Aspects of Differential Geometry [I, 1 ed.] 9781627056632

Table of contents :
Front Cover......Page 1
Half-Title......Page 3
Copyright......Page 6
Title......Page 7
ABSTRACT, KEYWORDS......Page 8
Contents......Page 11
Preface......Page 13
Acknowledgments......Page 15
1.1 Metric Spaces......Page 17
1.2 Linear Algebra......Page 19
1.3 The Derivative......Page 23
1.4 The Inverse and Implicit Function Theorems......Page 28
1.5 The Riemann Integral......Page 39
2.1 Smooth Manifolds......Page 69
2.2 The Tangent and Cotangent Bundles......Page 77
2.3 Stokes' Theorem......Page 86
2.4 Applications of Stokes' Theorem......Page 100
3.1 The Pseudo-Riemannian Measure......Page 107
3.2 Connections......Page 111
3.3 The Levi–Civita Connection......Page 116
3.4 Geodesics......Page 119
3.5 The Jacobi Operator......Page 126
3.6 The Gauss–Bonnet Theorem......Page 133
3.7 The Chern–Gauss–Bonnet Theorem......Page 142
Bibliography......Page 147
Authors' Biographies......Page 151
Index......Page 153

Citation preview

SSyntheSiS yntheSiS yntheSiSL LectureS ectureS ectureSon on on M atheMaticS atheMaticSand and andSStatiSticS tatiSticS tatiSticS MatheMaticS Series Series SeriesEditor: Editor: Editor:Steven Steven StevenG. G. G.Krantz, Krantz, Krantz,Washington Washington WashingtonUniversity, University, University,St. St. St.Louis Louis Louis

Aspects Aspectsof ofDifferential DifferentialGeometry GeometryII Peter Peter PeterGilkey, Gilkey, Gilkey,University University UniversityofofofOregon, Oregon, Oregon,Eugene, Eugene, Eugene,OR OR OR

JeongHyeong JeongHyeong JeongHyeongPark, Park, Park,Sungkyunkwan Sungkyunkwan SungkyunkwanUniversity, University, University,Suwon, Suwon, Suwon,Korea Korea Koreaand and and Institute Institute Institutefor for forAdvanced Advanced AdvancedStudy, Study, Study,Seoul, Seoul, Seoul,Korea Korea Korea

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ASPECTS OF DIFFERENTIAL DIFFERENTIAL GEOMETRY II ASPECTS ASPECTS OF OF DIFFERENTIAL GEOMETRY GEOMETRY I

Ramón Ramón RamónVázquez-Lorenzo, Vázquez-Lorenzo, Vázquez-Lorenzo,University University UniversityofofofSantiago Santiago Santiagode de deCompostela, Compostela, Compostela,Santiago Santiago Santiagode de deCompostela, Compostela, Compostela,Spain Spain Spain

GILKEY • PARK •• VÁZQUEZ-LORENZO VÁZQUEZ-LORENZO GILKEY GILKEY •• PARK PARK • VÁZQUEZ-LORENZO

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Aspects Aspects of of Differential Differential Geometry Geometry II

Peter PeterGilkey Gilkey JeongHyeong JeongHyeongPark Park Ramón RamónVázquez-Lorenzo Vázquez-Lorenzo

SSyntheSiS yntheSiS yntheSiSL LectureS ectureS ectureSon on on MatheMaticS atheMaticS atheMaticSand and andSStatiSticS tatiSticS tatiSticS M Steven Steven StevenG. G. G.Krantz, Krantz, Krantz,Series Series SeriesEditor Editor Editor

Aspects of Differential Geometry I

Synthesis Lectures on Mathematics and Statistics Editor Steven G. Krantz, Washington University, St. Louis

Aspects of Differential Geometry I Peter Gilkey, JeongHyeong Park, and Ramón Vázquez-Lorenzo 2015

An Easy Path to Convex Analysis and Applications Boris S. Mordukhovich and Nguyen Mau Nam 2013

Applications of Affine and Weyl Geometry Eduardo García-Río, Peter Gilkey, Stana Nikčević, and Ramón Vázquez-Lorenzo 2013

Essentials of Applied Mathematics for Engineers and Scientists, Second Edition Robert G. Watts 2012

Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations Goong Chen and Yu Huang 2011

Matrices in Engineering Problems Marvin J. Tobias 2011

e Integral: A Crux for Analysis Steven G. Krantz 2011

Statistics is Easy! Second Edition Dennis Shasha and Manda Wilson 2010

iii

Lectures on Financial Mathematics: Discrete Asset Pricing Greg Anderson and Alec N. Kercheval 2010

Jordan Canonical Form: eory and Practice Steven H. Weintraub 2009

e Geometry of Walker Manifolds Miguel Brozos-Vázquez, Eduardo García-Río, Peter Gilkey, Stana Nikčević, and Ramón Vázquez-Lorenzo 2009

An Introduction to Multivariable Mathematics Leon Simon 2008

Jordan Canonical Form: Application to Differential Equations Steven H. Weintraub 2008

Statistics is Easy! Dennis Shasha and Manda Wilson 2008

A Gyrovector Space Approach to Hyperbolic Geometry Abraham Albert Ungar 2008

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ISBN: 9781627056625 ISBN: 9781627056632

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DOI 10.2200/S00632ED1V01Y201502MAS015

A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS Lecture #15 Series Editor: Steven G. Krantz, Washington University, St. Louis Series ISSN Print 1938-1743 Electronic 1938-1751

Aspects of Differential Geometry I Peter Gilkey University of Oregon, Eugene, OR

JeongHyeong Park Sungkyunkwan University, Suwon, Korea Institute for Advanced Study, Seoul, Korea

Ramón Vázquez-Lorenzo University of Santiago de Compostela, Santiago de Compostela, Spain

SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS #15

M &C

Morgan & cLaypool publishers

ABSTRACT Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. In Book I, we focus on preliminaries. Chapter 1 provides an introduction to multivariable calculus and treats the Inverse Function eorem, Implicit Function eorem, the theory of the Riemann Integral, and the Change of Variable eorem. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and Stokes’ eorem. Chapter 3 is an introduction to Riemannian geometry. e Levi–Civita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the Gauss–Bonnet eorem is proved. e material is appropriate for an undergraduate course in the subject. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the Chern–Gauss–Bonnet eorem for pseudo-Riemannian manifolds with boundary is new.

KEYWORDS Change of Variable eorem, derivative as best linear approximation, Fubini’s eorem, Gauss–Bonnet eorem, Gauss’s eorem, geodesic, Green’s eorem, Implicit Function eorem, improper integrals, Inverse Function eorem, Levi–Civita connection, partitions of unity, pseudo-Riemannian geometry, Riemann integral, Riemannian geometry, Stokes’ eorem

vii

is book is dedicated to Alison, Arnie, Carmen, Junmin, Junpyo, Manuel, Montse, Rosalía, and Susana.

ix

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1

2

3

Basic Notions and Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1

Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 1.3

Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 e Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4

e Inverse and Implicit Function eorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5

e Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.6

Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.7

e Change of Variable eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.1

Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2

e Tangent and Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3

Stokes’ eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.4

Applications of Stokes’ eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Riemannian and Pseudo-Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.1

e Pseudo-Riemannian Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2

Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.3

e Levi–Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.4

Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.5

e Jacobi Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.6

e Gauss–Bonnet eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.7

e Chern–Gauss–Bonnet eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

x

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Authors’ Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xi

Preface is two-volume series arose out of work by the three authors over a number of years both in teaching various courses and also in their research endeavors. e present volume (Book I) is comprised of three chapters. Chapter 1 provides an introduction to multivariable calculus. It begins with two introductory sections on metric spaces and linear algebra. Various notions of differentiability are introduced and the chain rule is proved. e Inverse and Implicit Function eorems are established. One then turns to the theory of integration. e Riemann integral is introduced and it is shown that a bounded function is integrable if and only if it is integrable almost everywhere. Compact exhaustions by Jordan measurable sets, mesa functions, and partitions of unity are used to define improper integrals. Chapter 1 concludes with a proof of the Change of Variable eorem; upper and lower sums defined by cubes (rather than rectangles) together with partitions of unity are the fundamental tools employed. Chapter 2 completes the discussion of multivariable calculus. e basic materials concerning smooth manifolds are introduced. It is shown any compact manifold embeds smoothly in Rm for some m. A brief introduction to fiber bundle theory and vector bundle theory is given and the tangent and cotangent bundles are introduced. is formalism is then combined with the results of Chapter 1 to establish the generalized Stokes’ eorem. e classical Green’s eorem, Gauss’s eorem, and Stokes’ eorem are then established. e Brauer Fixed Point eorem, the Fundamental eorem of Algebra, and the Combing the Hair on a Billiard Ball eorem are presented as applications. Chapter 3 presents an introduction to Riemannian and pseudo-Riemannian geometry. e volume form is introduced. e notion of a connection on an arbitrary vector bundle is presented and the discussion is then specialized to the Levi–Civita connection. Geodesics are treated and the classical Hopf–Rinow eorem giving various equivalent notions of completeness is established in the Riemannian setting. e Jacobi operator is introduced and used to establish the Myers eorem that if the Ricci tensor on a complete Riemannian manifold is uniformly positive, then the manifold is compact and has finite fundamental group. Riemann surfaces are introduced and the classical Gauss–Bonnet eorem is established. e Chern–Gauss–Bonnet eorem in higher dimensions is treated and analytic continuation used to establish an analogous result in the pseudo-Riemannian setting.

xii

PREFACE

We have tried whenever possible to give the original references to major theorems in this area. We have provided a number of pictures to illustrate the discussion, especially in Chapters 1 and 2. Chapters 1 and 2 are suitable for an undergraduate course on “Calculus on Manifolds” and arose in that context out of a course at the University of Oregon. Chapter 3 is designed for an undergraduate course in Differential Geometry. us Book I is suitable as an undergraduate text although, of course, it also forms the foundation of a graduate course in Differential Geometry as well. Book II can be used as a graduate text in Differential Geometry and arose in that context out of a second-year graduate course in Differential Geometry at the University of Oregon. e material can, however, also form the basis of a second-semester course at the undergraduate level as well. While much of the material is, of course, standard, many of the proofs are a bit different from those given classically and we hope provide a new viewpoint on the subject. ere are also new results in the book; our treatment of the generalized Chern–Gauss–Bonnet eorem in the indefinite signature context arose out of our study of Euler–Lagrange equations using perturbations of complex metrics (i.e., metrics where the gij tensor is C -valued). Similarly, our treatment of curves in Rm given by the solution to constant coefficient ODEs which have finite total curvature is new. ere are other examples; Differential Geometry is of necessity a vibrant and growing field – it is not static! ere are, of course, many topics that we have not covered – this is a work on “Aspects of Differential Geometry” and of necessity must omit more topics than can possibly be included. For technical reasons, the material is divided into two books and each book is largely selfsufficient. To facilitate cross references between the two books, we have numbered the chapters of Book I from 1 to 3, and the chapters of Book II from 4 to 8. Peter Gilkey, JeongHyeong Park, and Ramón Vázquez-Lorenzo February 2015

xiii

Acknowledgments We have provided many images of famous mathematicians in these two books; mathematics is created by real people and we think having such images makes this point more explicit. e older pictures are in the public domain. We are grateful to the Archives of the Mathematisches Forschungsinstitut Oberwolfach for permitting us to use many images from their archives (R. Brauer, H. Cartan, S. Chern, G. de Rham, S. Eilenberg, H. Hopf, E. Kähler, H. Künneth, L. Nirenberg, H. Poincaré, W. Rinow, L. Vietoris, and H. Weyl); the use of these images was granted to us for the publication of these books only and their further reproduction is prohibited without their express permission. Some of the images (E. Beltrami, E. Cartan, G. Frobenius, and F. Klein) provided to us by the MFO are from the collection of the Mathematische Gesellschaft Hamburg; again, the use of any of these images was granted to us for the publication of these books only and their further reproduction is prohibited without their express permission. e research of the authors was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014053413) and by Project MTM2013-41335-P with FEDER funds (Spain). e authors are very grateful to Esteban Calviño-Louzao and Eduardo García-Río for constructive suggestions and assistance in proofreading. It is a pleasure to thank Mr. David Swanson for constructive suggestions concerning the exposition of Chapter 1. e assistance of Ekaterina Puffini of the Krill Institute of Technology has been invaluable. Wikipedia has been a useful guide to tracking down the original references and was a source of many of the older images that we have used that are in the public domain. Peter Gilkey, JeongHyeong Park, and Ramón Vázquez-Lorenzo February 2015

1

CHAPTER

1

Basic Notions and Concepts In Chapter 1 we present some material from multivariable calculus that will form the foundation of our later discussion of manifold theory. We begin in Section 1.1 by reviewing the theory of metric spaces. Section 1.2 presents some preliminary material concerning linear algebra. Normed vector spaces and inner product spaces are treated as is the norm of a linear map between normed linear spaces. e derivative as the best linear approximation is discussed in Section 1.3 and the chain rule is established. e Inverse and Implicit Function eorems are proved in Section 1.4. A brief sketch of the theory of Riemann integration is given in Section 1.5. Sets of measure zero and content zero are discussed and Fubini’s eorem is proved. In Section 1.6, improper integrals are introduced using mesa functions and partitions of unity. We conclude Chapter 1 by proving the Change of Variable eorem in Section 1.7. e calculus was founded by Gottfried Wilhelm von Leibniz and Sir Isaac Newton.

Gottfried Wilhelm von Leibniz (1646–1716)

1.1

Sir Isaac Newton (1643–1727)

METRIC SPACES

We present some basic material concerning metric spaces to establish notation; a good reference would be Baum [6] for point set topology and Rudin [36] for metric spaces. Let X be a set and let O be a collection of subsets of X . We say that O is a topology on X if the following axioms are satisfied: 1. If fOi gniD1 is a finite collection of elements of O then O1 \


\ On 2 O: 2. If fO˛ g˛2A is an arbitrary collection of elements of O, then [˛2A O˛ 2 O. 3. e empty set and all of X belong to O. A set O is said to be open if and only if it is a member of O; O is the collection of open sets. We say a map f from a topological space X to a topological space Y is continuous if O 2 OY implies f 1 .O/ 2 OX . A set C is said to be closed if and only if the complement C c is open. Consequently, we have equivalently that f is continuous on all of X if and only if f 1 .C / is

2

1. BASIC NOTIONS AND CONCEPTS

a closed subset of X for any closed set C  Y . A bijective map f from X to Y is said to be a homeomorphism if f and f 1 are continuous. A pair .X; d / is said to be a metric space and d is said to be the distance function if d is a non-negative function on X  X which satisfies:

1. d.x; y/ D d.y; x/  0 for all x; y 2 X: 2. d.x; y/ D 0 if and only if x D y: 3. d.x; z/  d.x; y/ C d.y; z/ for all x; y; z 2 X: Let Br .x/ WD fy 2 X W d.x; y/ < rg be the open ball of radius r about a point x of X . A subset O of X is said to be open if given any point x 2 O, there exists ı D ı.x; O/ > 0 so Bı .x/  O; this defines a topology on X . A map f from a metric space .X; dX / to a metric space .Y; dY / is continuous at x 2 X if given  > 0 there exists ı D ı.f; x; / > 0 so f .Bı .x//  B .f .x//; f is continuous on all of X if it is continuous at every point of X . If S is a subset of a metric space X , then the restriction of the distance on X to S makes S a metric space; the inclusion map of S in X is then a continuous map. e Euclidean distance function on Rm is given by: ˚ d..x 1 ; : : : ; x m /; .y 1 ; : : : ; y m // WD .x 1

y 1 /2 C


C .x m

y m /2

12

:

Let .X; d / be a metric space. Given a sequence of points xn 2 X , we say xn ! x as n ! 1 and write limn!1 xn D x if given  > 0, there exists N D N./ so n  N implies d.x; xn / <  ; x is said to be the limit point of the sequence. A point x is said to be a cluster point of a set S if there exists a sequence of points xn 2 S so limn!1 xn D x . Let S 0 be the set of cluster points of S . e set S is said to be closed if S 0  S . We let SN D S [ S 0 be the closure of S ; SN is a closed set since .S [ S 0 /0 D S 0 . For example, the closure of the open ball is the closed ball BN r .x/ WD fy 2 X W d.x; y/  rg :

We say that a set C is bounded if there exists .r; x/ so C  Br .x/. We say that a subset C of X is compact if given any open cover fO˛ g of C , there exists a finite subcover. e following is well-known: Lemma 1.1

Let X and Y be metric spaces.

1. If C is a compact subset of X , then C is closed and bounded. 2. If C is a closed subset of a compact set, then C is compact. 3. If C is a closed and bounded subset of Rm , then C is compact. 4. If fxn g is a sequence of points in a compact set, then there is a convergent subsequence. 5. A continuous real-valued function on a compact set attains its maximum and minimum values. 6. If X is compact and if f is a continuous map from X onto Y , then Y is compact.

1.2. LINEAR ALGEBRA

7. If f is a continuous bijective map from X to Y and if X is compact, then f is a homeomorphism.

1.2

LINEAR ALGEBRA

1.2.1 THE REAL, COMPLEX, AND QUATERNION FIELDS. We shall let R, C and H WD fx0 C ix1 C jx2 C kx3 g be the real, complex, and the quaternion numbers, respectively, where we have i 2 D j 2 D k 2 D 1, ij D j i D k , j k D kj D i , and ki D ik D j . Both C and H have a conjugation operator. We set x C iy D x

iy

and x0 C ix1 C jx2 C kx3 D x0

ix1

jx2

kx3 :

Let k
k be the usual Euclidean norm. Let z 2 C D R2 and let w 2 H D R4 . en: kzk2 D z zN D zN z D x 2 C y 2

kwk2 D w wN D ww N D x02 C x12 C x22 C x32

so

z

so

w

1 1

D kzk D kwk

2

zN ; 2

wN :

If zi 2 C and wi 2 H, then z1 z2 D zN1 zN2 and w1 w2 D wN 2 wN 1 . 1.2.2 VECTOR SPACES. Let V be a finite-dimensional F vector space of dimension m; if we omit the field, it is assumed to be R. For example, in what follows GL.V / D GLR .V /. e dual bundle V  is the vector space of linear maps from V to F . If fe1 ; : : : ; em g is a basis for V , we can expand any element x 2 V in the form x i ei D x 1 e1 C


C x m em where we adopt the Einstein convention and sum over repeated indices. Let e i .x/ WD x i . e fe i g form a basis for V  that is called the dual basis and the fx i g are called the coordinate functions. e map x ! .x 1 ; : : : ; x m / defines a vector space isomorphism from V to F m . 1.2.3 NORMED VECTOR SPACES. A pair .V; k
k/ is said to be a normed vector space if V is a real vector space and if k
k is a map from V to R with: kxk  0 and kxk D 0 if and only if x D 0 for all x 2 V; kcxk D jcj kxk for all x 2 V and for all c 2 R; kx C yk  kxk C kyk for all x; y 2 V :

(1.2.a) (1.2.b) (1.2.c)

We may give V the structure of a metric space by setting d.x; y/ D kx yk and thereby define a topology on V . Let h
;
i be a positive definite symmetric bilinear form on V ; the pair .V; h
;
i/ is said to be a Euclidean vector space and we set: 1

kxk WD hx; xi 2 :

3

4

1. BASIC NOTIONS AND CONCEPTS

e properties of Equation (1.2.a) and Equation (1.2.b) are then immediate. One has the Cauchy– Schwarz–Bunyakovsky inequality for all

jhx; yij  kxk
kyk

V. Bunyakovsky (1804–1889)

x; y 2 V :

A. Cauchy (1789–1857)

K. Schwarz (1843–1921)

e triangle inequality of Equation (1.2.c) is immediate from this identity; this completes the verification that .V; k
k/ is a normed linear space in this setting.

Let .V; h
;
i/ be a Euclidean vector space of dimension m over R. We can use the Gram– Schmidt process to find a basis fe1 ; : : : ; em g for V so that hei ; ej i D ıij where ı is the Kronecker symbol ( ) 0 if i ¤ j ıij WD : 1 if i D j Such a basis is said to be an orthonormal basis. Let x D x i ei . en: i

x D hx; ei i;

hx; yi D

X

i

x y

i

and

(

X .x i /2 kxk D

) 12

:

i

i

us the map T W x ! .x 1 ; : : : ; x m / is a linear isomorphism from V to Rm which preserves the inner products, i.e., hx; yiV D hT x; T yiRm ; such a map is called an isometry. Note that: jx i j  kxk

for

1  i  m:

(1.2.d)

We have the polarization identity which expresses the inner product in terms of the norm: hx; yi D 14 fkx C yk2

kx

yk2 g :

We say that two norms k
k1 and k
k2 on V are equivalent norms if there exist positive constants C1 and C2 so that kxk1  C1 kxk2 and kxk2  C2 kxk1 for all x 2 V ; equivalent norms define the same topology. e following is a useful observation:

1.2. LINEAR ALGEBRA

Lemma 1.2

Let V be a finite-dimensional real vector space.

1. Any two norms on V are equivalent. 2. Let k
k1 be a norm on V . Let B D fe1 ; : : : ; em g be a basis for V . ere exists a constant C D C.k
k1 ; B / so that if we expand x D x 1 e1 C


C x m em , then jx i j  C kxk1 . Proof. Let k
k be an auxiliary norm on V which is given by a positive definite inner product h
;
i on V . We will show k
k1 is equivalent to k
k; a similar argument shows k
k2 is equivalent to k
k and hence k
k1 is equivalent to k
k2 . Let fe1 ; : : : ; em g be an orthonormal basis for V with respect to h
;
i; we identify V with Rm . We may use Equation (1.2.d) to estimate: kx

yk1

D k.x 1 y 1 /e1 C


C .x m y m /em k1  jx 1 y 1 j ke1 k1 C


C jx m y m j kem k1  kx ykfke1 k1 C


C kem k1 g  kx yk

for  WD ke1 k1 C


C kem k1 . Given  > 0, let ı D =. C 1/. If d.x; y/ D kx kx yk1 < ı <  . By the triangle inequality,

yk < ı , then

kxk1  kyk1 C ky xk1 and kyk1  kxk1 C kx yk1 : ˇ ˇ Consequently, ˇkxk1 kyk1 ˇ  kx yk1 <  . is shows that the map x ! kxk1 is continuous with respect to the topology defined by k
k (which is the usual Euclidean topology on Rm ). We apply Lemma 1.1. e unit sphere in Rm is compact. A continuous function on a compact set attains its minimum and maximum values. Let CQ 1 WD minkxkD1 kxk1

and CQ 2 WD maxkxkD1 kxk1 .

Since kxk1 D 0 if and only if x D 0, CQ 1 and CQ 2 are positive. e desired inequalities of Assertion 1 for all x now follow from the following identity by rescaling: CQ 1 kxk  kxk1  CQ 2 kxk

for all x 2 V

with

kxk D 1 :

Let B be a basis for V . We may define a positive definite inner product on V by requiring the basis to be orthonormal and obtain an auxiliary inner product h
;
i with associated norm k
k2 . We use Equation (1.2.d) to estimate jx i j  kxk2 . e desired estimate using k
k1 now follows from Assertion 1 as k
k1 is equivalent to k
k2 . u t If .V; k
k/ is a finite-dimensional real normed linear space, let dk
k .x; y/ WD kx yk. By Lemma 1.2, any two norms are equivalent. Consequently, the underlying topology on V is independent of the particular norm which is chosen; the identity map is a homeomorphism from .V; dk
k1 / to .V; dk
k2 / for any pair of norms on V . Furthermore, the coordinate functions x i relative to a basis for V are continuous with respect to this topology.

5

6

1. BASIC NOTIONS AND CONCEPTS

Let .V; k
kV / and .W; k
kW / be finite-dimensional normed vector spaces. Let A be a linear map from V to W , and let fe1 ; : : : ; em g be a basis for V . By Lemma 1.2: kAxkW D kx 1 Ae1 C


C x m Aem kW

 jx 1 j kAe1 kW C


C jx m j kAem kW

 C kxkV fkAe1 kW C


C kAem kW g

D C kxkV

for

(1.2.e)

 WD kAe1 kW C


C kAem kW :

If  > 0 is given, let ı D =.2C  C 1/. By Equation (1.2.e), if kx ykV < ı , then we have that kAx AykW D kA.x y/kW  C kx ykV <  . Consequently, A is a continuous map from V to W . Equation (1.2.e) also shows that kAk WD sup

0¤x

kAxkW  C kxkV

(1.2.f )

is well-defined and finite. We then have kAxkW  kxkV kAk

for all

x2V :

Let Hom.V; W / be the set of all linear maps from V to W ; .Hom.V; W /; k
k/ is a normed vector space with dim.Hom.V; W // D dim.V / dim.W / : If we choose a basis, we may identify V with Rm and End.V; V / with the set of m  m matrices Mm .R/. If A 2 Mm .R/, let det.A/ be the determinant of A. Lemma 1.3

1. (Cramer’s Rule [13]) Let A 2 Hom.Rm ; Rm / be an m  m matrix. (a) en A is invertible if and only if det.A/ ¤ 0.

(b) Let Cij be the matrix of cofactors formed by crossing out row i and column j . If det.A/ ¤ 0, then .A 1 /ij D . 1/iCj Cj i det.A/ 1 . 2. Let A 2 Hom.V; W /. If limx!0

kAxkW kxkV

D 0, then A D 0.

Proof. Assertion 1 is well-known so we will omit the proof. Suppose A ¤ 0. Choose v 2 V so Av ¤ 0. Let vn WD v=n. en vn ! 0 so 1 kAvkW kAvkW kAvn kW D lim n1 D ¤ 0: n!1 kvn kV n!1 kvkV kvk V n

0 D lim

is contradiction establishes the lemma.



1.3. THE DERIVATIVE

1.2.4 THE GENERAL LINEAR GROUP. If F 2 fR; C; Hg, then the general linear group GLF .V / is the set of all invertible F linear transformations from V to V . Suppose that F D R or that F D C . Since the determinant function det is continuous, GLF .V / is an open subset of HomF .V; V /; GLF .V / forms group under composition. We use Cramer’s rule to see that the group operation and inverse are continuous with respect to the induced topology. If F D H, then we cannot use the determinant directly as H is non-commutative. We can forget the quaternion structure to define an underlying real vector space VR . If A 2 HomH .V; V /, then A defines an element AR of HomR .VR ; VR /. We have A is invertible , A is injective , AR is injective , AR is invertible , det.AR / ¤ 0 :

us GLH .V / D fA 2 HomH .V; V / W det.AR / ¤ 0g. is shows GLH .V / is an open subset of HomH .V; V /. Furthermore, since AR ! AR1 is continuous on GLR .VR /, the map A ! A 1 is continuous on the subgroup GLH .V /  GLR .VR /.

1.3

THE DERIVATIVE

In this section, we will establish the basic facts concerning the differential calculus that we shall need. A good auxiliary reference would be Spivak [38]. e appropriate extension of the notion of a derivative to multivariable calculus is as the best linear approximation. We begin by presenting several different notions of differentiability: Definition 1.4 Let .V; k
k/ and .W; k
k/ be finite-dimensional real normed vector spaces, with dim.V / D m and dim.W / D n. Let O be an open subset of V , let F W O ! W , and let P be a

point of O.

1. F is said to be differentiable at P if there exists a linear map A W V ! W so that lim

u!0

kF .P C u/

F .P / kuk

Auk

D0

or, in other words, F .P C u/ D F .P / C Au C o.kuk/. We note that if such an A exists, then it is unique by Lemma 1.3, and thus we shall call A the derivative of F at P and write A D F 0 .P /. Define EF .P1 ; P / by the identity: F .P1 / D F .P / C A.P1

P / C EF .P1 ; P / :

en F 0 .P / D A if and only if for any  > 0, there exists ı D ı./ so kP kEF .P1 ; P /k  kP1

P1 k < ı implies

Pk:

2. F has a directional derivative .Du F /.P / at P in the direction u 2 V , if the following limit exists: F .P C t u/ F .P / : .Du F /.P / WD lim t !0 t

7

8

1. BASIC NOTIONS AND CONCEPTS

3. Fix a basis fe1 ; : : : ; em g for V and let .x 1 ; : : : ; x m / be the dual coordinates on V . If the directional derivative .Dei F /.P / exists for 1  i  m, then we shall denote this direc@F tional derivative by the partial derivative @x i .P /. We shall also use the notation .@x i F /.P /. P Choose a basis feQk g for W and expand F D k F k eQk . If all the partial derivatives of F exist at P , we shall denote the Jacobian matrix by 0 1 @x 1 F 1 : : : @x m F 1 B C ::: : : : A .P / : @ ::: @x 1 F n : : : @ x m F n 4. We say that F is C k if F has continuous partial derivatives up to order k . Note that F is C k for k  2 if and only if F 0 is C k 1 where we regard F 0 as a map from O to Hom.V; W /. If f is C k for all k , then f will be said to be smooth or C 1 . e following result is basic in this subject; we fix a system of coordinates and work in Euclidean space for simplicity. eorem 1.5

Let O be an open subset of Rm and let F W O ! Rn .

1. If F has continuous partial derivatives at each point P 2 O, then F is differentiable at each point P of O. 2. If F is differentiable at some point P 2 O, then: (a) F is continuous at P . (b) All the partial derivatives of F exist at P and .Du F /.P / D F 0 .P /.u/. (c) F 0 .P / is given by the Jacobian matrix.

3. If all the directional derivatives of F exist at some point P 2 O, then all the partial derivatives of F exist at P . Proof. It is immediate from the definition that F W Rm ! Rn is differentiable if and only if each of the component functions is differentiable. us we shall take n D 1 in the proof of Assertion 1. We shall also take m D 2 in the interests of notational simplicity; the general case is no more difficult. We assume O is an open rectangle as this is a local result. We apply the Mean Value eorem; it was for this reason that we assumed n D 1. Let P D .a; b/ and let P1 D .c; d / be points of O. As @x 1 F and @x 2 F are continuous, F .P1 /

F .P / D F .c; d / F .a; b/ D F .c; d / F .a; d / C F .a; d / D @x 1 F .˛; d /.c a/ C @x 2 F .a; ˇ/.d b/

F .a; b/

1.3. THE DERIVATIVE

where  D .d / is between a and c and where  D .a/ is between b and d . Let A be the Jacobian; A.u; v/ D @x 1 F .a; b/u C @x 2 F .a; b/v . en: F .P1 /  F .P /  A.P1  P / D EF .P1 ; P / where EF .P1 ; P / WD f@x 1 F .; d /  @x 1 F .a; b/g.c  a/ C f@x 2 F .a; /  @x 2 F .a; b/g.d  b/ :

We estimate: kEF .P1 ; P /k  E1 .P1 ; P / kP1  P k where E1 .P1 ; P / WD fj@x 1 F .; d /  @x 1 F .a; b/j C j@x 2 F .a; /  @x 2 F .a; b/jg :

Since the partial derivatives are continuous, E1 can be made arbitrarily small if P1 is close to P . Assertion 1 now follows. Suppose F is differentiable at P . We may express

E1 kP1  P k

kF .P1 /  F .P /  F 0 .P /.P1  P /k 

where E1 D E1 .P1 ; P / can be made arbitrarily small for P1 close to P . In particular, by shrinking O if necessary, we may assume E1 is at most 1. We may then estimate kF .P1 /  F .P /k 

kF 0 .P /.P1  P /k C E1 kP1  P k 

is shows that F is continuous at P . Let 0 ¤ u. en n o F .P / 0 lim t!0 F .P Ct u/  F .P /.u/ D kuk lim t!0 t

fkF 0 .P /k C 1gkP1  P k :

F .P Ct u/ F .P / F 0 .P /.t u/ tkuk

D 0:

is shows that .Du F /.P / D F 0 .P /.u/. us F has all directional derivatives and in particular partial derivatives at P . Since F 0 .P / is linear and F 0 .P /.ei / D .Dei F /.P / D .@x i F /.P /, F 0 is u t the Jacobian matrix. Assertion 2 now follows. Assertion 3 is immediate. 1.3.1 EXAMPLES. None of the implications of eorem 1.5 are reversible. We first give an example of a function which is differentiable but not continuously differentiable when m D 1. f .x/ D

(

10x 2 sin.x  0

We compute: f 0 .x/ D

(

1

/ if x ¤ 0 if x D 0

20x sin.x 

limx!0

1

)

Picture:

:

/  10 cos.x 

10x 2 sin.x  x

1 /

0

1

/ if x ¤ 0

D0

if x D 0

)

:

us f is differentiable at every point x 2 R. However since limx!0 f 0 .x/ does not exist, f is not continuously differentiable.

9

10

1. BASIC NOTIONS AND CONCEPTS

Next we give an example of a real-valued function with directional derivatives at each point of R but which is nevertheless not differentiable at 0. Set 2

f .x; y/ WD

(

xy 2 x 2 Cy 2

0

)

if .x; y/ ¤ .0; 0/ if .x; y/ D .0; 0/

Picture:

:

It is clear that f has continuous partial derivatives for .x; y/ ¤ .0; 0/ and thus is continuously differentiable for .x; y/ ¤ .0; 0/. It is also clear f is continuous at .0; 0/. Clearly .D0 f /.0; 0/ D 0. If u D .x; y/ ¤ .0; 0/, then t 3 xy 2 D f .x; y/ : t !0 t 3 .x 2 C y 2 /

.Du f /.0; 0/ D lim

us f has directional derivatives at .0; 0/. We have .De1 f /.0; 0/ D .De2 f /.0; 0/ D 0 :

If f was differentiable at 0, then the map which sends u ! .Du f /.0; 0/ would be a linear function of u. is implies .Du f /.0; 0/ D x.De1 f /.0; 0/ C y.De2 f /.0; 0/ D 0. Since D.1;1/ f .0; 0/ D f .1; 1/ D 12 ¤ 0, this is false. us f is not differentiable at the origin. We could also consider the function

f .x; y/ WD

(

xy 2 x 2 Cy 4

0

if .x; y/ ¤ .0; 0/ if .x; y/ D .0; 0/

)

:

Picture:

We see that f is not continuous as 1 t4 D ¤ f .0; 0/ : t!0 t 4 C t 4 2

lim f .t 2 ; t / D lim

t!0

If x D 0, then .D.0;y/ f /.0; 0/ D 0. If x ¤ 0, then .D.x;y/ f /.0; 0/ D lim

y2 t 3 xy 2 D : 5 4 Ct y x

t !0 t 3 x 2

us all the directional derivatives exist and yet f is not continuous. Finally, we can give an example of a function all of whose partial derivatives exist but does not have directional derivatives in all directions. Let

1.3. THE DERIVATIVE

f .x; y/ D

(

xy x 2 Cy 2

0

if .x; y/ ¤ .0; 0/ if .x; y/ D .0; 0/

)

:

11

Picture:

en all the partial derivatives of f exist for .x; y/ ¤ .0; 0/ and it is an easy direct calculation .0; 0/ D @f .0; 0/ D 0. us the partial derivatives of f also exist at the origin. However that @f @x @y t2 t!0 2t 3

D.1;1/ f .0; 0/ D lim

does not exist. us the partial derivatives can exist without f having directional derivatives at .0; 0/. In view of eorem 1.5, all the usual rules of calculus pertain; the derivative of the sum is the sum of the derivatives, multiplying by a constant rescales the derivative, etc. Furthermore, if f .x/ D Ax where A is linear, then f is differentiable and f 0 .P / D A for any P . ere is a delicate point here. If f is an R-valued function of one variable, then f 0 .P / is usually thought of as a number  2 R. However, in our context we shall think of f 0 .P / D m being the linear transformation defined as multiplication by  . It is a minor point, but an important one. 1.3.2 THE CHAIN RULE. We now establish the chain rule; to simplify the exposition, we assume the functions in question have domains all of Rm and Rn ; this is inessential as all the computations are purely local. eorem 1.6 (Chain Rule). Let f W Rm ! Rn be differentiable at P and let g W Rn ! Rp be differentiable at Q D f .P /. en h WD g f W Rm ! Rp is differentiable at P and the derivative of h at P is given by h0 .P / D g 0 .Q/ f 0 .P /.

Proof. Define Ef , Eg , and Eh by the equations:

f .P1 / D f .P / C f 0 .P /.P1  P / C Ef .P1 ; P /; g.Q1 / D g.Q/ C g 0 .Q/.Q1  Q/ C Eg .Q1 ; Q/; h.P1 / D h.P / C g 0 .Q/f 0 .P /.P1  P / C Eh .P1 ; P / :

e assumption that f is differentiable at P and that g is differentiable at Q D f .P / implies that given f > 0 and g > 0, there exist f D f .f / > 0 and g D g .g / > 0 so that kEf .P1 ; P /k  f kP1  P k if kP1  P k < f ; kEg .Q1 ; Q/k  g kQ1  Qk if kQ1  Qk < g :

12

1. BASIC NOTIONS AND CONCEPTS

We have h.P1 / D g.f .P1 // and h.P / D g.f .P //. Consequently, we may expand

Eh .P1 ; P / D g.f .P1 // g.f .P // g 0 .Q/f 0 .P /.P1 P / D g 0 .Q/.f .P1 / f .P // C Eg .f .P1 /; f .P // g 0 .Q/f 0 .P /.P1 D g 0 .Q/.f 0 .P /.P1 P / C Ef .P1 ; P // CEg .f .P1 /; f .P // g 0 .Q/f 0 .P /.P1 P / D g 0 .Q/Ef .P1 ; P / C Eg .f .P1 /; f .P // :

P/

Let h > 0 be given. Choose 0 < f < 1 and 0 < g < 1 so kg 0 .Q/kf < 21 h

and g fkf 0 .P /k C 1g < 12 h :

Let ıf D ıf .f / and ıg D ıg .g /. Since f is continuous at P , we may choose 0 < ı < ıf so kP1

Let kP1

Pk < ı

implies kf .P1 /

f .P /k < ıg :

P k < ı . We estimate

kg 0 .Q/Ef .P1 ; P /k  kg 0 .Q/k
kEf .P1 ; P /k  kg 0 .Q/kf kP1 P k < 21 h kP1 P k; kEg .f .P1 /; f .P //k  g kf .P1 / f .P /k  g fkf 0 .P /.P1 P /k C kEf .P1 ; P /kg  g fkf 0 .P /k C f gkP1 P k  g fkf 0 .P /k C 1gkP1 P k  21 h kP1 P k; kEh .P1 ; P /k  kg 0 .Q/Ef .P1 ; P /k C kEg .f .P1 /; f .P //k  h kP1 P k :

is shows that h is differentiable at P and that h0 .P / D g 0 .Q/ ı f 0 .P /.

u t

1.3.3 INDEX NOTATION. We introduce the standard index notation. Let x D .x 1 ; : : : ; x m /;

y D .y 1 ; : : : ; y n /;

and

z D .z 1 ; : : : ; z p /

be coordinates on Rm , on Rn , and on Rp , respectively. If A is a linear map from Rp to Rq , let Aji be the components of A so that the j th component of Ax is Aji x i summed over i . e chain rule takes the form: .f

1.4

0 j /i

@y j ; D @x i

.g 0 /jk

@z k D j; @y

.h0 /ki

n

n

j D1

j D1

X X @z k @y j @z k 0 k 0 j D : D .g / .f / D j i @x i @y j @x i

THE INVERSE AND IMPLICIT FUNCTION THEOREMS

is section deals with the Inverse Function eorem and with the Implicit Function eorem. A good auxiliary reference would be Spivak [38]. e Implicit Function eorem is also known as Dini’s eorem, owing to the seminal contribution of the Italian mathematician Ulisse Dini.

1.4. THE INVERSE AND IMPLICIT FUNCTION THEOREMS

13

Ulisse Dini (1845–1918) Recall that if A is a linear transformation from Rm to Rn , we define the operator norm kAk by setting kAxk kAk WD sup < 1: 0¤x2Rm kxk We then have kAxk  kAk
kxk for all x 2 Rm . Also, recall that in Rm there is a canonical notation for the inner product; thus, we set x
y D x 1 y 1 C


C x m y m to define the inner product of .x 1 ; : : : ; x m / and .y 1 ; : : : y m /. We begin our discussion with: Lemma 1.7

Let O be an open subset of Rm . Let F W O ! Rn be continuously differentiable.

1. Let C be a convex compact subset of O. Let  WD supP 2C kF 0 .P /k < 1. en kF .P1 /

F .P /k  kP1

Pk:

2. If F is real-valued and if F 0 .P / ¤ 0, then F does not have a local minimum at P . 3. Suppose n D m and that F 0 is invertible on O. (a) Let U be an open set with compact closure in O. Let  WD supP 2UN k.F 0 .P //

1

k < 1.

If P 2 U , then there exists ı D ı.P; F / > 0 so BN ı .P /  U and: i. If P1 2 Bı .P / and P2 2 Bı .P /, then kP1 P2 k  2 kF .P1 / ii. F is an injective map from Bı .P / to Rn . iii. ere exists  > 0 so F .Bı .P // contains B .F .P //.

F .P2 /k.

(b) F is an open map. Proof. If F .P1 / D F .P /, the inequality of Assertion 1 is obvious. Consequently we may assume that F .P1 / ¤ F .P / and set v D .F .P1 / F .P //=kF .P1 / F .P /k; kvk D 1. Since the region is convex, we can consider the straight line segment tP1 C .1 t/P from P to P1 . We use the

14

1. BASIC NOTIONS AND CONCEPTS

Fundamental eorem of Calculus and the Chain Rule to see Z 1 kF .P1 / F .P /k D v
fF .P1 / F .P /g D @ t fv
.F .tP1 C .1 0 Z 1 D v
fF 0 .tP1 C .1 t/P /.P1 P /gdt :

t/P //g dt

0

We apply the Cauchy–Schwarz–Bunyakovsky inequality to establish Assertion 1 by estimating: Z 1 kF .P1 / F .P /k  jv
fF 0 .tP1 C .1 t/P /.P1 P /gjdt 0 Z 1  kvk kF 0 .tP1 C .1 t/P /.P1 P /kdt 0 Z 1  kP1 P k kF 0 .tP1 C .1 t /P /kdt  kP1 P k : 0

0

If F is real-valued and if F .P / ¤ 0, choose u so F 0 .P /u ¤ 0. Let g.t / WD F .P C t u/. By the Chain Rule, g 0 .0/ D F 0 .P /u ¤ 0. us g does not have a local minimum at 0. is shows that F does not have a local minimum at P which proves Assertion 2. Let n D m and let F 0 be invertible on O. Let U be an open set with compact closure in O. Fix P 2 U and let A D F 0 .P /. Let 1 WD .2/ 1 and let G.x/ WD F .x/ Ax . Since G is continuously differentiable, since G 0 .x/ D F 0 .x/ A, and since G 0 .P / D 0, we can choose ı > 0 so Bı .P /  O and so kG 0 .x/k < 1 on Bı .P /. us by Assertion 1, if Pi 2 Bı .P / k.F .P1 /

F .P2 //

A.P1

P2 /k D kG.P1 /

G.P2 /k  1 kP1

P2 k :

(1.4.a)

Since A is invertible, kP1

P2 k D kA

1

.A.P1

P2 //k   kA.P1

(1.4.b)

P2 /k :

We use Equation (1.4.a), Equation (1.4.b), and the triangle inequality to estimate: kA.P1

P2 /k  kF .P1 /  kF .P1 /  kF .P1 /

F .P2 /k C 1 kP1 P2 k F .P2 /k C 1 kA 1 k
kA.P1 F .P2 /k C 12 kA.P1 P2 /k :

P2 /k

is shows 1 kA.P1 2

P2 /k  kF .P1 /

F .P2 /k :

(1.4.c)

Assertion 3-a-i now follows from Equation (1.4.b) and Equation (1.4.c). Assertion 3-a-ii follows from Assertion 3-a-i. Let  WD 21 ı and let S .P / WD fP1 2 Rm W kP

P1 k D g

1.4. THE INVERSE AND IMPLICIT FUNCTION THEOREMS

15

be the sphere of radius  about P . Since F is injective on BN  .P /  Bı .P /, F .P / does not belong to F .S .P //. Since a continuous function attains its minimum on a compact subset and since, by Lemma 1.1, S .P / is compact, we can choose  > 0 so kF .P1 /

F .P /k  3

Q1 k  kF .P1 /

P1 2 S .P / :

F .x/k2 . We use the triangle inequality to see

Let Q1 2 B .F .P //. Let f .x/ WD kQ1 kF .P1 /

for

F .P /k

kF .P /

Q1 k  3

 D 2

for

P1 2 S .P / :

Since kF .P / Q1 k   , f does not attain its minimum on S .P /. Because a continuous function attains its minimum on a compact set, we can choose P1 2 BN  .P / so f .P1 / is minimal. As noted above, P1 … S .P / so P1 is in the open ball B .P /. By Assertion 2, f 0 .P1 / D 0. Let .x 1 ; : : : ; x m / be the usual coordinates on Rm . We express m X f .x/ D .F i .x/

Q1i /2

iD1

so

m X @x j f .x/ D 2 .@x j F i .x//.F i .x/

Q1i / :

iD1

Setting @x j f .P1 / D 0 for 1  j  m gives rise to the equations: 0D

m X .@x j F i .P1 //.F i .P1 /

Q1i /

for

iD1

1j m

which can be written in matrix form as: 0 D F 0 .P1 /.F .P1 / Q1 /. Since, by assumption, F 0 .P1 / is invertible we have F .P1 / D Q1 . is proves Assertion 3-a-iii. us F .O/ contains an open neighborhood of F .P /. Since P was arbitrary, F .O/ is open. Applying the same argument to an arbitrary open subset of O then establishes Assertion 3-b. t u

Let O be an open subset of Rm and let F be a continuously differentiable function mapping O to R with F 0 invertible on O. Fix P 2 O. ere exists an open neighborhood U of P with U  O and there exists an open neighborhood V  F .O/ of Q D F .P / so F W U ! V is bijective. Let G be the inverse map. en G is continuously differentiable and G 0 .Q/ D fF 0 .G.Q//g 1 .

eorem 1.8 (Inverse Function eorem).

m

Proof. e Inverse Function eorem is a local result. Let O1 be an open neighborhood of P which has compact closure in O. Because .F 0 / 1 is continuous on ON 1 ,  WD max k.F 0 .P1 // P1 2ON 1

1

k

is well defined. us by replacing O by O1 if necessary, we may assume without loss of generality that k.F 0 / 1 k   on all of O. Let U D Bı .P /  O be given by Lemma 1.7. By Lemma 1.7, V WD F .U / is open and F is a bijective map from U to V . We restate the estimate kP1

P2 k  2kF .P1 /

F .P2 /k

16

1. BASIC NOTIONS AND CONCEPTS

in the form: kG.Q1 /

G.Q2 /k  2kQ1

Q2 k

for

Qi 2 V :

(1.4.d)

is proves G is continuous. We use the following equations to define EF and EG : F .P1 / D F .P2 / C F 0 .P2 /.P1 P2 / C EF .P1 ; P2 / for Pi 2 U ; G.Q1 / D G.Q2 / C fF 0 .G.Q2 //g 1 .Q1 Q2 / C EG .Q1 ; Q2 / for

We set P1 D G.Q1 / and P2 D G.Q2 / and use Equation (1.4.e) to see: Q1 D Q2 C F 0 .G.Q2 //.G.Q1 /

G.Q2 // C EF .G.Q1 /; G.Q2 //

for

(1.4.e) Qi 2 V : (1.4.f ) Qi 2 V :

(1.4.g)

We multiply Equation (1.4.g) by fF 0 .G.Q2 //g 1 and perform some algebraic rearrangements to see G.Q1 / G.Q2 / D fF 0 .G.Q2 //g 1 f.Q1 Q2 / EF .G.Q1 /; G.Q2 //g : (1.4.h) We compare Equation (1.4.f) with Equation (1.4.h) to see:

EG .Q1 ; Q2 / D fF 0 .G.Q2 //g 1 EF .G.Q1 /; G.Q2 // :

(1.4.i)

Let  > 0 be given. Fix Q1 2 V and let P1 D G.Q1 / 2 U . Since F is differentiable at P1 , we may choose ı1 D ı1 .Q1 / > 0 so kP1 P2 k < ı1 implies  kEF .P1 ; P2 /k  2 kP1 P2 k : 2 Since G is continuous, we may find ı > 0 so kQ1 Q2 k < ı implies kG.Q1 / G.Q2 /k < ı1 . Since k.F 0 / 1 k   on O, Equation (1.4.d), and Equation (1.4.i) imply: kEG .Q1 ; Q2 /k  kfF 0 .G.Q2 //g 1 k
kEF .G.Q1 /; G.Q2 //k     2 kG.Q1 / G.Q2 /k  2 2 2 kQ1 Q2 k 2 2  kQ1 Q2 k :

It now follows G 0 .Q1 / D fF 0 .G.Q1 //g 1 for Q1 2 V . Since F is continuously differentiable, we can use Cramer’s rule to see .F 0 / 1 is continuous. Since G is continuous, it follows that G is continuously differentiable.  1.4.1 REMARK. If F is smooth, i.e., has continuous partial derivatives of all orders, then we can use the equation G 0 D .F 0 / 1 ı G to conclude that the local inverse G is smooth as well; one simply differentiates this equation recursively and uses the Chain Rule.

If O and U are open subsets of Rm , if F is a continuous map from O to U , and if F 1 is continuous, then F is said to be a homeomorphism. If F is smooth and if F 1 is smooth, then F is said to be a diffeomorphism. It follows from the Chain Rule that the composition of diffeomorphisms is a diffeomorphism. It follows from the Inverse Function eorem that the inverse of a diffeomorphism is a diffeomorphism. We now present several examples of canonical coordinate systems.

1.4. THE INVERSE AND IMPLICIT FUNCTION THEOREMS

1.4.2 REMARK. We consider the function ( 0 f .x/ D x C x 2 sin.x  1 / 2

)

if x D 0 if x ¤ 0

17

:

is function is differentiable everywhere and f 0 .0/ D 12 . e derivative is not continuous at x D 0. is function is not injective on any neighborhood of 0; thus the assumption that f has continuous partial derivatives is crucial in the Inverse Function eorem.

1.4.3 POLAR COORDINATES. Let F .r; / WD .r cos./; r sin.//, i.e., x D r cos./;

y D r sin./

for

r > 0;  2 R :

en j det.F 0 /j D r and a local inverse near x D 1 and y D 0 is given by 1

r D .x 2 C y 2 / 2

and  D arctan



y x



:

e natural domain of the local inverse is the right half plane y > 0 where we choose the branch of the arctan function to have range . 2 ; 2 /. is is not globally invertible as we have the relation F .r; / D F .r;  C 2/. We say that this is an admissible change of coordinates since we can express .r; / in terms of .x; y/ and .x; y/ in terms of .r; / at least locally. is permits us to express certain curves very simply. For example, one has: Region bounded by Cardioid r D sin./  1

Archimedes spiral r D

1.4.4 CYLINDRICAL COORDINATES. Let F .r; ; z/ WD .r cos./; r sin./; z/, i.e., x D r cos./;

y D r sin./;

zDz

for

r > 0; .; z/ 2 R2 :

18

1. BASIC NOTIONS AND CONCEPTS

en j det.F 0 /j D r so again this is an admissible change of coordinates, i.e., we can write .r; ; z/ in terms of .x; y; z/, locally. Again, the inverse is not globally injective but can be used to provide a local parametrization of R3 minus the z -axis. us for example,

z D r yields the cone

1.4.5 SPHERICAL COORDINATES. We let x D r cos./ sin. /; y D r sin./ sin. /; z D r cos. /

for

r > 0;  2 R; 0 <  <  :

Since j det.F 0 /j D r 2 sin. /, F is locally injective; as with polar and cylindrical coordinates it is not globally injective but F  1 is well-defined and smooth locally. e parameter r measures the distance to the origin, the parameter  is the angle in the xy plane, and the parameter  is the angle down from the z -axis. Again, we get wonderful surfaces. For example,

r D  and 0 



 yields:

e Inverse Function eorem can be regarded as a change of coordinates. We use this point of view to establish the following result; note the assumption of differentiability in the following result is in fact not needed and methods of algebraic topology (Invariance of Domain) can be used to prove it in the C 0 category.

Let O be an open non-empty subset of Rm and let H W O ! Rn be continuously differentiable. If H is injective, then n m.

eorem 1.9

Proof. We suppose the theorem fails. By padding out the coordinates if necessary, we may assume n D m  1. Choose a counter example where m 2 is minimal. If @x 1 H  0, then Z  Z  H.x C e1 /  H.x/ D @ t H.x C te1 /dt D @x 1 H.x C t e1 /dt D 0 : 0

0

is is impossible as H is injective. us there exists a point P 2 O so @x 1 H.P / ¤ 0. By reordering the components if necessary, we may assume @x 1 H1 .P / ¤ 0. Let F .x/ WD .H1 .x/; x 2 ; : : : ; x m / :

1.4. THE INVERSE AND IMPLICIT FUNCTION THEOREMS

19

0

en det.F / D @x 1 H1 . us by the Inverse Function eorem, F is invertible near P . We set HQ .y 1 ; : : : ; y m / WD H.F 1 .y 1 ; : : : ; y m //; this is defined near F .P / and is continuously differentiable. Since the composition of injective functions is injective, HQ is injective. Tracing through the definitions yields HQ 1 .y 1 ; : : : ; y m / D H1 .F 1 .y 1 ; : : : ; y m // D F1 .F 1 .y 1 ; : : : ; y m // D y 1 HQ .y 1 ; : : : ; y m / D .y 1 ; HQ 2 .y 2 ; : : : ; y m /; : : : ; HQ m 1 .y 1 ; : : : ; y m // :

and

If m D 2, then HQ .y 1 ; y 2 / D y 1 and this is not an injective map. us m  3 and fixing y 1 gives an injective map from Rm 1 to Rm 2 FQ .y 2 ; : : : ; y m / WD .HQ 2 .y 1 ; y 2 ; : : : ; y m /; : : : ; HQ m

is gives a counter example in dimension m

1 .y

1

; y 2 ; : : : ; y m // :

1 contradicting the minimality of m.



What is going on, of course, is that one is gradually changing the coordinate system to put H into a particularly simple form. is is a typical application of the Inverse Function eorem. We will also use the Inverse Function eorem to prove the Implicit Function eorem; this permits one to solve equations to determine certain variables (called the dependent variables) in terms of other variables (the independent variables). is is best motivated by examples so we shall first present several examples and then summarize our algorithm with the Implicit Function eorem. 1.4.6 EXAMPLE. Consider the equation x5 C x C y5 C y D 4 :

(1.4.j)

Suppose P D .x0 ; y0 / solves Equation (1.4.j); for example we could take P D .1; 1/. We let x be the independent variable and y the dependent variable. We try to solve this equation for y D y.x/ near the point P . Let F .x; y/ WD .x; x 5 C x C y 5 C y/. Here and subsequently let “?” be a term which is not of interest. We have: ! 1 0 0 det.F .P //.x0 ; y0 / D det D 5y04 C 1 ¤ 0 : ? 5y04 C 1 is is invertible. If F .x; y/ D .u; v/, then the inverse function G.u; v/ D .x; y/ has the property that G1 .x; v/ D x and y.x/ WD G2 .x; 4/

satisfies y.x0 / D y0 and x 5 C x C y 5 .x/ C y.x/ D 4. In other words, we have found the unique solution to Equation (1.4.j) where y.x0 / D y0 and where y.x/ is close to y0 . e map x ! y.x/ is continuously differentiable. We implicitly differentiate Equation (1.4.j) to see: y 0 .x/ D

5x 4 C 1 : 5y 4 .x/ C 1

(1.4.k)

20

1. BASIC NOTIONS AND CONCEPTS

e Inverse Function eorem implies y 2 C 1 . We use Equation (1.4.k) to see y 0 2 C 1 so y 2 C 2 has two continuous derivatives. We continue in this fashion to see that in fact y 2 C 1 . (In fact y is real analytic).

e curve x 5 C x C fy.x/g5 C y.x/ D 4.

1.4.7 EXAMPLE. Consider the equations: x2 C y4 C z8 D 3

and x 4 C y 6 C z 4 D 3 :

Let x be the independent variable and .y; z/ the dependent variables. We wish to solve y D y.x/ and z D z.x/ near the point .1; 1; 1/. We set F .x; y; z/ D .x; x 2 C y 4 C z 8 ; x 4 C y 6 C z 4 / :

e Jacobian is

1 1 0 0 C B F 0 .1; 1; 1/ D @ ? 4 8 A : ? 6 4 0

Since det.F 0 .1; 1; 1// ¤ 0, we can apply the Inverse Function eorem. Let G be the inverse. en y.x/ D G2 .x; 3; 3/ and z.x/ D G3 .x; 3; 3/ satisfies the equations x 2 C y 4 C z 8 D 3 and x 4 C y 6 C z 4 D 3. We differentiate to see 2x C 4y 3 @x y C 8z 7 @x z D 0

and

4x 3 C 6y 5 @x y C 4z 3 @x z D 0 :

Expressing these equations in matrix notation and then inverting yields: ! ! ! @y 4y 3 8z 7  2x @x D @z  4x 3 6y 5 4z 3 @x ! ! 1 ! @y 4y 3 8z 7  2x @x D : @z 6y 5 4z 3  4x 3 @x e right hand side is a C 1 function of x . us the left hand side is C 1 . is implies y.x/ and z.x/ are C 2 . We proceed inductively to see these functions are in fact C 1 .

1.4. THE INVERSE AND IMPLICIT FUNCTION THEOREMS

21

x 2 C y 4 C z 8 D 3 and x 4 C y 6 C z 4 D 3 yields:

1.4.8 EXAMPLE. Consider the equations: x 5 C 5x C y 7 C 7y C u6 C v 8 D 16;

(1.4.l)

x 3 C 2x C y 5 C 2y C u4 v 6 D 7 :

Let P D .1; 1; 1; 1/; this solves these equations. We first let fx; yg be the independent variables and fu; vg be the dependent variables and try to solve for .u; v/ in terms of .x; y/. Let F .x; y; u; v/ D .x; y; x 5 C 5x C y 7 C 7y C u6 C v 8 ; x 3 C 2x C y 5 C 2y C u4 v 6 / :

We have: 0

B B F 0 .1; 1; 1; 1/ D B @

1 0  

0 0 0 1 0 0  6u5 8v 7  4u3 v 6 6u4 v 5

1

0

 

C C C A  

B B DB @ .1;1;1;1/

1 0  

0 1  

0 0 6 4

0 0 8 6

1

C C C: A

us det.F 0 .1; 1; 1; 1// D 36  32 ¤ 0. e Inverse Function eorem shows G WD F  defined and C 1 . en x D G1 .x; ?; ?; ?/ and y D G2 .?; y; ?; ?/. If we set u.x; y/ WD G3 .x; y; 16; 7/

1

is well

and v.x; y/ WD G4 .x; y; 16; 7/ ;

then .x; y; u.x; y/; v.x; y// is the unique solution to Equation (1.4.l) that is near .1; 1; 1; 1/. e pair .u; v/ are C 1 with u.1; 1/ D 1 and v.1; 1/ D 1. If we differentiate implicitly, we obtain the equations: ! ! ! @u @u 6u5 5x 4 C 5 7y 6 C 7 8v 7 @x @y : D @v @v 3x 2 C 2 5y 4 C 2 4u3 v 6 6u4 v 5 @x @y is yields @u @x @v @x

@u @y @v @y

!

D

6u5 4u3 v 6

8v 7 6u4 v 5

!

1

5x 4 C 5 7y 6 C 7 3x 2 C 2 5y 4 C 2

!

:

If the right hand side is C k for k 1, then the left hand side is C k . is implies the right hand side is C kC1 and hence the functions u. ;  / and v. ;  / are C 1 .

22

1. BASIC NOTIONS AND CONCEPTS

1.4.9 EXAMPLE. We consider the same relations as those given in Equation (1.4.l) but instead let .u; v/ be the independent variables, we let .x; y/ be the dependent variables, and we try to solve for x D x.u; v/ and y D y.u; v/. We now set F .x; y; u; v/ D .u; v; x 5 C 5x C y 7 C 7y C u6 C v 8 ; x 3 C 2x C y 5 C 2y C u4 v 6 / :

e Jacobian is given by 0 B B F 0 .1; 1; 1; 1/ D B @

1 0 0 0 0 1 0 0 4 6 ? ? 5x C 5 7y C 7 ? ? 3x 2 C 2 5y 4 C 2

1ˇ ˇ ˇ Cˇ Cˇ Cˇ Aˇ ˇ ˇ

0

B B DB @ .1;1;1;1/

1 0 0 0 0 1 0 0 ? ? 10 14 ? ? 5 7

1

C C C: A

Since det.F 0 .1; 1; 1; 1// D 0, we cannot employ the Inverse Function eorem. Nevertheless, we still assume we could somehow find C 1 solutions so x D x.u; v/ and y D y.u; v/. We differentiate implicitly to get: ! ! ! @x @x 5 7 6u 8v 5x 4 C 5 7y 6 C 7 @u @v : D @y @y 4u3 v 6 6u4 v 5 3x 2 C 2 5y 4 C 2 @u @v At the point .1; 1; 1; 1/ this would yield: ! @x 10 14 @u @y 5 7 @u

@x @v @y @v

!

D

6 8 4 6

!

:

e determinant of the left hand side is zero; the determinant of the right hand side is 4. is is not possible and thus the equations cannot be solved in a C 1 fashion for .x; y/ as a function of .u; v/ near .1; 1; 1; 1/. 1.4.10 IMPLICIT FUNCTION THEOREM. With these examples in mind, we can now establish the following result: eorem 1.10 (Implicit Function eorem). Let U be an open subset of Ra and let V be an open subset of Rb . Let x D .x 1 ; : : : ; x a / belong to U and let y D .y 1 ; : : : ; y b / belong to V ; x are the independent variables and y are the dependent variables. Let H D H.x; y/ be a continuously differentiable map from U  V to Rb . Let .P; Q/ 2 U  V . Let A be the b  b matrix .@y i H j /.P; Q/. Assume det.A/ ¤ 0. ere exist open neighborhoods U1 of P and V1 of Q so that:

1. If x 2 U1 , there exists a unique y 2 V1 so H.x; y/ D H.P; Q/. 2. e map f W x ! y.x/ is a C 1 map from U1 to V1 and ) ( X @H j @f i @H j C .x; f .x// D 0 for k @y i @x k @x i

1  k  a; 1  j  b :

1.5. THE RIEMANN INTEGRAL k

23

k

3. Let k  2. If H is C , then f is C . Proof. As before, we define an auxiliary function F .x; y/ WD .x; H.x; y// from U  V to RaCb and compute: ! Id 0 a F 0 .P; Q/ D : ? A

Since det.F 0 .P; Q// D det.A/ ¤ 0, we have a local inverse G D .G1 ; G2 / where G1 takes values in Ra and G2 takes values in Rb . We have G1 .x; ?/ D x . Let f .x/ D G2 .x; H.P; Q//. en H.x; f .x// D H.P; Q/. e Assertion 1 of the Lemma follows; we use the Chain Rule to es@y tablish Assertion 2. We invert the equation of Assertion 2 to express @x in terms of . @H / 1 and @y @f . @x

We differentiate recursively and apply the Chain Rule once again to obtain Assertion 3.

u t

We note that if H is C 1 , then f is C 1 . With a bit more work, one can show that if H is real analytic (i.e., given by a convergent Taylor series), then f is real analytic. In Example 1.4.9, we gave an example where the hypotheses of the Implicit Function eorem failed and where one cannot in fact solve the equations. On the other hand, if one considers the quadratic equation x 2 2xy C y 2 D 0, then again the hypotheses of the Implicit Function eorem fail since all derivatives vanish at x D y D 0. is equation is equivalent to the equation .x y/2 D 0, i.e., y D x . us when the hypotheses of the Implicit Function eorem fail, little can be said in general although there are certain methods which pertain and which these two examples illustrate.

1.5

THE RIEMANN INTEGRAL

In this section, we shall establish the results concerning the Riemann integral needed subsequently; this integral was created by the German mathematician Georg Friedrich Bernhard Riemann.

G. Riemann (1826–1866) e theory closely parallels that of the 1-dimensional case and a good auxiliary reference would be Rudin [36]. A rectangle R  Rm is the Cartesian product R D Œa1 ; b1  


 Œam ; bm 

for

a1 < b1 ; : : : ; am < bm :

24

1. BASIC NOTIONS AND CONCEPTS

We do not consider degenerate rectangles where ai D bi for some i . e volume of R is then defined to be: vol.R/ WD .b1 a1 /




.bm am / > 0 : (1.5.a) A partition of an interval Œa; b  R is simply a collection of intermediate points

P WD fa D c0 <


< c D bg : By an abuse of notation we may identify P with a dissection of Œa; b of the form:

P WD fŒa; b D Œc0 ; c1  [ Œc1 ; c2  [


[ Œc

1 ; c g

:

More generally, a partition P of a rectangle R  Rm is a collection ˚ P D fa1 D c01 <


< c11 D b1 g; : : : ; fam D c0m <


< cmm D bm g which we identify with a dissection of R of the form: ˚ P D R D [1i1 1 ;:::;1im m fŒci11 1 ; ci11  


 Œcimm



m 1 ; cim g

:

C C

C C C Lemma 1.11

C

C

C

C

If P is a partition of a rectangle R  Rm , then

X P 2P

vol.P / D vol.R/.

Proof. If m D 1, this is nothing but the fact that we have a telescoping series X vol.P / D .c1 c0 / C


C .c c 1 / D c c0 D b a D vol.R/ : P 2P

In higher dimensions, we use this argument together with the distributive law: X P 2P

vol.P / D

n D .c11

D .b1

1 X i1 D1






m X

.ci11

ci11

im D1

c01 / C


C .c11 a1 /




.bm

c11

1/

o

1/






.cimm

˚




.c1m

am / D vol.R/ :

cimm

1/

c0m / C


C .cmm

cmm



1/

u t

1.5. THE RIEMANN INTEGRAL

25

Let f be a bounded real-valued function on a rectangle R  R. If P  R, set: M.f; P / WD sup f .x/;

m.f; P / WD inf f .x/ : x2P

x2P

Clearly m.f; P /  M.f; P /. If P is a partition of R, we define the lower sum L.f; P / and the upper sum U.f; P / to be: X X L.f; P / WD m.f; P / vol.P / and U.f; P / WD M.f; P / vol.P / : P 2P

P 2P

We may estimate: vol.R/m.f; R/ D vol.R/M.f; R/ D

X P 2P

vol.P / inf f .x/ 

X

P 2P

x2R

vol.P / sup f .x/  x2R

X P 2P

vol.P / inf f .x/ D L.f; P /; x2P

X

vol.P / sup f .x/ D U.f; P / :

(1.5.b)

x2P

P 2P

A partition Q of R is said to be a refinement of a partition P of R if all the rectangles of Q are contained in some rectangle of P . Since Q induces a partition of every rectangle of P , the inequalities of Equation (1.5.b) can be summed over the rectangles P 2 P to see: L.f; P /  L.f; Q/  U.f; Q/  U.f; P / :

Given any two partitions P1 and P2 , we can form a common refinement Q and conclude: L.f; P1 /  L.f; Q/  U.f; Q/  U.f; P2 / :

is inequality shows that L.f;
/ is uniformly bounded from above and U.f;
/ is uniformly bounded from below. We therefore define the lower integral and the upper integral by setting: Z R

We then have

f WD Z R

Lemma 1.12

sup P is a partition of R

f 

Z

L.f; P /

and

Z R

f WD

inf

P is a partition of R

U.f; P / :

f . e following observation is immediate:

R

Let R be a rectangle in Rm .

1. e following assertions are equivalent and if any is satisfied, then f is said to be integrable on R: R R (a) R f D R f . (b) Given  > 0, there exists a partition P of R so U.f; P /  L.f; P / C  .

26

1. BASIC NOTIONS AND CONCEPTS

2. Let f be integrable on R. Set

R

R

f WD

R

R

f D

R

R

f. R

of R satisfying Assertion 1-b, then  (a) If   R P is a partition   R f  U.f; P /   .

   f  L.f; P / R

 and

(b) Let P be a rectangle which is contained in R. en f jP is integrable on P . R R P (c) Let P be a partition of R. en R f D P 2P P f jP . R R (d) If c 2 R, then cf is integrable on R and R cf D c R f . 3. Let f and g be integrable on R. R R C g/ D R f C R g . R R (b) If f .x/  g.x/ for all x 2 R, then R f  R g . R We will sometimes simply write f when region of integration is clear and introducing additional notation would only complicate matters unnecessarily. If R D Œa; b R, then we will Rb also use the standard notation a f .x/dx .

(a) f C g is integrable on R and

R

R .f

e following example is instructive. 1.5.1 EXAMPLE. Let frn g be an enumeration of the rationals in Œ0; 1. Let

f .x/ WD

(

1 n

0

if x D rn for some n if x ¤ rn for all n

)

.

Graph:

Fix n and let Pn be the partition of Œ0; 1 into n2 intervals of length

1 ; n2



 1 2 n2  1 Pn D 0; 2 ; 2 ; : : : ; ;1 : n n n2

Let Pn1 be the collection of rectangles which contain rk for some k  n and let Pn2 be the remaining rectangles. Since any rational number is in at most two of the intervals, jPn1 j  2n. Furthermore, jf .x/j  n1 for any x 2 P 2 Pn2 . Since 0  f  1, we may estimate: 0

L.f; Pn / 

U.f; Pn / 

X P 2Pn1

1  vol.P / C R

X 1 vol.P /  n2 2

P 2Pn

2n 1 C 2: n2 n

Since this tends to 0 as n ! 1, f is integrable and Œ0;1 f D 0. Note that f is discontinuous at every rational point of Œ0; 1 and continuous at every irrational point of Œ0; 1.

1.5. THE RIEMANN INTEGRAL

27

m

Let f be a bounded real-valued function on a rectangle R  R . We have used closed rectangles to define M.f; P / and m.f; P /. Occasionally, it is convenient to use open rectangles. Let P be a partition of R. Set: P L m.f; L P / WD infx2int.P / f .x/; L.f; P / WD P 2P vol.P /m.f; L P /; P L L L M .f; P / WD sup f .x/; U .f; P / WD P 2P vol.P /M .f; P / : x2int.P /

Let f be a bounded real-valued function on a rectangle R  Rm . Let  > 0 be given and let P be a partition of R.

Lemma 1.13

1. ere exists a rectangle Q contained in the interior of R so vol.R/

vol.Q/ <  .

L 2. L.f; P /  L.f; P /  UL .f; P /  U.f; P /.

L 3. ere is a refinement Q of P so L.f; P /  L.f; Q/ C  and U.f; Q/

Proof. Let ı < 21 min1im .bi ai /. Let Qı WD Œa1 C ı; b1 contained in the interior of R. Assertion 1 now follows since

lim vol.Qı / D lim .b1

ı!0

ı!0

a1

2ı/




.bm

  UL .f; P /.

ı 


 Œam C ı; bm

am

ı be

2ı/ D vol.R/ :

Assertion 2 is immediate. Suppose first that the partition P consists only of the rectangle R. Let Qı be the partition: Qı WD ffa1 ; a1 C ı; b1 ı; b1 g; : : : ; fam ; am C ı; bm ı; bm gg : Let C D supx2R jf .x/j. Let Qı D Œa1 C ı; b1

ı 


 Œam C ı; bm

be the big central rectangle of the partition. Let Qı WD Qı the partition. We have m.f; L R/  m.f; Qı /. We estimate:

Qı be the remaining rectangles of

m.f; L R/ vol.R/  m.f; Qı / vol.R/ D m.f; Qı / vol.Qı / C D m.f; Qı / vol.Qı / C 

m.f; L Qı / vol.Qı / C

X Q2Qı

X

Q2Qı

L D L.f; Q/ C 2C fvol.R/

m.f; Q/ vol.Q/ C m.f; L Q/ vol.Q/ C

X Q2Qı

X

(1.5.c)

ı

X

m.f; Qı / vol.Q/

Q2Qı

fm.f; Qı /

m.f; Q/g vol.Q/

2C vol.Q/

Q2Qı

vol.Qı /g :

If P is a more complicated partition, we perform a similar construction on each rectangle of P to construct a partition Qı and for each P 2 P , we let Qı .P / be the corresponding rectangle defined in Equation (1.5.c). We have: X L L.f; P /  L.f; Qı / C 2C fvol.P / vol.Qı .P //g : P 2P

28

1. BASIC NOTIONS AND CONCEPTS

L e error can be made uniformly small as ı ! 0. is shows L.f; P /  L.f; Q/ C  ; the proof of the remaining assertion is similar. u t

1.5.2 SETS OF CONTENT AND MEASURE ZERO. In view of Lemma 1.13, we may either take the inf and sup over all of a rectangle or over just the interior of a rectangle in future arguments involving integrals. We shall not belabor the point, but simply use Lemma 1.13 as necessary to simplify an argument. We now introduce just a bit of additional terminology from measure theory. Let S be a subset of Rm .

1. We say S has content zero if given  > 0, there exist a finite number of rectangles fR1 ; : : : ; Rn g so S  R1 [


[ Rn and so vol.R1 / C


C vol.Rn / <  . 2. We say S has measure zero if given  > 0, there exist a countable (or finite) number of rectangles fRi g so S  R1 [ R2 [


and so vol.R1 / C vol.R2 / C


<  . By using an argument similar to that used to prove Lemma 1.13, we may work with open rectangles rather than with closed rectangles and define equivalently: 1. S has content zero if and only if given  > 0, there exist a finite number of rectangles fR1 ; : : : ; Rn g so S  int.R1 / [


[ int.Rn / and so vol.R1 / C


C vol.Rn / <  . 2. S has measure zero if and only if given  > 0, there exist a countable (or finite) number of rectangles fRi g so S  int.R1 / [ int.R2 / [


and so vol.R1 / C vol.R2 / C


<  . Lemma 1.14

1. A finite union of sets of content zero has content zero. 2. A countable (or finite) union of sets of measure zero has measure zero. 3. Any countable (or finite) set has measure zero. 4. Any compact set of measure zero has content zero. 5. e closure of any set of content zero is a compact set of content zero. Proof. Suppose sets fS1 ; : : : ; Sn g have content zero. Let  > 0 be given. Choose rectangles Rijj for 1  ij  j so j j X [ j j vol.Rij / < 3  and Sj  Rijj : ij D1

ij D1

We show S1 [


[ Sn has content zero and establish Assertion 1 by observing: j n X X j D1 ij D1

vol.Rijj /


0 be given. As noted above, we can use open rectangles rather than closed rectangles to find a countable collection of rectangles Ri so 1 X

vol.Ri / < 

iD1

and C 

1 [

int.Ri / :

iD1

Since fint.Ri /g is an open cover of the compact set C , there is a finite subcover fint.Ri1 / [


[ int.Rik /g

of C . is shows that C has content zero and establishes Assertion 4. Finally let S have content zero. Let  > 0 be given. We find a finite collection of closed rectangles so n X iD1

vol.Ri / < 

and

S  R1 [


[ Rn :

Since closure.S/  closure.R1 [


[ Rn / D R1 [


[ Rn , we conclude the closure of S has content zero. A finite union of closed rectangles is compact. A closed subset of a compact set is compact and hence the closure of S is compact. t u If f is a real-valued function on a rectangle R  Rm , define the graph of f by: Gf WD f.x; f .x// 2 RmC1 W x 2 Rg .

Lemma 1.15

If f is integrable, then Gf has content zero.

Proof. Let  > 0 be given. Since f is integrable, we may choose a partition P D fPi g of the defining rectangle R  Rm so that U.f; P / L.f; P / < ". If P 2 P is an m-dimensional rectangle, define an m C 1-dimensional rectangle Q.P / by setting: Q.P / WD P  Œm.f; P /; M.f; P / :

We have Gf  [P 2P Q.P /. is implies the following inequality from which the Lemma follows:

30

1. BASIC NOTIONS AND CONCEPTS

X

P 2P

vol.Q.P // D

X

P 2P

fM.f; P /

m.f; P /g vol.P / D U.f; P /

L.f; P / <  .

u t

1.5.3 REMARK. e converse of Lemma 1.15 is false. Let f be the characteristic function of the rational numbers in Œ0; 1; f .x/ D 1 if x 2 Œ0; 1 is rational and f .x/ D 0 if x 2 Œ0; 1 is not rational. By eorem 1.17 below, f is not integrable since f is discontinuous at every point of Œ0; 1. On the other hand Gf  Œ0; 1  f0g [ Œ0; 1  f1g has content zero in R2 .

Let f be a bounded real-valued function defined on a set S  Rm . If 0 < r < s , then ( ) ( ) sup

f .x/

x2S \Br .P /

inf

x2S\Br .P /

f .x/ 

sup

f .x/

x2S\Bs .P /

inf

x2S\Bs .P /

f .x/

so this difference is a monotonically decreasing non-negative function of r . We define the oscillation of f around a point P of S by setting: o.f; P / WD lim

r!0

(

sup

x2S\Br .P /

f .x/

inf

x2S \Br .P /

)

f .x/ :

(1.5.d)

e following result follows immediately from the definition: Lemma 1.16

Let f be a bounded real-valued function defined on a set S  Rm .

1. f is continuous at P 2 S if and only if o.f; P / D 0. 2. Let  > 0. If S is closed, then fx 2 S W o.f; x/  g is closed. We can now give a useful characterization of what it means for a function to be Riemann integrable: eorem 1.17

Let R be a rectangle which is contained in Rm .

1. Let f be a bounded real-valued function on R. Let D be the set of all points of R where f is discontinuous. en f is integrable on R if and only if D has measure zero. 2. If f and g are integrable on R, then fg is integrable on R. 3. If f is integrable on R and if g is continuous on the range of f , then g ı f is integrable on R. 4. If f is a non-decreasing function on R D Œa; b, then f is integrable.

1.5. THE RIEMANN INTEGRAL

31

Proof. Before proving Assertion 1, we establish some additional notation. Since f is bounded, we may choose k so jf j  k on the closed rectangle R. Let o.f; x/ be the oscillation of f about x . Let Dn WD fx 2 R W o.f; x/  n1 g. By Lemma 1.16, Dn is closed. Since Dn  R, Dn is bounded and hence compact. We have D D D1 [ D2 [


. Suppose first that D has measure zero. en each Dn has measure zero and hence, being compact, has content zero by Lemma 1.14. Let  > 0 be given. Choose 1 > 0 and n 2 N so 2k1 < 31 

and

1 n

vol.R/ < 13  :

Since Dn has content zero, we may find a finite number of rectangles fR1 ; : : : ; R g so Dn  R1 [


[ R

and

vol.R1 / C


C vol.R / < 1 :

By considering all the hyperplanes which form the sides of the rectangles Ri , we can construct a partition P of R and decompose P D P1 [ P2 as the disjoint union of two families of rectangles so [ X Dn  P; vol.P / < 1 ; int.P / \ Dn D ; for P 2 P2 : P 2P1

We then have

P 2P1

X P 2P1

fM.f; P /

m.f; P /g vol.P /  2k1 < 13  :

(1.5.e)

For each rectangle P 2 P2 , use Lemma 1.13 to find a rectangle Q.P /  int.P / so that X fvol.P / vol.Q.P //g < 1 : P 2P2

We use the same argument which was used to establish Equation (1.5.e) to show that: X fM.f; P / m.f; P /gfvol.P / vol.Q.P //g  2k1 < 13  :

(1.5.f )

P 2P2

Since Q.P /  int.P / and Dn \ int.P / D ;, for each x 2 Q.P /, we can find r.x/ > 0 so that: jy

xj < r.x/

)

jf .x/

f .y/j
0 be given. Fix n. Choose 1 > 0 so that n1 < 13  . Choose a partition P so that U.f; P /  L.f; P / < 1 . Let P1 be the set of rectangles P 2 P so that Dn \ int.P / ¤ ;. e following picture is perhaps useful: e rectangles of P1 where o.f; x/  n1

We then have 1 X n

P 2P1

vol.P / 

X P 2P1

fM.f; P /  m.f; P /g vol.P / 

U.f; P /  L.f; P / < 1 :

P Consequently P 2P1 vol.P / < 13  . ere can, of course, be points of Dn which are not in the interior of any rectangle of P . But such points necessarily lie on the boundary of some rectangle. e boundary of any rectangle necessarily has content zero. us Dn has content zero and hence measure zero. Since D D [n Dn , D has measure zero by Lemma 1.14. is establishes Assertion 1; Assertion 2 and Assertion 3 then follow immediately. Let f be a monotone non-decreasing real-valued function on an interval Œa; b and let

P D fa D c0 < c1 <    < cm D bg P be a partition of Œa; b. One may verify that one has i o.f; ci /  f .b/  f .a/. From this it follows that the set of discontinuities of f is countable and thus of measure zero. is implies f is integrable which establishes Assertion 4. u t We illustrate eorem 1.17 by considering the function constructed in Example 1.5.1. is function is integrable and the set of discontinuities is the set of rational points in Œ0; 1. Since this set is countable, the set of discontinuities has measure zero. Let S be a subset of Rm . We define the characteristic function of S by setting: ) ( 1 if x 2 S S .x/ D . 0 if x … S Lemma 1.18

Let R be a rectangle in Rm and let S  R.

1. e following conditions are equivalent and if either is satisfied, then S is said to be Jordan Z measurable and we set vol.S / D (a) S is integrable.

S :

R

1.5. THE RIEMANN INTEGRAL

33

(b) e boundary of S has content zero. 2. S is Jordan measurable if and only if given any  > 0, there exists a partition P of R so that if P1 WD fP 2 P W P  Sg and if P2 WD fP 2 P W P \ S ¤ ;g, then [ [ X X X P S  P and vol.P /  vol.P /  vol.P / C  : P 2P1

P 2P2

In this setting, we have

X

P 2P1

P 2P1

vol.P /  vol.S / 

P 2P2

X

P 2P2

P 2P1

vol.P / 

X

P 2P1

vol.P / C  .

3. If vol.R/ > 0, then R does not have measure zero. Proof. Assertion 1 follows from eorem 1.17 since the set of discontinuities of S is exactly the boundary of S . Let P be a partition of R and let P 2 P . We have 3 cases:

1. If P \ S D ;, then m.S ; P / D M.S ; P / D 0. 2. If P \ S ¤ ; and P 6 S , then m.S ; P / D 0 and M.S ; P / D 1. 3. If P  S , then m.S ; P / D 1 and M.S ; P / D 1. is shows that

X P 2P1

vol.P / D L.S ; P /  U.S ; P / D

X

vol.P / :

P 2P2

If S is Jordan measurable, then S is integrable. us given  > 0, there exists a partition P so U.S ; P / L.S ; P / <  and the desired inequality holds. Conversly, if given  > 0 we can find a suitable partition P , then U.S ; P / L.S ; P / <  and S is integrable. is proves Assertion 2. Suppose that R is a rectangle in Rm with Vol.R/ > 0 but, to the contrary, that R has measure zero. Since R is compact, R has content zero. Let R1 , …, R` be a cover of R by closed rectangles with vol.R1 / C


C vol.R` / < 12 vol.R/. We have R1 C


C R` ; Z Z Z Z vol.R/ D R  fR1 C


C R` g D R1 C


C R` R



D vol.R1 / C


C vol.R` /
0. u t is is, of course, exactly the approach taken by Euclid to determine the area of a circle by examining inscribed and circumscribed polygons; if we set S1 WD [P 2P WP S P and S2 WD [P 2P WP \S ¤; P ;

then S1 and S2 are unions of rectangles (and hence can be regarded as polygons) which satisfy the relations S1  S  S2 and vol.S1 /  vol.S /  vol.S2 /  vol.S1 / C  .

34

1. BASIC NOTIONS AND CONCEPTS

Let S be a Jordan measurable subset of a rectangle R and let f be integrable on R. By eorem 1.17, S f is integrable and one sets Z Z f WD S f : S R R In particular, if we take f D 1, we see that S 1 D vol.S/. Clearly Jordan measurable sets are bounded. e following 1-dimensional example shows that not every bounded open set is Jordan measurable. 1.5.4 EXAMPLE. Let frn g be an enumeration of the rationals in .0; 1/. Let

On WD B5 n .rn / \ .0; 1/ and O WD

1 [

On :

nD1

Note that O is an open subset of .0; 1/. We assume O is Jordan measurable and argue for a contradiction. Since O is Jordan measurable, the boundary bd.O/ has measure zero. Let C be the complementary closed set of Œ0; 1, C D Œ0; 1 \ Oc . Since O is a dense subset of Œ0; 1, bd.O/ D ON \ Oc D Œ0; 1 \ Oc D C :

is implies C has measure zero. en we can find a countable collection of rectangles Ri so P C  int.R1 / [


and so i vol.Ri / < 51 . e collection fOn ; int.Ri /g is an open cover of Œ0; 1. Since Œ0; 1 is compact, there is a finite subcover Œ0; 1  O1 [


[ On [ R1 [


[ Rn :

erefore Œ0;1  O1 C


C On C R1 C


C Rn . We derive the desired contradiction and show O is not Jordan measurable by integrating this inequality to see: Z Z Z Z Z 1 D Œ0;1  O1 C


C On C R1 C


C Rn D vol.O1 / C


C vol.On / C vol.R1 / C


C vol.Rn / 

n X 2i iD1

5i

C

1 < 1: 5

1.5.5 FUBINI’S THEOREM. Fubini’s eorem will permit us to evaluate multivariable integrals as iterated integrals. It is due to the Italian mathematician Guido Fubini.

Guido Fubini (1879–1943)

1.5. THE RIEMANN INTEGRAL a

35

b

eorem 1.19 (Fubini’s eorem). Let A  R and B  R be rectangles. Let f be a bounded real-valued integrable function on A  B . Set fx .y/ WD f .x; y/ for x 2 A and y 2 B . Let L and U be R R the lower and upper integrals, i.e., L.x/ WD B fx and U .x/ WD B fx . en L and U are integrable

and:

Z

AB

f D

Z

A

LD

Z

A

U.

Proof. Let P  Q be a partition of A  B where P is a partition of A and Q is a partition of B . Let P  Q 2 P  Q. For any x 2 P , we have m.f; P  Q/  m.fx ; Q/. Consequently: Z X X m.f; P  Q/ vol.Q/  m.fx ; Q/ vol.Q/ D L.fx ; Q/  fx D L.x/ : Q2Q

B

Q2Q

We take the inf over x 2 P to see X m.f; P  Q/ vol.Q/  m.L; P / : Q2Q

Consequently, summing over P 2 P yields: X X L.f; P  Q/ D m.f; P  Q/ vol.Q/ vol.P / P 2P Q2Q



X

P 2P

m.L; P / vol.P / D L.L; P / 

Z A

L:

R R Taking the sup over all partitions P  Q then yields AB f  A L. A similar argument yields R R A U  AB f . us we have Z Z Z Z Z L L U f : f  AB

A

A

A

AB

Since all these inequalities must have been equalities, L is integrable and ilar argument can be used to deal with U .

R

AB

f D

R

A

L. A simu t

It is necessary to introduce several caveats. ese are best illustrated by a series of examples. If f is in fact continuous, then it is not necessary to go through the technical fuss of considering L and U as the double integral can simply be computed as the iterated integral. However in the general setting, it is necessary to be more careful. We have the following illustrative examples: 1. Let A D B D Œ0; 1. Define

8 C1 ˆ ˆ ˆ < 0 f .x; y/ WD ˆ 0 ˆ ˆ : C1

if x if x if x if x

is rational and y  21 is rational and y > 12 is irrational and y  21 is irrational and y > 21

9 > > > = > > > ;

:

36

1. BASIC NOTIONS AND CONCEPTS

R1 en fx is integrable for any fixed x and 0 fx .y/dy D 21 . We have that f is discontinuous on all of Œ0; 1  Œ0; 1. By Lemma 1.18, Œ0; 1  Œ0; 1 does not have measure zero and thus f is not integrable.

2. Let S be a countable dense subset of R WD Œ0; 1  Œ0; 1 so that any horizontal or vertical line contains at most one point of S . en S is discontinuous at every point of R and R1 R1 hence is not integrable. However, 0 fx .y/dy D 0 for all x and 0 fy .x/dx D 0 for all y so the iterated integrals exist. 3. Let R WD Œ0; 1  Œ0; 1. Let f be as defined in Example 1.5.1, i.e., if frn g is an enumeration of the rationals in Œ0; 1, then f .x/ WD

(

Let g.x; y/ WD

1 n

0 (

if x D rn for some n if x ¤ rn for all n f .x/ if y is rational 0 if y is irrational

)

)

:

:

e R argument given in Example 1.5.1 extends immediately to show R g is integrable and g D 0 . However, if x is rational, then f .x/ ¤ 0 so the integral R Œ0;1 gx .y/dy does not R exist if x is rational and the iterated integral R g.x; y/dydx does not exist and it is necessary to dealR with U .x/ D f .x/ and L.x/ D 0 in examining the R iterated integrals. On the other hand Œ0;1 gy .x/dx D 0 for all y so the iterated integral R g.x; y/dxdy can be evaluated directly. 1.5.6 APPLICATIONS OF FUBINI’S THEOREM. eorem 1.20

1. (Clairaut’s eorem [12]) Let O be an open subset of R2 . Let f W O ! R be C 2 . en @x @y f D @y @x f . 2. (Cavalieri’s Principle) Let Ai be Jordan measurable subsets of Rm . For xE 2 Rm 1 , consider the slice Ai .c/ WD fxE W .x; E c/ 2 Ai g. Suppose for any c that Ai .c/ are Jordan measurable subsets of m 1 R with vol.A1 .c// D vol.A2 .c//. en vol.A1 / D vol.A2 /. Cavalieri’s principle was introduced by the Italian mathematician Bonaventura Francesco Cavalieri.

1.5. THE RIEMANN INTEGRAL

37

B. Cavalieri (1598–1647) Proof. Suppose Assertion 1 fails; we argue for a contradiction. By interchanging the roles of x and y if necessary, we may assume that f@x @y f  @y @x f g.P / > 0 for some point P 2 O. Since the second partial derivatives are continuous, we can find a rectangle R so that P 2 R  O and so that @x @y f  @y @x f  on R for some  > 0. Let R D Œa; b Œc; d . Consequently: Z .@x @y f  @y @x f /  vol.R/ > 0 : (1.5.h) R

We use eorem 1.19 and the Fundamental eorem of Calculus to see: Z Z dZ b Z d @x @y f D @x f@y f g.x; y/dxdy D f@y f .b; y/  @y f .a; y/gdy c

R

Z

@y @x f R

a

c

D f .b; d / C f .a; c/  f .a; d /  f .b; c/; Z b Z bZ d @y f@x f g.x; y/dydx D f@x f .x; d /  @x f .x; c/gdy D a

c

a

D f .b; d / C f .a; c/  f .a; d /  f .b; c/ :

ese two integrals are equal, which contradicts Equation (1.5.h). We have the following picture illustrating Assertion 2:

Introduce coordinates .x; E t/ on Rm . We compute: Z Z Z vol.A1 / D .A1 /.x; E t /d xdt E D vol.A1 .t//dt Z Z Z .A2 /.x; E t /d xdt E D vol.A2 / . D vol.A2 .t //dt D

u t

38

1. BASIC NOTIONS AND CONCEPTS

e assumption that the second partial derivatives are continuous in Assertion 1 is necessary as the following example shows. Let

f .x; y/ WD

(

xy.x 2  y 2 / x 2 Cy 2

0

if .x; y/ ¤ .0; 0/ if .x; y/ D .0; 0/

)

.

is is clearly C 1 away from the origin. Suppose x D 0. en f .0; y/ D 0 so: f .0 C  x; y/  f .0; y/ f . x; y/ D lim  x!0 x x 2 2 1  x  y. x  y / D lim D y  x!0  x  x2 C y2

.@x f /.0; y/ D

lim

 x!0

and thus .@y @x f /.0; 0/ D  1. A similar calculation shows .@x @y f /.0; 0/ D C1.

1.6

IMPROPER INTEGRALS

If  is a real-valued function on Rm , then the support of  is the closure in Rm of the set of points where  ¤ 0. 1.6.1 MESA FUNCTIONS AND PARTITIONS OF UNITY. Let O  Rm be open.

1. A compact exhaustion of O is a countable collection of compact subsets Cn  O so that O D [n Cn and Cn  int.CnC1 /. 2. Let C be a compact subset of O. A mesa function for C which is supported in O is a smooth real-valued function  on Rm so that 0    1, so that   1 on C , and so that supportf g  O. 3. Let fO g be open sets with O  O and [ O D O. A locally finite partition of unity for O which is subordinate to the cover fO g is a countable collection of smooth real-valued functions  n on Rm which take values in Œ0; 1 so: (a) For each point x 2 O, there is a neighborhood U .x/ of x so that only a finite number of the functions  n are non-zero on U .x/ (locally finite). P (b) If x 2 O, then n  n .x/ D 1 (partition of unity). (c) For each n, there exists .n/ so that supportf n g  O.n/ (subordinate to the cover).

1.6. IMPROPER INTEGRALS

39

Lemma 1.21 Let C  O , where C is compact and O is a non-empty open subset of R , and let fO g be a cover of O. m

1. Let  > 0 be given. ere exist smooth non-negative functions on R so that: (a) f0 .x/ D 0 for x 

0 and f0 .x/ > 0 for x > 0.

(b) f1;.x/ D 0 for x 

0 or x  and f1;.x/ > 0 for 0 < x <  .

(d) f3;.x/ D 1 for x 

 , f3;.x/ D 0 for x 2 , and 0 

(c) f2;.x/ D 0 for x 

0, 0 

1 for 0 

f2;.x/ 

 , and f2;.x/ D 1 for x  .

x

f3;.x/ 

1 for  

x

2 .

2. ere exists a compact exhaustion of O by Jordan measurable sets. 3. ere exists a mesa function  for C supported in O. 4. ere exists a locally finite partition of unity for O subordinate to the cover fO g. Proof. To focus our discussion in the proof of Assertion 1, we give graphs of the 4 functions:

f0

0

f1; 0



Let  > 0 be given. Let f0 .t / WD

(

f2; 0

if t 

0 e

t

1

 0

if t > 0

)

f3; 0



2

:

Clearly f0 has the properties desired in Assertion 1-a for t ¤ 0 so it suffices to show f0 is smooth n  1 1  n t and consequently at 0. Since e t  tnŠ for t > 0, we have e t  nŠ e

t

1



nŠt n

for t > 0 and for any n 2 N :

We assume inductively that there is some polynomial pk . / so that ) ( if t  0 0 .k/ : f0 .t/ D  1 if t > 0 pk .t  1 /e  t We set p0 D 1 and k D 0 to begin the induction. We use Equation (1.6.a) to see lim pk .t  1 /e 

t !0

t

1

D0

(1.6.a)

(1.6.b)

40

1. BASIC NOTIONS AND CONCEPTS

and thus f0.k/ is continuous at t D 0. Differentiating Equation (1.6.b) yields ( ) 0 if t < 0 .kC1/ f0 .t/ D 1 pkC1 .t 1 /e t if t > 0 where pkC1 .t 1 / D t 2 fpk0 .t 1 / pk .t 1 /g. e left hand derivative of f0.k/ vanishes at 0. We complete the inductive step and establish Assertion 1-a by using Equation (1.6.a) to compute the right hand derivative: f0.k/ .t/

lim

f0.k/ .0/

D lim t

t

t!0C

t !0

1

pk .t

1

/e

t

1

D 0:

e function f1; .t/ WD f0 .t /f0 .

t/ has the properties of Assertion 1-b. e function Z 1  1Z t f2; .t / D f1; .s/ds f1; .s/ds (1.6.c) 1

1

has the properties of Assertion 1-c and the function f3; .t/ WD 1 of Assertion 1-d. is establishes Assertion 1. If O D Rm , we set Ck D Bk .0/. Otherwise, we set:

f2; .t

2/ has the properties

Ck WD fx 2 O W kxk  k and dist.x; Oc /  k1 g :

e collection fCk g forms a compact exhaustion of O; however the sets Ck need not be Jordan measurable. As C1 is compact, we can find a finite number of rectangles R1i contained in O so C1  [i int.R1i /. Set D1 WD [i R1i . Because bd.D1 /  [i bd.R1i /, D1 is Jordan measurable. We have C1  int.D1 /. Apply the same argument to the compact set C2 [ D1 to find a number of rectangles R2i contained in O so D1 [ C2  [i int.R2i /. Set D2 WD [i R2i . We continue in this fashion to construct a sequence of compact Jordan measurable sets Dk contained in O so that Dk [ Ck  int.DkC1 /  DkC1 . Since [k Ck D O, we have [k Dk D O. is establishes Assertion 2. Let C be a compact subset of O. Find a finite number of points xi and associated radii i > 0 for 1  i  ` so B3i .xi /  O and so C  [i int.Bi .x//. Set .x/ WD

` X iD1

f3;2 .kx i

x i k2 / I

is smooth and non-negative, has support in O and is at least 1 on C . We cut off to have maximum value 1 and establish Assertion 3 by defining .x/ WD f2;1 . .x//. Let fCn g be a compact exhaustion of O. Let Rn WD Cn int.Cn 1 /; the “ring” Rn is a compact subset of O. Fix n. We cover Rn by a finite number of balls Brni .xni / so B3rni .xni /  Rn

1

[ Rn [ RnC1

and B3rni .xni /  O˛ni

for some

˛ni :

1.6. IMPROPER INTEGRALS

41

e following picture will focus our discussion in the proof of Assertion 4 where for the purposes of illustration we suppose Cn D fx W kxk  ng and Rn D fx W n  1  kxk  ng; we show two small balls B3rni .xni / contained in the larger ring Rn 1 [ Rn [ RnC1 :

For each .i; n/, let ni be a mesa function for the closure of Brni .xni / which are supported in B2rni .xni /. If jn  kj 5, then ki vanishes on Rn 1 [ Rn [ RnC1 which is a neighborhood of Rn . P us the collection f ni g is a locally finite family and  .x/ D n;i ni is smooth on O (although it need not be defined on Oc ). Since the fBrni .xni /g covers Rn and since every x belongs to Rn for some n, we have  .x/ 1 on O. Set  ni WD ni   1 . Since ni has support in B2rni .xni /  O, we can extend  ni to B3rni .xni /c to be identically zero to obtain a function which is smooth on all of Rm . As each B2rni .xni / is contained in O ni , the collection is subordinate to the given cover. P Assertion 4 now follows as n;i  ni D 1 on O.  Let f be a real-valued function on an open subset O of Rm . We say that f is locally bounded if given any compact set C  O, there exists a constant  C so that jf .x/j   C for all x 2 C . We say that f is continuous almost everywhere if the set of discontinuities of f has measure zero.

Lemma 1.22 Let f be a non-negative real-valued function on an open set O which is locally bounded and continuous almost everywhere. Let f n g be a partition of unity subordinate to the cover of O by the interior of all the closed rectangles contained in O, let f k g be a collection of mesa functions for a compact exhaustion fAk g of O, and let B` be a compact exhaustion of O by Jordan measurable sets. en: Z Z 1 Z X  n f D lim f D lim f . k nD1

k!1

`!1 B`

Proof. By definition  n f has support in the interior of a rectangle Rn which is contained in O. Consequently,  n f is bounded on Rn , and  n f is continuous almost everywhere on Rn . R us  n f is well-defined and non-negative. Consequently, the infinite sum in question is welldefined. It can, of course, be infinite. Next note that k f has R support in a compact set and hence f is bounded and continuous almost everywhere. us k k f is well-defined.R Furthermore, R R f is non-negative and k  kC1 implies kf  kC1 f and thus limk!1 k f is welldefined; this limit can, of course, be infinite. Finally, note that B` is Jordan measurable, that f is

42

1. BASIC NOTIONS AND CONCEPTS

R bounded on B` , and that f is continuous almost R R everywhere on B` so BR` f is well-defined. Since B`  B`C1 and f is non-negative, B` f  B`C1 f and thus lim`!1 B` f is well-defined; the limit can, of course, be infinite. We establish the proof in three steps:

Step 1. Fix N . en the support of 1 C


C N is contained in a finite union of rectangles and hence is compact. Since O D [ int.Ak /, there exists K so the support of 1 C


C N (which is a compact set) is contained in AK (a compact set covered by an increasing union of open sets is eventually contained in a single set). us 1 C


C N  K so: N Z X nD1

n f D

Z

N X nD1

!

n f 

Z

Kf

 lim

Z

k!1

kf

:

As N was arbitrary, we may take the limit as N ! 1 to show: 1 Z X nD1

n f  lim

Z

kf

k!1

(1.6.d)

:

Step 2. Fix k . As AkC1 is compact, the previous argument shows there exists L so AkC1  BL . Since k has support in AkC1 , k  AkC1  BL and thus Z Z Z Z BL f D f  lim f : kf  `!1 B`

BL

As k was arbitrary, we may take the limit as k ! 1 to see Z Z lim f : k f  lim k!1

(1.6.e)

`!1 B`

Step 3. Fix `. Since B` is a compact set and the fn g are locally finite, only a finite number of the n are non-zero on B` . In particular, we can choose N so n D 0 for n > N . Since fn g is a partition of unity, 1 C


C N D 1 on B` . us B`  1 C


C N so: Z B`

f D

Z

B` f 

Z

N X nD1

!

n f D

N Z X nD1

n f 

1 Z X

n f :

nD1

As ` was arbitrary, we may take the limit as ` ! 1 to see that: lim

Z

`!1 B`

f 

1 Z X

n f :

(1.6.f )

nD1

e Lemma now follows from Equations (1.6.d), (1.6.e), and (1.6.f).



1.6. IMPROPER INTEGRALS

43

1.6.2 INTEGRABILITY IN THE EXTENDED SENSE. Let f be continuous except on a set of measure zero and locally bounded. If any (and hence all) of the 3 terms in Lemma 1.22 are finite, we say that f is integrable in the extended sense and we define Z

e

O

f WD

1 Z X nD1

n f D lim

Z

kf

k!1

D lim

Z

`!1 B`

f :

e value is independent of partition of unity fn g, of the mesa functions f n g, and of the compact exhaustion by Jordan measurable sets fB` g. We have had to deal with a fair amount of technical fuss. e difficulty is, of course, that for the Riemann integral one does not have in general that the integral of an infinite sum is the infinite sum of the integrals nor is it true in general that the limit of the integrals is the integral of the limit; such theorems are, of course, the reason the Lebesgue integral was developed. We now show that the integral in the extended sense R e agrees with R the ordinary integral if O is Jordan measurable and we shall subsequently replace simply by . Lemma 1.23

Let f be a non-negative real-valued functionZwhich isZbounded and continuous

almost everywhere on a Jordan measurable open set O. en

e

O

f D

O

f.

Proof. Choose a rectangle R so O  R. Choose  so jf j   . Let  > 0 be given. Choose 1 > 0 so 1 <  . By Lemma 1.18, there exists a partition P of R so if K WD [P 2P WP O P ;

then K is a compact Jordan measurable subset of O with vol.O K/ < 1 . Since we can embed K as the first term of a compact exhaustion of O by Jordan measurable sets, Z Z e f : f  O

K

is shows that: Z O

f D

Z K

f C

Since  is arbitrary, we have

Z O K

f  Z O

Z

e

O

f C  vol.O

f 

Z

K/ 

Z

e

O

f C:

e

f : O

On the other hand if fC` g is aRcompactRexhaustion of O by Jordan measurable sets, then C`  O implies C`  O and hence C` f  O f . Since this holds for any `, we may take the sup over Z e Z ` to obtain the reverse inequality and complete the proof; f  f.  O

O

44

1. BASIC NOTIONS AND CONCEPTS

1.6.3 ABSOLUTELY INTEGRABLE FUNCTIONS. Let f be continuous almost everyR where and locally bounded on an open set O. We say that f is absolutely integrable if O jf j < 1. We set fC WD 12 .f C jf j/ and f WD 12 . f C jf j/. By eorem 1.17, the functions f are continuous almost everywhere; they are clearly locally bounded as well. ) ) ( ( if f .x/ 0 f .x/ if f .x/ 0 0 : fC .x/ D and f .x/ D 0  f .x/ if f .x/  0 if f .x/  0

We then have f 

jf j and hence the f are integrable in the extended sense. Define: Z Z Z f WD fC  f : O

O

O

Let f be absolutely integrable in the extended sense on O. Adopt the notation of Lemma 1.22. en Z Z Z 1 Z X  n f D lim f D f D lim f . k

Lemma 1.24

O

k!1

nD1

`!1 B`

We remark that then all the usual properties of the integral continue to hold. We shall not belabor the point. e advantage of the extended integral is that we can assign a volume (possibly infinite of course) to an open subset of Rm . For example, let O  Œ0; 1 be the open subset of ExampleR 1.5.4 that is not Jordan measurable. Since O is a bounded open set, we may define vol.O/ WD O 1 < 1. e following are other examples which further illustrate improper integrals phenomena. 1. Let f .x/ D

p1 x

on .0; 1/; f is continuous, non-negative, and locally bounded. We consider

the compact exhaustion defined by Bn WD Πn1 ; 1  Z 0

1

f .x/dx D lim

Z

n!1

y D x

1 1 n

1 2

1 n

x

1 2

for

1  for n 1





dx D lim 2x 2  n!1

1 n

0, we use Lemma 1.25 to find n so that X vol.S/  vol.P /  vol.S/ C 1 for n  N : P 2Pn WP \S ¤;

We note that if A and B are Jordan measurable sets, then vol.A/ C vol.B/ D vol.A [ B/ C vol.A \ B/ : In particular, if A \ B has content zero, then the volume is additive and we may ignore the possible overlap of .P1 / \ .P2 / for rectangles Pi 2 Pn in what follows. We can use the inclusion S .S/  P 2Pn WP \S ¤; .P / to see that: ˚ vol. .S//  vol [P 2Pn WP \S ¤; .P / D  j det.T0 /j.1 C  j det.T0 /jf1 C

p

mkT0 1 k/m

X

vol. .P //

P 2Pn WP \S ¤;

X

vol.P /

P 2Pn WP \S ¤;

p mkT0 1 kgm fvol.S / C 1 g :

e desired inequality now follows as 1 was arbitrary. We can now establish a result which is dual in a certain sense to eorem 1.9.



50

1. BASIC NOTIONS AND CONCEPTS

Let O be an open non-empty subset of Rm and let H W O ! Rn be continuously differentiable. If H.O/ contains a non-empty open subset of Rn , then n  m. eorem 1.27

Proof. Suppose to the contrary that n > m. Let .x; y/ WD H.x/ for y 2 Rn m . Let S be a rectangle in Rm . en S  f0g has content zero in Rm and hence by Lemma 1.26, .S / has content zero in Rn . Consequently H.S / has content zero in Rn so H.O/ has measure zero in Rn and cannot contain any non-empty open subset of Rn . u t

1.7.1 CHANGE OF VARIABLE THEOREM. e following result will be central to our discussion of the generalized Stokes’ eorem:

Let O be an open subset of Rm . Let ˚ W O ! Rm be continuously differentiable and injective with det.˚ 0 .x// ¤ 0 on O. Let f be integrable on ˚.O/ and let ˚  f WD f ı ˚ . en .˚  f /
j det.˚ 0 /j is integrable on O and Z Z f D .˚  f /
j det.˚ 0 /j . eorem 1.28 (Change of Variable eorem).

˚.O/

O

Proof. We shall divide the proof into several steps. We suppose f  0 for the majority of the proof. In Step 1, we work locally and establish an inequality related to integrating over cubes. In Step 2, we extend this inequality to Jordan measurable sets and we reverse the roles of the domain and range to establish the Change of Variable eorem for integrating over a cube; it is necessary to consider Jordan measurable sets rather than just cubes since ˚.cube/ need not, of course, be a cube. In Step 3, we pass to extended integrals using a partition of unity to complete the proof. Step 1. Assume f  0. Let C be a cube which is contained in O. Let o.f; x/ be the oscillation of f that was defined in Equation (1.5.d). Set D` D fx 2 ˚.C / W o.f; x/  1` g

and D D [`2N D` D fx 2 ˚.C / W o.f; x/ > 0g :

Since f is integrable on the Jordan measurable set ˚.C /, D has measure zero and the sets D` have content zero. Applying Lemma 1.26 to ˚ 1 yields ˚ 1 .D` / has content zero and hence ˚ 1 .D/ D [`2N ˚ 1 .D` / has measure zero. is implies ˚  f is continuous almost everywhere on C . Since ˚.C / is compact, f is bounded on ˚.C / and thus ˚  f is bounded on C . is shows that ˚  f is integrable. We wish to show: Z Z f  .˚  f /
j det.˚ 0 /j if f  0 : (1.7.g) ˚.C /

C

Let " > 0 be given. Let  WD supx2C fk˚ 0 .x/ 1 k; k˚ 0 .x/kg. Since ˚ 0 is continuous on the cube C and since C is compact, ˚ 0 is uniformly continuous on C . We may therefore choose  > 0 so that if xi 2 C with kx1 x2 k < ı , then k˚ 0 .x1 / ˚ 0 .x2 /k < ". Choose a partition P of C which has mesh less than ı (i.e., if x1 and x2 are two points of any rectangle P 2 P , then jx1 x2 j < ı ). We then have Z
 0 U .˚ f /
j det ˚ j; P  " C .˚  f /
j det ˚ 0 j : C

1.7. THE CHANGE OF VARIABLE THEOREM 0

51

˚ .x2 /k <  for x1 ; x2 2 P 2 P , we apply Lemma 1.26 to see p vol.˚.P //  .1 C m/m j det.˚ 0 .y//j
vol.P / for any y 2 P : 0

Since k˚ .x1 /

is permits us to estimate vol.˚.P //  .1 C Clearly f .z/  Z

p m/m inf .j det.˚ 0 .y//j/
vol.P / : y2P

P

˚.C /

M.f; ˚.P //˚.P / .z/ for any z 2 ˚.C /. Consequently: Z X X f  M.f; ˚.P // ˚.P / D sup ff .˚.x//g
vol.˚.P // P 2P

˚.C /

P 2P

p m"/m

 .1 C

X

P 2P x2P

sup f.˚  f /.x/g
inf fj det.˚ 0 .y//jg
vol.P /

P 2P x2P

y2P

P 2P x2P

y2P

  X p m  0 sup .˚ f /.x/
inf fj det.˚ .y//jg
vol.P / D .1 C m"/ p m"/m

 .1 C

X

sup f.˚  f /.x/
j det.˚ 0 .x//jg
vol.P /

P 2P x2P

p m"/m U ..˚  f /
j det ˚ 0 j; P / ˚ R p  .1 C m"/m " C C .˚  f /
j det ˚ 0 j . D .1 C

Taking the limit as " ! 0 establishes Equation (1.7.g). Step 2. Let S be a Jordan measurable subset of C . Replacing f by ˚.S / f then yields: Z Z f  .˚  f /
j det.˚ 0 /j : (1.7.h) ˚.S /

S

Assume additionally there exists a cube CQ so ˚.S /  CQ  ˚.O/. Set ˚Q WD ˚

1

;

fQ WD .˚  f /
j det.˚ 0 /j;

Applying Equation (1.7.h) to this setting yields: Z Z ˚ ˚  0 .˚ f /
j det.˚ /j  .˚ 1 / Œ˚  f 
.˚ S

˚.S /

SQ WD ˚.S / :

/ Œj det.˚ 0 /j
j det.˚

1 

1 0

/ j:

As det.AB/ D det.A/ det.B/, the chain rule implies: ˚ .˚ 1 / Œj det.˚ 0 /j
j det.˚ 1 /0 j D j detf.˚ 0 ı ˚ 1 /
.˚ 1 /0 gj D j detf.˚ ı ˚ 1 /0 gj D j det.Id/j D 1 :

(1.7.i)

52

1. BASIC NOTIONS AND CONCEPTS

Since .˚

1 

/ Œ˚  f  D f ı ˚ ı ˚ Z S

1

D f , we may rewrite Equation (1.7.i) as: Z .˚  f /
j det.˚ 0 /j  f : ˚.S /

Combining this inequality with Equation (1.7.h) shows Z Z f D .˚  f /
j det.˚ 0 /j : ˚.S /

(1.7.j)

S

Step 3. Assume that f  0. Choose a cover fO˛ g of O by Jordan measurable open sets so that each O˛ is contained in some cube C˛  O and so that each ˚.O˛ / is contained in some cube CQ ˛  ˚.O/. Let fQ n g be a locally finite partition of unity on ˚.O/ subordinate to the cover ˚.O˛ /. en n WD ˚  Q n is a locally finite partition of unity on O subordinate to the cover O˛ . us we may use Equation (1.7.j) to see Z

f ˚.O /

D D

1 Z X nD1 ˚.O˛.n/ /

Z

O



Q n
f D

1 Z X nD1

O˛.n/

n
.˚  f /
j det.˚ 0 /j

0

.˚ f /
j det.˚ /j :

is establishes the Change of Variable eorem if f  0. Decomposing f D f C f as the difference of two positive functions then establishes the Change of Variable eorem in complete generality. 

53

CHAPTER

2

Manifolds In Chapter 2, we will discuss the basic theory of smooth manifolds. In Section 2.1, we define what it means for M to be a smooth manifold or for M to be a submanifold of another manifold. We discuss immersions, fiber bundles, and vector bundles. In Section 2.2 we treat the tangent bundle, submersions, and the cotangent bundle. Section 2.3 deals with the exterior algebra and Stokes’ eorem. In Section 2.4, we present some applications of Stokes’ eorem.

2.1

SMOOTH MANIFOLDS

In this section, we present the basic foundational material concerning smooth manifolds that we shall need. 2.1.1 LOCALLY EUCLIDEAN SPACES. Let M be a locally compact metric space; the particular metric is not essential as only the underlying topology is relevant at this point. We say that M is locally Euclidean of dimension m if there exists an open cover fO˛ g of M and homeomorphisms ˛ from each O˛ to open subsets U˛ of Rm . If .x 1 ; : : : ; x m / are the usual coordinates on Rm , then we set x˛i WD x i ı ˛ to express ˛ D .x˛1 ; : : : ; x˛m / on O˛ :

We call the pair .O˛ ; ˛ / a coordinate chart. We have transition functions ˛ˇ WD ˛ ı ˇ 1 ; domainf˛ˇ g D ˇ fO˛ \ Oˇ g  Rm

and

rangef˛ˇ g D ˛ fO˛ \ Oˇ g  Rm :

If all the functions f˛ˇ g are diffeomorphisms (i.e., if the functions ˛ˇ are smooth and satisfy 0 det.˛ˇ / ¤ 0), then M is said to be a manifold or to have a smooth structure. e collection f.O˛ ; ˛ /g is said to be a coordinate atlas; the atlas can always be chosen to be maximal. We shall only work in the smooth setting and shall not consider locally Euclidean spaces with C k structures or piecewise-linear structures. us the word “manifold” shall always mean “locally Euclidean with a smooth structure.” 2.1.2 SMOOTH MAPS. If f is a map from a manifold M to Rn , then f is smooth if f ı ˛ 1 W U˛ ! Rn is smooth for all ˛ :

Let C 1 .M / be the infinite-dimensional real vector space of all smooth functions on M . If f.O˛ ; ˛ /g is a coordinate atlas for a manifold M and if f.OQ ˇ ; Q ˇ /g is a coordinate atlas for a

54

2. MANIFOLDS

manifold MQ , then the Cartesian product f.O˛  OQ ˇ ; ˛  Q ˇ /g

is a coordinate atlas which gives M  MQ a smooth structure. A function f from M to MQ is said to be smooth if Q ˇ ı f ı ˛ 1 is a smooth map from U˛ to UQˇ for all ˛ , ˇ whenever Q ˇ ı f ı ˛ 1 is defined. If f is bijective, then f will be said to be a diffeomorphism if f and f 1 are both smooth; M and MQ will be said to be diffeomorphic if there exists a diffeomorphism between them. ere can be inequivalent smooth structures on the same underlying locally Euclidean metric space; there are manifolds which are homeomorphic but not diffeomorphic. 2.1.3 PULLBACK. Let F W M ! N be smooth. For a function f 2 C 1 .N /, the pullback is defined by F  .f / WD f ı F 2 C 1 .M /; F  is a unital ring homomorphism from C 1 .N / to C 1 .M /, i.e., F  .f C g/ D F  .f / C F  .g/;

F  .1 N / D 1M ;

F  .f
g/ D F  .f /
F  .g/ ;

where 1 denotes the constant function identically equal to 1. If G is a smooth map from a manifold L to a manifold M , we note that .F ı G/ D G  ı F  and Id D Id. 2.1.4 SUBMANIFOLDS. If W is a k -dimensional subspace of Rm and if P 2 Rm , then the translated subspace A WD W C P is said to be an affine subspace of Rm of dimension k . Let MQ be a closed subset of a manifold M . We say that MQ is a submanifold of M of dimension k if there exists a coordinate atlas f.O˛ ; ˛ /g for M so that either O˛ \ MQ is empty or ˛ .O˛ \ MQ / D U˛ \ Ak where Ak is an affine subset of dimension k in Rm . Setting OQ ˛ WD O˛ \ MQ , and Q ˛ WD ˛ jOQ˛ then gives MQ the structure of a manifold so the inclusion of MQ in M is a smooth map. If O is an open subset of Rm and if V is a k -dimensional affine subspace of Rm , then O \ V is a k -dimensional submanifold of Rm . ere are, however, less trivial examples. Let 1  k < m. Let M be an m-dimensional manifold and let F W M ! Rm k be smooth. We say that F is regular at a point P 2 M if there exists a coordinate chart .O; / with P 2 O so that if we set F WD F ı  1 and if we set Q D .P /, then F0 .Q/ is a surjective map from Rm to Rm k . Note that this condition is independent of the particular coordinate chart chosen. We use Inverse Function eorem to establish: eorem 2.1 Let 1  k < m. Let M be an m-dimensional manifold and let F W M ! Rm k be smooth. Fix c 2 Rm k and let Mc WD F 1 .c/ be the level set. Suppose F is regular at every point P 2 Mc . en Mc is a smooth submanifold of M of dimension k .

Proof. Choose a coordinate chart .O; / so P 2 O. By replacing M by U D .O/ and by replacing F by F ı  1 , we may reduce to the special case F W U ! Rm k . Let Aij WD @x i Fj for 1  i; j  m k . Since F 0 .P / is surjective, by permuting the coordinates if necessary, we may suppose that det.A/ ¤ 0. Let u D .x 1 ; : : : ; x m k / be the dependent variables and

2.1. SMOOTH MANIFOLDS m kC1

55

m

let v D .x ; : : : ; x / be the independent variables. Let G.u; v/ WD .F .u; v/; v/. Since 0 det.G .P // D det.A/ ¤ 0, G is an admissible change of coordinates and we have a new coorQ where Q WD G ı  . Since F .G.u; v// D u is just projection on the first m k dinate chart .O; / coordinates, this gives rise to a new coordinate atlas where the level sets of F ı G are linear subspaces of Rm k .  e following example is illustrative. Let .V; h
;
i/ be an m-dimensional inner product space; we impose no restriction on the signature. Let Sr .V; h
;
i/ WD fv 2 V W hv; vi D rg be the pseudo-sphere of radius r ¤ 0. Let fx 1 ; : : : ; x m g be the coordinates on V defined by an orthonormal basis fe 1 ; : : : ; e m g for V . en hx; xi D 1 .x 1 /2 C


C m .x m /2 where i D ˙1. e derivative of the defining function is .21 x 1 ; : : : ; 2m x m /. is is non-zero on V f0g. us Sr .V; h
;
i/ is a smooth submanifold. 2.1.5 REGULAR COVERING SPACES. Let G be a finite group which acts smoothly on a manifold M without fixed points, i.e., g
x ¤ x for g 2 G and for x 2 M , and g ¤ Id. Let d be a metric on M . We give the orbit space MQ WD M=G the structure of a metric space by defining dQ .xG; yG/ WD min d.g1 x; g2 y/ : gi 2G

Choose local coordinate charts .O˛ ; ˛ / on M so that g O˛ \ O˛ D ; for g ¤ Id. e coordinate charts then descend to give MQ the structure of a smooth manifold so that the natural projection  W M ! MQ is a smooth map. e triple .M; ; MQ / is called a regular covering projection. We refer to Spanier [37] for further details. 2.1.6 REAL PROJECTIVE SPACE. ere are several natural manifolds which arise as quotients of orthogonal actions by finite groups on the sphere. e antipodal map on the sphere is given by a.x/ D x . Since a2 D Id, G WD fId; ag is a group of order 2 that acts smoothly and without fixed points on the sphere S m 1  Rm . e quotient manifold RP m 1 WD S m 1 =Z2 is called real projective space. e map  W  ! 
R defines a map from the sphere to the set of lines through the origin in Rm . Since  ./ D ./ if and only if  D ˙. e map  identifies RP m 1 with the space of lines through the origin in Rm . If  2 S m 1 , then orthogonal projection on  is defined by  W x ! .x; / where .
;
/ is the usual Euclidean inner product. us we may also identify RP m 1 with the space of orthogonal projections of rank 1 in Hom.Rm ; Rm /. 2.1.7 LENS SPACES AND SPHERICAL SPACE FORMS. If m D 2n is even, we may regard S 2n 1 as the unit sphere in C n . Identify the cyclic group Zk of order k with the k th roots of unity in C . Let E WD .1 ; : : : ; n / be a collection of integers coprime to k . Let E ./
.z 1 ; : : : ; z n / WD .1 z 1 ; : : : ; n z n / define a fixed point free action E of Zk on S 2n 1 . Let L.kI E / WD S 2n 1 =E .Zk /; this is called a lens space. More generally the spherical space forms arise by setting M D S m 1 =G where G is a finite subgroup of the orthogonal group O.m/ so that det.g Id/ ¤ 0 for g ¤ Id. We refer to Wolf [41] for further details.

56

2. MANIFOLDS

2.1.8 LIE GROUPS. We say that a manifold M is a Lie group if M is also a group such that the group multiplication m.g; h/ D g
h from M  M to M is smooth and so that the map g ! g 1 from M to M is smooth. Cramer’s rule (Lemma 1.3) can be used to show that the general linear group GL.V / of invertible linear transformations of V is a Lie group. Sophus Lie initiated the study of continuous transformation groups and was a pioneer in this area.

Sophus Lie (1842–1899) Let .V; h
;
i/ be an inner product space. e associated orthogonal group is defined to be:

O.V; h
;
i/ WD fA 2 GL.V / W hAv; Awi D hv; wi for all v; w 2 V g : We may verify that O.V; h
;
i/ is a Lie group as follows. If A belongs to Hom.V; V /, then the adjoint A , which belongs to Hom.V; V /, is characterized by the identity hAv; wi D hv; A wi for all v; w 2 V . Clearly A 2 O D O.V; h
;
i/ if and only if A A D Id. It now follows that O is closed under composition, and that A D A 1 . Let f .A/ WD A A map Hom.V; V / to the linear space of self-adjoint matrices S WD fB 2 Hom.V; V / W B D B  g. en O D f 1 .Id/. Let A 2 O and let B 2 S . Let .t/ WD A.1 C tB/. en f . .t// D .1 C tB  /A A.1 C tB/ D 1 C t.B  C B/ C O.t 2 / D 1 C 2tB C O.t 2 /

and thus @ t f . .t//j t D0 D 2B . is shows f 0 .A/ is surjective. We apply eorem 2.1 to see O is a submanifold of GL.V /. e group operation and the inverse map are the restriction of smooth maps on GL.V / and hence smooth on O. us O is a Lie group. We will discuss more examples subsequently in Book II. 

2.1.9 FIBER BUNDLES. We say that F ! E !M is a fiber bundle with fiber F over M if E is a manifold and if  is a smooth surjective map from E to M with F D  1 .P / where P is the base point of M . We assume there is a coordinate atlas f.O˛ ; ˛ /g for M and diffeomorphisms 1 .O˛ / so that P D . ˛ .P; v// for any point P 2 O˛ and any element ˛ from O˛  F to  v of the fiber F . Such a map is said to be fiber preserving since ˛ defines a diffeomorphism from fxg  F to the fiber Ex WD  1 fxg. e transition or gluing functions are ˛ˇ WD ˛ 1 ı ˇ . e maps ˛ˇ are fiber preserving diffeomorphisms of fO˛ \ Oˇ g  F . Note that we have the cocycle condition: ˛˛

D Id on O˛  F

and

˛ˇ

ı

ˇ

D

˛

on .O˛ \ Oˇ \ O /  F :

(2.1.a)

2.1. SMOOTH MANIFOLDS

57

2.1.10 IMMERSIONS. Let F be a smooth map from a manifold M to a manifold N . Let .O;  / be local coordinates near P 2 M , and let .OQ ; Q / be local coordinates near F .P / in N . We say that F is an immersion if .Q F   1 /0 . .P // is an injective linear transformation for every P 2 M ; this condition is independent of the particular coordinate systems chosen. We say that F is an embedding if F is an immersion which is injective. Clearly if M is a submanifold of N , then the inclusion map is an embedding of M into N . e embedding is said to be a proper embedding if the inverse image of a compact set in N is a compact set in M . 2.1.11 THE KLEIN BOTTLE. is surface was first described by the German mathematician Felix Klein.

Christian Felix Klein (1849–1925) Let M D S 1 be the unit circle. We regard S 1 WD Œ0; 2 where we identify 0  2 . We take Œ0; 2 S 1 and glue 0    2   N (where  N is the complex conjugate) to define the Klein bottle K. e natural projection  W K ! S 1 is a fiber bundle with fiber S 1 . e Klein bottle can be immersed, but not embedded in R3 . We present below an immersion due to S. Dickson [14]; the surface is created in two pieces S1 and S2 which glue together smoothly but which intersect each other. Let 0  u   and 0  v  2 . Set 1 0 1 0 6 cos.u/.1 C sin.u// C 4.1  :5 cos.u// cos.u/ cos.v/ x C B C B S1 @ y A D @ 16 sin.u/ C 4.1  :5 cos.u// sin.u/ cos.v/ A, z 4.1  :5 cos.u// sin.v/ 1 0 1 0 6 cos.u C /.1 C sin.u C // C 4.1  :5 cos.u C // cos.v C / x C B C B S2 @ y A D @ 16 sin.u C / A. z 4.1  :5 cos.u C // sin.v/

58

2. MANIFOLDS

We could also glue 0    2   to define the torus T 2 WD S 1  S 1 . e natural projection  W T 2 ! S 1 is also a fiber bundle with fiber S 1 . We shall see presently that the torus T 2 is orientable and the Klein bottle K is not orientable. us T 2 is not diffeomorphic to K. e torus can be embedded in R3 . For 0  u  2 and 0  v  2 , set: 1 0 1 0 .3 C cos.u// cos.v/ x C B C B @ y A D @ .3 C cos.u// sin.v/ A ; 0  u; v  2 z sin.v/

Let M and MQ be connected smooth manifolds. 1. If F W M ! MQ is a proper embedding, then F .M / is a submanifold of MQ and F is a diffeomorphism from M to F .M /.

eorem 2.2

2. (Whitney Embedding eorem [40]) ere exists a proper embedding of M into Rk for some k and, consequently, M is diffeomorphic to a closed proper submanifold of some Euclidean space. Proof. It suffices to establish Assertion 1 in the following special case; the assumption that F is proper then lets one pass from the local to the global setting without topological difficulties. Let O be an open subset of Rk and let F W O ! Rm be smooth. Let P 2 O and assume F 0 .P / is an injective linear map from Rk to Rm . If k D m, then the Inverse Function eorem shows that F is a local diffeomorphism so we assume k < m. By making a linear change of coordinates on Rm , we may assume that rangefF 0 .P /g D Rk viewed as a subset of Rm . Let G.x; y/ WD .F .x/; y/ for x 2 Rk and for y 2 Rm k define the germ of a map from Rm to itself. en G 0 .P / is injective and hence bijective as a map from Rm to itself. is provides the needed change of coordinates on Rm and establishes Assertion 1. We shall assume that M is compact in proving Assertion 2; it is then automatic that the embedding is proper. Extending our proof to the general case requires dimension theory which is beyond the scope of this book. We refer to Munkres [32] for further details. Let Br .0/ be the open ball of radius r about the origin. We can choose a coordinate atlas f.O ;   /g for M so that   is a diffeomorphism from O to B3 .0/ and so that the open sets   1 .B1 / still cover M . Since M is compact, there is a finite subcollection f.Oi ;  i /g for 1  i  ` so that f i 1 .B1 /g covers M . Use Lemma 1.21 to find a smooth mesa function so that D 1 on B1 .0/ and so that has compact support inside B2 .0/. Set: i .x/ i .x/

WD

WD 0;

. i .x//;  ij .x/ WD  ij .x/

. i .x//xij .x/

WD 0

if x 2 Oi ;

if x 62 Oi :

Since the supports are contained in  i 1 .B2 .0//, these functions are smooth on all of M . We define a smooth map  from M to R.mC1/` by setting:  .x/ WD .

1 m 1;  1 ; : : : ;  1 ; : : : ;

1 m `;  ` ; : : : ;  ` / :

2.1. SMOOTH MANIFOLDS

59

1

Suppose ˚.P / D ˚.Q/. Choose  so P 2  fB1 .0/g. is implies  .P / D 1 and since ˚.P / D ˚.Q/, we also have  .Q/ D 1. Consequently, P and Q both belong to O . Since j j  .P / D  .Q/ D 1, x .P / D x .Q/ for 1  j  m so P D Q. is shows that ˚ is injective. 1 Furthermore, the coordinates .x ; : : : ; xm / are among the components of ˚ . us the Jacobian is injective so ˚ is an embedding. Assertion 2 now follows from Assertion 1. t u 2.1.12 VECTOR BUNDLES. We now discuss vector bundles. Atiyah [4] or Karoubi [23] are  good additional references. Let F ! V !M be a fiber bundle whose fiber F is a real or complex vector space of dimension r . If the transition functions ˛ˇ preserve the vector space structure, then the fiber bundle is said to be a vector bundle. In this setting, we may regard ˛ˇ as a smooth map from O˛ \ Oˇ to the general linear group GL.F /. Conversely, given a collection of smooth maps ˛ˇ from O˛ \ Oˇ to GL.F / which satisfy the cocycle condition given in Equation (2.1.a), we can recover the vector bundle in question by an appropriate identification of the disjoint union of the O˛  F . e vector bundle is said to be a line bundle if r D 1. e trivial bundle of fiber dimension r is the Cartesian product 1 r D M  Rr . 

2.1.13 TRIVIALIZATIONS AND BUNDLE MAPS. Let us consider F ! V !M and  Q FQ ! VQ !M two vector bundles over M modeled on vector spaces F and FQ . By an abuse of notation we shall in the future simply say V and VQ are vector bundles over M where the projections  and Q are implicit. A trivialization of V over an open subset O of M is a fiber preserving diffeomorphism W O  F !  1 .O/ which is linear on the fibers; V is said to be trivial over O if there exists a trivialization. e ˛ give trivializations of V over the open sets O˛ of the cover. We say that  is a bundle map from V to VQ if  is a smooth map from V to VQ which is a linear map from the fibers VP to the fibers VQP for each P 2 M . We say  is a bundle isomorphism if  is a diffeomorphism. We refer to Atiyah [4] for the proof of the following:

If .O; / is a coordinate chart so that .O/ is an open convex subset of Rm , then any vector bundle over O is trivial. If  is a bundle map from V to VQ such that Rank.x / is constant on M , then kerf g is a subbundle of V and rangef g is a subbundle of VQ . Lemma 2.3

2.1.14 SECTIONS, FRAMES, TRANSITION FUNCTIONS, AND FIBER METRICS. Let V be a vector bundle over M of fiber dimension r . A section to V over an open subset O of M is a smooth map s from O to V so that .s.P // D P for all P 2 O. Fix a basis fe1 ; : : : ; er g for the model vector space F . Let f.O˛ ; ˛ /g be a coordinate atlas for M so that V is trivial over each O˛ . Over each O˛ , we have local sections si;˛ .x/ WD ˛ .x; ei /. e collection sE˛ WD .s1;˛ ; : : : ; sr;˛ / gives a local frame for V over O˛ , i.e., fs1;˛ ; : : : ; sr;˛ g is a basis for the fiber Vx of V over x . Conversely, given such a frame over an open set O, we obtain a local trivialization of V over O by setting O .x; / WD 1 s1 .x/ C


C r sr .x/. By an abuse of notation, let C 1 .V / be the set of smooth sections to V . Fiberwise addition and multiplication give C 1 .V / the structure of module over the ring of smooth functions on M .

60

2. MANIFOLDS

Let V and W be vector bundles over M . We choose a coordinate atlas f.O ;   /g so that V W V and W are trivial over each O and are defined by transition functions   and   , respectively. e direct sum vector bundle V  W and the tensor product vector bundle V  W are V W V W    and      . e dual vector bundle V  is defined using defined, respectively, by   V   1 g and the vector bundle the induced dual transition functions on the dual vector space f  V W  Hom.V; W / WD V  W is defined by      . Other bundles may be defined similarly. If V has fiber dimension r and if W has fiber dimension s , then V  W has fiber dimension r C s , V  W has fiber dimension r  s , and V  has fiber dimension r . A fiber metric on a vector bundle V is a positive definite inner product . ;  / on the fibers Vx which varies smoothly. Given such a metric, the unit sphere bundle S.V / and the unit disk bundle D.V / are defined by: S.V / D f 2 V W .; / D 1g

and D.V / D f 2 V W .; / 

1g :

Note that S.V / is the boundary of D.V /. By using the Gram–Schmidt process, one can choose orthonormal frames; we then have  is an orthogonal matrix in the real setting and a unitary matrix in the complex setting. 2.1.15 THE TORUS AND THE MÖBIUS STRIP. e torus T 2 is the unit circle bundle of the trivial bundle 1 2 over S 1 and the Möbius strip M WD Œ0; 2 Œ 1; 1=  is defined by gluing 0    2  .  /. One embeds the Möbius strip M in R3 by setting: x.u; v/ D .3 C u cos.:5v// cos.v/;

y.u; v/ D .3 C u cos.:5v// sin.v/;

z.u; v/ D u sin.:5v/

for u 2 Œ 1; 1 and v 2 Œ0; 2. One glues .u; 0/ to .1  u; / to obtain:

Replacing Œ 1; 1 by R yields a non-trivial line bundle L over the circle. Define an open cover of the circle by setting: O1 WD S 1  f.1; 0/g and O2 WD S 1  f. 1; 0/g. en O1 \ O2 consists of two disjoint open sets. e gluing function is C1 on one and  1 on the other of these sets. e Möbius strip is the unit disk bundle of L and the Klein bottle is the unit circle bundle of 1  L over the circle. e transition function for L  L is given by . 1/2 D 1 and thus L  L D 1 2 is the trivial line bundle. One can also show that L  L is the trivial 2-plane bundle although this takes more work. We have L has transition functions f.C1/ 1 ; . 1/ 1 g; L is isomorphic to L .

2.2. THE TANGENT AND COTANGENT BUNDLES

61

2.1.16 TENSORS. Let V and W be vector bundles over a manifold M . We say that a linear map T from C 1 .V / to C 1 .W / is a tensor or is tensorial if T is a C 1 .M / module homomorphism; that means that T .f s/ D f T .s/ for s 2 C 1 .V / and f 2 C 1 .M /. Lemma 2.4 Let T be a tensorial map from C 1 .V / to C 1 .W /. en T arises from a bundle map from V to W .

Proof. Let fs1 ; : : : ; sr g be a local frame field for V over a coordinate neighborhood .O; .x 1 ; : : : ; x m //. It suffices to show that if s 2 C 1 .V / satisfies s.P / D 0, then we have that P .T .s//.P / D 0. Express s.x/ D i f i .x/si .x/. Use Lemma 1.21 to find a smooth function with compact support in O so is identically 1 near P . Choose  with compact support in O so that  is identically 1 on the support of . en si and f i extend smoothly to all of M . Furthermore  D . We compute: .T .s//.P / D .T ..1 /s//.P / C .T . s//.P / D .1 /.P /.T .s//.P / C .T . s//.P / P P i i D 0 C i .T . f si //.P / D i . f .P //
.T .si //.P / D 0 : P Suppose given X.P / in the fiber of V over P . Expand X.P / D i ai si .P /. Define a smooth P global section to V by defining XL WD i ai si .P /. en define a map T .P / from VP to WP by P setting T .P / WD i .T . ai si ///.P /. is is independent of the choices made. We see that T is P smooth by noting that T . ai si / D i ai si where D 1. 

2.2

THE TANGENT AND COTANGENT BUNDLES

In this section, we shall adopt the Einstein convention and sum over repeated Roman indices; indices ˛ , ˇ , and will index coordinate charts and are not summed over. Let f.O˛ ; ˛ /g be a coordinate atlas for M . Express ˛ D .x˛1 ; : : : ; x˛m / where the fx˛i WD ˛ x i g are the coordinate functions on O˛ . Let f 2 C 1 .M /. en the pullback .˛ 1 / f WD f ı ˛ 1 is a smooth function on U˛ D ˛ .O˛ / and we define @x˛i f WD @x i f.˛ 1 / f g. By the Chain Rule, @x˛i D

@xˇj @x˛i


@x j

ˇ

on

O˛ \ Oˇ :

(2.2.a)

e transition functions ˛ˇ are given by the Jacobian matrix ˛ˇ ji WD @x˛i xˇj and we have the cocycle condition of Equation (2.1.a): ˛ˇ ji
ˇ jk D ˛ ki

on O˛ \ Oˇ \ O :

We denote the resulting vector bundle over M by the tangent bundle TM ; the fiber TP M is called the tangent space at P . Given a local system of coordinates xE˛ D .x˛1 ; : : : ; x˛m /, the f@x˛i g form a local frame. A global section to TM is said to be a tangent vector field. By an abuse of notation, we let C 1 .TM / be the infinite-dimensional vector space consisting of all smooth tangent vector fields. is is a module under pointwise scalar multiplication over the space of smooth functions on M .

62

2. MANIFOLDS

2.2.1 DERIVATIONS. We say that a linear map  W C 1 .M / ! R is a derivation based at P if .fg/ D .f /g.P / C f .P /.g/

f; g 2 C 1 .M / :

for all

Similarly we say that a linear map  W C 1 .M / ! C 1 .M / is a derivation if for all

.fg/ D .f /g C f .g/

f; g 2 C 1 .M / :

If X D ai @x i 2 TP M is a tangent vector at P , let X.f /.P / WD ai @x i f .P / where we suppress the role of the coordinate chart index ‘˛ ’ to simplify the notation. is is a derivation based at P . Similarly if X is a global tangent vector field, then we obtain a derivation of C 1 .M /. If  is a derivation of C 1 .M / which is based at P 2 M , then there exists a unique tangent vector X 2 TP M so that .f / D X.f /.P / for any f 2 C 1 .M /. If  is a derivation of C 1 .M /, then there exists a unique tangent vector field X so .f / D X.f /. Lemma 2.5

Proof. Let Br be the open ball of radius r about the origin in Rm . Let  be a derivation of C 1 .M / based at P 2 M . Choose a coordinate system .O; / so P 2 O, so  is a diffeomorphism from O to B3 , and so .P / D 0. To avoid notational complexities, we identify O with B3 , we identify P with 0, and we ignore the role of  . Let 1 be the constant function identically equal to 1. We observe .1 / D .1
1 / D .1 /
1 .0/ C 1 .0/
.1 / D 2.1 / :

Consequently, .1 / D 0. Fix f 2 C 1 .M /. Let x 2 B3 . Set gi .x/ WD

Z

f .x/

f .0/ D

0

.@x i f /.tx/dt :

0

Note that gi .0/ D .@x i f /.0/. We have Z

1

1

@ t ff .tx/gdt D

Z 0

1

x i f.@x i f /.tx/gdt D x i gi .x/ :

(2.2.b)

Use Lemma 1.21 to find a smooth mesa function  taking values in Œ0; 1, so that  D 1 on B1 and so  D 0 on B2c . Note that 2 also is a mesa function. Since 2 gi and 2 x i have support in B2 , we can extend them to be zero on B2c and regard these functions as belonging to C 1 .M /. We decompose f D f0 C f1 C f2 where the fi 2 C 1 .M / are defined by f0 WD f .0/1 ;

f1 D .1

2 /.f

f0 /;

f2 D 2 .f

f0 / :

We consider the tangent vector X WD .x i /
@x i 2 TP M . Because .1 / D 0, we have .f0 / D .f .0/1 / D 0.

2.2. THE TANGENT AND COTANGENT BUNDLES

Because .1

2

f0 /.0/ D 0,

.f1 / D ..1 D .1 2 /
.f

2 /.f f0 // f0 /.0/ C .1 2 /.0/
.f

 /.0/ D 0 and .f

Equation (2.2.b) implies that 2 .f .f2 / D .2 .f

63

f0 / D 0 C 0 D 0 :

f0 / D 2 .x i gi / D .x i /.gi /. Consequently

f0 // D ..x i /.gi // D .x i /
.gi /.0/ C .x i /.0/
.gi /

D .x i /gi .0/ D .x i /
.@x i f /.0/ D .Xf /.0/.

is shows .f / D .Xf /.0/ and represents  as a tangent vector; since the coefficients are .x i /, the representation is unique. If  is a derivation of C 1 .M /, then we define a derivation based at P by setting P .f / WD .f /.P /. If X D .2 x i /@x i , then .f / D X.f / in the coordinate chart O; since X is globally defined,  D X . t u 2.2.2 THE LIE BRACKET. Let X and Y be vector fields on a manifold M . By Lemma 2.5, we can identify X and Y with derivations of C 1 .M /. e commutator ŒX; Y , characterized by the identity ŒX; Y .f / D .XY YX /f is a linear map from C 1 .M / to C 1 .M / which is called the Lie bracket. We verify that it is a derivation by computing: ŒX; Y .fg/ D D D

X.Y.fg// Y.X.fg// D XfY.f /g C f Y .g/g Y fX.f /g C f X.g/g X.Y.f //g C Y .f /X.g/ C X.f /Y .g/ C f X.Y .g// Y.X.f //g X.f /Y .g/ Y .f /X.g/ f Y .X.g// ŒX; Y .f /
g C f
.ŒX; Y .g// :

If X D ai @x i and Y D b j @x j , then ŒX; Y  D faj @x j .b i / isfies the Jacobi identity:

b j @x j .ai /g@x i . e Lie bracket sat-

ŒŒX; Y ; Z C ŒŒY; Z; X  C ŒŒZ; X ; Y  D 0 :

2.2.3 THE FLOW OF A VECTOR FIELD. If  is a derivation of C 1 .M / which is based at P and if F W M ! N is smooth, define .F  /.f / D .F  f / for f 2 C 1 .M /; this is a derivation based at F .P /. e pushforward is a linear map F from TP M to TF .P / N given locally by: F .@x i / D @x i y j
@y j I

.G ı F / D G ı F

and

Id D Id :

(2.2.c)

We say that a map ˚ W M  R ! M has the semigroup property if ˚.y; 0/ D y for all y and if ˚.˚.y; t/; s/ D ˚.y; t C s/ for all y , t , and s . Let X be a smooth vector field on a compact manifold M . ere exists a unique smooth map ˚ W M  R ! M , called the flow of X , which has the semigroup property so that ˚ .@ t / D X.˚.y; t// for all y , t . Lemma 2.6

64

2. MANIFOLDS

Proof. Suppose first that X is a smooth vector field on an open subset O of Rm . Expand X D ai @x i to regard X D .a1 .x/; : : : ; am .x// as a smooth map from O to Rm ; this is, of course, just the canonical identification of T O D O  Rm . Let C be a compact subset of O. e Fundamental eorem of Ordinary Differential Equations shows there exists  > 0 and a unique smooth map W C  Œ ;  ! Rm so that .@ t /.y; t / D X. .y; t //

on

C  Œ ;  :

is map has the semigroup property for jtj C jsj   . Let f.O˛ ; ˛ /g be a coordinate atlas on M so that the corresponding subsets U˛  Rm are all the open ball B2 .0/ of radius 2 about the origin. Let C˛ WD ˛ 1 fBN 1 .0/g. Since fint.C˛ /g is a cover of M , we can choose a finite subcover fint.C1 /; : : : ; int.C` /g of M . e argument quoted above permits us to construct flows i for X with domains C  Œ  ;   which take values in O . One sets  D min.1 ; : : : ;  /. e uniqueness of the flow shows that i D j on their common domain. is constructs a flow W M  Œ ;  ! M satisfying the semigroup property for jsj C jtj <  . We iterate the process and use the semigroup property to construct the desired flow. u t 2.2.4 THE FROBENIUS THEOREM. We begin our discussion of the Frobenius eorem with the following technical result: Lemma 2.7 Let fX1 ; : : : ; Xk g be smooth vector fields on M satisfying ŒXi ; Xj  D 0 for all i; j and so that fX1 .P /; : : : ; Xk .P /g are linearly independent at a point P of M .

1. We can choose local coordinates xE D .x 1 ; : : : ; x m / so that @x 1 D X1 near P . Xj

2. We have ˚ tXi ˚s

X

Xi

D ˚s j ˚ t

near P .

3. We can find local coordinates on M centered at P so Xi D @x i for 1  i  k . Proof. Let .x 1 ; : : : ; x m / be local coordinates on M centered at P . Make a linear change of coordinates, to assume X1 .P / D @x 1 .P /. Let T .x 1 ; x 2 ; : : : ; x m / WD ˚xX11 .0; x 2 ; : : : ; x m /. is is a smooth map and T .0/ D Id. us T is an admissible change of coordinates and clearly in this new coordinate system, @x 1 D X1 . is proves Assertion 1. By Assertion 1, we may assume the coordinates are chosen so X1 D @x 1 . Sum over repeated indices to expand X2 D ai .x/@ E x i . Since the bracket of X1 and X2 is zero, ai .x/ E is independent 1 of x . Let .e1 ; : : : ; em / be the standard basis for Rm : e1 D .1; 0; : : : ; 0/;

e2 D .0; 1; 0; : : : ; 0/;

::::

en a point xE 2 Rm can be represented as xE D x 1 e1 C


C x m em . Consequently the flow for X1 is the linear map sX1 W e1 ! e1 C s while ˚sX1 W ei ! ei for i  2 or, equivalently, sX1 .x/ E D

2.2. THE TANGENT AND COTANGENT BUNDLES

xE C se1 . Let

65

n o t;s .x/ E WD sX1 tX2 .x/ E D tX2 .x/ E C se1 ; ˚  t;s .x/ E WD tX2 sX1 .x/ E D tX2 .xE C se1 / :

Fix s . We have the initial conditions 0;s .x/ E D xE C se1 and 0;s .x/ E D xE C se1 . Since the coefficients ai are independent of x 1 , we have the evolution equations: @ t t;s .x/ E D @ t tX2 .x/ E D X2 . tX2 .x// E D X2 . tX2 .x/ E C se1 / D X2 . t;s .x//; E X2 @ t  t;s .x/ E D X2 . t .xE C se1 // D X2 . t;s .x// E :

Assertion 2 now follows from the Fundamental eorem of Ordinary Differential Equations for i D 1 and j D 2 while general case then follows by permuting the indices. We now make a linear change of coordinates to assume Xi .P / D @x i for 1  i  k . We define X

T .x 1 ; : : : ; x k ; x kC1 ; : : : ; x m / D .˚xX11 ı


ı ˚x kk /.0; : : : ; 0; x kC1 ; : : : ; x m / : X1 D @x 1 in these coordinates. As we can commute the ˚ 0 s , Xi D @x i for 1  i  k .

t u

Let V be a subbundle of TM of dimension k . e following conditions are equivalent and if either is satisfied, then V is said to be integrable. eorem 2.8 (Frobenius [16]).

1. If X 2 C 1 .V / and Y 2 C 1 .V /, then ŒX; Y  2 C 1 .V /. 2. ere are local coordinates .x 1 ; : : : ; x m / around any point P 2 M so that the collection f@x 1 ; : : : ; @x k g forms a local frame for V . Proof. Suppose Assertion 2 holds. Let X 2 C 1 .V / and Y 2 C 1 .V /. Fix P 2 M . Choose local coordinates .x 1 ; : : : ; x m / so V D spanf@x 1 ; : : : ; @x k g. Expand X X XD ai @x i and Y D b j @x j : ik

j k

We may show that Assertion 2 implies Assertion 1 by computing: X ˚ ŒX; Y  D ai @x i b j b i @x i aj
@x j 2 C 1 .V / : i;j k

Suppose Assertion 1 holds. Let fX1 ; : : : ; Xk g be smooth sections to V which are linearly independent at P . ey then form a local frame for V near P . Expand Xi D ai1 @x 1 C


C aim @x m :

66

2. MANIFOLDS

Since X1 .P / ¤ 0, we have a1i .P / ¤ 0 for some i . By renumbering, we assume a11 .P / ¤ 0. By replacing X1 by .a11 / 1 X1 and shrinking the neighborhood under consideration, we may assume a11 D 1. By replacing Xi by Xi ai1 X1 , we can normalize the frame so that X1 D 1
@x 1 X2 D 0
@x 1 :::

C a12 @x 2 C a22 @x 2 :::

C : : : C a1m @x m ; C : : : C a2m @x m ; ::: :::

By permuting the indices, we may assume a22 ¤ 0. Multiplying X2 by .a22 / 1 and shrinking the neighborhood permits us to assume a22 D 1. Replacing Xi by Xi ai 2 X2 for i ¤ 2 then permits us to assume ai 2 D 0 for i ¤ 2 and thus our frame has the form: X 1 D 1
@x 1 X 2 D 0
@x 1 X 3 D 0
@x 1 :::

C 0
@x 2 C 1
@x 2 C 0
@x 2 :::

C a13 @x 3 C a23 @x 3 C a33 @x 3 :::

C : : : C a1m @x m ; C : : : C a2m @x m ; C : : : C a3m @x m ; :::

Continue in this way to choose a local frame so Xi D @x i C ŒXi ; Xj  D

X fXi .aj ` /

X

ai ` @x ` . us

`>k

Xj .ai` /g@x ` :

`>k

Since ŒXi ; Xj  2 C 1 .V /, ŒXi ; Xj  D 0. e theorem now follows from Lemma 2.7.

u t

2.2.5 SUBMERSIONS. Let F be a smooth surjective map from a manifold M to a manifold MQ . We say that F is a submersion if F W TP M ! TF .P / MQ is surjective for every point P of M . By the Implicit Function eorem, the fiber Fx WD  1 .PQ / is a smooth submanifold of M for every point PQ 2 MQ . We have: eorem 2.9

Let  W M ! MQ be a smooth surjective map from M to MQ .

1. If  W M ! MQ is a fiber bundle, then  is a submersion. 2. If  W M ! MQ is a submersion, if M is compact, and if MQ is connected then .M; ; MQ / is a fiber bundle. Proof. Assertion 1 is immediate from the definition. Assume  is a submersion and M is compact; this implies MQ is compact. Use a partition of unity to put a Riemannian metric on M , i.e., a positive definite inner product on the tangent bundle TM . By assumption, the map  W TP M ! T.P / MQ is surjective. We may split TM D H ˚ V where the vertical space V D kerf g is the tangent space of the fibers and H D V ? . We then have  is an isomorphism

2.2. THE TANGENT AND COTANGENT BUNDLES

67

from HP to T.P / MQ . us, if XQ is a smooth vector field on MQ , we can lift XQ to a unique vector field X on M by requiring that X 2 H and  .X / D XQ . If .t / is an integral curve for X in M , then . .t// is an integral curve for XQ in MQ . Let PQ 2 MQ and let .OQ ; .xQ 1 ; : : : ; xQ m // be local coordinates centered at PQ . Let XQi WD @xQ i be vector fields on OQ and let Xi be the lifted vector fields on O D  1 .OQ /. If tE 2 Rm , let XtE be a coordinate vector field, i.e., XtE WD t 1 X1 C


C t m Xm

and XQ tE WD  .XtE/ D t 1 XQ 1 C


C t m XQ m :

Let ˚.s; tE; y/ be the integral curve for XtE starting at a point y in the fiber over PQ . Since the fibers are compact, the Fundamental eorem of Ordinary Differential Equations shows that there exists ı > 0 so these curves extend for 0  s  1 if ktE k  ı . Let ˚1 .tE; y/ WD ˚.1; tE; y/. Because we have that  .XtE/ D XQtE, .˚.s; tE; y// D s tE and, consequently, .˚1 .tE; y// D tE. is constructs a diffeomorphism from Bı .PQ /  F to  1 .Bı .PQ // and shows  is a fiber bundle. It shows all the fibers over Bı .PQ / are diffeomorphic and thus, as MQ is connected, all the fibers are diffeomorphic. t u 2.2.6 THE COTANGENT BUNDLE. e cotangent bundle T  M is the dual of the tangent bundle. Let 
;
 denote the natural paring between these bundles. Let .O; / be local coordinates where  D .x 1 ; : : : ; x m /. If f 2 C 1 .M /, then the exterior derivative of f at P is the element of TP .M / characterized by the identity

for all X 2 TP M

 df; X D X.f /.P /

or equivalently

df D @x i f
dx i :

A smooth (local) section to T  M is called a 1-form; the collection fdx 1 ; : : : ; dx m g is a local frame for T  O which is dual to the coordinate frame f@x 1 ; : : : ; @x m g for the tangent bundle T O. If .y 1 ; : : : ; y m / is another set of coordinates, then Equation (2.2.a) dualizes to become: dy i D

@y i
dx j : @x j

If F is a smooth map from M to N , then Equation (2.2.c) dualizes to give a pullback map F from TF.P / N to TP M so that 

 !; F X D F  !; X 

for

X 2 TP M

and ! 2 TF.P / .N /;

(2.2.d)

i

 @y j    or, equivalently, F  .dy i / D @x j dx . We have Id D Id and .F ı G/ D G ı F . e Chain Rule yields the intertwining formula:

dM .F  f / D F  dN .f / :

2.2.7 THE GEOMETRY OF THE COTANGENT BUNDLE. e cotangent bundle has, in many respects, a richer geometry than does the tangent bundle. We refer to Jost [22] and Yano and Ishihara [42] for further details concerning this material. Let  W T  M ! M be the natural

68

2. MANIFOLDS

projection from the cotangent bundle to the base manifold. We may express any point PQ of T  M in the form PQ D .P; !/ where P WD  .PQ / belongs to M and where ! belongs to TP M . Let .O; / be a coordinate chart on M . Expanding ! D xi 0 dx i defines a system of local coordinates .x 1 ; : : : ; x m I x10 ; : : : ; xm0 / on OQ WD  1 .O/  T  M . e dual coordinates are written with the index down as they transform covariantly rather than contravariantly; this permits us to retain the formalism of summing over repeated indices. Let X 2 C 1 .TM / be a smooth vector field on M . e evaluation map X is a smooth function on the cotangent bundle T  M which is defined by the identity: X.P; !/ D !.XP / :

We may express X D X i @x i to define the coefficients X i D X; dx i . We then have that X.x i ; xi 0 / D xi 0 X i D xi 0  X; dx i  :

Vector fields on T  M are characterized by their action on the evaluation maps X (see Yano and Ishihara [42]): Suppose that YQ and ZQ are smooth vector fields on the cotangent bundle T  M . Q Suppose that YQ .X/ D Z.X / for all smooth vector fields X on M . en YQ D ZQ .

Lemma 2.10

Proof. Let YQ D ai .x; E xE 0 /@x i C bi 0 .x; E xE 0 /@xi 0 be a smooth vector field on T  M with YQ .X / D 0 for all smooth vector fields on M . We must show this implies that YQ D 0. Let X D X j .x/@ E xj . Since X D xi 0 X i , we have: 0 D YQ .X / D xj 0 ai .x; E xE 0 /.@x i X j /.x/ E C bi 0 .x; E xE 0 /X i .x/ E :

Fix j and do not sum. If we take X D @x j , then X i D ıji so X D xj 0 and 0 D YQ .X/ D bj 0 .x; E xE 0 /

so bj 0 D 0

and YQ D ai .x; E xE 0 /@x i :

If we take X D x j @x j , then we have similarly 0 D YQ .X / D xj 0 aj .x; E xE 0 /. us aj .x; E xE 0 / D 0 j 0 j when xj 0 ¤ 0. Since the functions a .x; E xE / are smooth, this implies a vanishes identically and hence YQ D 0 as desired. t u Let X be a smooth vector field on M . e complete lift X C is the vector field on the cotangent bundle T  M characterized via Lemma 2.10 by the identity: X C .Z/ D ŒX; Z

for all smooth vector fields Z on M .

Let ! 2 TP M with ! ¤ 0. en the tangent space T.P;!/ T  M is spanned by the complete lifts of all the smooth vector fields on M .

Lemma 2.11

2.2. THE TANGENT AND COTANGENT BUNDLES

69

Proof. We first compute the complete lift in a system of local coordinates. Let X D X j .x/@ E xj

and X C D aj .x; E xE 0 /@x j C bj 0 .x; E xE 0 /@xj 0 :

Let Z D Z i .x/@ E x i . en X C .Z/ D X C .xi 0 Z i / D xi 0 aj .x; E xE 0 /.@x j Z i /.x/ E C bj 0 .x; E xE 0 /Z j .x/ E i j i j D f.X @x j Z Z @x j X /@x i g j D xi 0 .X .x/.@ E x j Z i /.x/ E Z j .x/.@ E x j X i /.x// E :

Since Z is arbitrary, we conclude aj .x; E xE 0 / D X j .x/ E and bj 0 .x; E xE 0 / D xi 0 @x j X i so that X C D X j .x/@ E xj

E xj 0 : xi 0 .@x j X i /.x/@

Taking X D @x j then yields X C D @x j . Since ! ¤ 0, we have xi 0 ¤ 0 for some i . Furthermore, taking X D x j @x i then yields X C D x j @x i xi 0 @xj 0 which completes the proof. We note that this fails on the 0-section. Since the zero section is defined by the relation xi 0 D 0, =X C D X j .x/@ E x j does not involve @xi 0 . Consequently on the zero section, spanfX C g D spanf@x j g .  A crucial point is that since X and X C are invariantly characterized, we do not need to check the local formalism transforms correctly; on the other hand, the local formalism shows that X C in fact exists. We shall apply similar arguments subsequently. Two smooth tensor fields 1 , 2 of type .0; s/ on T  M coincide with each other if and only if we have the following identity for all vector fields Xi on M :

Lemma 2.12

1 .XiC1 ; : : : ; XiCs / D 2 .XiC1 ; : : : ; XiCs / .

Proof. e Lemma is immediate if ! ¤ 0 by Lemma 2.11; we now use continuity to extend the result to the 0-section. u t

Let T 2 C 1 .Hom.TM // be a smooth .1; 1/-tensor field on M . Define a corresponding lifted 1-form T 2 C 1 .T  .T  M // on the cotangent bundle characterized by:  T; X C D .TX / :

We compute in local coordinates as follows. Expand T D ai .x; E xE 0 /dx i C b i .x; E xE 0 /dxi 0 and j T @x i D Ti @x j . Let X D X i @x i . We compute: .T /.X C / D ai .x; E xE 0 /X i .x/ E b j .x; E xE 0 /xi 0 .@x j X i /; i j i j .TX/ D .X Ti @x j / D xj 0 X Ti :

is shows that b j D 0 and that ai D xj 0 Tij , i.e., that T D xj 0 Tij dx i where T D Tij @x j ˝ dx i . We also refer to Kowalski and Sekizawa [25] for additional information concerning natural lifts in Riemannian geometry. e following result summarizes the formalisms which we have established:

70

2. MANIFOLDS

Lemma 2.13

1. e evaluation map X for X 2 C 1 .TM /: (a) Invariant formalism: X.P; !/ D !.XP /.

(b) Coordinate formalism: X D xi 0 X i for X D X i @x i . 2. e complete lift of a vector field X 2 C 1 .TM /. (a) Invariant formalism: X C .Z/ D ŒX; Z for Z 2 C 1 .TM /.

xi 0 .@x j X i /@xj 0 for X D X j @x j .

(b) Coordinate formalism: X C D X j @x j

3. Let T 2 C 1 .Hom.TM //; T is the 1-form on T  M : (a) Invariant formalism:  T; X C D .TX /.

(b) Coordinate formalism: T D xj 0 Tij dx i .

2.3

STOKES’ THEOREM

is section is devoted to the proof of the generalized Stokes’ eorem. 2.3.1 THE EXTERIOR ALGEBRA. Let V  be the dual space of an r -dimensional real vector space V . e exterior algebra ..V  /; ^/ is the universal real unital algebra generated by V subject to the relations v ^ w C w ^ v D 0. More formally, let

T WD R ˚ V  ˚ .V  ˝ V  / ˚


D ˚n0 f˝n V  g be the complete tensor algebra. Let I be the 2-sided ideal of T generated by all tensors of the form v 1 ˝ v 2 C v 2 ˝ v 1 for v i 2 V  . Set .V  / D T =I . e ideal I is homogeneous; let

I k WD spanfv i1 ˝


˝ v i` ˝ .v i`C1 ˝ v i`C2

v i`C2 ˝ v i`C1 / ˝ v i`C3 ˝


˝ v ik g

(2.3.a)

to decompose .V  / D ˚k0 k .V  / where k .V  / WD ˝k V  =I k . Note that:

I 0 D f0g and I 1 D f0g so 0 .V  / D R and 1 .V  / D V  .

Let fe1 ; : : : ; er g be a basis for V and let fe 1 ; : : : ; e r g be the associated dual basis for V  . Since fe i g spans V  , fe i1 ^


^ e ik g spans k .V  / as a vector space over R where fi1 ; : : : ; ik g is an arbitrary collection of k indices between 1 and r . e defining relation shows e i ^ e j D e j ^ e i and hence we have e i ^ e i D 0. If I D .i1 ; : : : ; ik / is a collection of strictly increasing indices 1  i1 <


< ik  r , set e I WD e i1 ^


^ e ik . en k .V  / D spanfe I gjI jDk :

k



r k

2.3. STOKES’ THEOREM




k

71



is shows dim. .V //  if 0  k  r and that  .V / D f0g if k > r . If jI j D k and jJ j D `, we must pass k indices over ` indices to compare e I ^ e J with e J ^ e I . Consequently !k ^ !Q ` D . 1/k` !Q ` ^ !k

if

!k 2 k .V  /

and !Q ` 2 ` .V  / :

ere is a natural evaluation of k .V  / on k V that will play a central role in our subsequent discussion. Let vE WD .v1 ; : : : ; vk / 2 k V and let vE WD .v 1 ; : : : ; v k / 2 k V  . en  vi ; v j 1i;j k is a k  k real matrix. Let vE .E v  / WD det. vi ; v j / 2 R. e map from k   vE to vE .E v / is multilinear and, consequently, extends to a map vE W ˝k V  ! R. Let  be the k  k  k  k natural projection from ˝ V to  .V / D ˝ V =I . Since the determinant changes sign if we interchange two adjacent columns,  vanishes on the elements of Equation (2.3.a) and extends to a map  W k .V  / ! R. If ! 2 k .V  /, we define !Œv1 ; : : : ; vk  WD vE .!/. We then have a well-defined evaluation (which is linear in ! ) that we use subsequently: .v 1 ^


^ v k /Œv1 ; : : : ; vk  D det. vi ; v j / :

(2.3.b)

ere is always the danger when constructing an algebra using generators and relations that everything collapses; that there are somehow unforeseen relations that make everything trivial. Fortunately that is not the case in the setting at hand. Adopt the notation established above. en fe I gjI jDk is a basis for k .V  /. Con
r k  sequently, dim. .V // D k if 0  k  r and k .V  / D f0g if k > r .

Lemma 2.14

Proof. We have already noted k .V  / D f0g if k > r . If k  r , then the elements fe I gjI jDk form a spanning set for k .V  /. Let J D .j1 ; : : : ; jr / for 1  j1 <


< jk  r . We use the defining relation of Equation (2.3.b) to see that ( ) 0 if I ¤ J e I Œej1 ; : : : ; ejk  D det. ej ; e i ; / D : 1 if I D J

Since the pairing ! ! !ŒeJ  is a linear map from k .V  / to R, this shows the fe I gjI jDk form a basis for k .V  /. u t e pairing !Œ
 described in Equation (2.3.b) identifies k .V  / with the set of multilinear totally alternating bilinear forms on V and embeds k .V  / as a subspace of ˝k V  . Let ˘.k/ be the set of all permutations of k elements. Let k  .v 1 ˝


˝ v k / WD

1 X .1/ v ˝


˝ v .k/ : kŠ 2˘.k/

k 2 k k k en . / D  and rangef g D k .V  / so  provides a projection of ˝k V  onto k .V  /. k k We add a note of caution. With our notational conventions,  D kŠ  .

72

2. MANIFOLDS

2.3.2 THE PULLBACK. If belongs to End.V; W /, then the dual map  belongs to End.W  ; V  / and extends to a map T .  / from T .W  / to T .V  /. By Equation (2.3.a), T .  / maps the defining ideal for W to the defining ideal for V . Consequently, we have naturally defined linear maps k .  / from k .W  / to k .V  /. Let fe i g be a basis for V  , let ff a g be a basis for W  , and let  .f a / D ia e i . Let e ij WD e i ^ e j for i < j and e ij k WD e i ^ e j ^ e k for i < j < k . We have, for example:  .f a / D ia e i ;

 .f a ^ f b / D ia jb e i ^ e j D . ia jb

 .f a ^ f b ^ f c / D ia jb kc e i ^ e j ^ e k D . ia jb kc C ja kb ic C ka ib jc

ia kb jc

ka jb ic

ja ib /e ij ; ja ib kc /e ij k :

e coefficients are, of course, just the determinant of the minors  for  2 fi1 ; : : : ; ik g and  2 fa1 ; : : : ; ak g. us, for example, if V D W and if k D r , we have  .e 1 ^


^ e r / D det. /e 1 ^


^ e r :

We have . 1 ı 2 / D 2 ı 1 and Id D Id. In the language of algebraic topology, the association V k .V / is a contravariant functor from the category of finite-dimensional vector spaces to the category of graded skew-commutative unital rings. It is not necessary to fuss unduly about this notation. We refer to Section 8.1.1 of Book II for further details concerning category theory. Let V be a smooth vector bundle over a manifold M . e construction of k .
/ is natural, i.e., it is well-defined and is independent of the basis chosen. us the vector spaces k .Vx / patch  together to define a smooth vector bundle k .V  / over M . If ˛ˇ are the transition functions of   k k  V , then  . ˛ˇ / are the transition functions of  .V /. Let Vi be vector bundles over M . If W V1 ! V2 is a bundle map, then k .  / is a bundle map from k .V2 / to k .V1 /. 2.3.3 DIFFERENTIAL FORMS. We take V to be the cotangent bundle T  M and simply the notation by setting k M WD k .T  M /. e space of smooth k -forms over M is the space of smooth sections to k M . Let ! 2 C 1 .k M /. If xE D .x 1 ; : : : ; x m / is a system of local coordinates on M , expand ! D aI .x/dx I

where dx I WD dx i1 ^


^ dx ik

and

I D f1  i1 <


< ik  mg :

If yE D .y 1 ; : : : ; y m / is another system of local coordinates on M , then on the common domain of definition, we have:  i @y I @y I @y dy I D J dx J where WD det : (2.3.c) @x @x J @x j i2I;j 2J In particular, if k D m, then the transition function is the Jacobian determinant:  i @y 1 m dy ^


^ dy D det dx 1 ^


^ dx m : @x j

(2.3.d)

2.3. STOKES’ THEOREM

73

If F is a smooth map from a manifold M to a manifold N , then we have the pushforward map F W TP M ! TF .P / N and the pullback map F  W TF.P / N ! TP M . is extends to linear maps F  W kF .P / N ! Pk M . We have linear maps F  W C 1 .k N / ! C 1 .k M /

so F  .!k ^ !Q ` / D F  .!k / ^ F  .!Q ` / :

If .x 1 ; : : : ; x m / are local coordinates on M and if .y 1 ; : : : ; y n / are local coordinates on N , then Equation (2.3.c) generalizes immediately to become:   i @y I @y I @F y  I J F .aI .y/dy / D aI .F .x// J dx where WD det : J @x @x @x j i2I;j 2J If G is a smooth map from N to S , then we have that .G ı F / D F  ı G  ; Id D Id. Lemma 2.15 If f is a smooth function on M , let df be the smooth 1-form characterized by  df;  D .f / for any smooth vector field  on M ; if xE D .x 1 ; : : : ; x m / is a system of local coordinates on M , then df D @x i f
dx i .

1. ere is a unique extension of d to map C 1 .k M / to C 1 .kC1 M / for k  1 satisfying d 2 D 0 and d.!p ^ !Q q / D d!p ^ !Q q C . 1/p !p ^ d !Q q for !p 2 C 1 .p M / and !Q q 2 C 1 .q M /. 2. d.fI dx I / D dfI
dx I . 3. If F is a smooth map from M to N , then dM F  D F  dN . Proof. Since  df;  D .f /, df D @x i f
dx i . Suppose there exists an operator d which satisfies the properties of Assertion 1. We first establish that d is local. Suppose O is an open subset of M and that !1 D !2 on O. Let P 2 O. Choose a smooth mesa function  so  is identically 1 near P and so  has support in O. Since .!1 !2 / vanishes identically, 0 D d..!1 !2 //.P / D d.P /.!1 .P / D 0 C d!1 .P / d!2 .P / :

!2 .P // C .P /.d!1 .P /

d!2 .P //

us !1 D !2 on O implies d!1 D d!2 on O. Let xE D .x 1 ; : : : ; x m / be a system of coordinates defined on an open subset O of M . Fix P 2 O. Choose a mesa function 1 which is identically 1 near P with support in O. Choose a second mesa function 2 which is identically 1 on the support of 1 . en d.2 x i / D dx i near P . We may then expand: 1 ! D 1 f.i1 ;:::;ik / d.2 x i1 / ^


^ d.2 x ik /; d.1 !/ D d.1 f.i1 ;:::;ik / /d.2 x i1 / ^


^ d.2 x ik / C

k X

. 1/ .1 f.i1 ;:::;ik / /d.2 x i1 / ^


^ dd.2 x i / ^


^ d.2 x ik / :

D1

74

2. MANIFOLDS

Since dd.2 x i / D 0, since d.1 /.P / D 0, and since 1 .P / D 1, we have: ˚ d.!/.P / D d.1 !/.P / D d.f.i1 ;:::;ik / /dx i1 ^


^ dx ik .P / : is establishes the formula of Assertion 2 and shows that if d exists, then it is unique. To establish existence, we work in a coordinate chart. We use Assertion 2 to define dx on O. We show dx2 D 0 by computing:

dx2 fI dx I D dx @x i fI
dx i ^ dx I D @x i @x j f
dx j ^ dx i ^ dx I :

is vanishes since @x i @x j f D @x j @x i f but dx i ^ dx j D dx j ^ dx i . If f; g 2 C 1 .M /, then dx .fg/ D gdx f C f dx g . We complete the proof of Assertion 1 by computing:
dx .!p ^ !Q q / D dx fI dx I ^ gJ dx J D dx .fI gJ / ^ dx I ^ dx J D gJ dx .fI / ^ dx I ^ dx J C fI dx .gJ / ^ dx I ^ dx J



D dx f ^ dx I ^ gJ ^ dx J C . 1/p fI dx I ^ dx gJ ^ dx J :

is shows d WD dx exists on a coordinate chart. On the intersection of two coordinate charts, we must have dx D dy by the uniqueness assertion. We can now define d globally with the desired properties using a partition of unity. is establishes Assertions 1 and 2. Let F W M ! N . We complete the proof by computing: dF  .dy i / D d dF  .y i / D 0, P dF  .dy I / D  . 1/ 1 F  .dy i1 / ^


^ dF  .dy i / ^


^ F  .dy ik / D 0, d.F  .fI dy I // D d.F  fI / ^ F  .dy I / C F  .fI / ^ dF  .dy I /

D F  .dfI / ^ F  .dy I / D F  .dfI ^ dy I / D F  d.fI dy I / .



2.3.4 THE EXTERIOR DERIVATIVE AND THE LIE BRACKET. ere is a useful relationship between the Lie bracket and the exterior derivative that can be used to give an invariant definition of the action of d on 1-forms: eorem 2.16 Let ! be a 1-form on a manifold M and let X and Y be vector fields on M . en d!.X; Y / D X.!.Y // Y.!.X // !.ŒX; Y /.

Proof. Let xE D .x 1 ; : : : ; x m / be a system of local coordinates on M . Expand ! D ai dx i ;

X D b j @x j ;

Z D c k @x k :

Let f=a WD @x a f . We complete the proof by comparing the first line with the sum of the next three lines in the following display: d!.ŒX; Y / D ai=` dx ` ^ dx i .b j @x j ; c k @x k / D ai=` .b ` c i i , X.!.Y // D X.ai c i / D b j ai=j c i C b j ai c=j i , Y.!.X// D Y .ai b i / D c k ai=k b i C c k ai b=k

k @x k !.ŒX; Y / D !.b j c=j

j i @x j / D ai b j c=k c k b=k

b i c ` /,

i . ai c k b=k

u t

2.3. STOKES’ THEOREM

75

We note that the bracket can be used to give an invariant definition of the exterior derivative. We shall omit the verification as it plays no role in our subsequent development. X d!.X0 ; X1 ; : : : ; Xk / D . 1/ X .!.X0 ; : : : ; XO i ; : : : ; Xk // 

C

X

 0/. M is said to be orientable if it is oriented by some coordinate atlas. e Klein bottle is not orientable. On the other hand, the torus T 2 D S 1  S 1 is orientable. If Mc is the level set of a smooth map f from RmC1 to R with df non-vanishing on Mc , then Mc is orientable. us the sphere S m is orientable for any m.

A connected manifold M of dimension m is orientable if and only if there exists a non-vanishing m-form on M , i.e., if the bundle m M is trivial. Lemma 2.17

Proof. Suppose that there exists a non-vanishing m-form m on M . Fix a coordinate atlas f.O˛ ; .x˛1 ; : : : ; x˛m //g for M ; we may assume without loss of generality that the O˛ are connected. Express m D f˛ dx˛1 ^


^ dx˛m on O˛ . Since m is nowhere vanishing and O˛ is connected, f˛ does not change sign. If f˛ < 0, we interchange the first two coordinates to createa new  i

coordinate system where f˛ > 0. us we may assume f˛ > 0 for all ˛ . Let ˛ˇ D det @x˛j ; @xˇ express dx˛1 ^


^ dx˛m D ˛ˇ dxˇ1 ^


^ dxˇm (2.3.e)

and hence fˇ D ˛ˇ f˛ . is implies ˛ˇ > 0; we use m to adjust the local coordinate systems appropriately and create an oriented atlas. Conversely, suppose M is orientable. Let f.O˛ ; .x˛1 ; : : : ; x˛m //g be a coordinate atlas giving the orientation. Let  ˛ be a partition of unity P subordinate to the cover O˛ of M . Let m WD ˛  ˛ .dx˛1 ^


^ dx˛m / on M ; m is well defined since  ˛ D 0 on O˛c where dx˛1 ^


^ dx˛m is not defined. Fix ˇ . We use Equation (2.3.e) to express m in terms of dxˇ1 ^


^ dxˇm on Oˇ . e convex combination of positive functions is P positive. Consequently ˛  ˛ ˛ˇ > 0 and ( ) X ˛ ˇ  ˛ dxˇ1 ^


^ dxˇm ¤ 0 on Oˇ . u t m D ˛

2.3.6 PROJECTIVE SPACE. We continue the discussion of Section 2.1.6. Let F be the field of real numbers R, the field of complex numbers C , or the skew-field of quaternion numbers H as discussed in Section 1.2.1. Let F  WD F f0g be the group (with respect to multiplication) of non-zero elements of F . Let F  act on F mC1 f0g by scalar multiplication from the left and let FP m WD fF mC1

f0gg=F 

76

2. MANIFOLDS

be the quotient space; this is the set of all F -lines through the origin in F mC1 . Let  be the natural projection from F mC1 f0g to F P m . We give F P m the quotient topology; a set O in FP m is open if and only if  1 .O/ is open in F mC1 f0g;  is then an open map. Let X be a topological space. A map fQ W FP m ! X is continuous if and only if fQ ı  is continuous. Correspondingly, if f W F mC1 f0g ! X is continuous and if f .x/ D f .x/ for all  2 F  and all x 2 F mC1 f0g, then f descends to a continuous map fQ W FP m ! X . Let S be the unit sphere in F mC1 . Since S is compact and  W S ! FP m is surjective, FP m is compact. Let xN be the conjugation operator of Section 1.2.1; xx N D x xN D kxk2 and uv D vN uN ; the order is important for the quaternions as quaternionic multiplication is not commutative. e map x ! xN is R linear. Let .x 1 ; : : : ; x mC1 / 2 F mC1 f0g. We define the matrix T .x/ 2 MmC1 .F/ by setting Tij .x/ WD kxk 2 xN i x j : N j D jj 2 jj2 kxk 2 xN i x j D Tij .x/, T extends to a continuous Since Tij .x/ D kxk 2 xN i x m map from FP to MmC1 .F/. Let x 2 S and let TQ .x/ be orthogonal projection on the line through x . en TQ .x/y D .y; x/
x where .y; x/ D y 1 xN 1 C


C y mC1 xN mC1 . us .TQ .x/ei ; ej / D .ei ; x/.x; ej / D xN i x j

so T D TQ :

us if  2 FP m , then rangefT  g D  . is shows that T is an injective map from FP m to MmC1 .F/. Since FP m is compact, T is a homeomorphism to its image; we may identify FP m with the orthogonal projections of rank 1 in MmC1 .F /. We give FP m the structure of a smooth manifold as follows. Let Ui WD fx 2 F mC1

f0g W x i ¤ 0g

be an open cover of F mC1 f0g for 1  i  m C 1. Since F 
Ui D Ui , Oi WD .Ui / is an open cover of FP m . Let ˚i .x/ WD .xi 1 x 1 ; : : : ; xi 1 x mC1 / on Ui . Note that ˚i .x1 / D ˚i .x2 / if and only x1 D x2 and thus ˚i descends to a continuous injective map from Oi to F m where we omit the i t h coordinate since it is identically 1. We set yik WD .x i / 1 x k to obtain coordinates on Oi . On Oi \ Oj , we have yik D .x i /

1 j

x .x j /

1 k

x D f.x j /

1 i

x g

1

.x j /

1 k

x D .yji /

1 k yj

:

us the transition functions are smooth and give F P m the structure of a smooth manifold. We have that: Tab

D kxk 2 xN a x b D jx i j2 kxk D .1 C jyi j2 / 1 yNia yib :

2 a

xN .xN i /

1

.x i /

1 b

x D fjx i j

2

kxk2 g

1 a b yNi yi

is is the quotient of two polynomials in the real and imaginary parts of y and hence is a smooth function of yi D .yi1 ; : : : ; yim /. It is not difficult to verify that the Jacobian of T is injective so that FP m is a smooth submanifold of MmC1 .F /. If F D R, the coordinate systems defined above are

2.3. STOKES’ THEOREM

77

real analytic; if F D C , the coordinate systems defined above are holomorphic; we postpone a further discussion of complex projective space until Section 4.3.3 in Book II. We now turn to the case of real projective space. RP m is orientable if and only if m is odd.

Lemma 2.18

Proof. Let !i D dyi1 ^


^ dyii 1 ^ dyiiC1 ^


^ dyimC1 be a non-zero m-form on Oi . We examine Oi \ Oj . We suppose i < j . We have: yik D .yji / 1 yjk . Consequently 8 i k i 1 ˆ < .yj / dyj .yj / dyik D .yji / 2 dyji ˆ : 0

2 k yj dyji

if k ¤ i; j if k D j if k D i

9 > = > ;

:

Consequently, !i D

.yji /

D

.yji /

m 1 m 1

dyj1 ^


^ dyji

. 1/iCj

1

1

^ dyjiC1 ^


^ dyjj

dyj1 ^


^ dyji

1

1

^ dyji ^ dyjj C1 ^


^ dyjmC1

^ dyji ^


^ dyjj

1

^dyjj C1 ^


^ dyjmC1 D . 1/iCj .yji /

m 1

!j .

Suppose m is odd. We let i WD . 1/i !i . We then have: i D . 1/i !i D . 1/i . 1/iCj .yji /

m 1

!j D .yji /

m 1

j :

Since m C 1 is even, .yji / m 1 > 0 and RP m is orientable. If m is even, then .yji / m 1 changes sign on Oi \ Oj and thus there is no possibility of arranging the orientations to agree. us RP m is not orientable if m is even.  2.3.7 MANIFOLDS WITH BOUNDARY. We consider the lower half space Rm WD fx D .x 1 ; : : : ; x m / 2 Rm W x 1  0g :

e boundary of Rm is the hyperplane Rm 1 WD fx 2 Rm W x 1 D 0g. Let O be an open subset of Rm regarded as a metric space in its own right; O need not be an open subset of Rm . We say that F is smooth on O if there is an open subset OQ of Rm which contains O and a smooth function FQ on OQ such that F is the restriction of FQ to O : Let bd.M / be a closed subspace of a metric space M . Let fO˛ g be an open cover of M . If O˛ \ bd.M / is empty, we assume given homeomorphisms from O˛ to an open subset U˛ of Rm . If O˛ \ bd.M / is non-empty, we assume given homeomorphisms from O˛ to open subsets U˛ of Rm such that ˛ .O˛ \ bd.M //  Rm 1 . We assume that the transition functions ˛ˇ WD ˛ ı ˇ 1 are smooth on the appropriate domain. In this setting, the restriction of the coordinate charts to bd.M / gives bd.M / the structure of

78

2. MANIFOLDS

a smooth embedded submanifold of M and M bd.M / is a smooth open manifold. e pair .M; bd.M // is said to be a manifold with boundary. e unit sphere S m 1 is the boundary of the unit disk D m . Every manifold N is the boundary of the non-compact manifold . 1; 0  N . However, there are compact manifolds N which are not the boundary of any compact manifold. For example, complex projective space CP 2 does not bound any smooth 5-dimensional manifold. is leads naturally to the study of the cobordism groups; we refer to Stong [39] for further details. Lemma 2.19

1. Let f be a smooth map from a manifold M without boundary to R. Let Lc WD f 1 .c/ be the level sets and let Mc WD f 1 . 1; c. Suppose that f is regular at every point of Lc . en Mc is a manifold with boundary Lc . 2. Let M be a compact manifold with boundary bd.M /. Take two copies M˙ of M . Join them at the common boundary to define the double DM WD MC [bd.M / M . (a) e collar Œ0; 1  bd.M / is diffeomorphic to a neighborhood of bd.M / in M .

(b) e double DM is a manifold.

(c) M˙ are smooth manifolds with boundary bd.M / D MC \ M .

(d) ere exists a smooth function f W DM ! R so bd.M / D f 1 .0/, so every point of bd.M / is a regular value, so MC D f 1 Œ0; 1/, and so M D f 1 . 1; 0. 3. Let M be a manifold with boundary bd.M /. If f.O˛ ; ˛ /g is a coordinate atlas for M which gives M an orientation, then the restriction of the atlas to bd.M / induces a natural orientation on bd.M /. Proof. Assertion 1 is an immediate consequence of the proof given of eorem 2.1. We now prove Assertion 2, which is in essence the converse of Assertion 1. Using a partition of unity, construct a vector field  which is non-zero and points inward along bd.M /. Let ˚ t .x/ be the associated flow discussed in Lemma 2.6. e map ˚.t; y/ WD ˚ t .y/ provides a diffeomorphism between the collar Œ0;   bd.M / to a neighborhood of bd.M / in M for some  > 0. Rescaling the parameter t permits us to take  D 1 and establish Assertion 2-a. Gluing Œ 1; 0  bd.M / in M to Œ0; 1  bd.M / in MC along f0g  bd.M / gives the required smooth structure to DM and establishes Assertion 2-b and Assertion 2-c. We have an identification of Œ 1; 1  bd.M / with a neighborhood of bd.M / in DM . We use the collection of functions constructed in the proof of Lemma 1.21 to find a smooth monotonically increasing function taking values in Œ 1; 1 so that 8 2 9 2 ˆ > < 3 if t  3 = 1 1 .t / D : t if t 2 Œ 2 ; 2  ˆ > : 2 ; 2 if t  3 3

2.3. STOKES’ THEOREM

79

˙ 32

We now define f .t; y/ D .t/ for .y; t / 2 bd.M /  Œ 1; 1. We extend f .x/ D on M˙ minus the appropriate collar to construct the desired function f and complete the proof of Assertion 2. Let M be an oriented manifold. Let O˛ be a coordinate atlas where the coordinate functions 0 satisfy det.˚˛ˇ / > 0. If O˛ intersects the boundary of M , we may assume O˛ is defined by the 1 1 1 relation x˛  0. Since the boundary is preserved, ˚˛ˇ .0; y/ D 0. Furthermore, @x 1 ˚˛ˇ .0; y/ > 0. We have: 1 0 1 0 1 1 1 @x 1 ˚˛ˇ @x 2 ˚˛ˇ ::: @x 1 ˚˛ˇ 0 ::: C B C B 2 2 2 2 0  det @ @x 1 ˚˛ˇ @x 2 ˚˛ˇ : : : A D det @ @x 1 ˚˛ˇ @x 2 ˚˛ˇ ::: A ::: ::: ::: ::: ::: ::: 0 1 2 2 @x 2 ˚˛ˇ @x 3 ˚˛ˇ ::: B C 1 3 3 D @x 1 ˚˛ˇ
det @ @x 2 ˚˛ˇ @x 3 ˚˛ˇ ::: A : ::: ::: ::: 2 m e transition functions for bd.M / are given by y ! .˚˛ˇ .0; y/; : : : ; ˚˛ˇ .0; y//. Since 1 @x 1 ˚˛ˇ .0; y/ > 0, we conclude from the expression above that 0 1 2 2 @x 2 ˚˛ˇ @x 3 ˚˛ˇ ::: B C 3 3 det @ @x 2 ˚˛ˇ t u @x 3 ˚˛ˇ ::: A > 0. ::: ::: :::

If ˚ is a local diffeomorphism from an open set U1 of Rm to another open set U2 of Rm and if f is a smooth function with compact support in U2 , then f is integrable on U2 and the Change of Variable eorem (see eorem 1.28) yields Z Z f D .f ı ˚ /
j det ˚ 0 j : (2.3.f ) U2

U1

Let M be compact and oriented; let f.O˛ ; ˛ /g be a coordinate atlas giving the orientation for 1  ˛  `. Let ˛ be a partition of unity subordinate to this cover. Let ! be a smooth m-form on M . In each coordinate chart, express ! D !˛ .dx˛1 ^


^ dx˛m /. en: ! @x˛ !˛ D det !ˇ : @xˇ Since M is oriented, the Jacobian determinant is positive and the absolute value of the determinant plays no role. us we may use Equation (2.3.f) to show that the following integral is independent of the choices made: Z XZ ! WD ˛ !˛ : M

˛

U˛

80

2. MANIFOLDS

2.3.8 GREEN’S THEOREM. e generalized Stokes’ eorem will include the classic theorems of vector calculus as special cases. Let W Œa; b ! Rm be a piecewise smooth curve and let ! D f1 dx 1 C


C fm dx m be a 1-form. e line integral is given by: ) Z Z b (X m dx i fi . .t // ! WD dt : dt

tDa iD1

is is independent of the parametrization as long as the orientation is preserved. In this special instance, df D f 0 .t/dt . e scalar curl of a 1-form in the plane ! D pdx C qdy is: sc.!/ WD @x q

@y p :

We then have d! D sc.!/dx ^ dy . e following result will follow from eorem 2.22: (Green’s eorem) Let R be a compact submanifold of R2 with smooth boundary. Orient the boundary of R to keep R on the left. Let ! be a smooth 1-form defined on all of R. en Z Z !D sc.!/dxdy .

eorem 2.20

bd.R/

R

2.3.9 THE OPERATORS OF 3-DIMENSIONAL VECTOR CALCULUS. We now introduce the classical operators from vector calculus: div, curl, and grad. Let .x; y; z/ be the usual coordinates on an open subset O of R3 and, at least formally, let r WD .@x ; @y ; @z /. Let
and  denote the Euclidean inner product and the cross product on R3 . If f 2 C 1 .O/, then the gradient of f is the smooth vector field given formally by grad.f / D r.f /, i.e.,

grad.f / WD .@x f; @y f; @z f / : If F D .f1 ; f2 ; f3 / is a smooth vector field on O, then the divergence of F is the smooth function which is given formally by div.F / D r
F ; more precisely: div.F / WD @x f1 C @y f2 C @z f3 : e curl of F is the smooth vector field which is given formally by curl.F / D r  F , i.e., curl.F / WD .@y f3

@z f2 ; @z f1

@x f 3 ; @ x f 2

@y f1 / :

2.3.10 TRANSLATION TABLE. ere is an equivalence of terminology between the language of differential forms and the language of vector calculus in R3 :

1. We identify a 1-form !1 D f1 dx C f2 dy C f3 dz with the vector field F D .f1 ; f2 ; f3 /. 2. We identify a 2-form !2 D g1 dy ^ dz C g2 dz ^ dx C g3 dx ^ dy with the vector field G D .g1 ; g2 ; g3 /.

2.3. STOKES’ THEOREM

81

3. We identify a 3-form !3 D gdx ^ dy ^ dz with the function g . 4. e exterior derivative d0 W C 1 .O/ ! C 1 .1 .O// corresponds to the gradient: d0 .f / D @x f dx C @y f dy C @z f dz ,

grad.f / D .@x f; @y f; @z f / . 5. e exterior derivative d1 W C 1 .1 .O// ! C 1 .2 .O// corresponds to the curl: d1 .f1 dx C f2 dy C f3 dz/ D df1 ^ dx C df2 ^ dy C df3 ^ dz D .@y f3

@z f2 /dy ^ dz C .@z f1 0 i j k B curl.f1 ; f2 ; f3 / D det @ @x @y @z f1 f2 f3 D .@y f3

@z f 2 ; @ z f 1

@x f3 ; @x f2

@x f3 /dz ^ dx C .@x f2 1

@y f1 /dx ^ dy ,

C A

@y f1 / .

6. e exterior derivative d2 W C 1 .2 .O// ! C 1 .3 .O// corresponds to the divergence: d.g1 dy ^ dz C g2 dz ^ dx C g3 dx ^ dy/ D .@x g1 C @y g2 C @z g3 /dx ^ dy ^ dz ,

div.g1 ; g2 ; g3 / D @x g1 C @y g2 C @z g3 . 7. e relation d1 d0 D 0 corresponds to the fact that curl ı grad D 0. 8. e relation d2 d1 D 0 corresponds to the fact that div ı curl D 0. 9. If ˚ W R2 ! R3 parametrizes a piece of a smooth surface S , a local unit normal  can be ˚@v ˚ taken to be given by  WD k@@uu ˚@ and the corresponding element of surface area can be v ˚k taken to be given by dA D k@u ˚  @v ˚ kdudv . 0 1 i j k B C @u ˚  @v ˚ D det @ @u x @u y @u z A @v x @v y @v z D .@u y@v z F
dA D g1 .@u y@v z

@u z@v y; @u z@v x

@u z@v y/ C g2 .@u z@v x

Cg3 .@u x@v y

@v z@u x; @u x@v y

@v x@u y/,

@v z@u x/

@v x@u y/dudv ,

dy ^ dz D .du ydu C dv ydv/ ^ .du zdu C dv zdv/ D .@u y@v z

@u z@v y/du ^ dv ,

dz ^ dx D .du zdu C dv zdv/ ^ .du xdu C dv xdv/ D .@u z@v x

@u x@v z/du ^ dv ,

82

2. MANIFOLDS

dx ^ dy D .du xdu C dv xdv/ ^ .du ydu C dv ydv/ D .@u x@v y  @u y@v x/du ^ dv , g1 dy ^ dz C g2 dz ^ dx C g3 dx ^ dy D g1 .@u y@v z  @u z@v y/du ^ dv C g2 .@u z@v x  @v z@u x/du ^ dv Z S

G   dA D

Cg3 .@u x@v y  @v x@u y/du ^ dv ,

Z

!2 . S

We say S is orientable if we can choose a consistent normal. is defines the “outside” of S ; we orient the boundary to keep S on the left standing outside. 2.3.11 THE MÖBIUS STRIP. e Möbius strip is not orientable. We present below two views. e normal line through the center is in gray. It is the dual Möbius strip where we have only drawn the positive part. e Möbius strip itself is striped. e outward unit normal bundle is in gray. Only the positive direction is shown so it does not close.

2.3.12 STOKES’ THEOREM AND GAUSS’S THEOREM. e following results will also follow from eorem 2.22: eorem 2.21

1. (Stokes’ eorem) Let S be a smooth compact oriented surface in R3 . Orient the boundary of S to keep S on the left. Let ! D f1 dx C f2 dy C f3 dz 2 C 1 . 1 S/. en Z Z ! D fcurl.f1 ; f2 ; f3 /   gdS : bd.S /

S

2. (Gauss’s eorem) Let R be a compact region in R3 with smooth boundary oriented using the outward normal. Let F be a smooth vector field defined on R. en Z Z .F   /dS D div F dxdydz . bd.R/

R

2.3. STOKES’ THEOREM

83

e classical Stokes’ eorem seems to be due to Lord Kelvin (William ompson) and to Sir George Stokes.

Lord Kelvin (1824–1907)

Sir George Stokes (1819–1903)

2.3.13 GENERALIZED STOKES’ THEOREM. eorem 2.22 (Generalized Stokes’ eorem). Let M be a compact smooth oriented manifold of dimension m with smooth boundary bd.M /. Let !m 1 2 C 1 .m 1 M /. Give the boundary bd.M /

the orientation discussed in Lemma 2.19. en Z !m bd.M /

1

D

Z

d!m

1

.

M

Proof. Let O˛ be a cover of M by coordinate charts. By taking a partition of unity subordinate to this cover, we may restrict to the case that !m 1 is compactly supported in just one coordinate chart O. us there exist fi which are smooth functions compactly supported in O so that: !m

1

d!m

We may suppose O 

D 1

Rm C.

m X

fi dx 1 ^


^ dx i

iD1 m X

D

. 1/i

iD1

1

1

^ dx iC1 ^


^ dx m ;

(2.3.g)

@x i fi dx 1 ^


^ dx m :

ere are two cases that must be considered:

Case I. e coordinate chart O does not intersect the boundary. is implies Z !m 1 D 0 : bd.M /

(2.3.h)

R Furthermore, we may let M d!m 1 range over all of Rm since the support of !m 1 is contained in int.Rm C /. To simplify the notation, we examine the term with i D m in Equation (2.3.g); the other terms are handled similarly modulo an appropriate reordering of the indices. We use Fubini’s eorem (see eorem 1.19) to express Z m 1 .@x m fm /dx 1 ^


^ dx m . 1/ O   Z 1  Z 1 1 m m m 1 m m ::: .@x fm /.x ; : : : ; x /dx dx : : : dx 1 : D . 1/ 1

1

84

2. MANIFOLDS

We use the Fundamental eorem of Calculus together with the fact that fm has compact support to see: ˇxm D1 Z 1 ˇ 1 m m 1 m ˇ .@x m fm /.x ; : : : ; x /dx D fm .x ; : : : ; x /ˇ D 0: m 1

x D 1

is shows m 1

. 1/

Z O

.@x m fm /dx 1 ^


^ dx m D 0 :

(2.3.i)

We use Equation (2.3.h) and Equation (2.3.i) to see that both sides of Stokes’ eorem vanish in this instance and the result follows in this special case. Case II. e coordinate chart O intersects the boundary Rm 1 . In this case, the integral over O ranges over x 1 2 . 1; 0 and the boundary of M corresponds to O \ Rm 1 ; this changes the analysis slightly. If i > 1, then dx 1 ^


^ dx i 1 ^ dx iC1 ^


^ dx m vanishes on O \ Rm 1 . Consequently, we may use Fubini’s eorem to see:   Z Z 1  Z 1 !m 1 D ::: f1 .0; x 2 ; : : : ; x m /dx 2 : : : dx m : (2.3.j) O \Rm 1

R

1

1

e analysis of Case I shows O .@x i fi /dx ^


^ dx m vanishes for i > 1. Consequently, after applying Fubini’s eorem and the Fundamental eorem of Calculus, we have:    Z Z 1  Z 1 Z 0 1 2 m 1 2 .@x 1 f1 .x ; x ; : : : ; x //dx dx : : : dx m ::: d!m 1 D 1 1 1 O ) ) ˇx 1 D0 ) Z 1 ( (Z 1 ( ˇ 2 1 2 m ˇ dx : : : dx m f1 .x ; x ; : : : ; x /ˇ D ::: 1 1 1 x D 1   Z 1  Z 1 f1 .0; x 2 ; : : : ; x m /dx 2 : : : dx m : ::: D 1

1

1

is agrees with the integral of Equation (2.3.j) which completes the proof.

2.4



APPLICATIONS OF STOKES’ THEOREM 2

Since d D 0, we may define the de Rham cohomology groups by setting: k HdR .M / WD

kerfd W C 1 .k M / ! C 1 .kC1 M /g : rangefd W C 1 .k 1 M / ! C 1 .k M /g

(2.4.a)

 k We may use Lemma 2.15 to see that wedge product gives HdR .M / WD ˚k HdR .M / the structure   of a graded unital skew-commutative ring. Since dF D F d , pullback induces a natural map   F  W HdR .N / ! HdR .M / that makes de Rham cohomology into a contravariant functor. (We refer to Section 8.1.1 of Book II for further details concerning category theory and functors; the reader need not fuss unduly about this notation.) In Lemma 2.23, we use Stokes’ eorem to

2.4. APPLICATIONS OF STOKES’ THEOREM

85

m HdR .RmC1

exhibit a non-trivial element in the f0g/ that will play a central role in our proof of the Fundamental eorem of Algebra, of the Brauer Fixed Point Formula, and of the Billiard Ball eorem subsequently. Lemma 2.23

Let m 2 C 1 .m .RmC1 m 1

m .x/ WD kxk

f0g// be given by

mC1 X iD1

. 1/iC1 x i dx 1 ^


^ dx i

1

^ dx iC1 ^


^ dx mC1 :

(2.4.b)

1. e restriction of  to S m is never vanishing. us S m is orientable. 2. dm D 0. R 3. S m m ¤ 0. m 4. Œm  ¤ 0 in HdR .RmC1

m f0g/ and in HdR .S m /.

Proof. Let m WD 12 d.k
k2 / D x 1 dx 1 C


C x mC1 dx mC1 . We compute: m ^ m

D kxk

m 1

D kxk

mC1

mC1 X iD1 1

. 1/iC1 xi2 dx i ^ dx 1 ^


^ dx i

1

^ dx iC1 ^


^ dx mC1

dx ^


^ dx mC1 :

is is non-zero on RmC1 f0g and hence m is nowhere vanishing on RmC1 f0g. Let fX1 ; : : : ; Xm g be a basis for TP S m for some point P in S m . Choose XmC1 so that fX1 ; : : : ; Xm ; XmC1 g is a basis for TP RmC1 . Since kxk2 is constant on S m , the restriction of m to S m is zero and hence  m ; Xi D 0 for i  m. Extend 
;
 to a pairing between arbitrary covectors and vectors. We compute: 0 ¤  m ^ m ; X1 ˝


˝ XmC1  mC1 X . 1/i 1  m ; Xi 
 m ; X1 ˝ : : : XO i


˝ XmC1  D iD1

D . 1/m  m ; XmC1   m ; X1 ˝


˝ Xm  :

Assertion 1 now follows. To prove Assertion 2, we compute: (mC1 )  ˚  X .mC1/=2 @x i x i .x 1 /2 C


C .x mC1 /2 dm D dx 1 ^


^ dx mC1 iD1

(mC1 X D kxk n

.mC1/=2

iD1

D .m C 1/kxk D 0.

.mC1/=2

i 2

.m C 1/.x / kxk

.mC3/=2

.m C 1/kxk2 kxk

) 

.mC3/=2

dx 1 ^


^ dx mC1

o

dx 1 ^


^ dx mC1

86

2. MANIFOLDS

Let Q m D

mC1 X iD1

. 1/iC1 x i dx 1 ^


^ dx i Z Sm

1

^ dx iC1 ^


^ dx mC1 . Since kxk D 1 on S m ,

m D

Z Sm

Q m :

Since Q m is smooth on the unit disk D mC1 , we may use Stokes’ eorem to compute Z Z Z Q m D d Q m D .m C 1/dx 1 ^


^ dx mC1 Sm

D mC1

D .m C 1/ vol.D

D mC1 mC1

(2.4.c)

(2.4.d)

/ ¤ 0:

is proves Assertion 3. Suppose m D d m 1 on RmC1 f0g or on S m . We can apply Stokes’ eorem to compute Z Z m D m D 0 : (2.4.e) Sm

bd.S m /

We combine Equations (2.4.c), (2.4.d), and (2.4.e) to obtain the desired contradiction and complete the proof. u t 2.4.1 HOMOTOPY. We now introduce just a bit of additional notation that will be useful in our discussion of the Fundamental eorem of Algebra (eorem 2.25), of the Brauer Fixed Point Formula (eorem 2.26), and of the Billiard Ball eorem (eorem 2.27) subsequently. As these concepts are properly those of algebraic topology, we only present the basic definitions and refer to Spanier [37] for further details. We say that two smooth maps F0 and F1 from a manifold M to a manifold N are homotopic if there exists a smooth map  from M  Œ0; 1 to N so that .x; 0/ D F0 .x/ and so that .x; 1/ D F1 .x/. is is an equivalence relation. Suppose N is connected. Fix a base point PN of N and a base point PS of the sphere S k . We say that a map F from S k to N is base point preserving if F .PS / D PN . A homotopy  between two such maps is said to be base point preserving if .PS ; t/ D PN for all t 2 Œ0; 1. e k th homotopy group k .N / is the set of base point preserving homotopy classes of maps from S k to N . is has a natural group structure and up to group isomorphism, k .N / is independent of the base point. If f is a smooth base point preserving map from M to N , then there is a natural group homomorphism from k .M / to k .N /. 2.4.2 THE WINDING NUMBER. Let 1 2 C 1 .C p If we set z D x C 1y , then 1 D

f0g/ be as defined in Equation (2.4.b).

  xdy ydx dz D= : x2 C y2 z

2.4. APPLICATIONS OF STOKES’ THEOREM 1

87

Let be a smooth map from S ! C

f0g. We define: Z 1 W . ; 0/ WD

 1 : 2 S 1

is is the winding number of about 0, and is a very classical object. Lemma 2.24

1. If 0 and 1 are smooth maps from S 1 to C W . 0 ; 0/ D W . 1 ; 0/.

f0g which are homotopic, then we have that

2. If .z/ D z n , then W . ; 0/ D n. 3. e map ! W . ; 0/ is a surjective map from 1 .C

f0g/ to Z.

We note that the winding number is in fact a group homomorphism from 1 .C f0g/ to Z. is group is Abelian and the base point plays no role. A similar computation using m would yield a surjective group homomorphism from m .RmC1 f0g/ to Z which is, in fact, an isomorphism. e homotopy groups k .S m / vanish for k < m. Studying the homotopy groups k .RmC1 f0g/ for k > m is a more difficult problem. Proof. e maps 0 and 1 are homotopic maps from S 1 to C f0g implies there is a smooth map W S 1  Œ0; 1 ! C f0g so that .z; 0/ D 0 and so that .z; 1/ D 1 . We use Stokes’ eorem to prove Assertion 1 by computing: Z Z Z Z     2fW . 1 ; 0/ W . 0 ; 0/g D

1 1

0 1 D 1 1 1 1 1 1 S S f1g ZS Z Z S f0g  D d  1 D d1 D 0 D 0: S 1 Œ0;1

S 1 Œ0;1

S 1 Œ0;1

p

If .z/ D z n , then ./ D e 1n D .cos.n/; sin.n// for 0    1. We prove Assertion 2 by computing: Z 2 Z 2 1 cos.n/d sin.n / sin.n /d cos.n / 1 W . ; 0/ D

 1 D 2 0 2 0 cos2 .n/ C sin2 .n/ Z 2 Z 2 1 1 D fn cos2 .n/d C n sin2 .n/d g D nd D n .  2 0 2 0 2.4.3 THE FUNDAMENTAL THEOREM OF ALGEBRA. We can use Stokes’ eorem to establish the following result:

88

2. MANIFOLDS

Let f .z/ be a complex polynomial of degree

eorem 2.25 (Fundamental eorem of Algebra). n  1. en there exists z 2 C so f .z/ D 0.

Proof. Assume to the contrary that f .z/ ¤ 0 for all z 2 C . We argue for a contradiction. We may assume without loss of generality that f .z/ D z n C an 1 z n 1 C


C a0 is monic. Consider the 1-parameter family of curves with values in C f0g:

R D .; R/ WD f .Re

p

1

/ W S1 ! C

f0g :

By Lemma 2.24, (2.4.f )

W . 0 ; 0/ D W . R ; 0/

is independent of R. Since 0 .; 0/ D a0 is the constant path, 0 1 D 0. Consequently (2.4.g)

W . 0 ; 0/ D 0 :

Let R WD ja0 j C


C jan 1 j C 2. Define Q W Œ0; 2  Œ0; 1 ! C by setting: Q .; t / WD t.Re D Rn e

p

p

We verify that Q takes values in C k Q .; t/k  Rn

jan

1 jR

1 n 1n

/ C .1 C .1

t/f .Re t/an

1R

p

n 1

1 / p

e

1n

C


C a0 :

f0g by using the triangle inequality to estimate

n 1

C


C ja0 j  Rn

1

fR

jan

1j






ja0 jg > 0 :

Consequently, R is homotopic to the curve Rn z n . is curve is homotopic to z n so W . R ; 0/ D n

(2.4.h)

by Lemma 2.24. We use Equations (2.4.f), (2.4.g), and (2.4.h) to see that 0 D n which provides the desired contradiction.  2.4.4 BRAUER FIXED POINT THEOREM. is result is due to the Dutch mathematician Luitzen Egbertus Jan Brauer; it follows from Stoke’s eorem.

L. Brauer (1881–1966)

2.4. APPLICATIONS OF STOKES’ THEOREM m

89

m

eorem 2.26 (Brauer Fixed Point eorem). Let D WD fx 2 R W kxk  1g be the unit disk in Rm . If F is a smooth map from D m to D m , then there exists a point P 2 D m so F .P / D P .

Proof. We suppose to the contrary that F .P / ¤ P for all P 2 D m . Let

.t; P / WD F .P / C t .P

F .P //

be the ray from F .P / to P . Since kF .P /k  1 and kP k  1, .t; P / belongs to D m for t 2 Œ0; 1. Let T .P / be the first t  1 so that k .t; P /k D 1; one may use the quadratic formula to see that the map P ! T .P / is smooth. Let r.P / WD .T .P /; P /; r.P / is obtained by drawing the halfline from F .P / to P and finding where it intersects the boundary S m 1 . is gives a smooth map from D m to S m 1 so that r.P / D P if P 2 S m 1 . We use Lemma 2.23 and Stokes’ eorem to derive a contradiction by computing: Z Z Z Z   0¤ m 1 D Id m 1 D r m 1 D dr  m 1 D

Z

Sm

Dm

1

Sm

r  dm

1

D

1

Z

Dm

Sm

1

Dm

0 D 0.



2.4.5 BILLIARD BALL THEOREM. Somewhat fancifully, one thinks of “combing the hair on a billiard ball” as constructing a non-zero tangent vector field on the unit sphere S m ; the vector field gives the direction in which the hair is supposed to lie. is is possible if and only if m is odd. eorem 2.27 (Billiard Ball eorem). S m if and only if m is odd.

ere exists a smooth nowhere vanishing vector field on

Proof. Suppose m is odd. We define F .x 1 ; : : : ; x mC1 / D . x 2 ; x 1 ; : : : ; x mC1 ; x m /; this is possible, of course, only if there are an even number of coordinates, i.e., if m is odd. p is gives a nowhere vanishing vector field on S m ; this vector field is, of course, nothing but 1Ez if we mC1 m N identify R with C where 2m N D m C 1. We adopt the notation of Lemma 2.23. Let m 2 C 1 .m .RmC1 f0g// be given by m .x/ WD kxk

m 1

mC1 X iD1

. 1/iC1 x i dx 1 ^


^ dx i

1

^ dx iC1 ^


^ dx mC1 :

R We showed that dm D 0 and S m m ¤ 0. Suppose m is even and that there exists a nowhere vanishing vector field. We argue for a contradiction. By replacing F by kF k 1 F , we may assume without loss of generality that F is a unit vector field. Let G.x;  / WD cos. /
x C sin./
F .x/ :

90

2. MANIFOLDS

Since x ? F .x/, this takes values in S m  RmC1 and provides a homotopy as  2 Œ0;  between the identity map and the antipodal map. By Stokes’ eorem: Z Z Z  m a m D bd.S m  Œ0; /G  m Sm Sm Z Z D dG  m D G  dm D 0 : S m Œ0;

As m C 1 is odd, a m D m . is implies 2 establishes the eorem.

R

Sm

S m Œ0;

m D 0 which contradicts Lemma 2.23 and 

91

CHAPTER

3

Riemannian and Pseudo-Riemannian Geometry In Section 3.1, we introduce pseudo-Riemannian geometry. We show that the spheres of even dimension do not admit Lorentzian metrics. We define the pseudo-Riema�are perpendicular nnian measure, verify this agrees with the surface measure dS defined previously in Stokes’ eorem, and compute the volume of spheres and disks. In Section 3.2, we examine connections and their curvature. In Section 3.3, we specialize to the case of the Levi–Civita connection; this is the unique torsion-free Riemannian connection on the tangent bundle. In Section 3.4, we use the Levi–Civita connection to study geodesics. In Section 3.5, we use the Jacobi operator to establish some basic results concerning manifolds of constant sectional curvature. Section 3.6 is devoted to the proof of the Gauss–Bonnet eorem and the study of Riemann surfaces. In Section 3.7, the generalization to higher dimensions and also to the pseudo-Riemannian setting is exhibited.

3.1

THE PSEUDO-RIEMANNIAN MEASURE

3.1.1 INNER PRODUCT SPACES. We say that the pair .V; h
;
i/ is an inner product space if h
;
i is a non-degenerate symmetric bilinear form on a finite-dimensional real vector space V . We say that 0 ¤ v 2 V is spacelike (resp. timelike or null) if hv; vi > 0 (resp. hv; vi < 0 or hv; vi D 0). A subspace W of V is said to be spacelike (resp. timelike or null) if the restriction of h
;
i to W is positive definite (resp. negative definite or identically zero). Lemma 3.1

Let .V; h
;
i/ be an inner product space.

1. ere exists an element of V so hv; vi ¤ 0. 2. ere exists an orthogonal direct sum decomposition V D V ˚ VC where V is timelike and VC is spacelike. Moreover, if V D VQ ˚ VQC is another such decomposition, then we have that dim.V˙ / D dim.VQ˙ / and we shall say that .V; h
;
i/ has signature .p; q/ where we set p D dim.V / and q D dim.VC /. 3. ere exist bases fei g1ip for V and feaC g1aq for VC so hei ; ej i D ıij , hei ; eaC i D 0, and heaC ; ebC i D ıab . e set fe1 ; : : : ; ep ; e1C ; : : : ; eqC g is said to be an orthonormal basis. Let a C a C a a v D xC ea C y i ei and vQ D xQ C ea C yQ i ei . en it follows that hv; vi Q D xC xQ C y i yQ i .

92

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

4. Any two inner product spaces with the same signature are isomorphic.

Proof. Let .V; h
;
i/ be an inner product space of dimension m. Suppose Assertion 1 fails, i.e., hv; vi D 0 for all v . en 0 D hv C w; v C wi D hv; vi C hw; wi C 2hv; wi D 2hv; wi

for all

v; w 2 V :

is contradicts the assumption that h
;
i was non-degenerate; Assertion 1 now follows. If m D 1, then h
;
i is definite and we are done. We may therefore establish Assertion 2 by induction on m. By Assertion 1, choose y 2 V so hy; yi ¤ 0. Let ˚.x/ WD hx; yi define a non-trivial linear map from V to R; this is non-trivial as hy; yi ¤ 0. us W WD kerf˚ g is a linear subspace of V of dimension m 1. Let 0 ¤ x 2 W . Suppose hx; wi D 0 for all w 2 W . Since hx; yi D 0 and since V D y
R ˚ W , we conclude hx; xi Q D 0 for all xQ 2 V . is contradicts the assumption h
;
i was non-degenerate. Consequently, the restriction of h
;
i to W is nondegenerate. By induction, we may decompose W D WC ˚ W . We obtain a suitable orthogonal decomposition of V by setting: VC WD

(

yR ˚ WC WC

if y is spacelike if y is timelike

)

and V WD

(

yR ˚ W W

if y is timelike if y is spacelike

)

:

e decomposition V D V ˚ VC defines projections  . Suppose given another decomposition V D VQ ˚ VQC . Suppose there exists 0 ¤  2 VQ so that  ./ D 0. en  2 VC so h; i > 0 which is false. us  W VQ ! V is injective and dim.VQ /  dim.V /. Reversing the argument gives dim.V /  dim.VQ /. us dim.V / D dim.VQ / and similarly dim.VC / D dim.VQC /. is proves Assertion 2. Let h
;
i˙ be the restriction of h
;
i to V˙ . We prove Assertion 3 by applying the Gram–Schmidt process to definite inner product spaces .V˙ ; h
;
i˙ /; Assertion 4 follows from Assertion 3.  INDEFINITE FIBER METRICS ON VECTOR BUNDLES. Let V be a vector bundle over M . A fiber metric on V of signature .p; q/ is a smooth section h to the bundle of symmetric 2cotensors S 2 .V  / so that the restriction of h to each fiber VP is a non-degenerate symmetric inner product of signature .p; q/ on VP ; the pair .V; h/ is then said to be an orthogonal bundle. Since the convex combination of positive definite inner products is positive definite, we can use a partition of unity to construct positive definite inner products on V . If V D TM , then h is said to be a pseudo-Riemannian metric on M . We say h is a Riemannian metric if h is positive definite and that h is a Lorentzian metric if h has signature .1; m 1/. ere always exist Riemannian metrics on M but there are, however, topological restrictions to the construction of indefinite inner products.

3.1. THE PSEUDO-RIEMANNIAN MEASURE

93

Lemma 3.2

1. Let .V; h/ be an orthogonal bundle. ere exist smooth subbundles V˙ of V so that h restricts to a spacelike (resp. timelike) fiber metric on VC (resp. on V ), so that VC is perpendicular to V with respect to h, and so that V D VC ˚ V . 2. If m is even, then T S m does not admit a fiber metric of indefinite signature. If m is odd, then T S m admits a Lorentzian metric. Proof. Let he be an auxiliary positive definite fiber metric on V . Since we can diagonalize any quadratic form with respect to a positive definite one, we can find a smooth section ˚ to Hom.V; V / so that h.v; w/ D he .˚v; w/. For each P 2 M , we may diagonalize P ; let ˙ .P / be the projections on the positive and negative eigenspaces. e ˙ vary smoothly with P and we use Lemma 2.3 to see that V˙ WD rangef˙ g are smooth subbundles of V ; these have the desired properties of Assertion 1. Let m be even. Suppose to the contrary that T S m admits a pseudo-Riemannian metric of signature .p; q/ for p > 0 and q > 0. We apply Assertion 1 to decompose T S m D VC ˚ V as the orthogonal direct sum of a spacelike and a timelike bundle. Let H˙m be the upper and lower hemispheres of sphere and let PC be the north pole and P the south pole of S m . Stereographic projection shows that S m fP˙ g is diffeomorphic to Rm and hence contractible. Consequently, by Lemma 2.3, the bundles V˙ are trivial over S m fP˙ g so we can find non-vanishing smooth sections ˙ to V˙ over S m fP˙ g. Use Lemma 1.21 to find smooth functions ˙ on S m which are identically 1 on H˙m and which vanish identically near P . en ˙ ˙ are smooth tangent vector fields over all of S m which are non-zero on the hemispheres H˙ . Let  WD C C ˚  ; this is a non-vanishing vector field on S m . is contradicts eorem 2.27. Suppose m D 2m N 1 is odd. Let .
;
/ be the ordinary Euclidean inner product on T S m . Apply eorem 2.27 to construct a non-vanishing vector field  on S m . Let V WD 
R and let VC WD V ? be the complementary vector subbundle. Take .
;
/ on VC and .
;
/ on V to construct a Lorentzian metric on S m . u t

3.1.2 VOLUMES OF SPHERES. Let f.O˛ ; .x˛1 ; : : : ; x˛m //g be a coordinate atlas on M . Define the symmetric tensor product by setting: dx˛i ı dx˛j WD 21 .dx˛i ˝ dx˛j C dx˛j ˝ dx˛i / :

(3.1.a)

Express the metric g on O˛ in the form g D g˛;ij dx˛i ı dx˛j where g˛;ij WD g.@x˛i ; @x j /. We ˛ define a measure dg;x on each coordinate chart O by setting: 1

dg;˛ WD j det.g˛;ij /j 2 dx˛1




dx˛m :

If g is Riemannian (or equivalently, is positive definite), then det.g˛;ij / > 0 and it is not necessary to take the absolute value. Let Drm be the ball of radius r in Rm and let Srm 1 D bd.Drm / be the associated sphere of radius r . Give these manifolds the canonical Riemannian metrics induced from the Euclidean metric on Rm . Define the Gamma function for s > 0 by setting:

94

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

.s/ WD Lemma 3.3

Z

1

ts

1

e t dt .

0

(e pseudo-Riemannian measure) Adopt the notation established above.

1. dg;˛ D dg;ˇ on O˛ \ Oˇ . us these local measures patch together to yield an invariantly defined measure dg called the pseudo-Riemannian measure on M . 2. Let ˚.u; v/ WD .x.u; v/; y.u; v/; z.u; v// parametrize a surface in R3 . en dg D dS where dS WD k@u ˚  @v ˚ kdudv is as in Stokes’ eorem.

4. vol.S1m

rm m

vol.S1m 1 / and vol.Srm 1 / D r m 1 vol.S1m 1 /. ( .2/m=2 ) m 1 if m is even 2
4




.m 2/ 1 m=2 D 2.2/.m 1/=2 / D 2 . 2 if m is odd 1
3




.m 2/

3. vol.Drm / D

Proof. Let Jji WD

i @x˛ j

@xˇ

be the Jacobian matrix. We use the Change of Variable eorem to see that

dx˛1




dx˛m D jdet.J /j dxˇ1




dxˇm . Since @x j D Jji @x˛i , gˇ;j k D Jj` Jkn g˛;`n and conseˇ

1

1

quently det.gˇ / D det.J /2 det.g˛ / so j det.gˇ /j 2 D j det.J /j j det.g˛ /j 2 . us we may establish Assertion 1 by computing: dg;˛

1

1

D j det.g˛ /j 2 dx˛1




dx˛m D j det.g˛ /j 2 j det.J /j dxˇ1




dxˇm 1

D j det.gˇ /j 2 dxˇ1




dxˇm D dg;ˇ :

If ˚ parametrizes a surface in R3 , then: @u ˚  @v ˚ D .@u y
@v z

@u z
@v y; @u z
@v x

k@u ˚  @v ˚ k2 D .@u y
@v z g11 g22

g12 g12

@u x
@v z; @u x
@v y

@u z
@v y/2 C .@u z
@v x

@u x
@v z/2

@u y
@v x/,

C .@u x
@v y @u y
@v x/2 , ˚ ˚ D .@u x/2 C .@u y/2 C .@u z/2
.@v x/2 C .@v y/2 C .@v z/2 f@u x
@v x C @u y
@v y C @u z
@v zg2 .

Assertion 2 follows after a bit of algebraic computation. Let F1 .1 ; : : : ; m 1 / be a local parametrization of S m 1 . en F .r;  / WD rF1 . / is a local parametrization of RmC1 . We have: g.@i ; @j /.; r/ D r 2 f.@i F1 /
.@j F1 /g. / D r 2 g.@i ; @j /.; 1/; g.@i ; @r /.; r/ D rf@i F1
F1 g./ D 12 r@i .F1
F1 /./ D 21 r@i .1/ D 0; g.@r ; @r /.; r/ D F1 ./
F1 ./ D 1 :

3.2. CONNECTIONS

It now follows that dRm D r

m 1

95

drdS m 1 ; Assertion 3 now follows. Let Z 1 2  WD e x dx : 1

In polar coordinates (see Example 1.4.3), we have dx dy D rdr d . Consequently, ˇ1 Z 1Z 1 Z 1 Z 2 ˇ x2 y2 r2 r2 ˇ 2  D e dxdy D re ddr D e D. ˇ 1

1

0

0

rD0

We use this identity and Assertion 3. We integrate (setting  D r 2 ) to see Z Z 1 2 E 2  m=2 D m D e kxk dRm D r m 1 e r dS m 1 dr m R 0 Z 1 Z 1 m 1 m 1 r2 D vol.S / r e dr D 12 vol.S m 1 /  .m 2/=2 e  d  0 0 m D 12 vol.S m 1 / : 2 is establishes the first identity of Assertion 4; the second then follows from the functional p equation s .s/ D .s C 1/, the fact that .1/ D 1, and the fact that . 12 / D  . u t

3.2

CONNECTIONS

We refer to Besse [8] and to Kobayashi and Nomizu [24] for further details concerning the material of this section. Let V be a vector bundle over a manifold M . A connection r on V is a first order partial differential operator from C 1 .V / to C 1 .T  M ˝ V / satisfying the Leibnitz formula: r.f s/ D df ˝ s C f rs for s 2 C 1 .V / : (3.2.a)

Connections always exist locally; if sE is a local frame for V , then r.f i si / WD df i ˝ si defines a connection locally. Since the convex combination of connections is again a connection, we can construct connections using a partition of unity. e associated directional covariant derivative rX s is defined by setting rX s WD X; rs 

for

s 2 C 1 .V /

and X 2 C 1 .TM / :

If fei g is a basis for TM and if fe i g is the associated dual basis for T  M , the total covariant derivative can be recovered from the directional covariant derivatives by setting rs D e i ˝ rei s :

We extend r to a dual connection r  on V  by requiring that d  s; s  D rs; s   C  s; r  s  

for

s 2 C 1 .V /

and s  2 C 1 .V  / :

If r and rQ are connections on vector bundles V and VQ , respectively, then we can define connections on V ˚ VQ and on V ˝ VQ by setting, respectively: Q s / and r.s ˝ sQ / D r.s/ ˝ sQ C s ˝ r.Q Q s/ . r.s ˚ sQ / D r.s/ ˚ r.Q

96

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

3.2.1 HOLONOMY. Let be a smooth curve in a manifold M , i.e., a smooth map from some interval I D Œa; b to M . Let V be a vector bundle over M which is equipped with a connection r . Let V0 be the fiber of V over the initial point .a/. A section to V along is a smooth map s W I ! V so that s.t/ 2 V .t / . e section is said to be parallel if r@ t s.t / D 0. Given a basis fe1 ; : : : ; er g for V0 , we can use the Fundamental eorem of Ordinary Differential Equations to find a frame fe1 .t/; : : : ; er .t/g for V along so r@t ei D 0 and so ei .0/ D ei . If is a closed curve, i.e., if .a/ D .b/, then parallel translation along defines an invertible linear map T which belongs to End.V0 ; V0 / that is called holonomy. e holonomy group is the subgroup of GL.V0 / consisting of all the T as ranges over the smooth closed paths in M . 3.2.2 THE CHRISTOFFEL SYMBOLS. Let sE D .s1 ; : : : ; sk / be a local frame for V and let xE D .x 1 ; : : : ; x m / be a system of local coordinates on M . We may expand r@xi sa D

ia

b

sb :

Here the index i ranges from 1 to m WD dim.M / and the indices a; b range from 1 to k , i.e., to the fiber dimension of V . e i a b D r ia b are referred to as the Christoffel symbols of the first kind or sometimes simply as the Christoffel symbols of the connection r . ey are not tensorial. In view of the Leibnitz formula given in Equation (3.2.a), r is determined by the Christoffel symbols as we see by computing: r i @ i .f a sa / D  i ff a x

ia

b

sb C @x i .f a /sa g :

Let sE  D .s 1 ; : : : ; s k / be the local dual frame field for the dual bundle V  . e dual Christoffel symbols for the dual connection on V  are given by the identity: r@ i s b D

ia

x

b a

s :

Similarly, if V and VQ are vector bundles which are equipped with connections r and rQ , then the Christoffel symbols of the natural connections on V ˚ VQ and V ˝ VQ are given by ia

b

Q ˚ Qi aQ b

and

ia

b

Q ˝ Id C Id ˝ Qi aQ b :

Let V be equipped with a non-degenerate fiber metric h; r is a Riemannian connection if dh.s1 ; s2 / D h.rs1 ; s2 / C h.s1 ; rs2 /

for si 2 C 1 .V / :

Equivalently, r is Riemannian if and only if r agrees with r  when we use h to identify V with V  . We define the Christoffel symbols of the second kind by using h to lower indices: iab

WD h.r@xi sa ; sb / :

If sE is a local orthonormal frame for V , then r is Riemannian if and only if iab

C

i ba

D0

for all i; a; b :

3.2. CONNECTIONS

97

3.2.3 THE CURVATURE OPERATOR. e curvature operator R D R of a connection r is given by: R.X; Y /s WD frX rY rY rX rŒX;Y  gs : r

Let R.@x i ; @x j /sa D Rija b sb be the components of the curvature operator in a system of local coordinates xE D .x 1 ; : : : ; x m / and relative to a local frame sE D .s1 ; : : : ; sk / for V . en: Rija b D @x i

ja

b

@x j

ia

b

C

ic

b

ja

c

jc

b

ia

c

:

(3.2.b)

We can use a fiber metric to lower indices and define R.X; Y; s; sQ / WD h.R.X; Y /s; sQ / and Rijab WD h.R.@x i ; @x j /sa ; sb /. Lemma 3.4

Let r be a connection on a vector bundle V over M .

1. R.X; Y / D R.Y; X/. 2. e curvature is a bundle map from TM ˝ TM to Hom.V; V / which is given by R. i @x i ;  j @x j /f a sa D  i  j f a Rija b sb . 3. If h is a fiber metric on V and if r is Riemannian with respect to h, then R is skewsymmetric with respect to h, i.e., h.R.X; Y /s1 ; s2 / C h.s1 ; R.X; Y /s2 / D 0. Proof. e symmetry Assertion 1 is immediate. We will show that R is a C 1 .M / module homomorphism, i.e., that

R.f X; Y /s D R.X; f Y /s D R.X; Y /f s D f R.X; Y /s : We may then apply Lemma 2.4 to see that R is a tensor, i.e., its value only depends on the values of X , Y , and s at the point in question. Assertion 2 will then follow. We compute:

R.f X; Y / D rf X rY rY rf X rŒf X;Y  D f rX rY f rY rX Y.f /rX C f rŒX;Y  C Y.f /rX D f R.X; Y / : Since R.X; Y / D R.Y; X/, it follows that R.X; f Y / D f R.X; Y / as well. We complete the proof of Assertion 2 by verifying:

R.X; Y /f s D frX rY rY rX rŒX;Y  g.f s/ D rX .f rY s/ C rX .Y .f //s rY .f rX s/ rY .X.f /s/ ŒX; Y .f /s f rŒX;Y  s D f R.X; Y /s C X.f /rY s C .X Y /.f /s C Y.f /rX s Y .f /rX s .YX /.f /s X.f /rY s ŒX; Y .f /s D f R.X; Y /s :

98

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

Suppose that r is Riemannian. Let sE be an orthonormal frame field. We must show Rijab C Rijba D 0. We use Equation (3.2.b) to compute: Rijab C Rijba

D

@x i C@x i

jab jba

@x j @x j

C iba C iab

i cb

ja

i ca

jb

Since r is Riemannian and the frame field is orthonormal, derivative terms cancel off and we have Rijab C Rijba D

i cb

ja

Let c D h.sc ; sc / D ˙1. We then have indices to complete the proof: P Rijab C Rijba D c c f icb

c

ia

jcb c

jac

D c

ia

c

C

i ca

c c

jab

jb

D

c

c

jcb

ia

jca

ib

jba .

Consequently, the

jca

ib

c

c

:

:

i ac . We use the anti-symmetry in the last two

jcb

iac

C

ica

jca

jbc

i bc g

D 0.

u t

3.2.4 THE TORSION TENSOR. If r is a connection on TM , then we may define the torsion tensor T D r T by:

T .X; Y / D rX Y

rY X

ŒX; Y 

for X; Y 2 C 1 .TM / :

One has the following useful fact; it permits one to normalize the choice of the frame so that only the second derivatives of the Christoffel symbols enter into the computation of the curvature: Lemma 3.5

(e Torsion Tensor) Let r be a connection on TM and let P 2 M .

1. e torsion is a tensor. It defines a bundle map T W TM ˝ TM ! TM which satisfies the k identity T .X; Y / D T .Y; X /. It is given by T . i @x i ; j @x j / D  i j . ij k j i /@x k . 2. e following conditions are equivalent and if either condition is satisfied at all points of M , then r is said to be an affine connection or a torsion-free connection. (a) T .XP ; YP / D 0 for all XP ; YP 2 TP M .

(b) ere exist local coordinates for M centered at P so that

ij

k

.P / D 0.

3. (Bianchi Identity) If r is torsion-free, R.X; Y /Z C R.Y; Z/X C R.Z; X /Y D 0. Proof. It is immediate that T .X; Y / D T .Y; X / and that T .@x i ; @x j / D . ij k apply Lemma 2.4 to show T is a tensor and establish Assertion 1 by computing:

T .X; Y / D X./Y C rX Y Y./X rY X ŒX; Y  X./Y C Y./X D T .X; Y / :

ji

k

/@x k . We

3.2. CONNECTIONS

99

By Assertion 1, T .P / D 0 if and only if ij .P / D j i .P /. In particular, if there exists a coordinate system where .P / D 0, then necessarily T vanishes at P . us Assertion 2-b implies Assertion 2-a. Conversely, assume that Assertion 2-a holds. Choose any system of coordinates xE D .x 1 ; : : : ; x m / on M which are centered at P . Define a new system of coordinates by setting z i D x i C 12 cj k i x j x k where cj k i D ckj i remains to be chosen. As @x j D @z j C cj i ` x i @z ` , k

k

r@xi @x j .P / D r@zi @z j .P / C cj i ` @z ` .P / :

Set cij ` WD ij ` .P /; the fact that cij ` D cj i ` is exactly the assumption that the torsion tensor of r vanishes at P . us we conclude that the new coordinate system has vanishing Christoffel symbols at P . Suppose that r is torsion-free. Since R is a tensor, it suffices to establish the Bianchi identity on a basis. Since r is torsion-free, we may compute:

R.@x i ; @x j /@x k C R.@x j ; @x k /@x i C R.@x k ; @x i /@x j D r@xi r@xj @x k

r@xj r@xi @x k C r@xj r@xk @x i

r@xk r@xj @x i C r@xk r@xi @x j D r@xi .r@xj @x k Cr@xk .r@xi @x j

r@xi r@xk @x j

r@xk @x j / C r@xj .r@xk @x i

r @ x i @x k /

r @ x j @x i / .

Since r@xj @x k r@xk @x j D Œ@x j ; @x k  D 0, we have r@xi .r@xj @x k argument shows the remaining terms vanish as well.

r@xk @x j / D 0; a similar 

3.2.5 AFFINE GEOMETRY. An affine manifold is a pair .M; r/ where r is a torsion-free connection on TM . We say that a parametrized curve .t / in M is a geodesic if it satisfies the geodesic equation: r P P D 0

or equivalently

R i C

jk

i

. .t // P j P k D 0

(3.2.c)

if we express D . 1 ; : : : ; m / in some coordinate frame. We say that .M; r/ is a complete affine manifold if every geodesic extends for infinite time. e following is a useful observation: eorem 3.6

Let .M; r/ be an affine manifold.

1. If 0 ¤ c 2 R and if c .t/ WD .ct / is the reparametrized curve, then is a geodesic if and only if c is a geodesic. 2. Given any point P 2 M and any tangent vector X 2 TP M , there exists a unique geodesic defined on an interval . ; / for some  > 0 with .0/ D P and P .0/ D X . Proof. Assertion 1 is immediate from the definition; Assertion 2 follows from the Fundamental eorem of Ordinary Differential Equations. 

100

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

3.3

THE LEVI–CIVITA CONNECTION

Pseudo-Riemannian geometry is to a large extent the study of the Levi–Civita connection. In this section, we will establish the basic facts that we shall need subsequently. 3.3.1 THE FUNDAMENTAL THEOREM OF RIEMANNIAN GEOMETRY. We begin by giving an abstract characterization of the Levi–Civita connection [31]. is is named after the Italian mathematician Tullio Levi–Civita.

Tullio Levi–Civita (1873–1941) eorem 3.7

Let .M; g/ be a pseudo-Riemannian manifold.

1. (Koszul formula) If r is a torsion-free Riemannian connection on TM and if fX; Y; Zg are vector fields on M , then 2g.rX Y; Z/ D X.g.Y; Z// C Y.g.X; Z// Cg.ŒX; Y ; Z/

Z.g.X; Y //

g.ŒX; Z; Y /

g.ŒY; Z; X / .

2. ere exists a unique connection r on the tangent bundle of M (which is called the Levi–Civita connection) so that r is torsion-free and Riemannian. 3. e curvature of the Levi–Civita connection has the symmetries: (a) R.X; Y / C R.Y; X / D 0.

(b) R.X; Y /Z C R.Y; Z/X C R.Z; X /Y D 0.

(c) g.R.X; Y /Z; W / C g.Z; R.X; Y /W / D 0. Proof. We prove Assertion 1 by computing: g.rX Y; Z/ D g.rY X; Z/ C g.ŒX; Y ; Z/

(torsion-free)

D

g.rY Z; X / C Yg.X; Z/ C g.ŒX; Y ; Z/

(Riemannian)

D

g.rZ Y; X /

(torsion-free)

D g.rZ X; Y /

g.ŒY; Z; X / C Y g.X; Z/ C g.ŒX; Y ; Z/ Zg.X; Y /

Cg.ŒX; Y ; Z/

g.ŒY; Z; X / C Yg.X; Z/

(Riemannian)

3.3. THE LEVI–CIVITA CONNECTION

D g.rX Z; Y / C g.ŒZ; X ; Y /

Zg.X; Y /

101

g.ŒY; Z; X /

(torsion-free)

Zg.X; Y /

(Riemannian)

CYg.X; Z/ C g.ŒX; Y ; Z/

D

g.rX Y; Z/ C Xg.Z; Y / C g.ŒZ; X ; Y / g.ŒY; Z; X/ C Yg.X; Z/ C g.ŒX; Y ; Z/ .

Assertion 1 shows that if the Levi–Civita connection exists, then it is unique. We introduce the notation gij=k WD @x k g.@x i ; @x j / for the first derivatives of the metric. Applying the Koszul formula to the coordinate frame, then yields the Christoffel identity for the Christoffel symbols: ij k

D 12 fgj k= i C gik=j

gij=k g :

(3.3.a)

To show that a torsion-free Riemannian connection exists, we use Equation (3.3.a) to define the Christoffel symbols. It is then immediate that ij k D j i k and hence r is torsion-free. Similarly we have that ij k C ikj D gj k= i and hence r is Riemannian. Since r is unique, this local definition is globally consistent. Assertion 2 follows; Assertion 3 now follows from Lemma 3.4 and from Lemma 3.5. u t 3.3.2 GEOMETRIC REALIZABILITY. Let .V; h
;
i/ be an inner product space. A 4-tensor A 2 ˝4 V  is said to be an Riemannian algebraic curvature tensor if A satisfies the symmetries given in eorem 3.7, i.e., A.x; y; z; w/ C A.y; x; z; w/ D 0 8 x; y; z; w 2 V; A.x; y; z; w/ C A.y; z; x; w/ C A.z; x; y; w/ D 0 8 x; y; z; w 2 V; A.x; y; z; w/ C A.x; y; w; z/ D 0 8 x; y; z; w 2 V :

We say that A is geometrically realizable if there exists the germ of a pseudo-Riemannian manifold .M; g/, there exists a point P of M and there exists an isometry  from .TP M; gP / to .V; h
;
i/ so that g.RP .x; y/z; w/ D A.x; y; z; w/ for all x; y; z; w 2 TP M . Every Riemannian algebraic curvature tensor on an inner product space .V; h
;
i/ is geometrically realizable. Lemma 3.8

Proof. Fix an auxiliary Euclidean metric on V to define kxk. Let M WD fx 2 V W kxk < g and let P D 0. Let .x 1 ; : : : ; x m / be the system of local coordinates on V induced by a basis fei g for V , let "i k WD hei ; ek i, and let gi k WD "i k 13 Aij `k x j x ` :

Clearly gi k D gki . As gik .0/ D "ik is non-singular, there exists  > 0 so that g is non-singular on M . Let gij=k D @x k g.@x i ; @x j / and gij=k` WD @x k @x ` gij :

102

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

We use Equation (3.2.b) and Equation (3.3.a). Since g D " C O.kxk2 /, gij=k D O.kxk/. Consequently we have that D O.kxk/. is implies that: Rij k ` D @x i

jk

`

@x j

ik

`

C O.

2

/ D @x i

jk

`

@x j

ik

`

C O.

2

/:

Furthermore, @x i .g`n j k n / D g`n @x i j k n C O.kxk/. We lower indices to complete the proof: Rij k` D @x i j k` @x j i k` C O.kxk2 /, D 21 fgj `= i k C gi k=j ` gj k= i ` gi `=j k g C O.kxk2 / D 16 f Aj i k` Aj ki ` Aij `k Ai`j k CAj i `k C Aj `i k C Aij k` C Ai kj ` g C O.kxk2 / D 16 f4Aij k` 2Ai `j k 2Aik`j g C O.kxk2 / D Aij k` C O.kxk2 / .  is result shows, as a byproduct, that the symmetries of eorem 3.7 generate the universal curvature symmetries of the Riemann curvature tensor; there are no additional universal symmetries to be found. 3.3.3 THE SECOND FUNDAMENTAL FORM. Let M be a submanifold of dimension m of a pseudo-Riemannian manifold .N; gN / of dimension m C n. Let gM be the restriction of the metric gN to the submanifold. We suppose that gM is non-degenerate on M ; if this happens, we say M is a non-degenerate submanifold of N . is condition is automatic, of course, if gN is positive definite. Let M be orthogonal projection from TP N to TP M for P 2 M . Choose local coordinates .x; E y/ E on an open neighborhood O of P in N so xE D .x 1 ; : : : ; x m / gives local coordinates on M and so M is defined by yE D 0 where yE D .y 1 ; : : : ; y n /. Let r N be the Levi– Civita connection of N . Let X D ai .x/@ E x i 2 C 1 .TM / be a tangent vector field along M . Extend X to a vector field defined on O by setting VQ .x; E y/ E D ai .x/@ E x i . If VQ1 is another extension of X to a vector field on O so that VQ1 D V on M , then the coefficients of VQ1 VQ relative to the coordinate frame vanish identically on M . Consequently, if U 2 C 1 .TM /, then rUN .VQ1 VQ / vanishes identically on M . is shows that rUN V WD rUN VQ jM is well-defined and independent of the particular extension chosen. Note that rbNj .x/@

xj

.ai .x/@x i / D b j .@x j ai /@x i C ai b j

ij

k

@x k C a i b j

ij

a

@y a :

us if ij a is non-trivial, then rUN V will in general have components in @y a and thus r N need not induce a connection on TM directly. eorem 3.9 Let .M; gM / be a non-degenerate submanifold of a pseudo-Riemannian manifold .N; gN /. We may decompose r N jM D rQ M C II where the projected connection rQ M WD M r N is the Levi–Civita connection of M and where the second fundamental form II.U; V / WD .1 M /rUN V is a symmetric bilinear form taking values in the normal bundle. If U , V , W , and X are tangent to M , then the curvature of N and of M are related by:

3.4. GEODESICS

RN .U; V; W; X/ D RM .U; V; W; X /

103

g.II.V; W /; II.U; X // C g.II.U; W /; II.V; X // .

Proof. Let X , Y , and Z be tangent vector fields along M . Clearly rQ fMY X D f rQ YM X and clearly rQ M is bilinear. We show that rQ M is a connection on TM by checking that the Leibnitz rule is satisfied: M rXN .f Y / D M .X.f /Y C f rXN Y / D X.f /Y C f M .rXN Y / :

If  2 TP N , then gN .Z; / D gM .Z; M .//. us gN .Z; rY X/ D gN .Z; rQ YM X /. We show rQ M is Riemannian by computing: gM .rQ XM Y; Z/ C gM .Y; rQ XM Z/ D gN .rXN Y; Z/ C gN .Y; rXN Z/ D XgN .Y; Z/ D XgM .Y; Z/ :

We show rQ M is torsion-free by using the fact that r N is torsion-free to compute:   rQ XM Y rQ YM X D M rai @ i b j @x j rb j @ j ai @x i D M fŒX; Y g D ŒX; Y  . x

x

As rQ M is a torsion-free and Riemannian connection on TM , eorem 3.7 implies that rQ M is the Levi–Civita connection of M . We show that II is a symmetric tensor by computing: II.U; V / II.V; U / D .1 M /.rU V rV U / D .1 M /.ŒU; V / D 0; II.f U; V / D .1 M /.rf U V / D f .1 M /.rU V / D f II.U; V /; II.U; f V / D II.f V; U / D f II.V; U / D f II.U; V / : We complete the proof by computing: RN .U; V; W; X/ D g.rUN rVN W; X / D g.rUM .rVM W C II.V; W //; X / M g.rŒU;V W; X/ 

g.rVN rUN W; X /

N g.rŒU;V W; X / 

g.rVM .rUM W C II.U; W //; X /

D RM .U; V; W; X/ C Ug.II.V; W /; X /

g.II.V; W /; rUM X/

Vg.II.U; W /; X / C g.II.U; W /; rVN X /

D RM .U; V; W; X/

3.4

g.II.V; W /; II.U; X // C g.II.U; W /; II.V; X // .

u t

GEODESICS

Let .M; g/ be a pseudo-Riemannian manifold. A diffeomorphism T W M ! M is said to be an isometry if T  g D g . Let r be the Levi–Civita connection of .M; g/. Recall that is a geodesic if r P P D 0.

104

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

Lemma 3.10

Let .M; g/ be a pseudo-Riemannian manifold.

1. If is a geodesic in M , then j P j is constant. 2. If M is a non-degenerate submanifold of an inner product space .V; h
;
i/ and if g is the restriction of h
;
i to M , then a curve in M is a geodesic if and only if R ? T .t / M for all t in the parameter space. 3. Let T be an isometry of .M; g/. If a unit speed curve is the fixed point set of T , then the curve is a geodesic. 4. Let .V; h
;
i/ be an inner product space and let .M˙ ; g/ D fv 2 V W hv; vi D ˙1g be a pseudo-sphere. Let P 2 M˙ and let Q be a unit vector orthogonal to P . Let ( ) cos.t/P C sin.t/Q if hP; P i D hQ; Qi

.t/ WD : cosh.t /P C sinh.t /Q if hP; P i D hQ; Qi en is a geodesic in M˙ with .0/ D P and .0/ P D Q. Proof. Let be a geodesic. Since the Levi–Civita connection is Riemannian, we may prove Assertion 1 by computing: @ t g. ; P / P D 2g.r P P ; P / D 0 :

If M is a submanifold of N , eorem 3.9 shows that the Levi–Civita connection of M is obtained by projecting the Levi–Civita connection r N of N onto the tangent space of M . If N D .V; h
;
i/, then the Christoffel symbols of N vanish. us r N

P D R ; Assertion 2 now follows. P Adopt the notation of Assertion 3. Let P WD .0/ and  WD P .0/. Let 1 be the unique geodesic in M with 1 .0/ D P and P 1 .0/ D  . Since T fixes  , TP D P and T  D  . Since T 1 is a geodesic starting at P with initial direction  , we have T 1 D 1 . Since  is the fixed point set of T ,  is a reparametrization of 1 . Since  has constant non-zero speed,  is a geodesic. Adopt the notation of Assertion 4. It is immediate that has constant speed and takes values in M˙ . Choose an orthonormal basis fe1 ; : : : ; em g for V so P D e1 and Q D e2 . Define an isometry T of .V; h
;
i/ by setting T .x i ei / WD x 1 e1 C x 2 e2 x 3 e3


x m em . en T restricts to an isometry of M˙ fixing pointwise. Assertion 4 now follows from Assertion 3.  3.4.1 GEODESIC COORDINATES AND THE EXPONENTIAL MAP.

Let P be a point of a Riemannian manifold .M; g/. ere exists a diffeomorphism expP from a neighborhood U of 0 in TP M to a neighborhood O of P in M so that the curves t ! expP .t/ are geodesics in M starting at P with initial direction  for 0  t  1 and  2 U . Lemma 3.11

3.4. GEODESICS

105

Proof. Let .O; .x ; : : : ; x // be a system of local coordinates on M centered at P . We may then write the geodesic equation in the form given in Equation (3.2.c). e Fundamental eorem of Ordinary Differential Equations shows that there exists a neighborhood U of 0 in TP M and there exists  > 0 so that there is a smooth map 1

m

F W U  Œ0;  ! O

so that the curves t ! exp.; t / are geodesics starting at P with initial direction  . Because the curves t ! F .; ct/ are geodesics starting at P with initial direction c , F .; ct / D F .c; t /. us by shrinking U if need be, we may assume without loss of generality that  D 1 and the curves extend for 0  t  1. Let expP ./ WD F .; 1/. is is a smooth map from U to M with expP .0/ D P such that the curves t ! expP .t/ are geodesics from P with initial direction  . It now follows that .expP /0 .0/ D  so by the Inverse Function eorem (eorem 1.8), expP is a diffeomorphism if we shrink U .  We now restrict to the Riemannian setting. Fix an orthonormal basis fe1 ; : : : ; em g for TP M . ere exists  > 0 so that the map .x 1 ; : : : ; x m / ! expP .x i ei / is a diffeomorphism from the ball of radius  about the origin in TP M to a neighborhood of P in M . We use this map to define geodesic coordinates on M . Straight lines through the origin are geodesics under this identification. Lemma 3.12 Let .x 1 ; : : : ; x m / be a system of geodesic coordinates on a Riemannian manifold M for kxk <  for some  > 0.

1. Let .v/ be a curve in TP M with k k D 1. Let F .r; v/ WD r.v/. en F @r ? F @v . 2. Let  2 B . Let .t/ be a curve from 0 to  in B and let  .t / D t for t 2 Œ0; 1. en L. /  L./ D kk with equality if and only if is a reparametrization of  . 3. Let P and Q be two points of a connected Riemannian manifold .M; g/. We define the geodesic distance dg .P; Q/ D inf W .0/DP and .1/DQ L. /. is is a metric on M which gives the same underlying topology. 4. Let K be a compact subset of M . ere exists  D .K/ > 0 so that if P 2 K and if dg .P; Q/ < .K/, then there exists a geodesic  from P to Q of length dg .P; Q/. Furthermore, if Q is any curve from P to Q with dg .P; Q/ D L./ Q , then Q is a reparametrization of  . Proof. Adopt the notation of Assertion 1. As the radial curves from the origin are geodesics, rF @r F @r D 0. Furthermore, g.F @r ; F @r / D 1. Finally, Œ@r ; @v  D 0. us: @r g.F @r ; F @v / D g.rF @r F @r ; F @v / C g.F @r ; rF @r F @v /

106

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

D 0 C g.F @r ; rF @v F @r / D 21 @v g.F @r ; F @r / D 12 @v .1/ D 0 .

When r D 0, F @v D 0 and thus g.F @r ; F @v /jrD0 D 0. us g.F @r ; F @v / D 0, i.e., the radial curves are perpendicular to the angular curves. is proves Assertion 1. Adopt the notation of Assertion 2. Let D r.t /.t / where r.t / 2 Œ0; 1 and k.t /k D 1. Here we may assume r.t/ > 0 for t > 0; if the curve returns to the origin, we simply cut off this part of the curve. We use Assertion 1 to see: k k P 2 D g.r 0 .t/@r ; r 0 .t/@r / C r 2 g. 0 .t /;  0 .t //  .r 0 .t//2 Z Z Z 0 L. / D k k P  jr j  r 0 D kk D L./ .

so

If equality holds, there is no angular variance and r 0  0. Assertion 2 now follows. Suppose Q 2 expP .B .0//. If is any curve from P D expP .0/ to Q D expP ./ in M , then either stays entirely within expP .B .0//, in which case L. /  jj, or goes outside expP .B .0//, in which case the length of the part of which goes to the boundary is at least  which is greater than  . us if Q 2 expP .B .0//, we conclude dg .P; Q/ D kk. If Q lies outside expP .B .0//, then any curve from P to Q must first reach the boundary of B and by Assertion 2 this has length at least  . is shows dg .P; Q/  0 with equality if and only if P D Q. It is clear dg .P; Q/ D dg .Q; P / and dg .P; Q/  dg .P; R/ C dg .R; Q/. us dg is a metric on M . For any P , there exists  D .P / > 0 so that if ı <  is given, then fQ W dg .P; Q/ < ıg is diffeomorphic to Bı .0/ in TP M . is shows that the original topology on M and the topology defined by dg coincide; Assertion 3 now follows. Assertion 4 is immediate from the discussion above.  3.4.2 EXAMPLE. By Lemma 3.10, unit speed geodesics in S m can be parametrized in the form cos.t/P C sin.t/Q where P and Q are unit vectors with P ? Q. Fix P 2 S m . Let fe1 ; : : : ; emC1 g be an orthonormal basis for RmC1 where P D e1 . Let  D x 2 e2 C


C x mC1 emC1 2 TP S m :

e curve .t/ WD cos.t/P C sin.t /kk 1  is a geodesic in S m from P with initial direction kk 1  . us .t/ Q D cos.kkt /P C sin.kkt /kk 1  is a geodesic in S m from P with initial direction  . Setting t D 1 yields ( ) cos.kk/P C sin.kk/kk 1  if  ¤ 0 expP ./ D : P if  D 0 e ball of radius

 2

about the origin in TP M is the upper hemisphere in S m .

Lemma 3.13 If .x 1 ; : : : ; x m / are geodesic coordinates on M , then gij .0/ D ıij , gij=k .0/ D 0, and Rij k` .0/ D 12 f@x i @x k gj ` C @x j @x ` gi k @x i @x ` gj k @x j @x k gi` g.

3.4. GEODESICS

107

Proof. It is immediate from the definition that gij D ıij in geodesic coordinates. We take coordinate vector fields A WD ai @x i , B WD b j @x j , and C WD c k @x k where aE , bE, and cE belong to Rm . Since straight lines through the origin are geodesics, g.rA A; C /.0/ D 0

for all

A; C :

(3.4.a)

We polarize Equation (3.4.a). Let A.t/ D A C tB . If we differentiate with respect to t and set t D 0, we see g.rA B; C /.0/ C g.rB A; C /.0/ D 0 : Since r is torsion-free, g.rA B; C / D g.rB A; C / and hence g.rA B; C / D 0 for any coordinate vector fields. is implies ij k .0/ D 0. We show that the 1-jets of the metric vanish at 0 by computing: gj k= i .0/ D f ij k C i kj g.0/ D 0 : We lower an index in Equation (3.2.b) to compute: Rij k`

n p D g`n .@x i j k n C ip n j k p @x j ik n jp ik / D @x i .g`n j k n / @x j .g`n ik n / C O.kxk/ D .@x i j k` @x j ik` / C O.kxk/ D 21 f@x i @x k gj ` C @x i @x j gk` @x i @x ` gj k g C 21 f @x j @x k gi` @x j @x i gk` C @x j @x ` gi k g C O.kxk/ :

We cancel the term @x i @x j gk` to complete the proof.



3.4.3 GEODESIC CONVEXITY. We say that an open set O is geodesically convex if any two points of O can be joined by a unique length minimizing geodesic and if this geodesic is contained entirely in O. By Lemma 3.10 and Example 3.4.2, the great circles are the geodesics of the sphere S m . A hemispherical cap is a geodesic ball; such balls are geodesically convex if and only if the radius is less than 2 . More generally we have: Lemma 3.14

Let .M; g/ be a Riemannian manifold.

1. Let K be a compact subset of M . ere exists  D .K/ > 0 so that if P 2 K and for any 0 < ı <  , then the geodesic ball Bı .P / of radius ı about P is geodesically convex. 2. ere exists a coordinate atlas for M where the charts O˛ are geodesically convex open sets. In particular, if O˛1 \


\ O˛` is non-empty, then it is contractible. Such a cover is called a simple cover.

108

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

Proof. Since K is compact, we can choose a uniform Q > 0 so that if P 2 K , then geodesic coordinates .xP1 ; : : : ; xPm / which are centered at P are defined for kxk < Q on the ball B2Q.P / around P . Furthermore, given any two points Q1 and Q2 in BQ .P /, then there is a unique shortest geodesic from Q1 to Q2 lying in B2Q.P /. Again, as K is compact, we may choose C so that jı ij k j  C on B2Q.P /. Choose  > 0 so that  < Q

and  m3 C 3
0. ere exists ı > 0 so that .T ı; T /  B .P /. We may assume ı <  . Every geodesic in B .P / extends for time at least  and hence  extends to time T ; this contradiction establishes Assertion 2; it is immediate that Assertion 2 implies Assertion 3. Assume Assertion 3 holds. us all geodesics from P extend for infinite time. Let C be the set of all t 2 R so that if Q is a point of M with d.P; Q/  t , then there exists a geodesic  from P with L./ D dg .P; Q/. We use Lemma 3.12 to see C is non-empty. Since 0 < s < t and t 2 C implies s 2 C , either C D Œ0; 1/ or C D Œ0; T / for some T < 1 or C D Œ0; T  for some T < 1. We wish to rule out these latter two possibilities. Suppose C D Œ0; T / for T < 1. Suppose dg .P; Q/ D T but Q cannot be connected to P by a geodesic of length T . Let n be a sequence of curves from P to Q with L. n / ! T . Choosing appropriate points on the curves n constructs a sequence of points Qn with dg .P; Qn / < T with Qn ! Q. Express Qn D expP .n / for kn k D dg .Qn ; P / :

We then have kn k ! dg .Q; P /. Consequently, we can find a subsequence so that nk converges to some element  in TP M with kk D T . We then have expP ./ D lim expP .n / D lim Qn D Q : n!1

n!1

Since kk D dg .P; Q/, the requisite geodesic from P to Q is given by  .t / WD expP .t/ for t 2 Œ0; 1; this contradiction shows C ¤ Œ0; T /. Suppose C D Œ0; T . en the closed ball BN T .P / of radius T about P in M is the image of the closed ball BN T .0/ of radius T about 0 in TP M under the exponential map. Since BN T .0/ is compact, BN T .P / is compact. e argument of Lemma 3.12 shows we can find a uniform  > 0 so if Q is any point of BT .P / and if d.Q; R/ <  , then there is a unique shortest geodesic from Q to R of length d.Q; R/. Let dg .P; Q/ < T C  . Let i be a sequence of piecewise smooth curves so L. i / ! dg .P; Q/. By going sufficiently far out in the sequence, we may assume L. i / < T C  for all i . Let Ri be the first point on i so dg .P; Ri / D T . Decompose i into two curves; one from Q to Ri and one from Ri to P . e curve from Ri to P must have distance at least P and thus the curve from Q to Ri must have distance less than  . By replacing i by the geodesic from Q to Ri and the geodesic from Ri to P , we do not increase the length, and thus we may assume

110

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

that i is such a broken geodesic. Since the points of distance exactly T from P is a compact set, by passing to a subsequence, we may assume Ri ! R. is gives a (possibly) broken geodesic from Q to P of total length dg .P; Q/. If in fact the path is not smooth at R, we can “cut off the leg” and construct a shorter path from Q to P . Since this is not possible, we have constructed a geodesic from Q to P of length dg .Q; P /. is shows that every point of M can be joined to P by a shortest geodesic of length dg .P; Q/. Let K be a closed bounded subset of M . Since K is bounded, there exists r > 0 so that K  BN r .P /. Since Br .P /  expP .Br .0//, Br .P / is compact. Since K is closed, K is compact. us closed bounded subsets of .M; dg / are compact. It follows .M; dg / is complete. us Assertion 3 implies Assertion 1; the proof that Assertion 3 implies Assertion 1 then shows that if Assertion 2 holds, then any two points can be joined by a shortest geodesic giving the length between the two points.  It is worth making a few observations. e shortest geodesic need not be unique; any great circle in the sphere provides a shortest geodesic joining antipodal points. Furthermore, just because there exists a shortest geodesic giving the distance, the space need not be complete; any convex open subset of Rn has this property. is result can fail in the pseudo-Riemannian setting. e Lie group SL.2; R/ is connected and admits a bi-invariant Lorentzian metric. All geodesics extend for infinite time, but the exponential map is not surjective; we will show in Lemma 6.25 of Book II that there exist points in SL.2; R/ which cannot be connected by a Lorentzian geodesic.

3.5

THE JACOBI OPERATOR

If X; Y 2 TP M , let J .X/ W Y ! R.Y; X /X define the Jacobi operator. is operator is named after the German mathematician C. Jacobi.

Carl Gustav Jacob Jacobi (1804–1851) A vector field X along a geodesic is said to be a Jacobi vector field if it satisfies the Jacobi equation: XR C J ./X P D 0. One says that a smooth map T W Œ0; "  Œ0; " ! M is a geodesic spray if T is an embedding such that the curves t .s/ WD T .t; s/ are geodesics for all t . e following result (see, for example, do Carmo [9]) provides a geometric motivation for the study of Jacobi vector fields. Let T .t; s/ be a geodesic spray. en the variation T .@ t / is a Jacobi vector field along the geodesics t .s/ D T .t; s/. Lemma 3.16

3.5. THE JACOBI OPERATOR

111

Proof. Identify @ t with  @ t and @s with  @s . As the curves s ! t .s/ are geodesics, we prove the Lemma by computing: 0 D r@t r@s @s D R.@ t ; @s /@s C r@s r@ t @s D J .@s /@ t C r@s r@s @ t D J . /X P C XR :



3.5.1 CONSTANT SECTIONAL CURVATURE. Let fx; yg be a basis for a non-degenerate 2-plane   TP M . e sectional curvature ./ is defined by setting ./ WD

R.x; y; y; x/ : g.x; x/g.y; y/ g.x; y/2

(3.5.a)

Let .V; h
;
i/ be an inner product space of signature .p; q/. e pseudo-spheres are defined by setting S ˙ D S ˙ .V; h
;
i/ WD fv 2 V W hv; vi D ˙1g :

When considering S C , we assume that q > 0 so there are spacelike vectors; similarly when considering S , we assume p > 0 to ensure there are timelike vectors. Lemma 3.17

S ˙ is a symmetric space with constant sectional curvature ˙1.

Proof. Fix P 2 S ˙ . Choose an orthonormal basis fe1 ; : : : ; em g for V so that P D e1 . Let P ? WD f 2 V W hP; i D 0g D spanfe2 ; : : : ; em g

be the perpendicular space. Let x D .x 2 ; : : : ; x m / 2 Rm 1 . For x close to 0, define e.x/ WD x 2 e2 C


C x m em 2 P ?

1

and F .x/ WD .1  hx; xi/ 2 P C e.x/ :

en hF .x/; F .x/i D ˙.1  hx; xi/ C hx; xi D ˙1 so the map x ! F .x/ defines a system of local coordinates on S ˙ which are centered at P . e Levi–Civita connection of .V; h
;
i/ is flat. Since P is the normal vector to S ˙ at P , II.@x i ; @x j /.P / is the projection of @x i @x j F onto P . is shows that II.@x i ; @x j /.P / D h@x i ; @x j iP :

Consequently we have (after using eorem 3.9):

RS ˙ .U; V; X; W / D ˙fhU; W ihV; X i

hU; X ihV; W ig :

It now follows that S ˙ has constant sectional curvature ˙1. Let Q 2 TP S ˙ D P ? . By Lemma 3.10, geodesics in S ˙ from P take the form ( ) cos.t /P C sin.t/Q if hP; P i D hQ; Qi

.t/ WD : cosh.t/P C sinh.t/Q if hP; P i D hQ; Qi us the geodesic symmetry which interchanges Q and Q is induced by the isometry of V which is C1 on P
R and 1 on P ? . is shows the geodesic symmetry is a local isometry and hence S ˙ is a local symmetric space. 

112

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

We have the following examples: 1. Let .S m ; g0 / be the unit sphere of RmC1 with the standard positive definite metric. en R.X; Y; Z; W / D g.X; W /g.Y; Z/

g.X; Z/g.Y; W / :

is is a Riemannian manifold of constant sectional curvature C1. We shall see that any simply connected complete Riemannian manifold of constant sectional curvature C1 is isometric to this manifold. 2. Let .H m ; g0 / be the unit pseudo-sphere S in a Lorentzian manifold of signature .1; m/. en the induced metric g0 on H m is positive definite and R.X; Y; Z; W / D

fg0 .X; W /g0 .Y; Z/

g0 .X; Z/g0 .Y; W /g :

is has constant sectional curvature 1. We shall see presently that any simply connected complete Riemannian manifold of constant sectional curvature 1 is isometric to this manifold. 3. Let xE D .x 1 ; : : : ; x m / be the usual coordinates on Rm . e following metric on Rm (C) or on the open unit disk ( ) has constant sectional curvature ˙c : ds 2 WD 4

.dx 1 /2 C


C .dx m /2 : .1 ˙ kxk E 2 /2

Let .V; h
;
i/ be an inner product space. A Riemannian curvature model is a triple .V; h
;
i; A/ where A 2 ˝4 V  satisfies the identities of the Riemann curvature tensor given in eorem 3.7. ere are purely algebraic characterizations of constant sectional curvature. Let M WD .V; h
;
i; A/ be a Riemannian curvature model. e following conditions are equivalent and if any is satisfied, then M is said to have constant sectional curvature c. Lemma 3.18

1. A.x; y; z; w/ D cfhx; wihy; zi 2.

A.e1 ;e2 ;e2 ;e1 / he1 ;e1 ihe2 ;e2 i he1 ;e2 i2

hx; zihy; wig.

D c for any non-degenerate 2-plane  D spanfe1 ; e2 g in V .

3. If fe1 ; e2 g is an orthonormal set, then J .e1 /e2 D che1 ; e1 ie2 . Proof. It is immediate that Assertion 1 implies Assertion 2. Conversely assume Assertion 2 holds. Let fx; y; zg be an orthonormal set of vectors in V . Let " D hx; xihy; yi D ˙1 and ( ) cos.t/x C sin.t /y if " D 1 .t / WD : cosh.t /x C sinh.t /y if " D 1

3.5. THE JACOBI OPERATOR

113

We then have h.t/; .t/i D hx; xi. As M has constant sectional curvature c , chx; xihz; zi D ch.t /; .t /ihz; zi D A..t /; z; z; .t // ( cos.t/ sin.t/ D chx; xihz; zi C 2A.x; z; z; y/ cosh.t/ sinh.t/

if " D 1 if " D 1

)

:

is shows that A.x; z; z; y/ D 0. Next suppose that fx; y; z; wg is an orthonormal set. Next, polarize the identity A.x; z; z; y/ D 0 to see A.x; z; w; y/ C A.x; w; z; y/ D 0. us: 0 D A.x; y; z; w/ C A.y; z; x; w/ C A.z; x; y; w/ D A.x; y; z; w/ A.y; x; z; w/ A.x; z; y; w/ D 3A.x; y; z; w/ :

We have shown that A.x; z; z; y/ D 0 if fx; y; zg is an orthonormal set. We have also shown that A.x; y; z; w/ D 0 if fx; y; z; wg is an orthonormal set. Let fe1 ; : : : ; em g be an orthonormal basis for V . By the curvature symmetries, A.ei ; ej ; ek ; e` / D 0 unless fi; j g and fk; `g are distinct. We have A.ei ; ej ; ek ; e` / D 0 unless .i; j / D .k; `/ or .i; `/ D .j; k/. Assertion 1 now follows from Assertion 2. Let fe1 ; e2 ; e3 g be an orthonormal set of vectors in V . We have shown that A.e2 ; e1 ; e1 ; e3 / D 0 so J .e1 /e2 ? e3 . Since A.e2 ; e1 ; e1 ; e1 / D 0, J .e1 /e2 is perpendicular to e1 and e3 . Since e3 was an arbitrary unit vector perpendicular to e1 and e2 , we conclude J .e1 /e2 is some multiple  of e2 . We show  D c and show Assertion 3 holds by computing: che1 ; e1 ihe2 ; e2 i D A.e2 ; e1 ; e1 ; e2 / D hJ .e1 /e2 ; e2 i D he2 ; e2 i :

Let fe1 ; e2 g be an orthonormal set. If Assertion 3 holds, we establish Assertion 1 by checking that A.e1 ; e2 ; e2 ; e1 / D hJ .e2 /e1 ; e1 i D che1 ; e1 ihe2 ; e2 i.  e sectional curvature is a continuous function on the Grassmannian Gr02 .V / of nondegenerate 2-planes in V . In the positive definite setting, this space is compact and, consequently, the sectional curvature is bounded. is fails in the indefinite setting since Gr02 .V / is an open subset of the full Grassmannian Gr2 .V / and, in fact, the sectional curvature is bounded if and only if it is constant. Furthermore, R.x; y; y; x/ D 0 if x , y span a degenerate plane if and only if the sectional curvature is constant; see O’Neill [34] for further details. Let .M; g/ be a pseudo-Riemannian manifold. If .TP M; gP ; RP / has constant sectional curvature cP at each point P of M and if m  3, then a Schur type lemma shows that cP is constant. Such manifolds are often also-called space forms; if c > 0, the manifold is said to be a spherical space form while if c < 0, it is said to be a hyperbolic space form. If c D 0, then the manifold is flat since the curvature tensor vanishes identically. We refer to Wolf [41] for further information and content ourselves with using Lemma 3.18 to establish the following result:

114

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

Let .M; g/ and .MQ ; g/ Q be two connected pseudo-Riemannian manifolds of constant sectional curvature c and signature .p; q/. Let P and PQ be points of M and MQ . en there is a local isometry  from M to MQ with .P / D PQ . Lemma 3.19

Proof. We use expP to identify a neighborhood of 0 in TP M with a neighborhood of P in M . Let g0 D gP . Fix v 2 TP M so that g0 .v; v/ D " D ˙1. Let w 2 TP M be such that fv; wg forms an orthonormal set relative to g0 . We form the geodesic .s/ WD sv . Let e.s/ be a parallel vector field along with e.0/ D w ; we regard e as a vector-valued map from a neighborhood of 0 in R to TP M . Set 8 9 if c" > 0 = ˆ jcj 1 sin.jcjs/e.s/ > < : .v; s/ WD se.s/ if c D 0 ˆ > : ; 1 jcj sinh.jcjs/e.s/ if c" < 0 Let fe1 ; : : : ; em g be an orthonormal basis for TP M where e1 D v . We wish to show that: 8 9 if i ¤ j ˆ > < 0 = : (3.5.b) gsv .ei ; ej / D g0 .v; v/ if i D j D 1 ˆ > : 2 2 ; s  .v; s/g0 .ei ; ei / if i D j > 1

is determines the metric away from the origin and off the light cone of null vectors; the metric on the light cone can then be determined by continuity. e Lemma will then follow by choosing an isometry between TP M and TPQ MQ ; this is possible as M and MQ have the same signature. We establish Equation (3.5.b) as follows. Examine the Jacobi vector field Y WD sw on

arising from the geodesic spray T .t; s/ WD s.v C t w/. We use the Levi–Civita connection to covariantly differentiate vector fields. us, for example, R D 0 since is a geodesic and YR D J . /Y P since Y is a Jacobi vector field. We compute: @s @s g .s/ . P .s/; Y .s// D @s g .s/ . .s/; R Y .s// C @s g .s/ . P .s/; YP .s// D 0 C @s g .s/ . .s/; P YP .s// D g .s/ . R .s/; YP .s// C g. P .s/; YR .s// D0

g .s/ . .s/; P J . P /Y .s// D

R.Y .s/; P .s/; P .s/; P .s// D 0 :

Since g .s/ . .s/; P Y.s//.0/ D 0 and f@s g .s/ . .s/; P Y .s//g.0/ D g0 .v; w/ D 0, we may conclude that Y.s/ ? .s/ P for all s . By Assertion 2 of Lemma 3.18, we have: YR D

J . P /Y D c"Y

with Y .0/ D 0 and YP .0/ D w :

(3.5.c)

en Z.s/ WD .v; s/e.s/ satisfies the same ordinary differential equation given in Equation (3.5.c) that is satisfied by Y and, consequently, Z.s/ D Y.s/ D sw . Note that g .s/ .e.s/; e.s// D g.e.0/; e.0// D g0 .w; w/ :

us we have g .s/ .sw; sw/ D  2 .v; s/g0 .w; w/ if g0 .v; w/ D 0. Equation (3.5.b) now follows by polarization. 

3.5. THE JACOBI OPERATOR

115

We have actually proved a bit more. By applying the previous lemma to M itself, we have shown that .M; g/ is a local two-point homogeneous space, i.e., the local isometries of .M; g/ act transitively on the pseudo-sphere bundles S ˙ .M; g/ WD fX 2 TM W g.X; X / D ˙1g :

Let .V; h
;
i/ be an inner product space of signature .p; q/. Let c > 0. e pseudo-sphere S ˙ .V; h
;
i; c/ WD f 2 V W h; i D ˙c 2 g discussed previously provide local models for spaces of constant sectional curvature ˙c in all possible signatures. 3.5.2 THE RICCI TENSOR. Let r be the Levi–Civita connection of a Riemannian manifold .M; g/. Define the Ricci tensor by setting: .X; Y / WD TrfZ ! R.Z; X /Y g :

e curvature symmetries yield .X; Y / D .Y; X /. e following result relates the geometry of M to the topology of M through the Ricci tensor: eorem 3.20 (Myers’ eorem [33]) Let .M; g/ be a complete connected Riemannian manifold of dimension m  2.

1. Let .t/ be a unit speed geodesic in M from P to Q for 0  t  b . Assume that we have the inequality .; P / P  .m 1/.=L/. If b > L, then dg .P; Q/ < b . 2. Suppose .X; Y /  .m 1/g.X; Y / for some  > 0. en M is compact, the diameter of M is at most p , and the fundamental group of M is finite. Proof. Let .t/ be a unit speed geodesic for 0  t  b so dg .P; Q/ D b . Assume . b /2 < L i.e., b is larger than predicted by the Lemma so the diameter is larger than expected (or possibly infinite). We argue for a contradiction. Let Y .t / be a perpendicular vector field along  with Y .0/ D 0 and Y.b/ D 0. Consider the variation T .s; t / D exp .t / .sY .t //. For simplicity of notation, we identify @ t with T @ t and @s with T @s since no confusion is likely to ensue. Let Z b 1 LY .s/ D g.T .@ t /; T .@ t // 2 dt t D0

be the length of the curves s W t ! T .s; t /. ese curves all go from P to Q since Y.0/ D 0 and Y.b/ D 0. We compute the first variation of arc length: Z b Z b 1 1 @s LY .s/ D @s g.@ t ; @ t / 2 dt D g.r@s @ t ; @ t /g.@ t ; @ t / 2 dt : t D0

tD0

Since Y ? P and since  is a geodesic when s D 0: g.r@s @ t ; @ t /jsD0 D g.r@ t @s ; @ t /jsD0 D f@ t g.@ t ; @s /

g.r@ t @ t ; @s /gjsD0 D 0 :

(3.5.d)

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3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

us, the two relations above show that @s LY .s/jsD0 D 0 :

Next, we study the second variation of arc length. We compute: Z

@2s LY .s/ D C C

b

g.r@s @ t ; @ t /2 g.@ t ; @ t /

t D0 Z b t D0 Z b t D0

3=2

g.r@s r@s @ t ; @ t /g.@ t ; @ t / g.r@s @ t ; r@s @ t /g.@ t ; @ t /

dt 1 2

1 2

dt dt :

By Equation (3.5.d), g.r@s @ t ; @ t /jsD0 D 0. Since the curves s ! T .s; t / are geodesics, @2s LY .s/jsD0

D D D

Z

Z

b t D0 b

t D0 Z b t D0

˚ g.r@s r@t @s ; @ t / C g.r@ t @s ; r@t @s / dt fR.@s ; @ t ; @s ; @ t / fg.YP ; YP /

g.r@ t r@s @s ; @ t / C g.YP ; YP /gdt

R.Y; P ; ; P Y /gdt :

Let fe1 ; : : : ; em 1 g be parallel perpendicular vector fields along  . Consider the vector fields Yi .t/ D sin. t=b/ei .t/. We compute m X1 iD1

@2s LYi .s/jsD0

D 
0g the metric ds 2 D y 2 .dx 2 C dy 2 /. Let z D i wC1 ; we 1 w z i . We solve the equation z.1 w/ D iw C i for z to see the inverse map is given by w D zCi compute: dz D 2i.1 w/ 2 dw; d zN D 2i.1 w/ N 2 d w; N dz ı d zN D 4j1 wj 4 dw ı d w; N   wC1 yD= i D j1 wj 2 0, jwj > 1 $ y < 0, and the circle jwj D 1 goes to R [ 1. Furthermore the Jacobian is non-singular for w ¤ 1. us this provides an isometry of the interior of the unit disk with the metric 4.1 jwj2 / 2 dw ı d wN to the upper half plane with the metric y 2 .dz ı d z/ N . e 1 2 isothermal parameter for the upper half plane model is given by setting h D 2 ln.y / D ln.y/ so h=11 D 0; h=22 D @y .y 1 / D y 2 ; K D e 2h .h=11 C h=22 / D 1 : e geodesics in this geometry are straight lines perpendicular to the x -axis and circles perpendicular to the x -axis. e special linear group SL.2; R/ is the group of 2  2 real matrices of determinant 1. Set: ! a b az C b TA z WD for A D 2 SL.2; R/ : cz C d c d e map z ! Az is an isometry of this geometry and every orientation preserving isometry arises in this fashion. e full group of isometries is then generated by adding the isometry z ! zN . 3.6.9 THE TRACTRIX MODEL FOR HYPERBOLIC SPACE. ere is a partial realization called tractrix of the hyperbolic plane in R3 with the Euclidean inner product as a surface of revolution that is singular along the line u D 0 (Beltrami 1868). It is given by

3.6. THE GAUSS–BONNET THEOREM

123

.x D sech.u/ cos.v/; y D sech.u/ sin.v/; z D u  tanh.u//

We have guu D tanh2 .u/, guv D 0, gvv D sech2 .u/ while IIuu D  sech.u/ tanh.u/, IIuv D 0, and IIvv D sech.u/ tanh.u/. us K D  1. 3.6.10 THE LORENTZ SPHERE. Let hx; yi D x 1 y 1 C x 2 y 2  x 3 y 3 be the Lorentzian inner product on R3 . We obtained a model for the hyperbolic plane by considering the pseudo-sphere hx; xi D  1; this had signature .0; 2/. We now consider the Lorentz sphere S D fx W hx; xi D 1g; this has signature .1; 1/ and is a hyperboloid of one sheet which may be parametrized by setting T .u; v/ WD .cosh.u/ cos.v/; cosh.u/ sin.v/; sinh.u//. We compute: @u T D .sinh.u/ cos.v/; sinh.u/ sin.v/; cosh.u//; @v T D . cosh.u/ sin.v/; cosh.u/ cos.v/; 0/;  D .cosh.u/ cos.v/; cosh.u/ sin.v/; sinh.u//; guu D  1; guv D 0; gvv D cosh2 u; IIuu D 1; IIuv D 0 :

Consequently, K D det.II/= det.g/ D C1.

e Lorentz sphere

3.6.11 THE GEODESIC CURVATURE. Let ˛.t / be a curve parametrized by arc-length so k˛k P D 1. Let  .t/ be a unit normal to the curve. e geodesic curvature is defined to be  g .˛/ WD g.rP ˛; P / :

Since ˛ has constant length, 0 D @ t g.˛; P ˛/ P D 2g.rP ˛; P ˛/ P and thus ˛R is perpendicular to ˛P . is implies ˛R D  g .˛/   and thus ˛ is a geodesic if and only if  g .˛/ D 0. Let ˛.t / D .r cos.t=r/; r sin.t =r// parametrize the circle of radius r about the origin. en ˛.t/ P D . sin.t=r/; cos.t=r// is a unit vector so ˛ is parametrized by arc length. e (inward) unit

124

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

normal to the circle is given by .t / D . cos.t=r/; sin.t=r// and the geodetic curvature is given by g D .t/ R
.t / D 1r . cos.t =r/; sin.t=r//
. cos.t =r/; sin.t=r// D 1r : If is an arbitrary curve in the plane, and if Cr is the best circle approximating , then jg j D ˙ 1r where the ˙ sign reflects the choice of normal. 3.6.12 ANGLE SUM FORMULAS. A unit speed curve in the plane is a geodesic if and only if

R D 0, i.e., is a straight line. If T is a geodesic triangle with angles i at the vertices, then one has from Euclid that: 1 C 2 C 3 D  : (3.6.b)

Let S 2 be the unit sphere in R3 with the induced metric. Let P be a plane through the origin and let D P \ S 2 be a great circle in S 2 . Reflection in P defines an isometry of R3 whose fixed point set is P ; Lemma 3.10 then shows is a geodesic. us the great circles comprise the geodesics of S 2 . In 1603 omas Harriot showed that if T is a triangle in the unit sphere with angles f1 ; 2 ; 3 g, then Area.T / D 1 C 2 C 3  : (3.6.c) His result is unpublished; the first published result seems to be due to Albert Girard. In other words, the sum of the angles of a spherical triangle exceeds  and the excess measures the area. In hyperbolic geometry (i.e., on a surface of constant sectional curvature 1), the sign in Equation (3.6.c) is reversed. If T is a triangle whose edges are geodesics in the hyperbolic plane, then Lampert [30] showed in 1766 that Area.T / D 

1

2

3 :

(3.6.d)

3.6.13 THE GAUSS–BONNET THEOREM. Equations (3.6.b), (3.6.c), and (3.6.d) are consequences of the Gauss–Bonnet eorem which is named after Carl Friedrich Gauss (who first investigated the Gaussian curvature K ) and Pierre Ossian Bonnet (who published a special case of the Gauss–Bonnet eorem in 1848); the general case seems to be due to von Dyck (1888) who discussed the global version: eorem 3.23 Let .M; g/ be a 2-dimensional Riemannian manifold with piecewise smooth boundary @M . Let K D R1221 be the Gaussian curvature, let g be the geodesic curvature, and let ˛i be the interior angles at which consecutive smooth boundary components meet. en Z Z X 2
.M / D KC g C . ˛i / . M

@M

i

is result is due to the German mathematician Johann Carl Friedrich Gauss (1777–1855), to the French mathematician Pierre Ossian Bonnet (1819–1892), and to Walther Franz Anton von Dyck (1856–1934).

3.6. THE GAUSS–BONNET THEOREM

Gauss

125

Bonnet

If we specialize ˝ to a triangle in the plane, then K D 0 and g D 0 and we have Z

˝

0 dA C

3 X . iD1

i / D 2;

or equivalently, 1 C 2 C 3

 D0

and we recover the result of Euclid given in Equation (3.6.b). If we specialize ˝ to a spherical triangle with angles i , then K D C1 and g D 0 and we have Z

˝

1 dA C

3 X . iD1

i / D 2;

or equivalently,

Area.˝/ D 1 C 2 C 3

;

so we recover the result of Harriott given in Equation (3.6.c). On the other hand if ˝ is a hyperbolic triangle, then K D 1, then we may establish Equation (3.6.d): Z

˝

3 X . 1/ dA C . iD1

i / D 2;

or equivalently,

Area.˝/ D 1 C 2 C 3

:

Proof. We first examine the local theory and take M D R2 . We have global coordinates .x; y/ on M which we use to fix the orientation. Let ˝ be a subdomain of M with piecewise smooth boundary. Let fe1 ; e2 g be an orthonormal frame. Expand re1 D !e2

and re2 D !e1

for

! D !x dx C !y dy :

We take the usual orientation dx ^ dy for R2 and assume e 1 ^ e 2 is the orientation as well. 1 Recall that g D det.gij / 2 so the Riemannian measure is gdxdy . e Gaussian curvature is then characterized by the equation Kj dvol j D g 1 g..rx ry ry rx /e2 ; e1 /
gdx ^ dy D . @x !y C @y !x /dx ^ dy D d! :

Assume that @˝ is smooth. Use arc length to parametrize the boundary, oriented counterclockwise, by .t/. We may express P and the normal  in the form:

.t/ P D

sin..t//e1 C cos..t //e2

and .t / D

cos..t //e1

sin..t //e2 :

126

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

(is is modeled on taking the circle .t / D .cos.t /; sin.t// and using the inward unit normal in flat space). We compute

us

Z @˝

r P P D f cos..t //e1 sin..t //e2 gP .t / C f cos..t //e1 sin..t //e2 g!. /; P g D g.r P ; P / D P C !. / P : Z Z P C g dt D dt ! so we may apply Stokes’ eorem (which holds even if the @˝

@˝

boundary has corners) to see: Z Z Kj dvol j C g ds D ˝

Z

@˝

˝

d! C

Z

!C

@˝

Z @˝

P D dt

Z @˝

P D 2 : dt

Now if the boundary has corners, then the angular parameter  has jumps at the corners. If Pi is such a corner and if the exterior angle is ˇi D  ˛i , we have Z X P C dt . ˛i / D 2 @˝

i

is a full turn. is gives rise to the desired formula Z Z X Kj dvol j C g ds C . ˝

@˝

i

˛i / D 2 ;

since .˝/ D 1. e fact that ˝ was geodesically convex plays a role only in seeing that the total turn is 2 ; it works equally well if ˝ is compact and if @˝ has only one component. If ˝ has holes, then the outer component gives rise to 2 and each inner component gives rise to 2 . Since the Euler characteristic is 1 minus the number of inner boundary components, the desired result follows in this special case. e general case now follows by a cutting and pasting argument; we omit the details in the interests of brevity. 

3.7

THE CHERN–GAUSS–BONNET THEOREM

eorem 3.23 has been generalized to the higher-dimensional setting by Chern [10]. For the moment, we shall assume that .M; g/ is a smooth compact Riemannian manifold without boundary; we shall discuss the boundary correction term presently. Let m be even and let fe1 ; : : : ; em g be a frame for .M; g/. Let 2` be even and set Em WD

1 R i i j j




R i` .8/` `Š 1 2 2 1 1

1 i` j` j` 1

g.e i1 ^


^ e i` ; e j1 ^


^ e j` / :

(3.7.a)

Set j dvol j D det.g˛;ij / 2 dx 1




dx m . We refer to Chern [10] for the proof of the following result. We also refer to contemporaneous related work by Allendoerfer and Weil [2].

3.7. THE CHERN–GAUSS–BONNET THEOREM

127

S. Chern (1911–2004) Many proofs have been given subsequently of this result. eorem 3.24 Let .M; g/ be a compact Riemannian manifold without boundary of dimension m. If m is odd, the Euler–Poincaré characteristic .M Z / vanishes. If m is even,

.M / D

Em j dvol j .

M

3.7.1 THE CHERN–GAUSS–BONNET THEOREM IN INDEFINITE SIGNATURE. Chern [11] and Avez [5] subsequently extended eorem 3.24 to the indefinite setting. In the pseudo-Riemannian setting, the volume element is given by 1

j dvol j D j det.gij /j 2 dx 1 ^


^ dx m :

is is a measure and not a differential form; no assumption of orientability is made.

Let .M; g/ be a compact pseudo-Riemannian manifold of signature .p; q/ without boundary of even dimension m. If p is odd, then .MZ / vanishes. If p is even, then eorem 3.25

.M / D . 1/p=2

M

Em j dvol j .

Proof. We follow the discussion in Gilkey and Park [18]. Let g 2 S 2 .T  M / ˝ C be a complexvalued symmetric bilinear form on the tangent space. We assume det.g/ ¤ 0 as a non-degeneracy condition. e Levi–Civita connection and curvature tensor may then be defined. We set 1 dvol.g/ WD det.gij / 2 dx 1 ^


^ dx m ; we do not take the absolute value. ere is a subtlety here since, of course, there are two branches of the square root function. We shall ignore this for the moment in the interests of simplifying the argument and return Rto this point in a moment. We use Equation (3.7.a) to define the Euler form and consider M Em .g/ dvol.g/. Let h 2 S 2 .T  M / ˝ C give a perturbation of the metric. Consider the 1-parameter family g WD g C h and differentiate with respect to the parameter  . We then integrate by parts to define the Euler–Lagrange equations: Z ˇ Z ˇ D g..TEm /.g/; h/ dvol.g/ @ Em .g C h/ dvol.g C h/ ˇˇ M

D0

M

128

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

where TEm .g/ 2 S 2 .T  M / ˝ C is the transgression of the Pfaffian Em . is is defined in any dimension. Although a priori TEm can involve the 4th derivatives of the metric, Berger [7] conjectured it only involved curvature; this was subsequently verified by Kuz’mina [26] and Labbi [27– 29]. We also refer to related work in Gilkey, Park and Sekigawa [19]. What is important for us, however, is that eorem 3.24 shows that TEm .g/ D 0Rif dim.M / D m and if g is Riemannian, i.e., if g and h are real since the integrated invariant M Em .g C h/ dvol.g C h/ D .M / is independent of .; h/. e invariant TEm is locally computable. In any coordinate system, it is 1 polynomial in the derivatives of the metric, the components of the metric, and det.gij / 2 . us it is holomorphic in the metric. us vanishing identically when g is real implies it vanishes identically when g is complex. Consequently, the Euler–Lagrange equations vanish. In particular, if g is a smooth 1-parameter family of complex metrics starting with a real metric, then we can define a branch of the square root function along this family so that the following integral is independent of  : Z Em .g / dvol.g / : M

We apply Lemma 3.2 to find an orthogonal direct sum decomposition TM D V ˚ VC where V is timelike with dim.V / D p and where VC is spacelike with dim.VC / D q . is decomposes g D g ˚ gC where g is negative definite on V and gC is positive definite on VC . For  2 Œ0; , define: p g D e  1 g ˚ gC : en g0 D g ˚ gC is a Riemannian metric while g D g ˚ gC is the given pseudoRiemannian metric. We have det.g / D e 

p

1p

det. g / det.gC / D e 

p

1p

det.g0 /

sopthe branch of the square root function along the deformation from g0 to g is given by e  1p=2 . us if p is odd, this is purely imaginary for g and hence the Euler characteristic 1 1 vanishes. If p is even, then det.g0 / D det.g / so . 1/p=2 det.g0 / 2 D . 1/p=2 det.g / 2 . is completes the proof. u t 3.7.2 EXAMPLES. Let .M; g0 / be a Riemann surface. Let g D g0 have signature .2; 0/. en g

r D g0 r

so

g

Rij k ` D g0 Rij k `

and

.g/ D g j k g Rij k i D

g0j k

g0

Rij k i D

.g0 / :

As j dvol j.g0 / D j dvol j.g/, one must change the sign in the Gauss–Bonnet eorem: Z 1 .M / D .g/j dvol j.g/ : 4 M In dimension four, if .M; g/ D .M1 ; g1 /  .M2 ; g2 /, then the Gauss–Bonnet eorem decouples and we have .M / D .M1 /.M2 / and E4 .g/ D E2 .g1 /E2 .g2 /. us we will not need to change

3.7. THE CHERN–GAUSS–BONNET THEOREM

129

the sign in signature .4; 0/ or .0; 4/ but we will need to change the sign in signature .2; 2/. e fact that the Euler characteristic vanishes if p and q are both odd is not, of course, new but follows from standard characteristic class theory. It is possible to generalize the Gauss–Bonnet eorem to manifolds with boundary; in the Riemannian setting, this is due to Chern [10] and in the pseudo-Riemannian setting to Alty [3]. We introduce a bit of additional notation. Let .M; g/ be a pseudo-Riemannian manifold of signature .p; q/ with smooth boundary @M . We assume the restriction of g to the boundary is non-degenerate and let  be the inward pointing unit normal. Let f; e2 ; : : : ; em g be a local orthonormal frame field. Let Lab WD g.rea eb ; / be the second fundamental form. We sum over indices ai and bi ranging from 2 to m to define:   Ra1 a2 b2 b1




Ra2 1 a2 b2 b2 1 La2C1 b2C1




Lam 1 bm 1 Fm; WD .8/ Š Vol.S m 1 2 /.m 1 2/Š g.e a1 ^


^ e am 1 ; e b1 ^


^ e bm 1 / : Note that if m is odd, then .M / D 12 .@M / so we may apply eorem 3.25 to compute .@M / and thereby .M / in terms of curvature. We therefore assume m is even. eorem 3.26 Let .M; g/ be a compact smooth manifold pseudo-Riemannian manifold of even dimension m and signature .p; q/ which has smooth boundary @M . If p is odd, then .M / D 0. Other-

wise

p=2

.M / D . 1/

(Z M

Em .g/j dvol j.gM / C

XZ 

@M

)

Fm; j dvol j.g@M / .

Proof. If .M; g/ is Riemannian, this is well-known (see, for example, the discussion in Gilkey [17] using heat equation methods and extending the work of Patodi [35] to the case of manifolds with boundary). So the trick is to extend it to the pseudo-Riemannian setting. Again, we will use analytic continuation. But there is an important difference. We fix a non-zero vector field X which is inward pointing on the boundary. We consider a smooth 1-parameter of complex variations g so that g .X; X/ ¤ 0 and so that X ? T .@M / with respect to g . We may then set 1  D X
g .X; X/ 2 . In the expression for Fm; , there are an odd number of terms which contain 1 L and hence g .X; X/ 2 . Given a pseudo-Riemannian metric g , we construct g0 as above. Let @M @M gab and g0;ab be the restriction of the metrics g and g0 , respectively, to the boundary. If X is spacelike, then @M @M / / D . 1/p det.g0;ab det.gab

and the analysis proceeds as previously; the whole subtlety arising from the square root of the determinant. If X is timelike, then @M / D . 1/p det.gab

1

@M /: det.g0;ab

130

3. RIEMANNIAN AND PSEUDO-RIEMANNIAN GEOMETRY

However g.X; X/ D g0 .X; X / and thus once again, we must take the square root of . 1/p in the analytic continuation. Apart from this, the remainder of the argument is the same as that used to prove eorem 3.25. u t

131

Bibliography [1] J. F. Adams, “Vector fields on spheres,” Bull. Amer. Math. Soc. 68 (1962), 39–41. DOI: 10.1090/S0002-9904-1962-10693-4. [2] C. Allendoerfer and A. Weil, “e Gauss-Bonnet theorem for Riemannian polyhedra,” Trans. Am. Math. Soc. 53 (1943), 101–129. DOI: 10.1090/S0002-9947-1943-0007627-9. 126 [3] L. J. Alty, “e generalized Gauss-Bonnet-Chern theorem,” J. Math. Phys. 36 (1995), 3094–3105. DOI: 10.1063/1.531015. 129 [4] M. F. Atiyah, “K-theory. Lecture notes by D. W. Anderson,” W. A. Benjamin, Inc., New York-Amsterdam (1967). 59 [5] A. Avez, “Formule de Gauss-Bonnet-Chern en métrique de signature quelconque,” C. R. Acad. Sci. Paris 255 (1962), 2049–2051. 127 [6] J. Baum, “Elements of point set topology,” Prentice Hall, Englewood Cliffs, N. J. (1964). 1 [7] M. Berger, “Quelques formulas de variation pour une structure riemanniene,” Ann. Sci. Éc. Norm. Supér. 3 (1970), 285–294. 128 [8] A. L. Besse, “Einstein manifolds,” Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 10. Springer-Verlag, Berlin (1987). 95 [9] M. P. do Carmo, “Geometria riemanniana,” Projeto Euclides 10. Instituto de Matematica Pura e Aplicada, Rio de Janeiro (1979). 110 [10] S. Chern, “A simple proof of the Gauss-Bonnet formula for closed Riemannian manifolds,” Ann. Math. 45 (1944), 747–752. DOI: 10.2307/1969302. 126, 129 [11] S. Chern, “Pseudo-Riemannian geometry and the Gauss-Bonnet formula,” An. Acad. Brasil. Ci. 35 (1963), 17–26. 127 [12] A. Clairaut, “éorie de la figure de la terre, tirée des principes de l’hydrostatique,” Paris (1743). 36 [13] G. Cramer, “Introduction à l’Analyse des lignes Courbes algébriques,” Genève: Frères Cramer & Cl. Philbert (1750) (see pages 656–659). 6

132

BIBLIOGRAPHY

[14] S. Dickson, “Klein Bottle Graphic” (1991) 57 http://library.wolfram.com/infocenter/MathSource/4560/. [15] L. Eisenhart, “Riemannian Geometry,” Princeton University Press, Princeton, N.J. (1949). [16] G. Frobenius, “Über das Pfaffsche probleme,” J. für Reine und Agnew. Math. 82 (1877), 230–315. 65 [17] P. Gilkey, “Invariance theory, the heat equation, and the Atiyah-Singer index theorem,” Studies in Advanced Mathematics, CRC Press, Boca Raton (1995). 129 [18] P. Gilkey and J.H. Park, “Analytic continuation, the Chern-Gauss-Bonnet theorem, and the Euler-Lagrange equations in Lovelock theory for indefinite signature metrics”, Journal of Geometry and Physics (2015), pp. 88–93. DOI: 10.1016/j.geomphys.2014.11.006. 127 [19] P. Gilkey, J. H. Park, and K. Sekigawa, “Universal curvature identities,” Differ. Geom. Appl. 29 (2011), 770–778. DOI: 10.1016/j.difgeo.2011.08.005. 128 [20] A. Gray, “e volume of a small geodesic ball of a Riemannian manifold,” Michigan Math. J. 20 (1974), 329–344. DOI: 10.1307/mmj/1029001150. [21] H. Hopf and W. Rinow, “Über den Begriff der vollständigen differentialgeometrischen Fläche,” CommentarII Mathematici Helvetici 3 (1931), 209–225. DOI: 10.1007/BF01601813. 108 [22] J. Jost, “Riemannian geometry and geometric analysis,” Universitext, Springer-Verlag, Berlin (2002). DOI: 10.1007/978-3-662-04745-3. 67 [23] M. Karoubi, “K-theory, an introduction,” Grundlehren der mathematischen Wissenschaften 226, Springer Verlag (Berlin) (1978). 59 [24] S. Kobayashi and K. Nomizu, “Foundations of Differential Geometry vol. I and II,” Wiley Classics Library. A Wyley-Interscience Publication, John Wiley & Sons, Inc., New York (1996). 95 [25] O. Kowalski and M. Sekizawa, “Natural lifts in Riemannian geometry,” Variations, geometry and physics, 189–207, Nova Sci. Publ., New York (2009). 69 [26] G. M. Kuz’mina, “Some generalizations of the Riemann spaces of Einstein,” Math. Notes 16 (1974), 961–963; translation from Mat. Zametki 16 (1974), 619–622. DOI: 10.1007/BF01104264. 128 [27] M.-L. Labbi, “Double forms, curvature structures and the .p; q/-curvatures,” Trans. Am. Math. Soc. 357 (2005), 3971–3992. DOI: 10.1090/S0002-9947-05-04001-8. 128

BIBLIOGRAPHY

133

[28] M.-L. Labbi, “On Gauss-Bonnet Curvatures,” SIGMA, Symmetry Integrability Geom. Methods Appl. 3, Paper 118, 11 p., electronic only (2007). DOI: 10.3842/SIGMA.2007.118. [29] M.-L. Labbi, “Variational properties of the Gauss-Bonnet curvatures,” Calc. Var. Partial Differ. Equ. 32 (2008), 175–189. DOI: 10.1007/s00526-007-0135-4. 128 [30] J. Lambert, “eorie der parallillinien,” Leipzig (1766). 124 [31] T. Levi-Civita, “Nozione di parallelismo in una varietà qualunque e consequente specificazione geometrica della curvatura Riemanniana,” Rend. Circ. Mat. Palermo 42 (1917), 73–205. DOI: 10.1007/BF03014898. 100 [32] J. Munkres, “Elementary differential topology,” Lectures given at Massachusetts Institute of Technology, Fall, 1961. Annals of Mathematics Studies, No. 54 Princeton University Press, Princeton, N.J. (1963). 58 [33] S. B. Myers, “Riemannian manifolds with positive mean curvature,” Duke Mathematical Journal 8 (1941), 401–404; DOI: 10.1215/S0012-7094-41-00832-3. 115 [34] B. O’Neill, “Semi-Riemannian geometry. With applications to relativity,” Pure and Applied Mathematics 103, Academic Press, Inc., New York (1983). 113 [35] V. K. Patodi, “Curvature and the eigenforms of the Laplace operator,” J. Differential Geometry 5 (1971), 233–249. 129 [36] W. Rudin, “Principles of mathematical analysis,” McGraw-Hill Book Co., New YorkAuckland-Düsseldorf (1976). 1, 23 [37] E. Spanier, “Algebraic topology,” Berlin: Springer-Verlag (1995). 55, 86 [38] M. Spivak, “Calculus on manifolds. A modern approach to classical theorems of advanced calculus,” W. A. Benjamin, Inc., New York-Amsterdam (1965). 7, 12 [39] R. Stong, “Notes on cobordism theory”. Mathematical notes Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1968). 78 [40] H. Whitney, “Differentiable manifolds,” Annals of Math. 37 (1936), 645–680. DOI: 10.2307/1968482. 58 [41] J. Wolf, “Spaces of constant curvature,” American Mathematical Society, Providence, RI (2011). 55, 113 [42] K. Yano and S. Ishihara, “Tangent and cotangent bundles,” Pure and Applied Mathematics 16, Marcel Dekker Inc., New York (1973). 67, 68

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Authors’ Biographies PETER B GILKEY Peter B Gilkey¹ is a Professor of Mathematics and a member of the Institute of eoretical Science at the University of Oregon. He is a fellow of the American Mathematical Society and is a member of the editorial board of Results in Mathematics, J. Differential Geometry and Applications, and J. Geometric Analysis. He received his Ph.D. in 1972 from Harvard University under the direction of of L. Nirenberg. His research specialties are Differential Geometry, Elliptic Partial Differential Equations, and Algebraic topology. He has published more than 250 research articles and books.

JEONGHYEONG PARK JeongHyeong Park² is a Professor of Mathematics at Sungkyunkan University and is an associate member of the KIAS (Korea). She received her Ph.D. in 1990 from Kanazawa University in Japan under the direction of H. Kitahara. Her research specialties are spectral geometry of Riemannian submersion and geometric structures on manifolds like eta-Einstein manifolds and H-contact manifolds. She organized the geometry section of AMC 2013 (e Asian Mathematical Conference 2013) and the ICM 2014 satellite conference on Geometric analysis. She has published more than 71 research articles and books.

¹Mathematics Department, University of Oregon, Eugene OR 97403 U.S. email: [email protected] ²Mathematics Department, Sungkyunkwan University, Suwon, 440-746, Korea email: [email protected]

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AUTHORS’ BIOGRAPHIES

RAMÓN VÁZQUEZ-LORENZO Ramón Vázquez-Lorenzo³ is a member of the research group in Riemannian Geometry at the Department of Geometry and Topology of the University of Santiago de Compostela (Spain). He is a member of the Spanish Research Network on Relativity and Gravitation. He received his Ph.D. in 1997 from the University of Santiago de Compostela under the direction of E. GarcíaRío. His research focuses mainly on Differential Geometry with special emphasis on the study of the curvature and the algebraic properties of curvature operators in the Lorentzian and in the higher signature settings. He has published more than 50 research articles and books.

³Department of Geometry and Topology, Faculty of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain. email: [email protected]

137

Index absolutely integrable, 44 admissible change of coordinates, 17 affine connection, 98 affine manifold, 99 affine subspace, 54 antipodal, 55 base point preserving, 86 Bianchi identity, 98 bounded, 2 Brauer Fixed Point eorem, 89 bundle isomorphism, 59 bundle map, 59 category, 72 Cauchy–Schwarz–Bunyakovsky inequality, 4, 14 center of the cube, 46 chain rule, 11 Christoffel identity, 101 Christoffel symbols, 96, 98, 99, 101 closed, 1, 2 closed ball, 2 closure, 2 cluster point, 2 cocycle condition, 56, 61 collar, 78 compact, 2 compact exhaustion, 38 complete affine manifold, 99 complete lift, 68

conjugation operator, 3 content zero, 28 continuous, 1, 2 continuous almost everywhere, 41 contractible, 93 contravariant functor, 72 coordinate atlas, 53 coordinate chart, 53 coordinate vector fields, 107 cotangent bundle, 67 Cramer’s rule, 6, 16 cube, 45 curl, 80 curvature model, 112 curvature operator, 97 cylinder, 118 de Rham cohomology, 84 dependent variable, 19 derivation, 62 diffeomorphic, 54 diffeomorphism, 16 diffeomorphisms, 53 directional covariant derivative, 95 distance function, 2 div, 80 divergence, 80 double, 78 dual basis, 3, 95 dual bundle, 3, 96 dual Christoffel symbols, 96

138

INDEX

dual connection, 95, 96 dual frame, 96

Grassmannian, 113 Green’s eorem, 80

Einstein convention, 3, 61 Ekaterina Puffini, xiii embedding, 57 equivalent norms, 4 Euclidean spherical geometry, 119 Euclidean vector space, 3 Euler–Poincaré characteristic, 127 evaluation map, 68 exterior algebra, 70 exterior derivative, 67

holonomy, 96 holonomy group, 96 homeomorphism, 2, 16 homogeneous, 115 homotopic, 86 homotopy group, 86 hyperbolic space form, 113

fiber, 56, 59, 66 fiber bundle, 56 fiber metric, 60, 92, 96 fiber preserving, 56 flat manifold, 113 Fundamental eorem of Algebra, 88 Gamma function, 93 Gauss–Bonnet eorem, 124 general linear group, 7, 56 Generalized Stokes’ eorem, 83 geodesic, 99, 103, 110, 114 geodesic coordinates, 105 geodesic curvature, 123 geodesic distance, 105 geodesic spray, 110, 114 geodesically complete, 108 geodesically convex, 107 geometrically realizable, 101 gluing functions, 56 grad, 80 graded skew-commutative, 72 gradient, 80 Gram–Schmidt process, 4, 60 graph, 29

immersion, 57 Implicit Function eorem, 22 implicitly differentiate, 19 independent variable, 19 index notation, 12 inner product space, 91 integrable, 25, 65 Inverse Function eorem, 15 isometry, 4, 103 isothermal, 117 Jacobi equation, 110 Jacobi identity, 63 Jacobi operator, 110 Jacobi vector field, 110, 114 Jacobian matrix, 8, 61 Jordan measurable, 32 Klein bottle, 57 Krill Institute of Technology, xiii Kronecker symbol, 4 Leibnitz formula, 95, 96 lens space, 55 Levi–Civita connection, 100, 114 Lie bracket, 63 Lie group, 56 light cone, 114 limit point, 2

INDEX

line bundle, 59 Lobachevskian geometry, 120 local frame, 59 local trivialization, 59 locally bounded, 41 locally Euclidean, 53 locally finite, 38 Lorentz sphere, 123 Lorentzian metric, 92 lower half space, 77 lower integral, 25 lower sum, 25 manifold, 53 matrix of cofactors, 6 measure zero, 28 mesa function, 38 metric space, 2 module homomorphism, 61 Myers’ eorem, 115 natural projection, 68 non-degenerate submanifold, 102 normed vector space, 3 null, 91 open, 1, 2 open ball, 2 operator norm, 13 orientable, 75 oriented, 75 orthogonal bundle, 92 orthogonal group, 56 orthonormal basis, 4, 91 oscillation, 30 partial derivative, 8 partition, 24 partition of unity, 38 polar coordinates, 17

polarization identity, 4 projected connection, 102 proper embedding, 57 pseudo-Riemannian measure, 94 pseudo-Riemannian metric, 92 pseudo-sphere, 55, 104, 111, 115 pullback, 54, 61 pullback map, 67 pushforward, 63 real projective space, 55 rectangle, 23 refinement, 25 regular, 54 regular covering projection, 55 Ricci tensor, 115 Riemann surface, 117 Riemannian algebraic curvature tensor, 101 Riemannian connection, 96 Riemannian metric, 92 scalar curl, 80 second fundamental form, 102 section, 59 sectional curvature, 111 semigroup property, 63 signature, 91 simple cover, 107 smooth, 8, 16, 53, 54 smooth k -forms, 72 smooth structure, 53 space form, 113 spacelike, 91 special linear group, 122 spherical space form, 113 stereographic projection, 119 submanifold, 54 submersion, 66 subordinate to the cover, 38

139

140

INDEX

support, 38 symmetric tensor product, 93 tangent bundle, 61 tangent space, 61 tensor, 61 tensorial, 61 timelike, 91 topology, 1 torsion tensor, 98 torsion-free, 98 torus, 58 tractrix, 122 transgression, 128

transition functions, 53, 56 trivial bundle, 59 trivialization, 59 unit disk bundle, 60 unit sphere, 119 unit sphere bundle, 60 unital rings, 72 upper integral, 25 upper sum, 25 vector bundle, 59 volume, 24 winding number, 87