M
is not injective. Since|v0| ≤r, this contradicts the fact thatr <inj(p0;M).
This proves Step 2.

7.2. THE INJECTIVITY RADIUS* 339
Step 3.There exists aδ >0such that the mapexp
p:Br+δ(p)→Mis
injective for everyp∈Mwithd(p, p0)< δ.
Suppose, by contradiction, that such a constantδdoes not exist. Then there
exist sequencespi∈Mandui, vi∈Tpi
Msuch that
d(pi, p0)<1/i,|ui|< r+ 1/i,|vi|< r+ 1/i
and
ui̸=vi,exp
pi
(ui) = exp
pi
(vi) =:qi.
Passing to a subsequence, if necessary, we may assume that the limits
u0= lim
i→∞
ui∈Tp0
M, v 0:= lim
i→∞
vi∈Tp0
M
exist. These limits satisfy
|u0| ≤r,|v0| ≤r,exp
p0
(u0) = exp
p0
(v0).
Sincer <inj(p0;M), this impliesu0=v0and hence
lim
i→∞
ui= lim
i→∞
vi=v0.
Now define
wi:=
vi−ui
τi
∈Tpi
M, τ i:=|vi−ui|>0.
Passing to a further subsequence, we may assume that the limit
w0:= lim
i→∞
wi∈Tp0
M
exists. Then|w0|= 1 and hence
0 = lim
i→∞
exp
pi
(vi)−exp
pi
(ui)
τi
= lim
i→∞
Z
1
0
dexp
pi
(ui+t(vi−ui))
vi−ui
τi
dt
=dexp
p0
(v0)w0
̸= 0.
This contradiction proves Step 3. Now choose a constantδ >0 that satisfies
the requirements of Step 1, Step 2, and Step 3. Then, for everyp∈M
withd(p, p0)< δwe haveBr+δ(p)⊂Vpand the map exp
p:Br+δ(p)→M
is a diffeomorphism onto its image. Hence inj(p;M)≥r+δ > rfor every
elementp∈Mwithd(p, p0)< δ, and this proves Lemma 7.2.2.

340 CHAPTER 7. TOPICS IN GEOMETRY
Lemma 7.2.3.The setAr:={p∈M|r≤inj(p;M)}is closed for each real
numberr >0and hence also forr=∞.
Proof.Letpi∈Arbe a sequence that converges to an elementp∈M. We
prove in five steps thatr≤inj(p;M).
Step 1.Br(p)⊂Vp.
Choose a tangent vectorv∈TpMwith|v|< r, choose a constantε >0 such
that|v|+ε < r,and choose an integerisuch thatd(p, pi)≤ε.Define
K:= exp
pi
(
¯
B
|v|+ε(pi)) =
Φ
q∈M

d(q, pi)≤ |v|+ε

.
Here the second equality follows from Theorem 4.4.4 and the fact that
|v|+ε < r≤inj(pi;M). By definition,Kis the image of a compact set
under a continuous map, and so is a compact subset ofM. Hence
e
K:=
Φ
(q, w)∈T M

d(q, pi)≤ |v|+ε,|w|=|v|

is a compact subset ofT M. We claim thatv∈Vp. Suppose, by contradic-
tion, that this is not the case. ThenIp,v∩[0,∞) = [0, T) with 0< T≤1.
Denote byγ: [0, T)→Mthe geodesicγ(t) := exp
p(tv). Then
d(γ(t), pi)≤d(exp
p(tv), p) +d(p, pi)≤ |tv|+ε≤ |v|+ε
and|˙γ(t)|=|v|, and hence (γ(t),˙γ(t))∈
e
Kfor 0≤t < T. By Lemma 4.3.3
and Corollary 2.4.15, this implies that there exists a constantδ >0 such
that [0, T+δ)⊂Ip,v, in contradiction to the definition ofT. This contra-
diction shows that our assumption thatvis not an element ofVpmust have
been wrong. Thusv∈Vpand this proves Step 1.
Step 2.Letq∈Msuch thatd(p, q)< r. Then there exists a vectorv∈TpM
such that|v|=d(p, q)andexp
p(v) =q.
Sincepiconverges top, we have limi→∞d(pi, q) =d(p, q)< rand so
there exists an integeri0∈Nsuch thatd(pi, q)< rfor alli≥i0. Then,
for eachi≥i0, there exists a tangent vectorvi∈Tpi
Mthat satisfies the
conditions exp
pi
(vi) =qand|vi|=d(pi, q)< r(Theorem 4.4.4). Passing to
a subsequence, if necessary, we may assume that the limitv:= limi→∞vi
exists inR
n
. Thenv∈TpM,|v|= limi→∞|vi|= limi→∞d(pi, q) =d(p, q),
sov∈Br(p)⊂Vpand exp
p(v) = limi→∞exp
pi
(vi) =q.This proves Step 2.
Step 3.Letv∈TpMsuch that|v|< r. Thend(p,exp
p(v)) =|v|.
Defineq:= exp
p(v), sod(p, q)≤ |v|< r. Choose a sequencevi∈Tpi
Msuch
that|vi|=|v|for alliand limi→∞vi=v. Then the sequenceqi:= exp
pi
(vi)
converges toq= exp
p(v) and satisfiesd(pi, qi) =|vi|by Theorem 4.4.4.
Henced(p, q) = limi→∞d(pi, qi) = limi→∞|vi|=|v|and this proves Step 3.

7.2. THE INJECTIVITY RADIUS* 341
Step 4.The mapexp
p:Br(p)→Mis a local diffeomorphism.
Letv∈Br(p) and letbv∈TpMbe a nonzero vector. Choose any real num-
ber 1< λ < r/|v|so that|λv|< rand define the geodesicγ: [0, λ]→Mand
the vector fieldX∈Vect(γ) by
γ(t) := exp
p(tv), X(t) :=
∂
∂s

s=0
exp
p
Γ
t(v+sbv)
∆
=dexp
p(tv)tbv
for 0≤t≤λ. By Lemma 6.1.18,Xis a Jacobi field alongγand
X(0) = 0,∇X(0) =bv̸= 0, X(1) =dexp
p(v)bv.
SinceL(γ) =d(p,exp
p(λv)) =d(γ(0), γ(λ)) by Step 3, it follows from
part (i) of Theorem 7.1.2 that the geodesicγ: [0, λ]→Mhas no con-
jugate pointsτin the open interval 0< τ < λ. In particular,τ= 1 is not
a conjugate point, and soX(1)̸= 0. Hence the derivativedexp
p(v) of the
exponential map is bijective at every pointv∈Br(p) and this proves Step 4.
Step 5.The mapexp
p:Br(p)→Mis injective.
This is a covering argument. Letv0, v1∈Br(p) with exp
p(v0) = exp
p(v1),
choose a smooth pathv: [0,1]→Br(p) such thatv(0) =v0andv(1) =v1,
and defineγ(t) := exp
p(v(t)) for 0≤t≤1, soγ(0) =γ(1) =q:= exp
p(v0).
Chooseρ < randi∈Nsuch that|v(t)|< ρfor alltandd(pi, p)< r−ρ.
Thend(pi, γ(t))≤d(pi, p) +d(p, γ(t))< d(pi, p) +ρ < rfor allt. Define
β(s, t) := exp
pi
ı
sexp
−1
pi
(q) + (1−s) exp
−1
pi
(γ(t))
ȷ
,0≤s, t≤1.
This map takes values in the setUr(p) ={p
′
∈M|d(p, p
′
)< r}and satisfies
β(0, t) =γ(t) = exp
p(v(t)), β(1, t) =qfor 0≤t≤1,
andβ(s,0) =β(s,1) =qfor alls. Since the map exp
p:Br(p)→Ur(p)
is surjective by Step 2 and a local diffeomorphism by Step 4, a path lifting
argument shows that there exists a smooth mapu: [0,1]
2
→Br(p) such that
u(0, t) =v(t),exp
p(u(s, t)) =β(s, t)
for 0≤s, t≤1. This map satisfiesu(s,0) =v0andu(s,1) =v1for alls
and, moreover, the curvet7→u(1, t) must be constant. Hencev0=v1. This
proves Step 5 and Lemma 7.2.3.
Proof of Theorem 7.2.1.The set{p∈M|a <inj(p;M)< b}=Ua\Ab
is open for all nunbers 0< a < b≤ ∞by Lemma 7.2.2 and Lemma 7.2.3.
Hence the functionM→(0,∞] :p7→inj(p;M) is continuous.

342 CHAPTER 7. TOPICS IN GEOMETRY
7.3 The Group of Isometries*
This section is devoted to the Myers–Steenrod Theorem which asserts that
the groupI(M) of isometries of a connected smooth Riemannian mani-
foldMadmits the natural structure of a finite-dimensional Lie group [52].
7.3.1 The Myers–Steenrod Theorem
Assume throughout thatM⊂R
n
is a nonempty connected smoothm-
manifold. To state the main result, it is convenient to introduce the space
G:=
æ
(q,Φ, p)

p, q∈Mand Φ :TpM→TqM
is an orthogonal isomorphism
œ
. (7.3.1)
This space is agroupoid, i.e. a category in which every morphism is an
isomorphism. The space of objects is the manifoldM, the morphisms
fromp∈Mtoq∈Mare the triples of the form (q,Φ, p)∈ G, the identity
morphism frompto itself is the triple (p,1l, p), the inverse of (q,Φ, p)∈ Gis
the triple (p,Φ
−1
, q), and the composition of a morphism (q,Φ, p)∈ Gfromp
toqwith a morphism (r,Ψ, q)∈ Gfromqtoris the triple (r,ΨΦ, p).
The spaceGis a smooth manifold (in the intrinsic sense). To see this,
consider the diagonal action of the orthogonal group O(m) on the prod-
uct of the orthonormal frame bundleO(M) with itself (Definition 3.4.3).
This action is free and the mapπ:O(M)× O(M)→ Gwhich sends the
pair ((q, e
′
),(p, e)) to the triple (q, e
′
◦e
−1
, p)∈ Gdescends to a bijection
from the quotient (O(M)× O(M))/O(m) toG. By Theorem 2.9.14 there is
a unique smooth structure onGsuch that the mapπ:O(M)× O(M)→ G
is a submersion. With this structure the mapss, t:G →Mdefined by
s(q,Φ, p) :=p, t(q,Φ, p) :=q
are smooth, the mape:M→ Gdefined by
e(p) := (p,1l, p)
is an embedding, the inverse mapi:G → Gdefined by
i(q,Φ, p) := (p,Φ
−1
, q)
is a smooth involution, the maps×t:G × G →M×Mis a submersion,
and the composition mapm: (s×t)
−1
(∆)→ Gdefined by
m((r,Ψ, q),(q,Φ, p)) := (r,ΨΦ, p)
is smooth. (Here ∆ :={(q, q)|q∈M}is the diagonal inM×M.) These
properties assert that the tuple (G, s, t, e, i, m) is asmooth groupoid.

7.3. THE GROUP OF ISOMETRIES* 343
For eachp∈Mdefine
Gp:=
æ
(q,Φ)

q∈Mand Φ :TpM→TqM
is an orthogonal isomorphism
œ
. (7.3.2)
This space is a submanifold ofGbecauses:G →Mis a submersion. The
spaceGpis also a smooth submanifold ofR
n
× L(TpM,R
n
) and is dif-
feomorphic to the orthonormal frame bundleO(M)⊂R
n
×R
n×m
via the
mapGp→ O(M) : (q,Φ)7→(q,Φ◦e) for any orthonormal frameeofTpM.
SinceMis connected, Lemma 5.1.10 asserts that the map
ιp:I(M)→ Gp, ιp(ϕ) := (ϕ(p), dϕ(p)), (7.3.3)
is injective for everyp∈M. Denote the image of this map by
Ip:=ιp(I(M)) =
Φ
(ϕ(p), dϕ(p))

ϕ∈ I(M)

⊂ Gp. (7.3.4)
In the following theorem we do not assume thatMis complete. For a space
of smooth functions or maps onMwe use the termC
∞
topologyto mean
the topology of uniform convergence with all derivatives on each compact
subset ofM. Likewise we use the termC
0
topologyto mean the topology of
uniform convergence on each compact subset ofM. The latter is also called
thecompact-open topology(because a basis of the topology consists of
sets of maps, one for each compact subset of the source and each open subset
of the target, that send the given compact subset of the source into the given
open subset of the target).
Theorem 7.3.1(Myers–Steenrod).LetM⊂R
n
be a nonempty con-
nected smoothm-manifold. Then the following holds.
(i)There exists a unique smooth structure on the isometry groupI(M)such
that the mapιp:I(M)→ Gpin(7.3.3)is an embedding for everyp∈M.
The topology induced by this smooth structure agrees with theC
0
topology
and with theC
∞
topology onI(M)anddim(I(M))≤m(m+ 1)/2.
(ii)With the smooth structure in part (i) the maps
I(M)× I(M)→ I(M) : (ψ, ϕ)7→ψ◦ϕ
and
I(M)→ I(M) :ϕ7→ϕ
−1
are smooth. ThusI(M)is a finite-dimensional Lie group.
(iii)The Lie algebra ofI(M)is the spaceVectK,c(M)of complete Killing
vector fields (defined below).
Proof.See§7.3.4.

344 CHAPTER 7. TOPICS IN GEOMETRY
7.3.2 The Topology on the Space of Isometries
The next lemma shows that for eachp∈Mthe setIpin (7.3.4) is a closed
subset ofGpand that the mapιp:I(M)→ Ipin (7.3.3) is a homeomorphism
with respect to theC
∞
topology onI(M).
Lemma 7.3.2.Fix two elementsp0, q0∈M, letΦ0:Tp0
M→Tq0
Mbe an
orthogonal isomorphism, and letϕi:M→Mbe a sequence of isometries.
Then the following are equivalent.
(i)The sequenceιp0
(ϕi)∈ Ip0
converges to the pair(q0,Φ0)∈ Gp0
, i.e.
lim
i→∞
ϕi(p0) =q0, lim
i→∞
dϕi(p0) = Φ0. (7.3.5)
(ii)There exists an isometryϕ:M→Msuch thatϕ(p0) =q0,dϕ(p0) = Φ0,
andϕiconverges toϕin theC
∞
topology.
Proof.That (ii) implies (i) follows directly from the definitions. We prove in
three steps that (i) implies (ii). Forp∈Mandr >0 denote byVp⊂TpM
the domain of the exponential map ofMatp(Definition 4.3.5), and de-
fineUr(p) :={q∈M|d(p, q)< r}andBr(p) :={v∈TpM| |v|< r}.
Step 1.Assume(7.3.5)holds and choose a real number0< r≤inj(p0;M).
Thenr≤inj(q0;M)andϕiconverges on the open setUr(p0)in theC
∞
topology to the isometryϕ0:= exp
q0
◦Φ0◦exp
−1
p0
:Ur(p0)→Ur(q0).
Sincer≤inj(p0;M), Corollary 5.3.3 asserts that
Br(ϕi(p0)) =dϕi(p0)Br(p0)⊂dϕi(p0)Vp0
⊂V
ϕi(p0)
andϕi◦exp
p0
= exp
ϕi(p0)◦Φi:Br(p0)→Ur(ϕi(p0)).Thus the map
exp
ϕi(p0)=ϕi◦exp
p0
◦Φ
−1
i
:Br(ϕi(p0))→Ur(ϕi(p0))
is a diffeomorphism and sor≤inj(ϕi(p0);M) for eachi∈N. Sinceϕi(p0)
converges toq0, this impliesr≤inj(q0;M) (Lemma 7.2.3). Hence the se-
quence of maps exp
ϕi(p0)◦Φiconverges to the map exp
q0
◦Φ0in theC
∞
topology onBr(p0). Hence the sequence of isometries
ϕi= exp
ϕi(p0)◦Φi◦exp
−1
p0
:Ur(p0)→M
converges in theC
∞
topology to the diffeomorphism
ϕ0:= exp
q0
◦Φ0◦exp
−1
p0
:Ur(p0)→Ur(q0).

7.3. THE GROUP OF ISOMETRIES* 345
The mapϕ0satisfies
d(ϕ0(p), ϕ0(q)) = lim
i→∞
d(ϕi(p), ϕi(q)) =d(p, q)
for allp, q∈Ur(p0) and hence is an isometry (Theorem 5.1.1). This proves
Step 1.
Step 2.The sequenceϕiconverges in theC
∞
topology on all ofMto a
smooth mapϕ:M→M.
Define the setM0:={p∈M|the sequence (ϕi(p), dϕi(p)) converges}.This
set is nonempty becausep0∈M0. We prove thatM0is open. Fix any
elementp∈M0and defineq:= limi→∞ϕi(p) and Φ := limi→∞dϕi(p).
Choose a real numberrsuch that 0< r <inj(p;M). Then Step 1 as-
serts thatr≤inj(q;M) and the sequenceϕi|
Ur(p)converges to the diffeo-
morphism exp
q◦Φ◦exp
−1
p:Ur(p)→Ur(q) in theC
∞
topology onUr(p),
and henceUr(p)⊂M0. This shows thatM0is open, thatϕi|M0
converges
in theC
∞
-topology to a smooth mapϕ:M0→M, and thatϕ(M0) is an
open subset ofM.
We prove thatM0is closed. Letpν∈M0be a sequence that con-
verges to an elementp∈M. Then there exists a real numberr >0 such
thatr <inj(pν;M) for allν. HenceBr(pν)⊂M0for allνby Step 1.
Chooseνso large thatd(pν, p)< rto obtainp∈M0. This shows thatM0is
closed. SinceMis connected, we deduce thatM0=M. This proves Step 2.
Step 3.The mapϕ:M→Min Step 2 is an isometry.
We claim that the triplep0, q0,Φ0in (7.3.5) satisfies
lim
i→∞
ϕ
−1
i
(q0) =p0, lim
i→∞
dϕ
−1
i
(q0) = Φ
−1
0
. (7.3.6)
Namely, the sequenced(ϕ
−1
i
(q0), p0) =d(q0, ϕi(p0)) converges to zero by as-
sumption, and by Step 1 the derivativesdϕiconverge todϕuniformly in
some neighborhoodU0⊂Mofp0. Sinceϕ
−1
i
(q0)∈U0forisufficiently
large, this implies that the sequencedϕi(ϕ
−1
i
(q0)) converges todϕ(p0) = Φ0
and hence the sequencedϕi(ϕ
−1
i
(q0))
−1
=dϕ
−1
i
(q0) converges to Φ
−1
0
. This
proves (7.3.6). By (7.3.6) and Step 2 the sequenceϕ
−1
i
converges in theC
∞
topology to a smooth mapψ:M→M. By uniform convergence on com-
pact subsets ofMwe haveψ◦ϕ= id andϕ◦ψ= id, soϕ:M→Mis a
diffeomorphism. Moreover, sinceϕiis an isometry for eachiand converges
pointwise toϕ, we haved(ϕ(p), ϕ(q)) = limi→∞d(ϕi(p), ϕi(q)) =d(p, q) for
allp, q∈M, soϕis an isometry. This proves Step 3 and Lemma 7.3.2.

346 CHAPTER 7. TOPICS IN GEOMETRY
The next goal is to verify that the spacesIpin (7.3.4) are diffeomorphic
to each other. Forp0, p1∈Mdefine the mapFp1,p0
:Ip0
→ Ip1
by
Fp1,p0
(ϕ(p0), dϕ(p0)) := (ϕ(p1), dϕ(p1)), ϕ∈ I(M). (7.3.7)
This map is well-defined by Lemma 5.1.10, becauseMis connected. Col-
lectively, these maps give rise to a mapF:M× I → Idefined by
I:=
Φ
(ϕ(p), dϕ(p), p)

p∈M, ϕ∈ I(M)

⊂ G,
F(p1,(ϕ(p0), dϕ(p0), p0)) := (ϕ(p1), dϕ(p1), p1)
(7.3.8)
forp0, p1∈Mandϕ∈ I(M). The next lemma uses the concept of a smooth
map on an arbitrary subset of Euclidean space as in the beginning of§2.1.
Lemma 7.3.3.The mapF:M× I → Iis smooth. In particular, for each
pair of pointsp0, p1∈M, the mapFp1,p0
:Ip0
→ Ip1
is a diffeomorphism.
Proof.Letp0, p1∈Mand choose a smooth pathγ:I= [0,1]→Mwith
the endpointsγ(0) =p0andγ(1) =p1. LetUγ⊂ Gp0
be the set of all
pairs (q0,Φ0)∈ Gp0
such that there exists a development (Φ, γ, γ
′
) on the
intervalIthat satisfies
γ
′
(0) =q0,Φ(t) = Φ0. (7.3.9)
By Remark 3.5.22, the setUγis open inGp0
and the mapUγ→ Gp1
that
sends the pair (q0,Φ0)∈ Uγto the pair (γ
′
(1),Φ(1))∈ Gp1
is smooth. This
shows that there is a unique diffeomorphism
Fγ:Uγ→ U
γ
−1, γ
−1
(t) :=γ(1−t),Uγ⊂ Gp0
,U
γ
−1⊂ Gp1
that satisfies the condition
Fγ(γ
′
(0),Φ(0)) = (γ
′
(1),Φ(1)) (7.3.10)
for every development (Φ, γ, γ
′
) ofMalongMon the intervalI. The in-
verse ofFγis the smooth mapF
γ
−1:U
γ
−1→ Uγ. Ifϕ∈ I(M), then by
Lemma 6.1.12 there exists a development (Φ, γ, γ
′
) onIsatisfying the ini-
tial conditionsγ
′
(0) =ϕ(p0) and Φ(0) =dϕ(p0), and it is given by
γ
′
(t) =ϕ(γ(t)),Φ(t) =dϕ(γ(t)) (7.3.11)
fort∈I. Hence (ϕ(p0), dϕ(p0))∈ Uγand by (7.3.10) and (7.3.11) we have
Fγ(ϕ(p0), dϕ(p0)) = (ϕ(p1), dϕ(p1)) =Fp1,p0
(ϕ(p0), dϕ(p0))
for everyϕ∈ I(M). ThusIp0
⊂ UγandIp1
⊂ U
γ
−1andFγ|Ip
0
=Fp1,p0
.
HenceFp1,p0
is smooth. The smoothness ofFfollows from the smooth
dependence of the mapFγon the curveγ, the verification of which we leave
to the reader. This proves Lemma 7.3.3.

7.3. THE GROUP OF ISOMETRIES* 347
7.3.3 Killing Vector Fields
Killing vector fields are defined as those vector fields onMwhose flows
are one-parameter families of isometries. Assume throughout thatMis a
nonempty connected smoothm-dimensional submanifold ofR
n
. LetXbe
a vector field onMand denote by
R×M⊃ D →M: (t, p)7→ϕ(t, p) =ϕ
t
(p)
the flow ofX(Definition 2.4.8). Then Theorem 2.4.9 asserts thatDis
an open subset ofR×Mandϕis smooth. Thus, for everyt∈R, the
setDt:={p∈M|(t, p)∈ D}is open inMand the mapϕ
t
:Dt→ D−tis a
diffeomorphism with the inverseϕ
−t
:D−t→ Dt.
Lemma 7.3.4.In this situation the following are equivalent.
(i)For everyt∈Rthe diffeomorphismϕ
t
:Dt→ D−tis an isometry.
(ii)For everyp∈Mand every pair of tangent vectorsu, v∈TpM, we have
⟨∇uX(p), v⟩+⟨u,∇vX(p)⟩= 0. (7.3.12)
Proof.Letp∈Mandv∈TpM. Choose a smooth curveα:R→Msuch
thatα(0) =pand ˙α(0) =v, let Ω :={(s, t)∈R
2
|(s, α(t))∈ D}, and define
the mapγ: Ω→Mbyγ(s, t) :=ϕ
t
(α(s)) for (s, t)∈Ω. Then
∂sγ(0, t) =dϕ
t
(p)v, ∂ tγ=X◦γ, ∇s∂tγ=∇∂sγX(γ).
Thus the formula∇t∂sγ=∇s∂tγin Lemma 3.2.4 shows that
∇tdϕ
t
(p)v=∇
dϕ
t
(p)vX(ϕ
t
(p)) (7.3.13)
for allt∈Rand this implies
d
dt

dϕ
t
(p)v

2
= 2

∇
dϕ
t
(p)vX(ϕ
t
(p)), dϕ
t
(p)v

. (7.3.14)
The right hand side vanishes for allp, v, tif and only ifXsatisfies (ii),
and the left hand side vanishes for allp, v, tif and only if the flow ofX
satisfies (i). This proves Lemma 7.3.4.
Definition 7.3.5.A vector fieldX∈Vect(M)is called aKilling vector
field(named after Wilhelm Karl Joseph Killing [39]) iff it satisfies equa-
tion(7.3.12)for allp∈Mand allu, v∈TpM. The space of Killing vector
fields will be denoted byVectK(M).

348 CHAPTER 7. TOPICS IN GEOMETRY
The space of Killing vector fields is a vector subspace of the Lie algebra of
all vector fields onM. The next lemma shows that it is a finite-dimensional
Lie subalgebra. Part (ii) is the linear counterpart of Lemma 5.1.10.
Lemma 7.3.6.LetM⊂R
n
be a nonempty connected smoothm-manifold.
Then the following holds.
(i)IfXis a Killing vector field onMandψ:M→Mis an isometry, then
the pullbackψ
∗
Xis a Killing vector field. IfXandYare Killing vector
fields onM, then so is their Lie bracket[X, Y].
(ii)IfXis a Killing vector field onMand there exists ap0∈Msuch that
X(p0) = 0,∇X(p0) = 0, (7.3.15)
thenX(p) = 0for allp∈M.
(iii)IfMis complete andXis a Killing vector field, thenXis complete.
Proof.We prove part (i). LetXbe a Killing vector field, letϕ
t
:Dt→ D−t
be the flow ofX, and letψ:M→Mbe an isometry. Then
ψ
−1
◦ϕ
t
◦ψ:ψ
−1
(Dt)→ψ
−1
(D−t)
is the flow ofψ
∗
X. Hence, by Lemma 7.3.4, the flow ofψ
∗
Xis a one-
parameter family of isometries and soψ
∗
Xis a Killing vector field. Now
assume thatYis another Killing vector field and denote its flow byψ
t
.
Then eachψ
t
is an isometry and so the pullback (ψ
t
)
∗
Xis a Killing vec-
tor field for eacht. Hence, by Lemma 2.4.18 and Exercise 5.2.9, the Lie
bracket [X, Y] =
d
dt

t=0
(ψ
t
)
∗
Xis also a Killing vector field. This proves (i).
We prove part (ii). Letϕ
t
:Dt→ D−tbe the flow of a Killing vector
fieldXand assume that there exists ap0∈Msuch that (7.3.15) holds. Then
it follows from (7.3.13) thatϕ
t
(p0) =p0anddϕ
t
(p0) = id for allt. Hence,
by Lemma 5.1.10 the isometryϕt:Dt→ D−tis the identity for eacht∈R.
ThusDt=D−t=Mfor alltandX(p) = 0 for allp∈M. This proves (ii).
We prove part (iii). Thus assume thatMis complete andXis a
Killing vector field. Letγ:I→Mbe an integral curve ofXon its
maximal existence intervalI⊂Rand denotep:=γ(0).Differentiate the
equation ˙γ(t) =X(γ(t)) to obtain
d
dt
|˙γ(t)|
2
= 2⟨∇
˙γ(t)X(γ(t)),˙γ(t)⟩= 0 for
allt∈I. Hence the functionI→R:t7→ |˙γ(t)|is constant and this implies
the inequalityd(p, γ(t))≤
R
t
0
|˙γ(s)|ds=t|X(p)|for allt∈I. SinceMis
complete, the closed ball of radiusRaboutpis compact for everyR >0
(Theorem 4.6.5). Hence the restriction ofγto any bounded subinterval
ofIis contained in a compact subset ofMamd by Corollary 2.4.15 this
impliesI=R. This proves (iii) and Lemma 7.3.6.

7.3. THE GROUP OF ISOMETRIES* 349
The proof of Theorem 7.3.1 is somewhat analogous to the proof of the
Closed Subgroup Theorem 2.5.27. The first parallel is in part (i) of the next
lemma, which asserts that the space of complete Killing vector fields is a Lie
subalgebra of Vect(M) and can be viewed as an anlogue of Lemma 2.5.29.
Part (ii) is the analogue of Lemma 2.5.28.
Lemma 7.3.7. The setVectK,c(M)of complete Killing vector fields
onMis a Lie subalgebra ofVectK(M).
(ii)LetR×M→M: (t, p)7→ψt(p)be a smooth map such thatψtis an
isometry for everyt∈Rand defineX(p) :=
d
dt

t=0
ψt(p)forp∈M. ThenX
is a complete Killing vector field.
Proof.The proof has three steps.
Step 1.LetU, V⊂Mbe nonempty open sets, letϕ:U→Vbe an isometry,
and letϕk:M→Mbe a sequence of isometries that converges uniformly
on every compact subset ofUtoϕ. Thenϕextends uniquely to an isometry
fromMtoMandϕkconverges in theC
∞
topology to this extension.
Fix an elementp0∈U. Then by part (i) of Exercise 5.1.11 we have
lim
k→∞
ϕk(p0) =ϕ(p0), lim
k→∞
dϕk(p0) =dϕ(p0).
Hence the assertion of Step 1 follows from Lemma 7.3.2.
Step 2.We prove part (ii).
ThatXis a Killing vector field follows from the same argument as in
Lemma 7.3.4 with time dependent vector fields. Denote byϕ
t
:Dt→ D−t
the flow ofX. Then the sequence of isometriesψ
k
t/k
:M→Mconverges
toϕ
t
uniformly on every compact subset ofDt(see Exercise 7.3.10 below).
Hence Step 1 asserts thatϕ
t
extends to an isometry on all ofMwhen-
everDtis nonempty, in particular for smallt. The extended isometries still
satisfyϕ
s+t
=ϕ
s
◦ϕ
t
and∂tϕ
t
=X◦ϕ
t
. ThusDt=Mfor smalltand so for
allt, becauseϕ
−s
(D−s∩ Dt)⊂ Ds+t(Theorem 2.4.9). This proves Step 2.
Step 3.We prove part (i).
LetX, Y∈VectK,c(M), letϕ
t
be the flow ofX, and letψ
t
be the flow ofY.
Then
d
dt

t=0
ϕ
t
◦ψ
t
(p) =X(p) +Y(p) for allp∈M, and soX+Yis a com-
plete Killing vector field by Step 2. Hence VectK,c(M) is a vector space. It is
finite-dimensional by Lemma 7.3.6. Moreover, (ψ
−t
◦ϕ
s
◦ψ
t
)s∈Ris the flow of
the pullback vector field (ψ
t
)
∗
X, hence (ψ
t
)
∗
Xis a complete Killing vector
field for eacht∈R, and hence so is the Lie bracket [X, Y] =
d
dt
|t=0(ψ
t
)
∗
X
by finite-dimensionality. This proves Step 3 and Lemma 7.3.7.

350 CHAPTER 7. TOPICS IN GEOMETRY
Exercise 7.3.8.LetM⊂R
n
be a smooth manifold and letXbe a vector
field onM. Consider the following condition.
(K)Ifγ:I→Mis a geodesic, thenX◦γ∈Vect(γ)is a Jacobi field alongγ,
i.e. it satisfies the differential equation∇∇(X◦γ) +R(X◦γ,˙γ) ˙γ= 0.
Prove that every Killing vector field satisfies (K) and use this to give an
alternative proof of part (ii) of Lemma 7.3.6. IfMis compact, prove that
a vector field satisfies (K) if and only if it is a Killing vector field. Find a
vector field on a noncompact manifold that satisfies (K) but is not a Killing
vector field.Hint:Differentiate the functiont7→ ⟨X(γ(t)),˙γ(t)⟩twice.
Exercise 7.3.9(Gr¨onwall’s inequality).Ifa, c≥0andσ: [0, T]→Ris
a continuous function satisfying
0≤σ(t)≤a+c
Z
t
0
α(s)ds for0≤t≤T, (7.3.16)
thenσ(t)≤ae
ct
for every real number0≤t≤T.Hint:The function
τ(t) :=
Z
t
0
(ae
cs
−σ(s))ds
satisfies ˙τ(t)≥cτ(t) for allt. Differentiate the functiont7→e
−ct
τ(t) to show
thatτis nonnegative and nondecreasing.
Exercise 7.3.10.LetT, c, εbe positive real numbers, letM⊂R
n
be a
smoothm-dimensional submanifold, and let [0, T]×M→R
n
: (t, p)7→Xt(p)
and [0, T]×M→R
n
: (t, p)7→Yt(p) be continuous maps that satisfy the
conditionsXt(p), Yt(p)∈TpMand
|Xt(p)−Yt(p)|R
n≤ε,|Xt(p)−Xt(q)|
R
n≤c|p−q|R
n (7.3.17)
for allp, q∈Mand allt∈[0, T]. If the curvesβ, γ: [0, T]→Mare solutions
of the differential equations
˙
β(t) =Xt(β(t)) and ˙γ(t) =Yt(γ(t)),prove that
|β(t)−γ(t)|R
n≤
ı
|β(0)−γ(0)|R
n+T ε
ȷ
e
ct
(7.3.18)
for 0≤t≤T. Relax the continuity requirement inton the vector fields to
piecewise continuity.Hint:Define the functionσ: [0, T]→Rby
σ(t) :=|β(t)−γ(t)|R
n.
Show that
σ(t)≤σ(0) +tε+c
Z
t
0
σ(s)ds
and use Gr¨onwall’s inequality.

7.3. THE GROUP OF ISOMETRIES* 351
7.3.4 Proof of the Myers–Steenrod Theorem
With these preparations we are ready to prove Theorem 7.3.1. The following
lemma is the heart of the proof. It is the analogue of Lemma 2.5.30 in the
proof of the Closed Subgroup Theorem 2.5.27.
Lemma 7.3.11.Fix an elementp0∈M, a tangent vectorv0∈Tp0
M, and a
linear mapA0:Tp0
M→Tp0
M. Letϕi∈ I(M)be a sequence of isometries
and letτibe a sequence of positive real numbers such that
lim
i→∞
τi= 0, lim
i→∞
ϕi(p0) =p0, lim
i→∞
dϕi(p0) = 1l, (7.3.19)
and, for allv∈Tp0
M,
lim
i→∞
ϕi(p0)−p0
τi
=v0, lim
i→∞
dϕi(p0)v−v
τi
=A0v. (7.3.20)
Then there exists a unique complete Killing vector fieldXonMsuch that
X(p0) =v0, dX(p0) =A0. (7.3.21)
For everyp∈Mand everyv∈TpMthis vector field satisfies
lim
i→∞
ϕi(p)−p
τi
=X(p), lim
i→∞
dϕi(p)v−v
τi
=dX(p)v, (7.3.22)
and the convergence is uniform on compact subsets ofT M.
Proof.By (7.3.19), (7.3.20), and Exercise 2.2.4 we have
(p0, A0)∈T
(p0,1l)Gp0
.
Letp∈Mand recall the definition of the mapFp,p0
:Ip0
→ Ipin (7.3.7).
By Lemma 7.3.3 this map extends to a diffeomorphismFγfrom an open sub-
setUγ⊂ Gp0
containing the setIp0
to an open subsetU
γ
−1⊂ Gpcontaining
the setIpand it satisfiesFγ(p0,1l) = (p,1l). Define
(X(p), A(p)) :=dFγ(p0,1l)(v0, A0)∈T
(p,1l)Gp.
SinceFγ(ϕi(p0), dϕi(p0)) = (ϕi(p), dϕi(p)) for alli, it follows from (7.3.20)
and Exercise 2.2.16 that, for allv∈TpM, we have
lim
i→∞
ϕi(p)−p
τi
=X(p), lim
i→∞
dϕi(p)v−v
τi
=A(p)v. (7.3.23)
This formula shows that the pair (X(p), A(p))∈T
(p,1l)Gpis independent of
the choice of the extensionFγused to define it. Moreover, it follows from the
smoothness of the mapFin Lemma 7.3.3 that the mapp7→(X(p), A(p))
is smooth and that the convergence in (7.3.23) is uniform as the pair (p, v)
varies over any compact subset ofT M.

352 CHAPTER 7. TOPICS IN GEOMETRY
Now letv∈TpMand defineγ(t) := exp
p(tv) fort∈Ip,v. Then
X(γ(t))−X(p) = lim
i→∞
ϕi(γ(t))−γ(t)
τi
−lim
i→∞
ϕi(p)−p
τi
= lim
i→∞
ϕi(γ(t))−ϕi(γ(0))−γ(t) +γ(0)
τi
= lim
i→∞
Z
t
0
dϕi(γ(s)) ˙γ(s)−˙γ(s)
τi
ds
=
Z
t
0
A(γ(s)) ˙γ(s)ds.
Here the last step uses uniform convergence on compact sets in (7.3.23).
Divide bytand use the continuity of the curvet7→A(γ(t)) ˙γ(t) to obtain
A(p)v= lim
t→0
1
t
Z
t
0
A(γ(s)) ˙γ(s)ds= lim
t→0
X(exp
p(tv))−X(p)
t
.
Hence, by Exercise 2.2.16, we deduce that, for allp∈Mand allv∈TpM,
A(p)v=dX(p)v.
By (7.3.20) and (7.3.23) this shows thatXsatisfies (7.3.21) and (7.3.22).
By (7.3.22) and Exercise 2.2.4 we have (X(p), dX(p))∈T
(p,1l)Gpand this
implies⟨v, dX(p)v⟩= 0 for allv∈TpM. Here is a more direct proof of this
crucial fact. Namely, by (7.3.22) we have
lim
i→∞
|dϕi(p)v−v|
2
τ
2
i
=|dX(p)v|
2
.
Hence
⟨v, dX(p)v⟩= lim
i→∞
⟨v, dϕi(p)v−v⟩
τi
= lim
i→∞
⟨v, dϕi(p)v⟩ − |v|
2
τi
= lim
i→∞
2⟨v, dϕi(p)v⟩ − |v|
2
− |dϕi(p)v|
2
2τi
=−lim
i→∞
|dϕi(p)v−v|
2
τ
2
i
τi
2
= 0
for allp∈Mand allv∈TpM. This shows thatXis a Killing vector field
and so it remains to prove thatXis complete.

7.3. THE GROUP OF ISOMETRIES* 353
Letϕ
t
:Dt→ D−tbe the flow ofXand, forp∈M, denote by
Ip:=
Φ
t∈R

p∈ Dt

the maximal existence interval for the solutionγof the initial value prob-
lem ˙γ(t) =X(γ(t)),γ(0) =p. We prove in three steps thatXis complete.
Step 1.Letp∈Mand letT >0such thatT|X(p)|<inj(p;M).Then
[−T, T]⊂Ip.
Defineγ(t) :=ϕ
t
(p) fort∈Ip. SinceXis a Killing vector field, the diffeo-
morphismϕ
t
:Dt→ D−tis an isometry by Lemma 7.3.4, and hence
|˙γ(t)|=|X(ϕ
t
(p))|=|dϕ
t
(p)X(p)|=|X(p)|,
for allt∈Ip. This impliesd(p, γ(t))≤t|X(p)|for allt∈Ip, and henceγ
cannot leave the compact setU
T|X(p)|(p) ={q∈M|d(p, q)≤T|X(p)|}on
any subintervalI⊂[−T, T]. Hence [−T, T]⊂Ipby Corollary 2.4.15 and
this proves Step 1.
Step 2.Letp∈M. Ift∈Rsatisfies|tX(p)|<inj(p;M)and the sequence
of integersmi∈Zis chosen such that
miτi≤t <(mi+ 1)τi, (7.3.24)
thenϕ
t
(p) = limi→∞ϕ
mi
i
(p).
LetT >0 such thatT|X(p)|<inj(p;M). Then [0, T]⊂Ipby Step 1 and
the set
K:=
Φ
ϕ
t
(p)

0≤t≤T

is compact and|X(q)|=|X(p)|for allq∈K. By Lemma 4.2.7 there exists
a constantδ >0 such that, for allq, q0, q1∈M,
q∈K, d(q, q0)≤δ, d(q, q1)≤δ =⇒
d(q0, q1)
|q0−q1|
≤2.(7.3.25)
Fix a real number 0< ε≤1 and choosei0∈Nsuch that, for alli≥i0,
τi|X(p)|< δ,(T+τi)|X(p)|<inj(p;M),sup
q∈K
d(q, ϕi(q))< δ,(7.3.26)
sup
q∈K

ϕi(q)−q
τi
−X(q)

< ε, sup
q∈K

ϕ
τi
(q)−q
τi
−X(q)

< ε.(7.3.27)

354 CHAPTER 7. TOPICS IN GEOMETRY
Then, for alli≥i0and allq∈K, we haved(q, ϕ
τi
(q))≤τi|X(p)|< δ
andd(q, ϕi(q))< δby (7.3.26), and hence by (7.3.25) and (7.3.27),
sup
q∈K
d(ϕi(q), ϕ
τi
(q))≤2 sup
q∈K
|ϕi(q)−ϕ
τi
(q)| ≤4τiε. (7.3.28)
Now choose a real number 0≤t≤Tand choosemi∈N0as in (7.3.24).
Then, fork= 1, . . . , mi−1 we haveϕ
kτi
(p)∈Kand hence
d
ı
ϕ
k+1
i
(p), ϕ
(k+1)τi
(p)
ȷ
≤d
ı
ϕi
Γ
ϕ
k
i(p)
∆
, ϕi
Γ
ϕ
kτi
(p)
∆
ȷ
+d
ı
ϕi
Γ
ϕ
kτi
(p)
∆
, ϕ
τi
Γ
ϕ
kτi
(p)
∆
ȷ
=d
ı
ϕ
k
i(p), ϕ
kτi
(p)
ȷ
+d
ı
ϕi
Γ
ϕ
kτi
(p)
∆
, ϕ
τi
Γ
ϕ
kτi
(p)
∆
ȷ
≤d
ı
ϕ
k
i(p), ϕ
kτi
(p)
ȷ
+ 4τiε.
Here the last inequality holds fori≥i0by (7.3.28). By induction this implies
d(ϕ
mi
i
(p), ϕ
miτi
(p))≤4miτiε≤4T ε
for alli≥i0. Hence
lim
i→∞
ϕ
mi
i
(p) = lim
i→∞
ϕ
miτi
(p) =ϕ
t
(p)
and this proves Step 2 fort≥0. Fort≤0 the argument is similar.
Step 3.Ip=Rfor everyp∈M.
Fix an elementp∈Mand real numberT >0 such thatT|X(p)|<inj(p;M).
Choose the sequencemi∈N0such thatτimi≤T <(mi+ 1)τi. Then Step 2
asserts thatϕ
T
(p) = limi→∞ϕ
mi
i
(p).Sinceϕ
mi
i
:M→Mis an isometry, it
follows thatT|X(p)|<inj(ϕ
mi
i
(p);M) for alli∈Nand hence
T|X(p)|<inj
Γ
ϕ
T
(p);M
∆
.
By Step 1 this implies [0, T]⊂I
ϕ
T
(p). Hence [0,2T]⊂Ipand, by Step 2,
ϕ
2T
(p) = lim
i→∞
ϕ
mi
i
Γ
ϕ
T
(p)
∆
.
Continue by induction to obtain for allk∈Nthat
T|X(p)|<inj
Γ
ϕ
kT
(p);M
∆
and hence [0,(k+ 1)T]⊂Ipand
ϕ
(k+1)T
(p) = lim
i→∞
ϕ
mi
i
Γ
ϕ
kT
(p)
∆
.
Thus [0,∞)⊂Ipand the same argument shows that (−∞,0]⊂Ip. HenceX
is complete. This proves Step 3 and Lemma 7.3.11.

7.3. THE GROUP OF ISOMETRIES* 355
The next lemma establishes the smooth structure onI(M), in analogy
to the proof of the Closed Subgroup Theorem 2.5.27.
Lemma 7.3.12.For everyp∈Mthe setIpis a smooth submanifold ofGp
and its tangent space at(q,Φ)∈ Ipis given by
T
(q,Φ)Ip=
Φ
(X(q), dX(q)Φ)

X∈VectK,c(M)

. (7.3.29)
Proof.By Lemma 3.4.5 withR
m
replaced byTpMthe tangent space ofGp
at an element (q,Φ)∈ Gpis given by
T
(q,Φ)Gp=





(bq,
b
Φ)

bq∈TqM,
b
Φ∈ L(TpM,R
n
),
Γ
1l−Π(q)
∆
b
Φ =hq(bq)Φ,and
⟨Φu,
b
Φv⟩+⟨
b
Φu,Φv⟩= 0∀u, v∈TpM.





.(7.3.30)
If (q,Φ)∈ GpandX∈Vect(M), then (1l−Π(q))dX(q)Φ =hq(X(q))Φ by
the Gauß–Weingarten formula in Remark 5.2.5. Moreover, ifXis a Killing
vector field, then⟨Φu, dX(q)Φv⟩+⟨dX(q)Φu,Φv⟩= 0 for allu, v∈TpM, so
it follows from (7.3.30) that the pair (bq,
b
Φ) withbq=X(q) and
b
Φ =dX(q)Φ
is a tangent vector ofGpat (q,Φ).
Now abbreviate
k:= dim
Γ
VectK,c(M)
∆
≤
m(m+ 1)
2
= dim(Gp) =:ℓ.
Fix an isometryϕ0:M→Mand define
(q0,Φ0) := (ϕ0(p), dϕ0(p)) =ιp(ϕ0)∈ Ip.
We must construct a coordinate chart onGpin a neighborhood of the
point (q0,Φ0)∈ Ipwith values in an open set Ω⊂R
ℓ
that sendsIpto the
intersection of Ω withR
k
× {0}. For this we first choose a basisY1, . . . , Yk
of VectK,c(M) and then a basisη1= (bq1,
b
Φ1), . . . , ηℓ= (bqℓ,
b
Φℓ) of the tangent
spaceT
(q0,Φ0)Gpsuch that
ηj= (bqj,
b
Φj) =
ı
Yj
Γ
ϕ0(p)
∆
, dYj
Γ
ϕ0(p)
∆
dϕ0(p)
ȷ
, j= 1, . . . , k.
Next we choose any smooth map
R
ℓ−k
→ Gp: (t
k+1
, . . . , t
ℓ
)7→ι(t
k+1
, . . . , t
ℓ
)
such thatι(0) = (q0,Φ0) and
∂ι
∂t
j
(0, . . . ,0) =ηj= (bqj,
b
Φj), j=k+ 1, . . . , ℓ.

356 CHAPTER 7. TOPICS IN GEOMETRY
For a complete Killing vector fieldX∈VectK,c(M) letψX∈ I(M) be the
time-1 map of the flow ofXand define the map Θ :R
ℓ
→ Gpby
Θ
Γ
t
1
, . . . , t
ℓ
∆
:=
ı
ψX(q), dψX(q)Φ
ȷ
,
X:=t
1
Y1+· · ·+t
k
Yk,
(q,Φ) :=ι
Γ
t
k+1
, . . . , t
ℓ
∆
(7.3.31)
for
Γ
t
1
, . . . , t
ℓ
∆
∈R
ℓ
. Then
Θ(0) =ι(0) = (q0,Φ0) =ιp(ϕ0)∈ Ip
and
dΘ(0)
Γ
ˆt
1
, . . . ,ˆt
ℓ
∆
=ˆt
1
η1+· · ·+ˆt
ℓ
ηℓ
for
Γ
ˆt
1
, . . . ,ˆt
ℓ
∆
∈R
ℓ
. Thus the derivativedΘ(0) :R
ℓ
→T
(q0,Φ0)Gpis a bi-
jective linear map and so the Inverse Function Theorem asserts that the
map Θ restricts to a diffeomorphism from a sufficiently small open neigh-
borhood Ω⊂R
ℓ
of the origin onto its imageι(Ω)⊂ Gp, which is an open
neighborhood inGpof the point Θ(0) =ιp(ϕ0)∈ Ip. With this understood,
the assertion thatIpis a smooth submanifold ofGpis a direct consequence
of the following Claim.
Claim.There exists an open setΩ0⊂R
ℓ
such that
0∈Ω0⊂Ω,Θ
Γ
Ω0∩(R
k
× {0})
∆
=U0∩ Ip,U0:= Θ(Ω0).(7.3.32)
Suppose, by contradiction, that such an open set Ω0does not exist. Then
there exist sequencesti=
Γ
t
1
i
, . . . , t
ℓ
i
∆
∈R
ℓ
andϕi∈ I(M) such that
lim
i→∞
ti= 0, ti∈Ω\
Γ
R
k
× {0}
∆
,Θ(ti) =ιp(ϕi)∈ Ip.
Define
Xi:=t
1
iY1+· · ·+t
k
iYk,(qi,Φi) :=ι
Γ
t
k+1
i
, . . . , t
ℓ
i
∆
.
Then
Γ
ψXi
(qi), dψXi
(qi)Φi
∆
= Θ(ti) =ιp(ϕi) =
Γ
ϕi(p), dϕi(p)
∆
and hence
qi= (ψ
−1
Xi
◦ϕi)(p),Φi=dψXi
(qi)
−1
dϕi(p) =d(ψ
−1
Xi
◦ϕi)(p).
Thusιp(ψ
−1
Xi
◦ϕi) = (qi,Φi) = Θ
Γ
0, . . . ,0, t
k+1
i
, . . . , t
ℓ
i
∆
is still a sequence in
the set Θ(Ω\(R
k
× {0}))∩ Ipthat converges toιp(ϕ0). Thus we may as-
sume without loss of generality thatt
1
i
=t
2
i
=· · ·=t
k
i
= 0 and so
ι(ti) =ιp(ϕi), lim
i→∞
ιp(ϕi) =ιp(ϕ0). (7.3.33)

7.3. THE GROUP OF ISOMETRIES* 357
Thenϕi◦ϕ
−1
0
converges to the identity in theC
∞
topology by Lemma 7.3.2.
Sinceϕi̸=ϕ0, we have
τi:=

(ϕi◦ϕ
−1
0
)(p)−p

+ sup
0̸=v∈TpM
|d(ϕi◦ϕ
−1
0
)(p)v−v|
|v|
>0 (7.3.34)
for alliby Lemma 5.1.10, and limi→∞τi= 0. Hence, by Exercise 2.2.4 there
exists a tangent vector (v0, A0)∈T
(p,1l)Gpand a subsequence, still denoted
byϕi, such that, for allv∈TpM,
lim
i→∞
(ϕi◦ϕ
−1
0
)(p)−p
τi
=v0, lim
i→∞
d(ϕi◦ϕ
−1
0
)(p)v−v
τi
=A0v.(7.3.35)
It follows from (7.3.35) and Lemma 7.3.11 that there exists a complete
Killing vector fieldX∈VectK,c(M) such that
X(p) =v0, dX(p) =A0.
This vector field is nonzero because (v0, A0)̸= 0 by (7.3.34). Moreover, by
equation (7.3.22) in Lemma 7.3.11, withϕireplaced byϕi◦ϕ
−1
0
and the
pair (p, v) replaced by the pair (ϕ0(p), dϕ0(p)v), we obtain
X(ϕ0(p)) = lim
i→∞
ϕi(p)−ϕ0(p)
τi
,
dX(ϕ0(p))dϕ0(p)v= lim
i→∞
dϕi(p)v−dϕ0(p)v
τi
forv∈TpM. Since by (7.3.33) the sequence
(ϕi(p), dϕi(p)) =ιp(ϕi) =ι(ti)
converges to (ϕ0(p), dϕ0(p)) =ι(0), the pair (X(ϕ0(p)), dX(ϕ0(p))dϕ0(p))
belongs to the image of the derivativedι(0) by Exercise 2.2.4, and hence
is a nonzero linear combination of the vectorsηk+1, . . . , ηℓ. It is also a
linear combination of the vectorsη1, . . . , ηk, becauseXis a complete Killing
vector field and so is a linear combination of the vector fieldsY1, . . . , Yk.
This is a contradiction, and this contradiction proves the Claim. Thus there
does, after all, exist an open set Ω0⊂R
ℓ
that satisfies (7.3.32), and the
map Θ
−1
:U0→Ω0is then a coordinate chart onGpwhich satisfies
Θ
−1
(U0∩ Ip) = Ω0∩(R
k
× {0}).
HenceIpis a submanifold ofGpand this proves Lemma 7.3.12.

358 CHAPTER 7. TOPICS IN GEOMETRY
Proof of Theorem 7.3.1.By Lemma 7.3.3 and Lemma 7.3.12 there exists
a unique smooth structure onI(M) such that the mapιp:I(M)→ Gp
in (7.3.3) is an embedding for everyp∈M. That the topology induced
by this smooth structure agrees with theC
∞
topology was established in
Lemma 7.3.2. That it also agrees with the compact open topology (i.e. with
theC
0
topology of uniform convergence on conpact sets) is the content of
Exercise 5.1.11. This proves part (i).
We prove part (ii). Recall the definition of the map
F:M× I → I
in (7.3.8) and the target mapt:G →M, the inverse mapi:G → G, and the
composition mapm: (s×t)
−1
(∆)→ Gin the beginning of§7.3.1. These
maps are all smooth and turnIinto a smooth subgroupoid ofG. For
eachp∈Mthey endow the submanifoldIp⊂ Gpwith the structure of a Lie
group as follows. The unit is the element
ep:= (p,1l)∈ Ip (7.3.36)
The product is the mapmp:Ip× Ip→ Ipdefined by
(mp(η, ξ), p) :=m
ı
F
Γ
t(ξ, p),(η, p)
∆
,
Γ
ξ, p
∆
ȷ
(7.3.37)
forξ, η∈ Ip, and the inverse map is the involutionip:Ip→ Ipdefined by
(ip(ξ), p) :=F
Γ
p, i(ξ, p)
∆
(7.3.38)
forξ∈ Ip. The mapsmpin (7.3.37) andipin (7.3.38) are smooth by
definition. To show that they define a group structure and that the map
ιp:I(M)→ Ip
in (7.3.3) is a group isomorphism, we must verify the identities
ιp(id) =ep,
mp(ιp(ψ), ιp(ϕ)) =ιp(ψ◦ϕ),
ip(ιp(ϕ)) =ιp(ϕ
−1
)
(7.3.39)
for allϕ, ψ∈ I(p). The first equation in (7.3.39) follows directly from the
definitions. To prove the remaining equations, fix two isometriesϕandψ
and defineξ, η, ζ∈ Ipby
ξ:=ιp(ϕ), η:=ιp(ψ), ζ:=ιp(ψ◦ϕ).

7.3. THE GROUP OF ISOMETRIES* 359
Then, by the chain rule,
(ζ, p) =
ı
ψ(ϕ(p)), dψ(ϕ(p))dϕ(p), p
ȷ
=m
ı
Γ
ψ(ϕ(p)), dψ(ϕ(p)), ϕ(p)
∆
,
Γ
ϕ(p), dϕ(p), p
∆
ȷ
=m
ı
F
Γ
ϕ(p),(ψ(p), dψ(p), p)
∆
,
Γ
ϕ(p), dϕ(p), p
∆
ȷ
=m
ı
F
Γ
t(ξ, p),(η, p)
∆
,
Γ
ξ, p
∆
ȷ
= (mp(η, ξ), p).
Here the last equality follows from the definition of the mapmpin (7.3.37).
This proves the second equation in (7.3.39).
To verify the third equation in (7.3.39), we compute
Γ
ιp(ϕ
−1
), p
∆
=
Γ
ϕ
−1
(p), dϕ
−1
(p), p
∆
=F
Γ
p,(ϕ
−1
(ϕ(p)), dϕ
−1
(ϕ(p)), ϕ(p))
∆
=F
Γ
p,(p, dϕ(p)
−1
, ϕ(p))
∆
=F
Γ
p, i(ϕ(p), dϕ(p), p)
∆
=F
Γ
p, i(ξ, p)
∆
= (ip(ξ), p).
Here the last equality follows from the definition of the mapipin (7.3.38).
This proves the third equation in (7.3.39) and part (ii).
That the Lie algebra ofIpis isomorphic to the space VectK,c(M) of
complete Killing vector fields under the Lie algebra isomorphism
VectK,c(M)→TepIp:X7→(X(p), dX(p))
follows from Lemma 7.3.12. This proves part (iii) and Theorem 7.3.1.
Corollary 7.3.13.Let(p, e)∈ O(M). Then there exists a constantε >0
with the following significance. Ifϕ:M→Mis an isometry that satisfies
d(p, ϕ(p))< ε, |e−dϕ(p)e|
R
n×m< ε,
then there exists a complete Killing vector fieldX∈VectK,c(M)whose flow
has the time-1-mapψX=ϕ.
Proof.Use the construction of the map Θ : Ω→ Gpand the Claim in the
proof of Lemma 7.3.12 withϕ0= id to obtainϕ∈Θ(Ω∩(R
k
× {0})).

360 CHAPTER 7. TOPICS IN GEOMETRY
7.3.5 Examples and Exercises
Throughout we denote byI0(M)⊂ I(M) the connected component of the
identity in the group of isometries of a Riemannian manifoldM.
Example 7.3.14.The isometry group ofR
m
is the group of all affine trans-
formations ofR
m
with orthogonal linear part (Exercise 5.1.4).
Example 7.3.15.We have seen in Example 6.4.16 that the isometry group
of them-sphereS
m
⊂R
m+1
is the groupI(S
m
) = O(m+ 1) of orthog-
onal transformations of the ambient space. In the casem= 2 a theorem
of Smale [72] asserts that the inclusion O(3) =I(S
2
),→Diff(S
2
) of the
isometry group of the 2-sphere into the infinite-dimensional group of all
diffeomorphisms of the 2-sphere is a homotopy equivalence.
Example 7.3.16.In§6.4.3 we have introduced the hyperbolic spaceH
m
. Its
isometry group is the groupI(H
m
) = O(m,1) of all linear transformations
ofR
m+1
that preserve the quadratic formQin (6.4.9) (Exercise 6.4.25). In
the casem= 2 the identity component of this group is isomorphic to the Lie
group PSL(2,R) = SL(2,R)/{±1l}.Geometrically this can be understood by
examining the upper half space model ofH
2
(Exercise 6.4.27).
The preceding three examples are the constant sectional curvature man-
ifolds discussed in§6.2 and§6.4. In all three cases the isometry group has
the maximal dimension dim(I(M)) =m(m+ 1)/2 and is diffeomorphic to
the orthonormal frame bundleO(M), so these examples satisfy the con-
ditionIp=Gpin the notation of§7.3.1. By Corollary 6.4.13 a complete,
connected, simply connected manifoldMsatisfiesIp=Gpif and only if it
has constant sectional curvature.
Exercise 7.3.17.Consider the incomplete 2-manifolds
M0:=R
2
\ {(0,0)}, M 1:=R
2
\Z
2
.
Prove that fori= 0,1 every isometry ofMiextends to an isometry ofR
2
and
soI(Mi) is a subgroup of the Lie groupI(R
2
) of affine maps with orthogonal
linear part (Example 7.3.14). The isometry group ofM0is isomorphic to
the Lie group O(2). The isometry group ofM1is discrete and is an example
of a so-calledwallpaper group(of which there are 17). The Lie algebra
of Killing vector fields in both cases has dimension 3. Which Killing vector
fields are not complete?Hint:A Killing vector field on a connected open
setM⊂R
2
is a smooth mapM→R
2
: (x, y)7→(u(x, y), v(x, y)) that
satisfies the equations∂xu=∂yv= 0 and∂yu+∂xv= 0. Deduce that the
map (u, v) is affine and has a skew-symmetric linear part.

7.3. THE GROUP OF ISOMETRIES* 361
Example 7.3.18.The identity component of the isometry group of the
complex projective spaceCP
n
(Example 2.8.5) with the Fubini–Study metric
(Example 3.7.5) is the group
I0(CP
n
) = PSU(n+ 1) = SU(n+ 1)/Z(SU(n+ 1))
∼
=U(n+ 1)/U(1).
HereZ(SU(n+ 1)) =
Φ
λ1l|λ∈S
1
, λ
n+1
= 1

∼
=Z/(n+ 1)Zis the center
of the group SU(n+ 1). We emphasize that the dimensionn(n+ 2) of
the isometry groupI(CP
n
) is smaller than the dimensionn(2n+ 1) of the
orthonormal frame bundleO(CP
n
) unlessn= 0 orn= 1.
In the casen= 1 the projective lineCP
1
is isometric to the 2-sphere
by stereographic projection (Exercise 2.8.13 and Example 3.7.5). Hence the
identity component PSU(2) of the isometry group ofCP
1
is isomorphic to
the identity component SO(3) of the isometry group ofS
2
. An explicit
isomorphism is discussed in Exercise 2.5.22.
The full isometry groupI(CP
n
) has two connected components. In the
casen= 2 it is an open question whether the inclusionI(CP
2
),→Diff(CP
2
)
of the isometry group ofCP
2
into the group of all diffeomorphisms ofCP
2
is
a homotopy equivalence. By deep results of Gromov [21] and Taubes [74] a
positive answer to this question is equivalent to the assertion that the space
of symplectic forms onCP
2
in a fixed cohomology class is contractible (the
symplectic uniqueness conjecture forCP
2
). It is not even known whether this
space is connected or, equivalently, whether every diffeomorphism ofCP
2
that induces the identity on cohomology is isotopic to the identity. For a
more detailed discussion see [67, Example 3.4] and [49, Example 13.4.1].
Example 7.3.19.The identity componentI0(S
2
×S
2
) of the isometry group
of the product manifoldM=S
2
×S
2
is the product group SO(3)×SO(3).
In contrast to Smales’ Theorem the inclusionI0(S
2
×S
2
),→Diff0(S
2
×S
2
)
into the group of diffeomorphisms ofS
2
×S
2
that are isotopic to the identity
is not a homotopy equivalence. For example, ifϕx,θ:S
2
→S
2
denotes the
rotation about the axis throughx∈S
2
by the angleθ∈R/2πZand the
diffeomorphismψθ:S
2
×S
2
→S
2
×S
2
is defined by
ψθ(x, y) := (x, ϕx,θ(y)), x, y∈S
2
,
then the loopR/2πZ→Diff0(S
2
×S
2
) :θ7→ψθis not contractible and
neither is any of its iterates. Thus Diff0(S
2
×S
2
) has an infinite fundamental
group while the fundamental group of SO(3)×SO(3) is finite. ForS
2
×S
2
it
is an open question whether every diffeomorphism that induces the identity
on cohomology is isotopic to the identity. For a more detailed discussion
see [67, Example 3.5] and [49, Example 13.4.2].

362 CHAPTER 7. TOPICS IN GEOMETRY
7.4 Isometries of Compact Lie Groups*
In the following theorem we denote byI0(M) the connected component of
the identity in the group of all isometries of a manifoldM.
Theorem 7.4.1.LetG⊂GL(n,R)be a compact connected Lie group
equipped with a bi-invariant Riemannian metric. Ifϕ∈ I0(G), then there
exist elementsa, b∈Gsuch that
ϕ(g) =ϕa,b(g) :=agb
−1
for allg∈G. (7.4.1)
The proof of Theorem 7.4.1 makes use of the Killing form introduced in
Example 5.2.25. Recall the definition of the centerZ(g) in Exercise 2.5.34
and of the commutant [g,g]⊂gin Exercise 5.2.24. The heart of the proof
is Lemma 7.4.3. The case where the Lie algebra has a nontrivial center is
then dealt with in Lemma 7.4.5.
Lemma 7.4.2.Letgbe a finite-dimensional Lie algebra that admits an
invariant inner product and has a trivial center. Then the Killing form ong
is nondegenerate.
Proof.Exercise 5.2.27.
Lemma 7.4.3.Letgbe a finite-dimensional Lie algebra and assume that
the Killing form ongis nondegenerate. Then the following holds.
(i)Z(g) ={0}and[g,g] =g.
(ii)The Lie algebra homomorphismad :g→Der(g)is bijective.
(iii)Every derivationδ:g→ghas trace zero.
(iv)Letδ:g→gbe a linear map such that, for allξ, η, ζ∈g,
δ[[ξ, η], ζ] = [[δξ, η], ζ] + [[ξ, δη], ζ] + [[ξ, η], δζ]. (7.4.2)
Then there exists a unique elementξδ∈gsuch thatδξ= [ξδ, ξ]for allξ∈g.
Proof.We prove part (i). Ifξ∈Z(g), then ad(ξ) = 0, henceκ(ξ, η) = 0 for
allη∈g, and henceξ= 0 by nondegeneracy of the Killing form. To prove
that [g,g] =g, assume that Λ :g→Ris any linear functional that vanishes
on the commutant [g,g]. Since the Killing form is nondegenerate, there exists
an elementζ∈gsuch that Λ =κ(ζ,·). Hence 0 =κ(ζ,[ξ, η]) =κ([ζ, ξ], η)
for allξ, η∈g. Since the Killing form is nondegenerate, this implies [ζ, ξ] = 0
for allξ∈g, henceζ∈Z(g), henceζ= 0, and hence Λ = 0. This proves (i).

7.4. ISOMETRIES OF COMPACT LIE GROUPS* 363
We prove part (ii). The map ad :g→Der(g) is injective by part (i).
Choose a basisξ1, . . . , ξkofg. Since the Killing form is nondegenerate,
there exists a dual basisη1, . . . , ηkofgsuch thatκ(ξi, ηj) =δijfor alli, j.
These bases satisfyζ=
P
i
κ(ηi, ζ)ξifor allζ∈g. Now letδ∈Der(g) and
defineξ:=
P
i
trace
Γ
δad(ξi)
∆
ηi∈g.Then, for allζ∈g, we have
trace
Γ
δad(ζ)
∆
=
X
i
κ(ηi, ζ)trace
Γ
δad(ξi)
∆
=κ(ξ, ζ) = trace
Γ
ad(ξ)ad(ζ)
∆
.
Thus the derivationε:=δ−ad(ξ) satisfies trace(εad(ζ)) = 0 for allζ∈g.
Since [ε,ad(η)] = ad(εη),this implies
κ
Γ
εη, ζ
∆
= trace
Γ
ad(εη)ad(ζ)
∆
= trace
Γ
[ε,ad(η)]ad(ζ)
∆
= trace
Γ
ε[ad(η),ad(ζ)]
∆
= trace
Γ
εad([η, ζ])
∆
= 0
for allη, ζ∈g. Henceε= 0 by nondegeneracy of the Killing form and
henceδ= ad(ξ). This proves (ii).
We prove part (iii). Since trace(ad([ξ, η])) = trace([ad(ξ),ad(η)]) = 0 for
allξ, η∈gand [g,g] =gby part (i), we have trace(ad(ξ)) = 0 for allξ∈g.
Hence (iii) follows from part (ii).
We prove part (iv). Define the bilinear mapBδ:g×g→gby
Bδ(ξ, η) :=δ[ξ, η]−[δξ, η]−[ξ, δη] (7.4.3)
forξ, η∈g. Then equation (7.4.2) can be expressed in the form
Bδ([ξ, η], ζ) + [Bδ(ξ, η), ζ] = 0 (7.4.4)
forξ, η, ζ∈g. By part (i) there exists a basis ofgconsisting of vectors of the
formei= [ξi, ηi]. Define the linear mapAδ:g→gbyAδei:=−Bδ(ξi, ηi).
ThenBδ(ei, ζ) = [Aδei, ζ] for alliandζby (7.4.4). Hence
Bδ(ξ, η) = [Aδξ, η] = [ξ, Aδη] (7.4.5)
for allξ, η∈g. Here the second equality holds by the skew-symmetry ofBδ.
By the Jacobi identity and equations (7.4.4) and (7.4.5) we have
2[Bδ(ξ, η), ζ] = [[ξ, Aδη], ζ] + [[Aδξ, η], ζ]
=−[[ζ, ξ], Aδη]−[[Aδη, ζ], ξ] + [[Aδξ, η], ζ]
=−Bδ([ζ, ξ], η)−[Bδ(η, ζ), ξ] + [[Aδξ, η], ζ]
= [Bδ(ζ, ξ), η] +Bδ([η, ζ], ξ) + [[Aδξ, η], ζ]
= [[ζ, Aδξ], η] + [[η, ζ], Aδξ] + [[Aδξ, η], ζ] = 0
for allξ, η, ζ∈g. SinceZ(g) ={0}by part (i), we find thatBδ(ξ, η) = 0
for allξ, η∈g, henceδis a derivation, and henceδis in the image of the
map ad :g→Der(g) by part (ii). This proves (iv) and Lemma 7.4.3.

364 CHAPTER 7. TOPICS IN GEOMETRY
Lemma 7.4.4.The assertion of Theorem 7.4.1 holds under the additional
assumption that the center of the Lie algebrag= Lie(G)is trivial.
Proof.Letϕ∈ I0(G) and choose a smooth isotopy [0,1]→ I0(G) :t7→ϕt
joining the identityϕ0= id toϕ1=ϕ. For 0≤t≤1 define the diffeomor-
phismψt: G→G byψt(g) :=ϕt(1l)
−1
ϕt(g). Then eachψtis an isometry
and the path [0,1]→ I0(G) :t7→ψtsatisfies
ψ0= id, ψ t(1l) = 1l
for allt. By Theorem 5.3.1 the derivatives Ψt:=dψt(1l) :g→gpreserve
the Riemann curvature tensor of G at 1l and by Example 5.2.18 (for the
standard metric on Lie subgroups of O(n)) and Exercise 5.2.22 (for general
bi-invariant Riemannian metrics) this translates into the condition
Ψt[[ξ, η], ζ] = [[Ψtξ,Ψtη],Ψtζ] (7.4.6)
for alltand allξ, η, ζ∈g. Hence the endomorphismδt:= Ψ
−1
t
d
dt
Ψt:g→g
satisfies (7.4.2) for eacht. Moreover, by Lemma 7.4.2 the Killing form ong
is nondegenerate. Hence it follows from Lemma 7.4.3 that there exists a
smooth path [0,1]→g:t7→ξtsuch that Ψ
−1
t
d
dt
Ψt= ad(ξt) for allt. Thus
d
dt
Ψt= Ψtad(ξt),Ψ0= 1l. (7.4.7)
Now let [0,1]→G :t7→btbe the solution of the differential equation
d
dt
bt=btξt, b0= 1l, (7.4.8)
and define Φt:= Ad(bt) (Example 2.5.23). Then
d
dt
Φt= Φtad(ξt) for allt
and Φ0= 1l. Thus Φt= Ψtand so
Ψtη=btηb
−1
t
(7.4.9)
for alltandη. Taket= 1, defineb:=b1, and use Lemma 5.1.10 (uniqueness
of local isometries) to obtainψ1(g) =bgb
−1
for allg∈G. Hence
ϕ(g) =ϕ(1l)ψ1(g) =ϕ(1l)bgb
−1
=agb
−1
for allg∈G witha:=ϕ(1l)b. This proves Lemma 7.4.4.
We will denote byZ0(G)⊂Z(G) the connected component of 1l in the
center of G. ThenZ0(G)⊂G is a compact connected abelian Lie subgroup
of G (Exercise 2.5.34). Since G is connected, its Lie algebra is the center
ofg, i.e. Lie(Z0(G)) =Z(g).

7.4. ISOMETRIES OF COMPACT LIE GROUPS* 365
Lemma 7.4.5.LetG⊂GL(n,R)be a compact connected Lie group with a
bi-invariant Riemannian metric and letϕ∈ I0(G). Then
ϕ(hg) =ϕ(gh) =ϕ(g)h=hϕ(g) (7.4.10)
for allg∈Gand allh∈Z0(G).
Proof.The proof relies on the following basic observations.
(a)Z0(G) is a compact connected Lie subgroup of G and Lie(Z0(G)) =Z(g).
(b)exp(ξ+η) = exp(ξ) exp(η) for allξ, η∈Z(g).
(c)The exponential map exp :Z(g)→Z0(G) is surjective.
(d)Λ :={ξ∈Z(g)|exp(ξ) = 1l}is a discrete additive subgroup ofZ(g)
which spansZ(g).
Part (a) was noted above, part (b) follows from Exercise 2.5.39 because the
Lie algebraZ(g) is commutative, and part (c) follows from the Hopf–Rinow
Theorem 4.6.6. That the set Λ is an additive subgroup ofZ(g) follows
directly from (b). Moreover, by (a) and (b) the exponential map restricts to
a local diffeomorphism exp :Z(g)→Z0(G). Hence Λ is discrete and by (c)
the exponential map descends to a proper map fromZ(g)/Λ ontoZ0(G).
SinceZ0(G) is compact, the lattice Λ spans the vector spaceZ(g).
Now assumeϕ(1l) = 1l and define Φ :=dϕ(1l) :g→g.Then
ϕ(exp(ξ)) = exp(Φξ) (7.4.11)
for allξ∈gby Corollary 5.3.3 and Example 5.2.18. Moreover, Φ is an
orthogonal transformation ofgthat preserves the Riemann curvature tensor
(Theorem 5.3.1). Thus|[Φξ,Φη]|=|[ξ, η]|for allξ, η∈gby Example 5.2.18.
So, ifξ∈Z(g), then [ξ, η] = 0 for allη, hence [Φξ,Φη] = 0 for allη, and
hence Φξ∈Z(g). This shows that
ΦZ(g) =Z(g). (7.4.12)
By (7.4.11) and (7.4.12) we haveξ∈Λ if and only ifϕ(exp(ξ)) = 1l if and
only if exp(Φξ) = 1l if and only if Φξ∈Λ, so that ΦΛ = Λ.Sinceϕis isotopic
to the identity through isometries, by assumption, there exists a smooth
path of orthogonal transformations Φtofgfrom Φ0= 1l to Φ1= Φ that
satisfy ΦtΛ = Λ for allt. Thus Φξ=ξfor allξ∈Λ and so for allξ∈Z(g)
because the lattice spans the subspaceZ(g) by (d). Hence, by (7.4.11)
and (c), we obtainϕ(h) =hfor allh∈Z0(G) wheneverϕ(1l) = 1l.
To prove equation (7.4.10) in general, fix an elementg∈G and define the
diffeomorphismϕg: G→G byϕg(h) :=ϕ(g)
−1
ϕ(gh) forh∈G. Thenϕgis
an isometry andϕg(1l) = 1l. Moreover,ϕg∈ I0(G), because G is connected.
Henceϕg(h) =hfor allh∈Z0(G) and this proves Lemma 7.4.5.

366 CHAPTER 7. TOPICS IN GEOMETRY
Proof of Theorem 7.4.1.Letϕ∈ I0(G). By Lemma 7.4.5,ϕdescends to a
diffeomorphism
¯
ϕ:
¯
G→
¯
G of the quotient group
¯
G := G/Z0(G).
The Lie algebra of
¯
G is the quotient space
¯g:=g/Z(g)
∼
=Z(g)
⊥
and the invariant inner product ongrestricts to an invariant inner product
on the orthogonal complement ofZ(g). This defines a bi-invariant Rie-
mannian metric on
¯
G. The diffeomorphism
¯
ϕis an isometry with respect
to this metric, because the derivativedϕ(g) :TgG→T
ϕ(g)Gis an orthog-
onal transformation, which by (7.4.10) sends a tangent vectorgη∈TgG
withη∈Z(g) todϕ(g)gη=ϕ(g)η∈T
ϕ(g)G,and hence it sends the sub-
spacegZ(g)
⊥
⊂TgG toϕ(g)Z(g)
⊥
⊂T
ϕ(g)G. Apply Lemma 7.4.4 to the
isometry
¯
ϕto obtain elementsa, b∈G whose equivalence classes ¯a,
¯
b∈
¯
G
satisfy
¯
ϕ(¯g) = ¯a¯g
¯
b
−1
for all ¯g∈
¯
G.This implies
α(g) :=a
−1
ϕ(g)bg
−1
∈Z0(G) (7.4.13)
for allg∈G. Moreover, it follows from Lemma 7.4.5 that
α(gh) =α(g) (7.4.14)
for allg∈G and allh∈Z0(g). Now define the isometryψ: G→G by
ψ(g) :=a
−1
ϕ(g)b=α(g)g (7.4.15)
forg∈G. Fix an elementg∈G and define the linear mapsA,Ψ :g→gby
Aξ:=α(g)
−1
dα(g)gξ,Ψξ:=ψ(g)
−1
dψ(g)gξ (7.4.16)
forξ∈g. Then Ψ = id +Aby (7.4.15) and the mapAvanishes onZ(g)
by (7.4.14) and takes values inZ(g) by (7.4.13). Since Ψ is an orthogonal
transformation, this implies
Ψ = id, A= 0.
Hence it follows from the definition of the linear mapAin (7.4.16) that the
mapα: G→Z0(G) in (7.4.13) is constant. Thus
ϕ(g) =α(1l)agb
−1
for allg∈G, and this proves Theorem 7.4.1.

7.4. ISOMETRIES OF COMPACT LIE GROUPS* 367
Corollary 7.4.6.LetGbe a compact connected Lie group equipped with
a bi-invariant Riemannian metric. Then the map(a, b)7→ϕa,bin Theo-
rem 7.4.1 descends to a Lie Group isomorphism
ρG: (G×G)/Z(G)→ I0(G). (7.4.17)
Proof.The map G×G→ I0(G) : (a, b)7→ϕa,bis a group homomorphism by
definition, is smooth by Theorem 7.3.1, and is surjective by Theorem 7.4.1.
Moreover,ϕa,bis the identity if and only ifa=b∈Z(G). Hence the mapρG
in (7.4.17) is a Lie group isomorphism, with the smooth structure on the
quotient group (G×G)/Z(G) determined by Theorem 2.9.14. This proves
Corollary 7.4.6.
Example 7.4.7.Theorem 7.4.1 establishes a one-to-one correspondence
between isometriesϕ∈ I0(G) that satisfyϕ(1l) = 1l (the casea=b) and Lie
group automorphisms of G in the identity component. This correspondence
does not extend to other connected components.
For example, in the case G =T
n
=R
n
/Z
n
the group of automorphisms
ofT
n
is the infinite group Aut(T
n
) = GL(n,Z) of integer matrices with
determinant±1, while the group of isometries that fix the origin is the
finite subgroup O(n,Z)⊂GL(n,Z) of orthogonal integer matrices. Also,
if G is not abelian, then the isometry G→G :g7→g
−1
is not a Lie group
automorphism, but a Lie group anti-automorphism.
Exercise 7.4.8.Examine the case G = SU(2) (Example 2.5.21) and deduce
that there exists a Lie group isomorphism
SO(4)
∼
=
Γ
SU(2)×SU(2)
∆
/{±1l}.
Exercise 7.4.9.Let G be a compact Lie group equipped with a bi-invariant
Riemannian metric and letg:= Lie(G). Show that the formula
Θ
(ζ, δ),(ζ
′
, δ
′
)
Λ
:=
ı
δζ
′
−δ
′
ζ,
Θ
δ, δ
′
Λ
+
1
4
ad
ΓΘ
ζ, ζ
′
Λ∆
ȷ
(7.4.18)
forζ, ζ
′
∈gandδ, δ
′
∈Der(g) defines a Lie bracket ong×Der(g). Show
that the map VectK(G)→g×Der(g) :X7→(X(1l),∇X(1l)) identifies the
Lie algebra VectK(G) = Lie(I(G)) of Killing vector fields on G with the
Lie algebrag×Der(g). Show that the homomorphism (7.4.17) induces the
surjective Lie algebra homomorphism
g×g→g×Der(g) : (ξ, η)7→
ı
ξ−η,
1
2
ad(ξ+η)
ȷ
, (7.4.19)
whose kernel consist of all pairs (ξ, η)∈g×gthat satisfyξ=η∈Z(g).
Show that dim(Der(g)) = dim(g)−dim(Z(g)).

368 CHAPTER 7. TOPICS IN GEOMETRY
7.5 Convex Functions on Hadamard Manifolds*
The last two sections of this book are devoted to Donaldson’s beautiful
paper [17] in which he develops a differential geometric approach to Lie
algebra theory. The results of [17] will be explained in reverse order. The
first subsection examines thesphere at infinityof a Hadamard manifoldM
and contains a proof of [17, Theorem 4], which asserts that every convex
functionf:M→Rthat is invariant under the action of a Lie group G
by isometries must attain its minimum whenever the G-action has no fixed
point at infinity (§7.5.1). The next subsection deals with the special case
of [17, Theorem 3], whereMis the manifold of inner products on a vector
spaceVon which a Lie group G⊂SL(V) acts irreducibly (§7.5.2). IfGis
the identity component of the isotropy subgroup of a nonzero vectorw∈W
under a representation of the special linear groupρ: SL(V)→SL(W), then
by [17, Theorem 2] there exists an inner product onVfor which the Lie
algebragof G is symmetric (§7.5.3). This is used in [17, Theorem 1] in
the case whereV=gis a simple Lie algebra,wis the Lie bracket, and G
is the identity component of the group of automorphisms ofg, to establish
the existence of symmetric inner products ongand deduce various standard
results in Lie algebra theory. These applications to Lie algebra theory are
deferred to the next section.
7.5.1 Convex Functions and The Sphere at Infinity
Assume throughout thatMis a Hadamard manifold, i.e. a nonempty, con-
nected, simply connected, complete Riemannian manifold with nonpositive
sectional curvature (Definition 6.5.1). Forp∈Mdenote the unit sphere in
the tangent spaceTpMby
Sp:={v∈TpM| |v|= 1}.
Their union determines a submanifoldSM:={(p, v)|p∈M, v∈Sp}of the
tangent bundle, called theunit sphere bundle.
Definition 7.5.1.Define an equivalence relation∼onSMby
(p, v)∼(q, w)
def
⇐⇒ sup
t≥0
d(exp
p(tv),exp
q(tw))<∞ (7.5.1)
for(p, v),(q, w)∈SM. The equivalence class of a pair(p, v)∈SMwill be
denoted by[p, v] :={(q, w)∈SM|(q, w)∼(p, v)}and the quotient space
S∞(M) :=SM/∼=
Φ
[p, v]

(p, v)∈SM

is called thesphere at infinityofM.

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 369
The following lemma shows that the mapSp→S∞(M) :v7→[p, v] is
a homeomorphism with respect to the quotient topology onS∞(M) for
everyp∈M(see [17, Lemma 3]).exp (tw(R)/R)
qv
p
w(R)/R
q
q
p
v
γ(R)
p exp (tv)
v
Figure 7.1: The sphere at infinity.
Lemma 7.5.2.There exists a unique collection of mapsFq,p:Sp→Sq,
one for each pair of pointsp, q∈M, such that
Fq,p(v) = lim
R→∞
exp
−1
q(exp
p(Rv))
|exp
−1
q(exp
p(Rv))|
(7.5.2)
for allp, q∈Mand allv∈Sp. Moreover, the convergence in(7.5.2)is
uniform onSp, the mapsFq,pare homeomorphisms, and they satisfy
w=Fq,p(v) ⇐⇒ sup
t≥0
d(exp
p(tv),exp
q(tw))<∞ (7.5.3)
for all(p, v),(q, w)∈SMand
Fr,q◦Fq,p=Fr,p, F p,p= id (7.5.4)
for allp, q, r∈M.
Proof.Letp, q∈Mand define the mapsFR,q,p:Sp→Sqby
FR,q,p(v) :=
exp
−1
q(exp
p(Rv))
|exp
−1
q(exp
p(Rv))|
(7.5.5)
forR >0 andv∈Sp. We claim that, for allR > d(p, q) and allv∈Sp,

∂
∂R
FR,q,p(v)

≤
d(p, q)
R(R−d(p.q))
. (7.5.6)

370 CHAPTER 7. TOPICS IN GEOMETRY
To see this, fix an elementv∈Spand define the geodesicγ:R→M
byγ(t) := exp
p(tv). Define the curvew:R→TqMbyw(t) := exp
−1
q(γ(t))
fort∈R. ThenFR,q,p(v) =w(R)/|w(R)|and hence
∂
∂R
FR,q,p(v) =
˙w(R)
|w(R)|
−
ø
˙w(R)
|w(R)|
,
w(R)
|w(R)|
Æ
w(R)
|w(R)|
.
This is the orthogonal projection of the vector ˙w(R)/|w(R)|onto the or-
thogonal complement ofw(R). Hence its length connot decrease by adding
to it a scalar multiple ofw(R), and so

∂
∂R
FR,q,p(v)

≤

˙w(R)
|w(R)|
−
w(R)
R|w(R)|

=
|˙w(R)−w(R)/R|
|w(R)|
. (7.5.7)
Next we use the expanding property of the exponential map in Theorem 6.5.2
twice, namely first the inequality for the derivative in part (ii) at the pointq
and then the inequality in part (iii) at the pointγ(R) (see Figure 7.1).
Define the tangent vectorsvp, vq∈T
γ(R)Mby
vp:=−˙γ(R) =−dexp
q(w(R)) ˙w(R),
vq:=−
d
dt

t=R
exp
q(tw(R)/R) =−dexp
q(w(R))w(R)/R.
Then Theorem 6.5.2 yields the estimate
|˙w(R)−w(R)/R| ≤ |dexp
q(w(R))( ˙w(R)−w(R)/R)|=|vp−vq|.(7.5.8)
Also, exp
γ(R)(svp) = exp
p((R−s)v),exp
γ(R)(svq) = exp
q((R−s)w(R)/R).
Takes=R−tand use Lemma 6.5.5 to obtain, for 0≤t < R,
|vp−vq| ≤
d
Γ
exp
p(tv),exp
q(tw(R)/R)
∆
R−t
≤
d(p, q)
R
. (7.5.9)
Sinced(p, γ(R)) =Randd(q, γ(R)) =|w(R)|, the triangle inequality yields
R−d(p, q)≤ |w(R)| ≤R+d(p, q). (7.5.10)
Combinig the inequalities (7.5.7), (7.5.8), (7.5.9), and (7.5.10) we find that

∂
∂R
FR,q,p(v)

≤
|˙w(R)−w(R)/R|
|w(R)|
≤
|vp−vq|
|w(R)|
≤
d(p, q)
R(R−d(p, q))
.
This proves the estimate (7.5.6).

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 371
It follows from (7.5.6) by integrating from a fixed numberR > d(p, q)
to infinity that the mapsFR,q,p:Sp→Sqconverge to a mapFq,p:Sp→Sq
asRtends to infinity and that
sup
v∈Sp
|Fq,p(v)−FR,q,p(v)| ≤
Z
∞
R
d(p, q)
r(r−d(p, q))
dr= log
`
R
R−d(p, q)
´
for allR > d(p, q). Thus the convergence is uniform and hence the limit
mapFq,pis continuous.
Next we prove (7.5.3). Letp, q∈Mandv∈Sp. Then
w:=Fq,p(v) = lim
R→∞
FR,q,p(v) = lim
R→∞
w(R)
|w(R)|
= lim
R→∞
w(R)
R
.
Here the third equality follows from (7.5.5) and the definition of the vec-
torw(R) = exp
−1
q(exp
p(Rv)), and the last equality follows from (7.5.10).
Hence exp
q(tw) = limR→∞exp
q(tw(R)/R),and so it follows from (7.5.9) by
taking the limitR→ ∞thatd(exp
p(tv),exp
q(tw))≤d(p, q) for allt≥0.
This proves “ =⇒” in (7.5.3). The converse implication follows from the
fact that, by Theorem 6.5.2, there can be at most one tangent vectorw∈Sq
satisfying sup
t>0d(exp
p(tv),exp
p(tw))<∞. Thus we have proved that the
mapsFq,p:Sp→Sqin (7.5.2) satisfy (7.5.3). That they also satisfy (7.5.4)
follows directly from (7.5.3) and the fact that (7.5.1) defines an equivalence
relation onSM. Hence each mapFq,p:Sp→Sqis a homeomorphism with
the inverseFp,q:Sq→Spand this proves Lemma 7.5.2.
Lemma 7.5.3.Ifϕ∈ I(M),p, q∈M, andv∈Sp, then
F
ϕ(q),ϕ(p)◦dϕ(p) =dϕ(q)◦Fq,p:Sp→S
ϕ(q). (7.5.11)
Thus the group of isometries ofMacts continuously on the sphere at infinity
viaI(M)×S∞(M)→S∞(M) : (ϕ,[p, v])7→ϕ∗[p, v] := [ϕ(p), dϕ(p)v].
Proof.Sinceϕis an isometry it satisfiesϕ◦exp
q= exp
ϕ(q)◦dϕ(q) for
allq∈Mby Corollary 5.3.3. Hence
dϕ(q)
ı
exp
−1
q
Γ
exp
p(Rv)
∆
ȷ
= exp
−1
ϕ(q)
ı
ϕ
Γ
exp
p(Rv)
∆
ȷ
= exp
−1
ϕ(q)
ı
exp
ϕ(p)
Γ
R dϕ(p)v
∆
ȷ
and sodϕ(q)◦FR,q,p=F
R,ϕ(q),ϕ(p)◦dϕ(p) for allp, q∈Mand allR >0.
Divide by the norm and take the limitR→ ∞to obtain (7.5.11). This
proves Lemma 7.5.3.

372 CHAPTER 7. TOPICS IN GEOMETRY
Definition 7.5.4.LetG⊂GL(n,R)be a Lie group. AG-action onM
by isometriesis a Lie group homomorphism
G→ I(M) :g7→ϕg,
i.e. the mapG×M→M: (g, p)7→ϕg(p)is a smooth group action (Defi-
nition 2.5.40) and the mapϕg:M→Mis an isometry for eachg∈G. In
this situation we say that theGaction has afixed pointiff there exists an
elementp∈Msuch that
ϕg(p) =p
for allg∈G. We say that theG-action hasa fixed point at infinityiff
the inducedG-action on the sphere at infinity has a fixed point, i.e. there
exist elementsp∈Mandv∈Spsuch that
dϕg(p)v=F
ϕg(p),p(v)
for allg∈G. If such a pair(p, v)∈S(M)does not exist, we say that theG-
action hasno fixed point at infinity.
Definition 7.5.5.A smooth functionf:M→Ris calledconvexiff the
functionf◦γ:R→Ris convex for every geodesicγ:R→M.
With these preparations in place we are ready to state the following exis-
tence theorem for critical points of a convex function (see [17, Theorem 4]).
Theorem 7.5.6(Donaldson).LetMbe a Hadamard manifold equipped
with a smooth actionG→ I(M) :g7→ϕgof a Lie groupGby isometries,
letKbe a compact subgroup ofG, and letf:M→Rbe a convex function
such that
f◦ϕg=f
for allg∈G. Assume that theG-action has no fixed point at infinity. Then
there exists an elementp0∈Msuch that
f(p0) = inf
p∈M
f(p), ϕ u(p0) =p0 for allu∈K. (7.5.12)
As pointed out in [17], similar results can be found in the works of
Bishop–O’Neill [7] and Bridson–Haefliger [11, Lemma 8.26]. We remark
that the compactness of the subgroup K⊂G is only needed for an appeal to
Cartan’s Fixed Point Theorem 6.5.6. If we assume instead that the action
of K onMhas a fixed point, compactness is not required. The proof of
Theorem 7.5.6 is based in the following lemma (see [17, Lemma 5]).

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 373
Lemma 7.5.7.Letpibe a sequence inMsuch thatlimi→∞d(p, pi) =∞
for some (and hence every)p∈M. LetI ⊂ I(M)be a collection of
isometries ofMsuch thatsup
id(pi, ϕ(pi))<∞for allϕ∈ I. Then the
isometries inIhave a common fixed point at infinity, i.e. there exists an
element[p, v]∈S∞(M)such thatϕ∗[p, v] = [p, v]for allϕ∈ I.
Proof.Fix an elementp∈Mand define
vi:=
exp
−1
p(pi)
Ri
, R i:=|exp
−1
p(pi)|=d(p, pi), (7.5.13)
for eachi∈Nsuch thatpi̸=p. Passing to a subsequence, if necessary, we
may assume thatpi̸=pfor alli∈Nand that the limitv:= limi→∞vi∈Sp
exists. We will prove thatdϕ(p)v=F
ϕ(p),p(v) for allϕ∈ I. To see this,
letϕ∈ I, choosec >0 such thatd(pi, ϕ(pi))≤cfor alli, and define
v
′
i:=
exp
−1
p(ϕ(pi))
R
′
i
, R
′
i:=|exp
−1
p(ϕ(pi))|=d(p, ϕ(pi)).
Since exp
p(R
′
i
v
′
i
) =ϕ(pi) and exp
p(Rivi) =pi, Theorem 6.5.2 asserts that
|Rivi−R
′
iv
′
i| ≤d(pi, ϕ(pi))≤c
for alli. Since|Ri−R
′
i
| ≤cby the triangle inequality, it follows that
|vi−v
′
i| ≤
2c
Ri
and hence
lim
i→∞
R
′
i=∞, lim
i→∞
v
′
i= lim
i→∞
vi=v.
Since exp
p(R
′
i
v
′
i
) =ϕ(pi), this implies
Fq,p(v) = lim
i→∞
exp
−1
q(exp
p(R
′
i
v
′
i
))
|exp
−1
q(exp
p(R
′
i
v
′
i
))|
= lim
i→∞
exp
−1
q(ϕ(pi))
|exp
−1
q(ϕ(pi))|
for allq∈M. Takeq=ϕ(p) and use the identity exp
−1
ϕ(p)
◦ϕ=dϕ(p)◦exp
−1
p
and equation (7.5.13) to obtain
F
ϕ(p),p(v) = lim
i→∞
dϕ(p) exp
−1
p(pi)
|dϕ(p) exp
−1
p(pi)|
=dϕ(p) lim
i→∞
vi=dϕ(p)v.
Henceϕ∗[p, v] = [p, v] for allϕ∈ Iand this proves Lemma 7.5.7.

374 CHAPTER 7. TOPICS IN GEOMETRY
The next step in the proof of Theorem 7.5.6 is to examine the gradient
flow of the convex functionf:M→R. The flow equation has the form
˙γ(t) =−∇f(γ(t)), (7.5.14)
where the gradient vector field is defined by⟨∇f(p), v⟩:=df(p)vforp∈M
andv∈TpM. An important consequence of convexity is that the distance
between any two solutions of the gradient flow equation is nonincreasing.
This is the content of the next lemma (see [17, Lemma 6]).
Lemma 7.5.8.Letf:M→Rbe a smooth function. Thenfis convex if
and only if it satisfies the condition
⟨∇v∇f(p), v⟩ ≥0 (7.5.15)
for allp∈Mand allv∈TpM. Iffis convex, then the following holds.
(i)Equation(7.5.14)has a solutionγ: [0,∞)→Mon the entire positive
real axis for every initial conditionγ(0) =p0.
(ii)Letγ0: [0,∞)→Mandγ1: [0,∞)→Mbe two solutions of(7.5.14).
Then the function[0,∞)→R:t7→d(t) :=d(γ0(t), γ1(t))is nonincreasing.
Proof.Letγ:R→Mbe a geodesic. Thenf◦γ:R→Ris convex if and
only if
0≤
d
2
dt
2
f(γ(t)) =
d
dt
⟨∇f(γ(t)),˙γ(t)⟩=

∇
˙γ(t)∇f(γ(t)),˙γ(t)

for allt. This holds for all geodesics if and only iffsatisfies (7.5.15). In the
remainder of the proof we assume thatfis convex.
Letp0∈Mand forT >0 define
KT:={p∈M|d(p0, p)≤T|∇f(p0)|}.
This set is compact becauseMis complete. Now letγ: [0, T)→Mbe the
solution of (7.5.14) withγ(0) =p0on some time interval [0, T). Then
d
dt
|∇f(γ)|
2
= 2⟨∇˙γ∇f(γ),∇f(γ)⟩=−2⟨∇˙γ∇f(γ),˙γ⟩ ≤0
by (7.5.15). Thus the functiont7→ |∇f(γ(t))|=|˙γ(t)|is nonincreasing and
soγ(t)∈KTfor 0≤t < T. SinceKTis a compact subset ofM, the
solutionγextends to a longer time interval [0, T+δ) for someδ >0 by
Corollary 2.4.15. SinceT >0 was chosen arbitrary, this proves (i).

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 375
We prove part (ii). Assume without loss of generality thatγ0(0)̸=γ1(0)
and soγ0(t)̸=γ1(t) for allt. Fort≥0 let [0,1]→M:s7→γ(s, t)
be the unique geodesic that satisfiesγ(0, t) =γ0(t) andγ(1, t) =γ1(t).
Thend(t) =d(γ0(t), γ1(t)) =L(γ(·, t)) =|∂sγ(s, t)|for alls, tand hence
˙
d(t) =
Z
1
0
⟨∂sγ(s, t),∇t∂sγ(s, t)⟩
|∂sγ(s, t)|
ds=
1
d(t)
Z
1
0
∂
∂s
⟨∂sγ(s, t), ∂tγ(s, t)⟩ds
=−
1
d(t)
ı
⟨∂sγ(1, t),∇f(γ(1, t))⟩ − ⟨∂sγ(0, t),∇f(γ(0, t))⟩
ȷ
=−
1
d(t)
Z
1
0
⟨∂sγ(s, t),∇
∂sγ(s,t)f(γ(s, t))⟩ds≤0
by (7.5.15). This proves (ii) and Lemma 7.5.8.
Lemma 7.5.9.Letf:M→Rbe a convex function that has a critical
pointp∞. Thenf(p)≥f(p∞) =:cfor allp∈M, and the setCf:=f
−1
(c)
of minima offis geodesically convex.
Proof.Letp∈Mand letγ: [0,1]→Mbe the unique geodesic with the
endpointsγ(0) =p∞andγ(1) =p. Thenβ:=f◦γ: [0,1]→Ris a convex
function satisfyingβ(0) =f(p∞) =cand
˙
β(0) = 0, henceβ(t)≥cfor allt,
and sof(p) =β(1)≥c. Thusfattains its minimum atp∞.
Now letp0, p1∈Cfand letγ: [0,1]→Mbe the unique geodesic with the
endpointsγ(0) =p0andγ(1) =p1. Then the functionβ:=f◦γ: [0,1]→R
is convex, satisfiesβ(0) =β(1) =c, and takes values in the interval [c,∞).
Henceβ≡cand soγ(t)∈Cffor allt. This proves Lemma 7.5.9.
Proof of Theorem 7.5.6.Choose an elementp0∈Mand letγ: [0,∞)→M
be the unique solution of equation (7.5.14) that satisfies the initial condi-
tionγ(0) =p0. Assume first that
sup
t≥0
d(p0, γ(t)) =∞ (7.5.16)
and choose a sequenceti→ ∞such that limi→∞d(p0, γ(ti)) =∞.Now
letg∈G. Sinceϕg:M→Mis an isometry andf◦ϕg=f, it follows
thatdϕg(p)∇f(p) =∇f(ϕg(p)) for allp∈M(Exercise:Prove this.) Hence
the curve [0,∞)→M:t7→ϕg(γ(t)) is another solution of equation (7.5.14)
and henced(γ(ti), ϕg(γ(ti)))≤d(p0, ϕg(p0)) for alliand allg∈G by part (ii)
of Lemma 7.5.8. Hence Lemma 7.5.7 asserts that there exists a (p, v)∈SM
such thatdϕg(p)v=F
ϕg(p),p(v) for allg∈G, in contracdiction to our as-
sumption that the G-action has no fixed point at infinity.

376 CHAPTER 7. TOPICS IN GEOMETRY
This shows that our assumption (7.5.16) must have been wrong. Thus
sup
t≥0
d(p0, γ(t)) =:R <∞,
and so our solutionγ: [0,∞)→Mof (7.5.14) takes values in the compact
setB:={p∈M|d(p0, p)≤R}. Since the functiont7→f(γ(t)) is non-
increasing, this implies that the limit
c:= lim
t→∞
f(γ(t))≥min
p∈B
f(p) (7.5.17)
exists and is a real number (and not−∞). Since
d
dt
f(γ(t)) =−|∇f(γ(t))|
2
and the functiont7→f(γ(t)) is bounded below byc, there must exist a
sequenceti→ ∞such that
lim
i→∞
∇f(γ(ti)) = 0. (7.5.18)
Sinceγ(ti)∈Bfor alli, we may also assume that the limit
p∞:= lim
i→∞
γ(ti) (7.5.19)
exists (after passing to a subsequence, if necessary). This limit is a critical
point offby (7.5.18), andf(p∞) =cby (7.5.17). Hence, by Lemma 7.5.9,f
attains its minimum atp∞and the setCf:={p∈M|f(p) =c}of minima
offis geodesically convex. We must find an element ofCfthat is fixed
under the action of K. By Cartan’s Fixed Point Theorem 6.5.6 there exists
aq∈Msuch thatϕu(q) =qfor allu∈K. SinceMis complete andCfis
a nonempty closed subset ofM, there exists an elementp0∈Cfsuch that
d(q, p0) = inf
p∈Cf
d(q, p) =:δ. (7.5.20)
We claim thatϕu(p0) =p0for allu∈K. To see this, fix an elementu∈K,
letγ: [0,1]→Mbe the geodesic joiningγ(0) =p0toγ(1) =ϕu(p0), and
denote bym:=γ(1/2) the midpoint of this geodesic. Then Lemma 6.5.7
asserts that
2d(q, m)
2
+
d(p0, ϕu(p0))
2
2
≤d(q, p0)
2
+d(q, ϕu(p0))
2
. (7.5.21)
Sinceϕu(q) =qandϕuis an isometry, we haved(q, ϕu(p0)) =d(q, p0) =δ,
and sinceCfis geodesically convex, we havem∈Cfand sod(q, m)≥δ.
Hence it follows from (7.5.21) thatp0=ϕu(p0). This shows thatp0∈Cfis
a fixed point for the action of K onMand proves Theoren 7.5.6.

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 377
Example 7.5.10.To illustrate the argument in the proof of Theorem 7.5.6,
takeM=Cwith the standard flat metric. Then the equivalence relation
on the sphere bundleSM=C×S
1
is given by translation inCand so the
sphere at infinity isS∞(M) =S
1
. The orthogonal group G = O(2) acts by
isometries onMand has no fixed point at infinity. The subgroup K =Z/2Z
acts by complex conjugation and its fixed point set is the real axis. Choose
a smooth convex functionh:R→Rthat vanishes on the interval [−1,1]
and is positive elsewhere. Then the functionf(z) :=h(|z|) is convex and G-
invariant and the setCfof minima offis the closed unit disc inC. Ifq= 2
is the fixed point of the K-action chosen in the proof, thenp0= 1∈Cf.
If G = K =Z/2Z, then±1 are the fixed points at infinity andf(z) :=e
Re(z)
is convex and G-invariant, but does not take on its infimum.
Example 7.5.11([17]).Consider the case whereM=D
m
is the Poincar´e
model of hyperbolic space (Exercise 6.4.22). Then the sphere at infinity
is the boundary∂D
m
=S
m−1
(Exercise 6.4.24). If the convex functionf
extends continuously to the closed ball and does not take on its minimum
inM, then it attains its minimum at a unique point on the boundary,
because any two boundary points are the asymptotic limits of a geodesic
inM. Hence the minimum on the boundary is fixed under the action of any
Lie group onMby isometries, that leavefinvariant. This is reminiscent of
the Kempf Uniqueness Theorem in GIT (see [37] and [20, Theorem 10.2]),
whereM= G/K is associated to the complexification G of a compact Lie
group K andfis the Kempf–Ness function (see [38, 53] and [20,§4]).
7.5.2 Inner Products and Weighted Flags
We will now turn to a specific example, where the Hadamard manifold is the
space of positive definite symmetric matrices with determinant one (§6.5.3).
Following [17], we choose a finite-dimensional real vector spaceVequipped
with a fixed inner product⟨·,·⟩. Then every inner product onVhas the
form⟨v, v
′
⟩P:=⟨v, P
−1
v
′
⟩for some self-adjoint positive definite automor-
phismP. Denote the set of such automorphisms with determinant one by
P0(V) :=
Φ
P∈End(V)

P
∗
=P >0,det(P) = 1

. (7.5.22)
HereP
∗
∈End(V) is defined by⟨v, P
∗
v
′
⟩:=⟨P v, v
′
⟩forv, v
′
∈V, and the
notation “P >0” means⟨v, P v⟩>0 for allv∈V\ {0}. ThusP0(V) is a
codimension-1 submanifold of the space of self-adjoint endomorphisms ofV.
Its tangent space atP∈P0(V) is given by
TPP0(V) :=
Φ
b
P∈End(V)

b
P=
b
P
∗
,trace
Γ
b
P P
−1
∆
= 0

. (7.5.23)

378 CHAPTER 7. TOPICS IN GEOMETRY
The Riemannian metric onP0(V) is defined by
|
b
P|P:=
q
trace
Γ
b
P P
−1b
P P
−1
∆
(7.5.24)
forP∈P0(V) and
b
P∈TPP0(V) as in§6.5.3, and soP0(V) is a Hadamard
manifold by Theorem 6.5.10. For
b
P∈TPP0(V) the endomorphism
b
P P
−1
is self-adjoint with respect to the inner product⟨·, P
−1
·⟩onV.
The group SL(V)⊂GL(V) of automorphisms ofVwith determinant
one acts onP0(V) by the isometriesϕg(P) :=gP g
∗
forg∈SL(V). The
isotropy subgroup of 1l∈P0(V) is the special orthogonal group SO(V).
The action of a subgroup G⊂SL(V) onVis calledirreducibleiff there
does not exist a linear subspaceE⊂V, other thanE={0}andE=V,
such thatgE=Efor allg∈G. This notion can be used to carry over the
general existence theorem in§7.5.1 for critical points of a convex function
to the present setting (see [17, Theorem 3]).
Theorem 7.5.12(Donaldson).LetG⊂SL(V)be a Lie subgroup such
that the action ofGonVis irreducible. LetK⊂Gbe a compact subgroup
and letf:P0(V)→Rbe a convex function such thatf(gP g
∗
) =f(P)for
allg∈Gand allP∈P0(V). Then there exists aP0∈P0(V)such that
f(P0) = inf
P∈P0(V)
f(P), uP 0u
∗
=P0for allu∈K. (7.5.25)
The goal will be to deduce Theorem 7.5.12 from Theorem 7.5.6. Thus
we must understand the sphere at infinity of the spaceP0(V). This will be
accomplished with the help of the following definition.
Definition 7.5.13.Aweighted flaginVis a pair(F, µ), whereFis a
finite sequence of linear suspaces
{0}=F0⊂F1⊂F2⊂ · · · ⊂Fr=V
such thatni:= dim(Fi)/dim(Fi−1)>0fori= 1, . . . , r, andµis a finite
sequence of real numbersµ1> µ2>· · ·> µrsatisfying the conditions
r
X
i=1
niµi= 0,
r
X
i=1
niµ
2
i= 1. (7.5.26)
LetF=F(V)be the set of weighted flags. The groupSL(V)acts onF(V)
byg·(F, µ) := (gFi, µi)ifor(F, µ) = (Fi, µi)i∈F(V)andg∈SL(V).

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 379
ForP∈P0(V) the unit sphere in the tangent spaceTPP0(V) is the set
SP:=
Φ
b
P=
b
P
∗
∈End(V)

trace
Γ
b
P P
−1
∆
= 0,trace
Γ
b
P P
−1b
P P
−1
∆
= 1

.
LetP∈P0(V) and
b
P∈SP. Then the endomorphism
b
P P
−1
is self-adjoint
with respect to the inner product⟨·, P
−1
·⟩and hence has only real eigenval-
uesµ1> µ2>· · ·> µr. For eachiletEi⊂Vbe the eigenspace forµiand
defineni:= dim(Ei). Since trace
Γ
b
P P
−1
∆
= 0 and trace
Γ
b
P P
−1b
P P
−1
∆
= 1,
theµi, nisatisfy (7.5.26). Theweighted flag of(P,
b
P) is defined by
ιP(
b
P) := (F, µ) =
Γ
(F1, µ1), . . . ,(Fr, µr)
∆
∈F(V), (7.5.27)
whereFi:=E1⊕E2⊕ · · · ⊕Eifori= 1, . . . , r. For eachP∈P0(V) the
mapιP:SP→F(V) defined by (7.5.27) is bijective, and thus induces
a (compact, metrizable) topology onF(V). This topology is independent
ofPas the next lemma shows. The lemma also shows that the space of
weighted flags is the sphere at infinity (see [17, Lemma 4]).
Lemma 7.5.14.The equivalence relation in Definition 7.5.1 on the unit
sphere bundleSP0(V)is given by
(P,
b
P)∼(Q,
b
Q) ⇐⇒ ιP(
b
P) =ιQ(
b
Q) (7.5.28)
forP, Q∈P0(V),
b
P∈SP, and
b
Q∈SQ. Thus the mapFQ,P:SP→SQ
as defined in Lemma 7.5.2 is given byFQ,P=ιQ◦ι
−1
P
. Moreover, the
mapSP0(V)→F(V) : (P,
b
P)7→ιp(
b
P)isSL(V)-equivariant, i.e.
ιgP g
∗(g
b
P g
∗
) =g·ιP(
b
P) (7.5.29)
for allP∈P0(V), all
b
P∈SP, and allg∈SL(V).
Proof.Since (g
b
P g
∗
)(gP g
∗
)
−1
=g(
b
P P
−1
)g
−1
for all
b
P∈SPandg∈SL(V),
equation (7.5.29) follows directly from the definitions. The proof that the
equivalence relation satisfies (7.5.28) rests on the following claims.
Claim 1.LetS∈S1l, let(Fi, µi)
r
i=1
=ι1l(S)be the flag associated toS,
and leth∈SL(V)be an automorphism ofVwith determinant one such that
hFi=Fifori= 1, . . . , r. (7.5.30)
Then(1l, S)∼(hh
∗
, hSh
∗
).
Claim 2.Letg∈SL(V)and fix any flag{0}=F0⊊F1⊊· · ·⊊Fr=V.
Then there exist elementsh∈SL(V)andu∈SO(V)such thathsatis-
fies(7.5.30)andg=hu.

380 CHAPTER 7. TOPICS IN GEOMETRY
We prove, that these two claims imply (7.5.28). Fix any elementS∈S1l,
let (Fi, µi)
r
i=1
=ι1l(S) be the weighted flag ofS, and letP∈P0(V). Choose
an elementg∈SL(V) such thatgg
∗
=P, and chooseuandhas in Claim 2.
Thenhh
∗
=P, and the pairs (P,
b
P) := (hh
∗
, hSh
∗
) and (1l, S) have the
same flag by (7.5.30) and are equivalent by Claim 1. Hence any pair (P,
b
P)
is equivalent to (1l, S) if and only if it has the same flag. By transitivity of
the equivalence relation we deduce that (7.5.28) holds for allP, Q∈P0(V).
We prove Claim 2. LetF
′
i
:=g
−1
Fiandmi:= dim(Fi) fori= 1, . . . , r.
Then choose orthonormal basese1, . . . , emande
′
1
, . . . , e
′
mofVsuch that for,
eachi, the vectorse1, . . . , emi
form a basis ofFiand the vectorse
′
1
, . . . , e
′
mi
form a basis ofF
′
i
. Define the orthogonal transformationubyue
′
i
:=ei
fori= 1, . . . , m. It satisfiesF
′
i
=u
−1
Fiand hencegu
−1
Fi=Fifor alli.
Thush:=gu
−1
satisfies the requirements of Claim 2.
We prove Claim 1, following [17, Lemma 4]. Define the geodesicsγ0, γ1
inP0(V) byγ0(t) = exp(tS) andγ1(t) =hexp(tS)h
∗
(see Lemma 6.5.18).
By equation (6.5.12) the square of their distance is given by
ρ(t) :=d(exp(tS), hexp(tS)h
∗
)
2
= trace
ı
Γ
log (M(t)M(t)
∗
)
∆
2
ȷ
,
M(t) := exp
Γ
−tS/2
∆
hexp
Γ
tS/2
∆
.
(7.5.31)
In the eigenspace decompositionV=E1⊕ · · · ⊕Erof the self-adjoint endo-
morphismSthe automorphismshand exp(tS/2) have the form
h=






h11h12· · ·h1r
0h22
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.hr−1,r
0· · ·0hrr






,
exp(tS/2) = diag
Γ
e
tµ1/2
1lE1
, e
tµ2/2
1lE2
, . . . , e
tµr/2
1lEr
∆
.
(7.5.32)
Here the upper triangular form ofhfollows from (7.5.30). Hence
M(t) =






h11h12(t)· · ·h1r(t)
0h22
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.hr−1,r(t)
0· · ·0 hrr






, (7.5.33)
wherehij(t) :=e
−t(µi−µj)/2
hijfor 1≤i < j≤r. Sinceµi> µjfori < j, it
follows that the limitM∞:= limt→∞M(t) = diag(h11, . . . , hrr) exists and
is an invertible endomorphism ofV. Hence the functionρ: [0,∞)→R
in (7.5.31) is bounded. This proves Claim 1 and Lemma 7.5.14.

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 381
Proof of Theorem 7.5.12.Let G⊂SL(V) be a Lie subgroup which acts ir-
reducibly onV. Then G acts onP0(V) by isometries. By Lemma 7.5.14
the induced action on the sphere at infinityS∞(P0(V))
∼
=F(V) is given
by G×F(V)→F(V) : (g,(Fi, µi)i=1,...,r)7→(gFi, µi)i=1,...,r.This action
has no fixed points because the action of G onVis irreducible andr≥2
for each weighted flag (Fi, µi)i=1,...,r∈F(V). Hence all the assertions of
Theorem 7.5.12 follow directly from Theorem 7.5.6 withM=P0(V).
7.5.3 Lengths of Vectors
The material in this section goes back to ideas in geometric invariant theory
developed by Kempf–Ness [38], Ness [53], and Kirwan [40] in the complex
setting and by Richardson–Slodowy [59] and Marian [48] in the real setting.
We assume throughout thatV, Ware finite-dimensional real vector spaces
andρ: SL(V)→SL(W) is a Lie group homomorphism. Note that every
Lie group homomorphism from GL(V) to GL(W) restricts to a Lie group
homomorphism from SL(V) to SL(W), because every Lie group homomor-
phism from GL(V) to the multiplicative group of nonzero real numbers is
some power of the determinant.
Fix a nonzero vectorw∈Wand denote by Gw⊂SL(V) the connected
component of the identity in the isotropy subgroup ofw, i.e.
Gw:=



g∈SL(V)

∃a smooth pathγ: [0,1]→SL(V)
such thatγ(0) = 1l, γ(1) =g,and
ρ(γ(t))w=wfor 0≤t≤1



.(7.5.34)
By Theorem 2.5.27 this is a Lie subgroup of SL(V) with the Lie algebra
gw:=
Φ
ξ∈sl(V)

˙ρ(ξ)w= 0

.
There are many examples of this setup that are related to interesting ques-
tions in geometry. The vector spaceWcan be the space of all symmetric
bilinear forms onVandwcan be an inner product, in which case Gwis
the special orthogonal group associated to the inner product, orwcan be
the quadratic form (6.4.9), in which case Gwis the identity component of
the isometry group of hyperbolic space. OrWcan be the space of skew-
symmetric bilinear forms onVandwa symplectic form, in which case Gw
is the symplectic linear group. OrWcan be the space of skew-symmetric
bilinear maps onVwith values inV. Thenwcan be a cross product in
dimension three or seven, orwcan be the Lie bracket of a Lie algebrag=V
and then Gwis the identity component in the group of automorphisms ofg.
The latter example will be examined in detail in§7.6.2. Of particular inter-
est are the cases where the group Gwis noncompact.

382 CHAPTER 7. TOPICS IN GEOMETRY
Definition 7.5.15.An inner product⟨·,·⟩onVis called(ρ, w)-symmetric
iff the Lie subalgebragw⊂sl(V)is invariant under the involutionA7→A
∗
,
defined by⟨v, A
∗
v
′
⟩:=⟨Av, v
′
⟩forv, v
′
∈V.
Exercise 7.5.16.Let⟨·,·⟩be a (ρ, w)-symmetric inner product onV. Prove
thatg∈Gwimpliesg
∗
∈Gw.Hint:Choose a smooth pathg: [0,1]→Gw
with the endpointsg(0) = 1l andg(1) =g, and defineξ(t) :=g(t)
−1
˙g(t).
Show that the initial value problem
˙
h(t) =ξ(t)
∗
h(t),h(0) = 1, has a unique
solutionh: [0,1]→Gw(Exercise 2.5.36) and thath(t) =g(t)
∗
for allt.
The following theorem asserts the existence of a (ρ, w)-symmetric inner
product onVunder an irreducibility assumption (see [17, Theorem 2]).
Theorem 7.5.17.Assume that the groupGwin(7.5.34)acts irreducibly
onV. Then there exists a(ρ, w)-symmetric inner product onVwith the
following properties. The subgroup
Kw:= Gw∩SO(V)
is connected and is a maximal compact subgroup ofGw. Moreover, every
compact subgroup ofGwis conjugate inGwto a Lie subgroup ofKw. Thus,
ifKis any maximal compact subgroup ofGw, there exists an elementh∈Gw
such thatK =hKwh
−1
.
Proof.See Lemma 7.5.23.
Example 7.5.18.The hypothesis that the group Gwacts irreducibly onV
cannot be removed in Theorem 7.5.17. Consider the case whereW=V
has dimension at least two, the homomorphismρ: SL(V)→SL(V) is the
identity, andw∈Vis any nonzero vector. Then the one-dimensional linear
subspaceRw⊂Vis evidently invariant under the action of Gw, and there
does not exist any (ρ, w)-symmetric inner product onV.
To begin with, we will fix any inner product⟨·,·⟩VonVand define
the spaceP0(V) of self-adjoint positive definite automorphisms ofVwith
determinant one in terms of this fixed inner product. We will then use
Theorem 7.5.12 to find an elementP∈P0(V) such that the inner product

v, v
′

V,P
:=

v, P
−1
v
′

V
(7.5.35)
onVsatisfies the requirements of Theorem 7.5.17. The proof is based on
three lemmas. The fourth lemma restates the theorem in a modified form.

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 383
Lemma 7.5.19.There exists an inner product⟨·,·⟩WonWsuch that
ρ(SO(V))⊂SO(W)and˙ρ(A
∗
) = ˙ρ(A)
∗
for allA∈sl(V).
Proof.The proof follows the argument in [17,§3]. Assume without loss
of generality thatV=R
m
is equipped with the standard inner product,
and thatW=R
n
and thatρ: SL(m,R)→SL(n,R) is a Lie group homo-
morphism. We prove first that there exists a unique Lie group homomor-
phismρ
c
: SL(m,C)→SL(n,C) (thecomplexification ofρ) such that
ρ
c
|
SL(m,R)
=ρ, ˙ρ
c
(A+iB) = ˙ρ(A) +i˙ρ(B) (7.5.36)
for allA, B∈sl(m,R). Since SL(m,C) is connected, we can defineρ
c
(g)
forg∈GL(m,C) by choosing a smoth pathα: [0,1]→SL(m,C) with the
endpointsα(0) = 1lmandα(1) =g, and taking
ρ
c
(g) :=β(1), β(s)
−1˙
β(s) = ˙ρ
c
Γ
α(s)
−1
˙α(s)
∆
, β(0) = 1ln.(7.5.37)
To verify thatβ(1) is independent of the choice ofα, one can choose a smooth
map [0,1]
2
→SL(m,C) : (s, t)7→α(s, t) satisfyingα(0, t) = 1lm,α(1, t) =g,
defineS:=α
−1
∂sαandT:=α
−1
∂tα, and defineβ: [0,1]
2
→SL(n,C)
as the solution of the initial value problemβ
−1
∂sβ= ˙ρ
c
(S),β(0, t) = 1ln.
Since ˙ρ
c
is a Lie algebra homomorphism and∂tS−∂sT= [S, T], it follows
thatβ
−1
∂tβ= ˙ρ
c
(T) and soβ(1, t) is independent oft. Moreover, SL(m,C)
retracts onto SU(m) by polar decomposition and so is simply connected by a
standard homotopy argument. That the mapρ
c
: SL(m,C)→SL(n,C) thus
defined is smooth follows from the smooth dependence of solutions on the
parameter in a smooth family of differential equations. That it is a group
homomorphism follows by catenation of paths, and that it satisfies (7.5.36)
follows directly from the definition.
Now consider the action of the compact subgroup SU(m)⊂SL(m,C) on
the Hadamard manifoldQ0of positive definite Hermitiann×n-matricesQ
of determinant one (Remark 6.5.21) by (g, Q)7→ρ
c
(g)Qρ
c
(g)
∗
. This action
is by isometries and hence, by Cartan’s Fixed Point Theorem 6.5.6, there
exists an elementQ0∈Q0such thatρ
c
(g)Q0ρ
c
(g)
∗
=Q0for allg∈SU(m).
Differentiate this equation atg= 1lmto obtain ˙ρ
c
(B)Q0+Q0˙ρ
c
(B)
∗
= 0 for
every skew-Hermitian matrixB=−B
∗
∈sl(m,C) with trace zero. Now
letA∈sl(m,R), defineR:=
1
2
(A−A
T
),S:=
1
2
(A+A
T
), and takeB=R
andB=iS. Then ˙ρ(R)Q0+Q0˙ρ(R)
T
= 0 and ˙ρ(S)Q0=Q0˙ρ(S)
T
. Hence
the positive definite symmetricn×n-matrixQ:= Re(Q0) satisfies
˙ρ
Γ
A
T
∆
= ˙ρ
Γ
−R+S
∆
=Q˙ρ(R+S)
T
Q
−1
=Q˙ρ(A)
T
Q
−1
for allA∈sl(m,R). This shows that the inner product⟨w, w
′
⟩:=w
T
Q
−1
w
′
onR
n
satisfies the requirements of Lemma 7.5.19.

384 CHAPTER 7. TOPICS IN GEOMETRY
In the remainder of this subsection we will fix an inner product onW
as in Lemma 7.5.19. Recall also that we have already chosen a nonzero
vectorw∈W. The norm squared of this vector determines a function on
the space of inner products onV(see [17, Lemma 1]).
Lemma 7.5.20.Define the functionfw:P0(V)→Rby
fw(P) =

w, ρ(P
−1
)w

W
(7.5.38)
forP∈P0(V). Then an elementP∈P0(V)is a critical point offwif
and only if⟨˙ρ(A)w, ρ(P
−1
)w⟩W= 0for allA∈sl(V). Moreover, ifPis a
critical point offw, then the inner product(7.5.35)onVis(ρ, w)-symmetric.
Proof.Fix an elementP∈P0(V) and a tangent vector
b
P∈TpP0(V).
Then it follows from Lemma 7.5.19 that
dfw(P)
b
P=−
D
w,˙ρ(P
−1b
P)ρ(P
−1
)w
E
W
=−
D
˙ρ(
b
P P
−1
)w, ρ(P
−1
)w
E
W
.
Now letA∈sl(V) and take
b
P:=AP+P A
∗
to obtain
dfw(P)(AP+P A
∗
) =−

˙ρ(A+P A
∗
P
−1
)w, ρ(P
−1
)w

W
=−2

˙ρ(A)w, ρ(P
−1
)w

W
.
(7.5.39)
ThusPis a critical point offwif and only if the right hand side of (7.5.39)
vanishes for allA∈sl(V). To prove the last assertion, define the norm
|w
′
|W,P:=
p
⟨w
′
, ρ(P
−1
)w
′
⟩W
forw
′
∈W. Now letP∈P0(g) be a critical point offw, letξ∈sl(V), and
takeA:= [ξ, P ξ
∗
P
−1
]∈sl(V) in (7.5.39). Then
0 =

˙ρ([ξ, P ξ
∗
P
−1
])w, ρ(P
−1
)w

W
=|˙ρ(P ξ
∗
P
−1
)w|
2
W,P− |˙ρ(ξ)w|
2
W,P.
Ifξ∈gw, then ˙ρ(ξ)w= 0, hence ˙ρ(P ξ
∗
P
−1
)w= 0, and henceP ξ
∗
P
−1
∈gw.
But the endomorphismP ξ
∗
P
−1
is the adjoint ofξwith respect to the inner
product (7.5.35) onVand this proves Lemma 7.5.20.
Example 7.5.21.This example shows that the (ρ, w)-symmetry of the
inner product (7.5.35) does not imply thatPis a critical point offw.
TakeW=V×Vand letρ: SL(V)→SL(W) be the diagonal action. As-
sume dim(V) = 2 and choosew= (u, v)∈Wsuch thatu, v∈Vare linearly
independent. Then Gw={1l}and so every inner product onVis (ρ, w)-
symmetric (and the assertions of Theorem 7.5.17 are satisfied), however, the
functionfw(P) =⟨u, P
−1
u⟩V+⟨v, P
−1
v⟩Vdoes not have any critical point.

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 385
LetMw⊂P0(V) be the set of minima of the functionfw:P0(V)→R
in Lemma 7.5.20 and let Crit(fw)⊂P0(V) be its set of critical points. The
next result shows thatfwis convex and that Gwacts transitively onMw
whenever this set is nonempty (see [17, Lemma 2]).
Lemma 7.5.22.The functionfwhas the following properties.
(i)fwis convex andGw-invariant, and thusMw= Crit(fw).
(ii)Fix two elementsP0∈MwandP∈P0(V). ThenP∈Mwif and only if
there exists an elementη∈gwsuch thatη=P0η
∗
P
−1
0
andexp(η) =P P
−1
0
,
or equivalentlyρ(P P
−1
0
)w=w.
(iii)The groupGwacts transitively onMw.
Proof.We prove part (i). Letγ:R→P0(V) be a geodesic and define
P:=γ(0),
b
P:= ˙γ(0), S:=P
−1/2b
P P
−1/2
.
Then, by Lemma 6.5.18,γ(t) =P
1/2
exp(tS)P
1/2
and hence
fw(γ(t)) =⟨ρ(P
−1/2
)w,exp(−t˙ρ(S))ρ(P
−1/2
)w⟩W.
This implies
d
2
dt
2
fw(γ(t)) =⟨˙ρ(S)ρ(P
−1/2
)w,exp(−t˙ρ(S)) ˙ρ(S)ρ(P
−1/2
)w⟩W≥0
for alltand hencefwis convex. HenceMw= Crit(fw) by Lemma 7.5.9.
Thatfwis Gw-invariant follows directly from the definition. This proves (i).
We prove part (ii). Ifη∈gwsatisfiesη=P0η
∗
P
−1
0
, exp(η) =P P
−1
0
,
thenρ(P P
−1
0
)w=w. Ifρ(P P
−1
0
)w=w, thenρ(P
−1
)w=ρ(P
−1
0
)w,hence
⟨˙ρ(A)w, ρ(P
−1
)w⟩W=⟨˙ρ(A)w, ρ(P
−1
0
)w⟩W= 0
for allA∈sl(V), and henceP∈Mwby Lemma 7.5.20 and part (i). Now
assumeP∈Mwand letγ: [0,1]→P0(V) be the unique geodesic with the
endpointsγ(0) =P0andγ(1) =P∈Mw. Thenγ(t)∈Mwfor alltby
Lemma 7.5.9. Hence⟨˙ρ(η)w, ρ(γ(t)
−1
)w⟩W= 0 for alltand allη∈sl(V),
by Lemma 7.5.20. Differentiate this equation att= 0 to obtain
⟨˙ρ(η)w, ρ(P
−1
0
) ˙ρ(
b
P P
−1
0
)w⟩W= 0,
b
P:= ˙γ(0)∈TP0
P0(V).
Takeη:=
b
P P
−1
0
to obtain ˙ρ(η)w= 0, and thusη∈gwandη=P0η
∗
P
−1
0
.
By Lemma 6.5.18 we also haveP=γ(1) = exp(
b
P P
−1
0
)P0= exp(η)P0and
this proves (ii).
We prove part (iii). LetP0, P∈Mw, choose an elementη∈gwas in (ii)
so thatηP0=P0η
∗
andP= exp(η)P0, and defineh:= exp(η/2)∈Gwto
obtainP=hP0h
∗
. This proves (iii) and Lemma 7.5.22.

386 CHAPTER 7. TOPICS IN GEOMETRY
With these preparations we are ready to prove Theorem 7.5.17.
Lemma 7.5.23. IfGwacts irreducibly onV, thenMw̸=∅.
(ii)IfP∈Mw, then the inner product⟨·, P
−1
·⟩VonVsatisfies all the
requirements of Theorem 7.5.17.
Proof.The functionfwis convex and Gw-invariant by Lemma 7.5.22. Hence
part (i) follows from Theorem 7.5.12. To prove part (ii), assume thatPis
a critical point offw. Then, by Lemma 7.5.20, the inner product⟨·, P
−1
·⟩V
is (ρ, w)-symmetric. We prove the remaining assertions in four steps. Define
KP:= Gw∩SO(V,⟨·, P
−1
·⟩V) =
Φ
u∈Gw|uP u
∗
P
−1
= 1l

.
Step 1.LetK⊂Gwbe any compact subgroup. Then there exists an ele-
menth∈Gwsuch thath
−1
Kh⊂KP.
By Theorem 7.5.12, there exists aP0∈P0(V) such thatfw(P0) = inffw
anduP0u
∗
=P0for allu∈K. By Lemma 7.5.22, there exists an ele-
menth∈Gwsuch thatP0=hP h
∗
. HenceuhP h
∗
u
∗
=hP h
∗
for allu∈K,
and hence the automorphismh
−1
uhis orthogonal with respect to the inner
product⟨·, P
−1
·⟩Vfor everyu∈K.
Step 2.KPis a compact connected subgroup ofGw.
By the Closed Subgroup Theorem 2.5.27 KPis a closed, and hence compact,
subgroup of SO(V,⟨·, P
−1
·⟩V) and so is a compact Lie subgroup of Gw. We
prove that KPis connected. Letu∈KP⊂Gw. Since Gwis connected,
there exists a smooth pathg: [0,1]→Gwsuch thatg(0) = 1l andg(1) =u.
Thus, by Exercise 7.5.16, we haveg(t)P g(t)
∗
P
−1
∈Gwfor allt. Hence, by
part (ii) of Lemma 7.5.22, there exists a smooth pathη: [0,1]→gwsuch
thatη(t) =P η(t)
∗
P
−1
, exp(η(t)) =g(t)P g(t)
∗
P
−1
,andη(0) =η(1) = 0.
Henceu(t) := exp(−η(t)/2)g(t) is a path in KPjoiningu(0) = 1l tou(1) =u.
Step 3.KPis a maximal compact subgroup ofGw.
Let K⊂Gwbe a compact subgroup containing KP. Then by Step 1 there
exists anh∈Gwsuch thath
−1
Kh⊂KP.Hence KP⊂K⊂hKPh
−1
and
so Lie(KP)⊂Lie(K)⊂hLie(KP)h
−1
.Since Lie(KP) andhLie(KP)h
−1
have the same dimension, this implies Lie(KP) =hLie(KP)h
−1
.Since KP
andhKPh
−1
are connected, this implies KP=hKPh
−1
and so KP= K.
Step 4.LetKbe a maximal compact subgroup ofGw. Then there exists an
elementh∈Gwsuch thatK =hKPh
−1
.
By Step 1 there exists anh∈Gwsuch thath
−1
Kh⊂KP, thus K⊂hKPh
−1
and so K =hKPh
−1
, because K is a maximal compact subgroup of Gw. This
proves Step 4, Lemma 7.5.23, and Theorem 7.5.17.

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 387
Lemma 7.5.23 shows that all minima of the functionfwin Lemma 7.5.20
give rise to inner products that satisfy the requirements of Theorem 7.5.17.
The next lemma shows that the set of minima offwis a Hadamard manifold.
Lemma 7.5.24.The setMwis a geodesically convex and totally geodesic
submanifold ofP0(V). Hence, if it is nonempty, it is a Hadamard manifold
and a symmetric space.
Proof.By Lemma 7.5.22 the functionfwis convex, and soMwis geodesi-
cally convex by Lemma 7.5.9. Now assumeMwis nonempty and fix any
elementP0∈Mw. Then the exponential map
TP0
P0(V)→P0(V) :
b
P7→exp(
b
P P
−1
0
)P0
is a diffeomorphism (Theorem 6.5.10 and Lemma 6.5.18) andMwis the
image of the linear subspace{
b
P∈TP0
P0(g)|
b
P P
−1
0
∈gw}under this diffeo-
morphism (Lemma 7.5.22). SinceP0can be chosen to be any element ofMw,
this shows thatMwis a totally geodesic submanifold ofP0(V). Moreover,
the isometryP0(V)→P0(V) :P7→ϕ0(P) =P0P
−1
P0in Step 2 of the
proof of Theorem 6.5.10 satisfies
ϕ0
ı
exp
Γ
b
P P
−1
0
∆
P0
ȷ
= exp
Γ
−
b
P P
−1
0
∆
P0
for all
b
P∈TPP0(V) and so restricts to an isometry ofMw. HenceMwis a
symmetric space and this proves Lemma 7.5.24.
We emphasize that Lemma 7.5.24 does not require the hypothesis that
the group Gwacts irreducibly onV. This hypothesis was only used to prove
that the spaceMwis nonempty. The next example shows thatfwcan have
critical points in cases where Gwacts reducibly onV.
Example 7.5.25.LetV=R
2
, letW=S⊂R
2×2
be the space of symmet-
ric matrices, equipped with the standard inner product and the standard
actionS7→gSg
T
of SL(2,R), and letw=S:= diag(1,−1)∈Sso that
GS=
æ`
a b
b a
´

a >0, a
2
−b
2
= 1
œ
.
The action of GSonR
2
is reducible, because the diagonal inR
2
is GS-
invariant, however, the functionfS(P) = trace(SP
−1
SP
−1
) attains its min-
imum on the setMS= GS⊂P0(R
2
), corresponding to the symmetric in-
ner products onR
2
. If one modifies this example by takingW=S×S
andw= (1l, S), thenfw(P) = trace(P
−2
+SP
−1
SP
−1
) has a unique critical
point atP= 1l, the group Gw={1l}acts reducibly onV, and the assertions
of Theorem 7.5.17 are trivially satisfied for every inner product onV.

388 CHAPTER 7. TOPICS IN GEOMETRY
Example 7.5.21 and Example 7.5.25 show that, in general, one cannot
expect there to be a one-to-one correspondence between the minima offw
and the (ρ, w)-symmetric inner products onV. However, we shall see below
that such a one-to-one correspondence does exist in many cases.
Remark 7.5.26.Once it is known, that the functionfw:P0(V)→Rhas
a critical pointP0∈P0(V), we can modify the entire setup as follows.
Replace the inner product onVby⟨·,·⟩V,0:=⟨·, P
−1
0
·⟩Vand the inner
product onWby⟨·,·⟩W,0:=⟨·, ρ(P
−1
0
)·⟩W.This pair of inner products sat-
isfies the requirements of Lemma 7.5.19. LetP00(V) be the space of self-
adjoint positive definite endomorphisms with respect to the new inner prod-
uct and define the functionfw,0:P00(V)→Rby the analogous formula.
ThenP∈P0(V) if and only ifP P
−1
0
∈P00(V) andfw,0(P P
−1
0
) =fw(P)
for allP∈P0(V). Thusfw,0attains its minimum atP= 1l.
In the next corollary we do not assume that Gwacts irreducibly onV.
Corollary 7.5.27(Cartan Decomposition).Assume the functionfwin
Lemma 7.5.20 has a critical point atP0= 1land define
Kw:= Gw∩SO(V),pw:={η∈gw|η=η
∗
}. (7.5.40)
Then the map
Kw×pw→Gw: (u, η)7→exp(η)u=:ϕw(u, η) (7.5.41)
is a diffeomorphism. Hence the mapGw→P0(V) :g7→
√
gg
∗
descends to
a diffeomorphism from the quotient spaceGw/KwtoMw= Gw∩P0(V).
Proof.By part (ii) of Lemma 7.5.22 withP0= 1l the functionfwattains its
minimum on the setMw= Gw∩P0(g). Now define the map
ψw: Gw→Kw×pw
byψw(g) := (u, η),where
η:=
1
2
exp
−1
(gg
∗
)∈pw, u:= exp(−η)g∈Kw
forg∈Gw. This map is well defined and smooth because, for everyg∈Gw,
we havegg
∗
∈Gw∩P0(V) =Mw(see Exercise 7.5.16), and the exponential
map descends to a diffeomorphism exp :pw→Mw(see Lemma 7.5.22).
Sinceϕw◦ψw= id andψw◦ϕw= id, it follows thatϕwis a diffeomorphism.
This proves Corollary 7.5.27.

7.5. CONVEX FUNCTIONS ON HADAMARD MANIFOLDS* 389
As a warmup for the main application of Theorem 7.5.17 in§7.6 it may
be useful to consider the following two examples.
Exercise 7.5.28. LetVbe anm-dimensional real vector space and
W:=S
2
V
∗
be the vector space of all symmetric bilinear formsQ:V×V→R. De-
fine the homomorphismρ: SL(V)→SL(W) by the standard action of the
group SL(V) onW, i.e.
Γ
ρ(g)Q
∆
(v, v
′
) :=Q(g
−1
v, g
−1
v
′
)
forg∈SL(V),Q∈S
2
V
∗
, andv, v
′
∈V. Assume thatVis equipped with
an inner product and an orthonormal basise1, . . . , em, and define an inner
product onWby⟨Q, Q
′
⟩:=
P
i,j
Q(ei, ej)Q
′
(ei, ej) forQ, Q
′
∈S
2
V
∗
. Show
that this inner product is independent of the choice of the orthonormal basis
and satisfies the requirements of Lemma 7.5.19.
(ii)The inner product onVis an elementQ0∈Wwhose isotropy sub-
group is the special orthogonal group SO(V). Show that the functionfQ0
in (7.5.38) is given byfQ0
(P) = trace(P
2
) forP∈P0(V) and that it has a
unique critical point atP= 1l.
(iii)Examine the case whereV=R
m+1
is equipped with the standard inner
product andQ∈Wis the quadratic form in (6.4.9). Relate this example to
the isometry group of hyperbolic space (§6.4.3). Find a maximal compact
subgroup of the identity component of O(m,1).
Exercise 7.5.29. LetVbe a 2n-dimensional real vector space and
W= Λ
2
V
∗
be the vector space of all skew-symmetric bilinear formsτ:V×V→R.
Define the homomorphismρ: SL(V)→SL(W) by the standard action of
the group SL(V) onW, i.e.
Γ
ρ(g)τ
∆
(v, v
′
) :=
Γ
g∗τ
∆
(v, v
′
) :=τ(g
−1
v, g
−1
v
′
)
forg∈SL(V),τ∈Λ
2
V
∗
, andv, v
′
∈V. Assume thatVis equipped with
an inner product and an orthonormal basise1, . . . , e2n, and define an inner
product onWby⟨τ, τ
′
⟩:=
P
i,j
τ(ei, ej)τ
′
(ei, ej) forτ, τ
′
∈Λ
2
V
∗
. Show
that this inner product is independent of the choice of the orthonormal
basis and satisfies the requirements of Lemma 7.5.19.

390 CHAPTER 7. TOPICS IN GEOMETRY
(ii)Letω:V×V→Rbe a nondegenerate skew-symmetric bilinear form.
Then the pair (V, ω) is called asymplectic vector spaceandωis called
asymplectic form onV. The isotropy subgroup ofωin GL(V) is called
thesymplectic linear group. Denote this group and its Lie algebra by
Sp(V, ω) :=
Φ
g∈GL(V)

ω(g·, g·) =ω

,
sp(V, ω) := Lie(Sp(V, ω)) =
Φ
A∈End(V)

ω(A·,·) +ω(·, A·) = 0

.
The group Sp(V, ω) is connected and contained in SL(V) (see [49]). An au-
tomorphismJ:V→Vis called alinear complex structureiffJ
2
=−1l.
A linear complex structureJis calledcompatible withωiff the bilin-
ear formω(·, J·) is an inner product onV. An inner product⟨·,·⟩onVis
calledcompatible withωiff there exists a linear complex structureJsuch
thatω(·, J·) =⟨·,·⟩. Prove that an inner product onVis compatible withω
if and only if it satisfies the conditions (for any basise1, . . . , e2nofV)
det
Γ
ω(ei, ej)
∆
= det
Γ
⟨ei, ej⟩
∆
, (7.5.42)
A∈sp(V, ω) = ⇒ A
∗
∈sp(V, ω). (7.5.43)
If an inner product onVis compatible withω, prove thatg∈Sp(V, ω)
impliesg
∗
∈Sp(V, ω) (without using the fact that Sp(V, ω) is connected).
Hint:DefineJ∈End(V) byω(·, J·) :=⟨·,·⟩and show thatJ+J
∗
= 0.
Show thatA∈sp(V, ω) if and only ifAJ+JA
∗
= 0. Use (7.5.43) to prove
thatJ
2
commutes with every self-adjoint endomorphism ofVand hence
satisfiesJ
2
=λ1l for someλ∈R. Use (7.5.42) to conclude thatλ=−1.
(iii)Fix an inner product onVand a symplectic formω:V×V→Rthat
satisfies (7.5.42). Then the functionfωin (7.5.38) is given by
fω(P) =
X
i,j
ω(ei, ej)ω(P ei, P ej), P ∈P0(V), (7.5.44)
wheree1, . . . , e2nis an orthonormal basis ofV. Prove that this is the norm
squared ofωwith respect to the inner product⟨·, P
−1
·⟩. Prove thatPis a
critical point offωif and only if the inner product⟨·, P
−1
·⟩is compatible
withω. Prove that the spaceJ(V, ω) ofω-compatible linear complex struc-
tures is a Hadamard manifold (Exercise 6.5.24). Prove that, ifJ∈ J(V, ω),
then the unitary group U(V, ω, J) :=
Φ
g∈Sp(V, ω)

gJg
−1
=J

is a maxi-
mal compact subgroup of Sp(V, ω) and every compact subgroup of Sp(V, ω)
is conjugate to a subgroup of U(V, ω, J). All this is of course well known, but
this exercise shows how these results can be derived from Theorem 7.5.17.
Moreover, it is not necessary to assume that Sp(V, ω) is connected. One
can start with the identity component of Sp(V, ω), prove that it is contained
in SL(V), and deduce the connectivity of Sp(V, ω) from that of U(V, ω, J).

7.6. SEMISIMPLE LIE ALGEBRAS* 391
7.6 Semisimple Lie Algebras*
This section discusses applications of the results in§7.5.3 to Lie algebra
theory, following the work of Donaldson [17]. It examines symmetric inner
products on Lie algebras (§7.6.1), establishes their existence on simple Lie
algebras (§7.6.2), and derives as consequences several standard results in Lie
algebra theory, such as the uniqueness of maximal compact subgroups up
to conjugation for semisimple Lie algebras (§7.6.3), and Cartan’s theorem
about the compact real form of a semisimple complex Lie algebra (§7.6.4).
Here are some basic definitions that will be used throughout this section.
Letgbe a finite-dimensional real Lie algebra (Definition 2.4.22). A vector
subspaceh⊂gis called anidealiff [ξ, η]∈hfor everyξ∈gand everyη∈h.
The Lie algebragis calledabelianiff the Lie bracket vanishes. It is called
simpleiff it is not abelian and does not contain any ideal other thanh={0}
andh=g. Examples of ideals in any Lie algebragare the centerZ(g)
(Exercise 2.5.34) and the commutant [g,g] (Exercise 5.2.24), defined by
Z(g) :={ξ∈g|[ξ, η] = 0 for allη∈g},
[g,g] := span{[ξ, η]|ξ, η∈g}.
Recall also the definition of the adjoint representation ad :g→Der(g) in
Example 2.5.23 by ad(ξ) := [ξ,·] forξ∈gand the definition of the Killing
formκ:g×g→Rin Example 5.2.25 byκ(ξ, η) := trace(ad(ξ)ad(η))
forξ, η∈g. Let Aut0(g) be the connected component of the identity in the
group Aut(g) of automorphisms ofg. This is a Lie subgroup of GL(g) whose
Lie algebra is the space Lie(Aut0(g)) = Der(g) of derivations ong.
7.6.1 Symmetric Inner Products
In [17] Donaldson introduced the following notion.
Definition 7.6.1.Letgbe a finite-dimensional real Lie algebra. An inner
product⟨·,·⟩ongis calledsymmetriciff it satisfies the condition
δ∈Der(g) = ⇒ δ
∗
∈Der(g). (7.6.1)
Hereδ
∗
:g→gdenotes the adjoint of the endomorphismδwith respect to
the inner product, i.e. it satisfies⟨ξ, δ
∗
η⟩=⟨δξ, η⟩for allξ, η∈g.
Exercise 7.6.2.Letgbe a finite-dimensional real Lie algebra equipped with
a symmetric inner product. Prove thatg∈Aut0(g) =⇒g
∗
∈Aut0(g).
Hint:See Exercise 7.5.16.
Some consequences of the existence of a symmetric inner product are
derived in Lemma 7.6.8 below. To begin with we discuss some examples.

392 CHAPTER 7. TOPICS IN GEOMETRY
Every vector space endomorphism of a Lie algebragthat takes values
in the centerZ(g) and vanishes on the commutant [g,g] is a derivation.
Conversely, every derivation that takes values inZ(g) necessarily vanishes
on [g,g]. The space of such derivations is an ideal in Der(g), denoted by
DerZ(g) :=
Φ
δ∈Der(g)

im(δ)⊂Z(g)

. (7.6.2)
Example 7.6.3.Consider the abelian Lie algebrag=R
m
. The adjoint
representation is trivial and Der(g) =gl(m,R) =R
m×m
is the Lie algebra
of all vector space endomorphisms ofg. Thus Der(g) = DerZ(g), the Killing
form ongvanishes, and every inner product ongis symmetric.
Example 7.6.4.Consider the Lie algebrag=gl(m,R) with [g,g]=sl(m,R)
andZ(g)=R1l. It satisfiesg= [g,g]⊕Z(g) and Der(g) = ad(g)⊕DerZ(g),
where DerZ(g)⊂Der(g) is the one-dimensional subspace generated by the
derivationδZ(A) = trace(A)1l (whose trace ism). Moreover, the standard
inner product⟨A, A
′
⟩= trace(A
T
A
′
) ongis symmetric and the kernel of the
Killing formκ(A, A
′
) = 2mtrace(AA
′
)−2trace(A)trace(A
′
) isZ(g).
Example 7.6.5.Consider the Lie algebrag=gl(m,R)×R
m
with the
Lie bracket [(A, v),(A
′
, v
′
)] := ([A, A
′
], Av
′
−A
′
v).This Lie algebra can be
identified with the space of all affine vector fields onR
m
. It has a triv-
ial center and the commutant [g,g] =sl(m,R)×R
m
has codimension one.
Moreover, trace(ad(A, v)) = trace(A) for every (A, v)∈g, the kernel of the
Killing formκ((A, v),(A
′
, v
′
)) = (2m+ 1)trace(AA
′
)−2trace(A)trace(A
′
) is
the abelian ideal{0} ×R
m
, and the adjoint representation ad :g→Der(g)
is a Lie algebra isomorphism.
Example 7.6.6.Consider the Heisenberg algebrah=V×Rof a symplectic
vector space (V, ω) with the Lie bracket [(v, t),(v
′
, t
′
)] = (0, ω(v, v
′
)) (see
Exercise 2.5.15). It satisfiesZ(h) = [h,h] ={0} ×Rand the Killing form
vanishes. Every derivation onhhas the formδ(v, t) = (Av+λv,Λ(v)+2λt),
whereλ∈R, Λ∈V
∗
, andA∈sp(V, ω). The subspace ad(h) = DerZ(h)
consists of all derivations of the formδ(v, t) = (0,Λ(v)).
Example 7.6.7.The Heisenberg algebra of a symplectic vector space (V, ω)
extends to a Lie algebrag=sp(V, ω)×V×Rwith the Lie bracket
[(A, v, t),(A
′
, v
′
, t
′
)] = ([A, A
′
], Av
′
−A
′
v, ω(v, v
′
)).
It satisfies [g,g] =gand has a one-dimensional centerZ(g) ={0} × {0} ×R.
The kernel of the Killing form is the Heisenberg algebra and the adjoint
representation ad :g→Der(g) is surjective.
The next lemma shows that the Lie algebras in Examples 7.6.5, 7.6.6,
and 7.6.7 do not admit symmetric inner products.

7.6. SEMISIMPLE LIE ALGEBRAS* 393
Lemma 7.6.8.Letgbe a finite-dimensional real Lie algebra equipped with
a symmetric inner product. Then
Der(g) = ad(g)⊕DerZ(g),g= [g,g]⊕Z(g), (7.6.3)
the kernel of the Killing formκ:g×g→Ris the center ofg, and there
exists an involutiong→g:ξ7→ξ
∗
such that, for allξ, η∈g,
ad(ξ
∗
) = ad(ξ)
∗
,[ξ, η]
∗
= [η
∗
, ξ
∗
]. (7.6.4)
Proof.Consider the orthogonal decomposition
Der(g) =A ⊕ B,A:= ad(g),B:=A
⊥
, (7.6.5)
with respect to the inner product⟨δ, δ
′
⟩= trace(δ
∗
δ
′
) forδ, δ
′
∈Der(g).
Since [δ,ad(ξ)] = ad(δξ) forδ∈Der(g) andξ∈g, the subspaceAis an ideal
in Der(g). Moreover, ifδ∈ Bandε∈Der(g), thenε
∗
∈Der(g), hence
trace([ε, δ]
∗
ad(ξ)) = trace(δ
∗
[ε
∗
,ad(ξ)]) = trace(δ
∗
ad(ε
∗
ξ)) = 0 for allξ∈g,
and hence [ε, δ]∈ B. ThusBis also an ideal in Der(g).
Next define DerZ(g)
∗
:={δ
∗
|δ∈DerZ(g)}. We prove that
B= DerZ(g) = DerZ(g)
∗
. (7.6.6)
SinceAandBare ideals we have [δ, δ
′
] = 0 for allδ∈ Bandδ
′
∈ A. Thus
ad(δξ) = [δ,ad(ξ)] = 0 for allδ∈ Bandξ∈g, hence im(δ)⊂Z(g) for
allδ∈ B, and soB ⊂DerZ(g). Now letδ∈DerZ(g)
∗
. Thenδ
∗
∈DerZ(g),
hence [g,g]⊂ker(δ
∗
), hence trace(δ
∗
ad(ξ)) = 0 for allξ∈g, and soδ∈ B.
Thus DerZ(g)
∗
⊂ B ⊂DerZ(g) and so (7.6.6) holds for dimensional reasons.
The first equation in (7.6.3) follows directly from (7.6.5) and (7.6.6). It fol-
lows also from (7.6.6) that trace(δ
∗
ad(ξ)
∗
) = trace(ad(ξ)δ) = 0 for allξ∈g
and allδ∈ B, and so ad(ξ)
∗
∈ B
⊥
=Afor allξ∈g. Thus
δ∈ A =⇒ δ
∗
∈ A. (7.6.7)
By (7.6.7), an elementζ∈gbelongs toZ(g) if and only if ad(ξ)
∗
ζ= 0 for
allξ∈gif and only if⟨ζ,ad(ξ)η⟩= 0 for allξ, η∈g, if and only ifζ∈[g,g]
⊥
.
Thus [g,g]
⊥
=Z(g) and this proves the second equation in (7.6.3).
By (7.6.3), the map ad :g→Der(g) restricts to a Lie algebra isomor-
phism from [g,g] toA. Hence, by (7.6.7) there exists a unique involution
[g,g]→[g,g] :ξ7→ξ
∗
that satisfies (7.6.4) for allξ, η∈[g,g]. By (7.6.3) this
involution extends uniquely to an involutiong→g:ξ7→ξ
∗
such thatζ
∗
=ζ
for allζ∈Z(g), and the extended involution satisfies (7.6.4) for allξ, η∈g.
Now letζ∈gbelong to the kernel of the Killing form, i.e.κ(ζ, ξ) = 0
for allξ∈g. Then|ad(ζ)|
2
=κ(ζ, ζ
∗
) = 0 and henceζ∈Z(g). Conversely,
it follows directly from the definitions thatZ(g) is contained in the kernel
of the Killing form and this proves Lemma 7.6.8.

394 CHAPTER 7. TOPICS IN GEOMETRY
7.6.2 Simple Lie Algebras
The goal of this section is to establish the existence of symmetric inner
products on simple Lie algebras. First, the following lemma derives some
immediate consequences of the definition of a simple Lie algebra.
Lemma 7.6.9.Letgbe a finite-dimensional simple real Lie algebra. Then
the center ofgis trivial, the adjoint representationad :g→Der(g)is
injective, the commutant is[g,g] =g, andtrace(ad(ξ)) = 0for allξ∈g.
Proof.The centerZ(g) is an ideal ing. It is not equal togbecausegis not
abelian, and henceZ(g) ={0}becausegis simple. The subspace [g,g] is
also an ideal ing. It is nonzero becausegis not abelian, and hence [g,g] =g
becausegis simple. The adjoint representation ad :g→Der(g) is injec-
tive because its kernel is the center ofg. The last assertion follows from
the fact that [g,g] =gand trace
Γ
ad([ξ, η])
∆
= trace
Γ
[ad(ξ),ad(η)]
∆
= 0 for
allξ, η∈g. This proves Lemma 7.6.9.
That the Killing form of a simple Lie algebra is nondegenerate is a deeper
result that does not follow directly from the definition. In [17, Theorem 1]
Donaldson deduces the existence of symmetric inner products on simple Lie
algebras from Theorem 7.5.17 and derives as corollaries various standard
results in Lie algebra theory, including nondegeneracy of the Killing form.
Theorem 7.6.10.Letgbe a finite-dimensional simple real Lie algebra.
Then the Killing form ongis nondegenerate, the adjoint representation
ad :g→Der(g)is bijective, every derivationδ:g→ghas trace zero, and
every automorphism in the identity componentAut0(g)has determinant one.
In particular, Theorem 7.6.10 establishes for every simple Lie algebrag
the existence of a connected Lie group Aut0(g)⊂SL(g) whose Lie algebra
is isomorphic tog.
Theorem 7.6.11(Donaldson).Every finite-dimensional simple real Lie
algebra admits a symmetric inner product. Moreover, ifSO(g)is the special
orthogonal group associated to a symmetric inner product ong, then
K := Aut0(g)∩SO(g) (7.6.8)
is connected and is a maximal compact subgroup ofAut0(g), every compact
subgroup ofAut0(g)is conjugate to a Lie subgroup ofK, and every maximal
compact subgroup ofAut0(g)is conjugate toK.

7.6. SEMISIMPLE LIE ALGEBRAS* 395
Given an inner product⟨·,·⟩on anym-dimensional Lie algebragand an
orthonormal basise1, . . . , emofg, define the functionfg:P0(g)→Rby
fg(P) :=
X
i,j

[ei, ej], P
−1
[P ei, P ej]

(7.6.9)
forP∈P0(g). The right hand side of (7.6.9) is independent of the choice
of the orthonormal basis and is the norm squared of the Lie bracket with
respect to the inner product⟨·, P
−1
·⟩ong.
Theorem 7.6.12(Donaldson).Letgbe a finite-dimensional simple real
Lie algebra equipped with a symmetric inner product. Then the set
Mg:=
Φ
P∈P0(g)

dfg(P) = 0

=
Φ
P∈P0(g)

fg(P) = inffg

=P0(g)∩Aut(g) =
Φ
exp(δ)

δ∈Der(g), δ=δ
∗

=
Φ
P∈P0(g)

the inner product⟨·, P
−1
·⟩is symmetric

(7.6.10)
of critical points offgis a geodesically convex and totally geodesic subman-
ifold ofP0(g). Hence it is a Hadamard manifold and a symmetric space.
The proofs require three preparatory lemmas. We do not assume that
every derivation has trace zero. Thus it is necessary as an intermediate step
to introduce the subspace Der0(g) :={δ∈Der(g)|trace(δ) = 0}.We will
consider inner products ongthat satisfy the condition
δ∈Der0(g) = ⇒ δ
∗
∈Der0(g). (7.6.11)
Lemma 7.6.13.Letgbe a finite-dimensional real Lie algebra satisfying
the conditionsZ(g) ={0}and[g,g] =g, and fix an inner product⟨·,·⟩ong.
Then the inner product is symmetric if and only if it satsfies(7.6.11). More-
over, if such an inner product exists, thenDer(g) = ad(g) = Der0(g).
Proof.Assume (7.6.11) and consider the decomposition Der0(g) =A0⊕ B0,
whereA0:= ad(g)⊂Der0(g) (because [g,g] =g) andB0:=A
⊥
0
are ideals
in Der0(g) as in the proof of Lemma 7.6.8. Then ad(δξ) = [δ,ad(ξ)] = 0 for
allδ∈ B0and allξ∈g. SinceZ(g) ={0}, this impliesB0= 0, and hence the
adjoint representation ad :g→Der0(g) is bijective. By (7.6.11), this implies
the existence of an involutionξ7→ξ
∗
such that ad(ξ
∗
) = ad(ξ)
∗
for allξ∈g.
Since the adjoint representation ad :g→Der(g) is injective, this in turn im-
plies that the Killing form is nondegenerate and so Der(g) = ad(g) = Der0(g)
by Lemma 7.4.3. Thus the inner product ongis symmetric. Conversely,
if the inner product ongis symmetric, then it satisfies (7.6.11) becauseδ
andδ
∗
have the same trace. This proves Lemma 7.6.13.

396 CHAPTER 7. TOPICS IN GEOMETRY
Lemma 7.6.14.Letgbe a finite-dimensional simple real Lie algebra with
a symmetric inner product, and letg→g:ξ7→ξ
∗
be the unique involution
that satisfies(7.6.4). Then there exists a constantc >0such that
κ(ξ
∗
, η) =c⟨ξ, η⟩for allξ, η∈g. (7.6.12)
Proof.By Lemma 7.6.9 the adjoint representation ad :g→Der(g) is injec-
tive. Hence the mapg×g→R: (ξ, η)7→κ(ξ
∗
, η) = trace(ad(ξ)
∗
ad(η)) is
an inner product ong. Thus there exists a self-adjoint positive definite vec-
tor space isomorphismA:g→gsuch thatκ(ξ
∗
, η) =⟨Aξ, η⟩for allξ, η∈g.
Letc >0 be an eigenvalue ofAand defineh:=
Φ
η∈g

Aη=cη

.We prove
thathis an ideal ing. Letξ∈gandη∈h. Then, for allζ∈g,
⟨A[ξ, η], ζ⟩=κ([ξ, η]
∗
, ζ) =κ([η
∗
, ξ
∗
], ζ) =κ(η
∗
,[ξ
∗
, ζ]) =⟨Aη,[ξ
∗
, ζ]⟩
=⟨Aη,ad(ξ
∗
)ζ⟩=⟨ad(ξ)Aη, ζ⟩=⟨[ξ, Aη], ζ⟩=c⟨[ξ, η], ζ⟩.
HenceA[ξ, η] =c[ξ, η] and so [ξ, η]∈h. This shows thathis a nonzero ideal
and henceh=g. ThusA=c1l and this proves Lemma 7.6.14.
Given any inner product ong, call an elementP∈P0(g)symmetric
iff the inner product⟨·, P
−1
·⟩ongis symmetric. Thus every symmetric
elementP∈P0(g) determines an involutionτP: Der(g)→Der(g) given
byτP(δ) :=P δ
∗
P
−1
. Denote its determinant byεP:= det(τP)∈ {−1,+1}.
Lemma 7.6.15.Letgbe a finite-dimensional simple real Lie algebra, choose
any inner product ong, and letP, P0∈P0(g)be symmetric elements.
ThenP P
−1
0
∈Aut(g).
Proof.The composition of the involutionsτPandτP0
is the Lie algebra au-
tomorphismτP◦τP0
(δ) =P
Γ
P0δ
∗
P
−1
0
∆
∗
P
−1
=P P
−1
0
δP0P
−1
forδ∈Der(g).
By Lemma 7.6.8 and Lemma 7.6.9 the very existence of a symmetric inner
product ongimplies that the adjoint representation ad :g→Der(g) is a
Lie algebra isomorphism. Hence there exists ag∈Aut(g) such that
ad(gξ) =P P
−1
0
ad(ξ)P0P
−1
for allξ∈g. (7.6.13)
Since ad(gξ) =gad(ξ)g
−1
, this automorphism satisfies the equations
g
−1
P P
−1
0
[ξ, η] = [ξ, g
−1
P P
−1
0
η] for allξ, η∈g, (7.6.14)
det(g) =εPεP0
∈ {−1,+1}. (7.6.15)

7.6. SEMISIMPLE LIE ALGEBRAS* 397
SinceP0is symmetric, it follows from Lemma 7.6.8 and Lemma 7.6.14 that
there exists an involutiong→g:ξ7→ξ
∗
and a constantc >0 such that
ad(ξ
∗
) =P0ad(ξ)
∗
P
−1
0
, κ(ξ
∗
, η) =c⟨ξ, P
−1
0
η⟩ (7.6.16)
for allξ, η∈g. By (7.6.13) and (7.6.16) we have
κ((gξ)
∗
, η) = trace
Γ
P0ad(gξ)
∗
P
−1
0
ad(η)
∆
= trace

P P
−1
0
ad(ξ)P0P
−1
∆
∗
P
−1
0
ad(η)P0
∆
= trace
Γ
P
−1
P0ad(ξ)
∗
P
−1
0
P P
−1
0
ad(η)P0
∆
= trace
Γ
ad(ξ
∗
)P P
−1
0
ad(η)P0P
−1
∆
= trace
Γ
ad(ξ
∗
)ad(gη)
∆
=κ(ξ
∗
, gη).
This shows thatgis self-adjoint and positive definite with respect to the
inner product⟨·, P
−1
0
·⟩, and so det(g) = 1 by (7.6.15). SinceP P
−1
0
is self-
adjoint and positive definite with respect to the same inner product, the
vector space isomorphismg
−1
P P
−1
0
=g
−1/2
(g
−1/2
P P
−1
0
g
−1/2
)g
1/2
has only
positive real eigenvalues. Letλ >0 be one such eigenvalue. Then the
eigenspace ker(λ1l−g
−1
P P
−1
0
) is a nonzero ideal ingby (7.6.14), and so
is equal tog, becausegis simple. Thusg
−1
P P
−1
0
=λ1l, and sinceg, P, P0
all have determinant one, it follows thatλ= 1 and soP P
−1
0
=g∈Aut(g).
This proves Lemma 7.6.15.
Proof of Theorems 7.6.10, 7.6.11, and 7.6.12.We use the results of§7.5.3
in the situation whereV:=gis the Lie algebra itself andW:= Λ
2
g
∗
⊗gis
the space of all skew-symmetric bilinear mapsτ:g×g→g. The Lie group
homomorphismρ: SL(g)→SL(W) is given by the standard action of the
group SL(g) onW, i.e.
(ρ(g)τ)(ξ, η) :=gτ(g
−1
ξ, g
−1
η)
forg∈SL(g),τ∈W, andξ, η∈g. Fix an inner product⟨·,·⟩ongand an
orthonormal basise1, . . . , emofg, and define
⟨σ, τ⟩W:=
X
i,j
⟨σ(ei, ej), τ(ei, ej)⟩ (7.6.17)
forσ, τ∈W. This inner product satisfies the requirements of Lemma 7.5.19,
i.e.ρ(A
∗
) =ρ(A)
∗
for allA∈sl(g). The vectorw:= [·,·]∈Wis chosen
to be the Lie bracket. This vector is nonzero becausegis not abelian. The
isotropy subgroup ofwis the group Aut(g)∩SL(g) of all automorphisms of
determinant one. Denote its identity component by G⊂Aut0(g)∩SL(g).
This is a Lie subgroup of SL(g) with the Lie algebra Lie(G) = Der0(g).

398 CHAPTER 7. TOPICS IN GEOMETRY
Claim 1.Gacts irreducibly ong.
Leth⊂gbe a subspace that is invariant under the action of G. Thenδh⊂h
for everyδ∈Der0(g). By Lemma 7.6.9 this implies ad(ξ)h⊂hfor allξ∈g,
sohis an ideal. Thush={0}orh=gbecausegis simple.
Claim 2.fgis convex andG-invariant and has a critical point. Moreover,
every critical pointP∈P0(g)is symmetric, andDer0(g) = Der(g).
In the present setting the functionfwin Lemma 7.5.20 agrees with the
functionfgin (7.6.9) and Gw= G. Hencefgis convex and G-invariant by
Lemma 7.5.22, and G acts irreducibly ongby Claim 1. Thus Lemma 7.5.23
asserts thatfghas a critical point. LetP∈P0(g) be a critical point offg.
Then by Lemma 7.5.23 the inner product⟨·, P
−1
·⟩satisfies (7.6.11) and so,
by Lemma 7.6.13, this inner product is symmetric and Der(g) = Der0(g).
Claim 3.Fix a critical pointP0∈P0(g)offgand any elementP∈P0(g).
Then the following are equivalent.
(a)Pis a critical point offg.
(b)fg(P) = inffg.
(c)P P
−1
0
∈Aut(g).
(d)There exists aδ∈Der(g)such thatδ=P0δ
∗
P
−1
0
,exp(δ) =P P
−1
0
.
(e)The inner product⟨·, P
−1
·⟩ongis symmeric.
Sincefgis convex by Claim 2, the equivalence of (a) and (b) follows from
Lemma 7.5.9. Since Der(g) = Der0(g) by Claim 2, the equivalence of (b),
(c), and (d) follows from part (ii) of Lemma 7.5.22. Moreover, (a) implies (e)
by Claim 2, and (e) implies (c) by Lemma 7.6.15. This proves Claim 3.
The existence of a symmetric inner product ongwas proved in Claim 2.
Thus the nondegeneracy of the Killing form follows from Lemma 7.6.8.
The remaining assertions of Theorem 7.6.10 are direct consequences of the
nondegeneracy of the Killing form. In particular, the adjoint represen-
tation ad :g→Der(g) is bijective and Der(g)⊂sl(g) by Lemma 7.4.3.
Hence Aut0(g)⊂SL(g) and so Gw= G = Aut0(g) in the notation of§7.5.3.
Now fix a symmetric inner product ong. ThenP0= 1l is a critical point
offgby (e) =⇒(a)” in Claim 3. Hence the assertions about the sub-
group K = Aut0(g)∩SO(g) in Theorem 7.6.11 follow from Lemma 7.5.23.
The equalities in (7.6.10) follow from the equivalence of (a), (b), (c), (d), (e)
in Claim 3 withP0= 1l, and the remaining assertions about the spaceMg
in Theorem 7.6.12 follow from Lemma 7.5.24. This completes the proof of
Theorems 7.6.10, 7.6.11, and 7.6.12.

7.6. SEMISIMPLE LIE ALGEBRAS* 399
7.6.3 Semisimple Lie Algebras
The following theorem characterizes semisimple Lie algebras in terms of
symmetric inner products. First observe that, ifhis an ideal in a finite-
dimensional real Lie algebrag, then the subspace
h
′
:=
Φ
η
′
∈g|κ(η, η
′
) = 0 for allη∈h

(7.6.18)
is also an ideal, because everyη
′
∈h
′
satisfiesκ(η,[ξ, η
′
]) =κ([η, ξ], η
′
) = 0
for allξ∈gand allη∈h, and so [ξ, η
′
]∈h
′
for allξ∈g.
Theorem 7.6.16(Semisimple Lie algebras).Letgbe a finite-dimen-
sional real Lie algebra. Then the following are equivalent.
(i)ghas a trivial center and admits a symmetric inner product.
(ii)The Killing formκ:g×g→Ris nondegenerate.
(iii)Ifh⊂gis an ideal, theng=h⊕h
′
.
(iv)gis a direct sum of simple ideals.
(v)There exists an inner product⟨·,·⟩ong, an involutiong→g:ξ7→ξ
∗
,
and a constantc >0such that, for allξ, η∈g,
ad(ξ
∗
) = ad(ξ)
∗
,[ξ, η]
∗
= [η
∗
, ξ
∗
], κ(ξ
∗
, η) =c⟨ξ, η⟩.(7.6.19)
Definition 7.6.17.A real Lie algebragis calledsemisimpleiff it is finite-
dimensional and satisfies the equivalent conditions in Theorem 7.6.16.
Proof of Theorem 7.6.16.That (i) implies (ii) was shown in Lemma 7.6.8.
We prove that (ii) is equivalent to (iii). Assume first that the Killing
form is nondegenerate and leth⊂gbe an ideal. Then
η∈h, η
′
∈h
′
=⇒ [η, η
′
] = 0, (7.6.20)
becauseκ([η, η
′
], ξ) =κ(η,[η
′
, ξ]) = 0 for allξ∈g, allη∈h, and allη
′
∈h
′
.
Now letη∈h∩h
′
. Thenη
′
:= ad(ξ)ad(η)ζ= [ξ,[η, ζ]]∈h
′
for allξ, ζ∈g.
Hence, by (7.6.20) we have (ad(ξ)ad(η))
2
ζ= [ξ,[η, η
′
]] = 0 for allξ, ζ∈g.
Thus (ad(ξ)ad(η))
2
= 0 and soκ(ξ, η) = trace(ad(ξ)ad(η)) = 0 for allξ∈g.
This impliesη= 0 by nondegeneracy of the Killing form. Thush∩h
′
={0}
and sog=h⊕h
′
because dim(h) + dim(h
′
) = dim(g). This shows that (ii)
implies (iii). To prove the converse, takeh=gso thath
′
={0}is the kernel
of the Killing form.
It follows from (ii) and (iii) that, for every idealh⊂g, the Killing forms
ofhandh
′
are both nondegenerate. Hence an induction argument shows
that (iii) implies (iv).

400 CHAPTER 7. TOPICS IN GEOMETRY
We prove that (iv) implies (v). Assume thatg=g1⊕ · · · ⊕gris a di-
rect sum of simple idealsgj⊂g. Fix an indexj∈ {1, . . . , r}and denote
byκj:gj×gj→Rthe Killing form ofgj. By Theorem 7.6.11 the Lie alge-
bragjadmits a symmetric inner product⟨·,·⟩j. Hence, by Lemma 7.6.8
there exists an involutiongj→gj:ξ7→ξ
∗
that satisfies (7.6.4), and by
Lemma 7.6.14 there exists a constantcj>0 such thatκj(ξ
∗
, η) =cj⟨ξ, η⟩j
for allξ, η∈gj. Thus the inner product⟨ξ, η⟩:=
P
j
cj⟨ξj, ηj⟩jforξj, ηj∈gj
andξ=ξ1+· · ·+ξr,η=η1+· · ·+ηrsatisfies the requirements of part (v)
withc= 1 andξ
∗
:=ξ
∗
1
+· · ·+ξ
∗
r.
If (v) holds, then the Killing form is nondegenerate, henceZ(g) = 0
and Der(g) = ad(g) by Lemma 7.4.3, and hence the inner product in (v) is
symmetric. Thus (v) implies (i) and this proves Theorem 7.6.16.
Corollary 7.6.18(Cartan Involution).Letgbe a nonzero semisimple
real Lie algebra equipped with a symmetric inner product and letξ7→ξ
∗
be
the involution in Lemma 7.6.8. Then the map
g→g:ξ7→ −ξ
∗
(7.6.21)
is a Lie algebra homomorphism (called aCartan involution). The Cartan
involution(7.6.21)gives rise to a splitting
g=k⊕p,k:={ξ∈g|ξ+ξ
∗
= 0},p:={η∈g|η=η
∗
},(7.6.22)
such that
[k,k]⊂k,[k,p]⊂p,[p,p]⊂k. (7.6.23)
Moreover,kis nontrivial, the Killing formκ:g×g→Ris negative definite
onkand positive definite onp, andκ(ξ, η) = 0for allξ∈kand allη∈p.
Proof.That the map (7.6.21) is a Lie algebra homomorphism follows di-
rectly from (7.6.4). It follows also from (7.6.4) that the subspacesk,p⊂g
in (7.6.22) satisfy (7.6.23). Thatgis the direct sum of these subspaces,
follows from the identityζ
∗∗
=ζforζ∈g, which implies
ζ=ξ+η, ξ:=
1
2
Γ
ζ−ζ
∗
∆
∈k, η:=
1
2
Γ
ζ+ζ
∗
∆
∈p.
By (7.6.23) the summandkmust be nontrivial, becausegis nonzero and
hence is not abelian.
Next observe that the formula⟨A, B⟩:= trace(A
∗
B) defines an inner
product on End(g) with the norm|A|:=
p
trace(A
∗
A), and that
κ(ξ, η
∗
) =κ(ξ
∗
, η) = trace
Γ
ad(ξ)
∗
ad(η)
∆
=⟨ad(ξ),ad(η)⟩.
Forξ∈kandη∈pthis impliesκ(ξ, ξ) =−|ad(ξ)|
2
, κ(η, η) =|ad(η)|
2
,
andκ(ξ, η) = 0. This proves Corollary 7.6.18.

7.6. SEMISIMPLE LIE ALGEBRAS* 401
Lemma 7.6.19.Letgbe a semisimple real Lie algebra equipped with a
symmetric inner product, letξ7→ξ
∗
be the involution in Lemma 7.6.8,
letg=g1⊕ · · · ⊕grbe a decomposition into simple idealsgj, and letk,p⊂g
be as in Corollary 7.6.18. Then the following holds.
(i)The simple idealsgj⊂gare pairwise orthogonal.
(ii)Each idealgjis invariant under the involutionξ7→ξ
∗
.
(iii)pis the orthogonal complement ofk.
(iv)For eachjthe restriction of the inner product togjis symmetric
andgj=kj⊕pj, wherekj=k∩gj,pj=p∩gjare as in Corollary 7.6.18.
Proof.We prove part (i). For eachithe orthogonal complementg
⊥
i
is an
ideal, because⟨[ξ, η], ζ⟩=⟨η,[ξ
∗
, ζ]⟩= 0 for allξ∈g,η∈g
⊥
i
,ζ∈gi, and
so [ξ, η]∈g
⊥
i
for allξ∈g,η∈g
⊥
i
. This implies thathj:=
T
i̸=j
g
⊥
i
is an
ideal, and so is the subspacehj∩gj. So eitherhj∩gj={0}orhj=gj,
becausegjis simple. Ifhj∩gj={0}, then [ξ, η] = 0 for allξ∈gjand
allη∈hj, hencegj⊂Z(g), and this is impossible because the center ofgis
trivial. Thushj=gjfor alljand this proves (i).
We prove part (ii). The subspaceg
∗
j
:={η
∗
|η∈gj}is an ideal, be-
cause [ξ, η
∗
] = [η, ξ
∗
]
∗
∈g
∗
j
for allξ∈gand allη∈gj. By part (i) we
have⟨[ξ, η
∗
], ζ⟩=−⟨ξ,[η, ζ]⟩= 0 for allξ∈giandη∈gjwithi̸=j, and
allζ∈g. Hence [gi,g
∗
j
] = 0 fori̸=j. Hence the idealg
∗
j
∩gjcannot be
zero, because otherwise [gj,g
∗
j
] = 0 and sog
∗
j
⊂Z(g). Henceg
∗
j
=gj,
becausegjis simple. This proves (ii).
We prove part (iii). By (ii) and Lemma 7.6.14 the involutionξ7→ξ
∗
preserves the inner product ongj, and so by (i) it preserves the inner product
on all ofg. Hence⟨ξ, η⟩=⟨ξ
∗
, η
∗
⟩=−⟨ξ, η⟩for allξ∈kand allη∈p. Thusk
andpare orthogonal to each other and this proves (iii).
Part (iv) follows directly from (ii) and this proves Lemma 7.6.19.
Theorem 7.6.20.Letgbe anm-dimensional real Lie algebra that is not
abelian and fix an inner product ongand an orthonormal basise1, . . . , em
ofg. Then the following are equivalent.
(i)P= 1lis a critical point offg.
(ii)There exists a real numbercsuch that
m
X
i=1
ı
2ad(ei)
∗
ad(ei)−ad(ei)ad(ei)
∗
ȷ
=c1l. (7.6.24)
(iii)gis semisimple, the inner product is symmetric, and there exists an
involutiong→g:ξ7→ξ
∗
and a constantc >0such that(7.6.19)holds.

402 CHAPTER 7. TOPICS IN GEOMETRY
Proof.By Lemma 7.5.20 the elementP= 1l∈P0(g) is a critical point of
the functionfgin (7.6.9) if and only if, for allA∈sl(g),
0 =
m
X
i,j=1
⟨[ei, ej], A[ei, ej]−[Aei, ej]−[ei, Aej]⟩
=
m
X
i=1
trace
ı
ad(ei)
∗
Aad(ei)−2ad(ei)
∗
ad(ei)A
ȷ
.
This holds if and only if there exists a constantc∈Rthat satisfies (7.6.24).
Thus we have proved that (i) is equivalent to (ii).
We prove that (i) and (ii) imply (iii). SinceP= 1l is a critical point offg,
Lemma 7.5.20 asserts that the inner product ongis symmetric. Thus by
Lemma 7.6.8 there exists an involutiong→g:ξ7→ξ
∗
that satisfies (7.6.4).
Hence by (ii) there exists a real numbercsuch that
Qg:=
m
X
i=1
ı
2ad(e
∗
i)ad(ei)−ad(ei)ad(e
∗
i)
ȷ
=c1l. (7.6.25)
Sincegis not abelian, the endomorphismQghas a positive trace, soc >0.
This implies that the center ofgis trivial, becauseZ(g)⊂ker(Qg). Hence,
by Lemma 7.6.8, the Killing form ongis nondegenerate. By Lemma 7.6.19
this implies that the decompositiong=k⊕pin Corollary 7.6.18 is or-
thogonal. Hence the orthonormal basise1, . . . , emofgcan be chosen such
thate1, . . . , ekis a basis ofkandek+1, . . . , emis a basis ofp. Thuse
∗
i
=−ei
fori≤kande
∗
i
=eifori > k, and so it follows from (7.6.25) that
m
X
i=1
ad(e
∗
i)ad(ei) =c1l. (7.6.26)
Henceκ(ξ
∗
, η) =
P
i
⟨ei,ad(ξ
∗
)ad(η)ei⟩=
P
i
⟨ξ,ad(e
∗
i
)ad(ei)η⟩=c⟨ξ, η⟩for
allξ, η∈g. This proves (iii).
That (iii) implies (ii) follows by reversing this argument. By (iii) the
Killing form is nondegenerate and the inner product is symmetric and is
preserved by the involutionξ7→ξ
∗
. Thus the splittingg=k⊕pis orthog-
onal, and hence the orthonormal basise1, . . . , emofgcan be chosen such
thate1, . . . , ekis a basis ofkandek+1, . . . , emis a basis ofp. Moreover,
m
X
i=1
⟨ξ,ad(e
∗
i)ad(ei)η⟩=
m
X
i=1
⟨ei,ad(ξ
∗
)ad(η)ei⟩=κ(ξ
∗
, η) =c⟨ξ, η⟩
for allξ, η∈gby (7.6.19). This implies (7.6.26). Sincee
∗
i
=±eifor alli,
equation (7.6.24) follows from (7.6.26). This proves Theorem 7.6.20.

7.6. SEMISIMPLE LIE ALGEBRAS* 403
Corollary 7.6.21(Cartan Decomposition).Letgbe a semisimple real
Lie algebra equipped with a symmetric inner product, letg→g:ξ7→ξ
∗
be
the involution in Lemma 7.6.8, and define
K := Aut0(g)∩SO(g),Der
+
(g) :={δ∈Der(g)|δ=δ
∗
}.
Then the following holds.
(i)Kis connected and is a maximal compact subgroup ofAut0(g), every
compact subgroup ofAut0(g)is conjugate inAut0(g)to a Lie subgroup ofK,
and every maximal compact subgroup ofAut0(g)is conjugate toK.
(ii)The map
K×Der
+
(g)→Aut0(g) : (u, δ)7→exp(δ)u
is a diffeomorphism.
(iii)If there exists ac >0such thatκ(ξ
∗
, η) =c⟨ξ, η⟩for allξ, η∈g, then
Mg:= Crit(fg) =P0(g)∩Aut(g) ={exp(δ)|δ∈Der
+
(g)}
is a totally geodesic and geodesically convex submanifold ofP0(g)and so is
a Hadamard manifold and a symmetric space, and the map
Aut0(g)→P0(g) :g7→
p
gg
∗
descends to a diffeomorphism from the quotient spaceAut0(g)/KtoMg.
Proof.By Theorem 7.6.16 there exists a splittingg=g1⊕ · · · ⊕grinto
simple ideals, this splitting is preserved by every derivation ofg, and by
Lemma 7.6.19 it is also preserved by the involutionξ7→ξ
∗
. Thus Aut0(g) is
isomorphic to the product of the groups Aut0(gj), and K = Aut0(g)∩SO(g)
is isomorphic to the product of the subgroups Kj:= Aut0(gj)∩SO(gj).
Hence part (i) follows from Theorem 7.6.11. Moreover, by Lemma 7.6.19,
we havep=p1⊕ · · · ⊕pr, and so part (ii) follows from Corollary 7.5.27.
Under the assumptions of (iii) Theorem 7.6.20 asserts thatP0= 1l is a
critical point offg, so (iii) follows from Lemma 7.5.22, Lemma 7.5.24, and
Corollary 7.5.27. This proves Corollary 7.6.21.
Remark 7.6.22.The Lie algebra of the group K = Aut0(g)∩SO(g) in
Corollary 7.6.21 is given by Lie(K) ={ad(ξ)|ξ∈k}(see Corollary 7.6.18).
If the summandpin (7.6.22) is trivial, then Aut0(g) = K is a compact Lie
group. Ifpis nontrivial, then the quotient space Aut0(g)/K is a nontrivial
Hadamard manifold diffeomorphic top.

404 CHAPTER 7. TOPICS IN GEOMETRY
Remark 7.6.23.One can replace the spaceP0(g) of positive definite self-
adjoint vector space isomorphismsP:g→gof determinant one by the
spaceHgof inner products ongwith a fixed determinant (Remark 6.5.11).
This eliminates the dependence on the background inner product and there
is then only one functionfg:Hg→Rwhose set of minima is the totally
geodesic submanifoldMg⊂Hgof all symmetric inner products ongwith a
fixed determinant that satisfy part (iii) of Theorem 7.6.20. The main result
asserts that, whengis not abelian, the spaceMgis nonempty if and only
ifgis semisimple (Theorem 7.6.16 and Theorem 7.6.20).
7.6.4 Complex Lie Algebras
Acomplex Lie algebrais a complex vector spacegequipped with a Lie
bracketg×g→g: (ξ, η)7→[ξ, η] that is complex bilinear, i.e. it is a skew-
symmetric bilinear map that satisfies the Jacobi identity and
[iξ, η] = [ξ,iη] =i[ξ, η]
for allξ, η∈g. Thus every complex Lie algebra is also a real Lie algebra.
Letgbe a finite-dimensional complex Lie algebra. Acomplex idealing
is a complex linear subspaceh⊂gthat satisfies [ξ, η]∈hfor allξ∈gand
allη∈h. The complex Lie algebragis calledsimpleiff it is not abelian and
has no complex ideals other thanh={0}andh=g. It is calledsemisimple
iff it is finite-dimensional and thecomplex Killing formκ
c
:g×g→C,
defined byκ
c
(ξ, η) := trace
c
(ad(ξ)ad(η)) forξ, η∈g, is nondegenerate.
Sinceκ= 2Reκ
c
, a complex Lie algebra is semisimple if and only if it is
semisimple as a real Lie algebra. The next lemma shows that the analogous
assertion holds for simple complex Lie algebras (see [17, Lemma 7]).
Lemma 7.6.24.A finite-dimensional complex Lie algebragis simple if and
only if it is simple as a real Lie algebra.
Proof.Letgbe a simple complex Lie algebra of complex dimensionnand
leth⊂gbe a real linear subspace ofgthat satisfies [ξ, η]∈hfor allξ∈g
and allη∈h. Then the subspaces
h∩ih,h+ih
are complex ideals ing, and their real dimensions satisfy the equation
dim
R
(h∩ih) + dim
R
(h+ih) = 2 dim
R
(h).
Since both summands on the left are either 0 or 2n, the real dimension ofh
is either 0,n, or 2n. We claim that the dimension cannot ben.

7.6. SEMISIMPLE LIE ALGEBRAS* 405
Assume, by contradiction, that dim
R
(h) =n. Then
h∩ih={0},h+ih=g.
Hence, for allζ, ζ
′
∈gthere existξ, η∈hsuch thatζ=ξ+iη, and so
[ζ, ζ
′
] = [ξ, ζ
′
] + [η,iζ
′
] =i
Γ
[ξ,−iζ
′
] + [η, ζ
′
]
∆
∈h∩ih={0}.
This contradicts the fact thatgis not abelian. Thush={0}orh=g, and
this proves Lemma 7.6.24.
Lemma 7.6.25.Letgbe a semisimple complex Lie algebra. Thengis a
direct sum of simple complex ideals.
Proof.Leth⊂gbe a real ideal and leth
′
⊂gbe as in (7.6.18). Then
it follows from (ii) =⇒(iii)” in Theorem 7.6.16 that [h+ih,h
′
] ={0}.
Henceh+ih⊂h
′′
=hand soih=h. Thus every real ideal ingis a complex
ideal. Hence the assertion follows from (ii) =⇒(iv)” in Theorem 7.6.16.
The next result is a theorem of Cartan [13] which asserts that every
semisimple complex Lie algebra has a compact real form. The proof given
here is due to Donaldson [17, Lemma 8].
Theorem 7.6.26(Cartan).Letgbe a nonzero semisimple complex Lie
algebra equipped with a symmetric inner product. Then
p=ik,g=k⊕ik, (7.6.27)
wherek,pare as in Corollary 7.6.18. Moreover, the groupAut0(g)is the
complexification of the maximal compact subgroupK = Aut0(g)∩SO(g), i.e.
Aut0(g) ={exp(iδ)u|u∈K, δ∈Lie(K)} (7.6.28)
and the mapK×Lie(K)→Aut0(g) : (u, δ)7→exp(iδ)uis a diffeomorphism.
Proof.Assume first thatgis simple. Thengis simple as a real Lie algebra
(Lemma 7.6.24), and so has a nondegenerate Killing form (Theorem 7.6.10).
By the inclusions in (7.6.23) the subspace
h:= (k∩ip) + (p∩ik) = (k∩ip) +i(k∩ip)
is a complex ideal ing. Hence it is either{0}org. Ifh=g, then it follows
from (7.6.22) thatp=ik. Assume, by contradiction, thath={0}. Then
k∩ip={0},g=k⊕ip.

406 CHAPTER 7. TOPICS IN GEOMETRY
Hence the mapσ:g→g, defined by
σ(ξ+iη) :=ξ−iη
forξ∈kandη∈pis a Lie algebra homomorphism and an involution. This
implies ad(σ(ζ)) =σad(ζ)σand soκ(σ(ζ), σ(ζ
′
)) =κ(ζ, ζ
′
) for allζ, ζ
′
∈g.
Henceκ(ξ,iη) = 0 for allξ∈kand allη∈p. Thusipis the orthogonal
complement ofkwith respect to the Killing form. Thus, by Corollary 7.6.18,
p=ip.
However, for allη∈pwe haveκ(iη,iη) =−κ(η, η) =−|ad(η)|
2
.Hence
the Killing form is negative definite onipand positive definite onp, and
hencep={0}. This impliesk=ik.Since the Killing form is positive definite
onikand negative definite onk, this is a contradiction. This contradiction
shows that our assumptionh={0}must have been wrong. Thush=gand
hencep=ik. This completes the proof of (7.6.27) in the simple case.
For general semisimple complex Lie algebras, the proof of the equations
in (7.6.27) reduces to the simple case by Lemma 7.6.19 and Lemma 7.6.25.
Equation (7.6.28) follows directly from (7.6.27) and Corollary 7.6.21. This
proves Theorem 7.6.26.
Remark 7.6.27.Letgbe a semisimple complex Lie algebra. Then Aut(g)
is acomplex Lie group, i.e. it admits the structure of a complex manifold
such that the structure maps
G×G→G : (h, g)7→hg, G→G :g7→g
−1
are holomorphic. The proof uses the fact that Der(g) is isomorphic togand
that the resulting almost complex structure on Aut(g) is integrable (as it is
preserved by the torsion-free connectiong
−1
∇bg=
d
dt
(g
−1
bg) + [g
−1
˙g, g
−1
bg]).
Theorem 7.6.26 asserts that the identity component G = Aut0(g) of the
group of automorphisms ofgis thecomplexificationof the maximal com-
pact subgroup K = Aut0(g)∩SO(g), i.e. its Lie algebra Der(g) = ad(g)
is the complexification of the Lie algebra Lie(K) = ad(k) and the quo-
tient space G/K is contractible. These conditions imply the universality
property that every Lie group homomorphismρ: K→Gwith values in a
complex Lie groupGextends to a unique holomorphic Lie group homo-
morphismρ
c
: G→Gsuch thatρ
c
|K=ρ.Such a complexification exists
for every compact Lie group K, whether or not it is semisimple. (For an
exposition see [20, Appendix B].)

7.6. SEMISIMPLE LIE ALGEBRAS* 407
The following exercise is inspired by a remark in [17,§5.1] concerning a
positive curvature manifold that is dual toMg.
Exercise 7.6.28.Letgbe a semisimple real Lie algebra equipped with
a symmetric inner product that satisfies condition (iii) in Theorem 7.6.20.
Consider the complexified Lie algebra
g
c
:=g⊕ig
with the Lie bracket
[ζ, ζ
′
] := [ξ, ξ
′
]−[η, η
′
] +i
Γ
[ξ, η
′
] + [η, ξ
′
]
∆
(7.6.29)
and the Hermitian form
⟨ζ, ζ
′
⟩
c
:=⟨ξ, ξ
′
⟩+⟨η, η
′
⟩+i
Γ
⟨ξ, η
′
⟩ − ⟨η, ξ
′
⟩
∆
(7.6.30)
forζ=ξ+iη∈g
c
andζ
′
=ξ
′
+iη
′
∈g
c
. With this convention the Hermi-
tian form (7.6.30) is complex anti-linear in the first variable and complex
linear in the second variable. Prove the following.
(a)g
c
is semisimple.Hint:g
c
has a trivial center and the real part
of (7.6.30) is a symmetric inner product ong
c
.
(b)Ifgis simple, theng
c
is simple.Hint:Ifh
c
is a complex ideal ing
c
,
then the linear subspace
h:={Re(ζ)|ζ∈h
c
}
is an ideal ing.
(c)Every real linear derivation ong
c
is complex linear and has complex
trace zero.
(d)The identity component Aut0(g
c
) of the group of real linear Lie algebra
automorphisms ofg
c
consists of complex linear automorphisms of complex
determinant one. (Complex conjugation is a Lie algebra automorphism ofg
c
not in the identity component.)
(e)The subgroup
K
c
:= Aut0(g
c
)∩SU(g
c
)
is connected and is a maximal compact subgroup of Aut0(g
c
). Its Lie algebra
Lie(K
c
)
∼
=k+ip
is the compact real form ofg
c
(Corollary 7.6.18 and Theorem 7.6.26).

408 CHAPTER 7. TOPICS IN GEOMETRY
(f)Let Φ +iΨ :g→g
c
be a Lie algebra homomorphism, i.e. for allξ, η∈g,
Φ[ξ, η] = [Φξ,Φη]−[Ψξ,Ψη],Ψ[ξ, η] = [Φξ,Ψη] + [Ψξ,Φη].
Assume Φ +iΨ is injective and denote its image by
l:={Φξ+iΨξ|ξ∈g}.
Then the following are equivalent.
(I)The real part of (7.6.30) restricts to a symmetric inner product onl.
(II)Φ
∗
Φ + Ψ
∗
Ψ∈Aut(g).
(III)lis a Lagrangian subspace ofg
c
with respect to the imaginary part
of (7.6.30), i.e. Φ
∗
Ψ−Ψ
∗
Φ = 0.
Hint:If Φ
∗
Φ + Ψ
∗
Ψ∈Aut(g), then there exists a derivationα∈Der(g)
such thatα=α
∗
and exp(2α) = Φ
∗
Φ + Ψ
∗
Ψ (Corollary 7.6.21). Prove that
δ:= exp(−α)(Φ
∗
Ψ−Ψ
∗
Φ) exp(−α)
is a derivation whose image is abelian. Prove that
κ(δξ, δη) = 0
for allξ, η∈gand deduce thatδ
∗
δ=−δ
2
= 0.
(g)The space of oriented Lagrangian Lie subalgebrasl⊂g
c
isomorphic tog
(that can be joined togby a path of such subspaces) is diffeomorphic to the
quotient space
Lg:= K
c
/K,
where K
c
:= Aut0(g
c
)∩SU(g
c
) and K := Aut0(g)∩SO(g).Hint:Choose
the embedding Φ +iΨ in (f) such that
Φ
∗
Φ + Ψ
∗
Ψ = 1l,Φ
∗
Ψ−Ψ
∗
Φ = 0,
and extend it to a unitary automorphism ofg
c
.
(h)The spaceLgin part (g) embeds as a totally geodesic submanifold
of dimension dim(Lg) = dim(p) into the symmetric space U(g
c
)/SO(g) of
all oriented Lagrangian subspaces ofg
c
. HenceLghas nonnegative sectional
curvature. (Hint:Example 5.2.18.) One can think of the positive curvature
manifoldLgof all Lagrangian Lie subalgebras ofg
c
that are isomorphic
togas dual to the negative curvature manifoldMgof all symmetric inner
products ongthat satisfy condition (iii) in Theorem 7.6.20.

7.6. SEMISIMPLE LIE ALGEBRAS* 409
Remark 7.6.29.The idea of minimizing the norm of the Lie bracket was
the approach to the existence of a compact real form of a simple complex
Lie algebra suggested by Cartan [14] and carried out by Richardson [58].
One significant difference in the method of Donaldson [17], which we follow
in the section, is that there is no need to assume that the Killing form is
nondegenerate, but that this result emerges as a byproduct of the proof
(Theorem 7.6.10). There is also no need to use the structure theory of Lie
algebras as in the work of Weyl [77]. Instead one can use the existence of a
symmetric inner product as a starting point to develop the structure theory
of Lie algebras.
Remark 7.6.30.As pointed out by Donaldson [17], a more direct approach
to Theorem 7.6.26 would be to carry over the entire program in the present
section and§7.5 to the complex setting, starting in§7.5.2 with convex func-
tions on the Hadamard manifoldM=Q0(V)
∼
=SL(V)/SU(V) of positive
definite Hermitian automorphisms with determinant one of a complex vector
spaceVequipped with a Hermitian inner product (Remark 6.5.20).
In the complex Lie algebra setting withQ0(g)
∼
=SL(g,C)/SU(g,C) the
logarithm of the functionfg:Q0(g)→Ris the log-norm function of Kempf
and Ness in geometric invariant theory [38, 20]. Thus the existence of a
critical point offgis the polystability condition in GIT. This approach
was developed by Lauret [44] and he proved that the polystable points are
precisely the semisimple Lie algebras. This is the content of Theorem 7.6.20
in the complex setting. Lauret’s proof uses Cartan’s theorem about the
compact real form of a semisimple complex Lie algebra.
One can also deduce the theorems in the real setting from those in the
complex setting by complexifying the relevant real inner product spaceV
to obtain a complex vector spaceV
c
=V⊕iVwith a Hermitian inner
product, and embedding the spaceP0(V)
∼
=SL(V)/SO(V) as a totally
geodesic submanifold intoQ0(V
c
)
∼
=SL(V
c
)/SU(V
c
).

410 CHAPTER 7. TOPICS IN GEOMETRY

Appendix A
Notes
This appendix explains some notations and standard results from first year
analysis that are used throughout this book.
A.1 Maps and Functions
The notationf:X→Ymeans thatfis a function which assigns to every
pointxin the setXa pointf(x) in the setY. WhenY=Rwe express this
by saying thatfis areal valuedfunction defined on the setXand, ifYis
a vector space, we may say thatfis avector valuedfunction. However in
general it is better to say thatfis amapfromXtoYand call the setX
thesourceof the map and the setYitstarget. Thegraphoffis the set
graph(f) :={(x, y)∈X×Y|y=f(x)}.
We always distinguish two maps with the same graph when their targets are
different.
A mapf:X→Yis said to be



injective
surjective
bijective



iff



f(x1) =f(x2) =⇒x1=x2
∀y∈Y∃x∈Xs.t.y=f(x)
it is both injective and surjective.



Then
(a)fis injective⇐⇒it has a left inverseg:Y→X(i.e.g◦f= idX);
(b)fis surjective⇐⇒it has a right inverseg:Y→X(i.e.f◦g= idY);
(c)fis bijective⇐⇒it has a two sided inversef
−1
:Y→X.
(Item (b) is theAxiom of Choice.)
411

412 APPENDIX A. NOTES
The analogous principle holds for linear maps: ifA∈R
m×n
, then the
linear mapR
n
→R
m
:x7→Axis
(a)injective⇐⇒BA= 1lnfor someB∈R
n×m
;
(b)surjective⇐⇒AB= 1lmfor someB∈R
n×m
;
(c)bijective⇐⇒ Ais invertible (i.e.m=nand det(A)̸= 0).
(Here 1lkis thek×kidentity matrix.) However, this principle fails com-
pletely for continuous maps: the mapf: [0,2π)→S
1
defined byf(θ) =
(cosθ,sinθ) is continuous and bijective but its inverse is not continuous.
(HereS
1
⊂R
2
is the unit circlex
2
+y
2
= 1.)
A.2 Normal Forms
TheFundamental Idea of Differential Calculusis that near a pointx0∈U
a smooth mapf:U→Vbehaves like its linear approximation, i.e.
f(x)≈f(x0) +df(x0)(x−x0).
TheNormal Form Theorem from Linear Algebrasays that ifA∈R
m×n
has
rankr, then there are invertible matricesP∈R
m×m
andQ∈R
n×n
such
that
P
−1
AQ=
`
1lr 0
r×(n−r)
0
(m−r)×r0
(m−r)×(n−r)
´
.
By the Fundamental Idea we can expect an analogous theorem for smooth
maps.
Theorem A.2.1(Local Normal Form for Smooth Maps). LetU⊂R
n
andV⊂R
m
be open,x0∈U, andf:U→Vbe smooth. Assume that the
derivativedf(x0)∈R
m×n
has rankr. Then there is an open neighborhood
U0ofx0inU, an open neighborhoodV0off(x0)inV, a diffeomorphism
ϕ:U1×U2⊂R
r
×R
n−r
, a diffeomorphismψ:V0→U1×V2⊂R
r
×R
m−r
,
such thatϕ(x0) = (0,0),ψ(f(x0)) = (0,0), and
ψ
−1
◦f◦ϕ(x, y) = (x, g(x, y))and dg(0,0) = 0
for(x, y)∈U1×U2.
The Local Normal Form Theorem is an easy consequence of the Inverse
Function Theorem.

A.2. NORMAL FORMS 413
Theorem A.2.2(Inverse Function Theorem).LetU⊂R
n
,V⊂R
m
,
x0∈Uandf:U→Vbe a smooth map. Ifdf(x0)is invertible, then (m=n
and) there are neighborhoodsU0ofx0inUandV0off(x0)inVso that the
restrictionf
|U0
:U0→V0is a diffeomorphism.
Here follow some other consequences of the Inverse Function Theorem.
The termssubmersionandimmersionare defined in§2.6.1 and Defini-
tion 2.3.2 of§2.3.
Corollary A.2.3(Submersion Theorem). Whenr=mthe diffeomor-
phismsϕandψin Theorem A.2.1 may be chosen so that the local normal
form is
ψ
−1
◦f◦ϕ(x, y) =x.
Corollary A.2.4(Immersion Theorem). Whenr=nthe diffeomor-
phismsϕandψin Theorem A.2.1 may be chosen so that the local normal
form is
ψ
−1
◦f◦ϕ(x) = (x,0).
Corollary A.2.5(Rank Theorem).If the rank ofdf(x) =rfor allx∈U,
then for everyx0∈Uthe diffeomorphismsϕandψin Theorem A.2.1 may
be chosen so that the local normal form is
ψ
−1
◦f◦ϕ(x) = (x1, . . . , xr,0, . . . ,0).
Corollary A.2.6(Implicit Function Theorem).LetU⊂R
m
×R
n
be
an open set, letF:U→R
n
be smooth, and let(x0, y0)∈Uwithx0∈R
m
andy0∈R
n
. Define the partial derivatived2F(x0, y0)∈R
n×n
by
d2F(x0, y0)v:=
d
dt

t=0
F(x0, y0+tv)
forv∈R
n
. Assume thatF(x0, y0) = 0and thatd2F(x0, y0)is invertible.
Then there exist neighborhoodsU0ofx0inR
m
andV0ofy0inR
n
and a
smooth mapg:U0→V0such that
U0×V0⊂U, g(x0) =y0
and
F(x, y) = 0 ⇐⇒ y=g(x)
forx∈U0andy∈V0.

414 APPENDIX A. NOTES
A.3 Euclidean Spaces
This is the arena of Euclidean geometry; i.e. every figure which is studied in
Euclidean geometry is a subset of Euclidean space. To define it one could
proceed axiomatically as Euclid did; one would then verify that the ax-
ioms characterized Euclidean space by constructing “Cartesian Coordinate
Systems” which identify then-dimensional Euclidean spaceE
n
with then-
dimensional numerical spaceR
n
. This program was carried out rigorously
by Hilbert. We shall adopt the mathematically simpler but philosophically
less satisfying course of taking the characterization as the definition.
We shall use three closely related spaces:n-dimensional Euclidean affine
spaceE
n
,n-dimensional Euclidean vector spaceE
n
, and the spaceR
n
of
alln-tuples of real numbers. The distinction among them is a bit pedantic,
especially if one views as the purpose of geometry the interpretation of
calculations onR
n
. The purpose for distinguishing these three spaces is the
same as in elementary vector calculus; it aids geometric intuition. Here is
the precise definition.
Definition A.3.1.Ann-dimensionalEuclidean vector spaceis a realn-
dimensional vector spaceE
n
equipped with a (real valued symmetric positive
definite) inner productE
n
×E
n
→R: (v, w)7→ ⟨v, w⟩.Ann-dimensional
Euclidean affine spaceconsists of a setE
n
and ann-dimensional Euclidean
vector spaceE
n
and maps
E
n
×E
n
→E
n
: (p, q)7→p−q,
E
n
×E
n
→E
n
: (p, v)7→p+v
satisfying the axioms
p+ 0 =p, p+ (v+w) = (p+v) +w, q+ (p−q) =p
for allp, q∈E
n
and allv, w∈E
n
. The vectorp−q∈E
n
is called the
vectorfromqtopand the pointp+vis called thetranslateofpbyv.
It follows easily that each choice of a pointo∈E
n
determines a bijection
v7→o+vfromE
n
ontoE
n
. The inner product onE
n
equips the spaceE
n
with a metric via the formula
|p−q|=
p
⟨p−q, p−q⟩, p, q∈E
n
.
Thestandard Euclidean spaceof dimensionnisE
n
=E
n
=R
n
with the
usual matrix algebra operations (x±y)
i
=x
i
±y
i
,⟨x, y⟩=
P
i
x
i
y
i
.

A.3. EUCLIDEAN SPACES 415
Lemma A.3.2.Any choice of an origino∈E
n
and an orthonormal ba-
sise1, . . . , enforE
n
determines an isometric bijection:
R
n
→E
n
: (x
1
, . . . , x
n
)7→o+
n
X
i=1
x
i
ei
(the inverse of which is called aCartesian coordinate systemonE
n
).
Lemma A.3.3.IfE
n
→R
n
:p7→(x
1
, . . . , x
n
),(y
1
, . . . , y
n
)are two Carte-
sian coordinate systems, thechange of coordinates maphas the form
y
j
(p) =
n
X
j=1
a
j
i
x
i
(p) +v
i
,
where the matrixa= (a
j
i
)∈R
n×n
is an orthogonal matrix andv∈R
n
.
Example A.3.4.Anyn-dimensional affine subspace of some numerical
spaceR
k
(withk > n) is an example of a Euclidean space. The corre-
sponding vector spaceE
n
is the unique vector subspace ofR
k
for which:
E
n
=o+E
n
foro∈E
n
. This subspace is independent of the choice ofo∈E
n
. Note
thatE
n
contains the “preferred” point 0 whileE
n
has no preferred point.
Such spacesE
n
andE
n
would arise in linear algebra by takingE
n
to be
the space of solutions ofk−nindependent inhomogeneous linear equations
inkunknowns whileE
n
is the space of solutions of the corresponding ho-
mogeneous equations. The correspondence betweenE
n
andE
n
illustrates
the mantra
The general solution of an inhomogeneous system of linear equa-
tions is a particular solution plus the general solution of the cor-
responding homogeneous linear system.
This discussion shows that a Euclidean spaceE
n
is ann-dimensional
manifold with its Cartesian coordinate systems whose tangent space at each
point is naturally isomorphic toE
n
. Thus it is natural to introduce sub-
manifolds of Euclidean space as submanifolds ofE
n
whose tangent spaces
are then linear subspaces of the associated vector spaceE
n
. Instead we have
chosen in this book for simplicity of the exposition to describe manifolds as
subsets of the vector spaceR
n
equipped with its standard inner product.

416 APPENDIX A. NOTES

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manifolds.Osaka Journal of Mathematics12(1960), 21–37.

Index
abelian Lie algebra, 391
affine subspace, 159
atlas, 4, 87
maximal, 9, 11
smooth, 11
topological, 8
automorphism
of a Lie algebra, 58, 61
of a Lie group, 58
ofF(M), 69
Axiom of Choice, 411
basic vector field, 145, 173
Bianchi identity
first, 239
in local coordinates, 254
second, 276
in local coordinates, 278
Bonnet–Myers Theorem, 310
Brouwer, 8
bundle
frame, 138
principal, 115, 140
tangent, 74, 97
vector, 74
Cartan decomposition, 388, 403
Cartan Fixed Point Theorem, 299
Cartan involution, 400
Cartan–Ambrose–Hicks Theorem
global, 259
weak version, 264
local, 265
Cartan–Hadamard Theorem, 293
Cartesian coordinate system, 415
category, 12
catenoid, 272
chain rule, 8, 10, 16, 29
Christoffel symbols, 162, 172
Closed Subgroup Theorem, 62
compact-open topology, 343
compatible, 87
smoothly, 11
topologically, 8
complete, 145
geodesically, 209
vector field, 42
complex Lie algebra, 404
compact real form, 405
semisimple, 404
simple, 404
complex projective space, 89
conformal
chart, 2
class of metrics, 317
diffeomorphism, 321
invariance of Weyl tensor, 321
conformally flat, 321
conjufate point
multiplicity, 328
conjugate point, 328
connection onT M, 235, 240
Levi-Civita, 240
Riemannian, 235, 240
423

424 INDEX
torsion-free, 235, 240
constant
scalar curvature, 316
sectional curvature, 281
Convex Neighborhood Theorem, 205
coordinate chart, 4, 11, 16, 87
frame bundle, 173
geodesically normal, 206
Monge, 28
smoothly compatible, 11, 87
tangent bundle, 74, 97
topologically compatibe, 8
countable base, 101
covariant derivative
along a curve, 127, 171
of a global vector field, 234
of a normal vector field, 135
of the curvature, 275
partial, 232
critical point
of the energy functional, 179
of the length functional, 179
curvature
Gaußian, 250, 251
in local coordinates, 255
Ricci, 309
in local coordinates, 309
positive, 310
Riemann, 232
in local coordinates, 254
scalar, 313
constant, 316
in local coordinates, 315
positive, 316
sectional, 280
constant, 281
nonpositive, 293
positive, 312
Weyl, 319
in local coordinates, 321
Curve Shortening Lemma, 213
intrinsic setting, 220
derivation
on a Lie algebra, 61
onF(M), 71
derivative, 10, 28
of the energy functional, 179
of the length functional, 179
developable manifold, 269
osculating, 270
tangent, 271
development, 154
diffeomorphic, 15, 92
diffeomorphism, 10, 15, 30, 92
conformal, 321
local, 33
dimension, 4
distance function, 183
intrinsic setting, 217
Einstein metric, 318
ellipsoid, 5
elliptic hyperboloid, 271
embedding, 34
Minto Euclidean space, 107
Pl¨ucker, 96
RP
2
intoR
6
, 22
Veronese, 96
energy of a curve, 176
Euclidean space, 414
Euler’s curvature, 250
Euler–Lagrange equations, 196
exponential map, 192
distance increasing, 293
exponential matrix, 55
extremal
of the energy functional, 177
of the length functional, 177
fiber of a vector bundle, 74

INDEX 425
first fundamental form, 122
flat, 268
flow, 41
foliation, 85
closed leaf, 113
foliation box, 80
frame bundle, 138, 173
coordinate chart, 173
horizontal tangent vector, 173
orthonormal, 140
Frobenius’ theorem, 80
Fubini–Study metric, 169
functorial, 70
fundamental form
first, 122
second, 125
Gauß Lemma, 199
Gauß map, 123, 249
Gauß–Codazzi formula, 233
Gauß–Weingarten formula
along a curve, 127
for global vector fields, 234
for normal vector fields, 135
Gaußian curvature, 250, 251, 280
in local coordinates, 255
general linear group, 23, 57
geodesic, 177
broken, 204
conjugate point, 328
in local coordinates, 196
intrinsic setting, 219
minimal, 3, 197
global existence, 210
local existence, 199
spray, 191
geodesically
complete, 209
convex, 205
normal coordinates, 206
Gr¨onwall’s inequality, 350
Grassmann manifold, 7, 90
great circle, 3
group action, 68
contravariant, 137
finite isotropy, 114
fixed point, 299
free, 114, 137
infinitesimal, 68, 114
orbit space, 114
smooth, 68, 114
transitive, 137
groupoid, 342
smooth, 342
Hadamard manifold, 293
sphere at infinity, 368
Hausdorff space, 91
Heisenberg algebra, 58
helicoid, 272
homogeneous space, 119, 274
homomorphism
of algebras, 69
of Lie algebras, 58
of Lie groups, 58
homotopy, 257
with fixed endpoints, 258
Hopf fibration, 221
Hopf–Rinow Theorem, 210
horizontal
curve, 142
lift, 142
subbundle, 173
tangent space, 142, 173
hyperbolic space, 286
Poincar´e model, 2, 290
upper half plane, 292
hyperboloid, 271
hypersurface, 249

426 INDEX
ideal
in a complex Lie algebra, 404
in a Lie algebra, 391
inF(M), 70
immersion, 34
Implicit Function Theorem, 79, 413
injectivity radius, 194
inner product
symmetric, 382, 391
integrable subbundle ofT M, 80
integral curve, 39
intrinsic
distance, 183, 217
topology, 86, 88
Invariance of Domain, 8
Inverse Function Theorem, 31
involutive subbundle ofT M, 80
isometry, 224
group, 224, 343
local, 230
of a Lie group, 362
isomorphism, 12
of Lie algebras, 58
of Lie groups, 58
isotopy, 45
isotropy subgroup, 114
Jacobi field, 266
Jacobi identity, 49
Jacobian matrix, 10
Kato Selection Theorem, 333
Killing form, 242
nondegenerate, 243
Killing form complex, 404
Killing vector field, 347
complete, 348, 349
Klein bottle, 112
Kuiper’s Theorem, 322
Laplace–Beltrami operator, 317
leaf of a foliation, 85, 113
closed, 113
extrinsic topology, 86
intrinsic topology, 86
length of a curve, 176
Levi-Civita connection, 171, 234
axioms, 171
Lie algebra, 50
abelian, 391
automorphism, 58, 61
Cartan involution, 400
center, 66
commutant, 242
complex, 404
derivation, 61
Heisenberg, 58
homomorphism, 50, 58
ideal, 391
infinitesimal action, 68, 114
invariant inner product, 242
isomorphism, 58
of a Lie group, 55
of vector fields, 50
semisimple, 399
simple, 391
symmetric inner product, 391
Lie bracket of vector fields, 49, 99
Lie group, 52
automorphism, 58
bi-invariant metric, 242
Cartan decomposition, 403
center, 66
complex, 406
group action, 68
homomorphism, 58
complexification, 383
irreducible representation, 378
isomorphism, 58
right invariant metric, 243
symplectic, 53, 390

INDEX 427
Lie subgroup, 62
lift of a curve, 142
local coordinates, 4
local diffeomorphism, 33
local isometry, 230
local trivialization, 76
locally finite open cover, 101
locally finite sum, 106
locally symmetric space, 273
manifold, 8, 11, 16, 87
complex, 13
developable, 269
dimension, 8
flat, 268
oriented, 322
osculating developable, 270
real analytic, 13
smooth, 11, 16, 87
symplectic, 13
tangent developable, 271
topological, 8, 88
maximal existence interval, 41
for geodesics, 192
maximal ideal inF(M), 70
maximal smooth atlas, 11
maximal topological atlas, 9
mean curvature, 272
metrizable, 101
minimal geodesic, 197
existence, 199, 210
minimal surface, 272
M¨obius strip, 78, 123, 272
paper model, 271
Monge coordinates, 28
Morse index, 330
motion, 146
without sliding, 149
without twisting, 151
without wobbling, 152
moving frame, 139, 142
Myers–Steenrod Theorem, 343
normal bundle, 76
1-form with values inT M, 240
oriented manifold, 322
orthogonal group, 23
paraboloid, 271
elliptic, 6
hyperbolic, 6
paracompact, 101
parallel
transport, 131
vector field, 129
parametrized
by the arclength, 182
proportional to arclength, 177
partition of unity, 103
existence, 103
subordinate to a cover, 103
Pl¨ucker embedding, 96
Pl¨ucker’s conoid, 271
Poincar´e metric, 290
principal bundle, 115, 140
projection
obvious, 9
orthogonal, 75
projective space, 7
complex, 89
real, 89
proper, 34
pullback
of a vector bundle, 76
of a vector field, 46
pushforward
of a vector field, 46
quadric surface, 5
quaternions, 53

428 INDEX
quotient space, 9
quotient topology, 9
real projective space, 22, 89
refinement, 101
regular value, 21, 32
relative topology, 9
relatively closed, 9
relatively open, 9
Ricci tensor, 309
in local coordinates, 309
positive, 310
Riemann curvature tensor, 232
covariant derivative, 275
in local coordinates, 278
first Bianchi identity, 239
in local coordinates, 254
second Bianchi identity, 276
Riemann surface, 102
Riemannian manifold, 166
Einstein, 318
flat, 268
homogeneous, 274
locally symmetric, 273
symmetric, 273
Riemannian metric, 13, 166
bi-invariant, 242
conformal class, 317
distance function, 183, 217
existence, 168
Roman surface, 22
ruled surface, 271
scalar curvature, 313
constant, 316
Yamabe problem, 317
in local coordinates, 315
positive, 316
second Bianchi identity, 276
second countable, 101
second fundamental form, 125
section of a vector bundle, 74
sectional curvature, 280
constant, 281
nonpositive, 293
positive, 312
self-dual
four-manifold, 325
Weyl tensor, 325
Siegel upper half space, 308
simple Lie algebra, 391
simply connected, 258
singular value, 21
Smirnov Metrization Theorem, 102
smooth, 10
atlas, 11
group action, 68
groupoid, 342
manifold, 11, 16, 87
map, 10, 15, 92
structure, 12, 88
vector bundle, 74
smoothly compatible, 11
special
linear group, 23, 57
unitary group, 57
sphere, 5
Sphere Theorem, 312
spray, 191
geodesic, 191
stereographic projection, 3
structure group, 140
subbundle, 80, 99
integrable, 80
involutive, 80
submanifold, 34, 95
totally geodesic, 303
submersion, 72
surface, 251
quadric, 5

INDEX 429
symmetric space, 273
is homogeneous, 274
locally, 273
symplectic vector space, 390
compatible inner product, 390
compatible linear complex
structure, 390
tangent bundle, 74, 97
coordinate chart, 74, 97
integrable subbundle, 80
involutive subbundle, 80
of the frame bundle, 141
of the tangent bundle, 191
tangent space, 24, 93
horizontal, 142, 173
of the frame bundle, 141
of the tangent bundle, 191
vertical, 142
tangent vector, 24, 93
Theorema Egregium, 245, 253
topological atlas, 8
topological manifold, 8, 88
topological space
countable base, 101
Hausdorff, 91
locally compact, 102
metrizable, 101
normal, 102
paracompact, 101
regular, 102
second countable, 101
σ-compact, 101
topologically compatibe, 8
topology
compact-open, 343
intrinsic, 88
quotient, 9
relative, 9
torus, 90
totally geodesic submanifold, 303
transition map, 8, 11, 87
2-form, 322
anti-self-dual, 323
self-dual, 323
Tychonoff’s Lemma, 102
uniformization theorem, 316
unitary group, 57
Urysohn Metrization Theorem, 102
variation of a curve, 176
vector bundle, 74
canonical projection, 74
local trivialization, 76
section, 74
vector field, 38, 97
along a curve, 127
along a map, 170
basic, 145, 173
flow, 41
integral curve, 39
Killing, 347
Lie bracket, 49
normal, 135
parallel, 129
with compact support, 43
Veronese embedding, 96
vertical tangent space, 142
wallpaper group, 360
Weyl tensor, 319
in local coordinates, 321
self-dual, 325
Whitney Embedding Theorem, 112
Yamabe problem, 317

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