darling - differential forms

DIFFERENTIALFORMS ANDCONNECTIONS

R.W. R. DARLING

This book introduces the tools of modern differential geometry - exterior calculus, manifolds, vectorbundles, connections - to advanced undergraduates and beginning graduate students in mathematics,physics, and engineering. It covers both classical surface theory and the modern theory of connections andcurvature, and includes a chapter on applications to theoretical physics. The only prerequisites aremultivariate calculus and linear algebra; no knowledge of topology is assumed.

The powerful and concise calculus of differential forms is used throughout. Through the use of numerousconcrete examples, the author develops computational skills in the familiar Euclidean context beforeexposing the reader to the more abstract setting of manifolds. There are nearly 200 exercises, making thebook ideal for both classroom use and self-study.

Differential Forms and Connections

Differential Forms and Connections

R.W.R. Darling

University ofSouth Florida

CAMBRIDGEUNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGEThe Pitt Building. Trumpington Street, Cambridge, United Kingdom

CAMBRIDGE UNIVERSITY PRESSThe Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk40 West 20th Street, New York, NY 10011-4211, USA http://www.cup org10 Stamford Road, Oakleigh, Melbourne 3166, AustraliaRuiz de Alarc6n 13, 28014 Madrid, Spain

® Cambridge University Press 1994

This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 1994Reprinted 1995, 1996, 1999

Printed in the United States of America

A catalog record for this book is available from the British Library

Library of Congress Cataloging in Publication Data is available

ISBN 0 521 46259 2 hardbackISBN 0 521 46800 0 paperback

Contents

Preface ix

1 Exterior Algebra 1

1.1 Exterior Powers of a Vector Space 1

1.2 Multilinear Alternating Maps and Exterior Products 5

1.1 Exercises 7

1.4 Exterior Powers of a Linear Transformation

1.5 Exercises 2

1.6 Inner Products L1.7 The Hodge Star Operator 17

1.8 Exercises 2B

1.9 Some Formal Algebraic Constructions 21

1.10 History and Bibliography 23

2 Exterior Calculus on Euclidean Space 24

2.1 Tangent Spaces - the Euclidean Case 24

2.2 Differential Forms on a Euclidean Space 28

2.3 Operations on Differential Forms 31

2.4 Exercises 33

2.5 Exterior Derivative 35

2.6 Exercises 32

2.7 The Differential of a Map 41

2.8 The Pullback of a Differential Form 43

vi Contents

2.9 Exercises 47


2.11 Appendix: Maxwell's Equations 50

3 Submanifolds of Euclidean Spaces 533.1 Immersions and Submersions 53

3.2 Definition and Examples of Submanifolds 55

3.3 Exercises 60

3.4 Parametrizations 61

3.5 Using the Implicit Function Theorem to Parametrize a Submanifold 64

3.6 Matrix Groups as Submanifolds 69

3.7 Groups of Complex Matrices 71

3.8 Exercises 72

3.9 Bibliography 75

4 Surface Theory Using Moving Frames 76

4.1 Moving Orthonormal Frames on Euclidean Space 76

4.2 The Structure Equations 78

4 3 Fxercispc 79-4.4 An Adapted Moving Orthonormal Frame on a Surface 81

4.5 The Area Form m46 Exercises 82-4.7 Girvanire of n Surface _84.8 Explicit Calculation of Curvatures 91

49 Exercises 244.10 The Fundamental Forms: Exercises 25


5 Differential Manifolds5.1 Definition of a Differential Manifold

5.2 Basic Topological Vocabulary 100

5.3 Differentiable Mappings between Manifolds 102

5.4 Exercises 104

5.5 Submanifolds 105

5.6 Embeddings

.

107

Contents vii

5.7 Constructing Submanifolds without Using Charts 110

5.8 Submanifolds-with-Boundary 111

5.9 Exercises 114

5.10 Appendix: Open Sets of a Submanifold 116

5.11 Appendix: Partitions of Unity 117


6 Vector Bundles 120

6.1 Local Vector Bundles 120

6.2 Constructions with Local Vector Bundles 122

6.3 General Vector Bundles 115

6.4 Constructing a Vector Bundle from Transition Functions 130

6.5 Exercises 1312

6.6 The Tangent Bundle of a Manifold 134

6.7 Exercises 139


6.9 Appendix: Constructing Vector Bundles 141

7 Frame Fields, Forms, and Metrics 1447.1 Frame Fields for Vector Bundles 144

7.2 Tangent Vectors as Equivalence Classes of Curves 147

7.3 Exterior Calculus on Manifolds 148

7.4 Exercises 151

7.5 Indefinite Riemannian Metrics 152

7.6 Examples of Riemannian Manifolds 153

7.7 Orthonormal Frame Fields 156

7.8 An Isomorphism between the Tangent and Cotangent Bundles 160

7.9 Exercises 161


8 Integration on Oriented Manifolds 1648.1 Volume Forms and Orientation 164

8.2 Criterion for Orientability in Terms of an Atlas 167

8.3 Orientation of Boundaries 169

8.4 Exercises 172

viii Contents

8.5 Integration of an n-Form over a Single Chart 174

8.6 Global Integration of n-Forms 178

8.7 The Canonical Volume Form for a Metric 181

8.8 Stokes's Theorem 183

8.9 The Exterior Derivative Stands Revealed 184

8.10 Exercises 187


8.12 Appendix: Proof of Stokes's Theorem 189

9 Connections on Vector Bundles 194

9.1 Koszul Connections 194

9.2 Connections via Vector-Bundle-valued Forms 197

9.3 Curvature of a Connection 202

9.4 Exercises 206

9.5 Torsion-free Connections 212

9.6 Metric Connections 216

9.7 Exercises 219


10 Applications to Gauge Field Theory 223

10.1 The Role of Connections in Field Theory 223

10.2 Geometric Formulation of Gauge Field Theory 225

10.3 Special Unitary Groups and Quaternions 231

10.4 Quaternion Line Bundles 233

10.5 Exercises 238

10.6 The Yang-Mills Equations 242

10.7 Self-duality 244

10.8 Instantons 247

10.9 Exercises 249


Bibliography 251

Index 253

Preface

PurposeThis book represents an extended version of my lecture notes for a one-semester courseon differential geometry, aimed at students without knowledge of topology. Indeed theonly prerequisites are a solid grasp of multivariate calculus and of linear algebra. Thegoal is to train advanced undergraduates and beginning graduate students in exteriorcalculus (including integration), covariant differentiation (including curvaturecalculations), and the identification and uses of submanifolds and vector bundles. It ishoped that this will serve both the minority who proceed to study advanced texts indifferential geometry, and the majority who specialize in other subjects, includingphysics and engineering.

Summary of the Contents

Every generation since Newton has seen a richer and deeper presentation of thedifferential and integral calculus. The nineteenth century gave us vector calculus andtensor analysis, and the twentieth century has produced, among other things, theexterior calculus and the theory of connections on vector bundles. As the title implies,this book is based on the premise that differential forms provide a concise and efficientapproach to many constructions in geometry and in calculus on manifolds.

Chapter 1 is algebraic; Chapters 2, 4, 8, and 9 are mostly about differential forms;Chapters 4, 9, and 10 are about connections; and Chapters 3, 5, 6, and 7 are aboutunderlying structures such as manifolds and vector bundles. The reader is not mistakenif he detects a strong influence of Harley Flanders's delightful 1989 text. I would alsolike to acknowledge that I have made heavy use of ideas from Berger and Gostiaux[ 1988], and (in Chapters 6 and 9) of my handwritten Warwick University 1981 lecturenotes from John Rawnsley, as well as other standard differential geometry texts.Chapter 9 on connections is in the spirit of S. S. Chern [ 1989], p. ii, who remarks that"the notion of a connection in a vector bundle will soon find its way into a class on

x Preface

advanced calculus, as it is a fundamental notion and its applications are widespread";these applications include the field theories of physics (see Chapter 10), the study ofinformation loss in parametric statistics, and computer algorithms for recognizingsurface deformation. Regrettably the Frobenius Theorem and its applications, and deRham cohomology, are among many other topics which could not be included; seeFlanders [ 19891 for an excellent treatment of the former, and Berger and Gostiaux[ 1988] for the latter.

Prerequisites

Linear Algebra: finite-dimensional vector spaces and linear transformations,including the notions of image, kernel, rank, inner product, and determinant.

Vector Calculus: derivative as a linear mapping; grad, div, and curl; line, surface,and volume integrals, including Green's Theorem and Stokes's Theorem; implicitfunction theorem; and the concept of an open set in Euclidean space.

Advice to the Instructor

In the diagram below, a solid arrow denotes dependency of chapters, and a fuzzy arrowdenotes a conceptual relationship. In one semester, an instructor would probably be hardpressed to cover more than six chapters in depth. Chapters 1 and 2 are essential. Someinstructors may choose to emphasize the easier and more concrete material in Chapters3 and 4, which is used in the sequel only as a source of examples, while others mayprefer to move rapidly into Chapters 5 and 6 so as to have time for Chapter 8 onintegration and/or Chapter 9 on connections. Alternatively one could deemphasizeabstract differential manifolds (i.e., skip over Chapter 5), cover only the "local vectorbundle" part of Chapter 6, and treat Chapters 7 to 10 in a similarly "local" fashion. Asalways, students cannot expect to master the material without doing the exercises.

1No

4

i 10

Acknowledgments and Comments

I wish to thank my Differential Geometry class of Spring 1992 for their patience, andalso Suzanne Joseph, Professor Ernest Thieleker, Greg Schreiber, and an anonymousreferee for their criticisms. The courteous guidance of editor Lauren Cowles of theCambridge University Press is gratefully acknowledged. The design is based on atemplate from Frame Technology's program FrameMaker®. Lists of errors andsuggestions for improvement will be gratefully received at [email protected].

1 Exterior Algebra

Anyone who has studied linear algebra and vector calculus may have wondered whetherthe notion of cross product of vectors in 3-dimensional space generalizes to higherdimensions. Exterior algebra, which is a prerequisite for the study of differential forms,shows that the answer is yes. We shall adopt a constructive approach to exterior algebra,following closely the presentation given in Flanders [ 19891, and we will try toemphasize the connection with the vector algebra notions of cross product and tripleproduct (see Table 1.2 on page 19).

1.1 Exterior Powers of a Vector Space

1.1.1 The Second Exterior Power

Let V be an n-dimensional vector space over R. Elements of V will be denoted u, V. w,u., vi, etc., and real numbers will be denoted a, b, c, a., b,, etc. For p = 0, 1, ..., n, thepth exterior power of V, denoted APV, is a real vector space, whose elements arereferred to as `p-vectors." For p = 0, 1 the definition is straightforward: A°V = R,and A I V =, V, respectively. A2 V, consists of formal sums t

Ia; (u; A v), (1.1)i

where the "wedge product" U A V satisfies the following four rules:

(au+w) Av = a(uAV) +wAV; (1.2)

1 A rigorous construction of the second exterior power is given in Section 1.9.

2 Chapter 1 Exterior Algebra

UA (bv+w) =b(uAV)+uAw; (1.3)

uAU = 0; (1.4)

a basisforV {v'AV':15i<j5n}is abasis for A2V. (1.5)

Postponing to the end of this chapter the question of whether a vector space with theseproperties exists, let us note two immediate consequences of (1. 2), (1. 3), and (1. 4).Apply (1. 4) to (u + v) A (u + v), and then express the latter as the sum of four termsusing (1. 2) and (1. 3); two of these terms, namely, U A u and V A v, are zero, and whatremains shows that U A V +vAu = 0; hence

VAU = -UAV. (1.6)

Second (1.2), (1. 3), and (1.4) by themselves imply that, for any basis { v t, ..., V"} of V.the set of vectors { v' A vi: 1 5 i <j5 n) spans A2 V, because it spans the set of"generators" { u A w, u and w e V}; to check this, we express u and w in terms of thebasis {v(, ..., v"}, and apply (1. 2), (1.3), and (1.6) to obtain:

UAW = (Ia,v') A (Ibjd) = I:aibj(v'AvV)

_ (a,bj-ajb;) (v'Av1).i<j

The linear independence of { V' A v1: 15 i <j:5 n} cannot, however, be deduced from(1. 2), (1. 3), and (1.4), and is studied in Section 1.9.

1.1.2 Higher Exterior Powers

The description of APV for any 2:5 p 5 n follows the same lines; APV is the set offormal sums2

Ia.(UY(1) A... AUY(P)) (1.7)Y

of "generators" UY (1) A ... AU 7 (p) , where each coefficient aY is indexed by a

multi-index y = (y ( 1 ) . y (p) ); elements of APV are called `p-vectors," and aresubject to the rules (1. 8), (1. 9), and (1. 10):

(av+w) AU2A... AuP = a(VAU2A... AuP) +WAU2A... AUp, (1.8)

2 A rigorous construction is given in Section 1.9.

1.1 Exterior Powers of a Vector Space 3

and similarly if any of the u; is replaced by such a linear combination;

u1=u forsome i*j=* u1A...AUp=0; (1.9)

and for any basis { v), ..., v"} of V, the following set of p-vectors forms a basis for APV:

i(l) iA...AV(p)1Si(1)<...<i(p)Snj(v (1.10)

The expression u 1 A ... A U,_ 1 A (v + w) A u,+, A ... A (v + w) A ... Aug. which iszero by (L 9), can be expanded using (1. 8) into four terms, two of which are zero; whatremains shows that

U1 A ... A up changes sign if any two entries are transposed. (1.11)

Also it follows from (1. 8) and (1. 9) that, for any basis { v1, ..., v"} of V the set ofvectors (1. 10) spans APV; in order to demonstrate this, we shall need the language ofpermutations.

1.1.3 Permutations

Let Ep denote the set of permutations of the set 11, 2, ..., p}. For example, E, can bewritten as{e, (1,2), (3, 1), (2,3), (1,3,2), (1,2,3) }, wheren = (3, 1) meansfor example that n (1) = 3, it (3) = 1. A transposition is an element n of EP thatswitches i and j for some i * j, but leaves k fixed for all k e { i, j}; thus in the list for E,above, the second, third, and fourth elements are transpositions. A result in algebrastates that any permutation can be expressed as a composition of transpositions, and thatthe number m of transpositions is unique modulo 2; we define the signature sgn(n) ofthe permutation It by

sgn (n) = (-1)'. (1.12)

It is also true, in the case of the composition n n' of two permutations, thatsgn (n it') = sgn (n) sgn (n'). It follows from (1. 11) that

up[ (1) A ... A UR(p) = sgn (n) (u, A ... A un). (1. 13)

Now we will show how to express an arbitrary generator of APV as a linearcombination of the set of vectors (1. 10). We may write

U I A ... A Up = b1.j(1)v1(1)) A... A (I bp. I (p)

Vj (p)

1(1) 1(p)

Cj(V1(1) A... Ato(p)LL!


where J = (j(1), ...J (p)(p) ).and c. = bl,1(1)...bp.J(p). For any I, there is a uniquemulti-index I = (i (1) , ..., i (p) ) such that i (1) < ... < i (p), and a unique n e EPsuch that J = n (1), meaning that ( j ( 1 )1 ) ,. . . ,j(_ (n (i (1)) - - n (i (P))Hence by (1. 13), we deduce

V(I) A ... AV (P) = sgn (n) ( 1 , x ( 1 ) A ... A V'(p))I

and therefore

ul A ...nun = (1: sgn (n) cx(o) (v(1) A ... A V'tp>), (1.14)E

where the first summation is over multi-indices I such that i (1) < ... < i (p), and thesecond summation is over EP. This completes the proof that the vectors (1. 10) spanAPV.

1.1.4 Calculating the Dimension of an Exterior Power

dim(APV)=111

' 05p5n. (1.15)(n_p)lpl,

Proof: For any basis { v1, ..., v"} of V the set ofp-vectors

{v'(1) A...AV'(P), 15i(1) <...<i(p) <_n}. (1.16)

forms a basis for AP V, by (1. 10). The number of elements of this set is the number ofways of choosing p objects from n distinct objects, which is the expression shown. tx

Let us illustrate these ideas by writing down bases for the exterior powers of R3.

P Basis for APV Dimension

0 (1) 1

I {e1, e2, e3} 3

2 {el Ae2,el Ae1,e2Ae3} 3

3 {e1Ae2Ae3} 1

Table 1.1 Exterior powers of Euclidean 3-space

1.2 Multilinear Alternating Maps and Exterior Products 5

1.2 Multilinear Alternating Maps and Exterior Products

For any set V, the set-theoretic product V x ... x V (p copies) simply means the set ofordered p-tuples (u1, ..., up) where each u1 E V. If V and W are vector spaces, amapping h: V X ... x V -4 W is called:

Multilinear if h (au + bu', u2, ..., up) = ah (u, u ..., up) + bh (u', u2, ..., up),and similarly for the other (p - 1) entries of h; h is called bilinear if p is 2;

Antisymmetric (or alternating) if

h (u. ('" ..., u,r(p)) = sgn (n) h (u1, ..., up), 1t E Ep, (1.17)

which implies h (u 1, ..., up) = 0 if u; = u,, some i *j; for when u, = u1, some i *j,transposing the ith and jth entries shows that h (u1, ..., up) is the same as its negative.

The student will have encountered the following examples of multilinear alternatingmaps in linear algebra or vector calculus courses:

(u,v)- uxv,R3xR3-4 R3;

(u,v) -vv2

,R2xR2-4 R;

(u,v,w) -iu (vxw),R3xR3xR3-4 R.

'Ihe linear maps from V to W will be denoted L (V -+ W), and the multilinearalternating maps will be denoted Ap (V -i W). The following property of exteriorpowers will play a central role in the remainder of this chapter.

1.2.1 Universal Alternating Mapping Property

To every g E Ap (V -+ W), there corresponds a unique g E L (APV -4 W) such that

g(u1A...AUp) =g(u1,...,up),b'u1,...,up;

in other words, a unique g such that the following diagram commutes.]

3 A diagram is said to commute if following any sequence of arrows from one set to anotheryields the same mapping.


Vx...xV (u1,...,llp)-+UTA...AUPADV

W

Proof: Deferred to Section 1.9.

1.2.2 Exterior Products

There exists a unique bilinear map (LA) --* X A g from APV x A' V to AP+qV, whoseeffect on generators is that

(u, A... AUP) A (w, A... Awq) = U, A... AUpAw, A... Awq. (1.19)

To see that this is true, apply 1.2.1 twice: first to the multilinear, alternating map

(U1,...,UP) -4 UTA...AUPAW,A...AWq,

for fixed w, A ... A wq, so as to obtain a unique f e L (APV - AP+qV) such that

f(u,A...AUP) = U,A...AUPAWIA...AWq,

so that we may define

XA(WIA...Awq)

and second to the multilinear, alternating map

(W1,...,wq) -+ XA (w,A...Awq),

for fixed X, so as to obtain g). e L (AqV -+ AP+qV) such that

ga (wl A ...Awq) = X A (w, A ...Awq). (1.20)

Finally the exterior product of X a APV and t e AqV is defined by X A µ = gX (µ).The properties of the exterior product, the first two of which are immediate from thepreceding construction, are:

06,g) -a X A p. is distributive over addition and scalar multiplication;

associativity: (X A µ) A V = X A (g A v);

µ A X = (-1) pq (X A µ), so two vectors of odd degrees anticommute; otherwisethe vectors commute.

1.3 Exercises 7

The last property follows from Exercise 4 below, in the case where A., .t are generators,and in general from linearity. In order to obtain a practical grasp of exterior products, tryExercises 5 and 6 below.

1.2.3 Example

Suppose V is 4-dimensional with a basis { v', v2, v3, v4 }. Then

(a(v3AV4)+b(v)AV3)) A (c(VIAV2)+d(V1AV4)) = ac (V3AV4AV' Av2)

_ (-1) 2(2)ac(v' A V2 A V3 A v4).

1.3 Exercises

1. (a) Repeat Table 1.1 for the case of R4, using the basis { e,, e2, e1, e4}.

(b) Let u = ae, + ce3, v = bee + de4; express u A V in terms of your basis of A2R4.

(c) Let w = a'e, + b'e2 ; express u A V A w in terms of your basis of A3R4.

(d) Express U A V A W A e3 in terms of your basis of A4R4.

2. Verify that, when V = R3, the cross product (u.v) -+ u x v, R3 x R3 -+ R3, and thetriple product (u, v, w) - u (v x w), R3 x R3 x R3 -+ R, are multilinear, alternatingmaps.4

Reminder: The cross product of u = (a,, a2, a3) and v = (b), b2, b3) is

2 a3

b2 b3

and the triple product satisfies

e)-a a3bi b3

e2 + ai a2

,b b2e3, (1.21)

U. (vxw) = V. (wxu) = w. (uxv) = -(w (vxu)). (1.22)

3. Decompose the permutation (6, 4, 3, 2, 1, 5) a E6 into a product of transpositions intwo different ways, and show that the number of transpositions used is the same modulo2 in both cases.

4. Prove, by induction or otherwise, that a permutation which sends (1, 2, ..., p + q) into(q + 1, ..., q + p, 1, 2, ..., q) has signature

4 By the end of this chapter, the reader will realize that, in terms of the star operator discussed inSection 1.7 below, uxv = * (u A v), and u. (vxw) = * (u A v A w).


sgn (7t) = (-1)pq, (1.23)

Hint: A possible inductive hypothesis Hk is that whenever p 2 1. q 2 1, p +q 5 k. then theassertion above holds. To prove Hk + t from Hk, start by transposing q and q + p, and thenrearrange (q, 1, ..., q - 1) so that Hk can be applied to the first p + q - 1 entries.

5. Let V = R3, with any basis { vt, v2, v3}; show that

(a (v2 A v3) +b (v3 A vt) + C (VI A v2)) A (avt + bv2 +cv3)

_ (aa+bb+ce)v'AV2AV3. (1.24)

6. Suppose V is 4-dimensional with a basis { vt, v2, v3, v4}. Express the following asmultiples of Vt A V2 A v3 A v4:

(i) (a(vt Av3) +b(v2AV4)) A (c(vt Av3) +d(v2AV4));(ii) (avt+bv4) A (c(VIA V2Av3)+d (v2Av3Av4)).

7. The setting is the same as for Exercise 6. Suppose µ e A3 V, µ * 0. Characterize thevectors u e V such that u A 0, and show that the vector space consisting of such uis of dimension 3.

Hint: Write u = u t v t + ... + u4v4, and express µ similarly in terms of the four basis elements ofthe third exterior power. Obtain a linear relation on the coefficients of u.

8. This is a generalization of Exercise 7. Suppose V is n-dimensional, and µ is an arbitrarynonzero element of A" - t V. Prove that the subspace W of elements u of V such thatu A µ = 0 is of dimension n - 1, and deduce from this that there exist vectors

wt,...,wn-t in Vsuch that .t = wtA...Aw"-t.

Hint: For the last part, take a basis for Wt`, extend it to a basis for V, and express µ in terms of thecorresponding basis of A" - t V. Warning: This kind of representation does not generally hold forelements of the other exterior powers.

1.4 Exterior Powers of a Linear Transformation

1.4.1 Determinants

Given A e L (V -4 V), define gA: V" -t A" V a R by

8A (ut, ..., u") = (Aut) A ... A (Aug). (1.25)

It follows immediately from the last equation that gA is multilinear and antisymmetric,and so, by 1.2.1, there is a unique fA e L (A"V -. A"V) such that

1.4 Exterior Powers of a Linear Transformation 9

fA(u1 A ... Aun) = (Au,) A ... A (Aun). (1.26)

Since A" V is one-dimensional and f is linear. it follows that f is simply multiplication bya scalar, which we denote by Al,Ithe determinant of A. In other words,

Al I(u( A ... A u") = (Au)) A ... A (Au"). (1.27)

It is somewhat surprising to discover that this abstract formulation refers to the samenotion of determinant that the student has encountered in matrix algebra:

1.4.2 Formula for the Determinant of a Matrix

Suppose that, in terms of a basis { v), ..., v"} for V, A has the matrix representationA = (a,) (skis" (a11 may also be written a1 i). Then taking

ui _ a)iv'

gives, as in (1. 14),

u(A... A u" _ v'(()) A ... Af (() /(n)

a".,(,,) (v'()) A ... A v'(")),J

where J = (j (l) , ..., j (n)) . Any J with two entries the same makes no contribution tothe sum, by (1. 9). In all other cases there is a unique it E E" such that(j(1),...,j(n)) = (7t(1),...,7t(n)).Hence by(1. 13),wededuce

VV(') A ... A V ( R ) = sgn (7t) (v(A ... A V"),

III A...AU,, = ( Y sgn(7)ai.x(()...an.,(n))v)A...At". (1.28)R E I

Thus the formula for the determinant of the matrix is

Al I= sgn (7t)a,.,,(,)...a".RE E

(1.29)

For example, when n = 2,


= a11a22-a12a21 = sg(71)al.t[(1)a2.x(2)X G F.2

1.4.3 Other Exterior Powers of a Linear Transformation

A generalization of the notion of determinant is that of exterior powers of a lineartransformation A e L (V -a W). The map VP -* APW given by

(u1, ..., up) -4 (Au1) A ... A (Aup)

is multilinear and alternating, and so by 1.2.1 it defines an element of L (APV -> APW)denoted APA, called the exteriorpth power of A; in other words, APA is specified byits action on generators as follows:

APA (u1 n ... nap) = (Au1) A ... A (AuP). (1.30)

The matrix representation of APA may be obtained as follows. If {v1, ..., v"} is a basisfor V, and {w1, ..., w' I for W, then {a'} and {TK} are bases for APV and APW,respectively, where

01 =v'(1)A...nv'(P), 15i(1)<...<i(p)Sn; (1.31)

K=Wk(1) A...AWk(P), 15k(1) <...<k(p) 5m. (1.32)

If Av' = I:akwk, thenk

(APA)& = (A v'(')) A... A (Av'(p))

1(1)...1(p)

where J runs through the set of all multi-indices. As usual, summands wherej (r) = j (s) for some r * s are zero, and we express the other summands as in the stepspreceding (1. 14): there is a unique K = (k (1) , ..., k (p) ) such thatk (l) < ... < k (p), and a unique n e EP such that J = n (K), meaning that(j(l),...,j(p)) = (n(k(l)), ..., n(k(p))). Since

W/(I) A ... A gy(p) = sgn (n) (wk(1) n ... A Wk(p)),

we obtain:

5 This idea is needed in calculations related to the pullback of differential forms in Chapter 2, andis also relevant to Stokes's Theorem in Chapter 8.

1.4 Exterior Powers of a Linear Transformation 11

(APA) sgn (n) ai a,(P) 1 (wk(()R

(k A ... A Wk(P)) (1.33)(1))... R(k(P))JJIIK RE fp

aK CK.

K

and so APA is represented by the matrix (aK) of all the p x p minors of A, where

aK = Sgn (n)i(') r(P)

Q(kO))...Q(k(P))'

Rely(1.34)

An opportunity to evaluate this matrix when m = n = 3, p = 2, is provided inExercise 10 below. This construction generalizes the notion of determinant because,when V = W and p = n, then A"A has the effect of multiplication by JAI.

1.4.4 The lsomorphlsm AP (V`) _ (AP V)

Recall that the dual space V` = L (V -> R) of the n-dimensional vector space V isanother n-dimensional vector space, consisting of the linear mappings from V to R,which are called linear forms. It is often helpful, though not necessary, to conceptualizeelements of Vas n-dimensional column vectors, and elements of V` as n-dimensionalrow vectors which act on the column vectors by usual matrix multiplication.

Given linear forms ty), ..., ytP e V" , where p:5 it, the isomorphism (constructedbelow) will show that Y, A ... A tyP a AP (V") acts linearly on AP V as follows:

(W,A...Aw,) (u,A...AUp) = E sgn(n)W)(u,(,))...ww(UK(P)). (1.35)Re Ep

1.4.4.1 ExamplesWhen p = 2 and when p = 3, respectively,

((PAW) (uAV) = (P(u)W(v) -(P(v)W(u); (1.36)

(W) A W2 A W3) ' (Ut+ u2, u3) = I (W; (u) ) I . (1.37)

1.4.4.2 Constructing the IsomorphismGiven linear forms W,, ..., y,, E where p:5 n, consider the mappingA E L (V -+ RP) given by

Au = W,(u)e,+...+Wp(u)eP, (1.38)

where { e,, ..., e,,} is the standard basis for RP. Referring to (1. 30), we see that therange of APA is the one-dimensional space APRP spanned by e, A ... A eP ; thereforethere exists a unique linear form, temporarily denoted


W10...OVP E (APV)*, (1.39)

given by the equation

APA (a.) _ (W10 ... 0 WP) (X) (e1 A ... Aep), . E APV. (1.40)

The reader may verify using (1. 33) (see Exercise 11 below) that

(W10 ... O WP) (U1 A ... A UP) = E sgn (n) W1(ux(1)) ...WP (ux(p)), (1.41)XE 1:1

and also that the map W1 A ... A WP W10 ... 0 WP is linear and one-to-one (seeExercise 12). Since the dimension of AP (V`) is the same as that of (APV) * , thisestablishes an isomorphism from AP (Vs) to L (APV -4 R). >x

In subsequent chapters, we shall drop the 0 notation, and identify W1 0 ... 0 WP withW1 A ... A WP. Thus equation (1. 35) replaces (1. 41).

1.5 Exercises

9. (a) Show using (1.27) that if A, B E L (V -4 V), then IABI = JAI I BI .

(b) Show using (1. 30) that if B E L (V -+ W) and A E L (W - Y), then

AP (AB) = AP (A) AP (B).

10. Suppose V = R3, and A E L (R3 -+ R) is expressible in terms of the usual basis{ e1, e2, e3} as the matrix

cosrP sintp 0-sisn cost 0

0 0 1

for some real number cp. Express A2A as a 3 x 3 matrix with respect to the basis{ e2 A e3, e3 A e1, e1 A e2}.

11. Verify the formula (1. 41), using (1.40).

12. Show that the map W1 A ... A WP -+ W, 0 ... 0 WP, appearing in (1.40), is linear andone-to-one.

Hint: To show the map is one-to-one, note that by (1. 41), W, 0 ... 0 WP is zero if and only if{ W1, ..., WP} is linearly independent; now appeal to 0 . 10).

1.6 Inner Products 13

13. Show that, for general finite-dimensional vector spaces V and W, the spacesAPL (V - W) and L (APV -> W) are not necessarily isomorphic.

14. Show that the exterior powers of a linear transformation A E L (V -> W) satisfy

(AP+yA) (X A µ) = APA (A.) A AMA (µ) (1.42)

for any p + q:9 n, X E APV, µ E Ag V, by applying (1. 30) to generators, and using theassociativity of the exterior product.

15. For p, q 2 1, let EPq

denote the set of permutations n of (1, 2, ..., p + q) such thatn(1) <...<it(p),n(p+1) < ... < n (p + q) (think of splitting the top p cards from adeck of p + q cards, and shuffling them in the usual way into the bottom q cards -there are (p + q) ! / (p! q!) such permutations). Notice that associativity of the exteriorproduct implies that the image of the exterior product of

((PIA...A(PP) E AP (V )and (WIA...AWq) E Ag(V )

under the isomorphism 1.4.4 must satisfy

((PIA...A(PP) A (VIA...AWq) - (910...0(PP) 0 (yr10...0Wq)

_ T10...0(PPOW10...0Wq

(see (1. 39) for the notation). Prove that 0 extends to a map

0:L(APV-,R) xL(AgV-,R)

(h 0 l) (U1 A ... A UP+q) (1.43)

sgn(n)h(u,(1) A... AuR(P))l(uR(P+1) A... AuR(P+q)).R E 1:1-q

Flint: Use Exercise 14.

1.6 Inner Products

1.6.1 Definition of an Inner Product

An inner product on a vector space V is a map V X V -a R, denoted (.1.), with:

Bilinearity: u -+ (ulv) is linear for every v, and v -> (ulv) is linear for every u;

Symmetry: (ulv) = (vlu);

Nondegeneracy: If z satisfies (zlu) = 0, Vu, then z = 0.


Note that this definition is a little more general than the one often given in linear algebracourses, since it is not assumed that (ulu) 2 0.

1.6.1.1 Characterization of Nondegeneracyif { v', .... v"} is any basis for V, the nondegeneracy condition is equivalent to:

(v'Iv') ... (v'Iv

(v"Iv') ... (v"Iv")

;e 0. (1.44)

Proof: To check that this condition is sufficient, take any z which satisfies(zlu) = 0, `du. Let us expand z in terms of the basis as z = alv' + ... +a"v". Takinginner products with each v' in turn gives the system of linear equations:

lai(vrI v') = 0, i = 1, ..., n. (1.45)i

Condition (1. 44) implies that the matrix ((v'IO) is invertible, and hence the onlysolution to (1. 45) is for all the ai to be zero, showing that z = 0. Proof of the converseis left as an exercise.

1.6.2 Examples

The dot product in R".

((a1,...,a")I(bi,...,b")) = (a1,...,a") (b1,...,b") = a1b1+...+a"b".

The Lorentz inner product in R4: if c denotes the speed of light,

((a1,...,a4)I(bl,...,b4)) = a,b, +a2b2+a363-c2a4b4. (1.46)

1.6.3 Orthonormal Bases and Their Signatures

It follows from the axioms that every inner product space contains an element v suchthat

(vlv) = t 1; (1.41)

for if (zlz) = 0 for all z e V, then

2(ulw) = (u + wlu + w) - (ulu) - (wlw) = 0

for every u and w, which contradicts nondegeneracy; so take some z with a = (zlz) # 0,and let v = I aI -''2z . A basis { v', ..., v"} for V is called an orthonormal basis if

1.6 Inner Products 15

(v'Iv') = O,i# j ; (v'Iv')=±l,i = 1,...,n. (1.48)

An induction argument, suggested in Exercise 23 below, shows that every inner productspace has an orthonormal basis. Moreover if there are r plus signs and s = n - r minussigns in (1. 48), t = r - s is called the signature of the inner product space; this doesnot depend on the choice of orthonormal basis (see Exercise 24).

A useful property of inner product spaces is the following.

1.6.4 Linear Forms on an Inner Product Space

Every f e L (V -- R) is of the form f I u) for some u e V.

Proof: Take u = f (v') v' + ... +f (v") v", using the orthonormal basis in (1. 48); thenfor any w = a,V1+...+a"v",

f (w) = ja/(v') = I (a) (vMu) = (wlu).i i

tx

1.6.5 Inner Products on Exterior Powers

Suppose V has an inner product (.1.). Then there exists a bilinear mapping (.1.)p fromAP V x AP V to R, characterized by the formula

(u1 A... AUpIv1 A... Avp)P ..2

(u1Iv1) ... (u1Ivp)

upiv,) ... (uplvp)

(1.49)

To see that this is so, note that the determinant on the right is multilinear and alternatingin (u1, ..., up) and in (v1, ..., vp), respectively, and use 1.2.1 twice as in theconstruction of the exterior product in Section 1.2.2. Clearly (.I.)p is symmetric,because transposing the matrix in (1. 49) does not change the value of its determinant.

1.6.5.1 An Orthonormal Basis for an Exterior Power(.1.)p is an inner product on APV. If { v', ..., v"} is an orthonormal basis for V 1 is anascending multi-index (i.e., 1 5 i (1) < ... < i (p) <- n), and

a' = vi (1) A ... A V' (P), (1.50)

then { a'}, as I ranges over ascending multi-indices, is an orthonormal basis of APV.

Proof: To show that (.1.)p is an inner product on AP V, it only remains to show that it isnondegenerate. We know from (1. 10) that the {a'} form a basis for APV, where now{ v1, ..., v' j is an orthonormal basis for V Nondegeneracy follows from (1.44) once we


show that the determinant of the matrix ((6'1611)), as I and H run through ascending setsof multi-indices, is nonzero. Now if I* H, then some entry i (q) in I does not belong tothe set { h (1) , ..., h (p) }. It follows that the qth row of the matrix

(VM)IVh0)) ... (V 0)Ivh(P))

(Vi (0100)) ... (v'(P)Ivh(P))

is zero, hence its determinant is zero. Thus

(6'1611) = ±61.11,6 (1.51)

and so (4. )P is nondegenerate as desired. This also demonstrates that the { 6'} form anorthonormal basis for APV. ]a

1.6.5.2 Example in Dimension 3V = R3 with the Euclidean inner product, and the standard orthonormal basis{e1, e2, e3}. Then {e, A e2, e, A e3, e2 A e3} is an orthonormal basis of A2R3, and{ e i A e2 A e3} is an orthonormal basis of A3R3.

1.6.5.3 Example in Dimension 4V = R4 with the Lorentz inner product, and the standard orthonormal basis{ e e2, e3, e4}, taking c = I for convenience. Then

{et Ae2,el Ae3,e, Ae4,e2Ae3,e2Ae4,e3Ae4}

is an orthonormal basis of A2R4, with signature zero. To see that three of the basiselements give negative inner products with themselves, note that, for example,

(e1 A e41e, A e4) =

1.6.5.4 Example in n DimensionsFor a general n-dimensional inner product space V. 1.6.5.1 shows that the n-vector

6 = V1 A ... A V"

is by itself a basis for A"V, and

(616)" = (V'Ivt)...(v"Iv") = (1)'(-1)s = (-I) ("-:)/2

(1.52)

(1.53)

6 The "Kronecker delta" notation means that S' 11 = I if 1 = H. and = 0 otherwise.

1.7 The Hodge Star Operator 17

1.7 The Hodge Star Operator

Let V be an n-dimensional vector space with an inner product It follows alreadyfrom (1. 15) that dim (A"-PV) = dim (AP V), p = 0, 1, ..., n. This section willprovide a natural isomorphism, denoted *, from APV to A"-PV, which will finally

clarify the relationship of the wedge product in R3 with the familiar cross product.

An equivalence relation on the set of orthonormal bases of V can be defined as follows:{ v', ..., v"} is said to have the same orientation as { v., ..., v"} if the lineartransformation A, defined by Av' = v', i = 1, ..., n, has positive determinant. Thisdivides the set of orthonormal bases into two equivalence classes. The definition of theHodge star operator depends, up to a sign, on which of these two orientations isselected. So we select an orientation, and then take an orthonormal basis { vI, ..., v"}with this orientation; there is a corresponding basis vector a for A" V as in (1. 52).

For any a. E APV, the map µ -3 A. A µ from An -PV to A' V is linear, so there exists aunique f), e L (A"-PV -a R) such that

a.A11 = fx(l.t)(Y.

Now it follows from 1.6.4 that there is a unique element of A"-PV, denoted * A., suchthat f)L (µ) _ (* a.1µ)" _P: in other words,

(1.54)A. A µ = (* A1µ)" Pa, Vµ a An -PV.

The operation which sends A. to * A. is called the Hodge star operator.

1.7.1 Example: The Hodge Star Operator in the 3-Dimensional Case

This is a continuation of Example 1.6.5.2; here a = e, A e2 A e3 in the previousnotation. We shall calculate *X for A. = e A e3, which as we saw is one of theelements of an orthonormal basis for A2R. Clearly * A. E A'R3 = R3, since p = 2and n - p = 1, and so there are real numbers a, b, c, such that * A = ae, + bee + ce3.Equation (1. 54) tells us that

e2 A e3 A µ = (ae, + bee + ce)µ)e, A e2 A e3, Vg e R3.

Taking µ to be each basis vector in turn, we see that b = c = 0, while

e2Ae3Ae, = a(e, Ae2Ae3),

and two transpositions on the left side show that a = I. The same calculation can be3carried out for the other elements of this orthonormal basis for A2R, showing that


* (e2 A e3) = e,, * (e1 A e3) = -e2, * (e1 A e2) = e3, (1.55)

and so by linearity,

* (c, (e2Ae3) +c2(e, Ae3) +c3(e1 Ae2)) = cie1-c2e2+c3e3. (1.56)

Since e2 x e3 = e1, etc., the last line shows that, if u, v e {e,, e2, e3}, then

uxv = * (UAV) (1.57)

and by linearity, this extends to all u, v E R3, giving the exterior algebra interpretationof the cross product in vector algebra. Note that, if we had chosen a basis with theopposite orientation, such as {el, e2, e3} where e) = et, e2 = e3, e3 = e2, then theright side of (1.57) would be minus the cross product.

1.7.2 Effect of the Hodge Star Operator on Basis Vectors

Given an orthonormal basis { v1, ..., vn}, we shall now derive a general formula for * A.when A. = vI n ... AVP. In other words A. = am, where H = (1, 2, ..., p) . UsingSection 1.6.5.1, we can specify * A. by considering (*)JaK)n-P forK = (k(l), ..., k(n -p)), where 1:5k(l) <...<k(n-p) Sn.Theidentity (1.54)gives

X n aK = (* MaK)n _Pa, (1.58)

and the left side is zero unless K = (p + 1, ..., n) = H', in which case(* XIaK)n - P = 1. It follows that *X = bae' for some constant b, and (1. 58) showsthat a = b(a"Ia" )a, and therefore b = (awlaH) = t I. In other words, forH = ( 1 , ..., p) and H' = (p+ 1, ..., n),

* aH = (aH'Ie )n -PaH

. (1.59)

Referring back to the properties of the exterior product in Section 1.2.2, we observe that

aKAaH = (-1)P(n-P) (aHAaK) = (-1)P(n-P)a = (*(yKIaH)Pa,

which implies that

(*cr%h)p = (-I)P(n-P) 8K.H', (1.60)

in the notation of footnote 6, and the same reasoning as before shows that

*e, = (-1)P(n-P)(aHlaH)pam. (1.61)

Combining this with (1.59) gives

1.7 The Hodge Star Operator 19

* (*a") = (a"'Ia"')n_P(*aH,) = (-l)P(n-P)((yHIa")P(a"10"')n_Pa". (1.62)

However, (1. 53) implies that

(cla)n = (v'IV')...(Vnly") = (a"ITH)P(aH,kYH,)n_P = (_,)(n-t)/2, (1.63)

and the last two identities combine to give:

*(*X) = (-1)P(n-P)+(n-r)12a.. (1.64)

in the case where ? = a". This generalizes immediately to any p-element ascendingmulti-index set H, because we can simply relabel the basis so that H becomes(1, 2, ..., p) ; this may cause a change of orientation when p = n - 1, but this does notaffect (1. 64). By linearity this formula extends to the whole of APV.

1.7.3 Examples

For the 3-dimensional Euclidean case studied in Example 1.7.1. n = t = 3; so forevery p e {0, 1, 2, 3 }, p (n - p) + (n - t) /2 is even, and

**71. _ X,XE U APR3 . (1.65)O5p53

For the Lorentz inner product in Example 1.6.2, n = 4 and the signature t is 2, andso when p = 2 or p = 4, p (n - p) + (n - t) /2 is odd; thus

**

** (e,Ae) = -(e;Ae,), 1 Si<j54; (1.66)

(et Ae2Ae3Ae4) = -(et Ae2Ae3Ae4).

1.7.4 Formula

For any?.,LEAPV,%I,A*p =A o*.%_ (-1)(n-1)/2(A,lµ)Pa.

(1.67)

Proof: Consider first the case where t = an as in (1. 59); the only basis element ? forwhich ). A * µ * 0 is X = Off, and in that case (1. 59) and (1.63) give

Vector Algebra Expression Exterior Algebra Version

cross product u x v (u A v)

triple product u- (v x w) (uAVAw)

I u x v12 = 1 ul 21 v12 - (u . V) 2 (u A vlu A V)2 = (ulu)(vIv) - (UIV)2

ux (vxw) = (u- V)w U A * (vAw) = (ulw)(*v) -(ulv)(*w)

Table 1.2 Correspondence between exterior algebra and 3-dimensional vector algebra


A*µ = a"A ((a"'Ia")"-Pa,) = (a"la")n-Pa

= (-1) (n-t)/2 (a"IaH)Pa = (-1) (n-r)i2(Xlµ)Pa.

The result extends by linearity to all X, g e APV. tx

With the help of this formula, it is possible to show that the constructions and formulasof 3-dimensional vector algebra are special cases of ones in exterior algebra. Note thatin Table 1.2, the identities in the second column are valid in any inner product space(see Exercises 21 and 22 below).

1.8 Exercises

16. Calculate the signature of the induced inner products on A2R4 and A3R4 for theLorentz inner product of Example 1.6.2, taking c = I.

17. (Continuation) Find out the effect of the Hodge star operator (with respect to theLorentz inner product) on each of the basis elements of the exterior powers of R4 thatyou calculated in Exercise 1.

18. Let { e1, ..., e5} be the standard basis of R5, and give R5 the inner product such that(e,Je,) = 1, 15i53,(e)e) = -1,45j55.(i) Write down an orthonormal basis for A2R5, and calculate its signature.

(ii) Find * (e, A e4) and * * (e, A e4).

19. Suppose { vt, ..., v"} is an orthonormal basis of V. with signature t.

(a) Write down an orthonormal basis for A" -' V, and calculate its signature.

(b) Do the same for A2 V.

20. Prove that condition (1. 44) is necessary for the nondegeneracy of an inner product.

Hint: Suppose the determinant in (1.44) is zero; show that there exists a nonzero z such thatW) = 0, Vj.

21. (a) Show that the formula

(U A vlu A v)2 = (ulu)(vly) - (ulv)2,

which appears in Table 1.2, holds in any inner product space.

(1.68)

1.9 Some Formal Algebraic Constructions 21

(b) Show that, in the case of R3 with the Euclidean inner product, this formula isequivalent to the formula I u x v12 = I ul 2I

vl2 - (u v) 2 in vector calculus.

Hint: For part (a), use (1. 49). For part (b), use 1.7.4. (1. 65), and 1.7.4 again.

22. Repeat Exercise 21 for the pair of formulas appearing on the last line of Table 1.2.

Mat: Since U A * (v A w) = (ulw) (* v) - (ulv) (* w) is linear in u,v, and w, it suffices to verifyit for elements of an orthonormal basis. Try taking the exterior product of both sides with anotherbasis element, and use 1.7.4.

23. Prove, by induction on the dimension n, that every inner product space has anorthonormal basis.

Hint: (1.47) shows that the assertion is true when n = 1. In general, find a vector v such that (1.47) holds, and apply the inductive hypothesis to the (n - I)-dimensional subspace consisting ofvectors orthogonal to v.

24. Let { v', ..., v"} and {w', ..., w"} be two orthonormal bases of an inner product spaceV. arranged such that (v'lv') = I = (w'Iw') for 1 5 i:5 q, 1 :5j S r, but for no otherindices. Let H denote the set { v E V: (vlv) 2t 0}. Show that H is a subspace, and it hasboth { v', ..., v4} and { w', ... w'} as bases. Conclude that q = r, and so the signatureof an inner product space does not depend on the basis.

1.9 Some Formal Algebraic Constructions

This section is intended merely to fill in some of the logical gaps of earlier sections. Theconstructions given here will not play any part in later chapters, and may be omitted.

1.9.1 Formal Construction of the Second Exterior Power

There exists a vector space satisfying (1. 1), (1. 2), (1. 3), and (1. 4).

Proof: Let V x V denote the product set { (u.v): u, v e V}. Let F (V x V) be the vector spaceconsisting of all finite linear combinations of elements of V x V, and let S (V x V) be thesubspace of F ( V x V) generated by the set of all elements of the following types:

(u + v,w) - (u,w) - (v,w), (u,v + w) - (u.v) - (u,w), (1.69)

(au.v) - a(u.v), (u,av) - a(u.v), (1.70)

(u,u). (1.71)

Define A2V to be the quotient space F (V x V) IS (V x V); in other words an element of A2 V isan equivalence class of vectors in F (V x V), where two vectors are called equivalent if theirdifference lies in S (V x V). We define

u A V = [ (u.v) ] = equivalence class containing (u,v) e V x V. (1.72)


Properties (1. 2), (1. 3), and (1.4) follow immediately from (1. 69), (1.70), and (1.71). n

1.9.2 Formal Construction of the pth Exterior Power

There exists a vector space satisfying (1. 7). (1. 8), and (1. 9).

Proof: This is an obvious generalization of 1.9.1. The expressions V x ... x V, F(Vx ... x V),and S (V x ... x V) refer to the obvious p-factor versions of those in the proof of 1.9.1. Note thatin lines (1.69) and (1.70), we need p types of terms instead of just two, and (1.71) becomes

(u1, ..., up), where u; = ul for some i *j. (1.73)

Thus APV is defined to be the quotient space F (V x ... x V) IS (V x ... x V). and we may defineUIA ... A up = [ (u1, ..., up) ] = equivalence class of (u1, ..., up). n

1.9.3 Proof of the Universal Alternating Mapping Property

To every g e Ap (V -4 W), there corresponds a unique g e L (ApV -> W) such that

g(u1A...AUp) =g(u,,...,up),Vu.,.... up. (1.74)

Proof: Any g which is multilinear and alternating may be uniquely extended to a mapj e L (F (V x ... x V) -+ W) such that S (V x ... x V) is contained in the kernel of g, in theterminology of 1.9.1 and 1.9.2. Since APV is defined to be the quotient spaceF (V x ... x V) IS (V x ... x V), every 0 e F (V x ... x V) in the equivalence class [ ] a APVis mapped to the same element g in W. So define g ( [4]) = g (m). In particular,

(ulA...AUp) =g([(Up...,up)]) g(Ut,...,up).

If were another such map, then

g(UiA...AUP)

so g - g is zero on the generators of APV, and hence on all of APV.

1.9.4 Calculating a Basis for an Exterior Power

For any basis {v1....,v"} of V,

{v'(1) A ... A v'(p), 1 5i(1) <... <i(p) 5n}

n

(1.75)

forms a basis for AP V.

Proof: We saw already in (1. 14) that these vectors span AP V-, only the linear independenceremains to be proved. First take the case p = n; here it suffices to show that vI A ... A v" x 0 forevery basis { v1, ..., v"} for V. Select such a basis, and for any vectors { ut, ..., u"} in V, letA = (a11)15 be the n x n matrix of coefficients given by

Ui = 1 a,,vr.

Define a multilinear alternating map h: V" -, R by the formula:


h(ul.....u") = I sgn(n)a, (I)...a"x(n).Ire 1,.

Note that h (vt, ..., v") = 1, and hence the corresponding linear map h: A"V -+ R. as in 1.2.1.satisfies h (vt A ... A v") = 1; therefore vI A ... A v" x 0.

Now consider the case of an arbitrary p, 2!9 p!5 n - 1. Suppose we have some linear combinationof the vectors (1.75) which is equal to zero:

la,(v'(t)A...Av`(P)) = 0, (1.76)1

where the summation is over multi-indices I such that i (1) < ... < i (p). Pick a specific such 1,and let I' = (k (1) <

.t t< k (n - p)k be the complementary set of indices. Taking the exterior

product of the vector v A ... A V with the left side of (1. 14) makes all entries vanishexcept

k(I) k(n-p) i(I) i(P)al(v A...AV )A(v A...AV ),

= tal(vIA...AV"),

which must equal zero by (1. 14). Since vI A ... A v" * 0. it follows that a, = 0. Thus the linearindependence of the vectors (1. 75) is proved. IX

1.10 History and Bibliography

Exterior algebra is attributed to Hermann Grassmann (1809-77). Many books onmanifolds and geometry give a brief exposition of exterior algebra, usually in the moregeneral context of tensor algebra, and usually with greater emphasis on multilinearmappings; see, for example, Chapter 6 of Abraham, Marsden, and Ratiu [ 1988]. For afuller treatment of the subject, see Greub [ 1978].

2 Exterior Calculus on EuclideanSpace

The calculus of differential forms, known as exterior calculus, offers another approachto multivariable calculus, including line, surface, and volume integrals, which isultimately more powerful than vector calculus, and, being coordinate-free, is ideallysuited to the context of the "differential manifolds" we shall encounter later. In thischapter we shall adopt an algebraic approach to exterior calculus, which is efficient interms of proofs but lacking in intuitive content; Chapter 8 will attempt to remedy thisdeficiency by showing the role of differential forms in multidimensional integration.

2.1 Tangent Spaces - the Euclidean Case

In vector calculus, the distinction between the position vector of a point in space, anddirections of motion from a point in space, is not clearly drawn. Before commencing thestudy of differential geometry, it is necessary to formalize this distinction by puttingthese two kinds of vectors into different vector spaces.

A function on an opens subset U of R° into R" is said to be smooth if its partialderivatives of all orders exist and are continuous. Let C°° (U) denote the set of smoothfunctions from U to R.

The tangent space to R" at y, denoted Ty,R", is simply a copy of R" labeled with theelement y E U. An element 4 e TVR" is identified with the mapping which takes every

t We say that U is open when, for every y e U, the "open ball" B (y, i:) = (X: 11 x - Y11 < e} iscontained in U for all sufficiently small e; here II II denotes Euclidean length. An arbitrary unionof open sets is open, and a finite intersection of open sets is open.


real valued smooth function f, defined on any neighborhood2 of y, to its directionalderivative at y along t;, denoted V; in symbols,

V jf(Y+t4)J,_o. (2.1)

Take a basis for R", and let {x', ..., x"} denote the corresponding set of coordinatefunctions. Formally speaking, this means that x`: R" -+ R is the function such thatx' (a,, ..., a") = a.. Then we may express (2. 1) as

V= Of ,(Y)+...+f," L- (Y),ax"

_ (4'aII I)f.ax y ax" y

In view of the last line, we see that TYR" can be regarded as the vector space spanned bythe differential operators

Elements of TrR" are called tangent vectors at y. Let U be an open subset of R", andconsider a function X which assigns an element X (y) a TYR" to each y e U; define amapping Xf from U to R, by taking

Xf (Y) = X (Y) f.

X is said to be a (smooth) vector field on U if Xf e C- (U).

(2.2)

The set of smooth vector fields on U will be denoted s (U).The representation of X (y)in terms of the basis above determines functions 4': U -+ R, called the componentfunctions with respect to this coordinate system, by the formula:

X (Y) (Y) ai I + ... + 4" (Y) aI .ax y

ax"

y(2.3)

By taking fin (2. 2) to be each of the coordinate functions in turn, we see that X is asmooth vector field if and only if all the component functions are smooth. In that casewe may abbreviate (2. 3) to

2 A "neighborhood" of y means an open subset of R" containing y.

26 Chapter 2 Exterior Calculus on Euclidean Space

x = a

ax' ax"

Figure 2. 1 The vector field X = sin (xy)

a

+ cos (xy) ay on (0, n) x (0, n)

(2.4)

2.1.1 Derivations

It is useful to have a more abstract characterization of vector fields. We say thatZ: C°' (U) -4 C° (U) is a derivation of C"° (U) if, for all f, g E C° (U), a, b r= R, thefollowing two properties hold:

Z(af+bg) = aZf+bZg;Z (fg) = f Zg + Zf g, where (f Zg) (x) = f (x) Zg (x)

2.1.1.1 The Set of Vector Fields May Be Identified with the Set of Derivations3 (U) is identical with the set of derivations of C°° (U) .

Proof (may be omitted): It follows immediately from the product rule for differentiationthat every vector field is a derivation; conversely if Z is a derivation, we claim that Z canbe expressed, as in (2. 4), as the vector field:

Z=Zx'a+...+Zx"a.ax' ax"

This representation is obtained as follows; for ease of notation let y = 0, and noticethat a smooth function f, defined on a neighborhood of zero, has the following Taylorexpansion:


n

f (x) -f (0) 5 f (tx) dt = Jx'loa (tx) (dt) _ x'h; W.r=t r=I

For any derivation Z, Z (1 1) = Z (1) + Z (1) = Z (1) = 0, and so Z annihilatesconstants; applying Z to both sides of the equation above, and using the second of thetwo rules for derivations, shows that

n n

Zf (x) _ Zx' (x) h; (x) + x' (Zhj) (x).

Taking x = 0 removes the second term, and so

n n

Zf (0) _ Zx' (0) h; (0) _ Zx` (0) aL (0).

This shows that `3 (U) is identical with the set of derivations of C" (U). a

2.1.2 Lie Derivative with Respect to a Vector Field

Given vector fields X and Yon U, the mapping [X,Y] : Cam' (U) -* C° (U), defined by

[X,Y]f = X (Yf) - Y (X.f), (2.5)

is indeed a derivation, as the reader may verify in Exercise 1, and hence [X, Y] is avector field by 2.1.1.1; we call this the bracket of X and Y, or the Lie derivative of Yalong (or with respect to) X, also denoted LXY.

The bracket of vector fields is anticommutative, that is, [ Y,X] = -[X,Y] , and satisfiesthe Jacobi identity, verified in Exercise 2 below:

M[Y21] + [Y,[Z,X]] + [Z[X,Y]] = 0. (2.6)

The Lie derivative can also be applied to differential forms (see Exercises 15, 16, 17,22, and 23), and has an important dynamical interpretation in terms of the "flow" of avector field (see Exercise 23). It is useful in dynamical systems and in Riemanniangeometry. The Jacobi identity crops up in the study of curvature, and elsewhere.

2.1.3 Example of a Lie Derivative

Let U = R3\ {0}, and take the usual Cartesian coordinate system {x, y, z} . The"gravitational field" associated with a point mass at 0 is (up to a constant multiple):

X = -{x2+y2+z2}-3/2{xa +y y+za }.


One may visualize the value of X at a point (x, y, z) in U as a vector pointing in thedirection (-x, -y, -z), with a Euclidean length proportional to I/ {x2 +y2 + Z2}. Thevector field associated with linear flow parallel to the y-axis is:

Y=ay.

If r = {x2 +Y2 +Z2} 1/2, then the Lie derivative of Y along X is given by:

(LxY)f = -r (xax+yay+z)ay+sy{r 3(xax+yay+za) If.

= rs{-3xyax+(r2-3 y2)ay-3zyaz}f.

See Exercise 23 on p. 48 for the dynamical interpretation of Lie derivative.

2.2 Differential Forms on a Euclidean Space

In the last section we introduced the set 3 (U) of vector fields on an open set U c R",and showed that this notion is "intrinsic," that is, does not depend on the basis used forR". Before giving an intrinsic (but cryptic) definition of a differential form of order p onU, we shall first give a description, using the coordinate system {x', ..., x"}, whichmay be more illuminating for the reader.

Let us recall the notion of the dual space V* = L (V -, R) of an n-dimensional vectorspace V Given a basis for V, the dual basis3 {X, ..., k," } of V consistsof the linear mappings defined by

X. (v1) = S =1 if k = j, 0 otherwise.

7i .(atvu+...+a"v") = a,.

Therefore

(2.7)

The cotangent space at y e U is defined to be (TyR") *, that is, the n-dimensionalvector space of linear forms on the tangent space at y. Elements of (TyR") * are calledcotangent vectors at y. Define a basis for (TyR") * as follows:

3 To see that {Ri, ..., R"} is linearly independent, observe that if b1X1 + ... +b"A" = 0, then

applying the left side to each v, in turn shows that each b, = 0; to see that it spans, note that any

We V can be expressed as W = W(v1)XI+...+W(v")X".

2.2 Differential Forms on a Euclidean Space 29

{dx'(y),...,dx"(y)} is the dual basis to { a aaxtly ax"I,

Using (2. 7) gives that

dx'(y)(4'ax'I

+...+4n-LI ) = i.Y

axn Y

(2.8)

Suppose Co is an assignment of an element w (y) e (TYR") * to each y e U;expressing w (y) in terms of the basis {dx' (y), ..., dx" (y) } defines componentshi:U-*R:

w (y) = h, (y) dx' (y) + ... + h" (y) dx" (y). (2.9)

A (smooth) differential form of degree 1 on U, or 1-form, is such a mapping to with theproperty that every h; a C" (U) . We shall abbreviate (2. 9) to:

Co = h,dx'+...+h"dx".

Likewise a 2-form to is an assignment of an element w (y) a A2 ( (TT.R") *) to eachy e U, so that in terms of the basis { dx` (y) A dxr (y) : 1 S i <j:5 n } forA2 ( (TR") *), we have the abbreviated representation:

0) = Yh;i(dx`ndx);<j

where each hii: U -4 R is smooth. The reader may look ahead to (2. 28) for therepresentation of a p-form; but before the notation becomes too bloated, let's move intosomething a little more abstract.

2.2.1 The w X NotationA much neater way of expressing the smoothness condition for 1-forms, withoutreference to a coordinate system, is that

y - w (y) (X (y)) e U), b'X e 3 (U).

We shall henceforward use the notation

(w X) (y) = w (y) (X (y) ) (2.10)

to express the duality between differential forms and vector fields. Let us emphasize that(w X) (y) is a real number. To see why this notation makes sense, suppose X is avector field as in (2.4), and Co is a 1-form as in (2.9); then (2. 8) shows that the notationto X can be interpreted as follows:


((o X) (y) = (h,dx'+...+h"dx") - (41--

+...+F,"a ) (Y). (2.11)

= h, (Y) 4' (Y) + ... + h" (Y) .," (Y),

= h (Y) . (Y);

that is, the dot product of the vectors whose entries are the components of w and X,respectively.

To express the smoothness condition for p-forms, we shall use the fact, discussed inChapter 1, that an element t) (y) E AP ( (TR") *) can be considered as a linear formon AP (TR"), that is, on the pth exterior power of the tangent space. Following thepattern of (2. 10), we shall use the notation

t) (X, n ... A XP) (Y) = t) (Y) (XI (Y) n ... A XP (Y)) e R (2.12)

for X,, ..., XP E 3 (U). For the local coordinate version of (2. 12) when p = 2, seeExercise 7; in general, in the case of a monomial t) = hdx (') A ... A dx (P) (asalways, each x` (Jl is a coordinate function) the calculations presented in Chapter 1 showthat

71 (X, A... AXP) = L (sgnx)h(XR(1)xi(1))... (X.(P)x(P)). (2.13)Re Ep

Now we are ready for the formal, coordinate-free definition of a p-form, after which weshall exhibit some concrete examples in dimension 3.

2.2.2 Definition of Differential Forms

A mapping w which assigns to each y E U an element w (y) a AP ( (TYR") *), wherep e 11, ..., n }, is called a (smooth) differential form of degree p, or "p form, " if, forall X,, ..., XP a 3 (U), the mapping (see (2. 12))

y-)w-(XIA...AXP)(Y) (2.14)

belongs to C- (U). The set of p forms on U will be denoted fPU. In the case p = 0,define S2° U = C`° (U).

Note that it follows immediately from this that SIPU is nontrivial only for 0:5 p 5 n.

2.2.3 The Space of p-forms

An element of UP U, when evaluated at a specific y e U, takes values in ann! /p! (n - p) !-dimensional space of linear forms. However, LIP U itself is not

2.3 Operations on Differential Forms 31

finite-dimensional, because C` (U) is an infinite-dimensional vector space over thereals. Its vector space structure is defined in the obvious way: For p-forms w, n and realnumbers a, b, the p-form aw + btl is defined by

(aw+btl) (y) = aw(Y) +brl(Y)

Moreover4 for every p-form w and smooth function f on U, a p-form fw is defined by

(N) (Y) = f (y) (0 (Y)

For example, take the usual Cartesian coordinate system { x, y, z) on an open subsetU c R3. General expressions for differential forms of order 0, 1, 2, and 3 are shown inTable 2.1. Symbols F, A, B, C, P, Q, R denote smooth functions F = F (x, y, z) , etc.

Name Differential Form dim (AP ((T,.R3) *) )

0-form F = F (x, y, z) I

1-form Adx + Bdy + Cdz 3

2-form P(dyAdz) +Q(dzAdx) +R(dxAdy) 3

3-form F (dx A d y Adz) i

Table 2.1 General expressions for differential forms in dimension 3

2.3 Operations on Differential Forms

2.3.1 Exterior Product of Differential Forms

For any p, q e {0, ..., n) with p + q:5 n, and for any (o e QP U and 11 a Q" U, we

may define the exterior product W n 116 1;2P * 9U,

(w A rl) (Y) = w (Y) A rl (Y),

where the expression on the right is the exterior product developed in Chapter 1: therules for exterior product apply immediately here. In particular,

11AW = (_1)Pq(wnrl).

A special case of this formula is

(dxht'1 A ... Adxh(P)) n (dx`(') A... Adx`'Q))

4 The sophisticated reader will recognize that W U is a "C_ (U)-module." This point of view ispresented in Helgason [ 1978].


= (-1)P4(dx`(1) A ... Adx`te)) A (dxh (1) A ... Adxh(I)).

For example, using the differential forms in Table 2.1,

(Adx + Bdy + Cdz) A (P (dy A dz) + Q (dz A dx) + R (dx A dy) )

_ (AP+BQ+CR) (dxAdvAdz),

and in R4, with coordinates {x, y, z, t},

(A(dxAdz)) A (B(dyAdt)) = -AB(dxAdyAdzAdt).

2.3.2 Hodge Star Operator on Differential Forms

If each cotangent space (T,,R") * is equipped with an inner product (.1.) (which maydepend on y), then the Hodge star operator gives a mapping

*:QPU-+0'PU.

where * w is the (n -p) -form given by (* co) (y) = * ((o (y) ) . For example, theLorentz inner product with c = I may be applied to every cotangent space to R4, andmay be expressed in terms of coordinate functions { x, y, z, t} by saying that{dx, dy, dz, dr} is an orthonormal basis for each cotangent space, with

(dxldx) = 1, (dyldy) = 1, (dzldz) = 1, (dddt) = -1. (2.15)

The calculations performed at the end of Chapter 1 can be duplicated in the differentialform notation: for example,

* dx = -dy A dz A dt, * dy = dx A dz A dt, (2.16)

*dz = -dxAdyAdt,*dt = -dxAdyAdz. (2.17)

2.3.3 Tensor Product of Differential Forms

The well-informed reader will know that the exterior algebra described in Chapter 1 isusually presented within the broader context of tensor algebra. In this book we shallonly need the following simple case of a tensor product. If V and Ware realfinite-dimensional vector spaces, V ® W is a vector space consisting of formal sums

tai (v1® w.)

obeying the linearity conditions

(av, +v2) ®w = a(v, 0 w)+v2®w, v® (bw,+w2) = b(v®w,) +v®w2,

2.4 Exercises 33

and the condition that if { v', ..., v' I is a basis for V and { wt, ..., w" } is a basis forW, then { v` ®w': 1 5 i:5 m, 1 :5j:5 n) is a basis for V ®W. The construction of such aspace follows in the same way as that of the second exterior power at the end of Chapter1. Analogous to the isomorphism AP (V`) . (APV) * in Chapter 1, the duals V* andW* satisfy:

V` ® W' _- {bilinear maps V x W -* R);

V` ®W=L(V-+W);

for the second isomorphism, identify A. ® w E V ® W with the linear transformationv -* A. (v) w in L (V -4 W) , and extend to all of V` 0 W by linearity.

The tensor product of 1-forms co', w2, denoted co' 0 0)2, is the map which assigns toeach y e U the bilinear map

w' ®w2 (y) E (TYR") * ® (TYR") * .

w' ®w2 (y) (4, c) = co' (y) (4) (02 (y) (c).

Thus for vector fields X, Y, we may write:

w' ®w2 (X, y) = (co' . X) (w2 - Y) E C`° (U).

For example, dx ® xdy (eza-, ay + yaz) =

xe;. The relationship with the exteriorproduct is:

w1 AO)2 = w1 ®0)2-w20 w1.

A map which assigns to each y e U an element of (TYR") * 0 (T,R") * is called a(0,2)-tensor; a general (0,2)-tensor can be expressed in the form

Ih;i(dx'®dx')i.i

where the { h13 } are smooth functions on U.

2.4 Exercises

1. Verify that for any vector fields X and Yon an open set U Q R", [X, Y1 defined by (2. 5)is a derivation, and hence a vector field.


Hint: In verifying that MX,Y] is a derivation, the fact that (2. 5) is linear in f is clear, and to checkthe second property, expand the expression X (Y(fg)) - Y (X (fg)) into eight terms, four ofwhich cancel out, leaving f [X.Y]g + [X.}l f g.

2. Verify directly (i.e., without using the notion of a derivation), that for any vector fields Xand Yon an open set U c R", [X. Y1 defined by (2. 5) is indeed a vector field.

Hint: Since [X.YJ is clearly linear in X and y, it suffices to check this when

X- ax,Yax

where F, = F, (x1...., X"), C = C W, ..., x"). Show that the two second-derivative terms cancelout.

3. Verify the Jacobi identity (2. 6) for vector fields.

Hint: Work in terms of derivations; don't differentiate anything!

4. Show that if X and Y are the vector fields on U = R3\ { 0} given below, thenLXY = 0:

X = -{x2+y2+z2}-3/2{xa +ya +za}; Y = -ya +xa .ax ay az ax ay

-3 -3Hint: For brevity, take r = {x2 + y2 + z2 } 1 /2 and note that (xar - y ax ) = 0.

y

5. Repeat Table 2.1 for the case of R4 with coordinates {x, y, z, t}.

6. Express the 2-form (A dx + Bdy + Cdz) A (Fdx + Gdy + Hdz) as a linearcombination of three basis elements, as in the third row of Table 2.1.

7. Show that if t) = P1 (dy n dz) + P2 (dz A dx) + P3 (dx A dy) is a 2-form on an openset U g. R3, as in Table 2.1, and if

X = a +42a +43a , y = t a + r2 a + r3 aax ay az ax bay S Wz- '

then the coordinate expression for (2. 14) when p = 2 is

rj (XAY) =bKI

2 3

rl Y2 r3

Hint: Calculate (dy n dz) (X A Y) using methods of Chapter 1, and then use linearity.

2.5 Exterior Derivative 35

2.5

8. Suppose we apply the Lorentz inner product to every cotangent space to R4, as in(2. 15).

(a) Verify the formulas (2. 16) and (2. 17).

(b) Calculate * (E (dx n dt) + F (dy n dt) + G (dz n dt) ).

Exterior Derivative

The exterior derivative is an operation on differential forms which does not make sensefor exterior algebra in general. It is a concept of enormous power and wide application.and it plays a central role throughout the rest of the book. As we see in Table 2.1, all the"differentiation" concepts that a student encounters in calculus, such as grad. div, andcurl, are special cases of this one. Here we shall follow an "algebraic" approach to theconstruction of the exterior derivative; this means that we stipulate in advance thealgebraic rules that it must obey, and show that there is a unique operation satisfyingthose rules. The "geometric" significance of the exterior derivative will not becomeapparent until the end of Chapter 8.

For U open in R", there is a natural mapping d: S2°U -+ f2' U which associates to thesmooth function f the 1-form df, also called the differential of f, whose value at y isdefined by

df (y) CJ,, c E TT U, (2.18)

or, in the notation of (2. 10),

Xf,XE 3(U) . (2.19)

If df is expressed in terms of coordinates by df = h i dx1 + ... + h"dx", then (2. 11) and(2. 19) combine to show that

(h,dx'+...+h"dx") a = hj = a ,aAj

L

and therefore

df = of idx'+...+afdx".ax ax"

(2.20)

2.5.1 Theorem: Existence and Uniqueness of the Exterior Derivative

There is a unique R-linear mapping d: S)P U + QJ' + I U for p = 0, 1, ..., n - 1, whichagrees with (2. 19) when p = 0, such that for all differential forms w, il:


d(0AT1) = dwAT1+ (- I)d`8`D(wndT1), (2.21)

d(d(O) = 0, (2.22)

where deg w denotes the degree of co, and where f A T1 means the same as frl iff E S2° U.

This mapping d is called the exterior derivative. The proof of the theorem, whichdepends on nothing more than combining the rules of multivariable calculus with thoseof exterior algebra, will be given a little later. Condition (2. 21) is a sophisticated formof the "product rule" in calculus, designed to specify the relationship between theexterior derivative and the exterior product of differential forms, while condition (2.22)is tantamount to the equality of the mixed second partial derivatives, and could be calledthe Iteration Rule.

We are now going to relate exterior differentiation of differential forms in R3 to vectorcalculus. Given a vector field X = 4' a + 42a + 43a on U, there exist differentialforms y CIZ

ax a i2' U, ax = t 1 dx + 42dy + 43dz , (2.23)

called the work forms of X, and

4X 1E t' (dyAdz) +t2(dzAdx) +F,3(dxAdy), (2.24)

called the flux form of X. The logic of these terms will become more apparent inChapter 8, when we shall see that the "line integral" of the work form along a pathmeasures the work done by the vector field X in moving a particle along the path, andthe "surface integral" of the flux form across a surface measures the flux of the fieldthrough the surface.

Finally, given a smooth function F on U, define a differential form

PFE L 13 = F(dxAdyAdz) (2.25)

called the density form of F. To see how exterior differentiation relates to grad, div, andcurl of a vector field, consider the following extension of Table 2.1.

Let us indicate how the rules (2.21) and (2. 22) above are used to obtain the second rowof the table: Taking only the first summand, and using the fact that d (dx) = 0, andwritingA for 4',

5 1 am indebted to John Hubbard (Cornell) for this terminology.

2.5 Exterior Derivative

Differential Form co Exterior Derivative do)

37

FF = F(xYz) dF =-grad-

"work form" O\r3

42 "curl"

axx=(a az

)(dyAdz)+dax=0crlap' at 3 42 I

d ddd +( -- x) ay)( xA y)zAax)( )x(aa

"flux form" d$x = pmvx"div"

OX = i;l (dy Adz) +g2 (dz Adx)' -t23 d d d++

=+E3 (dandy)xA( yA z)

)( ax 5Y a

F(dxAdyAdz) 0

Table 2.2 Exterior derivatives of differential forms in dimension 3

d(Adx) =d(AAdx) = dAAdx+ (-1)°(AAd(dx))

_ (azdx+aAdy+a-dz) Adx

= ay (dyAdx) +az (dzAdx) _ -a4(dxAdy) +aZ (dzAdx).

The other four terms are obtained similarly.

2.5.1.1 Uniqueness Part of Theorem 2.5.1As a consequence of the discussion in Chapter 1, we know that the p-forms

{dx'(') A ... Adx`(P):1 Si(1) < ... <i(p) !5 n) (2.26)

provide a basis of AP (T R") * at each y E U. We shall begin by showing that if amapping d with the properties described in the theorem exists, then it must satisfy:

d(dx'(') A... Adx`(P)) = 0. (2.27)

We shall use induction on p. The assertion holds when p = I by (2. 22), since thecoordinate function P) is a 0-form. If it is true for p - I. then by (2. 21),

,d (x i(I)

A dxi(2) n ... A dxi(P) ) = dxi(1) n ... A dxi(P),

= d(dx'(')A...Adx`(P)) =d(d(x'(')Adx'(2) A...Adx'(P))) =0


by (2. 22), which completes the induction. Now we shall demonstrate the uniquenesspart of the theorem by showing that, for every w r= (ZP U, do) is uniquely specified bythe rules above. For, summing over ascending multi-indices 1 = (i (1) , ..., i (p) ), wemay express w E Y' U in terms of (2. 26) as:

w = g, (dx (') n ...ndx' (P) ),1

(2.28)

where each g, is a smooth function from U to R. Using (2.21), (2. 27), and (2. 20):

dw = Id(g,(dx'(') A... Adx'(r)))I

= Idg,AdX (') A... ndx'(P)I

_ V ndx'(') A...Adx'(p), (2.29)ax ax"

so now we see that dw is uniquely specified. tx

2.5.1.2 Existence Part of Theorem 2.5.1Note that (2. 19) specifies the action of d on 0-forms (and shows that df is indeed a1-form): let us define don the p-forms for p z 1 by (2. 29), which is clearly linear in w.We must now show that (2.21) and (2. 22) hold for all p-forms. By linearity over R, it issufficient to demonstrate these for "monomials"

w = gdx (» A ... ndx'(P),tl = hdx'(') A ... ndx'(e), (2.30)

where 1 and J are ascending multi-indices, as in (2. 28). Using only (2. 20), (2. 29), andthe rules for the exterior product from Chapter 1, we obtain:

d(wntl) = d(gh(dx'(')n...Adx'(P)) A (dx'(')A...ndxi(9)))

_ h)dx") n (dxA...Adzj (9))

_ {aLkh(dxkAdx'(1)A...Adx'(v))

+hkg (dxk AdX (') n ...ndx' (e)) }ax

= 1: (a8k)dxkA (dx'(') A...Adx'(P)) A71

k

2.6 Exercises

(dx' A ... Adx'(Q)))aXk;k

this says that d ((OA rl) = dw A n + (-1) d*8 0Dw A drl , which establishes (2. 21); asfor the "Iteration Rule,"

d (do)) = d agdxk A (dx' (1) A ... A dx' cPt) )

_ jag (dfAdxkA (dx't't A...Adx'tP>))k, M axmaxk

= E(ag -a8 I(dx"'ndxkn(dx'(l)n...Adxi(P))).k « axmaxk axkax'" J

39

which is zero, by equality of the mixed second partial derivatives. it

2.6 Exercises

9. (i) Verify the formula given in Table 2.2 for the exterior derivative of the flux formtQX = (dy A dz) + 42 (dz A dx) + 43 (dx A dy), where X is the vector field

X = 41 a +42 a +43aax ay az-

(ii) If F E C" (R3), calculate d (FOX) using (2. 21), and use the result to verify thefollowing vector calculus formula:

div (FX) = grad F X + Fdiv X.

10. When OX = 41 dx + 42dy + 43dz as in Table 2.1, and F e C" (R3), calculate d (Fox)using (2.21), and use the result to verify the following vector calculus formula:

curl (FX) = (grad F) x X + Fcurl X

11. Using the Iteration Rule (2. 22) and the ideas of Table 2.2, deduce the vector calculusidentities:

curl (grad F) = 0; div (curl X) = 0.

12. (a) In R4, with coordinates {x, y, z, t}, calculate the exterior derivative of the followingexpressions; here P, Q, F. G, and H are functions of (x, y, z. t).


Pdx + Qdt;

F(dyndz) +G(dzndx) +H(dxndy).

(b) Calculate * (df A dg) if f = f (x, z), g = g (y, t), with respect to the Lorentz innerproduct on every cotangent space to R4, as in (2. 15).

13. Calculate * (df) for f E C'° (R3), with respect to the Euclidean inner product.

14. (i) Prove the following identity holds for any 1-form o) and vector fields X and Y:

dW - (XAY) = (2.31)

Hint: Since the equation is linear in to, the case to = hdxk suffices.

(ii) Using induction starting from (2.31) and formula (2. 13), verify that for any p-formW and vector fields X°, ..., XP,

P

dW X°n...nX= 7,i=o

+ ,(-1)'+jt [Xi,Xj] nX°n...Akin...A jA...AXP,i<j

where the hat A denotes a missing entry.

15. Let U C. R" be an open set, and let X be a vector field on U. A set of linear mappingsLx: SYP U --+ i2P U for each p e 10, 1, ..., n) will be called a Lie derivative ofdifferential forms along X if it satisfies the following three rules:

Lxf = Xf, f e S2°U;

Lx(df) =d(L,r/),fE L20 U;

Lx (w A TI) = LxW n tl + W A Lxr), W E i2P U, r(E tZ4 U.

(2.32)

(2.33)

(2.34)

(i) Assuming that such a mapping exists, derive from (2. 32), (2. 33), and (2. 34) theformula Lx (gdf) = (Xg) df+gd (XJ), where f, g r= f2°U.

(ii) Using induction on p, or otherwise, show that LxW, if it exists, is uniquely definedfor all a) E LIP U, and show that the formula for the Lie derivative is

Lx(gdx'(') A ... Adx'(P)) _ (2.35)

A...Adx'tpl+g E dx`(1)A...Ad(Xx`t'"i) A...Adx' .

1SmSp

2.7 The Differential of a Map 41

Hint: It suffices, by linearity, to considerp-forms as in (2. 30). that is, monomials.

(iii) Now prove the existence of the Lie derivative of differential forms as follows;define LXw for w e QPU, p 2 1, using (2. 35), and show that it satisfies (2. 34).

(iv) Finally, using induction or otherwise, prove that

LX (d(o) = d (LXW), w e OP U.

16. (i) Show that the Lie derivative of a I-form, specified by (2. 32), (2. 33), and (2. 34),has the following representation when w = h , dx' +... + hdx" :

(Xhj) dx' + hkd?;k, (2-36)Lxw = 7j k

where X =' a + ... +"aax ax"

(ii) Let U be R2 without the x-axis, with coordinates {x, y}; calculateLX (y-1 dx - y-t dy) where X = yea/ay.

17. Show that the Lie derivative of a 1-form, specified by (2.32), (2. 33), and (2. 34),satisfies the following identity, for all vector fields X and Y:

LXw Y = Lx (w Y) - ((o Lx Y). (2- 37)

Hint: Since (2. 37) is linear in w, the case w = hdxk suffices.

2.7 The Differential of a Map

In this section we extend the notion of the differential of a function to the case offunctions with values in another Euclidean space. Suppose U c R" and V ( R' areopen sets, and ip: U -> V is a smooth map. If f e C (V) , then we have the followingdiagram:

UcR" iP VcR' f *R

The differential of cp at x r= U is the linear map dip (x) E L (TxR" -* TO (x) R'") givenby

TxR",f C (V). (2.38)

In other words, the effect of the differential operator dip (x) E, on f is given by applyingthe differential operator 4 to the composite function f ip E C (U). This is consistent


with (2. 18) when m = 1, provided we regard the real number df (x) 4 _ as anelement of the tangent space Tf(.) R.

An equivalent description of the differential, once a basis is chosen for V, is to say thatdip is the m-vector of 1-forms (dip', ..., d(pm), where W = ((p', ..., tpm). However, thebeauty of (2. 38) is that the chain rule is already encoded in it.

2.7.1 The Chain Rule

If yr: V - W Q R'q is another smooth map, then for y = ep (x),

(x) = dyr(y) dtp(x). (2.39)

Proof: Two applications of (2. 38) show that, for 4 E TR", f E C' ( W),

(d (W (p) (x) 4)f = 4 (f W (p) = (d(p (x) )f' W = (dW (y) . dip (x) F) f. n

2.7.2 Interpretation of the Differential in Terns of Multivarlable Calculus

Using coordinate systems {x', ..., x"} on U and l y', ..., ym} on V, the smooth maptp = ((p', ..., (pm) has a derivative Dip (x) E L (R" -a Rm) at each x E U, which maybe written as:

D 1 tp' ... D"tp' aW'

Dip = . ... ... Drm' ° ax'Dim ... D"Tml

We shall compute in terms of Dip the effect of applying dip (x) to the tangent vector

4= 'a I +...+V a

I

.

aX' = ax" s

2.7.2.1 The Derivative Matrix Is the Local Expression for the Differential

if = i',....f;"]T, _ b'....,cm]T,andc _ +...+rm al ,thenay " ay

mv

= dip (x) E = Dcp (x)

Proof: By the usual chain rule for differentiation,

(d(p(x)4)f = (4'a,I +...+V" a ()TX x ax" X

2.8

2.8 The Pullback of a Differential Form

e

So' = D(p (x) ?; is necessary and sufficient for d4p (x) 4 =

The Pullback of a Differential Form

S(a +...+r"aay t ,. aym

43

In 2.8.1, we shall give the definition of an operation on differential forms called thepullback, which is really nothing more than a change of variable formula. This is a veryimportant construction, because in later chapters it will allow the apparatus ofdifferential forms on Euclidean spaces to be transferred to differentiable manifolds.After the definition we give a slick characterization in a frequently encountered specialcase, and then the messy local coordinate version. Our suggestion to the novice is tocheck that one understands the syntax of the definition (i.e., check that a function,applied to an argument, really does lie in the intended domain), to use the localcoordinate version for purposes of reassurance, but to memorize only the slickcharacterization. By the way, the "*" in this section has nothing to do with the HodgeStar Operator of Chapter 1!

The notation is the same as in Section 2.7. If ? e L (Tv (x) R' -* R) for some x e U.then the composite linear map k d(p (x) : T.,R" -a R is well defined, as shown in thefollowing diagram:

T.,R"d(p

(x) Tm(x)R'X

e. R.

Now define (Px* e L ((To (x) R'") * -4 (T,,R") *) (note the direction!) by

cpx*x = (T,o(x)Rm)*

2.8.1 Definition of the Pullback

The pullback (p* : SiP V -a SZP U, for each p e 10, 1, ..., m}, is a linear map created outof the linear maps { cpx : x e U} as follows:

p = 0:forfe tZ°V = C°'(V),define (p*f =P Z, 1: for we QPV, define (,* (o) (x) = (AP(Px*) (w (y) ). where y = (p (x).


Here we have used the p- exterior power of a linear transformation, introduced inChapter 1.

2.8.2 The Slick Characterization

The following special case will be encountered frequently in later chapters. Whencp: U Q R" -i V Q R" is a smooth bijection with a smooth inverse (a "diffeomorphism"),then every vector field X on U induces a vector field cp. X on V, called thepush-forward of X under tp, by the formula

tp.X(y) =dcp(x)(X(cp'(y))). (2.40)

The formula is elucidated by the following diagram.

dcp (x)

The definition of pullback says that if w is a 1-form, then for every vector field X on U,

(tp` w) - X (x) = w cp.X (y).

In this notation, the pullback (e rl of a p-form rl on V is uniquely characterized whenpz 1 by

(p`il (XIA...AXp) = TI (tp.XIA...A(P.XP)

for all X1,...,X e!3(U).

2.8.3 Interpretation of the Pullback In Terms of Multivariable Calculus

Suppose CO =A dy' + ... +Amdy' is a 1-form on V, and n = tp' to has an expressionr) = BIdx' + ... +B"dx" as a 1-form on U. Let us represent co and n as the row vectors

w = [A1....,Am1,n = [B1,...,B"],

respectively.

2.8 The Pullback of a Differential Form 45

2.8.3.1 Formula for the Pullback In Terms of the Derivative Matrix71 = 4' w if and only if tj (x) = w (y) Dcp (x) for all y = cp (x). Written out in full.the condition says

Drcpr ... D"cpr

[Br, ..., B"] _ [A1, ..,Am] (2.41).........cp... cp

Proof: From the formula cp" w (x) (4) = w (y) (d(p (x) 4) and from 2.7.2.1. we see. ttthat r) = (p* co means that '94 = w (D(p) for every

Thus the differential and the pullback correspond to premultiplication andpostmultiplication by the m x n matrix Dcp, respectively. Clearly the pullback of ap-form can be expressed in terms of the pth exterior power of the matrix D4p, as wasdiscussed in Chapter 1. A specific example which will be important for the theory ofintegration is the following: When p = m = n

w=A(dyrA...Ady")= q'w=I DoA(dxrA...Adx"). (2.42)

2.8.4 Example: Spherical Coordinates

Let U = { (r, 0, 0) E R3: r > 0, 0 < 9 < 2n, 0 < 4, < n}, which is open in R3, anddefine 4): U -+ R3 to be the usual change of variable map from spherical coordinates toCartesian ones, namely,

4) (r, 0, 4,) = (rcosOsin4,, rsin0sin$, rcos4,) = (x, y, z).

This map is not onto (for example, (0, 0, 1) e 0 (U) ), but it is a diffeomorphism ontoits image V. As one may easily check,

cos0sin4 -rsin0sin4, rcos0cos4,sinOsin4, rcos0sin4, rsin9cos¢

cos4 0 -rsin$

and now it follows from (2. 41) that

4)* dx = cos0sin4dr-rsin0sin4d9+rcos0cos4d$; (2.43)

$# dy = sin9sin4dr+ rcos0sin4dO+ rsin0cos$d4; (2.44)

4)` dz = cos4dr-rsin4d4,. (2.45)

It follows from (2. 42) that


it* (dxndyndz) = -r?sin4(drAd9Ado), (2.46)

which is supposed to remind you of the Jacobian determinant which you use whenevaluating a volume integral in spherical coordinates; more of this in Chapter 8.

2.8.5 Formal Properties of the Pullback

The pullback defined in 2.8.1 has the following properties, V4 T) E S?P V, p e (q V:

cp* (W+71) = cp* w + cp* n ; (2.47)

cp*(tl A P) _ e rl A tP* P, (2.48)

and in particular cp* (fp) = (f (p) p for f c= ' (V);

d (cp* (o) = (p* (do)); (2.49)

and if W : V -> W c Rk is also smooth, then

tp* - W* (2.50)

Proof: (2. 47) is immediate from the definition, and (2. 50) is immediate from the chainrule 2.7.1. Property (2. 48) is simply an algebraic property of the exterior powers of thelinear map (p:. mentioned in Chapter 1. To prove (2.49), note first that for f r= C'° ( V)and X E 3 (U),

((p.X)f = X(p*f) = d((r f) .X;

in other words

cp* df = d (,* f), (2.51)

which proves that (2. 49) holds for 0-forms. It suffices by (2. 47) to prove (2. 49) whenw is a monomial, that is, to show that

d(cp* (hdy'(t)A...Ady;(a))) = 0' (d(hdy'(un...Ady'(v)))

Now it follows from (2. 21) and (2. 22) that

d(hd),'(') A ... Ady'(p)) = dhndy'(l) A ... Ady'(v).

and so by (2. 48) and (2. 51), the right side of (2. 52) is

(2.52)

(2.53)

= d(cp*h) Ad(tp*y'(`)) A... Ad(cp*y'(a))

2.9 Exercises 47

= d ((tp* h) d (,* yi(')) A ... A d (cp* y'(p)) )

= d((tP*h) (tP*dyr(i) A... A(*dy'(v)))

= d(tp* (hdy'(1) A... Adyr(c))),

where the second line uses (2. 53), the third uses (2. 48) and (2. 51), and the fourth uses(2. 48). This verifies (2. 52) and completes the proof. tt

2.8.6 Spherical Coordinates Example; Continued

It follows from (2. 48), for example, that

e (dx n dy) = ((D* dx) A (e dy),

and by substituting from equations (2.43) and (2. 44), and performing the usual exterioralgebra operations on the resulting nine terms, we obtain:

r(sin$) 2 (drAdA) - r2sin0cos0 (dO A do).

2.9 Exercises

18. In Example 2.8.4, calculate D* (dy n dz), and verify directly that

e (dx) AV* (dy A dz)

coincides with the formula (2. 46) for 0* (dx A dy n dz).

19. Let U = { (r, A) : r> 0, 0 < A < 2n} c R2, and take the usual radial coordinate map0D (r, 0) _ (rcosO, rsinO) = (x, y). Calculate 0* (dx), 4)* (dy), andV (dx n dy).

20. Let U = { (t, A) : -< t < c*, 0 < 0 < 2n} c R2, and map U into R3 using:

40 (t, 0) = (coshtcosf, coshlsinO, sinht) = (x, y, z)

Calculate V (dx), a (dy), and a (dx A dy).

Remark: This map is a parametrization of the "hyperboloid" x2 +Y 2 - z2 = 1.

21. Consider the map 0: R3 -a R4 given by

(x,y,z,w) = 0D(s,t,u) = (u(s+t),u(s-t),s(t+u),s(t-u)).


Calculate 4)* (dx), e (dy), e (dz) , and e (dx A dy A dz).

Remark: The image of this map is the set { (x, y, z, w) : x2 - y2 = z2 - w2}.

22. Suppose tp: U c R" -+ V C R' is a diffeomorphism between open sets U and V (i.e., asmooth bijection with a smooth inverse).

(i) Show that the push-forward of vector fields, defined in (2. 40), commutes with theLie derivative of vector fields and differential forms (see Exercise 15), in the followingsense: For all X, Y E 3 (U) and co E C2PV,

(P. (LXY) = La, x (tp. )'), (2.54)

N* (Lo. xco) = Lx ((p* w). (2.55)

(ii) Calculate the push-forward of the vector fields a/ar and a/ae under the sphericalcoordinate map 4) of Example 2.8.4, and verify directly that formula (2. 54) holds.

(iii) Using formula (2. 36), calculate the Lie derivative of e (dx) along a/ar inExample 2.8.4, and thus verify (2. 55) directly for co = dx and X = a/ar.

23. (Dynamical interpretation of the Lie derivative) A family of diffeomorphisms6 { m } ofR", where t takes values in some open interval of R including 0, is called the flowl of avector field X on R" if, for all x E R,

X ($, (x)) = a, ()-

(i) Using the fact that $: 0r = Os, t, prove that if Y is another vector field, then

at 1=0'

where the symbolism on the right side uses the push-forward defined in (2. 40).

(ii) By writing 4 * df as;(f 0,) (x) dx', prove that for every f E 920R",

Iax

a(0* df) Ipso = d (Xf) = Lxdf.

(2.56)

(2.57)

(iii) Using the properties of the pullback proved in 2.8.5, derive the formula for theLie derivative of an arbitrary differential form in terms of the flow:

6 A diffeomorphism means a smooth map with a smooth inverse.7 For the existence and uniqueness of the flow, and other technical matters, see books ondifferential equations, or Abraham, Marsden, and Ratiu [1988].


LXw k(0*(0)Ir_o,dwEQW. (2.58)

24. Given a p-form co and a vector field X, the interior product or contraction of X and (0is the (p -1)-form tXw defined as follows: tXw = 0 if p = 0, tX o = CO X ifp = 1, and

(2.59)

(i) Explain why the identities thXw = htXw and txdh = LXh = Xh are valid for everysmooth function h.

(ii) Show, using (2. 13), that for any p-form X and any q-form µ,

tX(A.nµ) = tXAnµ+ (-1)d`gXA1Xt. (2.60)

(iii) Prove, by induction on the degree of to, the identity

LX(O = tX (do)) +d (tX(o), (2.61)

where the left side is the Lie derivative of the form was discussed in Exercise 15.

Hint: It suffices to prove the result for monomials. Write a monomial p-form w as co = df A µfor a (p - l)-form t and use formulas (2. 21), (2.34), and (2. 60).

(iv) Prove that for a smooth function h, LhXw = hLXw + dh n tXw.

25. (Continuation) Suppose U c R" and V c R" are open sets, and tp: U -4 V is adiffeomorphism. Prove that, for every vector field X on U,

tX (tP* w) = (p* (t9. Xw). (2.62)


Differential forms in the sense discussed here originated in 1899 in an article bythe Cartan (1869-1951) and in the third volume of Les Methodes Nouvelles de laMecanique Clleste by Henri Poincard (1854-1912). The program of restating thelaws of physics in an invariant notation, including the use of differential forms, wasinitiated by Gregorio Ricci-Curbastro (1853-1925) and his student Tullio Levi-Civita(1873-1941), and helped to provide the framework in which Albert Einstein(1879-1955) developed the theory of relativity.


The account given here follows along the same lines as Flanders [ 1989]. For much moredetailed information about exterior calculus and its use in mathematical physics, seeEdelen [1985] and Curtis and Miller [1985).

2.11 Appendix: Maxwell's Equations

The purpose of this section is to illustrate the power of exterior calculus to expressphysical laws in a coordinate-free manner. Flanders [1989) gives many other examples.

As in the earlier part of this chapter, the Lorentz inner product may be applied to everycotangent space to R4, and may be expressed in terms of coordinate functions {x, y, z, t}by saying that { dx, dy, dz, dt} is an orthonormal basis for each cotangent space, with

(dxldx) = 1, (dyldy) = 1, (dzldz) = 1, c2(dtldt) = -1, (2.63)

where c denotes the speed of light. Assume that the following "fields" are smooth mapsfrom R4 to R3; according to electromagnetic theory, they are related by Maxwell'sequations in the table below:

Electric Field E = E1 ax + E2- -+ E3a ;cly

Magnetic Field B = B Iax

+ B2 Y + B3a -.

Electric Current Density J = Jt ax + J2ay +j3az

The symbol p in Table 2.3 refers to a smooth map from R4 to R called the chargedensity. For ease of comparison with our earlier calculations, let us subsume thespeed of light into the t variable, which has the effect of setting c = 1. The electricand magnetic fields may be encoded in "work forms"

I aB Faraday's LawcurlE _ -cat

(Electric field produced by a changing magnetic field)

4n I aE Ampere's Law'curlB = J+ pat (Magnetic field produced by a changing electric field)

divE = 4np Gauss's Law

div B = 0 Nonexistence of true magnetism

Table 2.3 Maxwell's equations

We omit the dielectric constant and the permeability, and assume we deal with bod-ies at rest.

2.11 Appendix: Maxwell's Equations 51

mE = E,dx + E2dy + E3dz, mB = B,dx+B2dy+B3dz

and also in "flux forms"

(2.64)

mE = EldyAdz+E2dzndx+E3dxndy, (2.65)

OB = B,dyndz+B2dzndx+B3dxndy,

while the "4-current" is the 1-form:

(2.66)

y = J,dx+J2dy+J3dz-pdt.

It follows from formulas above that

(2.67)

*y = -J,dyAdzndt-J2dxndzAdt-J3dxndyndt+pdxndyAdz. (2.68)

A restatement of the results of an exercise in Chapter 1 gives:

* (dxAdt) = dy A dz, * (dy A dt) = dzAdx, * (dzAdt) = dxAdy,

* (dyAdz) = -dxAdt, * (dzndx) _ -dyAdr, * (dxAdy) = -dzAdt;

and from these formulas, the reader may check that

* (mEAdt) =4E,*4B = dtAtB. (2.69)

Define a 2-form

T = G3EAdt+$B; (2.70)

dn = dti3EAdt+d4)B. (2.71)

Noting that d(E,dx) = aE,/ay(dyAdx) +aE,/az(dzndx) + aE, /at (dt A dx),etc., the reader is invited to verify, by the methods of this chapter, that the right side of(2. 71) is equal to

dmEAdt+d4)B = 4)cur,EAdt+pmvB+dtA B, (2.72)

where B; = aBi/at, using the work-form, flux-form, and density-form notation. Thusthe first and fourth of Maxwell's equations together are equivalent to

dt) = 0. (2.73)

On the other hand, (2. 69) implies that * Tj = E + dt A 3B, and so the rules for exteriordifferentiation give:


d(*r) =

and a calculation similar to the one above shows that the second and third of Maxwell'sequations together are equivalent to

*d(*Tl) = 4ny,

using the fact, deduced from the methods of Chapter 1, that * (* y) = y. It turns outthat a deeper understanding of the theory is obtained by replacing (2. 73) by a certainequation which implies it; Maxwell's equations can be solved by finding anelectromagnetic potential a, which means a 1-form A idx + A2dy + A3dz + Aodt, suchthat

da = Tl, * d (* TI) = 4ny. (2.74)

Note that, by properties of the exterior derivative, da = T1 dTl = d (da) = 0.Theconcise formulation (2. 74) is more than just a clever trick. It is an intrinsic formulationof natural law, independent of any coordinate system, whereas the original formulationof Maxwell's equations is specific to the {x, y, z, t} coordinate system. Moreover, as weshall see in Chapter 10, (2. 74) arises naturally as the solution of a variational problem,phrased in geometric language; indeed, it is a special case of so-called gauge fieldtheories in mathematical physics, which will be described in Chapter 10.

2.11.1 Exercise

26. (i) Verify in detail that (2. 72) is a correct expression for dTl.

(ii) Calculate in detail d OE - dt n dG 8, and check that this is 4n (* y) under Maxwell'sequations.

3 Submanifolds of Euclidean Spaces

The main purpose of this chapter is to introduce a class of concrete examples - thesubmanifolds of Euclidean space - in order to motivate the definitions of abstractmanifold, vector bundle, etc., in later chapters. The primary technical tool in much ofthis work will be the Implicit Function Theorem, which will be restated in geometriclanguage.

3.1 Immersions and Submersions

Before we can define submanifolds of R' +k we need to consider carefully the notion ofthe rank of the derivative of a mapping; to avoid constant mention of degrees ofdifferentiability, we work with smooth mappings throughout.

For W open in R", a C' map `F: W - * R" +k is called a (smooth) immersion if itsdifferential d`F (u) E L (T R" -> Ty, (.) R" +k) is a one-to-one map at every u e W.Here are two statements, each of which, according to linear algebra, is equivalent to thedefinition of immersion:

for every u e W, multiplication by the (n + k) x n derivative matrix

DIP (u) =a%pn+klau, ... a'F"+kiau"

av/au, ... avlau"(3.1)

induces a one-to-one linear map from R" to R" +k

for every u E W, DT (u) has rank n, which here means that it has n linearlyindependent rows (or equivalently, n linearly independent columns).

54 Chapter 3 Submanifolds of Euclidean Spaces

Now we introduce the dual notion of submersion. Let U be open in R"+k. and suppose

f: U -* Rk is a smooth map. To keep things in their proper place, think of the followingdiagram:

WcR' 41 Rn+k;2 U fRk

For the case n = 2 and k = 1, Figure 3.8 on page 64 may be suggestive.

We say that f is a submersion if its differential df (x) e L (T1Rn+k Tf(x) Rk) is anonto map; according to linear algebra, two equivalent statements are:

multiplication by the k x (n + k) derivative matrix Df (x), where

of /ax, ... af'/ax"+kDf(x) =

Laf/lax, ... af'Iax" +

(3.2)

induces an onto map from R" +k to Rk for every x e U.

for every x e U, the derivative matrix Df (x) has rank k, which here means that ithas k linearly independent columns (or equivalently, k linearly independent rows).

3.1.1 Examples of Immersions

Let us consider the case where n = 2 and k = 1. Let h e C" (R) be a strictly positivefunction, and consider the map T: W = R2 - R3 which rotates the curve x = h (z)infinitely many times around the z-axis, namely,

'P (u, 0) = (h (u) cosO, h (u) sinO, u) . (3.3)

This gives what is known as a (parametrized) surface of revolution. In this case

h' (u) cos0 -h (u) singD'Y(u,0) = h'(u)sing h(u)cosO

1 0

By inspection of the last row, the only way that the two columns of this matrix could belinearly dependent is for the second column to be zero; this is impossible becauselI [-h (u) sing, h (u) cos0, 0] T11 = h (u) > 0. Hence 'Y is an immersion. Note,incidentally, that neither the map '1' nor the map (u, 0) -+ D'Y (u, 0) is one-to-one; itis merely that, for fixed (u, 0), the linear transformation D'I' (u, 0) is one-to-one.


On the other hand the derivative of the map 4 (s, t) = (s, s, st) from R2 to R3 is a3 x 2 matrix with second column [0, 0, s] T, which is zero on { (s, r) : s = 0} . Thus (is not an immersion.

3.1.2 Examples of Submersions

When we consider the case where n = 2 and k = 1, and U is open in R3, thenf: U --4 R is a submersion if its derivative does not vanish on U. For example, let U bethe set { (x1, x2, x3) :x, +xZ+x3 >0}, and let f (x1, x2, x3) = x,+x2+x3. ThenDf (x1, x2, x3) = 2 [x1, x2, x3] , which is never zero on U, and so f is a submersion. Onthe other hand the map g (x,, x2, x3) = has derivative [x2x3, x,x3, x,x2], whichis zero, for example, at (0, 0, 1) E U, and so g is not a submersion on U.

The complementary notions of immersion and submersion are so important that wesummarize the information above in the following table, using the terminology ofdifferentials introduced in Chapter 2.

Domain and range

Defining property

Derivative matrix is

... whose rank should be

... or in other words

Immersions`Y:WgR"-_R"+k f:UcRn+k-4Rk

d`P (u) is 1-1 `due W df (x) is onto `dx E U

(n+k) xn kx (n+k)rank=n rank=k

columns are independent rows are independent

Submersions

Table 3.1 Immersions and submersions

3.2 Definition and Examples of Submanifolds

Our primary goal is to define a class of subsets of R"+k which are "smooth" and"locally n-dimensional," in some sense which we shall try to make precise. Whenn = 2 and k = 1, this class should include certain parametrized surfaces such as thesphere, torus, and ellipsoid; it should also include open subsets of the subspace R2, suchas the {x, y}-plane with (0, 0) deleted. On the other hand this class should excludeanything with sharp corners, such as the surface of a cube or of a cone, and anythingwhose "dimension" could be said to vary, such as

{(x,y,0):-x2<y<x2} v {(0,0,0)} cR3,

which collapses from being "2-dimensional" to "1-dimensional" at (0, 0, 0) . There isalso a topological subtlety in the definition which will be apparent a little later.


We have two main choices when it comes to giving the definition of an n-dimensionalsubmanifold M of R"+k The first choice is to try to generalize the vector calculusnotions of "parametrized curve" and "parametrized surface," and say that M has to bethe union of images of certain one-to-one immersions W1: W1 c R" -* R"+k, called

n-dimensional parametrizations, which are discussed in Section 3.4; this would give thesame end result as the definition below, but it requires a lot more work, because

it would put us under the obligation to construct the parametrizations in order tocheck that M is a submanifold;

we would have to include a topological condition on each of the immersions, whichis tedious to state and to verify.

Instead we choose a second way using submersions, which is slick but initiallyincomprehensible; we shall try to explain heuristically why it works for surfaces; thenwe shall show that it is equivalent to the first kind of definition, using the ImplicitFunction Theorem.

3.2.1 Definition of an n-Dimensional Submanifold of R"+k

A subset M of R" +k is called an n-dimensional submanifold of R" +k if, for each x e M,there is a neighborhood U ofx in R"+k and a submersion f: U -* Rk such that

UnM=f '(0). (3.4)

Note incidentally that in the trivial case k = 0, the condition that f be a submersion isvacuous, and so any open subset of R" is an n-dimensional submanifold of R".

Figure 3. 1 The level surface of a submersion

The figure above is supposed to help in understanding this definition, at least whenn = 2 and k = I. For f to be a submersion amounts to saying that gradf (x) is anonzero vector at every x in U. The reader may recall from multivariate calculus, orderive directly, that existence of gradf (x) implies that there exists a tangent plane tothe level surface f = 0 at x, namely, the plane through x normal to gradf (x). Thus the


Figure 3. 4 Graph of a function on an annulus in R2

3.2.6 A Nonexample of a Submanifold

0.5 0.5

- .5

Figure 3. 5 A curve that is not a submanifold of R2

Let P = { (sins, sin4t) a R2: 0 S t!5 27t}, which can also be expressed using theaddition formulas as P = g-1 (0), where

g (x, y) = y2 - 16x2 (1 - x2) (1 - 2x2) 2. (3.9)

A picture of P is presented Figure 3.5.

Warning! Showing that g fails to be a submersion at (0, 0) is not sufficient to provethat P is not a submanifold, because it does not address the possibility that there mightbe some other submersion f on a neighborhood U of (0, 0) such that P n U = f' (0).A correct argument to show that P is not a 1-dimensional submanifold of R2 is to notethat if f is a smooth function on any neighborhood U of (0, 0) , or of (± 1 / F, 0), with


P n U = f' (0) , then there are two tangent directions in R2 along which f is zero;hence f cannot be a submersion. The reader may work out the details of this argument inExercise 2.

3.3 Exercises

1. Determine whether each of the following functions is an immersion and/or asubmersion,' for the domain given:

(i) f (u, v) = (u/v, u, u2); domain (0, oo) x (0, co).

(ii) f (u, v, w) = (uv, vw, wu); domain (0, oo) x (0, co) x (0, co).

(iii) f(u, v, w, z) = (uv - wz, u - z); domain (0, 00) x (0, oo) x (0, cc) x (0, co).

2. Suppose M is made up of the edges of a nondegenerate triangle in R2. Prove that M isnot a 1-dimensional submanifold of R2.

Hint: Suppose that f is a smooth function on a neighborhood U of one of the vertices x such thatf- t (0) = U n M. By taking directional derivatives off in the directions of the two edgesmeeting at x, show that f cannot be a submersion.

3. Which of the following curves, pictured below, are 1-dimensional submanifolds of R2?Justify your answers carefully.

(i) The graph M = { (x, y) : y2 = x2 - x4} = { (sin t, sin2t) : 0 < t < 2n}.

(ii) The circle { (x, y) : x2 +y2 = 11 without the point (0,1).

t A mapping which is both an immersion and a submersion is called a regular mapping;obviously domain and range must have the same dimension, and for the map to be regular meansthat its derivative is a nonsingular matrix at every point.

3.4 Paremetrizations 61

4. Show that the torus 7", as in (3. 6). is an n-dimensional submanifold of R2".

3.4 Parametrizations

This notion is a generalization of the notions of parametrized curve and parametrizedsurface, which the reader may have encountered in multivariable calculus. Suppose M isa subset of R"+k. An n-dimensional parametrization of M means a one-to-oneimmersion `I': W -a U c R" +k, where W is an open subset of R", U is an open subset ofR"+k with UnM*O,and`l'(W) = UnM.IfyE UnM,werefertothisasann-dimensional parametrization of Mat y, to emphasize that only part of M is actuallybeing parametrized. The image of a 1-dimensional parametrization is called aparametrized curve, and the image of a 2-dimensional parametrization is called aparametrized surface.

For example 0 -t' (cos0, sin0), with domain (0, 2n), and range U = R2 - (1, 0),gives a 1-dimensional parametrization of the unit circle in R2.

3.4.1 Caution! The Image of a Parametrization Need Not Be a Submanifold2

The standard counterexample is the "figure eight" M = { (x, y) : y2 = x2 - x4},pictured in Exercise 3. This is not a submanifold of R2, but it is the image of theparametrization IF (t) = (sins, sin2t), with domain (0, 2n). The reader may checkthat the derivative of `l' never vanishes.

2 Note that some other authors use a stronger definition of parametrization, so that it is anembedding in the sense of Chapter 5. However, we have chosen the present definition for the sakeof consistency with multivariable calculus notions of "parametrized surface," etc. According tosome authors, a parametrization need not be I - I; for example the map from R to the unitcircle which takes x to (cos x, sin x) would be admissible.


Suppose M is a subset of R" +k and g: U -+ Rk is a submersion on some open subset Uof R"' k such that U n M = g-' (0). Given y E U n M, we can rearrange the labelingof the coordinates in R"+k in such a way that there exist open sets W C; R" and V Q Rkwith y e W x V c U, and a smooth function h: W - V (the "implicit function ") suchthat the map 'P: W -4 W x V given by

'P (x) = (x,h(x)) (3.14)

is an n-dimensional parametrization of M at y; that is, `P is a one-to-one immersion with'P (W) = (W X V) n M.

Proof (may be omitted): The k x (n + k) derivative matrix Dg (y) has rank k since g isa submersion. By relabeling the coordinates in R"+k if necessary, we can arrangematters so that the last k columns of Dg (y) are linearly independent. If we express anelement of R" +k = R" X Rk in the new labeling system as (x, z), where x E R, z e R',then we may group these last k columns of Dg (y) into a k x k matrix DZg (y); the firstn columns of Dg (y) are denoted Dsg (y) .

Consider the map F: U -+ R" X Rk given by F (x, z) = (x, g (x, z) ); differentiatinggives

DF(x, z) =LDxg

1]

(x, z),DZg

and therefore

IDF(y)I = IDZg(y)I *0.

Applying the Inverse Function Theorem 3.5.1, we see that there exists a neighborhoodU' of y = (x0, za) with U' g U, such that the restriction of F to U' is adiffeomorphism onto its image U" = F (U') 3 (xe, 0). Take open sets W c R" andV g Rk with W x {0} (Z U" and y E W x V c U', and define h: W-+ V by theequation

'P (x) = (x,h(x)) = F'(x,0). (3.15)

We see that 'P is a one-to-one immersion because F' is a one-to-one immersion on U".For every (x, z) E (W x V) n M, we have g (x, z) = 0, and thereforeF (x, z) = (x, 0); thus (x, z) = 'P (x) by (3. 15). n

3.5.3.1 CorollaryFor g and'P as above, Im dP (x) = Ker dg ('P (x)) 3 for all x E W.

3 The "kernel" Kerdg (y) = {, E T,R" +k: dg (y) 4 = 0}.


Proof: Apply the Chain Rule to the function g 'P, which is identically zero on W. Weobtain dg ('P (x) ) d'' (x) = 0, which implies Imd'P (x) Q Kerdg ('Y (x) ).However, since 'P is an immersion and g is a submersion, the vector subspacesImdW (x) and Kerdg ('P (x) ) of T,t,(X)R"+k satisfy

dim (Imd'P(x)) = n = dim (Kerdg(`Y(x)));

therefore they must be identical. ]a

The notion of the tangent plane to a surface is probably familiar from multivariablecalculus. Although one may continue to visualize a "tangent space at a pointy" in asubmanifold M as if it were actually resting against the submanifold at y, themathematical definition below of a tangent space implies that it is always a subspace ofR + k, that is, it passes through zero and not necessarily through y. The main idea is thatthis tangent space at y is something intrinsic to the manifold, and not dependent on thechoice of submersion or parametrization used to describe the submanifold around thepointy.

3.5.4 The Tangent Space to a Submanifold at a Point

If M is an n-dimensional submanifold of R" +k, and if f: U -> Rk and g: U' - Rk aretwo submersions defined on neighborhoods of y e M such that U r M = f' (0) andU' n M = g ' (0), then the kernel of df (y) and the kernel of dg (y) are the samen-dimensional subspace of TR" +k; in symbols,

Kerdf (y) = {4 a T)R"+k: df(y) = 0} = Kerdg(y) c TTR"+k. (3.16)

This subspace is called the tangent space to Mat y, denoted T,M; it may also beexpressed as Im dY' (u) = Span {D,'P (u), ..., D"'Y (u) }, where u = 'P'' (y).

Proof: Take an implicit function parametrization'P associated to f as in the proof of3.5.3. By the reasoning found in the proof of 3.5.3.1, we see that, since f 'Y and g 'Pare identically zero on their respective domains, we have

Ker df (y) = Im d'P (T-' (y) ) = Kerdg (y)

as desired. The final assertion follows from 3.5.3.1.

3.5.4.1 Example of Computing a Tangent Space"Find the tangent space to the torus T2 a R4 at the point x = (0, 1, 0.6, 0.8). "

Solution: Recall from (3. 6) that T2 = g-' (0) for the submersion

n

g(X1,x2,x3,x4) = (XI+X2-1,x3+X4-1).


One may easily compute that

Dg (x) = 02 0 00 0 1.2 1.6J

The tangent space at x = (0, 1, 0.6, 0.8) is the kernel of this linear transformation, thatis,

TTM = { (v1, v2, v3, v4) : v2 = 0, 1.2v3 + 1.6v4 = 0}.

Alternative Solution: In the parametrization `I': (0, 4)) -+ (cosO, sine, toss), sin4)),we have

a'P alp-sine 0cose 0

o -sin,0 cos41

At the specific (9, $) whose image under `P is i = (0, 1, 0.6, 0.8). the seconddescription of the tangent space shows that

a'PalpT;M = Span { a9

(9, ) , a (9, ) } = Span

-1

0

0

0

00

-.80.6 i

which is the same result as before. rx

The next assertion is important because it relates n-dimensional submanifolds of R"'kto the abstract manifolds discussed in Chapter 5. It says that, when we "changevariables" from one parametrization to another, this change is always smooth.

3.5.5 Switching between Different Parametrizations

Suppose M is an n-dimensional submanifold of R" +k, Ut and U2 are open sets in R"",and `P.: Ws Q R" -, U. c R"+k are n-dimensional parametrizations of M at y, for i = 1,2, such that U, n U2 n M is nonempty. Then both the domain and the range ofthe map

(3.17)

are open sets in R", and the map itself is a diffeomorphism (see Figure 3.9).


UtnU2nM

/-7 x,21.W, ZS1W, r9' (U2) W2n'P2' (U1)

Figure 3. 9 Switching from one parametrization to another

Proof (may be omitted): Since IF, is continuous and U2 is open, 'I' (U2) is open;therefore so is W, n'P r' (U2) , since a finite intersection of open sets is open and Wt isopen. The openness of W2 n'P2' (U,) follows similarly. By definition, 't and `P2 are1-1 maps, and therefore the map in (3. 17) is well defined and 1-1. It remains to provethat it is smooth at an arbitrary point x in the domain. Let y = `P, (x) a M. Accordingto the proof of 3.5.2, there exists a local parametrization IV: V c R" -> Uo n M Q R"+k

of M at y of the form (possibly after rearranging variables)

4) (u) = (u, h (u) ),

as in (3. 14). Note that the inverse of 1 is the smooth map which projects a vector inR" +k onto its first n components. Therefore if Q denotes Uo n U, n U2 n M, then4>-' TI: `l']' (Q) - $-' (Q) is a smooth map, and D ((b-' 'Ps) (x) is nonsingularbecause, by the chain rule,

D (4-' 'P1) (z) = D!-' (Y) DTI (x) ,

and both the matrices on the right are of rank n. The range of the first transformationis the domain of the second one. By the Inverse Function Theorem, the inverse(0-' 'P `Pr' 4' is a diffeomorphism on a neighborhood of 4) (y) , andconsequently

T21 'P' = (q'21 4') (40-1 'P1)

is smooth on a neighborhood of z being the composite of two smooth functions. rx

3.6 Matrix Groups as Submanifolds 69

3.6 Matrix Groups as Submanifolds

In the sequel, we shall use the notation R""m to denote the vector space of real n x mmatrices, which may be identified with R". Some of the most important examples ofsubmanifolds of Euclidean spaces are certain infinite groups of matrices known asLie groups,4 under the operation of matrix multiplication. Our first example is:

3.6.1 The General Linear Group over R

The general linear group GL" (R) c R"", and its subgroup GL* (R), consist of non-singular matrices; they are defined by

GLn (R) = {A E R""": IAI * 0}, GL+ (R) = {A E R"Xn: JAI > O}. (3.18)

Since the map A - Al Iis continuous, both of these groups are open subsets of R""",and hence are n2-dimensional submanifolds of R""". An important fact about thegeneral linear group is the following:

3.6.2 Smoothness of Matrix Multiplication and Inversion

The maps (A, B) - AB and A -,A-' are smooth on GL" (R) x GL" (R) andGL,, (R), respectively.

Proof: Matrix multiplication must be smooth, because it consists only of a sequence ofmultiplications and additions of matrix entries, and polynomials are smooth functions.As for inversion, we note that for any A e GL" (R) and any H e R""", and for allsufficiently small E, A - EH E GL,, (R), and the series expansion below converges

(A - eH) -' = (I - EA-' H) -'A-'

_ (I+EA-'H+ (EA-'H)2+...)A-';

here I denotes the n x n identity matrix, and convergence of a series of matrices refersto convergence of every entry. This gives an infinite-order Taylor expansion for the mapA -, A-, which verifies smoothness. tt

3.6.3 Lie Subgroups of GL" (R)

The study of abstract Lie groups is beyond the scope of this book. However, we say thatG is a Lie subgroup of the general linear group if the following two conditions hold:

4 Named after the Norwegian mathematician Sophus Lie (1842-99).


G is an algebraic subgroup of GL" (R), that is, the inverse of every matrix in G andthe product of two matrices in G are in G;

G is a submanifold of R" x"

Examples include the following.

The special linear group SL" (R) = {A a R" x":IAI = 11; this is an algebraicsubgroup because the determinant of the product of two matrices of determinant 1also has determinant 1; it is a submanifold of dimension n2 - I because it is theinverse image of 1 under the map A - IAI, which is a submersion (proof suggestedin Exercise 11).

The orthogonal group O (n) a R"'", and its subgroup the special orthogonalgroup SO (n), consist of orthogonal matrices; they are defined by

O(n) = {A a R"x":ATA =1}. (3.19)

SO(n) = {Ae R"x":IAI = 1,ATA=I}, (3.20)

where AT denotes A transpose. Of course, SO (n) = O (n) n GL* (R). It is clearthat the orthogonal and special orthogonal groups are algebraic subgroups ofGL (R); to show that they are Lie subgroups of GL" (R), we need:

3.6.4 The (Special) Orthogonal Group Is a Submanifold

O (n) and SO (n) are n (n - 1) /2-dimensional submanifolds ofRnx"

Proof: We shall carry out the proof for SO (n); Since a similar argument applies tomatrices in 0 (n) with determinant of -1, it follows that 0 (n) is ann (n - 1) /2-dimensional submanifold of R"" also.

Let Sym (n) denote the symmetric n x n matrices, that is, { B e R" x": B = BT }; this isa vector space of dimension n (n + 1) /2. Define

f:GLn (R) - Sym(n),f(A) = ATA-1. (3.21)

Recall that GL+ (R) is open in R"", and note that SO (n) = f' (0). Hence the resultfollows if we can show that f is a submersion. Observe that if II HII denotes the squareroot of the sum of squares of entries in the matrix H, and if we write a function g (x) aso(x) to mean that g(x)/x-+0 as x-a0,then

f(A+H) -f(A) = (A+H)T(A+H) -ATA = HTA+ATH+o(IIHII);

:.Df(A)H = HTA+ATH.

3.7 Groups of Complex Matrices 71

Since the tangent space to Sym (n) at B e Sym (n) is just another copy of Sym (n), toshow that df is onto it suffices to show that, for every S E Sym (n) and everyA E SO (n), there exists an H such that Df (A) H = S. Choosing H = AS/2 gives

Df(A)H = (1/2) (SATA+ATAS) = S,

which verifies that Df (A) is onto provided A E SO (n). Moreover it is also of full rankon a neighborhood of A, on which f is therefore a submersion. Since f maps from anopen subset of an n2-dimensional space into an n (n + 1) /2-dimensional space, itfollows that the dimension of SO (n) = f- (0) is n (n - 1) /2. tt

3.7 Groups of Complex Matrices

Since a complex number may be regarded as a pair of real numbers, the group ofnonsingular n x n matrices with complex entries, or complex general linear group,denoted

GL,, (C) = {AE C"': JAI * 01,

can be regarded as an open subset of R""" ® R' x ", with the associated differentiablestructure; here we use the fact that

AE C"""c A=X+iY,X,YER"x". (3.22)

As in the real case, matrix multiplication and inversion are smooth operations onGL" (C) x GL" (C) and GL" (C), respectively, and algebraic subgroups of GL" (C)that are also submanifolds of R` 9 R` are called Lie subgroups of the complexgeneral linear group. Two of the most important of these are:

The complex special linear group, SL" (C) = {A E C"": JAI = 1 }, which is asubmanifold of dimension 2n2 - 2; for the proof, see Warner [ 1983].

The unitary group, U (n) = (A E C"": AAT = !}, where A means the matrix inwhich every entry of A is replaced by its complex conjugate; this is a submanifold ofdimension n2 (see 3.7.1).

The special unitary group, SU (n) = U (n) n SL" (C), which is a submanifold ofdimension n2 - 1; this fact will be proved in the exercises of Chapter 5.

3.7.1 The Unitary Group Is a Submanifold

U (n) is an n2-dimensional submanifold of R" "" ® R" " ".

Proof: This follows along the same lines as 3.6.4. Let Sym (n) denote the realsymmetric n x n matrices, and let o (n) denote the real antisymmetric n x n matrices(this is not the same as the "tittle o" notation in the proof of 3.6.4!); these are vectorspaces of dimension n (n + 1) /2 and n (n - 1) /2, respectively. Let


Sym(n) e ( )o(n) = {S+1We C"":Se Sym(n),WE o(n)}, (3.23)

which is an n2 -dimensional space of matrices. Define a map

fR"x"®R"x"- Sym(n) ® (,/- 1)o(n);

f(X,Y) = (X-iY)T(X+iY) -1.

(The reader should check that the real part of the image off is symmetric, and thecomplex part antisymmetric.) Observe that f' (0) is identifiable as U (n) under theidentification between (X, Y) E R` (D R""" and X + iY a C"". A similarcalculation to the one in the proof of 3.6.4 shows that

Df(X,Y)(H,K) = (H-iK)T(X+iY)+(X-iY)T(H+iK). (3.24)

As before, it suffices to show that f is a submersion on the set where

(X-iY)T(X+iY) -1 = 0. (3.25)

Given a specific (X, Y) satisfying (3.25), and an arbitrary S r. Sym (n) , W e 0(n), wechoose H and K by the equation:

H+iK =(X+iY) (S+iW)

2

Using (3. 24), (3. 25), and the fact that ST = S, WT = -W, the reader may easily verifythat for this choice of H and K, Df (X, Y) (H, K) = S + iW, which verifies thatDf (X, Y) is onto as desired. Finally, the dimension of U (n) must be

dim (R""" ED R""") - dim (Sym (n) ®( ) o (n)) = 2n2 - n2,

giving n2 as claimed. tt

3.8 Exercises

5. (i) Using the notation of (3.8), verify that H,' is an n-dimensional submanifold of R"

for c # 0, but is not a submanifold when c = 0.

(ii) Sketch Hi , and find a parametrization for H H.

6. Verify that the mapping 'Y (0, $) = ((a + cos0) cos0, (a + cos0) sin0, sin0), usedto parametrize the 2-torus in R3, is indeed an immersion for a > 1. Use it to calculatethe tangent space to 7-2 a R3 at the point (-0.6 (a + 1), -0.8 (a + 1), 0).

3.8 Exercises 73

7. Let W = {(u,0):-1<u<l,0<0<2it}.(i) Verify that ((u, 8) = ((1 + u2) COs(), (1 + u2) sin0, u) is a 2-dimensionalparametrized surface obtained by rotating the graph of x = 1 + z2 about the z-axis inR3.

(ii) Show that 'Y (u, 0) = (u2cos0, u2sin0, u) fails to be an immersion at (u, 0) _(0, n/2).

8. The graph of the smooth function h : W ie R" -> V r. Rt, where Wand V are open sets, isthe set

M= { (x, z) E R" x Rk: z = h (x) } c R"+k (3.26)

Show that M is an n-dimensional submanifold of R"+k, and calculate its tangent space

at an arbitrary point (xo, zo).

Hint: Consider the function g (x, z) = z - h (x).

9. Suppose 'F: (a, b) c R -+ R2 is a one-to-one immersion, so C = IF ((a, b)) is aparametrized curve in R2 (Warning! C is not necessarily a 1-dimensional submanifoldof R2; see 3.4.1). Prove that, for every t E (a, b), there exists a subinterval (a', b')with a:5 d< t < V5 b, such that C' = 'Y ((a', b')) is a 1-dimensional submanifoldof R2.

Hint: Apply the Implicit Function Theorem 3.5.3 to the submersion

g (x. Y, S) = (`FI (s) -x, `Y2 (s) -Y)

to construct an implicit function parametrization of C of the form y = h (x) or x = h (y) nearIF (t) ; now g (x, y) = y - h (x) or x - h (y) is a submersion with which we can complete theproof.

10. (Continuation) Extend the method of Exercise 9 to show that, given a one-to-oneimmersion': W C R" -, R"+k, and given any u E W, there exists a neighborhood Wof u in W such that M = '1' (W) is an n-dimensional submanifold of R"+k

11. (i) Prove that SL2 (R) is a 3-dimensional submanifold of GL+ (R), and calculate thetangent space to SL2 (R) in R2 X 2 at the identity.

Hint: Consider f (x, y, z, w) = xw - yz.

(ii) By considering IA + HI - IAI when H = LA, prove that SL" (R) is a submanifoldof GL4 (R) of dimension n22- 1

12. Construct a 3-dimensional parametrization for SO (3) at the identity I using thefollowing steps:

(i) Denote the set of skew-symmetric n x n matrices by

74 Chapter 3 Submanffolds of Euclidean Spaces

o(n) = {KE R"x":K=-KT}. (3.27)

Define the Cayley transform of K E o (n) to be 'P (K) = (I + K) (I - K) -1; notethat (1- K) -1 is certainly well defined for K in a neighborhood of 0, since the mappingA -+ A-' is smooth.5 Prove that if B = `Y (K), then B is orthogonal, that is, BTB = 1.

Hint: Use the fact that (1 + K) (1- K) _ (I - K) (I + K), and that the inverse of A T is thetranspose of A-.

(ii) By comparing T (K+ H) with `P (K), find D`P (K) H and show that `P is animmersion, at least on a neighborhood of 0.

Hint: This is similar to the calculation in the proof of 3.6.4.

(iii) By finding a formula for K in terms of B = `P (K), show that `P-t is well defined,at least on U n SO (n), where U is a neighborhood of I in R"', and therefore ifW = `P-t (U n SO (n) ), then 'F: W -+ U n SO (n) is one-to-one and onto.

(iv) Now verify that the following composite mapping, restricted to a neighborhood of 0in R3, yields a 3-dimensional parametrization for SO (3) at the identity:

(u, v, w) E R3 -+O u

V

l

U

v l -u -v1[_Iu 0 w- -u l w

IUIW--v

-w 0 -v -w 1 v w

13. Show that the tangent space to SO (3) in R313 at the identity can be identified witho (3) (as defined in (3. 27)).

14. Let 1" denote the n x n identity matrix. The symplectic group Sp (2n) is defined by

Sp(2n) _ {AE R2nx2":ATJA =J},J = 0 1"-I" 0

(i) Prove that Sp (2n) is a submanifold of R2n x 2", and find its dimension.

Hint: Note that the map A -* ATJA - J takes values in the set o (2n) of 2n x 2nskew-symmetric matrices, that is, the matrices { K e R2n x 2n: K = -K} .(ii) Show that, in the case n = 1, Sp (2) is identical with SL2 (R), the set of 2 x 2matrices with determinant equal to 1.

15. The Lorentz group 0 (n, 1) is the set of real (n + 1) x (n + I) matrices that preservethe inner product (x{x) x2 + xi + ... + x, ; in other words,

5 In fact 1- K is nonsingular for all K e o (n); see Lancaster and Tismenetsky [1985]. p. 219.

3.8 Bibliography 75

O(n, 1) = {AE R("+i)xtn+l):ATJA=J},J= [_01

Ol

1"'(3.28)

where 1" denotes the n x n identity matrix. Show that 0 (n, 1) is an m-dimensionalsubmanifold of R ("

+ 1) x (" + 1)for some m, and find m.

16. Prove that SL2 (C) (i.e., the group of 2 x 2 matrices with complex entries and withdeterminant equal to 1) is a 6-dimensional submanifold of R2x2 G R2 x 2. Calculate thetangent space to SL2 (C) at the identity.

3.9 Bibliography

A fuller treatment of submanifolds of R" is given in Berger and Gostiaux [ 1988]. Ofcourse, all general results about differential manifolds (see Chapter 5) apply inparticular to this special case. Conversely, it turns out that any finite-dimensionalmanifold which can be covered by a countable number of charts can be expressed as asubmanifold of R" for sufficiently large n (Whitney's embedding theorem), andtherefore there is only a loss of elegance, not of generality, in studying submanifolds ofR" instead of abstract manifolds. For more on both Implicit Function Theorems and onLie groups, Warner [ 1983] is recommended. For a lively account of Lie groups andLie algebras, with a lot of applications, see Sattinger and Weaver [1986]; for a moreadvanced treatment, see Helgason [1978].

4 Surface Theory Using MovingFrames

The method of the repPre mobile (moving frame) - invented by G. Darboux and usedextensively by the Cartan - is an extremely powerful technique in differentialgeometry, which we shall not discuss in full generality. Instead we shall focus mainly onsurfaces in 3-space, with the intention of building intuition for the later work onconnections in vector bundles.

4.1 Moving Orthonormal Frames on Euclidean Space

Recall from Chapter 3 the notion of the special orthogonal group, SO (n). Suppose W isopen in Rk, and Y': W -* R" is a one-to-one immersion, in other words a k-dimensionalparametrization of M = Y' (W). A moving orthonormal frame on this parametrization(or, more loosely, a moving orthonormal frame on M) simply means a map

-: W-,SO(n) cR"' (4.1)

that is smooth, considered as a map into R""". A special case of this is where W is anopen subset of R" and `Y is the identity map, in which case we simply have a movingorthonormal frame on W Let us give this a geometric interpretation. In terms of a fixedorthonormal basis for R", the matrix

E11 (u) ...n(u)UE W, (4.2)

is orthogonal, that is, 8 (u) TE (u) = I, which is the same as saying that the columnvectors

4.1 Moving Orthonormal Frames on Euclidean Space 77

(u), .., (u) ] T, i = 1, 2, ..., n, (4.3)

satisfy (, J) = 6, ; in other words { l;j (u), ..., " (u) } forms an orthonormal basisfor R". For the sake of understanding, take the case where W is an open subset of R"and P is the identity map, and consider these vectors as an orthonormal basis of thetangent space at x by identifying with the vector field E,, where

(x) (x) ax- + ... + (x) ax". (4.4)

in terms of an orthonormal coordinate system {x" ..., x"} for R". In brief, a movingorthonormal frame should be conceptualized as a smooth assignment of an orthonormalcoordinate system for the tangent space at every point of M, as in Figure 4. 1.

x3 t

x2

Figure 4. 1 A moving orthonormal frame on R3

Let IF _ [x' (u), ..., x" (u) ] T denote the position vector of the point x = `I' (u) as afunction of u E W. Applying exterior differentiation to each of the functions x', andexpressing the results as a vector of 1-forms (or "vector-valued 1-form") gives

d4' _ [dxr, ..., dx"] T .

Define a column vector A = [Ar, ..., A"] T of 1-forms by

(4.5)

A

(u) = E (u)Td4' (u). (4.6)

or e = _T4 for short. Since (u) TE (u) = 1, it follows that . (u)T = E (u)-t, so(u) E (u) T = 1. It follows that d`I' can be expressed in terms of the orthonormal

basis {4t (u), ..., f;" (u) } as follows:

d4' = (u)E(u)TdP = -e. (4.7)

78 Chapter 4 Surface Theory Using Moving Frames

Likewise exterior differentiation of all n component functions of the basis vector j (u)yields a column vector dT; of 1-forms; arrange these in the form of a matrix to obtain

d° = [ d , . . . , dTn). (4.8)

Now define an n x n matrix of 1-forms,

known as connection forms, by the equation

ID= Td-= =

or, in other words,

r

41

[di,.....dr.j; (4.10)

(4.11)4Td4j = (Qd4j).

We often write the equation defining O in the format:

d===0.Since {1(u), ...," (u) } is an orthonormal basis, the dot product satisfies(;I = 6.,, and therefore

(4.12)

W + W = (,)d4j) + (d t) = d (b1,S1) = 0. (4. 13)

In other words, to is skew-symmetric: O + OT = 0.

4.2 The Structure Equations

The first and second structure equations are obtained by exterior differentiation) of theequations for d`Y and dfi, respectively. Applying d to (4. 7) gives:

1 To reassure the reader about exterior differentiation of wedge products of matrix-valueddifferential forms, note that if A = (ais an I x m matrix of p-forms, and B = (bjk) is anm x n matrix of q-forms, then

d(AAB)Ik = d(ya;jnbjk) = Ida,,Abjk+(-1)PY a;jndbjk.i i j

4.3 Exercises 79

a a a0 = d(d`I') = d(80) = (d8A0) +Ed0. (4.14)

Using (4. 12) and grouping the terms on the right gives

a a a a8mA0+!de = '(on0+do) = 0. (4.15)

Now premultiply both sides by =T, and use the identity E (u) TE (u) = 1, to obtain thefirst structure equation:

a a

de = -MAO. (4.16)

Similarly, applying d to (4. 12), and applying d? = Em again, gives

0 = d(dc) = d5Am+-dm = .oA o+Edo. (4.17)

Now premultiplication by `T as before gives the second structure equation:

do = -MAO. (4.18)

For ease of reference, let us express the six important equations so far in the form ofTable 4.1.

Definitions Identities Structure Equations

dY=E0 =TT=1 de= -MAO

d-= Em o+oT=0 do= -onoTable 4.1 Equations for a moving orthonormal frame in Euclidean space

4.3 Exercises

1. Let s: (a, b) -+ R2 be a smooth curve parametrized by arc length, which means thatII T' (s) I = 1 for all s. Let T (s) _ [TI (s) , T2 (s) j T, with exterior derivative

di = Ti'ds(4.19)

T2'ds

(1) Verify that

a a

4,1 (s) = t'(s),42(s) = t"(s)/IIT"(s)II (4.20)

defines a moving orthonormal frame on this parametrization (i.e., verify that these unitvectors are orthogonal).

Assume that the second derivative of r is non-vanishing on (a,b).


Hint: Differentiate the identity (t,') 2 + (t2') 2 = I to show the orthogonality.

(ii) Show that 01 = ds and 02 = 0.

(iii) Since the matrix of connection forms is skew-symmetric, it clearly takes the form:

0 =Cµ o1

(2W4-,) 00 [0(4.21)

Using the notation x (s) = II t" (s) II, calculate and show that µ = Kds.

Note: K (s) = II t" (s) II is the curvature of the curve.

2. The setup is the same as in the previous exercise, except that t: (a, b) -> R3.

(i) Verify that if 4t (s) = t' (s), 42 (s) = t" (s) /11r" (s) II, and 43 = i x1:,2, then

141, 2, 3] is a moving orthonormal frame2 on this parametrization.

Hint: Differentiate the identity (ti') 2 + (t21)2+ (t3') 2 = 1 to show the orthogonality.

(ii) In this case the matrix of connection forms takes the form:

0 0 ... ..

µO 0 -µ2 = (42ki 1) 0 ..

2 0 (EsldE1) (43Id42) 0

Using the notation K (s) = II t" (s) II, prove that

µ° = Kds, µ' = 0, µ2 = Tds;

(4.22)

(4.23)

(t' x t") t"'T = (4.24)-PP -00

Note: K and T are called the curvature and torsion3 of the curve, respectively.

2 The vectors in this basis are known as the tangent, normal, and binormal vectors, respectively,and the orthonormal frame is called a Frenet frame.3 See Struik [1961 ] for clarification of these terms, and more information about the geometry ofspace curves.

4.4 An Adapted Moving Orthonormal Frame on a Surface 81

(iii) Show that the Serret-Frenet formulas (below) follow from the equation definingthe connection forms:

d4, = tc42ds, d42 = (- u41 + T2 3) ds, d43 = -7%2ds. (4.25)

3. (Continuation) Calculate the matrix of connection forms explicitly for the Frenet frameassociated with the helix

cosu sinu uT (U)

,F2 52 F2

4.4 An Adapted Moving Orthonormal Frame on a Surface

Let us now specialize to the case where `l': W c R2 -a R3 is a 2-dimensional parametri-zation, in other words M = `l (W) is a parametrized surface. Let us review the notionof tangent plane to a surface, in the multivariable calculus sense. Let { u, v} be a coor-dinate system on W, and express x = `Y (u, v) as (x' (u, v), x2 (u, v), x3 (u, v) ). Thetangent plane4 at x is the plane through the point x spanned by the two vectors:

_ [ax' axe axi T ('ax' axe ax Tau' au ' - Lav , aV ' av

v4

(4.26)

w

.

Figure 4. 2 Tangent plane to a parametrized surface

Note that these two vectors are linearly independent because `P is an immersion, butthey are not necessarily orthogonal. Those who prefer to think in terms of differentials

4 Note that the same tangent plane is obtained for every parametrization. as was proved inChapter 3.


may note that `P and `1 are coefficients of the tangent vectors d`P (a/au) andd`P (a/av) in terms of the basis { a/ax', a/axe, a/ax3 } of the tangent space at x.Figure 4. 2 shows the situation.

4.4.1 Existence of an Adapted Moving Orthonormal Frame on a ParametrizedSurface

There exists a moving orthonormal frame E = [41, 42, 43] on a parametrized surfaceM such that 43 is normal to the tangent plane at every point of M, that is, so that

( 0 = (43i ). (4.27)

Such a frame is called an adapted moving orthonormal frame.

Proof. Let us emphasize again that, since `P is an immersion, `P and `P are linearlyindependent, and therefore neither is ever zero, and that these vectors are smoothfunctions of (u, v) a W. Hence the prescription

t, = 'Pu/H'ull' t3 = '.X'/11 Wu X wvll l t2 = 43X41 (4.28)

ensures that 43 is normal to and `Pr, that 14, (u, v) , 42 (u, v) , 43 (u, v) } is aright-handed orthonormal basis of R3 at every (u, v) E W, and that each basis vectoris a smooth function of (u, v). it

4.4.2 Structure Equations for an Adapted Orthonormal Frame

Let us calculate the form of the equations in Table 4.1 for this moving orthonormalframe. Since

dx' = audu+-dv, i = 1, 2, 3,

it follows that

(4.29)

d`P = `P,,dv; (4.30)

=ETd`P=(4,I`P)du + (f;,P)dv

(421T)du + (421` )dv

0I°

(4.31)

using (4. 6), (4. 27), and (4. 30); since 03 = 0 in the last line, we may write the first ofthe "Definitions" of Table 4.1 as:

4.4 An Adapted Moving Orthonormal Frame on a Surface

dY' = 91`1+0242.

83

(4.32)

This equation is equivalent in classical terminology to calculation of the "FirstFundamental Form" (see Exercise 14). Let us lighten the notation for the connectionforms (w) by writing instead

Q = (4.33)

Note that the skew-symmetry of to implies that just three 1-forms suffice to specify (8.These three 1-forms are defined by d°_ = E ; in other words:

(d4l dE2, dE3] = 141, 4v 431 T1° 0 -112

I111 112 0

«2TI° -51111-42't21- (4.34)

To calculate the 1-forms 11°, 711, 712, take the inner product of the first column with thecolumn vector S2, and of the first and second columns with 43, and exploit theorthogonality of the ,, to obtain:

.11° = (4041%11' = (31d1),112 = (43142)

As for the structure equations, d9 = -(1 A 0 implies6 6

del 0 -71° -711 eI 11° A e2

de2 = - 11° 0 -112 A 192 = -11° A 91

0 111 112 0 0 -111 Ae1-112A02

Finally, the second structure equation da = -m A W gives

0 -11° -111 0 -11° -111 0 -11° -111

d 11° 0 -112 = - T1° 0 -112 A 11° 0 -112

111112 0 111 112 0 111 112 0

(4.35)

(4.36)

(4.37)


The whole of surface theory is in some sense implicit in (4. 31), (4. 34), and thefollowing six identities, which summarize the information in (4. 36) and (4. 37).

First Structure Equation:

d0' = 11°n02,dO2 = -71 °A0'; (4.38)

111 A0'+112n02 = 0. (4.39)

Gauss's Equation5:

d7)° = -ill A T)2. (4.40)

Codazzi-Mainardi Equations:

dill = -712 A 71°, dtr2 = -ri° n 111. (4.41)

4.4.3 Example: An Adapted Moving Orthonormal Frame on the Sphere

the sphere of radius A, { (x i, x2, x3) : x1 +X2 + x3 = A 2 }, admits the parametrization

`f' (u, v) = (Acosusinv, Asinusinv, Acosv), defined for (u, v) a (0, 2rt) x (0, it).One may readily check that

F-i

-Asinusinv A cos u cos v

Acosusinv Asinucosv0 v-A sin

. (4.42)

-A2cosu(sin v)2

-A2 sinu (sin v) 2

A2sinvcosv J

The construction 4.4.1 for an adapted moving orthonormal frame gives

41 = 4'u I MuII' 3 = u X w,./II u X 3 X 1(4.43)

It follows that

I -sinu KK, cosucosv KK, -cosusinvb1 = Cosu +S2 = sinuCOsV'b3 = -sinusinV

0 -sin v -cos v

(4.44)

The reader should make a mental check that the three vectors in the previous line indeedform an orthonormal basis at every point. Observe that, in the case of the sphere, the

,''uX'Vv =

s This becomes more likele conventional form of Gauss's equation when the right side isidentified with -K (81 A 0'). where K is the Gaussian curvature, as in (4.67).

4.5 The Area Form 85

normal vector 43 (u, v) is parallel to the position vector `Y (u, v); here it appears as aninward normal.

Exterior differentiation of (4. 44) gives

-cosudu , - sinucosvdu - cosusinvdvd4, = -sinudu d;2 = cosucosvdu - sinusinvdv (4.45)

0 (-cosv) dv

From (4. 34) we may calculate the matrix of connection forms as follows:

4.5 The Area Form

0 A2141) APO0 -(1:,3Id42)

0

0 cosvdu -sinvdu-cosvdu 0 -dv (4.46)

sinvdu dv 0

The following notation will be convenient. If a = [a', a2, a3] T and are R3-valued1-forms, we define an R3-valued 2-forma x P' by

a2 A R3-a3 A tit

a3 A ' -a' AF

a'Aa2_a2AR'I

(4.47)

It follows from this that (At;) x (94j) _ (A A µ) T, x 4, for any 1-forms A, .t, andthat x distributes over addition. In particular6

dYxd'' _ (9'4I+e242) x (e' I+A242)

(0' A02) (4, xb2) + (02A0') (42x4,).

It follows that

d'xd`Y = 2(0' A02)43 (4.48)

6 Although v x v = 0 for any 3-vector v, it is not r1ecessarIy true that &x a = 0 forvector-valued I-forms, since, for example, a' n a = -a n a .


This contains important information about the 2-form 01 A 02, known as the area form.

4.5.1 Properties of the Area Form

The "area form" 0' A92 is never zero, and is the same for any adapted movingorthonormal frame with the same choice of direction? for the moving normal vector 43 ;indeed if 43 is pointing in the direction T. x 9',,, then

e' A e2 = II 1'N x IPJI du A dv. (4.49)

Proof: According to (4. 30), d`Y = ì'Ndu + `P,,dv; therefore (4. 48) gives

2 (0' A e2) 43 = (WNdu + Ì',,dv) x (Y'udu + `P dv),

(0' A 02) 3 = (du A dv) (Ì'N x `Y,). (4.50)

Now T. x 9',, * 0 since ì' is an immersion, and so the 2-form 0' A 02 is alwaysnonzero. Equation (4. 49) follows immediately. In equation (4. 48), the terms other than0' A 02 are the same for any choice of adapted moving orthonormal frame with thesame choice of normal direction, and hence so is 0' A 02. u

4.5.2 The Sphere Example, Continued

For the parametrization of the sphere considered in Example 4.4.3, we have from (4. 42)

TNII =

A A2sin A dv. (4.52)

4.5.3 Relationship to Surface Integrals

This refers to the integration theory of differential forms, which will be coveredthoroughly in Chapter 8. Suppose

Y =F'dx+FZay+F3a (4.53)

is a vector field on an open set U c R3 with U Q Ì' (W). The flux of Ythrough theparametrized surface Ì' (W) means the "surface integral" of the 2-form

7 On any parametrized surface, there are two choices for the "outward" normal.

4.6 Exercises 87

A = F, (dy A dz) + F2 (dz A dx) + F3 (dx A dy) (4.54)

over `P (W) , namely the integral

j = J4

IF (W) w

Clearly this is the same as

0 13 = J(F.,F2,F3) F,)dudv.w

(4.55)

J(F.,F2,F3) 43(0 A82). (4.56)

w

4.6 Exercises

4. Calculate the connection forms and the area form for an adapted moving orthonormalframe on the following parametrization of a cylinder, for some A > 0:

`Y(u,v) = (Acosv,Asinv,u), (u,v) e (--oo,eo) X (0,2x).

5. Consider a surface of revolution of the form

`Y(u,v) = (g(u)cosv,g(u)sinv,h(u)), (4.57)

on some domain W g. R2, where g and h are smooth functions, and g is strictly positive.Moreover we assume that this is a "canonical parametrization," which means

g'(u)2+h'(u)2 = 1. (4.58)

(i) Calculate an adapted moving orthonormal frame as in (4. 28), and show that (4. 31)gives

8' = du, 02 = gdv. (4.59)

(ii) Prove that the matrix of connection forms is

0 -11° -T1' 0 -g'dv - (h"g' - h'g") duT1° 0 -T12 = g'dv 0 -h'dv (4.60)

T)' 7'12 0 (h"g'-h'g")du h'dv 0

6. (Continuation) Now consider any surface of revolution of the form

T(t,v) = (r(t)cosv,r(t)sinv,q(t)) (4.61)


on some domain W Q R2, where r and q are smooth functions, and r is strictly positive.Let u (r) be a smooth reparametrization such that ti = au/ar> 0, and ifg(u(t)) = r(t) andh(u(t)) = q(1),then g'(u)2+h'(u)2 = 1, whereg' = ag/au, etc.(i) Using the fact that g = ag/at = g'u = i, show that

tit = 2 2+q (4.62)

(ii) By applying the chain rule, obtain formulas for g', g", h', h" in terms ofi,i,q,q,4,ii.(iii) Show that 8' = iidt, 82 = rdv.

(iv) By inserting these expressions into (4. 60), show that the matrix of connectionforms is

0 -71 ° -TI ' 0 -idv/u -(qi-4i)dt/tieTl° 0 -t12 = idv/ti 0 -qdv/u711

T12 0 (qi - 41i) dt/ti2 qdv/ti 0

(4.63)

7. A surface of the form z = h (x, y), with some domain W g. R2, may be expressed usingthe parametrization `Y (u, v) = (u, v, h (u, v)) . Find an adapted moving orthonormalframe as in (4. 28), and show that 0, Q, and the area form are given by the followingformulas:

°= h"h"hV

h=hYY hYY

11 du + dv, PQdu+ PQdv,QP2 QP2

n2 =I

- hp '2YY + QZ Idu + C

hP2Y" + -2")dv; (4.65)

Q

8' = Pdu+h hvdv,02 = Qdv,0'A 2=Q(duAdv), (4.66)

2h --where we use the notation hY=ah,hYYa ,P = 1+h!,Q = l+h2+h2.Fu au2

4.7 Curvature of a Surface

Let us proceed toward the definition of curvature. Since the domain W of theparametrization is 2-dimensional, the 2-forms on W at any point (u,v) of W comprise aspace with the same dimension as A2R2, which is 1. By 4.5.1, every 2-form on W is

4.7 Curvature of a Surface 89

simply the area form above multiplied by some smooth function. In particular, thereexist smooth functions K and Hon W defined by the equations

7)t n7)Z = K(0' ne2),

-02n11'+0' A112 = 2H(0' A02).

(4.67)

(4.68)

Abusing notation slightly, we may speak of K and H as functions on the surface `Y (W)by composing them with T-1; in that case, K is called the Gaussian curvature, and His called the mean curvature. The relevance of these definitions to intuitive ideas about"curvature" will be explained in the next section, when we express K and H in terms ofthe principal curvatures.

It appears from the construction we have given that the definitions of K and H dependon a specific choice of adapted moving orthonormal frame; the following calculationswill show that this is not so, although the sign of H depends on the choice of normaldirection on the surface. In fact Gauss's Theorema Egregium says that the values of Kand 1,M on `Y (W) are invariant under change of parametrization also 7 The best way toprove this is to use a more abstract definition of the surface and of curvature, so thesefunctions are defined without reference to any parametrization, as we shall do later.However, a proof "in bad taste" is also given here.

Let us continue to take cross products of vector-valued 1-forms. By (4. 34),

dPxd43 = (9t41+92 ) x (- t1)'--421')2)

_ -(8' nr12) (fit x 2) - (02A11') ('2x ,)

_ -(e' n112-e2n11'),3.

Consequently

d`Pxd,3 = -2H(8'

a similar argument shows that

(4.69)

d43 x 2 (11' n 112) 3 = 2K (8' n 92) 43. (4.70)

The last equation already gives some insight into the relationship between K and theintuitive notion of curvature. When a surface is "highly curved," the normal direction43 will change very quickly as a function of the parametrization (u,v), which will tendto make the left side of (4. 70) have large magnitude; this corresponds to a large value ofK on the right side.

7 The theorem really says more: that the apparent extrinsic invariant K is actually intrinsic,and could be detected by 2-dimensional persons living in the surface and running aroundwith surveyors rods.


4.7.1 invariance Property

The Gaussian curvature is the same for any two adapted moving orthonormal frames;the mean curvature is the same for any two adapted moving orthonormal frames withthe same choice of normal direction.

Proof: Any two adapted moving orthonormal frames, with the same choice of normaldirection, share the same vector-valued 1-form d`I' and normal vector 43. Now (4. 48)shows that the 2-form 01 A 02 is the same for both moving orthonormal frames, andhence so are H and K, by equations (4. 69) and (4. 70), respectively. These equationsalso show that reversing the normal direction 43 changes the sign of 01 A 02, and henceof H, but not of K. xx

4.7.2 Theorem

The value of the Gaussian curvature does not depend on the parametrization.

Proof (may be omitted): Suppose 4b: V (Z R2 -> R3 is another parametrization of thesurface M; for simplicity suppose 0 (V) = M = `P (W). Taking f: V -4 W to be`Y-I 0, which is smooth by a result in Chapter 3, we have the diagram

Vf=

-W

aMs

In view of 4.7.1. there is no loss of generality in choosing f as the adapted movingorthonormal frame on 0: V c R2 -+ R3. Define $ by dO _ (? f) 0; it suffices by(4. 70) to show

a a

(10 1

However, applying the pullback off to the equation d`Y = EO gives

d4p = f*d`Y =

(4.71)

Thus 19 = f*0, and so f* cet A 02) = 191 A 02. Finally we may apply f* tod43 x 43 = 2K (01 A e2) 43 to obtain (4.71). tt

4.8 Explicit Calculation of Curvatures 91

4.8 Explicit Calculation of Curvatures

Since 0' A 02 is never zero, it follows that the 1-forms 91 and 02 form a basis for the1-forms on W at every point; in particular there exist smooth functions a = a (u, v), b,c, and c such that

711 = a9'+ c92, r)2 = c9' + 692. (4.72)

However the third part of the first structure equation says 11' A 9' + 112 A 02 = 0, andtherefore

T1' n 9' = c92 n 91 = -112 A 02 = -c91 n 92,

giving c =c. Thus we may restate (4. 72) as:

T,2 = racl 9Z

cb9

(4.73)

(4.74)

Knowledge of the terms in this matrix equation is equivalent in classical terminology tocalculation of the "Second Fundamental Form"; see Exercise 16. The eigenvalucsK1, 1C2 of the 2 x 2 matrix above, which by composition with `l'-1 can be treated asfunctions on M = `Y (W), are called the principal curvatures.

We shall now derive expressions for the Gaussian and mean curvatures in terms of theprincipal curvatures. It is immediate from (4. 67) and (4. 74) that

K(9' n92) = 11' n112 = (a9'+c82) A (c9'+b92) = (ab-c2) (91 n92).

Now we see that

K=ab-c2= accb

= K1K2, (4.75)

using the fact that the determinant of a matrix is the product of the eigenvalues.Similarly, (4. 68) gives

2H(0'n92) _ -92A1'+91AT12

_ (-02A (a9'+c82) +0' n (c8'+b02))

_ (a+b)(01n82).

Since the trace of a matrix is the sum of the eigenvalues, this shows that

4.8 Explicit Calculation of Curvatures 93

As we saw in 4.5.1,

0' A92 = IIT-.xduAdv. (4.80)

To calculate the 1-forms 111, 712, recall equation (4. 35):

II = (43141), X12 = (4042)- (4.81)

Computation of all kinds of curvatures follows from evaluating the 2-forms on the leftside of the following matrix equation (deduced from (4. 74)), and reading off theexpressions for a. b. c:

T11A0'T1'A0 = [-cal(01A02)[12A01 T12n82 b c

For example, as noted above, K = ab - c2 and H = (a+ b) /2.

4.8.2 Example: The Sphere

This calculation is a continuation of Example 4.4.3. We calculate from (4. 78):

-A sin vdu .

'(4.82)

02 Adv

0'

From (4. 46) comes

A 92

Tl'

= A2 sinvdu

sin vdul

A d v. (4.83)

= [ (4.84)T12 dv J

Now we calculate the following matrix of 2-forms using (4. 82), (4. 83), and (4. 84):

I A 0: T)1 A 92 _ 0 A sin v du A dv = 0 I IA 0' A 02.112AO T)2AO2 -Asinv 0 -1/A 0 ]

It follows that

a = 1 /A, b = I /A, c = 0. (4.85)

We conclude that both the principal curvatures take the value I /A, while the Gaussiancurvature is K = 1 /A2, and the mean curvature is H = 1 IA. Note that in this case all


these quantities happen to take constant values; in a more general example they willdepend on u and v.

4.9 Exercises

8. (Continuation of Exercise 4) Calculate the principal curvatures, the Gaussian curvature,and the mean curvature of a cylinder of radius A > 0, using the parametrization

`Y (u, v) = (A cosv, A sinv, u), (u, v) a (-cc,oo) x (0,2tr).

9. On the helicoid, parametrized by Y' (u, v) = (ucosv, usinv, v), an adapted movingorthonormal frame is given by

cos v -u sinv l

Sin Vsinv 2 = P ucosv 3 = p -cosy0 1 u

..a a

where P = 1 + u2. Find the 1-forms 01, 02 so that d`P = 0141

+()242 and the matrix0 of connection forms, and show that the mean curvature of the helicoid is zero.

10. (Continuation of Exercise 5) For the surface of revolution

`Y(u,v) = (g(u)cosv,g(u)sinv,h(u)),

where g' (u) 2 + h' (u) 2 = I, and where g and h are smooth functions with g strictlypositive, show that the principal curvatures, the Gaussian curvature, and the meancurvature are given by

, - h,g",KZ =

h' K=

h' (h"g' - h'g") H = h"g' - h'g"+ h'/gKVg'

g g 2

11. (Continuation of Exercise 6) Consider the general surface of revolution

IP(t,v) = (r(t)cosv,r(t)sinv,q(t)), (4.86)

on some domain W c R2, where r and q are smooth functions, and r is strictly positive.As before, let u (t) be a smooth reparametrization such that a2 = r2 +q2 andu = au/at > 0. Show that the principal curvatures are

_ 4r-4r qK

(r2+4

2)

3/2 KZ =r

4.r2+ q2

4.10 The Fundamental Forms: Exercises 95

Warning: When applied to the sphere. Example 4.8.2. the principal curvatures will turn out to bethe negatives of the results we obtained before, because in switching the u and v variables wereversed the direction of the normal vector.

12. Using the results of the last exercise, calculate the principal curvatures, the Gaussiancurvature, and the mean curvature of a hyperboloid, with a scale parameter A > 0, usingthe parametrization

`P(u,v) = (Asinhucosv,Asinhusinv,Acoshu), (u,v) a (-oo,oo) x (0,2n).

13. (i) Suppose A, B, and C are real numbers. Using the results of Exercise 7, show that theprincipal curvatures of the quadric surface

1 A CZ --IA2 CB

= Axe/2 + Cxy + By2/2 (4.88)x

LYJ

at (x, y) = (0, 0) are the eigenvalues of the matrix A CIC B

(ii) Using this result, calculate the Gaussian curvature of the quadric surfacesz = 2x2 + xy/2 -y2 and z = -3x2-xy/2-5,2/2 shown in Figure 4. 3, at(x, y) = (0, 0).

4.10 The Fundamental Forms: Exercises

The terminology of "fundamental forms" is a restatement of the ideas of this chapter;study of these exercises may help the reader who plans to consult classical geometrybooks.

14. Given an adapted moving orthonormal frame on a parametrized surface`P: W Q R2 - R3, the First Fundamental Form is a (0,2)-tensor on W, in the sense ofChapter 2, defined by

i = 81®0t+02®02.

(i) Using the fact that 0' = (1,I `P )du + (l;;l`P)dv by (4. 31), show that

(6''a)(0''a)+(02. )(92.au au au

11

au ar,

Hint: P. = (4i 2

(4.89)

(ii) By evaluating I (a/au, a/av), etc., show that


1(Alau+A2av'BIau+B2av) = [A, A2] II II

2 [BJ(4.90)

Comment: The First Fundamental Form tells us how distances would be experienced on thesurface by someone who was operating in (u, v) coordinates; this is because the square root ofI (l;, ) is the length in R3 of the image under the differential of IF of the tangent vector l; at apoint in W. The matrix appearing in (4.90) will later be called the Riemannian metric tensor, withrespect to the (u, v) coordinate system, of the metric induced by the immersion IF. Knowledgeof the First Fundamental Form would enable us in principle to calculate "minimal geodesics," thatis, paths of shortest length between two points on the surface; see Klingenberg [1982].

15. (Continuation) Calculate the First Fundamental Form of the ellipsoid

x2 2 z2A+B2+C2 = 1

for the parametrization'F (u, v) = (Asinucosv, Bsinusinv, Ccosu) with domain(0, n) x (0, 2n) . Show that the Gaussian curvature at (x, y, z) is

x2 y2z2 -2

K A2B2C2(A4+B°+C'41

16. Given an adapted moving orthonormal frame on a parametrized surface'1': W C R2 - R3, the Second Fundamental Form is a (0,2)-tensor on W, in the senseof Chapter 2, defined by

(4.91)

11 = a(O' ®0') +c(0t ®02+02®01) +b(02®02). (4.92)

11 = 11 10 01+T12®e2.

(1) Using the matrix appearing in (4. 74), show that

(ii) Using the fact that (L,V = 6,,, show that (43ld41 a/au) = -((a43/au)141).

(iii) Using the formulas 0' = ( T)du + (4,$41)dv and it = (3Idf;;) for i = 1, 2,as in (4. 31) and (4. 34), and the fact that 0, show that

I1(a.a)au au au au au au

where 'F,,,, = DT/au.

(iv) By evaluating II (a/au, a/av), etc., show that


II(A,# +A2 ?_,Beau+BZav) = [A, A2] (FF31`Y (3I'f'[BJ (4.93)

(0)31Comment: The Second Fundamental Form is essentially equivalent to the matnx of functions

a c

cb

appearing in (4. 74), from which all the previous curvature expressions were computed. It isshown by (4. 93) to contain all the information about the components of the second partialderivatives of IF in the direction normal to the surface; note incidentally that the sign of theSecond Fundamental Form changes if the direction of the normal vector is reversed.

17. (Continuation) Calculate the Second Fundamental Form of the hyperbolic paraboloid

x2 y2Z = A2B2

for the parametrization 'Y (s, t) = (As, 0, s2) + t (A, B, 2s). Show that the Gaussiancurvature is

K=- 1 x2+

y2

+1

4 A A B

)-2.


Gaspard Monge (1746-1818) and C. F. Gauss (1777-1855) may be considered as thefounders of the differential geometry of surfaces. The equations bearing his name werediscovered by D. Codazzi (1824-75). A comprehensive treatise on surface theory waswritten by Gaston Darboux (1842-1917). Further historical information may be foundin Struik [1961], which is an excellent reference for the classical theory of surfaces; seealso Spivak [ 1979]. The treatment given here is based on Flanders [ 1989].

5 Differential Manifolds

Until now we have been working on submanifolds of Euclidean spaces, so as to showclearly how exterior calculus is linked to multivariable calculus. However, it is moreefficient in the long run to have an "intrinsic" theory and notation for the objects wework with, and to forget about the Euclidean space they may be embedded in. This isthe theory of differential manifolds and vector bundles.

5.1 Definition of a Differential Manifold

Suppose M is a set. A pair (U, cp) is called an n-dimensional chart on M, or simply achart, if U c M and cp: U -> R" is a one-to-one map onto an open set p (U) Q R". Ifp E U, we may call (U, (p) a chart for M at p.

Two charts (U, (p) and (V, yr) are called C"°-compatible, or simply compatible, ifeither U n V = 0, or else U n V * 0, yr (U n V) and p (U n V) are open in R",and (p yr 1: ty (U n V) -* (p (U n V) is smooth with a smooth inverse (i.e., is adiffeomorphism).

V U

I I

'yr(UnV) (P (UnV) \1-%

5.1 Definition of a Differential Manifold 99

A smooth atlas is a family of charts, any two of which are compatible, whose domainscover M. Two atlases are called equivalent if their union is an atlas. As the nameimplies, this is indeed an equivalence relation on atlases (transitivity comes fromapplication of the chain rule from calculus). An equivalence class of atlases on M iscalled a smooth differentiable structure on M. A set M with a smooth differentiablestructure is called a (smooth) differential manifold. We say that M has dimension n ifthe dimension of the range of all the chart maps in some (hence any equivalent) atlas isn. The definition above allows M to have separate components with differentdimensions, but for simplicity we shall assume henceforward that all our manifoldshave a unique dimension.

5.1.1 Example: An Atlas on the Circle

The circle S' is the set { (x, y) E R2: x2 + y2 - I = 0 }. Let us show that the followingpair of charts (U, (p) and (V, yr) constitute an atlas for the circle:

U=S'-(0,1),cp(x,y) -- X

11-y;

V=S'-(0,-1),W(x,y) x= +Y*

First, we show that cp is one-to-one. If x, / (1 - y,) = x2/ (1 - y2), then squaring bothsides and applying the fact that x2 = 1- y, gives

1-y; 1-y2 l+y, 1+y2

(1-y,)2 = (1-y2)2 1-y, - 1-y2'

Therefore y, = y2, x, = x2, since division by 1 - y; is legitimate on U, where y; * I.Similarly yr is one-to-one. Observe next that cp (U) = R = yr (V), an open set, and sois

(p (UnV) = (-00,0). (0,oo) = W(UnV).

Since clearly S' = U v V, it only remains to show that

cP W-': W (U n V) = (-00, 0) u (0, oo) ---) (p (U n V) 0) U (0, oo)

is smooth with a smooth inverse (the one-to-one and onto properties are automatic fromwhat we have proved already). To calculate cp W-i , suppose w = x/ (1 - y) andz = x/ (1 + y); then it is easy to check that wz = 1, and thus

I/z,ze (-oo,0) U (0,oo).

This is indeed smooth with a smooth inverse, so (U, (p) and (V, yr) constitute an atlas.

100 Chapter 5 Differential Manifolds

5.1.2 An Equivalent Atlas on the Circle

Start with the 1-dimensional parametrizations'Yj (0) _ (cos0, sin0), i = 1, 2, withdomains W, = (-it, it), W2 = (0, 21t), respectively. By performing the same steps asin the previous example, one may show that another atlas for the circle consists of thecharts

U, = S'- (-1,0),(P, =4P11;

U2 = S'- (1,0),82 ='F2'

To show that { (U, (p), (V, W) } is equivalent to { (U,, (,,) , (U2, (p2) }, it nowsuffices to show that p (pt T T2 W (PI W cp. are diffeomorphisms whereverdefined. This is easy because

_ cos0 cos01-sin0'l' 9'(0) = l+sin0

5.1.3 Submanifolds of Euclidean Space

Every n-dimensional submanifold M of R" + k, in the sense of Chapter 3, has adifferentiable structure induced by its n-dimensional local parametrizations, and istherefore an n-dimensional differential manifold.

Proof: Given p e M, a result in Chapter 3 shows that there exists a neighborhood U' ofp in R" +k and an implicit function parametrization P: W Q R" -4 U'; this is bydefinition one-to-one and onto U' n M. Taking (U, (p) = (U' n M, 'F-) gives achart for M at p, since W is open by definition. Evidently M can be expressed as theunion of domains of such charts. Now the Chapter 3 result on switching betweendifferent parametrizations shows that, if Ti: W. C R' -4 U;' (Z R" +k are n-dimensionalparametrizations of M at y for i = 1, 2, thenW1 n'F 1' (U2') = 'F 1' (U,' n U2' n M) andW2 r-1 %P2' (U,') = 'F2' (U,' n U2' n M) are open sets in R", and if the last two setsare nonempty then

412' 'F, : W, "1' (U2') -4 W2 n'P2' (U1')

is a diffeomorphism onto its range. This implies (Ul' n M,'Pr') and (U2' n M,'F21)are compatible charts. tt

5.2 Basic Topological Vocabulary

Topology can be loosely described as the study of properties of a space which areinvariant under "homeomorphisms," that is, continuous mappings with a continuous

5.2 Basic Topological Vocabulary 101

inverse. In order to know what the homeomorphisms are, we have to define the class ofopen sets, since in technical language the homeomorphisms are precisely the mappingswhich preserve the class of open sets. Although topology is outside the scope of thisbook, we include here some pieces of vocabulary which will be needed to state laterresults. Most important, we shall need to know what an open set in a differentialmanifold M means.

5.2.1 Open Sets in a Differential Manifold

Given a specific atlas { (Ua, (Pa) , a e 1} for M, we shall say that a subset V of M isopen in M if (pa (V n Ua) is open[ in R" for every chart (Ua, (pa) . It turns out that thesame class of open sets is obtained if { (Ua, (pa) , a e I} is replaced by an equivalentatlas; thus the class of open sets is determined by the differentiable structure on M, notthe specific choice of atlas within that structure. Moreover this class of open sets isclosed under finite intersections and arbitrary unions, and thus indeed forms a"topology" on M, making M into a "topological space." The proofs of these statementsare given in detail in Berger and Gostiaux [1988], pp. 55-6 1, which offers furtherinformation about the topology induced by a differentiable structure.

It is easy to check that every open subset V of a differential manifold M can be given adifferentiable structure, using the atlas { (V n U,, (pi I V), i e 1}, where

{ (U,, (pi), i e 1} is an atlas for M.

In the case of an n-dimensional submanifold M of R"+k, ifW: W Q R" -4 U' c R"+k is

a parametrization of M, our definition implies that `P (W n G) is open in M for everyopen set G in R". For example, in the case 5.1.1 of the circle, the segment{ (x, y) :0 < x < 1, 0 <y < 1, x2 +y2 = 1 }, which is the image of the open set (0, m/2)under the parametrization a -4 (cos0, sine), is open in S' .

5.2.2 Closed and Compact Sets in a Differential Manifold

As in any topological space, a subset H of M is called closed if its complement H` isopen; H is called compact if, for every collection { V. } of open sets in M whose unioncontains H (known as an "open cover" of H), there is a finite subcollection{ Va(t), ..., Va(m) } whose union contains H. In the case where M is a Euclidean space,a subset of M is compact if and only if it is closed and bounded (i.e., lies entirely withinsome finite ball centered at the origin); this is the Heine-Borel Theorem.

If M is a submanifold of R", then the compact subsets of M are precisely the sets of theform K n M, for K compact in R'; this can be deduced from 5.10.1 below. In particular

' W is open in R" when, f o r every y e W, the "open b a l l " B (y, e) = {x: 11 x - y I I < e j iscontained in W for all sufficiently small e; here ii ii denotes Euclidean length.


if the submanifold M is itself closed and bounded in R", then it is compact. Spheres,tori, and special orthogonal groups are all examples of compact manifolds. To see, forexample, that SO (n) is a compact subset of R""", note that a nonorthogonal matrix, oran orthogonal one with determinant of -1, lies inside a small open ball in R"" whichdoes not intersect SO (n), and so SO (n) is closed; moreover each row in anorthogonal matrix is a unit vector in Euclidean space, and so each entry is constrained tolie within [-I, I]-. thus SO (n) is closed and bounded.

In the case of an abstract differential manifold M of dimension n, the compact sets arenot so easy to characterize, partly because the notion of boundedness no longer makessense, since there is no canonical measure of distance.

5.3 Differentiable Mappings between Manifolds

A function from an open subset of Euclidean space into another Euclidean space iscalled C' if its derivatives exist up to order r and are continuous; it is called C" if it isC for all r z 1.

5.3.1 Definition of a Cr Mapping

Let r z I be an integer, or else -. If M and N are differential manifolds, withdimensions m and n, respectively, f. M -4 N is said to be a C' mapping (or C'immersion, or C' submersion) from M to N if, for every p E M, there is a chart (U, (p)for M at p and a chart (V, W) for N at f (p) with f (U) C V, such that

I:cp(U) F_ Rm -W(V) cR"

is a C' map (or a C' immersion, or C' submersion, respectively); a C map is usuallycalled a smooth map. Then for any charts (U,, (ps) for M at p and (V,, W,) for N atf (p), restricted if necessary so that f (U,) Q V1, yrl f cps' : cp (U,) -* yr (Vj) is alsosmooth by the chain rule, because

W f (p7' = (Wi W-) (W f `Y_) (cp (P-1).

These maps are illustrated below. Naturally the immersion case can only occur whenn = m + k for nonnegative k, and the submersion case only when m = n + k.

5.3.1.1 Definition of Cr DiffeomorphismA C' map f is called a C' diffeomorphism if it is one-to-one and onto, and its inversef-1 is also C'.

5.3 Differentiable Mappings between Manifolds 103

5.3.2 Examples

(i) In the case where M is an open interval J in R, a C' map r: J -4 N is called a C'curve in N; this map is an immersion if, at every s e J and some (hence every) chart(V, ti) at t (s), the derivative (W t)' (s) is nonzero. For example, for any n x nmatrix H, there exists e > 0 such that t (s) =1 + sH, -E < s < E, is a smooth curve inGL' (R), and in fact an immersion if H * 0.

(ii) Another simple example of a smooth map between two manifolds is the inclusionmap L: GL' (R) -4 GL+ + i (R) which sends the nonsingular n x n matrix A to thenonsingular (n + 1) x (n + 1) matrix

Since GL+ (R) is a subset of a Euclidean space, we can take the trivial chart(U, (p) = (GLn (R), identity), and similarly for GLn+I (R). The derivative oft issimply

Dt(A)H = t(A+H) -t(A) = 10F1 010.

which shows that Dt (A) is one-to-one, and hence t is an immersion.

(iii) To see an example of a submersion, consider the map f which sends a 3 x 3nonsingular matrix A to its first column A [ 1, 0, 0] T, which is a nonzero vector.Evidently we can treat this as a map f: GL3 (R) -. N = R3 - { (0, 0, 0) } from thegeneral linear group to R3 minus the origin. The reader is left to calculate Df (A) H asin the previous example, and check that H -* Df (A) H is onto.


5.4 Exercises

1. The n-sphere S" = { (xo, ..., x") E R" + 1: xo + ... + xn = 11 was discussed in Chapter3. Generalize Example 5.1.1 to show that the following pair of charts (U, (p) and(V, iy) , known as stereographic projections, constitute an atlas for the n-sphere:

U = S"- {(1,0,...,0)},tp(XO,...,X") =(x,1-XX")

0

V = S"- {(-1,0,...,0)},W(xo,...,x") =(xt, ..., x")

1 + xo

2. (Continuation) Let W c R2 be the open set (0, 2it) x (0, n), and define(b 1: W -* S2, i = 1, 2, by

4), (0, 0) _ (cos0sin4', sin0sin4', cos4'),

4)2(()'0) _ (-cos0sin4,, cos4', sin0sin4').

(i) Define charts on S2 by (U,, Wi) = (4b, (W), 0-1) for i = 1, 2. Draw a picture toconvince yourself that the domains of these charts cover S2, and show that they form anatlas.

(ii) For the 2-sphere, describe precisely the steps that have to be performed to show thatthe atlas { (U, (p), (V, 4f) } in the previous exercise is equivalent to the atlas{ (U,, yr i) , (U2, w2) }. You need not carry out the whole proof, but at least show thatcp We 1 is a diffeomorphism.

3. Suppose { (Ua, (pc,) , a E 1} is an atlas for a set M, and { (Vr VY) , Y E J} is an atlasfor a set N.

(i) Construct an atlas for M x N = { (p, q) : p e M, q e N} .

(ii) Show that for the associated differentiable structures on M, N, and M x N, theinclusion map p -* (p, q) from M to M x N, and the projection map (p, q) -> q fromM x N to N are smooth.

4. Suppose M is an n-dimensional submanifold of R" +,t If M is given the differentiablestructure described in 5.1.3, show that the identity map from M to R" +,t is smooth.

5. Let W = { (x, y, z) E S2 c R': z > 01, in other words the "Northern hemisphere" ofs2.

(i) Show that W is open in S2 for the differentiable structure described in Exercise 1.

(ii) Let B (0, 1) = { (x, y) E R2: x2 + y2 < 11. Show that the map

5.5 Submanifolds 105

f(x,y) = (X, Y, l -x2-y)

from B (0, 1) to S2 is a smooth diffeomorphism onto W

6. Prove that if f: P -a M and g : M -4 N are C' maps, then g f: P -a N is a C' map.

7. Fix a nonzero vector v E R3. Show that the map F: GL3 (R) -4 S2 a R3 given byF(A) = (Av) /IlAvll is smooth.Hint: Express F as the composite of A - A v and w -4 w/ II w 1l , and show that both of thesemaps are smooth.

5.5 Submanifolds

A subset Q of an (n + k)-dimensional manifold M is called an n-dimensionalsubmanifold of M if, for every q e Q, there exists a chart (U, (P) for M at q such that(p (Q n U) is an n-dimensional submanifold of R"+k It follows that Q is itself ann-dimensional manifold, using the atlas obtained by taking chart maps of the form`I'-r (p with suitable domain in Q, and image in R", where Y' is a parametrization forcp (Q n U). The integer k is called the codimension of the submanifold Q in M. In thecase where M = R"+k, this definition is seen to be internally consistent, by taking(U,(p) = (Rn+k, identity).

It could be tedious to use the definition as it stands to check that a subset of M is asubmanifold, because one may have to perform calculations in every single chart ofsome atlas. In the next section, we shall give quicker ways to identify a submanifold.First we develop a useful technical result.

5.5.1 Properties of Maps between Submanifolds

UcRm+' VcR"+k

I

Mm N

Suppose M is an m-dimensional submanifold of an open subset U of Rm+j, N is ann-dimensional submanifold of an open subset V of R" +k, and f: U -> V is a C' map (ora Cr immersion, respectively) such that f (M) g N. Then the restriction


off to M, denoted Am, is a Cr map (or a C' immersion, respectively) from M to N.

Proof: Take any p E M, and let q = f (p) e N. According to the Implicit FunctionTheorem of Chapter 3, there exist smooth parametrizations 4): W C R'° -, U' C R' *jand `I': Y S R" -> V' C R" +k at p and q, respectively, of the form

0(x) = (x,z(x)) E RmxR',`F(y) = (y,w(y)) E R"xRk, (5.1)

possibly after rearranging the order of variables. Let us express the map f as

f(P) =f(x,z) = (F(x,z),G(x,z)) E R"xRk. (5.2)

The situation is shown in the picture below.

U R" +'

x

V R"+k

Since (U' n M, 4-') and (V' n N, `P'') are charts at p and q respectively, it sufficesto show that if f: U -* V is a C' map (or a C immersion, or C' submersion,respectively), then so is

'i`-1 AM O = F 4): W -- Y,

where we use the notation of (5. 2).

If f is a C' map, then so is F and hence so is F 40, since 40 is a smoothparametrization. If f is a C' immersion, then all three components in the derivative

D (`l'-1 ftV 0) (x) =D`P_i

(q) Df (P) DO (x)

are one-to-one linear maps, and so the composite is one-to-one; this proves that F 0 isa C' immersion.

If f is a C' submersion, write the derivative off as the (n + k) x (m +j) matrix

5.6 Embeddings 107

Df (p) = Gx Gz (5.3)x

in the abbreviated notation Fx = DxF, etc., where necessarily m +j Z n + k. SinceDf (p) is onto, given E TyR" there exists a solution (41, 42) E Tx (R'" x RR) to thelinear equation

Fs F1 r _

Gx G 42 -Y-

(5.4)

However, (5. 1), (5. 2), and the constraint that f (M) c N imply by the chain rule that

(x) = DF(x,z(x)) = Fx+FZzr; (5.5)

(x) = (x) = w, (Fx+Fz,) = Gx+G.zx. (5.6)

On comparison of (5. 4), (5. 5), and (5. 6), matrix algebra shows that, once , isselected, then setting 42 = zx4l solves (5. 4). In other words, given l; E TyR" thereexists 4, e TXRM such that

(Fx+Fzzx)4, = c

5.6

By (5. 5), this is equivalent to saying that D (F 0) (x) is onto, which proves thatw-Iof asubmersion, as desired. tx

Embeddings

Figure 5. 1 Picture of an embedding


If P and M are differential manifolds, a one-to-one Cm immersion f: P -4 M is called anembedding if, for every open set U in P, there exists an open set Win M such thatf (U) = W n f (P), as shown in Figure 5. 1.

Note that a map which sends an open interval of R into the curve in R2 shown inFigure 5. 2 is not an embedding. To see why it is not an embedding, first convinceyourself that the image of the map is not a submanifold of R2, by looking at whathappens at the point in the center under a submersion; then apply 5.6.1..

Figure 5. 2 Nonexample of an embedding

Here is the first useful result for obtaining submanifolds without recourse to charts.

5.6.1 Submanifolds Obtained through Embeddings and Submersions

Either of the following conditions implies that Q is a submanifold of M.

f: P -* M is a C' embedding, and Q = f (P), or

g: M - N is a C° submersion, p e N, and Q = g-1 (p).

Proof: Suppose f: P -4 M is a C' embedding, and Q = f (P). We may suppose that Phas dimension n and M has dimension n + k. Given any q e Q, let p be the unique pointin P such that q = f (p) , and take a chart (U, (p) for P at p and a chart (V,,V) for Mat q with f (U) c. V, such that

1:cp(U) cR"-aW(V) gR"+k

is a one-to-one C' immersion; these charts exist by definition. By a result presented inone of the Exercises of Chapter 3, there exists an open set W c cp (U) which containstp (p) such that tat. f.-r (W) is an n-dimensional submanifold of tV (V). Since f isan embedding, and since .p1 (W) is open in P, there is an open set V c V c M suchthat f ((p-1 (W)) = V' n Q. Now (V, W) is a chart for Mat q such that W (V' n Q)is an n-dimensional submanifold of R"+k. This proves that Q is a submanifold of M.The proof of the other assertion, which is easier, is left as an exercise. tt

It would appear that to verify that a mapping is an embedding is not easy. A useful factfrom topology, whose proof (though not difficult) is outside the scope of this book, is:

5.6 Embeddings 109

5.6.2 Compactness Lemma

1f f: P -4 M is a one-to-one C immersion, and P is compact, then f is an embedding.

Another useful topological result is the following.

5.6.3 Relation of Open Sets In a Manifold to Those of a Submanifold

If Q is a submanifold of M, then the open sets of Q are precisely the sets

{ U' n Q: U' open in M}.

Proof: This proof is postponed to Section 5.10 because it is somewhat technical.

(5.7)

tt

5.6.4 Inclusion2 of a Submanifold Is an Embedding

If Q is a submanifold of M. then the inclusion map t: Q -4 M is an embedding.

Proof: The condition that, for every open set U c Q. there is an open set U' Q M suchthat t (U) = U' n t (Q), is verified in 5.6.3. Since the inclusion map is alreadyone-to-one, it only remains to prove that it is an immersion. First we claim that, forevery r e Q and every chart (U, (p) for Mat r, the inclusion

id:cp(UnQ) -+cp(U) g. R"

is an immersion; this is true by 5.5.1, since (p (U n Q) is a submanifold of q (U) byassumption, and id : cp ( U n Q) -+ cp (U) c R" is the restriction of the identity map,which is an immersion on R", to a submanifold. Now if we take a k-dimensional chart(V, ty) for ep (U n Q) at cp (r) , then (U n cp-' (V), t i (p) is a chart for Q at r, and

cP t (ty (p) -1 = ty-' is an immersion; this verifies the condition for t: Q -4 M tobean immersion.

The next result is included merely to show that the relation "is a submanifold of' isreasonably well behaved.

5.6.5 Composition of Embeddings

(i) The composition of two embeddings is an embedding.

(ii) M a submanifold of N and N a submanifold of P implies M is a submanifold of P.(Note: The dimensions of M, N, and P could be different.)

2 If S is a subset of a set T, then the inclusion map t: S - T simply means the identity maprestricted to S.


Proof: The proof of (i) is left as an exercise. Given (i), we may prove (ii) as follows: By5.6.4 the inclusion maps t : M - N and t2: N -* P are embeddings, hencet2 t 1: M -a P is an embedding, and so M is a submanifold of P by 5.6.1. tt

Finally, here is a result that will save us from a lot of calculation when dealing withsubmanifolds of R.

5.6.6 When a Subset of a Submanifold Is a Submanifold

If M and N are both submanifolds of R", and if M C N, then M is a submanifold of N.

Proof: The identity map t: R" -4 R" is an immersion, and hence its restriction to M is animmersion into N by 5.5.1; it is also one-to-one. To prove that M is a submanifold of N,it suffices by 5.6.1 to prove that t: M -4 N is an embedding. Given U open in M, weknow by 5.6.3 that U = U' n M for some U' open in R". So V = U' n N is open in Nby 5.6.3, and t (U) = V n M. Thus t: M - N is an embedding as desired. M

5.7 Constructing Submanifolds without Using Charts

The preceding results are especially useful for dealing with mappings and submanifoldsof matrix groups, such as SO (n), for which convenient charts are not available. Hereare some examples.

5.7.1 A Sphere as a Submanifold of a Hyperboloid

We may construct a sphere S" - 1 e R' + 1 as follows:

(x= (0,x1,...,x"):f(x) =x2+...+x., -1 =0}

Note that this sphere is a submanifold of R", and R" is a submanifold of R"+ 1. andtherefore this sphere is a submanifold of R"* 1. by 5.6.5. It is also a subset of thehyperboloid H", a R" + 1, which is the submanifold of R" + 1 defined by

{x = (x0,...,x"):h(x) 1 =x0}.

Think, for example, of the unit circle in the x, y plane inside the figure of a hyperboloidin 3-space in Chapter 3. Since S"- 1 is a subset of H", a R" + 1, it follows from 5.6.6that S- 1 is a submanifold of H", .


5.7.2 A Submersion of the Special Orthogonal Group

Let v e R", v * 0, and define f: GL; (R) - R" by f (A) = Av. Since Df (A) H = Hv,Df (A) is clearly onto, and so f is a submersion. Hence f restricted to SO (n) is also asubmersion by 5.6.1,

G = f-1 (v) = {Ae SO(n):Av=v}

is a submanifold of SO (n) . As a special case, take v to be the vector [0, ..., 0, 1] T.then A v = v implies that the last column of A is (0, ..., 0, 1 ) T, and by orthogonality ofthe columns of A e SO (n), we see that

G =I1; O1:Be SO(n-1)} =-SO (n-1).

In this particular case, it would have been easier to prove that G is a submanifold ofSO (n) by using 5.6.6, since G is clearly a submanifold of GL* (R), which is open in

Rnxn

5.7.3 An Embedding of the Special Orthogonal Group

In 5.3.2 we considered the smooth immersion t: GL* (R) -+ GL*+k (R) which sendsthe nonsingular n x n matrix A to the nonsingular (n + k) x (n + k) matrix

CA 01

O B]'

where B is any matrix in SO (k) . We also know from the construction of the SpecialOrthogonal Group in Chapter 3 that SO (n) is a submanifold of GL; (R), and that tmaps SO (n) into SO (n + k) . According to 5.5.1, the restricted map

t:SO(n) -*SO(n+k)

is also an immersion, and it is clearly one-to-one. Since SO (n) is compact, the"Compactness Lemma" 5.6.2 shows that t is an embedding. Now 5.6.1 shows that theimage of SO (n) under t, which can be identified with SO (n) itself, is a submanifoldof SO (n + k).

5.8 Submanifolds-with-Boundary

The material in this section will be needed in Chapter 8 in discussing orientation, and inthe statement and proof of Stokes's Theorem. A closed subset P of an n-dimensional


5.8.2 Characterization Using Submersions

P is an n-dimensional submanifold-with-boundary of M if and only if, for every r E P,there exists an open set U c M containing r so that either r e U c P or there is asubmersion f: U -> R with P n U = {q E U: f (q) 5 0}; in the second case

aPnU = {qE U:f(q) =0}. (5.8)

Proof: The "=*" part of the proof is immediate on taking, for the second case, (U, (p)to be a chart at r of the kind described in the definition, and f = x' (p to be thesubmersion. To prove the "4--" part, assume we have a submersion f: U -* R withP n U = { q e U: f (q) 5 0} where U 3 r; we need to find a chart at r of the specialkind mentioned in the definition of submanifold-with-boundary. For this, start with anychart (V, W) for M at r, with V Q U without loss of generality, and let x = Ni (r) .Observe that D (f Nr') (x) * 0. since f 11r' is a submersion on W (V) c R", andtherefore there exists a set consisting of (n - 1) of the basis vectors { e1, ..., e,,) (afterrelabeling, we can suppose the subset to be {e2, ..., e.) ) whose span does not includethe vector D (f W ') (x) . In other words, the following determinant satisfies:

D, (f Vr') (x) 0 ... 0D2 (f. XV- ') (x) I ... 0 * 0. (5.9)

(x) 0 ... I

If we define (p: V -9 R" by (p' = f, (p2 = W2, ..., (p" = yt", then (5.9) says that

D((p Vr') (x) I *0.

Now we may apply the Inverse Function Theorem (see Chapter 3) to assert that there isa neighborhood W C W (V) of x e R" on which p W-' is a diffeomorphism; if wetake U' = t}r' (W) a U, it follows from this that (U', (p) is a chart for M at r suchthat

(p (Pr U') = (p ({qE U':f(q) <0}) = {(x,,...,x") E (p(U'):X,50},

which verifies the condition for P to be an n-dimensional submanifold-with-boundary ofM. The assertion (5. 8) follows immediately. U

5.8.3 Examples

5.8.3.1 A Closed Ball In Eucildean SpaceTake M = R" +', and f: R" +' -* R to be the map f (xo, ..., x") = xo + ... + x -Since f restricted to U = R"+' - 10 } is a submersion, 5.8.2 shows that the closed ball


B(0, 1) = {(xe,...,x,1):f(xa,...,x") 51}

is an (n + 1)-dimensional submanifold-with-boundary of R"" t whose boundary is thesphere S.

5.8.3.2 A General Class of ExamplesSuppose U is an open set in R" + k, g: U - Rk and P. U -, R are submersions, anda E Rk and b E R are chosen so that M = g t (a) and { r e M: f (r) < b } are non-empty. As we saw in Chapter 3, M is an n-dimensional submanifold of R" +k By 5.5.1,the restriction off to M is a submersion, and 5.8.2 shows that P = { r e M: f (r) S b}is an n-dimensional submanifold-with-boundary of M, with boundary

DP = {re U:g(r) =a,f(r) =b}.

For a simple illustration, take n = k = 2, U = R4 - {01, and

g(x0,...,x3) _ (x +Xj,x2+Xj),f(X0,...,x3) = x0-2xt+3x2-4x3.

If a = (1, 1) and b = -2, then M = T2 a R4, and aP is the intersection of the2-torus with the hyperplane xo - 2x1 + 3x2 - 4x3 = -2.

5.9 Exercises

(5.10)

8. Prove the second part of 5.6.1, namely, that if g: M - N is a C'° submersion andp E N, then Q = g-1 (p) is an n-dimensional submanifold of M. where M and N areassumed to have dimensions n + k and k, respectively.

Hint: This follows straight from the definitions.

9. Suppose P is a product of two hyperboloids, and M is the product of two cylinders, asindicated by the following formulas:

P = {(Xp,...,XS) E 1,X3-X4-XS=-1};

M= {(xo,...,x5)ER6:Xi+x2=1,X4+x5=1}.

(i) Show directly that P and M are submanifolds of R6.

(ii) Define P n M, and determine whether it is a submanifold of R6, P, and M.respectively.

Hint: Once you have proved the first part by showing that certain maps are submersions, part (ii)can be done by the results proved in this chapter.

10. Show that the rescaled torus

5.9 Exercises 115

7" (xl,...,x2n) E R2":x2 +x2= 1/n,...,x2"_I+X2n= 1/n}1 2 2 2

is a submanifold of Stn- I (x1, ..., x2n) e R2n: xj + ... +x2" = 1 }.

11. Show that if t: R" _+ R"+k is the inclusion map which takes (x1, ..., x") to(x1, ..., x", 0, ..., 0), then the image of the sphere S" - I e R" under t is a submanifoldof the sphere S" +k - I c R" +k

12. Let us accept without proof that the map from U (n) (the Unitary Group discussed inChapter 3) to C which takes a complex matrix to its determinant is a submersion. Usethat fact to show that the Special Unitary Group S U (n) is a submanifold of U (n) , anda submanifold of GL" (C), and find its dimension.

13. Show that if f: M -> N and g: N -I' Pare embeddings, then g f is an embedding.Hint: 'Mere are three things to check; they follow from the definitions.

14. Find an example where M and P are submanifolds of R2. and f: P -a M is aimmersion, but f (P) is not a submanifold of M.

15. Let H be a symmetric n x n matrix of rank it, and J a symmetric m x m matrix of rankm. Define

GI = {A E GLn+m (R) :AT[0 H[]A = CO H] },10

G2

Determine whether GI and G2 are submanifolds of GLn+," (R), and whether G2 is asubmanifold of GI . If so, find the codimension of G2 in G1.

Remark: If H and J are both equal to the identity, then G2 is simply SO (n + m).

16. Fix a nonzero vector w e R", and let P = {A E GL" (R) : wTAw:5 11. Prove that P isan n2-dimensional submanifold-with-boundary of GL" (R) , and determine theboundary.

17. Let M = {(xe,...,x5) E 1,x4+x 5= 1}, and let

P = { (XO, ...,XS) E M:XO-Xi+X3-x450}.

Determine whether P is a 4-dimensional submanifold-with-boundary of M, and if sodetermine the boundary.


5.10 Appendix: Open Sets of a Submanifold

The following technical result was needed in the proof of 5.6.3.

5.10.1 Relation of Open Sets In a Manifold to Those of a Submanifold

If Q is a submanifold of M, then the open sets of Q are precisely the sets

{ U' n Q: U' open in M}. (5.11)

Proof: We shall prove the assertion in three steps.

Step I. Consider first the case where M = R"+k, and Q is an n-dimensionalsubmanifold of the form Q = IF (W) , where 'I': W c R" - Q c U is an n-dimensionalimplicit function parametrization, that is,

x") = (XI, ..., xn, zt (x) , ..., Zk (x) ), (5.12)

and Wand U are open in R" and R" +k, respectively. By Definition 5.2.1, V is open in Qif and only if V is of the form 'P (Wt) for some W, open in W; thus

V = U,nQ,

where U, a R" + k is the open set

U, _ {(x,y)ER"xRk:xE W1,z;(x)-1<y1<z;(x)+1,1<-i5k};

x = (x1, ..., xn), y = (y,, ..., yk).

Conversely, if V = U1 n Q and U, g R" +k is open, then V is the image of theprojection of U, onto the first n components, which is an open set W, in W, thusV = 'P (W,) is open in Q. Thus the assertion is proved in this special case.

Step II. Next consider the case where Q is any n-dimensional submanifold ofM = R" + k. By 5.1.3 and the Implicit Function Theorem of Chapter 3, we may take anatlas for Q consisting of charts of the form { (Vs,'F;') , a e 1) , where each

'Pa: WacR"-+ Va = UanQcRn+k

is a parametrization of the form (5. 12) above, maybe with the coordinates rearranged;by definition, T. is onto, Ua is open in R"+k, and Q = U Va. By definition 5.2.1.

V open in Q a V n Va open in 'Pa (WW), Va

5.11 Appendix: Partitions of Unity 117

g V n V. = UI a n Q, Va,

by Step I, where each U1 a is some open subset of R" +k. This is equivalent to

V = U (V n Va) = U (U,.a n Q) = (U U,, a) n Q = Ul n Q,a a a

where U, C R" +k is open, since it is the union of open sets. Thus the assertion is provedfor this case.

Step IIL Finally consider the general case. Let { (U7', (py) , y r= J} bean atlas for M.By the definition of submanifold, and of the differentiable structure of Q,

V open in Q c (Py ( V n Uy') open in 4py (Q n Uy') , Vy,

(py(VnUY) = U1,yn(Dy(QnUy) cR"+k, Vy,

for some open set UI, y in R" +k, by the result of Step II, since (py (Q n Uy') is asubmanifold of R"+k This is equivalent to

V n Uy' = (pY I (U, y) n Q, Vy,

where necessarily 0.1 (U, y) is open in M. This is equivalent to

V=U,'nQ,U,' = UpyI(U).y),y

and the last set is open in M, so the proof is complete. rz

5.11 Appendix: Partitions of Unity

This is a technical tool which will be needed in Chapter 8. Suppose { U , j E J} is anopen cover of a differential manifold M. A (smooth) partition of unity subordinate to{ Uj, j E J} means an open cover { Va, a e 11 of M together with a collection{ va: M -+ [ 0, 1 ] , a e 1} of smooth functions with the following four properties:

The support4 of va is contained in V.

For each Va there exists a j (a) such that Va g Uj (a1.

{ Va, a e 11 is locally finite, which means that every r E M has a neighborhood Usuch that U n Va = 0 for all but finitely many a.

4 The support of a function f means the smallest closed set containing the set on which f * 0, thatis, the complement of the union of all the open sets on which f = 0.


E va (r) = 1, Vr a M. (For fixed r, only finitely many summands are nonzero.)aEI

In applications we usually omit mention of the { Va, a E 1}, and refer simply to "a par-tition of unity { va, a e 11 subordinate to { Up j e J}."

5.11.1 Existence of a Partition of Unity

If M is a differential manifold with a countable atlas { (Uj, cp1) , j E J} , then thereexists a partition of unity subordinate to { Ul}.

Proof: The proof is essentially topological, and is therefore outside the scope of thesenotes; see Berger and Gostiaux [1988] or Spivak [1979], Volume I. Amoresophisticated treatment applicable to infinite-dimensional manifolds may be found inLang (1972). rX

5.11.2 Example

The idea of a partition of unity is intuitively simple. Take, for example, the circleS' = { (x, y) a R2: x2 + y2 - I =0) with the atlas { (U, (p), (V, W) } , where

U=S'-(0,1),tp(x,y) = 1xy;V=S,-(0,-1),W(x, y) = 1+y.

Take any smooth function x: R -+ [0, 11 such that x (t) = 1 for t e [-1, 1 ] , andsuch that, for some a > 1, x (t) = 0 for 14 > a; a formula for this so-called "bumpfunction" is given in Berger and Gostiaux [1988], p. 13.

ZFigure 5. 3 A 'bump function'

Define

v, (x, Y) = K(p(x,y)), (x,y) E U; v, (0, 1) = 0;

t

Since cp (x, y) = 1 /yr (x, y) on U n V, it follows that vt + v2 = 1, and one maycheck that the support of v, is contained in U, and the support of v2 is contained in V



The notion of a differential manifold emerged in the work of B. Riemann (1826-66),E. Betti (1823-92), H. Poincar6 (1854-1912), and others. The modern viewpoint isassociated with a 1936 paper by H. Whitney (1907- ). A much fuller treatment of thesetopics may be found in Berger and Gostiaux [1988]. The infinite-dimensional version isin Lang [1972], and in more detail in Abraham, Marsden, and Ratiu [19881.

6 Vector Bundles

The notion of vector bundle gives a powerful and flexible tool for all kinds of calculuson differential manifolds. Important special cases are the tangent bundle of a differentialmanifold, the cotangent bundle, exterior powers of these bundles, etc. This chapterconsists mostly of definitions and constructions.

6.1 Local Vector Bundles

Suppose M is a differential manifold of dimension n, for example, an open subset of R",and V is an arbitrary k-dimensional vector space. The product manifold' M x V,together with the "projection map" n: M x V -> M such that n (p, v) = p, is called alocal vector bundle of rank k over M. We call M the base manifold, and { p) x V iscalled the fiber over p, for any p e M; think of a fiber as a copy of the vector space V,sitting on top of the point p, as in the picture on the next page. A convenient generalnotation is to refer to such a vector bundle as E = M x V, and to refer to the fiber overpas EP= {p}xV.

6.1.1 Sections of Local Vector Bundles

A Cr (resp., C) section of a local vector bundle E over M means a C' mapping2a: M --. M x Rk such that a (p) a EP for every p. The idea is simple: a chooses a pointa (p) in the fiber over p for every p, in a way that varies r-times differentiably acrossthe fibers. We denote the set of Cr sections of E by t' (E) . Every vector bundle has azero section defined by a (p) = (p, 0) e { p} x V.

1 If M is an open subset of R", then M x Rk is an open subset of R" +k with the correspondingdifferentiable structure; for the general case, see the exercises of Chapter 5.2 See Chapter 5 for the meaning of this.

122 Chapter 6 Vector Bundles

6.2 Constructions with Local Vector Bundles

6.2.1 Morphisms of Local Vector Bundles

This section may clarify the discussion of differentials and pullbacks in Chapter 2.Suppose E = M x V is a local rank-k vector bundle over an ni-dimensional manifoldM, and E' = N x W is a local rank-q vector bundle over an n-dimensional manifold N.A C' map 4: E = M x V -4 E' = N x W is called a Cr (local) vector bundlemorphism if, for some map (p: M -4 N, it takes the form:

41 (p, v) = ((p (p), g (p) v) e N x W, (6.1)

where g (p) e L ( V -a W) for every p. In other words, 4' is said to be "linear on thefibers," where the linear transformation may vary from fiber to fiber. To clarify theformalism, note that

(p, v)E {p}xV=EP,4'(p,v)E {(p(p)}xW=E'lp (P),

and therefore

(v -0(p,v)) E L(EP-4 E'v(P)),Vp.

Multivariable calculus shows that an equivalent condition (in this finite-dimensionalcase) for a map 4': E - E' to be a C' local vector bundle morphism is that it takes theform (6. 1), and that the maps (p: M -a N and g: M L (V -a W) are Cr.

We call the map 4' a vector bundle morphism over the identity if M = N and the map(p is the identity.

6.2.1.1 Examples: The Tangent Map and Pullback for Mappings of Euclidean SpaceFrom our study of differentials and pullbacks in Chapter 2, some obvious examplesspring to mind. Suppose U c R' and U%; R" are open sets, and (p: U - U' is asmooth onto map. The tangent map Tp is the morphism from the tangent bundle TU tothe tangent bundle TU' which sends a tangent vector F, = (y, v) at y to

T(p(y,v) = ((p(y),D(p (y) v) = d9 (y)4E To(Y)R". (6.2)

Thus for these local tangent bundles, the distinction between the differential and thetangent map is essentially one of formalism: When we view the maps {dg (Y), y E U}as a single smooth map between TUand TU', then we obtain T(p.

The pullback of (p is, strictly speaking, a mapping from differential forms on U' (i.e.,sections of the exterior powers of the cotangent bundle over U') to differential forms on

6.2 Constructions with Local Vector Bundles 123

U. However we may abuse notation slightly and write cp* as a map from T* U' to T* Ugiven by

0* ((P (y), a) = (y, aD(p (y) ) a (TRm) * , (6.3)

where we may think of the n-dimensional row vector a as premultiplying the n x mderivative matrix Dtp (y) to give an m-dimensional row vector. As in Chapter 2, onemay easily write down the corresponding extension of cp' to any exterior powerAq (T* U'). Actually the pullback construction is even more general, as we shall see inExercise 3 in Section 6.5.

6.2.2 Local Vector Bundle Isomorphisms

Consider the special case of a C' morphism of local vector bundles C: E -a E', wherethe bundles E = M x V and E' = N X W have base manifolds which arediffeomorphic to each other, V and Ware isomorphic vector spaces (i.e., of the samedimension k), and where 1 takes the form (6. 1) where:

cp: M -4 N is a C' diffeomorphism, and (6.4)

g (p) e L (V -4 W) is invertible for all p; (6.5)

in other words g (p) is a linear isomorphism between the fiber over p and the fiber overtp (p) which varies in a C' fashion with p. Then we call 0: E -4 E' a C' local vectorbundle isomorphism (over the identity, if (p is the identity map), and we say that E andE' are C' isomorphic; the usual case is when r where we often say "E and E' areisomorphic."

Examples would be the tangent map Tcp and the pullback in Example 6.2.1.1, in thecase where cp: U -> U' is a diffeomorphism (such as the change of variables mapbetween two coordinate systems on R"), for in this case the differential of <p is aninvertible linear map between respective tangent spaces.

6.2.3 The Homomorphism Bundle

Suppose E = M x V and E' = M X W are local vector bundles over the same basemanifold M. Define the homomorphism bundle Horn (E, E') to be the local vectorbundle M x L ( V -* W) ; in other words the fiber over p is

Hom (E, E') P = L (EP -4 Ep'). (6.6)

Clearly the rank of Hom (E, E') is the product of the ranks of E and E', since thedimension of L (V -+ W) is dim V x dim W. Referring to Section 6.2.1, we see thatthere is a one-to-one correspondence between sections a of Hom (E, E') and the vectorbundle morphisms 0 over the identity from E to E', by the formula


IV (P, v) = (P, (I (P) v) e M X W. (6.7)

This concept of homomorphism bundle has many applications, of which we now list afew.

6.2.3.1 Dual BundleIn the case where E' = M X R, the local line bundle over M, the fiberHom (E, M X R)

vis simply the dual space EP* , and the bundle Hom (E, M x R) is

called the dual bundle to E, denoted E" . For example, the cotangent bundle T* Udescribed in Section 6.1.2 is the dual of the tangent bundle TU, and a result in Chapter 1shows that, more generally, the vector bundle A9 (T* U) is isomorphic to (A" (TU)) *.

6.2.3.2 The Differential of a Map as a Section of a Homomorphism BundleSuppose U C R' and U' g R" are open sets, and (p: U - U' is a C onto map. Define anew vector bundle

(p* (TU') = UxR", (6.8)

where the fiber ((p* (TU') ) r is identified with T.(y) R". This is known as the pullbackof the tangent bundle TU' under (p. Now the differential y --a d(p (y) is a C'-1 sectionof Hom (TU, (p* (TU') ), since

d(p(y) E L(TRm-aTc(V) R") = Hom(TU,C (TU'))v, VyE U, (6.9)

and since the map y -a D(p (y) has just one degree less of differentiability than (D has.

6.2.3.3 Differential Forms with Values in a Vector BundleIf E is the tangent bundle TU of an open set U C R, then sections of Hom (TU, E')can be called 1-forms on U with values in the vector bundle E'. In the previousexample, we see that the differential of (p can be regarded as a 1-form on U with valuesin the pullback of the tangent bundle of U'. A fuller discussion of bundle-valued formswill be given in Chapter 9.

6.2.3.4 Tensor Product Notation for a Homomorphism Bundle(May be omitted.) As we discussed in Chapter 2, the tensor product V" 0 W isisomorphic to L (V -a W) , for vector spaces V and W. Thus if E = M x V andE' = M x W are local vector bundles over the same base manifold M, the vector bundleHom (E, E') is isomorphic to the local vector bundle

E` ®E'=Mx (V* ®W), (6.10)

a notation which some authors prefer to the "Hom" notation. To illustrate thisterminology, one could write the differential in (6. 9) in yet another way:


d(P (x) = ID,p (x) (dx` ®ayr), (6.11)

i.i

where {x`} and {)'} are coordinate systems on U and U', respectively. (In Chapter 9,we shall use a wedge instead of the 0 notation.)

6.2.4 Other Constructions with Local Vector Bundles

Let E = M x V and E' = M x W be local vector bundles over the same base manifoldM, with ranks k and k', respectively. Table 6.2 summarizes some of the constructionsabove, and gives some new ones.

Name Rank Fiber over p e M Smooth Sections

dual bundle = Hom (E, M x R) k EP* -

homomorphism bundle

Hom (E, E') __ E* 0 E'

kk' L (EP - Ep') vector bundle mor-phisms over the

identity

tensor product bundle E 0 E' kk' E OE' -exterior power bundle A' (E) k! /r! (k - r) ! A' (E) -

direct sum bundle E ®E' k + k' E E ' -Table 6.2 Constructions using local vector bundles over the same base manifold M

6.3 General Vector Bundles

6.3.1 Definition

Suppose M is a differential manifold. A manifold E together with a smooth onto mapic: E - M (called the projection) is called a C' vector bundle of rank k over M (oftenwe refer to E itself as the vector bundle) if the following three conditions hold:

There exists a k-dimensional vector space V such that, for every p e M,EP = X-1(p) is a real vector space isomorphic to V, EP is called the fiber over p.

Each point in M is contained in some open set U C M such that there is a C'diffeomorphism

41U: n-1 (U) -4 U X V (6.12)

with the property that OU restricted to the fiber E. maps EP onto (PI x V.


Now consider instead the MSbius strip as a subset of R1; if one were to extend the sidesof the strip to infinity, then here too we have a copy of the real line attached to everypoint on a circle, and this too is a rank one vector bundle over S1, because if one looksonly at the part of the manifold consisting of fibers attached to some arc U of the circle,this part can be "unrolled" to look like the infinite rectangle U x R. In Section 6.4.2.1we shall give a formal construction of the so-called "Mobius bundle."

6.3.2 Transition Functions

Suppose DU and a'u, are two local trivializations, and p r= U n U'. The assumptions(6. 12) imply that the map d'u 4 , applied to (p, v) E (U n U') x V, is linear in vfor fixed p, and thus is of the form shown in the following diagram, where guu. (p) is alinear map from V to V for each p:

n'(UnU') cE

Moreover since `DU and `Iu, are C' diffeomorphisms, 4U 4 must be C' with a C'inverse, and therefore

P -' guu (P) (6.14)

is a C' map from U n U' into GL (V), the invertible linear transformations from V to V(a differential manifold isomorphic to GLk (R), the invertible k x k matrices); this mapis called the transition function from the local trivialization (DU to the localtrivialization QDu.. It follows straight from the definition that transition functions havethe properties:

guu(P) = 1; (6.15)

guu' (P) gu'u (P) = 1, P E U n U'; (6.16)

guu'(P)gu'u-(P)gu-u(P) = 1,pE UnU'nU". (6.17)


6.3.2.1 Example: Transition Functions for the M8blus Band

W

U'

Figure 6. 4 Domains of transition functions for the MObius bundle

Divide the base manifold, in this case the circle, up into two open arcs U and U' whichoverlap at both ends, as in Figure 6.4, and let Wand W denote their regions of overlap.The local trivialization maps identify the parts of the band sitting on top of U and U'with U x R and U' x R, respectively. The single "twist" observed in the Mobius strip isobtained by letting the transition function, which here takes values in the nonzero reals(i.e., invertible one-by-one matrices!), be -1 on one of the overlap regions, and 1 on theother. To understand this, think of how you would glue together two rectangular piecesof paper, with bases U and U', to make a Mobius band. Specifically:

guu' (P) = gu'u (P) = 1, p e W, and = -1, p E W;

guu (P) = (P) = 1.

The reader may easily verify that (6. 15), (6. 16), and (6. 17) are satisfied.

(6.18)

6.3.3 Complex Vector Bundles

If V is replaced by a complex vector space throughout this chapter, we obtain a class ofvector bundles called the complex vector bundles of rank k over M. Note that E and Mare still "real" differential manifolds (not complex manifolds), but the group ofinvertible linear transformations of V would, for example, become GL, (C), themanifold of nonsingular k x k matrices with complex entries, if V were a complexk-dimensional space. For example, a complex vector bundle of rank 1 is called acomplex line bundle, and the total space has dimension n + 2 (since a complex numberis represented by two real numbers) if the base has dimension n. Complex vectorbundles will figure prominently in Chapter 10.


6.3.4 Vector Bundle Morphisms and Isomorphisms

If n: E -> M and tt': E' -* M' are vector bundles with fibers isomorphic to vectorspaces V and V', respectively, and f: M -+ M' is a Cr map, then a Cr map F: E -+ E' iscalled a Cr vector bundle morphism over f if it maps the fiber EP linearly into the

fiber Ef(P) for each p e M. The following diagram encapsulates the relationship of fand F

E.

R

M

F

f

It is easy to see from the definition that the composition of two vector bundlemorphisms is a vector bundle morphism. To see how to verify using local trivializationsthat an arbitrary mapping F: E -+ E' is a vector bundle morphism, see Exercise 5 inSection 6.5.

The word "morphism" can be replaced by "isomorphism" if f is a diffeomorphism andF acts as a linear isomorphism on each fiber. Of course this can only occur if M and M'are manifolds of the same dimension, and if dim V = dim V'. If such an isomorphismexists, then the two vector bundles are called C' equivalent; as the name suggests, thissets up an equivalence relation on vector bundles over M.

6.3.5 Subbundles

We say that a C' vector bundle it': E' - M is a subbundle of a C' vector bundlen: E -4 M (note that the base manifolds are the same) if EP' is a vector subspace of EPfor every p e M, and if the inclusion map t: E' - E is a C' vector bundle morphism.An important example will be given in Exercises 11. and 12. on page 140.

6.3.6 Trivial and Nontrivial Bundles

A bundle equivalent to the local vector bundle proj : M x V -+ M (see Section 6.1) isalso called a trivial bundle; otherwise a bundle is called nontrivial. Examples of trivialbundles include:

a bundle for which a single trivialization suffices as a trivializing cover.

the tangent bundle to the circle, to be constructed in Exercise 9.in Section 6.7.

Some nontrivial bundles that will be defined later include:

the Mbbius bundle (see Section 6.4.2.1 and Exercise 4 in Section 6.5);

the tangent bundle to the sphere S2 (the proof uses a theorem in topology which says"you can't comb a hairy ball," i.e., there is no nonvanishing section).


6.3.7 Sections of Vector Bundles

Following Section 6.1.1, a Cs section of a Cr vector bundle n: E -4 M, where s:5 r, isa C mapping a: M -4 E such that n a (p) = p, p e M; in other words, a (p) a EPfor all p e M. The set of smooth sections is denoted FE. The graph of a section lookslike a slice through the vector bundle which cuts each fiber exactly once, as in the fol-lowing picture.

EP

graph of a section a

p

6.4 Constructing a Vector Bundle from Transition Functions

Frequently we would like to construct a vector bundle over a particular base manifoldM, knowing what vector space V the fibers must be modeled on, and knowing what thetransition functions must be, but without knowing in advance what kind ofdifferentiable structure a corresponding vector bundle may have. The following theoremguarantees existence and uniqueness of a differentiable structure of a correspondingvector bundle over M, in the sense described below.

6.4.1 Vector Bundle Construction Theorem

Let V be a k-dimensional vector space (the "fiber"), and let { Ua: a e 11 be an opencover of a differential manifold M. Suppose that, for every a, y e I with Ua n Uy * 0.

p -->g.7(P) =gU,U,(P) (6.19)

is a C" map from U. n U., into GL (V), the invertible linear transformations from V toV such that (6. 15), (6. 16), and (6. 17) hold. Then there exists a Cr vector bundlen: E -4 M of rank k with the mappings (6. 19) as transition functions, and any othersuch vector bundle is Cr isomorphic to E.

The proof is in the appendix to this chapter, and need not be mastered yet. The mostimportant example of this construction, namely, construction of the tangent bundle, is in

6.4 Constructing a Vector Bundle from Transition Functions 131

Section 6.6.2. The following version is sometimes more convenient in the context ofconstructing new vector bundles from existing ones.

6.4.2 Vector Bundle Construction Theorem - Alternative Version

Let V be a k-dimensional vector space, and for each point p in a manifold M. let EP be avector space isomorphic to V Let E = UEP (disjoint union), and 7t: E -) M be themap such that n-1 (p) = EP. Suppose also that { Ua: a r= 1} is an open cover of M,and for each a

0a: 7t'' (Ua) -* Ua X V (6.20)

is a bijection such that, for every a, y e I with U. n Uy * 0,

Oa.0Y1: (Uan Uy) X V-4 (Uan Uy) X V (6.21)

is a Cr local vector bundle morphism over the identity (see Section 6.2.2) such that(6. 15), (6. 16), and (6. 17) hold for the corresponding transition functions. Then thereexists a unique differentiable structure on E such that 7t: E - M is a Cr vector bundleof rank k with the maps (6. 20) as a trivializing cover.

The proof is simply Step III in the proof of the previous theorem, in the appendix to thischapter.

6.4.2.1 Example: The M6bius BundleHere is the abstract construction of the Mobius bundle, as distinct from the concreterealization in R3 shown in Figure 6. 3. The role of "V" in Theorem 6.4.1 is played by R,the open cover consists of sets U and U' as shown in Figure 6. 4. and the transitionfunctions are given by (6. 18). The Mobius strip in R3 with infinite sides is C'equivalent to this bundle, because it has the same transition functions.

6.4.3 Constructions with Vector Bundles

All the constructions with local vector bundles listed in Table 6.2 are valid for any C'vector bundles.

Proof: For the sake of brevity, we shall only do the construction of Hom (E, E'), whereE and E' are C' vector bundles over the same base manifold M. with fibers isomorphicto vector spaces V and W, respectively. By taking restrictions if necessary, we canassume that the transition functions {gay} for E and {ga' } for E' are defined on thesame open cover { Ua, a e 1} for M.


We wish to apply Theorem 6.4.1. taking L (V -+ W) to be the fiber, and takingtransition functions:

hay (P) A = gay (P) Agya (P), A e L (V -), W). (6.22)

The right-hand side of (6. 22) makes sense as the composition of three linear maps,which are respectively V -* V, V --p W, and W -+ W, giving an element of L (V -), W)as desired. We must check the conditions (6. 15), (6. 16), and (6. 17). It is clear thathas (p) A = A because (6. 15) holds for the {gay} and {gay'} . Using (6. 16) for the{gay} and {gay },

hya (P) hay (P) A = gya' (P) (gay (P) Agya (P)) gay (P) = A. (6.23)

Finally

hap (p) hpy (P) hya (P) A = gap' (p) gpy (p) (gya' (P) Agay (P)) gyp (p) gpa (P)'

which equals A since (6. 17) holds for the {gay} and {gay}. Moreover the invertibilityof the linear transformations { hay (p) } is shown by (6. 23). Thus the conditions ofTheorem 6.4.1 are satisfied, giving a construction of the C' vector bundleHom (E, E'). zx

6.4.3.1 Description of the FibersThere is a natural isomorphism Hom (E, E') P = L (EP -+ EP') , and we shall usuallyregard these vector spaces as identical. The equivalence class [p, a, A] inHom (E, E') P, in the sense of (6. 50), may be identified with the linear map[p, a, v] - [p, a, Av] from EP to EP'; this map does not depend on the choice of a.

6.5 Exercises

1. Suppose E = M x V and E' = M x W are local vector bundles over the same basemanifold M. Which of the following maps are C local vector bundle morphisms? Jus-tify your answer.

(i) s: E G) E -* E. (p, (vi, v2)) -> (p, VI + v2).

(ii) t: Hom (E, E') ® E -* E', (p, A ® v) -> (p, Av).

(iii) Take W = R; u: E ® E'-4 E, (p, v ® w) -9 (p, v/w) when w * 0, and (p, 0)otherwise.

C - ( p ,(p r=(v)

Take E = S2 x R3; u: E ®E - E. (p, v ®w) - (p, (p v) w), consideringS2cR3.

6.5 Exercises 133

2. Give an example of a Cl vector bundle morphism that is not a C" vector bundlemorphism.

Hint: Consider part (iv) of Exercise l with a suitable (p.

3. Let n: E -a M be a C vector bundle and let g: N -+ M be a C' map for some manifoldN.

(i) Using Theorem 6.4.1, or otherwise, show that

g* E = { (n,4):n(E) =g(n)} cNxE (6.24)

may be identified with the total space of a vector bundle (the pullback bundle under g)over N, with projection g* n: g* E -4 N, g* n (n, ) = n, and where the fiber over nmay be identified with Eg (rt).

(ii) Show that the map 0: g* E -* E, 45 (n, 4) = t;, is a C' vector bundle morphismwhich is the identity on every fiber.

(iii) Show that if two vector bundles over M are C' equivalent, then their pullbacksunder g: N - M are C' equivalent.

(iv) If h: Q -a N is also a C' map, show that the vector bundles

(g*n):h* (g*E) -+Q

are C' equivalent.This problem was adapted from Abraham, Marsden, and Ratiu [1988], p. 193.

4. (i) Prove that if F in the following diagram is a vector bundle morphism, and a is asection of the bundle n: E -+ M, then F a is a section of n': E' -a M.

E F E'

nn'

M

Moreover if F is a vector bundle isomorphism, then

a(r) *OVr4* F a(r) *OVr.

(ii) Prove using the transition functions (6. 18) that if a is a section of the Mobiusbundle, then a (r) = 0 for some r e St.

(iii) Prove using (i) and (ii) that the Mobius bundle is not equivalent to the trivial bundleproj:S'xR-4 S'.

5. Let n: E -* M and it': E' -* M' be C' vector bundles with fibers isomorphic to vectorspaces V and W, respectively, and let f: M -+ M' be a Cr map. Also suppose F: E -4 E'


is an arbitrary mapping, not known to be differentiable, but which takes the fiber EPinto the fiber E'f(p3 for each p e M. Prove that F is a C' vector bundle morphism overf if and only if the following condition holds:For each p e M, there exist trivializations (u for M and `Y u, for M' with U 3 p andU' 3 if (p), restricted if necessary so that f (U) Q U', such that

):UxV-4 U'xV', (6.25)

represented below, is a C' local vector bundle morphism over f (see Section 6.2.1 ).

F

UXVTV*F'Ou

6. Using 6.4.3 as a model, carry out the construction of the following C' vector bundles,starting from a C vector bundle 7t: E -4 M with fibers isomorphic to a vector space V:

(1) The dual bundle n : E"` -4 M, where the fiber over p is to be identified with EP .duW

(ii) The exterior product bundle n.: A2E -i M, where the fiber over p is to be identifiedwith A2Ep.

6.6 The Tangent Bundle of a Manifold

6.6.1 Tangent Vectors - Intuitive Ideas

When a curve M is a one-dimensional submanifold of R3, it turns out to be useful to puta differentiable structure on the set of (disjoint) tangent lines to the curve, such thatthese tangent lines are in some sense "smoothly related" to one another as one movesalong the curve. In the same way, one may wish to put a differentiable structure on theset of (disjoint) tangent planes to a parametrized surface Mat all points in the surface.and indeed to the union of all the tangent planes to an n-dimensional submanifold M ofR"''k. Instead of treating all these cases separately, it is most efficient to define a vectorbundle called the "tangent bundle" of an abstract differential manifold. Although theabstract symbolism we are going to use now may seem somewhat removed from theoriginal idea of tangent line or tangent plane, the reader may continue to think of tan-gent spaces in those terms.


The only way we can identify a "tangent vector" in a manifold is by taking a chart, andpicking some tangent vector in the range of the chart. However, we then need some wayof keeping track of the same "tangent vector" if we shift to a different chart. This isaccomplished by referring to an "equivalence class" of vectors in the ranges of variouscharts, as we shall now describe.

6.6.2 Formal Construction of the Tangent Bundle

Now we are going to work through the main ideas of the proof of the Vector BundleConstruction Theorem, as presented in Section 6.9.1, for the special case of the tangentbundle. Let { (Ua, tpa) , a E 1} be an atlas for an n-dimensional smooth differentialmanifold M. For any chart (U, cp) for M at p, cp (U) is an open set in R", and thereforethe tangent space TxR" at x = cp (p) is well defined in the sense of Chapter 2. Considerthe set of triples

T = { (p, a, f;):aE 1,pe Ua,4E 7',po(p)R"}. (6.26)

We shall say that two such triples are equivalent, written (p, cc,

p = q, = d(Wy`t ') (pa(p))

(q, y, ), if

(6.27)

where, as in Chapter 2, d (q y gal) is the differential of the map

cpy cpat : Ua n UY g R" 4 Ua n UY g R" ,

which is being evaluated at cpa (p), and applied to the tangent vector 4. To shift into thenotation of Theorem 6.4.1, we may write our "transition function" as

gya (p) = d (q'. tpa') (pa (p)) E GL" (R), (6.28)

which is C° on U. n UY * 0. Of the conditions (6. 15), (6. 16), and (6. 17), the first isimmediate, while the others follow from the chain rule for differentiation (seeChapter 2), which shows that

d(ga'gy') (y) _ (x)

d9-') (x) = d(tp6*V') (y) ad(gy'ga') (x),

where x = ga (p) , y = py (p). Thus the conditions of Theorem 6.4.1 are satisfied. Inparticular, (6. 27) defines an equivalence relation on T, as in the proof in Section 6.9.1.The set of equivalence classes is denoted

TM = { [p, a, 4] : a e 1, p e Ua, 4 E Tea(P) R" }, (6.29)


and the proof in Section 6.9.1 shows the existence of a C- differentiable structure onTM under which we call 1t: TM - M, [p, a, 4) - p, the tangent bundle of M.

For fixed p, an equivalence class of the form (p, a, l;) is called a tangent vector at p;the set of such vectors is called the tangent space at p, denoted TPM.3 It has thestructure of an n-dimensional vector space, induced by the bijection

[P, a, 4] -4 4, TPM -4 T P. (P) Rn = Rn

and this structure does not depend on whether we represent a tangent vector as [p, a, ]

or as (p, y, l;], because d ((py (pa') ((pa (p) ) is a linear isomorphism.

6.6.3 Computing an Atlas for the Tangent Bundle

Note that TM is a 2n-dimensional manifold, and according to (6. 51), it has a smoothatlas { (n-1 (U,), (Pa) ,a e 1}, based on the atlas { (Ua, (pa) , a e 1} for M, given by

(Pa ([P, a+ 4) ) = Na (p) , ) E Rn x Rn. (6.30)

Note that, when Ua n Uy * 0, cpy a (p-,' involves the differential of the map (Py a (PQ' :

(Py (Pat (x,) _ (y, d ((P7 (PQ') (x) ), (6.31)

for x = (pa (P) , y = (py (p) , as shown by (6. 27).

6.6.3.1 Example: Tangent Bundle to the Projective PlaneThe real projective plane P2 (R), here abbreviated to P2, can be thought of as amanifold consisting of the lines through the origin in R3, or as the sphere S2 withopposite points identified. Formally speaking, P2 is the set of equivalence classes inR3 - { 0}, under the equivalence relation - defined by:

x - y if and only if x and y are collinear, (6.32)

and with the differentiable structure induced by the atlas { (U;, (Pi), i = 0, 1, 21defined as follows. For i = 0, 1, 2, let

V; = {X= (x0,x1,X2) E R3:X1:A 0} (6.33)

3 The reader will have the opportunity in Exercise 7 to check that this notation is consistent withearlier definitions.


and define maps D.: V, -4 R2 by

4o (x) = (x1/xo, x2/xo), c1(x) = (xo/xt, x2/xt), 42 (x) = (xo/x2, xt/x2).

If [x] denotes the equivalence class of x under ", and if U. _ { [x] : x e V1 }, then4, (x) = 41, (y) r-* [x] = [y], and so we may take cp,: U1 - R2 to be the bijection

(Pi ([x]) = 4)1 W. (6.34)

The change-of-chart maps take the form

i poi (zo, z,) = U-0 , zo) (6.35)

etc. (see Exercise 8). Thus the tangent bundle TP2 to the real projective plane is afour-dimensional manifold with an atlas consisting of the three charts{ (n-1 ( U 1 ) , 1 ) , i = 0, 1, 2 }, where, for example,

(Po ([ [xl , 0, 4]) = ((x1 /xo, x2/xo) , 4), (6.36)

and where, for zo * 0,

Z14 4291 tpo' (z0, zi, 41, 42) =

I-, zo-t--Z02' - - tt + zo

(6.37)

6.6.3.2 Interpretation of a Tangent Vector on a Surface In Terms of ParametrizationsConsider two different parametrizations `P = (`YI (u, v) , `Y2 (u, v) , `P3 (u, v)) and

_ (4)1 (r, s), 4)2 (r, s), (b3 (r, s) ) of some surface Min R3, and suppose thatp = `Y (u, v) = 0 s) a M. Each parametrization gives a chart for M as describedin Chapter 5. According to (6. 27), to say that

(p, `Y-1, a1+ ba) - (p, t-', Aar+ Bas)

means that

(d(4-i "F)) (a#-+ba-) Aar+Ba-,

or, in the terminology of Chapter 2, that

D (4)-' ') (u, v) I oal = [Al]

The same expression can be written in terms of a Jacobian matrix, that is,


Car/au ar/avl [alas/au as/avJ b B'

where the Jacobian must be evaluated at (u, v) . Loosely speaking, we can say that the(Euclidean) tangent vectors

aau + ba-at (u, v) and Aa + Bas at (r, s)

correspond to the same tangent vector on M, because one transforms into the otherunder the change of variables formula. The abstract definition of tangent vector ismerely intended to formalize this notion.

6.6.4 The Cotangent Bundle

The cotangent bundle T' M is simply the dual bundle to TM, in the sense ofSection 6.4.3. A step-by-step construction is suggested in Exercise 15. The fiber over p,denoted TA* M, is called the cotangent space at p, and maybe identified with (TAM) * ,as shown in 6.4.3.1. Elements of Tp M are called cotangent vectors at p.

6.6.5 Example of a Vector Bundle Morphism: The Tangent Map

Given a C+' map f: M -a M', the tangent map

Tf: TM - TM', (6.38)

whose restriction to TAM is denoted TAf, is defined as follows: If (Uw T.) and( Uy', 4'Y) are charts at p E M and f (p) E M', respectively, and x = Ta (p), then

Tf ([p. (x, jl) = [f (p) , Y, d (WT f (eat) (x) 41, (6.39)

which does not depend upon the choice of charts. Beneath the cumbersome notation, theformula (6. 39) is saying the same thing as (6. 2); TAf e L (TAM -a Tf(A) M') is simplya more abstract version of the differential we encountered in Chapter 2, and coincideswith it when M is an open subset of R". Note that the following diagram commutes (i.e.,fen =

n n

M f M'

6.7 Exercises 139

In terms of local trivializations (><-' (U"), (Do) and (7c'-' (Ur'), T.) for TM andTM', respectively,

p

Thus it follows from Exercise 5 that Tf is a C` vector bundle morphism.

6.7 Exercises

(6.40)

7. In the case where U is an open subset of R", verify that the local vector bundle TU overU, described in Section 6.1.2, coincides with the definition of the tangent bundle inSection 6.6.2, using the identity map as the chart for U.

8. In the projective plane example 6.6.3.1, calculate cp2 (pI' and <p2 cp] .

9. Consider the circle S' c R2 with the atlas { (U, 9), (V, yr) } as follows; let`I' (8) = (cosO, sinO) E S', and let

U =''(0,2n),(p = `I'-'1";

V ='P(-it,tt),W ='r-11 V.

Construct an atlas for the tangent bundle TS' to the circle from this atlas, and show thatTS' is C' equivalent to the trivial bundle S' x R -* S'.

10. In Exercise 1 of Chapter 5, we saw that the pair of charts (U, (p) and (V, yr) belowform an atlas for S" (x0, ..., x") E R" +' : xo + ... + xn = 1 } .

(x1, ..., x")U = S"- {(1,0, ...,0) },(p(xo, ...,x") =

1 -x0

" (x1, ..., x")V = S - { (-1, 0, ..., 0) } , p (xa, ..., x") =

l +xo

Calculate the associated trivializations Ou and Ov, and transition functions guy and9VU, for the tangent bundle TS" in the way suggested in Section 6.6.2.

11. Suppose that M is an n-dimensional submanifold of R"+* Prove that rt: TM -a M is aC' rank-n subbundle (in the sense of Section 6.3.5) of the C" vector bundle710 : TR"+kIM -+ M (i.e., the tangent bundle of R"+k, restricted to M).


12. Extend the result of Exercise 11 to show that if M is a submanifold of N. thenn: TM - M is a subbundle of no: TNI u -+ M.

13. (Normal Bundle) Suppose M is an n-dimensional submanifold of R" +k Our goal is toconstruct a vector bundle of rank k over M, called the normal bundle, whose fiber overp E M can be identified with the "normal space"

(TTM)1 = {p} x {CE TpR"+k:(CIa)=0,VCE TpM}, (6.41)

where here we are interpreting TpM as a vector subspace of the Euclidean inner productspace TpR"+k. Let

T1M = U (TTM) (6.42)p

be the disjoint union of these normal spaces. and it: T1M -+ M be the map such that

n I (P) = (TTM)1.(i) Given p e M, there exists U' 3 p open in R"+ k and a submersion fu: U'--+ Rk suchthat U = U' n M = fu- (0) , by definition of a submanifold. Show that the map

(DU: n-1 (U) U x Rk, (6.43)

OU(q,C) = (q,Dfu(q) (6.44)

is a bijection.

(ii) Suppose that fv: V' - Rk is a submersion from another open set V' c R"+k withV 3 p, such that V = V n M = fvt (0). Show that there is a well-defined smooth func-tion guv: U n V -+ GLk (R) such that

`Au' w (q, C) = (q, guv(q) C)

Hint: The function must satisfy guv(q) Dfv(q) l; = Dfu (q) , e (TqM) 1. For smoothness,extend Dfv (q) to an invertible (n + k) x (n + k) matrix Av(q) depending smoothly on q, sothat guv(q) = Dfu(q)Av(q)-t(iii) Complete the construction of a vector bundle n: TLM -+ M with local trivializa-tions and transition functions of the form given in (i) and (ii) above.

14. Suppose M is an n-dimensional submanifold of R"'*, and n: TL M -+ M is the NormalBundle discussed in Exercise 13. Prove that, in the case where there exists an opensubset U' C; R" +k and a submersion f: U' --+ R such that M = f t (0) (spheres, tori,and hyperboloids, for example), then the Normal Bundle is trivial.

15. Carry out a step-by-step construction of the cotangent bundle 7* M (the dual of thetangent bundle) along the lines of Section 6.6.2, starting with the set of triples


7* = { (p, a, X) : a E 1, p E Ua, ?. E (T,. (P) R") * }, (6. 45)

where two such triples are equivalent, written (p, a, ?) - (q, y, µ) , if

p = q, _ (6.46)

referring to the pullback introduced in Chapter 2.


The theory of fibered spaces, of which vector bundles are a special case, is attributed toH. Hopf (1894-1971), E. Stiefel (1909-78), N. Steenrod (1910-71), and others. For amore "functorial" treatment of vector bundles, see Lang [19721; for the topologicalviewpoint, see Husemoller [ 1975].

6.9 Appendix: Constructing Vector Bundles

6.9.1 Proof of the Vector Bundle Construction Theorem

Let V be a k-dimensional vector space, and let { Ua, a e 1} be an open cover of adifferential manifold M. Suppose that, for every a, y e I with Ua n Uy * 0,

p->gay(p) =8U,uy,(P) (6.47)

is a Cr map from Ua n U. into GL (V), the invertible linear transformations from V toV such that (6. 15), (6. 16), and (6. 17) hold. Then there exists a C vector bundlett: E --a M of rank k with the mappings (6. 19) as transition functions, and any othersuch vector bundle is Cr isomorphic to E.

Step I. Constructing the Fibers. Suppose { Ua, a e I} is an open cover of the mani-fold M, and for every a, y E I such that Ua n Uy * 0, a map gay: Ua n Uy -4 GLk (R)(short for gU,U,) is a C' map with the properties stated in the theorem. Define

E = {(p,a.,z) E MxIxV:pE U.J. (6.48)

and define a relation - on E as follows:

(p, a, v) - (q, y, w) e* p = q and w = gya (p) v. (6.49)

Reflexivity of this relation follows from the condition gaa (p) = I, symmetry fromgay (p) gya (P) = 1, p E Ua n Uy, and transitivity from gap (p) gpy (p) gya (p) = 1


p e Ua n Up n Uy; therefore - is an equivalence relation. The equivalence class of(p, a, v) is denoted [p, a, v]. For p in M, define

EP = 1[-' (P) = {[p,a,z]:Ua)p,zE V}. (6.50)

Introduce a vector space structure on E, by taking

c[P,a,z] + [P, Y, w] = c[p,a,z] + [P, a,gay(P)w) = [P. a,cz+gay(P)w)

Clearly this structure does not depend on the specific a chosen, because each gay (p) isa linear isomorphism.

Step II. Local Ttiivializations for E. For each a, the map

Oct = 4U': 1t'' (Ua) - Ua X V, 0a ( [p, a, z]) _ (p, z)

is a bijection onto an open subset of M x V: It is one-to-one because any [p, y, w] canbe uniquely expressed as [p, a, gay (p) w] by nonsingularity of gay (p) . Also

1a (b7' : (Ua n Uy) x V -* (Ua n Uy) x V, (p, w) -> (R gay (p) w)

is a Cr onto map with a C' inverse (p, z) -. (p, gya (p) z) , and of course(UanUy) x V is open in M x V. In other words, in the diagram

X_' (UanUy)

(U r U7) X V (U ('i U7) X V(p, w) -+ (P, gay (P) W)

Oa 0y' is a diffeomorphism.

Step III. Differentiable Structure for E. It follows that for any atlas{(U',,Wj),iEJ} for M,

{ (7t-' (U'1 n Ua), (si, x identity) dta) : i E J, a e 1, U'1 n U. * 0) (6.51)

is an atlas for E, making E into an (n + k)-dimensional manifold. With the resultingdifferentiable structure, each Oft: R-1 (Ua) - Ua x V becomes a local trivialization,because it is a diffeomorphism onto its range with the desired linearity property.

6.9 Appendix: Constructing Vector Bundles 143

Moreover tt: E -* M is C' because it can be expressed as the composition proj (ba inthe following diagram.

Clearly the transition functions for this vector bundle are exactly the ones that we usedto construct it.

Step IV. Uniqueness up to Bundle Equivalence. It remains to prove the followingstatement:

Two vector bundles with the same transition maps { gay} (which refer to the same opencover { Ua} of M) are Cr equivalent.

Proof: Let n: E -4 M be as above, and let n': E' -, M be another vector bundle withthe same {gay} , but with local trivializations {'Pa} . It is necessary to construct a C'map F from E to E' such that F is a linear isomorphism from EP to E'P for each p. Forp E U. and v E V, define a map

Fa: t[ ' (Ua) -* 1t'-' (U.), [p, a, v] -*'P;I (p, v). (6.52)

Clearly F. is a linear isomorphism on each fiber. We are going to show that Fa = FYon n ' (Ua n U.). For if [p, a, v] = [p, y, w] , then w = gya (p) v, and therefore,using the fact that the two vector bundles have the same transition functions,

Fy ([p, y, w]) = 'Y (p, w) = `l'y' (p, gya (p) v)

_ gya' ('a 'ITy') (P, gya (P) v)

_ Wa' (P+ gay (P) gya (P) Y)

= Fa([P,a,v])

Hence it makes sense to define F to be the function E -* E' which equals Fa onn ' (U.). This function is C' since, for each a, Fa = 'Fa' $a, and a composition ofC' maps is C'. tt

Frame Fields, Forms, and Metrics

This chapter is intended mainly to supply constructions needed in Chapters 8, 9, and 10.The important subject of Riemannian manifolds will receive only a cursory introductionhere; see do Carmo [ 1992] and Gallot et al. [1990] for further information.

7.1 Frame Fields for Vector Bundles

Suppose DU: n'' (U) -4 U X Rk is a local trivialization of a rank-k vector bundlen: E -k M. We immediately obtain sections

s; (r) = 1ol (r, ei), i = 1, 2, ..., k, r e U, (7.1)

of the restricted vector bundle n U: n-' (U) -4 U, where (e,, ..., ek} is the standardbasis of R*. From the definition of a vector bundle, it follows that { s t (r), ..., sk (r) }forms a basis for the fiber E,, and we call the map

r-4 is,(r),...,sk(r))

a local frame field for E over U. Any section a E I'E has a local expression

a (r) = al (r) st (r) + ... + ak (r) sk (r), r c- U, (7.2)

where each a' E C (U) .

7.1.1 Local Expressions for Vector Fields and 1-Forms

The smooth sections of the tangent bundle it: TM -4 M are called vector fields, denotedrTM or more commonly 3 (M). The smooth sections of the cotangent bundlen: T* M -* M (no confusion arises from using the same n) are called first-degree

7.1 Frame Fields for Vector Bundles 145

differential forms, or simply 1-forms, denoted rT* M or more commonly f1' M. Weshall see shortly that these definitions are entirely consistent with those given in Chapter2 for the case when M is an open subset of a Euclidean space.

7.1.2 Frame Fields for the Tangent Bundle

In the case of the tangent bundle n: TM -, M, take a chart (Ua, tpa) for M, and let{xt, ..., x"} be the standard Euclidean coordinate system on tpa (Ua) c R". Recall thatthe associated trivialization 0a, as described in Chapter 6, is defined by

d'a([r,a,4]) = (tpa(r,4)),rE a,r,4E Tca(,)R". (7.3)

If we use the identification between a vector f; and the directional derivative in direction4 discussed in Chapter 2, it makes sense to write i a frame field for the tangent bundle as:

Si Qyat (., e.) = -L, i = 1...., n.ax

(7.4)

Thus a vector field X, which over Ua takes the form X (r) = [ r, a, 4a (r) ], where

4a (r) = 4a(r)e, +...+i:,a(r)e"E R",

can also be expressed over Ua in the familiar form

X = 41 a

a C'° (Ua) . This suggests that vector fields can be interpreted asfirst-order differential operators (i.e., derivations of C- (M)), as in the Euclidean case;this is confirmed in 7.1.4.

7.1.3 Frame Fields for the Cotangent Bundle

Given a chart (Ua, tpa) for M, the associated trivializations T. for the cotangentbundle are given by (see the exercises of Chapter 6 for the equivalence class notations):

T,, ([ r, a, X] ) = (tpa (r), X) , r e Ua, . e (TPQ (,) R") * . (7.6)

The duality between tangent and cotangent vectors is expressed by:

I An alternative and nonabusive notation, which is preferable in cases where we want a/ax` torefer to a vector field on tpa (Ua), using the "push-forward" defined in Chapter 2, would be

((P') a. instead of a..ax ax

146 Chapter 7 Frame Fields, Forms, and Metrics

[r,a,?] [r,a,2,] = X(4), (7.7)

and this does not depend on a, as discussed in Chapter 6. Let {et, ..., E"} be the dualbasis of (R") * to {e1, ..., a"}. It follows from (7. 7) that

is the dual basis to { (I) (r, e,), ...,bat (r, a") }, and so a frame field for the cotangentbundle is

Si `Ya' (., t:,) = dx',i = 1, ..., n.

Hence over U,, any 1-form w can be expressed either as w (r) = [ r, a, h (r) ] , whereh (r) is the row vector with entries (h, (r), ..., h" (r) ) , or in the familiar form

w = h,dx'+...+h"dx",

where each h, a C" (Us).

7.1.4 Action of Vector Fields on Functions

For every X e 5 (M) , that is, X is a vector field on M, and for every smooth function fon M, there exists a smooth function Xf on U, where (U, cp) is any chart on M. definedby

Xf(r) _ ax+...+4"- ) (f (?') (tp(r)) _ (d(f`(P-') - 4) (r). (7.8)axn

Clearly the right side depends linearly on the value of X at r, and therefore there exists a1-form, denoted df, on U, specified by

(df X) (r) = Xf(r). (7.9)

In fact Xf and df are well defined on the whole of M, because their meanings areunambiguous on the intersection of charts (Ua, (pa) and (Uv tpy). For suppose thatX(r) (r)] _ [r,Y,1y(r)],re UarCUy,so F,y= d(tp7'(Pa')4a;omitting the arguments of certain differentials,

4Y =

(d(f.gP')) a =

The mapping f -i Xf presents X as a differential operator on C- (M); let us formalizethis as follows.

7.2 Tangent Vectors as Equivalence Classes of Curves 147

7.1.5 Vector Fields as Differential Operators

(i) The set 3 (M) of vector fields on M is in one-to-one correspondence with the set ofderivations of C°° (M) (i.e., R-linear mappings Zfrom C`° (M) to C`° (M) such that

Z(fg)

(ii) Every smooth function f on M gives rise to a 1 -form df on M characterized byXf,forXe 3(M).

Proof: The first assertion follows from applying the corresponding result in Chapter 2,for vector fields on open subsets of Euclidean space, to (7. 8). The second assertion wasproved above. tx

7.2 Tangent Vectors as Equivalence Classes of Curves

The definition of tangent vectors as equivalence classes of triples has served us so far,but it is clumsy in many calculations. Here is an equivalent characterization which isboth concise and intuitively satisfying.

Consider the set of smooth maps y from open intervals about zero in R to M; such mapsare called smooth curves in M. We shall say that two such curves are equivalent,written y, - y2, if y1 (0) = y2 (0) e M and, for some chart (U, (p) at r = yj (0)

(0) = (0). (7.10)

Note that this equivalence does not depend on the choice of chart, because for any otherchart (U', W) at r, and writing x = cp (r),

D(we71) (0) = (0) = D(we(p') (0)

= (0) = (0).

What we are saying here is simply that two curves are equivalent if they pass throughthe same point, and with the same velocity, at "time" zero. It is left as an exercise for thereader to prove that - is an equivalence relation.

7.2.1 Characterization of Tangent Vectors as Equivalence Classes of Curves

Given an atlas { (Us, q),,): a e 11, equivalence classes [y] of curves y are inone-to-one correspondence with tangent vectors using the mapping

[Y] -* (0)), (7.11)

where of course the index a must satisfy Ua 3 y (0).


Proof: Note first that the mapping is unambiguous, because for any indices a, 8

(0)] _ [y(0),a,D(%* (0)]

_ (0)]

Second, the mapping is one-to-one because (7. 10) shows that two classes of curveswith the same image are the same class. To show that it is onto, pick any r E M, any awith U. 3 r, and any vector v r= R", and let y(t) = cpat ((pa (r) +tv) e M, for t in aninterval about 0 small enough so that cpa (r) + tv always belongs to tpa (UQ) . Then(7. 11) implies that [ y] -> [ r, a, v] is a one-to-one correspondence, as desired. ix

7.3 Exterior Calculus on Manifolds

The qth exterior power AqT* M of the cotangent bundle can be constructed using themethods of Chapter 6. As explained in Chapters 1 and 2, we make the identifications

(Aq7* M) . = Aq (T r* M) = (Aq (T,M)) *. (7.12)

The differential (or exterior) forms of degree q on M, or q-forms for short, are simplythe smooth sections t (AqT* M) of the qth exterior power of the cotangent bundle; weusually denote them by Qq (M). As before, S2° (M) = C" (M).There are several waysof representing a q-form oa An elegant but abstract way is to think of a q-form as amultilinear alternating map on q-tuples of vector fields in 3 (M): We can treat w as amap 3 (M) x ... x!3 (M) -* C'° (M), namely,

(XI,...,Xq) -4 (A) - (XIA...AXq), (7.13)

(CO (XI A ... A Xq) ) (r) = (0 (r) (X1 (r) A ... AXq (r) ), rE M. (7.14)

This makes sense by (7. 12), since w (r) is a linear form on the qth exterior product ofthe tangent space at r.

A local representation of a q-form may be given in terms of a local frame field, as in7.1.1. Using the notation of local frame fields, we may write

w(r) _ A ( r ) ,( q ) U., (7.15)

where as usual I = (i (1) < ... < i (q) ), and b, (.) is smooth.

7.3 Exterior Calculus on Manifolds 149

Recall from Chapter 2 that the operations of exterior product and exterior derivative fordifferential forms on open subsets of Euclidean space "commute with the pullback," inthe sense that if Vt: U c R" - V c R" is smooth, then

V do) = d (V,` w), W* (w n rl) = (W* o) A (W* r1). (7.16)

In the case where W is a diffeomorphism, we note that duality of forms and vector fields,and the Lie derivative of forms, commute with the pullback in the sense that

yt*w (XI A...AXq) = w (W.X, A...AW.Xq); (7.17)

Lx (W* co) = LW. xw (7.18)

(the first statement holds by definition; the second can be inferred from the formula forLie derivative of a form).

7.3.1 Transferring Exterior Calculus to Manifolds

Any operation applied to differential forms on open subsets of Euclidean space thatcommutes with the pullback in the sense above -for example, exterior product, exteriorderivative, and Lie derivative of forms - is well defined for differential forms onmanifolds, and conforms to the same calculus; thus for any q -form w and p -form T1,

T1ACO =(-1)pq(wAr1),ddw = 0; (7.19)

d(cAT1) = dtoATl+ (-1)°`B'O (wndrl); (7.20)

Lxf = Xf, Lx (df) = d (Lxj), f r= C"' (M), X E 3 (M);2 (7.21)

Lx(cA71) = Lxwnrl+CO ALxT1; (7.22)

and if g: P -4 M is a smooth map, the pullback g* :12"M - S2gP, defined by

g*w (XIA...AXq) = w (7.23)

satisfies

g*dw = d(g*w),g* (wAT1) = (g*w) A (g*Tl) (7.24)

Proof: All that needs to be checked is that if we perform any of these operations on thelocal representation (7. 15) of one or more forms, then the result does not depend on

2 See (7.9) for the meanings of df and Xf.


which local representation we chose. Take the case of exterior differentiation, forexample. On Ua n U. g M, a differential form w may be expressed as

r-+ [r,ct,).(r)] = [r,y,µ(r)],? =

and so it follows that we may define do) on U. to be r -4 [ r, a, d (r) ] without fearof ambiguity, because (7. 16) tells us that

[r, a,A(r)j = [r,a, (r)l = [r,y,d.t(r)],re Uar Uy.

The other cases follow the same pattern. The formulas (7. 19) to (7. 22) and (7. 24) holdbecause they hold in every local representation. ]a

7.32 ExampleTake U = R3 - {0} , which is a three-dimensional submanifold of R3, and considerthe differential form co a f22U given by

x (dy Adz) -y (dx Adz) +z (dx n dy)w (7.25)

(x2 +y2 +z 2 )3/2

The restriction of in to the sphere S2 gives a 2-form 11 a f2S2, which can berepresented in various ways; for example, since x2 + y2 + Z2 = 1, it follows that

dx = ydy - zdz (7.26)x

and so on the portion of the sphere where

x = - 1-y2-z2*0,

we may represent 1 in the (y, z) coordinate system by

dyndz _ dyndzTl = x - l _yz_z2

(7.27)

Similarly, on the part of the sphere where y * 0. we can write i = - (dx A dz) /y. Oneconsequence of 7.3.1 is that the exterior derivative dtl will always be the same (actuallyit must be zero in this case, because the third exterior power of a 2-dimensional space isalways trivial) whether we calculate it in the representation (7. 27) or some other one.For example, to compute dri from (7. 27), write

d (xt)) = dx n rl +xdtl = d (dy A dz) = 0.

7.4 Exercises 151

Now substitute (7. 26) and (7. 27) into dx A tj to see that it must be zero, and hence dtjis zero.

7.4 Exercises

1 . Suppose 4 :7E-1 (U) -) U x Rk and 45v: tt l (V) -+ V x Rk are local trivializations ofa rank-k vector bundle n: E -+ M, with U n V # 0. Consider the local frame fieldsr-4 Is, (r), ..., sk (r) } and r It, (r), ..., tk(r) } over U n V, where

s.(r) =c (r,e.),t,(r) =4}y (r,e.),i = 1, 2, ..., k, re Ur V.

Let gj (r) denote the (i, j) entry of the transition function guy (r) . Prove, using thedefinition of transition function in Chapter 6, that

tj = gis,+...+ggsk. (7.28)

2. Two smooth curves on a manifold M are called equivalent, written y, - y2, ifyj (0) = Y2 (0) e M and, for some (and hence any) chart (U, (p) at r = yj (0),

D ((p Yi) (0) = D ((q Y2) (0) .

Prove that - is an equivalence relation.

(7.29)

3. Suppose f: M - N is a smooth map between two manifolds. Prove that the definition ofthe tangent map Tf given in Chapter 6 is equivalent to

TM YD = (f Y] (7.30)

for every curve yon M, using the notation of Exercise 2 and the correspondence given in(7.11).

4. Let us endow the sphere S2 c R3 with the atlas { (U, (p), (V, iy) } as follows:

x,U = SZ- {(0,0, 1) },cp(x,y,z) = 1

z;

v=s2-{(o,0,-1)},w(x,y,z) =x,1+z

(i) Consider the curve y (t) = (I _t2/2, 0, t l - t2/4) a S2. Express the equiva-lence class [y] of this curve as a tangent vector [r, a, z] for a = 1, 2 (i.e., withrespect to both charts) where r = (1, 0, 0) .

(ii) Define a map f: S2 -* S2 by taking 0 (z) = n (1 - z2) and


f (x, y, z) = (xcose (z) - ysine (z) , xsine (z) +ycose (z), z).

Calculate the tangent vector Tf ([ y]) in the form [f (r) , a, z], for one or other choiceof a.

7.5 Indefinite Riemannian Metrics

From the discussion in Chapter 6 of the tensor product of vector bundles, the reader maydeduce that, for any vector bundle n: E - M, e ®E'` -+ M is a vector bundle whosefiber over r e M can be regarded as the set of bilinear maps from E, x E, to R. Asmooth section of this bundle, denoted r -4 is called an Indefinite metric if it hasthe following two properties at every r e M:

(1) Symmetry: = (SIC,),, VV, C e E,;

(ii) Nondegeneracy: 4 * 0 (414), * 0.

In other words, an indefinite metric gives an inner product on the fiber over r, whichvaries smoothly with r. We call r -+ a metric if it satisfies (i) and the followingstrengthening of (ii):

(iii) Positivity: 4 * 0 = (Wr> 0.

In the special case where E is the tangent bundle of M, we refer to the map r -4 asan indefinite Riemannian metric, or as a Riemannian metric if (iii) holds. A manifoldM with a Riemannian metric (resp., indefinite Riemannian metric) r -a is referredto as "the Riemannian manifold (resp., pseudo-Riemannian manifold) (M, (4))." IfX and Y are vector fields on Al, we customarily abbreviate (X (r) 1Y (r)), to (XI 1)r-

7.5.1 Local Expression for an Indefinite Riemannian Metric

Suppose (U, cp) is a chart for an n-dimensional pseudo-Riemannian manifold (M,and {x', ..., x"} is the standard coordinate system on R". The local frame field

for TM over U

{ ((P-1) 'azl, ..., ((P-1) a}'

which we already abbreviated in (7. 4) to {ax"

...,}, will be further abbreviated toaxn

{D1, .., D.J.

7.6 Examples of Riemannian Manifolds 153

The local form of the Riemannian metric tensor means the map which takes r e U tothe n x n nondegenerate symmetric matrix G (r) = (gig (r) )given by

gig (r) = (D1ID1),. (7.31)

Of course G (r) is a positive definite matrix in the case where the metric is Rieman-nian. This gives a convenient formula for the inner product between two vector fields:

X = 4'D1+...+4"D",Y = 'Dj+...+1;"D"

(XI Y), = Igif (r) 4'(r) t (r). (7.32)f.f

A more concise way to write this formula, using some notation from Chapter 2, is

(.1.) = I:g,,dx' ®dxi. (7.33)i.i

It is often most convenient to check the smoothness condition on the metric using(7. 31); the proof of the following statement is omitted.

7.5.2 Condition for a Collection of Inner Products to Give an Indefinite Metric

An assignment of an inner product to the tangent space at r for every r in M is anindefinite Riemannian metric if and only if there is an atlas for M in which all of themaps (gig in (7. 31) are smooth.

7.6 Examples of Riemannian Manifolds

7.6.1 The Euclidean Metric

Take M to be an open subset of R", and give each tangent space T.,R" the Euclideaninner product (vIw)E = vlwI + ... + v,,w", using the standard coordinate system. Inother words, using the chart map cp = identity . the local expression of the metric tensoris the identity matrix. Often we want to express the same metric in a different chart. Forexample, if U c R2 - 101 and 0-1 (r, 0) = (rcos 0, rsinO) (polar coordinate chart),then dx = cos0dr- rsin0d0 and dy = sin0dr+ rcos0d0, and so (7. 33) gives

dx®dx+dy®dy

= (cos 0dr - rsin 0d0) 0 (cos 0dr - rsin 0d0) + etc.


= dr®dr+r2d9®d9.

(Note the absence of dx 0 dy and dr 0 dO terms for this metric.) Thus in this chart themetric tensor is expressed by the matrix

01.

G(r'e) = I J110

7.6.2 Induced Metric on Submanifolds of R" +k

If M is an n-dimensional submanifold of R"+k, the inclusion map t: M -> R"+k induces

a Riemannian metric on M from the Euclidean metric (4)E on R"+k, namely,

(Xll')X _ (X1Y)z, X, Y e S (M) . (7.35)

When n = 2 and k = 1, the induced metric corresponds intuitively to the notions oflength and angle in two dimensions for an insect crawling around on the surface of M.

Typically we study such a metric using a parametrization'F: W c R" -+ U C R"+k forM. The metric tensor can be expressed in terms of a coordinate system { u1, ..., u"} onWas follows: If IF (u) = ('F1 (u), ...,'Y"+k (u) ), then

dxt ®dxt+...+dx"+k®df+k

awl

1

'3'P1a`F' 1

NOauldu +...+au"du)0(autdu +...+au"du") +....

It follows that

a`Yma`FmJ

7au' au!du'® du'.

Q m

(7.36)

Thus in the chart induced by the parametrization, the metric tensor is expressed by thematrix

G(u) = D'F(u)TD'F(u). (7.37)

7.6.2.1 Example: The 2-Sphere In R3As a special case of 7.6.2, parametrize the 2-sphere in R3 using the usual angularcoordinates, namely, 'F (¢, 0) = (cos0sin$, sinOsin4, cosio). Elementarycalculations using the formula (7. 37) show that, in this coordinate system, the metrictensor is expressed by the matrix


H ( X ,

by taking polar coordinates, namely, x = rcos8, y = rsin0, z = 1 +r2.One mayeasily compute that

dx®dx = (cos8)2dr®dr-rcosOsinO(dr®d8+d8®dr) +r2(sin8)2d8®d8,

and similarly for dy ® dy and dz ® dz. Using (7. 40), a routine calculation (takingz = xo) shows that the metric on the hyperbolic plane is

(.1.) = -dz®dz+dx®dx+dy®dy

dr ®dr +r2d8®d8.1+rz

A useful reparametrization is r = sinhs (i.e., hyperbolic sine), since a little calculusshows that ds ® ds = dr ® drl (1 + r2); thus in the (s, 0) parametrization,

( 1.) = ds ®ds + (sinhs) 'de ®de .

Thus the form of the metric tensor in the (s, 8) parametrization is

G (s, 8) = 1 0

0 (sinhs) 2(7. 44)

The similarities and the differences among (7. 34), (7. 38), and (7. 44) should becarefully noted.

7.7 Orthonormal Frame Fields

A local frame field { e1, ..., e. } for TM over U is called an orthonormal frame field if(e1ef)P = ±8,, for all i, j and for all p e U. We have already encountered a special casein Chapter 4, where { 4i, 42, 43} was an orthonormal frame field for TR3 over aparametrized surface M, and { 4t, 42} was an orthonormal frame field for TM.

7.7.1 Existence of Orthonormal Frame Fields

In every chart for a pseudo-Riemannian manifold there exists an orthonormal framefield for the tangent bundle.

Proof: Take a chart (U, (p) around the point p. Consider the inductive hypothesisH (m) that, for every set of vector fields { X1, ..., Xm} on U such that the tangent vectors{ X, (p), ..., X. (p) } e T,M


are linearly independent for every p e U, there exists another set of vector fields{e,, ..., em} on U, with (e,)e,) = ±8ij, that span the same subspace of T,,M for everyp E U. Obviously if M is n-dimensional, then H (n) is what we have to prove. Notefirst that H (1) holds because, given a nonvanishing vector field X,, we can take

e, = X1/( J1(X1IX1)1). (7.45)

Now assume that 2:5 m!5 n - 1 and that H (m - 1) is true. Let { X1, ..., X", } be vectorfields on U such that the tangent vectors {X1 (p), ..., X," (p) } E TTMare linearly independent for every p E U. Define eI as in (7. 45), and define vectorfields { Y2, ..., Ym} on U by

(X,)e, )YJ Xi

eileie,,

which has the consequence that (Y)ei) = 0 for j = 2, 3, ..., m. Any linear dependenceamong { Y2 (p), ..., Y," (p) } would imply the same for {X, (p), ..., X. (p) } , whichis contrary to the assumptions; hence { Y2, ..., Ym} satisfy the conditions for theapplication of H (m - 1) , which guarantees that there exists a set of vector fields{ e2, ..., e,,,) on U, with (ell e = ±S.j,such that, for every p E U,

Span { e2 (P) , ..., em (P) } = Span { Y2 (P) , ..., Y. (P) }.

It follows in particular that (e1le) = 0 for j = 2, 3, ..., m. This construction of the setof vector fields {eP ..., em} verifies H (m), thus completing the induction. tt

7.7.2 Orthonormal Coframe Fields

Recall that, in our introduction to differential forms on Euclidean space in Chapter 2,the frame field {dx', ..., dx"} for the cotangent bundle was defined by taking{ dx' (y), ..., dx" (y) } to be the dual basis of (T,,R") * corresponding to the basis

for the tangent space at Y 'E R". Another way to say this is that{dx', ..., dx"} is the dual frame to {a/ax', ..., a/ax"} .

Likewise, given an orthonormal frame field { e 1, ..., em } on an open set U in apseudo-Riemannian manifold (M, (.I.)), we immediately have an orthonormalcoframe field { 9', ..., 0"j, namely, the dual frame to { e1, ..., em }. This is the set of1-forms defined by

9' (A'e, + ... +A"e") = A`. (7.46)

This is a complete definition because every vector field on U can be expressed in theform A'e, + ... +A"e" for some functions {A'}. The smoothness of the sections101__011 follows easily from that of {eP ..., em}, after taking local coordinates.


7.7.3 Expressing a Frame Field in Terms of an Orthonormal One

Suppose (U, (p) is a chart for an n-dimensional pseudo-Riemannian manifold (M,(.0), and {xt, ..., x"} is the standard coordinate system on R. As before, the localframe field { ((p t). (a/ax'), ..., (9-1) . (a/ax") } for TM over U will beabbreviated to { D,, ..., D,,}, and {4' dx', ..., tp* dx" } will be abusively referred to as{dx...... dx"}.

Now suppose { e,, ..., em } is an orthonormal frame field on U. To keep track of theplus ones and minus ones, we introduce a diagonal n x it matrix

X1 0 0

A = 0 ... 0 = diag(1,..., 1,-1,...,-1), (7.47)

0 0 X,

where Xi = (e,{e). Note that, because of smoothness of the inner product, the sameconstant matrix A applies at all points in the chart. Of course in the Riemannian case, Ais the identity matrix.

If we try to express the frame field { D,, ..., D,, } in terms of the orthonormal framefield, we obtain a function E from U into the nonsingular it x n matrices, which can bewritten as Emi (p) = 4," (p), or else as

E (P) _ ( 1(P), ...> " (P) I. P E U. (7.48)

where the { 1;i} are column vectors (cf. Chapter 4), such that

D, _ 1 E "em .(7.49)

M

It follows that

&ij (P) = (D,{Dj)p = Y (p) Ej (p) (emleq)p = m (p)'m (p);m.q m

G (p) _ = (P) T AH (p).

Suppose ('-:'(p) -t) i4 = k. Postmultiplying (7. 49) by this matrix gives

mr Di.e,t= ib'em=e

(7.50)

(7.51)

Also from (7. 46) and (7. 49), we see that


0m . Di = 4m, :.0m = Di dx1 (7.52)i

Evidently the construction of orthonormal frame and coframe fields from the originalframe field {D1, ..., D,,} amounts to finding a "smooth factorization" of the metrictensor G over U of the form (7. 50), that is, a smooth map H: U -4 GL (R) such thatG (p) = E (p) T AE (p), for then we may define the orthonormal frame using (7. 51)and the coframe field using (7. 52).

7.7.3.1 Example: Orthonormal Frame Field on a Parametrized SurfaceConsider the case where `P: W S R2 -4 R3 is a 2-dimensional parametrization, that is,M = `1'(W) is a parametrized surface. Let { u, v } be a coordinate system on W, andexpress P (u, v) as (,pt (u, v), `P2 (u, v), `Y3 (u, v) ). As we saw in 7.6.2, theEuclidean metric on R3 induces a metric

E "2(.1.> = I `i'uII duOdu+(`Yul`Y (duOdv+dv0du] +Il`l'' II dv 0 dv,

where P,, = [a'P /au, ap2/au, ap3/au] T etc. Therefore, if D. = IF. (a/au),etc., then

(D.ID) = II T. 12, (D ID) =(-.,-)E,

(DvID) = II

'-j12.

As we saw in Chapter 4, an orthonormal frame field is given by

(7.53)

E

= II II-'Du, 2 = - II IID,,. (7.54)

11 UIIII41.x%PJI II T. X PII

The fact that this is orthonormal will be checked again in Exercise 10.

7.7.3.2 Example: Orthonormal Coframe Field on a Parametrized SurfaceIn Chapter 4, we used the following expressions for the orthonormal coframe (01, 02}to the orthonormal frame (4,' 421 given in (7. 54):

01 = II wu I du + dv, 02 =II IV.; I'VIi

dv, (7.55)

IIIPull

where du is short for (`P-t)* du, etc. We leave it to the reader in Exercise 11 to checkthat 0i i = 8

.


7.8 An Isomorphism between the Tangent and CotangentBundles

7.8.1 Switching between Vector Fields and 1-Forms Using the Metric

Let us now define a vector bundle isomorphism from the tangent bundle to thecotangent bundle by using the map that sends C E T,M to the linear form -+ (LIE),,

henceforward abbreviated to linearity is obvious, and nondegeneracy of the innerproduct ensures it is one-to-one and onto. Abusing notation slightly, we extend this to amap from vector fields to 1-forms, where for a vector field X, Qfl is the 1-form given by

(XI - Y = (Al Y), Y e 3 (M). (7.56)

The inverse to this mapping takes a 1-form co to the vector field 0, pronounced`omega sharp," where

(w"IY)=to -Y,Ye 3(M). (7.57)

By definition, (XI" = X and (w"I = w. These operations are already familiar in R3with the Euclidean metric; for a vector field V = vta/ax + v2a/ay + via/az,(V = 4v = v'dx + v2dy + v3dz, that is, the "work form" referred to in Chapter 2; for afunction f on R3, the vector field (df)" = gradf in the vector calculus sense (hencewe use this formula to define gradf for a function on a pseudo-Riemannian manifold).

7.8.2 Formulas for (XI , a?, and grad f

(i) (D,I _ g,1dxJ and (dxk)" _ YgkJD), where G (r) -' _ (gt k (r)).J

(ii) For any orthonormal frame field (e,, ..., em } with orthonorma! coframe field{01, ..,e"},

(e;I = 0' and (A')" = e;. (7.58)

(iii) For any smooth function f on M, the vector field grad f defined by grad f = (df)has the local expression

grad! = (DJ gkl) D. (7.59)J, k

Proof: The first and second formulas follow immediately by substituting X = D,,Y = Di, and co = dxk into (7. 56) and (7. 57), respectively. The formulas in (ii) followfrom (7. 46), and the formula in (iii) from the second formula in (i). it

7.9 Exercises 161

7.9 Exercises

5. Calculate the Riemannian metric tensor for the following 2-dimensional submanifoldsof R3, using the induced metric 7.6.2:

(i) A surface of the form { (x, y, z) : z = f (x, y) }, using (x, y) parametrization.

(ii) The cylinder { (x, y, z) : x2 + y2 = A2}, using (0, z) parametrization; here 0 is anangular variable in the (x, y) plane.

(iii) The hyperbolic paraboloid

z = -x2 -2A2 B2

with the parametrization `P (s, t) = (As, 0, s'-) + t (A, B, 2s).

(iv) The ellipsoid

X2 zz2A+BZ+C2 = 1,

for the parametrization `Y (u, v) = (Asinucosv, Bsinusinv, Ccosu).

6. Show that the metric dxo 0 dxo + dxt 0 dx, + ... + dx" 0 dx restricted to

H" _

is indeed positive definite, by using the chart cp (xo, xt, ..., x") = (xt, ..., x"), and thefact that

X 1 dx 1 + ... + xdx"dxo =

1 +x1+...+x.,

7. Calculate the Riemannian metric tensor for the 3-dimensional hyperbolic space H3 withthe metric described in 7.6.3; use a spherical coordinate parametrization (r, $, 0) tobegin with, and make the reparametrization r = sinhs at the end.

8. Derive an orthonormal frame field {et, e2} and the corresponding orthonormalcoframe field {g', e2} for the hyperbolic plane, starting from the frame field{ alas, a/ae} . Also derive the formula for grad in the (s, 0) coordinate system.

9. Express the Euclidean metric on R3 in terms of the spherical coordinate system(r, 0, 0) . Derive the formula for grad in terms of { a/ar, a/a4), a/ae} .


10. Verify that the frame field 141'421 in (7. 54) really is orthonormal under the metric(7. 53). Also calculate the 2 x 2 matrix-valued function H such that

G (p) = = (p) TE (p)Hint: Use the identity from Chapter 1 which says II v x w 112 = (vlv)(wlw) - (vlw)2 for theEuclidean metric on R3.

11. Show that {61,02} in (7.55) is indeed the coframe to the orthonormal frame {, S2 }

given in (7. 54), by showing that 0' 1 = S'.

12. Using (7. 55), calculate an orthonormal coframe field for a surface in R3 with a parame-trization of the type `Y (u, v) = (u, v, h (u, v) ).

13. (i) Given a metric (.1.) on a real or complex vector bundle it: E -+ M, with fibersisomorphic to a vector space V. and two orthonormal frame fields { s, } and { t1 }related by

t1 = 14s±'

show that the matrix g is orthogonal, that is, g't = 9T.

Hint: (t ltk) = 8k 8 k.

(ii) Suppose rt: E' - M is another vector bundle, with fibers isomorphic to a vectorspace V', and with a metric Given sections F and F of the vector bundleHom (E, E') (i.e., F is a smooth assignment to each p e M of a linear transformationF (p) e L (V -> V') ), let us define

(F1F)p0m _ J(Fs,IFs?P ,

i

(7.60)

where the {s1} form an orthonormal frame field for n: E - M. Prove that the left sideof (7. 60) does not depend on the choice of frame field, in the sense that if { t1} isanother orthonormal frame field, then

J(Ft)Ft4- = Y(Fs,{Fs)p.

Hint: jg,g = S"t, by part (i).

(iii) Deduce that (7. 60) defines a metric on Hom (E, E').

(iv) Suppose that, with respect to the orthonormal frame field { s; }, F (p) is expressedby the matrix (Fj), where F(p) s1 = j:F;s;, etc. Prove that


(FlF)Pom

Tr (FT F) (P). (7.61)

14. Let (U, (p) be a chart for a Riemannian manifold such that, for some b > a > 0, all theeigenvalues of the metric tensor G (p) lie in the set [a, b] for all p e U; note that allthese eigenvalues are real because the metric tensor is symmetric. Let c,, be the nthcoefficient in the power series expansion of (1 - x) around x = 0.

(i) Show that the series

= (p) = T b cn [1- b-1 G (p) ] n (7.62)n=0

converges to a symmetric matrix E (p).

Hint: II [1-b-1G (p)]nII5 (1-alb) n because, for all v e R',

05vT(1-b-1G(p))v=11 v112-b-1VT G(p)v5 (I - alb) 11 VII 2.

(ii) Show that G (p) _ (p) 2 (p) TE (p), and that U -4 GL,, (R) is smooth.

Hint: G (p)1 /2 = [b (1- (1- b-1 G (p)])1/2 . Also 8 (p) is an analytic3 function of G (p),

which is smooth in p.

(iii) Use these results to obtain an alternative proof of the existence of an orthonormalframe field at any point of a Riemannian manifold.


Basic ideas of Riemannian geometry are due to B. Riemann (1826-66), E. Cartan(1869-1951), and others. The books of do Carmo [ 1992], Klingenberg [ 1982], andGallot, Hulin, and Lafontaine [1990] are highly recommended.

3 A function is called analytic if it is given by a convergent power series at every point. The set ofanalytic functions is contained in the set of smooth functions.

s Integration on Oriented Manifolds

In this chapter we come to one of the main uses of differential forms, which is toprovide a multidimensional, coordinate-free theory of integration on manifolds. In theprocess, we shall prove a version of Stokes's Theorem, which is a general form of thefundamental theorem of calculus, and uncover the geometric meaning of the mysterious"exterior derivative."

8.1 Volume Forms and Orientation

A volume form on an n-dimensional differential manifold M simply means an n-form aon M such that Cr (r) * 0 at every r E M. or in other words a "nowhere vanishing"n-form. For example, the "area form" calculated from an adapted moving frame inChapter 4 is a volume form on a surface, and f (dx A dy A dz) is a volume form on R3for every smooth function f on R3 such that f never equals zero.

In vector algebra, the bases {e1, e2, e3 } and {e2, e3, e , } are said to have the "sameorientation" because they are both "right-handed," whereas {e2, e1, e3} has theopposite orientation because it is "left-handed." This corresponds in exterior calculus tothe fact that dr A dy A dz = dy A dz A dx = -dy A dr A dz. In higher dimensions, thebest way to describe the notion of orientation is in terms of the equality or change ofsign in going from one volume form to another, as we shall now delineate.

Two volume forms a and p on M are said to be equivalent if a = fp for somefE C_ (M) with f (r) > 0 at every rE M.

8.1.1 Example of Equivalence of Volume Forms

The volume forms dx A dy A dz and dr A dA A d4 are not equivalent on the manifold

8.1 Volume Forms and Orientation 165

U = R3-.{

where x = rcos8sin$, y = rsinOsin$, and z = rcoso (i.e., standard sphericalcoordinates, with (r, 0, $) e (0, oo) x (0, 2n) x (0, it)), because as we noted inChapter 2,

dxAdyAdz = -r2sin$(drAdOAd$)

and -r2 sin 0 < 0. On the other hand, dx A dy A dz and dr A do A d8 are equivalent.

8.1.2 Orlentable Manifolds

An orientation for a manifold M means an equivalence class of volume forms. If anyvolume form exists on M, then M is said to be orientable, and the choice of anequivalence class of volume forms makes M oriented. A nonorientable manifoldmeans one on which no volume form exists. In the language of vector bundles, ann-dimensional manifold M is orientable if and only if the nth exterior power bundleA" (7` M) is trivial, because a line bundle is trivial if and only if it has a nowherevanishing section (which in this case means a volume form); see the exercises forChapter 6.

Note that a manifold may be orientable even if the tangent bundle is nontrivial; forexample, the 2-form 71 in the example in 7.3.2. on p.150 is a volume form on thesphere S2, so S2 is orientable; however, S2 has no nowhere vanishing vector field(the "hairy ball" theorem in topology), and hence TS2 is not trivial.

A chart (U, (p) for an oriented n-dimensional manifold M is said to be positivelyoriented if cp* (dx' A ... A dx") is equivalent on U to the chosen class of volumeforms; as we shall see in Section 8.2, an oriented manifold has an atlas consisting ofpositively oriented charts.

We would like to have criteria to determine whether a given manifold is orientable. Thebest kind of criterion would be one that can be verified without taking charts, such as thefollowing:

8.1.3 Orientability of a Level Set of a Submersion

Every n-dimensional submanifold of R"+k of the form M = f' (0), wheref: U r. R" +k -+ Rk is a submersion on an open set U D M, is orientable.

This result shows, for example, that spheres, tori, and hyperboloids of arbitrarydimension, and Lie subgroups of the (real or complex) general linear group, areorientable.

166 Chapter 8 Integration on Oriented Manifolds

Proof: The Case k = 1. Take the Euclidean inner product on R"+' , and apply theassociated Hodge star operator to the 1-form df a 11t U to obtain an n-form* df a f1" U. We claim that the restriction of * df to M (technically the pullback of * dfunder the inclusion map from M into R"+ 1, but the proliferation of stars could beconfusing) is a volume form on M.

Fix r e M and let 'Y: W g R" -3 U' n M c R"+ be an n-dimensional parametrizationof M at r, in the sense of Chapter 3, with 'Y (0) = r. We know from an exercise in thatchapter that, in the tangent space to R"' ' at r, Im dP = Kerdf. Another way of sayingthis is that, in terms of coordinates (x0, ..., x") for R"* 1, the (n + 1) x (n + 1)matrix of derivatives

Dof (r) D1'I'o (0) ... D"'I'o (0)

D, f (r) D,'P" (0) ... D"'I?" (0)

has the property that the first column (i.e., the gradient off at r) is orthogonal to thetangent space at r (see Chapter 3), and hence to the space spanned by the last n columns.However the last n columns constitute the derivative D'I' (0), which has a column rankof n, since 'P is an immersion. It follows from this that the entire matrix is nonsingular,and therefore its determinant is nonzero, or in other words

(- 1)iD1f(r)ID'"(0)1 *0, (8.1)

i=o

where D'/ (0) means the n x n matrix obtained from D'I' (0) by deleting the jth row,for 0:5j:5 n. It is a simple exercise using the methods of Chapter I to verify that

"

*df(r) = I (-1)iDI(r) (dxon... Ad?A... A (8.2)

i=o

where dxo n ... A di n ... A dx" is short for dxo A ... A dx' -' A dx' +' A ... A dx". Onreviewing the section in Chapter 1 on exterior powers of a linear transformation, thereader may verify that (8. 1) says precisely that

* df (r) (A"`h.) (au A ... A ") *0,

where { u t, ..., u" I is the coordinate system on W. This proves that * df is a nowherevanishing n-form, as desired.

8.2 Criterion for Orlentability in Terms of an Atlas 167

The Case of Arbitrary k Express f (x) as (ft (x), .-t (x)) . Take the Euclideaninner product on R"+k, and apply the associated Hodge star operator to the k-formdit A ... A dfk E f2kU to obtain an n-form * (df' A ... A d/) E 0"U. The restrictionof this differential form to M is a volume form because, by reasoning similar to thatabove,

*(df'A...Adf)(r)(A9`I'.)(a A...Aau"

eD,/ (r) ... D,f (r) D I V (0) ... D"`P' (0)

k (r) ... Dn+pt (r) D,'Pn+k (O) ... DnTn+k (0)[D,+a

which can never be zero.

The previous result is not the most natural way to obtain the orientability of matrixgroups; another way is as follows.

8.1.4 Orientability of Matrix Groups

All the Lie subgroups of GLn (R) and of GLn (C) are orientable.

tt

Proof: Suppose G is a Lie subgroup of GL, (R) or GL. (C) of dimension m. Take achart (V, yr) for G at the identity 1, and define an m-form a on V bya = yr* (dx' A ... A dx). Now define an m-form p on the whole of G by takingp (A-t) = (LA* a) (A-'), where LA: G -+ G is the diffeomorphism induced by leftmultiplication by the element A E G, so LA (A-') = 1. Since a is nonzero at 1, andsince the differential (LA) . is of full rank, it follows that p is never zero, and hence is avolume form. zx

8.2 Criterion for Orientability in Terms of an Atlas

8.2.1 Positivity of Determinants of Derivatives of the Change-of-Charts Maps

Suppose M is an n-dimensional manifold with a countable atlas { (Uj, y'j), j E J}. M isorientable if and only if there is an equivalent atlas { (Uj, (pj), j E J} such that,whenever r E U; r1 Uj * 0,

((Dj(r))I>0; (8.3)

that is, the derivative of every change-of-charts map has a positive determinant.


Proof: Assume that M has a countable atlas satisfying (8. 3). By the theorem onpartitions of unity (Chapter 5), we may take a partition of unity{ va: M - [0, 1 ], a c- I} subordinate to the atlas { (U;, ;) , j e J}.

Let Cr be the canonical volume form dx( A ... A dx" on R. We assert that

p (r) = 1 va(r) (,P;(a)* a) (r)aEI

is a volume form on M. The fact that it is an n-form is assured by the second and thirdproperties of a partition of unity; given r E M there is a neighborhood U of r on whichp takes the form of a finite sum of n-forms multiplied by smooth functions, and hence pis a smooth section of the bundle of n-forms. It remains to show that p (r) * 0 for all r.

Fix r r= M, and select y E I such that vY (r) > 0. For any a * y, we have

va(r) ((P;(a)*a) (r) = va(r) (((P;(a) W;(Y) `PJ(»)*a) (r)

= va(r) ((P;(Y)* (((p; (a) i(Y))*a)) (r).

Hence if Xk = ((pj Y)) (a/axk) (vector field on U,(Y)

), then

((PI(Y)* a) (X, ... AX") = 1;

vY(r)+va(r) A...Aa"asY

= vy(r) P;(Y))) >0,a*y

where, in the last line, we used the formula for the nth exterior power of a linear trans-formation given in Chapter 1, and condition (8. 3).

As for the converse, assume that M has a volume form p. Let S: R" -k R" be the map;(X11"-1X.) = (-x,, x2, ..., x"), and let{ (Uj, tir;) , j E J} be any countable atlas forM. If 'q"a and p are equivalent as volume forms on U,, then let (p; = v j; if not, let(p1 = S W!. As a result, { (U;, (p),J E J} is an equivalent atlas satisfying (8. 3). it


8.2.2 Applications

8.2.2.1 M6blus StripIt is intuitively clear that the Mbbius strip k, the two-dimensional submanifold of R3depicted in Chapter 6, has an atlas consisting of two charts which fail to satisfy (8. 3). Itis possible to show that no atlas for M satisfies (8. 3); alternatively topologicalarguments can be used to show that M is indeed nonorientable.

8.2.2.2 Products of Orientable Manifolds Are OrientableIf M and N are manifolds with atlases { (U;, (pi) , i e 1} and { (V1, Vj) , j E J),respectively, both satisfying (8. 3), then the obvious "product atlas" for the productmanifold M x N also satisfies (8. 3); the details are left as an exercise.

For example, the torus T" is orientable because it is the product S' x ... x S' of circles,each of which is orientable by 8.1.3.

8.2.2.3 Tangent Bundles Are OrientableFor any manifold M (even a nonorientable one!), the tangent bundle TM is an orientablemanifold; see Exercise 8, and the hints attached.

8.3 Orientation of Boundaries

8.3.1 In an Oriented Manifold, a Boundary Has a Canonical Orientation

Suppose P is an n-dimensional submanifold-with-boundary (see Chapter 5) of ann-dimensional oriented manifold M, where n 2 2. If the boundary aP is nonempty, thenit is an orientable (n -1)-dimensional submanifold of M with a natural orientationdetermined by that of M.

Proof: We already proved in Chapter 5 that aP is an (n - 1)-dimensional submanifoldof M; it only remains to prove that it is orientable, etc. By definition of asubmanifold-with-boundary, we can take an atlas for M such that charts (U, cp) thatintersect aP are of the special form p (P o U) = { (x i, ..., E 4p (U) : x, 5 01, and(p (aP n U) = { (xi, ..., xn) E cp (U) : x, = 0}. Take a partition of unity{ ua: M -+ [0, 1 ], a e 1} subordinate to this atlas. Recall from the exercise onsubbundles in Chapter 6 that a section of the vector bundle

tt:TMlap -->aP (8.4)

means a smooth assignment of a tangent vector in TM to every r e aP: Note inparticular that this tangent vector need not be tangent to the submanifold aP, whosetangent spaces have one less dimension than those of M. We define a smooth section Sof this vector bundle as follows:


A glance at the section in Chapter 2 about the local expression for the differential of amap will verify that

0PJ(«)'(Pi(Y)).ax, _ (DIW')ax1+...+(D,W")a".

Now we see from (8. 5) that (tpl («)) . S is a strictly positive multiple of a/ax' , plussome terms which are linearly independent of a/ax' ; here of course we use the fact that

1.

Hence (cps t«)) S * 0. for every a.

The canonical orientation of aP is constructed as follows. If p is a volume form on Mwhich belongs to the orientation of M, the formula

a.- (XIA...AX"-,) =P' (SnXIA...AX"-r),X,,....X"-,E S(aP), (8.6)

(here 3 (aP) denotes the set of vector fields on aP) defines a volume form on aP,because q is nonvanishing. The equivalence class of this volume form is the canonicalorientation of aP. rx

8.3.2 Examples

8.3.2.1 Boundary of a Half-Space in Euclidean SpaceSuppose M = R" with the orientation given by dx' A ... A dx", and takeP = { (x,, ..., x") E R": x, 5 0} . Then S = a/ax' , and taking X, = a/ax' `' in(8. 6) shows that the orientation of the boundary aP = { (x,, ..., x") a R": x, = 0} isgiven by the volume form dx2 A ... Adx".

8.3.2.2 Boundary of a Disk in the PlaneLet M = R2 - {O} with the orientation given by dr A d0, where (r, 0) is the usualsystem of polar coordinates; of course we will need two charts, with different9-domains, to cover M. Let P = { (r, 0) a M: r:5 1 } , which is a 2-dimensionalsubmanifold-with-boundary because (r, 0) -> r is a submersion; see the criterion for asubmanifold-with-boundary in Chapter 5. Here we may write q = a/ar, and (8. 6)shows that the orientation of the boundary aP = S' (r, 0) E M: r = 1 } is givenby the 1-form A. where

X.-- =drAdO (a Aae) = 1;

8.4 Exercises 173

5. The special linear group SL2 (R) a R2 X 2 can be regarded as f r (0), where

f([Z w]J = xw-yz-l,

which is a submersion when restricted to GL2 (R).

(i) Calculate explicitly the volume form * df in terms of the (x, y, z) parametrizationand show * (df) (1) = -2 (dx n dy A dz).

(ii) Show that the "left-invariant" volume form p (A-') = (LA* (* df) ) (A-t) is-2((x2+yz) Ix) (dx A dy A dz) in terms of the (x, y, z) parametrization, where

Lx ([uv])[xY][stfA

zuv]=

[xJw

6. (i) Show that P = {A E GL2 (R) : IAI S 1} is a 4-dimensionalsubmanifold-with-boundary of the general linear group GL2 (R), with boundarySL2 (R) as described in Exercise 5.

(ii) If GL2 (R) is given the volume form dx A dy A dz A dw, for the coordinates givenin Exercise 5, find the orientation induced on SL2 (R) as the boundary of P.

Hint: As a first step, change from dx A dy A dz A dw to an equivalent form where the first entryin the wedge product is df.

(iii) Determine whether the orientation of part (ii) is the same as the one represented bythe volume form *df of Exercise 5.

7. If M and Mare manifolds with atlases { (U;, tp1) , i E 1} and{ (V,, Wj) , j e J},respectively, both satisfying (8. 3), show that the product manifold M x N is orientable,by considering the atlas { (U; x V1, (p, x W,) , (i, j) E 1 x J}.

8. Let { (Ua, tpa) ,a a 1} be an atlas for an n-dimensional differential manifold M.

Recall that the tangent bundle rt: TM -> M, as constructed in Chapter 6, has an atlasgiven by { (><t (U,), Wa) a E 1}, where

W.([r,a,v)) _ ((p.(r),v) a R"xR". (e. 7)

Also recall that [ r, a, v) r, y, w) w = D (q , (p- 1) ((pa (r) ) v, and so, forevery a, y e I such that U. n UY * 0, the change-of-chart map is given by

at(x,v) = (x)v). (8.8)

Show that the manifold TM is always orientable by showing that the derivatives of themaps (8. 8) always satisfy (8. 3).

Hint: If A. B, C e RDOwn rA Ol IAI I C1.

LB CJ


8.5 Integration of an n-Form over a Single Chart

Let (U, (p) be a positively oriented chart on an oriented manifold M of dimension n.Suppose that K is a subset of U such that (p (K) is a bounded subset of R", and is"measurable" in the sense that it qualifies as a domain of integration under the reader'spreferred theory of integration;' the boundedness condition is included so that we do nothave to be concerned about infinite integrals. Suppose also that p is an n-form on M. Inthat case ((p-')* p makes sense as an n-form on (p (U), and it may be written as

((P-') p = h (dx' A ... A dx"), for some smooth function h on (p (U). Let us define theintegral of the n-form p over K as:

Jp = ((p-') p = 5 h (x,, ..., x") dx,...dx".K (P (K) ip(K)

(8.9)

For example, if (p (K) is a rectangle such as (s,, t,) x ... x (s", t"), the last integralbecomes the multiple integral

J... Jh (x,, ..., x") dx, ...dx".

S, S.

Note how we have dropped the wedge symbols on the right side of (8. 9) in shiftingfrom differential forms to multiple integrals. The key idea which makes this possible isthat the change of variables formula for multiple integrals is entirely analogous to theway that differential forms transform under pullback, as we shall now demonstrate.

8.5.1 The Integral of an In One Chart Is Intrinsic

If P E S2"M, if (U, (p) and (V, w) are positively oriented charts, and if K g U n V isa set such that (p (K) and ty (K) are bounded and measurable, then

P.f ((P-,)* p = J (W-t)*W(K)

(8.10)

In other words, the definition of 1 p does not depend on the chart.K

Proof: Suppose W = U n V, and the coordinate system {x', ..., x"} on (p (W) ismapped to the coordinate system { y', ..., Y") on W (W) under the diffeomorphism

F = I:W(W) cR"-a(p(W) cR".Let

' Readers familiar with Riemann integration can think of (p (K) as a finite union of open orclosed rectangles. For readers who know measure theory, (p (K) can be any bounded, measurableset. You do not need to know measure theory to read this section.


((_1)*P = h(dx'A...Adx"),

for some function h = h (x', ..., x") on (p (W). The key idea is that, by the calculationson exterior powers of a linear transformation in Chapter 1 and on pullbacks inChapter 2,

(Vr')*P = ((p,tlr')* ((p-')*P = F*(h(dx'A...Adx"))

= (F*h) (n"DF) (dy' A... Ady")

= h s FIDF1 (dy' A... Ady") .

Using the change of variable formula for multiple integrals,

19(K)

* hdx dx9(K)

= 1 hsFIDFIdy,...dy",w(K)

1(f ')* P,w(K)

as desired. rx

The integral constructed above is already familiar in the guise of the line integral andsurface integral in vector calculus, which we shall now review.

8.5.2 Line Integrals of Work Forms

Figure 8. 2 Parametrized Ccurve

Suppose I is an open interval of the real line, and t:1-> U Q R3 is a parametrization ofan oriented 1-dimensional submanifold of R3, such that the induced chart (t (1), T-1)


is positively oriented. In vector calculus, T would be called a parametrized curve, andcan be expressed in (x, y, z) coordinates as T (t) = (t (t) , T2 (t) , t3 (t) ). As we sawin Chapter 2, to any vector field

XFlax +

FZa +F3ay z

on U there is associated a "work form" aX E SZ( U, namely,

GSX = F'dx+F2dy+F3dz,

and by the calculations of Chapter 2,

T* ti3X = (F', F2, F3] T2 dt.

T3

According to (8. 9), the "line integral" of 6X along the path T (1) is:

I WX = JT' ox = f (FITS'+F2T2'+F3t3')dt,t (1)

I I

(F t')dt.

(8.11)

(8.12)

(8.13)

(8.14)

which coincides with the vector calculus definition of the line integral of the vector field

X along the curve. The significance of the term "work form" is that its integral measures

the work done, in a precise physical sense, by the vector field in moving a particle along

the curve. The advantage of the differential-form expression is that it is coordinate-free

(as 8.5.1 proves), whereas the vector calculus version appears to depend on Cartesian

coordinates and the Euclidean inner product. Note that choosing a parametrization of

the opposite orientation, that is, running the opposite way along the curve, would

change the sign of the integral.


8.5.3 Surface Integral of a Flux Form

40

V

Figure 8.3 Parametrized surface

Suppose ): W c R2 - U c R3 is a parametrization of an oriented 2-dimensionalsubmanifold of R3, such that the induced chart (0 (W) , 4-') is positively oriented. Invector calculus, 0 (W) would be called a parametrized surface. As we saw in Chapter2, to the vector field X on U in (8. 11) there is associated a "flux form" Ox a C12 U.namely,

Ox = F'(dyAdz) +F2(dzAdx) +F3(dxAdy). (8.15)

If we use the notation 0 (u, v) = (m', (D2, 03) and CDu = a0'/au, etc., then we seeas in Chapter 2 that

c'dx=6 du+0dv,edy=42du+0dv,0 dz=0du+0dv; (8.16)

e x - dundv = P. (0Yx(D,,)dundv,

where P = (F', F2, F3) T. and ("D. = (C., 1D.2, 0.3) T, etc. By (B. 9), the "surfaceintegral" of the flux form Ox over the surface 0 (W) is

(8.17)

1 Ox =5

0x = JF (4-DY x 46,,) dudv, (8.18)

which coincides with the vector calculus notion of integrating the normal component ofthe vector field X over the surface 0 (W) ; note that 0Y and CD. are linearlyindependent (since 0 is an immersion), and are "tangent" to the surface (see Chapter 4),so their cross product is a nonzero vector normal to the surface. The choice of anorientation for this 2-dimensional submanifold is equivalent to a choice of normaldirection; a parametrization with the opposite orientation would reverse the sign of the


integral, because it would be measuring the flux in the opposite direction through thesurface.

Thus we now discover the geometric meaning of a flux form in R3; its integral across asurface measures the flux of the associated vector field through the surface, with apositive or negative sign determined by the orientation of the surface.

8.5.4 Computational Examples

8.5.4.1 Example of a Line IntegralSuppose t (t) = (0, I,1/t),te (0, 1); then

f ydz = J -ft(-1/t2)dt = -tcoa) (0.u

8.5.4.2 Example of a Surface IntegralIf 'b (r, 0) = (rcos0, rsin0, r2/2), for O< r< 1,0< 0 <?t/2, then (8. 17) impliesthat

c1 (dzndx) =0 1 0

cos0 sinO r-rsin0 rcos0 0

dr n d0 = -r2sin0 (dr n d0);

and so, if W = (0, 1) x (0, n/2), then

I x/2

J dzndx = -Jr2dr f sinod0 = -1/3.0(W) 0 0

8.6 Global Integration of n-Forms

Suppose that { (U'il (p) , j e J} is a positively oriented atlas on an oriented manifoldM of dimension n, and p is an n-form on M. By splitting the domains { U',} up intosmaller domains if necessary, while using the same maps {(p' } on suitably restricteddomains, we can obtain an equivalent positively oriented atlas { (U,, c) , j e J} suchthat iJ (UI ) is a bounded subset of R" for every j.

Suppose that K is a subset of M such that

K is contained in some compact subset K. of M, so there is some finite set of charts,whose domains can be relabeled { U1, ..., Um}, such that K c ICJ U,; and

tsism

8.6 Global Integration of n-Forms 179

(pi (K n Ui) is a measurable2 subset of R" for every i E (1, 2, ..., m }.

Notice that (p, (K n Ui) is a bounded subset of R" by construction, and therefore anexpression such as

m

I ( u, P)

makes sense. However it would not be an appropriate formula for the integral of then-form p over K, because, for example, if a region H is in the intersection of exactly r ofthese charts, then we will have integrated over H not once but r times. The easiest wayaround this problem for computational purposes is to appeal to the"inclusion-exclusion" formula from combinatorics, which we will presently justify.Define the integral3 of the n-form p over K as

m

Jp = A1-A,+ + I Aijk-...+ (-1)m+IA12.. m, (8.19)K i = I i<j i<j<k

A1(I)...1(r) _1

KnU,ign.. nU,<qP. (8.20)

Note that the integral (8. 20) is well defined by (8. 9), and the sum in (8. 19) makessense because there are only a finite number of summands. To see why (8. 19) is theonly possible formula consistent with the one when K is contained in a single chart,consider the case of a region H contained in the intersection of exactly r of(UI, ..., U. }; the integral over His included r times in the first sum of (8. 19), and

2 See footnote 1 for the interpretation of "measurable."3 Those who know measure theory and topology will find a more complete treatment ofintegration in Berger and Gostiaux [1988]. In particular, here is how to extend the integral tononcompact sets. A volume form 0 which belongs to the equivalence class of the orientationinduces a Radon measure t on M as follows. Given a continuous function j on M with compactsupport K, define

A V) = jfa.K

where the right side is defined by (8. 19). By the Riesz Representation Theorem, t is a Radonmeasure on M. For any other n-form p, there exists a smooth function g on M such that p = ga.If g is a µ-integrable function, we may define, for any measurable set H C M,

Jp = 1 gdg,H H

and it can be proved that the left side is the same whatever volume form a in the equivalenceclass of the orientation was selected at the beginning.


r!

(kr k! (r-k)!

times in the kth sum, for k = 1, 2, ..., r. Taking the signs into account, we see that thenumber of times it is included in the integral of p over K is

(1) l2/+...+(-1)k+l(k) +...+(-I).+t = 1-(1-I)'= 1.

using the binomial theorem. In other words, the integral over H has been countedexactly once, as desired, and so (8. 9) and (8. 19) are consistent in the case where K iscontained in one of { U1, ..., Um} . It is also true that for disjoint K and K' satisfyingthe conditions above,

j p= jp+Jp.Ku K' K K'

(8.21)

the details of which are left as Exercise 15. Now we need to check that the integral ofthe n-form p over K is not dependent on the choice of atlas. Let { (VY, WY) , y E J'} beanother positively oriented atlas with W. ( VY) bounded for every y, and supposeKg V1 v ... v Vq. We obtain a similar formula for the integral, namely,

9s+1

K $=I i'(1)<...<i'(s)

Bi'(1)...i'(s) =KnV,(1)n ..nV, (,)

8.6.1 The Integral of an n-Form Is Intrinsic

The value of the integrals (8. 19) and (8. 22) is the same.

Proof: The integral

C1(I) .

Knt)(u(" ...nU,v)nVru)n ..nV,-(,)P

(8.22)

(8.23)

does not depend on which of the maps { qi (1), ..., (Pi (r) , Wi (1), ..., Wi' (r) } is used toparametrize the domain of integration, by 8.5.1. It follows from the definition that

p

qs+1Ai(l) ..i(r) - (-1) CUI) . i(r).i'(1)...i'(s)

s= 1 i'(1) < <i'(s)

8.7 The Canonical Volume Form for a Metric

and similarly for Bi. (1) ... (,) . Hence the integral in (8. 19) is

m

I (-1)r+I A r(1)...i(r)

m q(-1)r+l

r=1E E (-1),+I I

i(1)<...<i(r)s=I i'(I)<...<i, (s)

qI (-I)s+1Bi'(1)...i'(s)'

S= l i, (I) <... <i'(s)

by rearranging the order of summation, which gives (8. 22), as desired.

8.7 The Canonical Volume Form for a Metric

181

n

Suppose (M, is an oriented pseudo-Riemannian manifold (see Chapter 7). In thedomain of each chart, we may construct an orthonormal frame field (see Chapter 7), andthe corresponding orthonormal coframe field, denoted {E', ..., E"}. This is a field of1-forms with the property that

Em = j:f;°'dx', (8.24)

where the matrix of functions satisfies G (p) (p) TA= (p);here G = (g11) is the local expression of the metric tensor, and A is a diagonal matrixwhose diagonal entries are all ±1, in which the number of minus ones is determined bythe signature of the inner product.

An orthonormal coframe field will be called positively oriented if a = EI A ... A E" isin the equivalence class of the orientation of M; by switching the sign of El if necessary,any orthonormal coframe field may be made positively oriented.

8.7.1 The Canonical Volume Form Is Intrinsic

If { EI, ..., E" } and I -ell ..., E" } are two positively oriented orthonormal coframefields on an open set U in an oriented pseudo-Riemannian manifold (M, thenEI A ... A E" = El A ... A E"; therefore the formula

a!U=ElA...AEn (8.25)


defines unambiguously a volume form a on the whole of M, called the canonicalvolume form on (M, (4)).)). The local expression for El A ... A E", in any chart (U, cp)such that cp* (dx' A ... A dx") is equivalent to the orientation of M, is

BIA...Ar." = IGIIAIcp* (dx'A...Adx"). (8.26)

Proof: As usual, we write dxI A... A dx" instead of cP (dxl A ... A dx"), etc. From(8. 24) and our discussion of determinants in Chapter 1, we see that

FE IA...AC = T.(!)dxi(i) A...A 4.n(")dxi (n) = IIdx ! A...Adxn

t(1) i(n)

where G (p) = E (p) TAE (p) . Taking determinants, we see that I GI = I EI 2) Al ,

which can be rewritten as 1-'712 = I GI I Al because I Al = t l . This gives (8. 26). Sincethe same equation holds for el A... A ei", the conclusion follows. rX

8.7.2 Examples

8.7.2.1 Euclidean 3-Space{ dx, d y, dz } is an orthonormal coframe field with respect to the Euclidean innerproduct. The canonical volume form dx A dy A dz can be expressed in spherical polarcoordinates (see Example 8.1.1) as - r2 sin¢ (dr A A A

8.7.2.2 4-Space with the Lorentz Inner ProductFor the Lorentz inner product, the metric tensor with respect to coordinates {x, y, z, t}is diagonal with diagonal entries (1, 1, 1, -c2), and here A = diag (1, 1, 1, -1).{dx, dy, dz, cdt} is an orthonormal coframe field, and dx A dy A dz A cdt is thecanonical volume form.

8.7.2.3 The 2-SphereAs in Chapter 7, the Euclidean inner product on R3 induces a Riemannian metric on the2-sphere. Using the "latitude-longitude" parametrization (see the example in Chapter 7),I GI = (sin¢) 2, and the canonical volume form is sino (do A d9) (with signdepending on the orientation). Note that this is the same as the area form discussed inChapter 4.

8.7.2.4 The Hyperbolic PlaneFor the hyperbolic plane with (s, 0) parametrization (see Chapter 7), the canonicalvolume form can be written, for example, as

rJI + r

or as sinhs (ds A d9) .

8.8 Stokes's Theorem 183

8.7.3 Volume of a Pseudo-Riemannian Manifold

As one might expect, the Riemannian volume of an oriented pseudo-Riemannianmanifold (M, (.I.)) is simply the integral of the canonical volume form over M,denoted vol M. In the case of a 1- or 2-dimensional submanifold of R3, this is simplythe Euclidean length or surface area, respectively. For example, let M be the portion ofthe hyperbolic plane for which 0 < s < I in the (s, 0) parametrization (see Chapter 7),which might be called the punctured unit disc in hyperbolic 2-space. Its Riemannianvolume (i.e., area) is (omitting the proper pullback notation in the first integral)

2n 1

J sinhs (ds n de) = J Of sinhsds = 2tt (cosh I - 1) = 1.0861613... X.M 0 0

It is interesting to note that M has an area greater than that of the unit disc in EuclideanR2 ; indeed the disc of radius r has an area increasing exponentially with r, whereas inEuclidean R2 the area increases as the square of r.

8.8 Stokes's Theorem

Here is the main theorem about integration of differential forms on an orientedsubmanifold-with-boundary. It plays a fundamental role in areas of applied mathematicssuch as fluid dynamics and electromagnetic theory; see the final chapter of Abraham.Marsden, and Ratiu [ 19881 and of Flanders [ 1989] for examples.

8.8.1 Stokes's Theorem for a Compact Submanifold-with-Boundary

Let M be an oriented n-dimensional manifold and P a compact n-dimensionalsubmanifold-with-boundary of M, whose boundary aP is oriented as described in 8.3.1.For every co E SN -I M,

Jdco = J t* W, (8.27)

P aP

where t" Co denotes the pullback of Co under the inclusion map t: aP - M, that is, therestriction of Co to aP. In particular if aP = 0 (e.g., if M is compact and P = M),then

Jdco = 0,VcoE S2"-IM.P

(8.28)

The proof will be found in Section 8.12. To understand Stokes's Theorem, it helps torecall three special cases. For the first, recall that 0-forms are simply smooth functions


on M; hence the n = 1 case of the theorem does not require proof, since it is containedin:

8.8.1.1 n = 1: Fundamental Theorem of CalculusFor every smooth function g on an open interval of the real line containing a closedinterval P = [a, b] (:.DP = {a, b}),

b b

Jdg =Jg' (x) dx = g (b) - g (a)

a a

The reader should glance at the review of line integrals and surface integrals in 8.5.2and 8.5.3 and verify that 8.8.1 includes the following vector calculus results:

8.8.1.2 n= 2: Green's TheoremFor every compact oriented 2-dimensional submanifold-with-boundary P of the plane,and every vector field X on a neighborhood of p4

f d°x = f Q,x= J (X. (8.29)

P P aP

8.8.1.3 n = 3: Divergence TheoremFor every compact oriented 3-dimensional submanifold-with-boundary P of R3, andevery vector field X on a neighborhood of P.

POX = Jdiv X (dx n dy n dz) = j4. (8.30)

For the generalization of the Divergence Theorem to the case of an arbitrary orientedpseudo-Riemannian manifold (M, with canonical volume form a, see Exercise19.

8.9 The Exterior Derivative Stands Revealed

This section attempts to describe, using Stokes's Theorem, what might be called thegeometric meaning5 of exterior differentiation. Let w be an (n - 1)-form on aneighborhood of 0 in R" +k. Given a set of linearly independent vectors { v i, ..., v" } inR k, we would like to know the geometric interpretation of do) (0) (v1 A ... A v").First of all, it suffices to restrict Co to the tangent space at 0 to the copy of R" spanned by

4 See (8. 12) and (8. 15) for the "work form" and "flux form" notation.5 These ideas come from unpublished notes of John Hubbard (Cornell).

8.9 The Exterior Derivative Stands Revealed 185

the vectors { vt, ..., v"}; therefore we shall now regard was an (n -1)-form on aneighborhood of 0 in R".

Next consider the parallelepiped P with edges prescribed by the vectors { vt, ..., v"}.See the picture below for the case n = 3. For e > 0, let eP stand for the parallelepipedwith edges prescribed by the vectors { ev,, ..., 8v"}, and let a (EP) denote the"boundary" of EP; of course, EP is not a submanifold-with-boundary of R" because ithas sharp edges, but there is a modified version of the theory (see Spivak [ 19791,Abraham, Marsden, and Ratiu [ 1988], etc.) which applies to the piecewise smooth case;certainly a (cP) is a finite union of oriented (n - 1)-dimensional submanifolds of R.

A fact of geometry is that the Riemannian volume of P is given by the formula

vtn...AV. = (volP)e1A...Ae".

where {e1, ..., a"} denotes the standard basis of Euclidean space.

Figure 8. 4 Parallelepiped in 3-space

(8.31)

Necessarily the constant n-form dw (0) obtained by evaluating dw at 0 is expressibleas dw (0) = a (dxI A ... A dx") (0) for some real number a, so

dw(0) (v,A...Av") =a (dxl A ... A dx') ((volP)e,A...Ae")

= a (vol P)

=afdx'A...Adx"P

8.10 Exerclses

e e

= lime 2 j(B(e,y) -B(0,y))dy- j(A(x,e) -A(x,0))dx0 0

= aB (0,0)- ay (0, 0),

after about two lines of calculation, which the reader of Chapter 2 will recognize asd(Adx+Bdy) (0,0) (etAe2).

8.10 Exercises

187

In the following exercises, a submanifold of Euclidean space is assumed to have theinduced Riemannian metric (see Chapter 7), unless otherwise stated.

9. Using the usual parametrization IF ($, 0) = (cos0sin4, cos$) of the2-sphere S2, take M to be the subset of e with 0 < $ < 1 (a sort of punctured unit disc).Find the Riemannian volume of M using the canonical volume form sin0 (do A d9).

10. Find the induced Riemannian metric on the ellipse x2/a2 + y2/b2 = 1, its canonicalvolume form, and its Riemannian volume (i.e., length).

11. Calculate the induced Riemannian metric of the hyperboloid x2 + y2 - 2 = 1 in R',and the canonical volume form, using the parametrization

0 (t, 0) = (coshtcos0, coshtsin9, sinht) = (x, y, z).

Calculate the Riemannian volume of the portion M of the hyperboloid on which0<z<1.Remark: This metric is not the same as that of the hyperbolic plane!

12. Following the model of Example 8.9.2, verify formula (8. 32) for the case where w is a2-form on an open subset of R'.

13. Let B" + 1 denote the submanifold-with-boundary in R"' 1 consisting of points whosedistance from the origin is :5 1; thus )B+' = V. By applying Stokes's Theorem to then-form x0 (dx' A ... A dx"), prove that

volS" = (n + 1) vo1B"+1. (8.33)

14. Suppose M and N are oriented submanifolds of R' and R", respectively. Prove thatM x N, which we know from Exercise 7 to be an oriented submanifold of R'satisfies vol (M x N) = vol M x vol N.


Hint: The canonical volume form on M x N can be taken to be the wedge product of thecanonical volume forms on M and N. respectively; then use Fubini's theorem to factor the integralinto two parts.

15. Suppose { (VT, iyy ), y e J'} is a positively oriented atlas on an oriented manifold M.and K and K' are disjoint measurable subsets of M whose union is contained in acompact subset of M. Prove that, for any n-form p on M,

f p= Jp+fp.K v K' K K'

(8.34)

16. Prove the following variant of Stokes's Theorem in an open subset U of R;: For everyconstant vector field W. with associated flux form OW a i22 U. and every smoothfunction It on U,

f (Wlgradh)a. (8.35)P P

for every 3-dimensional submanifold-with-boundary P Q U.

17. Suppose M is a compact orientable n-dimensional manifold, and to E i2" -' M. Provethat do) must vanish somewhere on M.

18. Let P be a compact n-dimensional submanifold-with-boundary of an orientablen-dimensional manifold M, and let oP - N be a smooth map into a manifold N.

(i) Show that if dD can be smoothly extended to a map F on a neighborhood of P in M,then for all w e a -' N such that dw = 0,

f * W = 0. (8.36)

aP

(ii) In the special case where M = R", N = )P, and ) is the identity, use this result toshow that no such extension F of 4 can exist; in other words, a map from asubmanifold-with-boundary onto its boundary which keeps the boundary points fixednecessarily involves "tearing."

Hint: Suppose there were such an F; take (o = F1 (dF2 A ... A dF"), which satisfies dw = 0since N = aP is (n - I)-dimensional. Obtain a contradiction to (8. 36) using Stokes's Theorem.

19. Let M be an orientable n-dimensional manifold M. oriented by a volume form p. If X isa vector field on M, then the Lie derivative LXp a iZ"M, as discussed in the exercises inChapter 2. must be some function multiplied by p, since the nth exterior power of eachcotangent space is one-dimensional; this function is called the divergence div µX of thevector field X. To summarize:


Lxg = (divMX) µ. (8.37)

(i) Derive from Stokes's Theorem the Divergence Theorem, which states that for everycompact n-dimensional submanifold-with-boundary P of M,

5 (div,X) µ = J3.XJ.L(8.38)

where txg a f2" -'M is the interior product, that is,

AX" = nX".

Hint: Look in the exercises of Chapter 2 for a formula for d (txg).

(ii) Find the explicit formula for the integrands on both sides of (8. 38) in the case whereµ is the canonical volume form sinhs (ds A d0) on the hyperbolic plane with (s, 0)parametrization (see Chapter 7), and

X = u(s,0)a +v(s,0)a .


Stokes's Theorem (in three dimensions) was used by George Stokes (1819-1903) in hisSmith Prize examination at Cambridge University in 1854. Fuller treatments of theintegration theory of forms, including de Rham cohomology, may be found in Abraham,Marsden, and Ratiu [ 1988], Berger and Gostiaux [ 1988], and Spivak [ 1979], among

others.

8.12 Appendix: Proof of Stokes's Theorem

Proof is needed only for n > 2, in view of 8.8.1.1.

8.12.1 Step I

Take an atlas for M of the special kind that was used for the definition of asubmanifold-with-boundary; in other words, for every chart (U, (p), eitheraP n U = 0, or else

tp(PnU) = {(xi,...,x") a cp(U):xt<_0}, (8.39)

with cp(aPnU) = {(x,,...,x") a cp(U):xi =0}.


Given an orientation for M, we can turn this atlas into a positively oriented atlas withoutdisturbing (8. 39). by simply replacing tp by s2cp for any chart map cp which is notpositively oriented, where s2 (xI, ..., xn) = (x1, -x2, x3, ..., xn). We will assume thatthis has been done. Next take an open cover { Va, a e I) of M together with a collectionof smooth functions {,o,,: M - [ 0, 1 ] , a e 1} that form a partition of unity subordinateto this atlas. P has an open cover each element of which intersects only finitely many ofthe { Va} (see the definition of a partition of unity). Since P is compact, it has a finitesubcover, and therefore P n V. * 0 for only finitely many a. Thus we can write

f dw = f IUadw = I f vadw, (8.40)P P a a P

where the sum is finite, and of course each n-form vadw has its support inside thedomain of a particular chart U, (a) . Thus the proof is complete as soon as we haveverified (8. 27) for every summand on the right side of (8.40); in other words, it sufficesto check, for every chart (U, (p) in our special atlas, and for every (n - 1)-form w withsupport contained in U, that

aPnU=O= fdw=0; also (8.41)P

cp(PnU) _ {(x1,.... xn) a p(U):x150} f dw= f 1*w. (8.42)Pnu aPnu

8.12.2 Proof Step II: Verification of (8.41)

If aP n U = 0, then either P n U = 0, in which case (8.41) is vacuously true, orelse U g P, in which case we have to prove that, for every (n - 1)-form co with compactsupport contained in U, the integral of dw over U is zero. Suppose that

n

((p 1)* w = hi (dx' n ... Add' n ... Adxn), (8.43)

using the notation of (8. 2). By the rules for exterior differentiation, it follows that

* n ah;

((p-')* dw = d(((p-') w) (dx' A ... Adxn), (8.44)

=1

where the functions { h; } and their derivatives have their support in the interior of someclosed finite rectangle C in Rn, and therefore are zero on the perimeter of C. Thedefinition of the integral of dw given in Section 8.5 gives the following equation; proofof (8. 41) amounts to showing the expression is zero:

8.12 Appendix: Proof of Stokes's Theorem 193

rah'J

dw=

o

dx, J ax(xi, ..., xn) dx2...dx

P n U a C. I '

f (h, (0, x2, ... , x") - h, (a, x2, ..., x")) dx2 ... dx,,C._,

= f h, (0,x2, ..., x") dx2...dxC,'_i

f h, (dx2 n ... A dx").m(apnu)

Let v: R""' -> R" be the inclusion (x2, ..., x") -* (0,x2, ..., x"), which can berestricted to be the embedding of (p (aP n U) into (p (U); then we have

*(((rI.v) (1))

((p')*w (V. a A...AV. a A...AV. 3-),ax' a' ax"

which is zero unless i = 1. since v. (a/ax') = 0; in that case, by (8. 43) the lastexpression is

0

_ hi(dx'A...Ad.'n...Adx") (v. a A...AV.a) = h,.ax2 ax"

Moreover t = cp ' v cp, and so ((p-')* (t* (o) = ((p-' v)* w. Thus it follows that

f dw = f h, (dx2 A ... A dx") = f ((V l v)* wPnU ,(aPnu) 9(aPnU)

= f t*w,aP n u

as desired. 11

9 Connections on Vector Bundles

The theory of connections can be presented in many equivalent ways. Here we present itas a theory of exterior differentiation of bundle-valued differential forms. First we willdevelop the theory for general vector bundles, and then specialize to the importantexample of the Levi-Civita connection on the tangent bundle of a Riemannian manifold.

9.1 Koszul Connections

If V is a vector space, with a basis { v1, ..., vm}, let C- (M --* V) denote the smoothmaps from a differential manifold M to V. There is a natural way to differentiate afunction F e C°° (M -> V) with respect to a vector field X e 3 (M); write F asF'v, + ... + rvm, and take the vector-valued Lie derivative

LXF = (dF' X)v,+...+ (XF')vi+...+ (XFm)vm. (9.1)

The left-hand side is independent of the basis chosen for V. What can be said of the casewhere F is replaced by a section of a vector bundle over M, with fibers isomorphic to V?Now{v,, ..., vm} is not a fixed basis, but a local frame field for the vector bundle, andF'v, + ... + rvm is the local expression for a section of the bundle.

9.1.1 Definition of a Kossul Connection

Suppose n: E -+ M is a smooth rank-m vector bundle, and a: M --> E is a section,expressed in terms of a local frame field {s1, ..., sm} by

a (r) = a' (r) s (r) + ... + am (r) sm (r) E E,,

where the {a'} are smooth functions on some open set in the manifold. Our goal is asfollows. Given a vector field X on M, we want to have some way (possibly not unique)

9.1 Koszul Connections 195

to "differentiate a section a with respect to X"; in other words to construct a newsection, denoted Vxa, which will satisfy, Vh e C°° (M), the "Leibniz rule"

Vx (ha) = h (Vxa) + (Xh) a, (9.2)

and the C (M)-linearity property

Vhxa = h (Vxa), (9.3)

as well as the obvious additivity properties

Vx,+x2a= Vx1a+Vx2a,Vx(aI+a2) = VxOI+Vxa,. (9.4)

An operation V satisfying (9. 2), (9. 3), and (9. 4) is called a Koszul connection on E,and Vxa is called the covariant derivative of a along X. The "naturalness" of theseconditions will become especially apparent when we present connections in terms ofvector-bundle-valued forms in the next section.

9.1.2 Example: The Euclidean Connection on the Cotangent Bundle

The cotangent bundle of R" has a natural connection V, which can be called the flat orEuclidean connection, and which the reader has been using for many years probablywithout knowing that it was a connection.' It is important here to take the standardcoordinate system {x', ..., x"} on R. The sections of 7* R" -+ R" are simply the1-forms, and so we can define dxm for an arbitrary 1-form w = h

idx' + ... + h"dx"

by taking dx (dx') = 0 for all vector fields X. and using (9. 2) and (9. 4), namely,

I xW = Ix (h;dx') _ hid, (dx') + (Xh;) dx'

= (XhI)dx' +... + (Xh")dx", (9.5)

which is the same as the situation described in (9. 1), with only a change of notation.This is clearly consistent with (9. 3) also. For more information about this connection,see Exercise I in Section 9.4.

9.1.3 Example: The Euclidean Connection on the Tangent Bundle

A general formula will be given in Exercise 3 in Section 9.4 for deriving a connectionon a vector bundle from a connection on a dual bundle; as a special case of this we

' One recalls Moliere's bourgeois gentilhonune, who discovered that all his life he had beenspeaking prose.

196 Chapter 9 Connections on Vector Bundles

create from (9. 33) the Euclidean connection on the tangent bundle of R" as follows;for vector fields

X=Y

on R", VXY is the vector field

VXY = (Xci) a/ax! _ 4' (acJ/ax;) a/ax;. (9.6)

The relationship of this connection to Example 9.1.2 is shown in Exercise 2 inSection 9.4.

9.1.4 The Obvious Generalization Is Not Intrinsic

Any attempt to generalize the (9. 1) construction of VXa to the case of a general vectorbundle, that is, "VX ((7 1 s I

+ ... + esm) = (X(Y 1) s i + ... + (Xam) sm, ' has limitedusefulness, because it is not intrinsic; in other words, if we take another local frame field{ti, ..., tm}, and write a = TI t, + ... +T"t,,, then in general

(X&1)SI+...+ (Xo")Sm,* (XT')tI +...+ (XTm)tm. (9.7)

For example, if M = R2 - 101, E is the tangent bundle, a = a/ar and X = a/a0 inthe usual (r, 0) system of polar coordinates, {s1, s2} = {a/ar, a/a0}, and{ r,, t2 } = {a/ax, a/ay }, the reader can easily check that

a = lar +0s- = cos0ax+sin0aay

and so applying X = a/a0 to the component functions shows that the two sides of(9. 7) are indeed unequal. The heart of the matter is that we need to take intoconsideration the way the frame changes with position as well as the variation in thecomponent functions. The extra ingredient can be understood by doing Exercise 5 inSection 9.4.

9.1.5 Origin of the Word "Connection"

The word "connection" has the following geometric origin. If V is a connection on thetangent bundle, then VXY is a vector field which is supposed to describe the rate ofchange of the vector field Y along the "flow" of the vector field X. The difficulty is thatthe values of Yat different points along the flow belong to different tangent spaces,which are not in any natural relationship in general. The role of the "connection" is toconnect these different tangent spaces. The Christoffel symbols (see Section 9.2.4.1)were devised as a way to describe how to do this, because they tell us how a given frame


9.2

field (in terms of which X and Y may be defined) changes along the flow of each of itsconstituent vector fields.

Connections via Vector-Bundle-valued Forms

In order to develop the theory of connections in an efficient way, we shall now extendthe theory of differential forms to cover vector-bundle-valued forms.

Suppose tt: E -, M is a rank-m vector bundle over an n-dimensional manifold M, and1:5 q:5 n. Combining two constructions from Chapter 6, we obtain a vector bundleHom (AgTM;E), whose fiber over r e M, denoted Hom (AQTM;E),, consists of thelinear maps from ATM to E. It may help the reader to recall the characterization ofthe exterior power of a vector space in terms of multilinear alternating maps (seeChapter 1), which identifies Hom (AMTM; E) , with the q-multilinear alternating mapsfrom TM x ... x T,M to E,.

An E-valued q-form means a section of the vector bundle Hom (AQTM; E) . The set ofE-valued q-forms is denoted 1Zq (M;E), and q is called the degree of such a form,abbreviated to deg t for a form i. In keeping with the terminology for 0-forms on amanifold, the sections of tt: E -4 M will be called E-valued 0-forms. For futurereference, we make the formal statement:

S2g (M; E) = r (Hom (AQTM;E)) , q = 1, 2, ..., n; 00 (M; E) = rE. (9.8)

The union of the sets fg (M; E) as q runs from 0 through n is called the set of E-valuedforms. The action of an E-valued q-form µ on the wedge product of q vector fields on Mwill be denoted in the same way as the action of an ordinary differential form, namely,

µ X 1 A ... A Xg a 02° (M; E). (9.9)

To find a local expression for such a form, take a chart (U, tp) for M, inducing a localframe field {dx1, ..., dx"} for the cotangent bundle over U. If {sr, .... sm} is a localframe field for E over U, then µ E (1" (M; E) has the local expression

m

µ u = si(dx'ct> A ... Adx'cgr), (9.10)

i=l I

where 1 = (1 5 i (1) < ... < i (q) 5 n), and where the { are smooth functions on U.Since


7 h!(dX (U n... ndX'(g)) E aq(M),

a suggestive way to write g e QQq (M; E) is as

m m

=ISj AT,'f E Qq(M), (9.11)

which is consistent with the notation hw = h A w when h is a 0-form and w is a p-form,since s1 E i-EI u = CIO (MI U; E) is an E-valued 0-form. However, the wedge symbolsare actually superfluous, and we shall sometimes omit them.

9.2.1 Examples of Bundle-valued Forms

Q q (M;R) = QQgM, the set of differential forms of degree q on M.

If E is the trivial bundle M x Rm, then Q'? (M; M x Rm), which is usuallyabbreviated to Qq (M;Rm), can be identified with the set of m-tuples (n(, ..., Tim)of q-forms on M; the vector-valued 1-forms and 2-forms encountered in Chapter 4are good examples.

If E is the tangent bundle TM, an obvious element of s2I (M; TM) is the identitymap t from TM to TM, which can be regarded as an element of L (TM; TM) foreach r e M.

For any fixed vector field X on M, the Lie derivative LX, acting on tangent vectors, isan element of III (M;TM).

We saw in Chapter 6 that the differential of a map from M to N can be viewed as abundle-valued 1-form on M with values in the pullback of the tangent bundle of N.

The primary motivation for this concept is that the all-important "Curvature tensor"(see below) for a connection on a vector bundle E is an element of112 (M;Hom (E; E) ).

9.2.2 Exterior Products

There are natural mappings

f2q (M; E) x fZPM ---) S2p + q (M; E) F fYM x f2q (M; E),

which generalize the exterior product of differential forms; the map on the left is written

(µ, CO) --* iAw = (_l)pgwAµ,

and the map on the right is w A µ +- ((o, µ). They can be defined locally in the notationof (9. 11), and the outcome is independent of the choice of local frame field for E;namely,


M m/w^µlo = (w^tr) ^sl,illu^w = sJ^ ('( ^w). (9.12)

1=1 1=1

Obvious associativity and distributivity properties, which may be deduced from those ofthe ordinary exterior product, will be used without further comment.

9.2.3 A Connection as an Exterior Derivative of Bundle-valued Forms

In the notation of (9. 8), a covariant exterior derivative (of bundle-valued forms) ona vector bundle n: E -* M means a map

dE:12P (M; E) SIP, t (M; E), p = 0, 1, ..., n, (9.13)

that satisfies the "Leibniz rule": That is, for every E-valued form .t and everydifferential form w on M

dE (µ ^ (0) = dEµ ^ w+ (-1) d`81+ 1 A dw. (9.14)

Remark: Note how this resembles the rule "d (71 A (0) = dt) ^ w + (-1) d`8 nq ^ dw"for exterior differentiation. Observe also that the analog of the rule "d (d(o) = 0" hasbeen omitted; the reality is that dE (dEµ) need not vanish, and the obstruction to itsvanishing will turn out to be the curvature of the connection.

How is a covariant exterior derivative related to the Koszul connection defined in theprevious section? They are in one-to-one correspondence, and for this reason we shalloften refer to dE itself as a connection.

9.2.3.1 Equivalence between a Koszul Connection and a Covariant Exterior DerivativeGiven a Koszul connection V on a vector bundle n: E -+ M. there is a unique covariantexterior derivative dE such that the following formula holds:

dEa X = VXa, (9.15)

for every vector field X and every a e 1'E = S2° (M; E).

Proof: Assume that we are given an operation V satisfying (9. 2), (9. 3), and (9. 4). Thenfor every vector field X and every a e I'E = 00 (M; E), the map which takes X (r) todEa (r) (X (r)) = (dE(; X) (r) = (VXa) (r) belongs to L (T,M;E,) by (9. 3) and(9. 4), and the image varies smoothly with r; this shows that dEa indeed belongs toS2i (M;E).

Next we shall verify that (9. 14) holds for the case p = 0. Translated into the newnotation, (9. 2) says that, for every smooth function h on M and every a e Q° (M;E),


dE((Y nh) = dE(h(;) = h(dE6)+adh=dEanh+(-1)°andh,

where the latter version is designed to show concordance with (9. 14). So far we haveproved that there is a unique map dE: 00 (M;E) SI 1 (M; E) that satisfies (9. 14). Itremains to show that there exists a unique extension to E-valued forms of higherdegrees consistent with (9. 14). We shall treat the uniqueness and existence questions inthat order.

Uniqueness: If we use the local expression (9. 11), the action of dE on .t E SIP (M; E)must be given by

m

dEtlU = IdE(sj ATl')i=1

(9.16)

_ (dESj n111+ (-1)°s1Adti1),

using (9. 14) and the fact that sJ has degree 0. Since d%, is uniquely specified by(9. 15), it follows that dEµ i U is uniquely specified by (9. 16).

Existence: One can use the local expression (9. 16) to show existence of dEp. for allµ E SIP (M;E) such that (9. 14) and (9. 15) are satisfied; however, this requires that wecheck that the same value for dE t is obtained regardless of which local frame field isused for the vector bundle E, which is quite a tedious calculation. A more sophisticatedway to prove existence is to find an intrinsic way to define dEp. starting from O. This issuggested by the formula, given in the exercises to Chapter 2, for the exterior derivativeof a differential form in terms of the Lie derivative of functions; namely, for any p-form(c and vector fields X0, ..., XP,

P

d(u X°n...nXP = Y, (-I)'LX(CO-X°n...nk,A...AXP)i=o

[X1,Xjj nX°n...Alt,A...AR,A...AXP,,<f

where the superscript A denotes a missing entry. Here we simply replace theLie derivative LX by V. and define dEp for all µ E SIP (M; E) by the formula

P

dEp. X°n...AXP = 1: (-1)`VX(µ X°n...AX,A...AXP) (9.17)=o


, (- 1)'+jg , (Xi,Xj] AX°n...AkiA...nkjA...AXp.+Yi<j

The reader should note that the case p = 0 is precisely the formula (9. 15), so this isconsistent with the earlier definition of dEµ for µ E S2° (M;E) . A few (long!) lines ofcalculation will verify that (9. 14) is satisfied also. rz

The version of (9. 17) when p = I is so useful that we state it separately.

9.2.3.2 Covariant Exterior Derivative of an E-valued 1-FormFor every µ E C I (M; E) and every pair X, Y of vector fields,

(9.18)

= (X. Y]. (9.19)

9.2.4 Local Expression for a Covariant Exterior Derivative

Suppose {s,, ..., s",} is a local frame field for the rank-m vector bundle tt: E - Mover some open set U in M. Since dES1 a S21 (M; E) for each i, there must exist anm x m matrix of 1-forms to i e f2'M such that

(9. 20)

The I-forms {t;y} constitute the connection matrix relative to the local frame field{sI, ..., s,"}, and the {m } are traditionally called the connection forms. For anysection a e i2° (M;E), expressed in terms of the local frame field {s,, ..., s",} by

a(r) = a0 (r)sI (r) +...+am(r)sm(r) = sina'(r),

the rule (9. 14) gives the local form of the covariant exterior derivative of a as

dEa = 1: sjna'tI +1: Sindal. (9.21)i, j i

9.2.4.1 Christoffel SymbolsConsider the specific case where E is the tangent bundle TM, and { e 1, ..., e" } is a localframe field for the tangent bundle, with a corresponding dual frame field { E 1. ..., E" } for

the cotangent bundle. For example, a chart (U, (p) for M induces a local frame field{DI, ...,D"},meaning { ((p'').a/ax1, ..., ((p'1).a/ax"}, for the tangent bundle,with dual frame field {(e dx', ..., (p' dx" }. By expressing the connection 1-forms {d.}in terms of the coframe field { E1, ..., E" }, we obtain a collection { rJk} of n3 smooth


functions on U, called the Christoffel symbols of the connection in terms of this localframe field, determined by

jrFkk

These are related to the Koszul connection V as follows:

(9.22)

Ve ei = d e. ek = ei A d. ek = rkiei. (9.23)

i i

9.3 Curvature of a Connection

Let us see what happens to a section a of a vector bundle when we apply the covariantexterior derivative twice. Unlike the case of the ordinary exterior derivative, for whichddw = 0 for all differential forms c°+ it will usually happen that dEdEa * 0. Thenonvanishing of dEdEa will be described by a quantity called the curvature of theconnection. Naturally the reader will be curious to know what this has got to do with thevarious kinds of curvature of a surface in R3, described in Chapter 4; this will beexplained in 9.3.5.

9.3.1 The Curvature as a Hom(E;E)-valued 2-Form

Given a connection on a rank-m vector bundle tt: E -> M, there exists a bundle-valued2 -form 9 E (22 (M; Hom (E; E) ), called the curvature of the connection, such that

dE (dEQ) = 9 A a, 0 E i2° (M; E). (9.24)

In terms of a local frame field { s,, .... S.) for E. 9t is expressed by the m x m matrixof ordinary 2 -forms (R;), called curvature forms, defined by:

dE (d ES;) = 7R n si.J-1

In terms of the connection matrix {d } appearing in (9. 20). we have

R; =k

Proof: Note first that for any smooth function h,

dE (dE (ha) ) = dE (hdEa+ dh n a)

(9.25)

(9.26)


= hdE(dEa) +dhAdEa-dhAdEa+d(dh) na

= hdE (dEa).

In other words, the map a -4 dE (dEa) is C°° (M)-linear. It follows that, if we take alocal frame field and define 2-forms (R;) by (9. 25), then for any section a = a's;,

dE (dE(;) = (R),a) n si,i.i

and the right side is exactly what is meant by 91 A a for 9t E 02 (M; Hom (E; E) ); thus(9. 24) is established. As for (9. 26), (9. 20) implies

dE(dES;) = dE(I:sintl) = {dESints+sindty};i i

consequently

LRkASk = ISkAQk j. k k

from which (9. 26) immediately follows. n

9.3.2 The Curvature Tensor

Classical differential geometry usually deals with the set of m2n2 real-valued functions.called the curvature tensor, given by

k = k

where { D1, ..., D } is a local frame field for the tangent bundle. Here are the gorydetails: If X = Yt `Da, Y = Il PDp, then

dE (dES;) X A Y = I VC -Rkap,sk. (9.27)k,a.p

9.3.3 Curvature in Terms of the V Operator

For every pair X, Y of vector fields, and every section a of the vector bundle n: E -, M.

VXVYa-VYVXa-VIXY1 a. (9.28)

Proof: This is a routine application of formula (9. 18) on page 201 to the bundle-valuedform dEa E f1 (M;E). We have


- X 1,

and this is exactly the right side of (9. 28) by (9. 15). it

9.3.4 Examples

9.3.4.1 The Euclidean Connection Has Zero CurvatureRecall from 9.1.3 that, for X = F, 'a/ax;, Y = the Euclideanconnection on the tangent bundle of R" is given by

VX

Y = (Xc) a/ax; _ 14' (at;'/ax;) a/ax;.

It follows from (9. 28) that the vector field

9t A z - o, ez, vi,;.

Thus the curvature of the Euclidean connection is zero.

9.3.4.2 A Case of Nonvanishing CurvatureConsider the 2-dimensional manifold M = R2 - { 0), on which alar,2 = h (r) -1 a/a$ form a frame field for the tangent bundle, where (r, Q) are polarcoordinates, and h is a nowhere-vanishing smooth function. In Example 9.5.3, we shallsee that a natural connection on TM is given by

2 0 -h'p2 W2 h'd$ 0

Using R i d k A t + dd , we see that the curvature is given byk

1 1

RI R2

2 2RI R2

Q^rhdQ-O'dQ+rh"d0 dQ-h"dOrndQl'= [h0'dQ-h0'd

= 0 -h"drndQh"drndQ 0

This means that, for example,

dTM (d'"42) = R2' A41+R'A42 = -h"drA doA41.

(9.29)


9.3.5 Why Is It Called "Curvature"?

In Chapter 4, we were dealing with a three-dimensional adapted moving frame for aparametrized surface M c R3. The vector bundle involved there was

E = TR3IM-4M,

that is, the tangent bundle of R3, restricted to M, which has rank 3. The connection wasthe Euclidean connection, suitably restricted. The matrix of connection forms (G3 }, interms of the orthonormal frame field2 {41, 42, 43} was precisely the matrix

which we examined in Chapter 4, and is related to the orthonormal frame field by

dE41 = L, A N.

The 1-forms {91, 02,()31 make up the orthonormal coframe field. Since the Euclideanconnection is flat, the curvature 9t = 0, so (9. 26) implies

At3q,9

which is nothing other than the "second structure equation" of Chapter 4, from whichGauss's equation and the Codazzi-Mainardi equations are obtained. Since the Gaussiancurvature K is characterized by dT1° = -K (9' A 9), we see that knowing the curvature9t tells us dGZ, which in turn leads to knowledge of the Gaussian curvature. In summary,the computation of 9t gives the data from which various concrete sorts of "curvature"can be obtained.

The next formula describes what happens when one takes an exterior derivative of acurvature form. A more concise and abstract version is derived in Exercise 14 inSection 9.4.

2 A point of notation: Here we prefer to use the vector field 4i rather than the vector !;; ofcomponent functions.


9.3.6 Bianchi's Identity In Terms of the Connection Matrix

Exterior differentiation of the curvatureforms (R,) appearing in (9. 25), in terms of theconnection forms in (9. 20), gives Bianchi's identity:

dR)i' _{Rlknmk-dkARk}.k

Proof: Equation (9. 26) can be rewritten as

dd = tNgA01.q

Exterior differentiation of (9. 26) gives

dR; = I:ddknmkI:GYkndQk.k k

Substituting for dtiY , etc., using (9. 31) gives

dR _ nmk-Zdkn {Rk-Em4nm9},k q k q

(9.30)

(9.31)

and two of the terms on the right cancel (just interchange indices k and q in the lastproduct), leaving (9. 30). u

The following table is intended to review the similarities and differences betweenordinary exterior differentiation and covariant exterior differentiation.

Exterior Derivative Covariant Exterior Derivative

Domain and fPM-AP*'M ( M ; E)

Range

Action on dh X = Xh = Lxh dEa . X = DXa

0-forms

Leibniz Rule d (11 A (0) = dE (1 A co) _

dt1 nw+ (-1)d`g'l71 ndw dEtn W+ (-1)degMµAdo)

Iteration Rule d (dw) = 0 dE (dEa) = 9i ACT

Table 9.1 Comparison between the exterior derivative and the covariant exterior derivative

9.4 Exercises

1. If f e C°° (R"), and X and Yaze the vector fields X = a ' Y = C'aX' let

9.4 Exercises 207

DZf (X, Y) (x) = DZf (x) (X (x) , Y(x)) = I:D,/ (x) l;' (x) Q (x). (9.32)i.i

Here D2f is the second derivative off, considered as a (0, 2)-tensor, and Al is theoperation of taking the second partial derivative with respect to x, and x,.

(i) Show that if V is the Euclidean connection on the cotangent bundle of R", as inExample 9.1.2, then

dxdf = X (DJ) dx', and '2 df - Y = DZf (X, Y). (9.33)

(ii) Verify that if df = hdg, then Vx(hdg) - Y = (Xh) (dg - Y) +h'xdg - Y.

Hint: Df = hDg, so DZf (X. Y) = Dh (X) Dg (Y) + hD2g (X, y).

2. Prove that (9. 5) and (9. 6) are related by

df VxY = X (df - Y) - d xdf - Y, (9.34)

for all vector fields X and Y, and smooth functions f.

3. (Dual Connections) Prove that, to every connection V on a vector bundle E -4 M, therecorresponds a unique connection V on the dual bundle E* - M such that, fors e 1'E, X e re, X e 3 (M), if ? s denotes the action of A. on s, then

(dx') s = -X Vxs, (9.35)

by taking the following steps:

(i) By canceling two terms of the type (Xh) A. s, show that the right side isC°' (M)-linear ins, that is, ('0xX) hs = h ('0xA) s for smooth functions h; thisshows that 'XA. E T'E'".

(ii) Check conditions (9. 2), (9. 3), and (9. 4) for

(iii) Also show that (9. 34) is a special case of (9. 35) in the case where E -+ M is thetangent bundle and e -+ M is the cotangent bundle.

4. Consider the special case of constructing a connection for the cotangent bundlerr: T* M - M. One wonders whether some judicious combination of the exteriorderivative, the Lie derivative of forms. and the interior product would do the trick. If wis a 1-form and X is a vector field, one candidate is VXw = tx (dw) E Sgt M, where theright side uses the interior product described in the exercises to Chapter 2, sotx (d(o) - Y = dw - X A Y. Another candidate is the Lie derivative VXw = Lxw, whoseeffect on 1-forms is given by Lx (gdh) = (Xg) dh+gd (Xh). Prove that onecandidate fails (9. 2), and the other fails (9. 3), and so neither is a connection.


5. Suppose bu: i<-t (U) -* U x Rk and 4v: R-' (V) - V x Rk are local trivializations ofa rank-m vector bundle it: E -* M, with U n V * 0, giving local frame fieldsr - { s , (r), ..., sm (r) } and r - { t t (r), ..., tm (r) } over U n V, where

s.(r) =OU (r,e1),t1(r) =0vt(r,e;),i = 1,2,...,m,rE UnV.

Suppose the connection forms with respect to these two local frame fields for the samecovariant exterior derivative dE are given by

m m

d5s1 _ SjAt dEtk = tindk.I=1 i=t

Derive the identity © = g-t dg +g-'Qfg, meaning that

,gj'dk = dgk+y(Dg'k, (9.36)i i

where gj (r) denotes the (i, j) entry of the transition function guv (r).

Hint: In the exercises to Chapter 7, we proved that ti = g1 st + ... +g sm.

6. (Continuation of Exercise 5) Suppose that the curvature forms with respect to these twolocal frame fields are given by (9. 25), namely,

m M

dE (dEs1) _ R A si, dE (dEt;) _ ,; n tj.J=1 i=t

(i) Prove that k = g tRg, meaning that

(9.37)i

(ii) Give an alternative proof of (9. 37), based on (9. 36) and the formulas

R = tSkAGi +d5 I:&, A& +d4Y.k k

Remark: In classical language, the curvature forms transform as "tensors" whereas theconnection forms do not.

7. Given a connection V for TM, and a chart (U, tp) for M, let {11'j,) be the Christoffelsymbols associated with the local frame field {D1, ..., D.} for the tangent bundle,where D. = ((P- 1) . (a/ax1), as in 9.2.4.1.

9.4 Exercises 209

The Christoffel symbols induce a section r of Hom (TR" ® TR"; TR") -4 (p (U),called the local connector, namely, the unique C" (R")-bilinear map which sends a pairF, G of vector fields on p (U) to a vector field r (F, G) on (p (U) such that

r(a/ax1,a/ax;) _ r a/aXk (9.38)k

(i) Verify that the following formula holds for every pair X, Y of vector fields on U andevery smooth function f on (p (U):

d((p*j) (VXY)-X(d(q*f) - Y) +D2f((p.X,(p.Y)

(ii) Conversely, show that every section r of Horn (TR" ® TR"; TR") -, (p (U) (i.e., avector bundle over (p ( U)) defines a connection V on the tangent bundle, restricted to U,by the formula

(VXY) ((p* J) = X (d ((e f) Y) - D2f ((p. X, (p. Y) + df r ((p. X, (p. Y).

(iii) Suppose another chart (V, 4t) induces a local coordinate system {11, ..., x"} anda corresponding local frame field { B1, ..., A,} for the tangent bundle, withcorresponding Christoffel symbols {f as) and local connector f. Prove that, onU n V, if h = (p w-1, then for every pair F, G of vector fields on W (U n V),

f (F, G) = (h"'). {r(h.F,h.G) +D2h(F,G)} (9.39)

in the notation of (9. 32).

8. The setup is the same as for Exercise 7. Suppose V is the "dual connection" on thecotangent bundle, given by (9. 34). Prove that if DO denotes D. D,, then

9Ddf D, = D,/-rkDkf. (9.40)k

9. Suppose the image of a curve y = y (t) is a one-dimensional submanifold N of amanifold M, and V is a Koszul connection on the tangent bundle TM. We may restrict Vto the vector bundle n: TM I N -+ N (the restriction of TM to N), of which y = ay/at isa section; thus it makes sense to refer to V X as a "vector field along y," for any sectionX of n: TMI N -* N. We define y to be a geodesic with respect to V if

V yY = 0, for all t. (9.41)

(i) Suppose y (t) = (y' (t), ..., r (t)) in some coordinate system for which theChristoffel symbols are { Prove that y satisfies the differential equation:


dr + rjkd dr =02j, k

(ii) Write down the specific form of this equation for Example 9.3.4.2.

(9.42)

10. Work out all sixteen components of the curvature tensor for the curvature 9; of 9.3.4.2(most are zero).

11. Calculate the curvature of a covariant exterior derivative dE on a rank-3 vector bundleE -> R2 with connection matrix

0 hdy 0-hdy 0 fdx0 -fdx 0

where f = f (x, y) and h = h (x, y) .

12. (Curvature of a line bundle) A connection V on a complex line bundle L -4 M isspecified by a complex-valued 1-form a such that

VXs = (a - X) s, s E rL. (9.43)

Prove that the curvature of such a connection is given by 9t = dot.

Hint: Calculate 91 A S X A Y using formula (9. 28).

13. (1) Suppose V is a connection on a vector bundle E -* M. Prove that the followingformula defines a connection VHom on the vector bundle Horn (E,E) - M:

(VXomF)a = VX(Fa) -F(VX(Y) (9.44)

for X E 3 (M), F E r (Hom (E,E) ), and a E rE.

(ii) Show that the associated covariant exterior derivative dHom satisfies

dE(F(Y) = (d HO41F)a+F(d E(y). (9.45)

(iii) Extend (9. 45) to A E Cl" (M;Ho

dE (A A i) = dHom

m (E,E)) and .t

A A g + (-1) P

E fIq (M; E) as follows:

A A dE).t. (9.46)

Hint: Write

A = Fknwk, SJA11J

9.4 Exercises 211

for Fk e r (Hom (E,E)) and s, E rE, as in (9. 11). Now apply (9. 14) and (9.45).

14. Let 9t e £12 (M; Hom (E,E)) be the curvature of a covariant exterior derivative dE ona vector bundle E -a M. Using the associated covariant exterior derivative dHom

defined in Exercise 13, prove the following concise form of Bianchi's identity:

dHom 9t = 0. (9.47)

Hint: Apply (9. 46) when A = 9t and t = a E r E. Since dE (dEdEG) = dEdE (dEa), theresult follows.

15. (Continuation of Exercise 13) Let 91 be the curvature of a covariant exterior derivativedE on a vector bundle E * M, and let A E f 1 (M; Horn (E,E) ).

(i) Show that the formula dE,Aµ = dEµ+A A.t, for t E S?P (M;E), defines anothercovariant exterior derivative dE. A on E -a M.

(ii) Let 9tA be the curvature of dE A. By computing 9tA A a for a E FE, show that

9tA = 9t+d4omA+AAA. (9.48)

Comment: (9. 48) plays a crucial role in deriving the Yang-Mills equations in Chapter 10.

16. The definition of the curvature 9t of a covariant exterior derivative dE says only thatdE (dE(;) = 91 A a, a E Sl° (M; E). Show that

dE (dE t) = 9t A µ, for all t E £" (M;E), (9.49)

by writing µ in terms of a local frame field {s,, ..., sm} as µ = Si A t , for some7)' E f"M.

17. The setup is the same as for Exercise 5. The local frame field { D 1, ..., D. } for thetangent bundle, where D. = (tp 1) . (a/ax1), induces a coframe field{tp* dx., ..., tp* dx"}, which will be abusively referred to as {dxt, ..., dx"}.

(i) By exterior differentiation of formula (9. 22), show

dd = I:drk; A dxk. (9.50)k

(ii) From the formula (9. 26) for the curvature forms { R; }, deduce that

RIm9, = R; DmnDq = Dmrg,-D9r ,.+V (9.51)k


9.5 Torsion-free Connections

We shall be concerned here only with connections on the tangent bundle. To restrict thisclass somewhat, we shall introduce the notion of a connection with "zero torsion."

9.5.1 Torsion of a Connection on TM

There is a canonical TM-valued I -form, namely. t E 0' (M; TM) defined by t (4) _for all tangent vectors . The torsion of a connection V on TM. and of the associatedcovariant exterior derivative d"", is the TM-valued 2-form

t=d"t.To evaluate t X A Y for any pair X, Yof vector fields, we use (9. 19):

d' (t - Y) X-d" (t-X) - Y - t [X, Y]

= d"' Y- X-d'"X Y-[X,Y]

= VxY- VrX- [X,YI.

(9.52)

(9.53)

A connection is called torsion-free if t = 0, that is, if VxY- Vt,X - [X, Y] = 0 forevery pair X, Y of vector fields. The condition of being torsion-free appears to be a verynatural one, because one would expect a "derivative" of an identity map (in this case, t)to vanish.

9.5.1.1 Example of a Torsion-free ConnectionThe Euclidean connection on the tangent bundle of R" (Example 9.1.3) is torsion-free,because if X = J:f,'a/ax;, Y = Y then

t X A Y = 14' (as;J/ax,) D, - Is;' (a4j/ax,) D; -1

,, lia/ax,l

and a routine calculation shows that all the terms on the right cancel out.

9.5.1.2 Interpretation In Terms of Christoffel SymbolsIn terms of a local frame field { D 1, ..., D" } induced by a chart, the Christoffel symbolsof a torsion free connection V, characterized by

(9.54)

s a t i s f y r-0.1Y = f .


Proof: Simply take X =Do and Y = DT in the equation VxY- V yX - [X,YJ = 0, andnote that [ DO, Dy] = 0. rX

9.5.2 A Method of Constructing Torsion-free Connections

Suppose is any local frame field for the tangent bundle, and {8', ..., 9" }

is the associated coframe field of ]-forms, that is, 9k ei = Sk.

(i) If { wk} is a matrix of I forms such that

dOk = O' A wk, (9.55)

where the matrix expression is d 0 = -W A 0, then the connection given by

dne, = ek n Wk (9.56)

(i.e., taking the {wk) as connection forms) is torsion free.

(ii) There is a unique matrix of 1 forms { wk } that satisfy both (9. 55) and (9. 57):

Wk + W4 = 0. (9.57)

where the matrix expression is W +W7 = 0.

Proof: (i) Observe that the canonical TM-valued 1-form, namely, t e il' (M; TM)defined by t (4) = 4, can also be expressed as

t = ek n 9k. (9.55)

To verify this, take an arbitrary vector field X = jtiei, and observe that

14'ekA D'ei = X.k i, k

Now suppose I" is the covariant exterior derivative on TM given by (9. 56). Using(9. 55), (9. 56), and (9. 14), we see that

d"t = d" {YekA6k} =7(d"ek) AOk+1ekAdOkk k k

{Yejnftyk} A9k+1ekn {0'AWk}k j k

_ YekAO'n {Wk - Wk} = 0.i, k


(ii)3 Uniqueness: Suppose that { wk} satisfy both (9. 55) and (9. 57). There are uniquefunctions { a k , . } and { b } such that

wk = a (Y; (9.59)

i

dok = 1 b A` A A', with b = -b . (9.60)J1 4j,.i

Then (9. 55) and (9. 57) imply

aki = -ai; (9.61)

-al;=b . (9.62)

Therefore

{by+bjkb'k;} = 2 aI].. (9.63)

This proves that the { cok} are uniquely determined by (9. 55) and (9. 57).

Existence: Starting from (9. 60), let us define {a J} by (9. 63), and define {wk} by(9.59).4 Then

ak = 2 { bki + blk} _ -a,

since b = -b 1; thus (9.61) and hence (9. 57) hold. To verify (9. 55), note that (9. 63)holds, and so, for any q and m,

e'A tok emA eq = {(0 .em) (wk-e,) - (0'-eq) (w; em)}

{ smak - Sgaim } = amq - aqm

= bk = 1 bk - bkmq 2 { mq qm}

3 The remainder of this proof follows Spivak [1979), Volume 2.4 Note for future reference that the Christoffel symbols of the connection are l". = 4.


= l7b O'A01-emAeqI, )

= dOk - emAeq,

which proves that dOk = O' A w .

i

9.5.3 Example of Constructing a Torsion-free Connection

Consider the 2-dimensional manifold M = R2 - {0}, on which 0' = dr,02 = h (r) do form a frame field for the cotangent bundle, where (r, 4') are polarcoordinates, and h is a nowhere-vanishing smooth function. Then do' = 0 andd02 = h' (r) dr n do = (h'/h) (0' A 02), so (9. 55) holds with

W = 0,o4 = 0,w = (h'/h)02,w2 = 0,

M

which gives a torsion-free connection, by part (i) of 9.5.2; however this connection doesnot satisfy (9. 57). The unique connection that satisfies (9. 57) is obtained using (9. 60);we see that

b h = 0, bit = h'/h = -b2I, b = 0;2 ii

I' =a.Z{bJ1 0

it follows that

W)= [0

-h'/hJ' (r) _[h'0/h

01.

Even though the connection is torsion-free, the Christoffel symbols are not symmetric in(i, j) because the frame field is not induced by a chart.

Since tax = r 0', the desired connection is given by

w,' = 0, tit = -(h'/h) 02, M = (h'/h) 02, 02 = 0, or

_=

-h'd' (D - ¢ (9.64)

63-

0 p2 [h'odo 0


9.6 Metric Connections

The "product rule" (uv)' = u'v + uv' is familiar to every calculus student, and leadseasily to the rule for differentiating the Euclidean inner product of two R"-valuedfunctions u (t) and v (t), namely, (uIv)' = (ui) + (iiIv'). We are now going togeneralize this idea in the obvious way to a connection V on TM, where the manifold Mhas an indefinite metric (.1.), as described in Chapter 7. We say that V is compatiblewith (.1.), or is a metric connection with respect to (.1.), if

X ((YIZ)) = (VXY1Z)+ (YIVXZ) (9.65)

for all vector fields X, Y, Z Note that the left side denotes the action of the vector field Xon the function formed by taking the inner product of Y and Z.

9.6.1 Example of a Metric Connection

The Euclidean connection on the tangent bundle of R' (Example 9.1.3) is compatiblewith the Euclidean metric, because if Y = Y,ytta/ax1, Z = Lr`'ka/azk, then

X ((YIZ)) = X

E (XV) a/ax1IZ)+(l11(Xc') a/ax,),

= ((VX)IZ)+()1VXZ)).

9.6.2 Fundamental Theorem of Riemannian Geometry

9.6.2.1 Short VersionGiven an indefinite metric on M, there is a unique torsion free metric connection on TM,called the Levi-Civita connection.

9.6.2.2 Elaborated VersionSuppose that M has an indefinite metric, and that { ell ..., em} is an orthonormal framefield for the tangent bundle, with orthonormal coframe field 10 .. ..., On } of 1 forms,that is, Ak e, = Sk (see Chapter 7). Then the unique torsion free connection on TMwhose connection forms satisfy (9. 55) and (9. 57) (see 9.5.2) is compatible with themetric, and is the only torsion free metric connection.

9.6.2.3 Corollary: Formula for the Christoffel Symbols of the Levi-Civita ConnectionIn terms of a local frame field { DP ..., DR} induced by a chart, the Christoffel sym-bols, characterized by

dTM DO Dy = 1, yDa, (9.66)a

9.6 Metric Connections 217

take the form

17

=9" { pygbo + Dpg6 - D8gHY}. (9.67)1"y

P -2

where (gaY) is the metric tensor, with inverse (g«b)

Proof of uniqueness: Given a torsion-free metric connection V, the lack of torsionimplies

(VXY- V),XIZ) = ([X,Y]IZ), (9.68)

and (9. 65) implies

X(YIZ) + Y(ZIX) -Z(XIY)

= (VXYIZ)+()'IVXZ)+(VYAX)+(ZIV,.X)-(VZXIv./-(XIVZ4

= (VXY+VYXIZ)+(Vyz-VZYIX)-(VZX-VXZI}).

It follows that

(VXY+ VA Z) = X(YIZ) + Y(ZIX) - Z(XI n - ([ Y, z] IX) + ([z, X] I Y). (9.69)

Adding (9. 68) and (9. 69) gives

2(VXYIZ) = X(YIZ)+Y(ZIX)-Z(XY)-([Y,Z]IX)+([Z,X]IY)+([X,Y]IZ), (9.70)

which proves that (VXYIZ), and hence VXY, are uniquely defined by the metric.

As for the existence part of 9.6.2.1, the truly industrious can verify that if VXY isdefined by (9. 70), then the axioms for a Koszul connection are satisfied. However,existence follows easily from the approach taken in 9.5.2, as we shall now see.

Proof of existence: We need only prove that the connection V whose connection matrix{ ak } is characterized by

d@k =@' A Qk, Qk + Qk = 0, (9.71)

is a metric connection. By (9. 23), VXe; _ e1 A t X, and so for any vector field X,

X(e,]ek) = X (±Srk) = 0.

On the other hand,


(VXe,lek) + (e,]VXet) _ (t X) (ejek) + (dk X)i i

(ok+Qk) X,

which is zero by (9. 71). Thus for an orthonormal frame field, we have shown that

X(e;Iek) = (VXe,Iek)+(e11VXek), Vi, k. (9.72)

By writing arbitrary vector fields Yand Zas Y = 1:ydei, Z = J:cmem, (9. 2), (9. 4).and (9. 72) imply

X(l1Z) = (VXYIZ)+(l1VXZ). (9.73)

Proof of 9.6.2.3: Taking X = Do, Y = Dy, and Z = D6 in (9. 70), and noting that forthese vector fields all the Lie brackets vanish, gives

2Y I"O g., = Dpgys + Dygps - Dsgpy,

from which (9. 67) immediately follows. u

9.6.3 Calculating a Levi-Civita Connection

Instead of using (9. 67), it is often better to write down an orthonormal coframe field,and then calculate the matrix of connection 1-forms as in Example 9.5.3. For example,consider, as in 9.5.3, the 2-dimensional manifold M = R2 - {0}, on which 0' = dr,02 = h (r) do form a frame field for the cotangent bundle, where (r, $) are polarcoordinates, and his a nowhere-vanishing smooth function. Evidently { 01, 021 is anorthonormal coframe field for the metric

(.I.) = 01®01+02®92 = dr®dr+h2 (d$0dO). (9.74)

Hence the Levi-Civita connection is precisely the one obtained in (9. 64), namely,

Q = 1 02 = 0 -(h'/h)02 = [h'd0 -h'dhQz Qz

L(h'/h) 02 0 01 2

(9.75)

Consider two special cases: Equation (9. 74) gives the Euclidean metric if h (r) = r,and then it follows that 9i = t'9 n 6 + do = 0, as we know already. As we saw inChapter 7, on the hyperbolic plane the metric is given by taking h (r) = sinhr, so

9.7 Exercises 219

9t=ana+da =

a = 0 - (cosh r) d$(cosh r) do 0

0 -sinhr (dr n d$)1 = 0 -6' n O2

Isinh r (dr A do) 0 J 8tAe2 0

9.6.4 Sectional Curvature

Suppose (M, (.0) is a Riemannian manifold of dimension at least 2. If {e1, ..., a"} isan orthonormal frame field with coframe field { e', ..., 0"}, then the sectionalcurvature of the 2-dimensional subbundle of TM -9 M spanned by the pair of vectorfields { e;, ej} is defined to be

K(e1, ej) = (9t n e; e, Aelef) = R;,1. (9.76)k

It can be proved that this depends only on the subbundle, not on the specific choice offrame field. In the case of a 2-dimensional Riemannian manifold, there is only one suchsubbundle, and hence only one sectional curvature

2 ' (9.77)121 - 212

For example, in the case of the hyperbolic plane, whose curvature was obtained in 9.6.3,

Ri = 9'n92= R121 = -1;

thus the hyperbolic plane has constant sectional curvature -1. As the notation suggests,if M is a 2-dimensional submanifold of R3 with the embedded metric, then the Gaussiancurvature K defined in Chapter 4 is the same as the sectional curvature; see Exercise 21below for the steps of the proof.

9.7 Exercises

18. Calculate the connection matrix of the Levi-Civita connection for R3 with a metric ofthe type

Fdx®dx+Gdy®dy+dz®dz,

where F = F (z) and G = G (z) are smooth functions, and calculate the curvature 91.The functions F and G are assumed to be nowhere vanishing.


19. Calculate the Levi-Civita connection for a 2-dimensional Riemannian manifold with alocal coordinate system (r, $) and the metric

dr ®dr+ u2 (do ®do),

where u = u (r, o) is a nowhere-vanishing smooth function, and calculate thecurvature 9t.

20. We saw in Example 9.6.3 that if a 2-dimensional Riemannian manifold has a local coor-dinate system (r, 4) and the metric

dr ®dr+ h2 (d$ ®do),

where h = h (r) is a nowhere-vanishing smooth function, then the Levi-Civitaconnection with respect to the orthonormal frame field 01 = dr, 02 = h (r) do isgiven by (9. 75). Derive the general formula for the sectional curvature of this manifold,and check that for a portion of the 2-sphere (here h (r) = sinr) we obtain K = I.

21. Suppose M is a 2-dimensional submanifold of R3 with the embedded metric. Prove thatthe Gaussian curvature K defined in Chapter 4 is the same as the sectional curvaturedefined in 9.6.4, using the following steps:

(i) Using the orthonormal coframe j ()I, 02} induced from a parametrization `N, andusing the first structure equation, show that the matrix of connection forms is

to = 0 -0,(9.78)° 0

where t)° is the 1-form appearing in Chapter 4.

Hint: No computations with ' are needed!

(ii) Using Gauss's equation dri° = -TI 'A 112 = -K0' A 02 and formula (9. 26) for 9t,show that Rig, = K.

22. Suppose M is a manifold, U is an open subset of R", and f: U x M -4 (0, oo) is a strictlypositive function, integrable in the first variable and C°' in the second variable, withsufficient regularity such that all the integrals which arise in this problem exist and arefinite, and that differentiation on M can be interchanged with integration over U. Alsoassume that f is sufficiently nondegenerate in p so that if I= (p) = logf (x, p), then thefollowing formula induces a Riemannian metric on M:

(WIM)P = $WIx(p)YI,,(p)f(x,p)dx, W,Ye 3(M). (9.79)U

Abbreviate the last expression to f (WI) (YI) fdx. Consider the formula, for real ct,U

9.7 Exercises 221

(aVYW1Z) = 1 {Y(WI) + 2 (Yl) (WI) } (ZI)fdx, Y, W,Ze 3 (M). (9.80)

U

(i) Prove that aV is a torsion-free connection on TM for all real a.

Hint: For any vector field Y, Yf = f (Yl).

(ii) Prove that °V is the Levi-Civita connection for this metric.

Hint: You only need to prove that the connection OV is compatible with the metric.

Remark: This family of connections, whose curvature is related to information loss in statistics,is discussed in the article by Amari, in Amari et al. [1987), and in Murray and Rice [ 19931. Thefunction f is the likelihood associated with the observation x and the parameter p.

23. Let d be the connection on the cotangent bundle corresponding, via (9. 35), to theLevi-Civita connection V on the tangent bundle of a pseudo-Riemannian manifold M.Prove that d satisfies the formula, for every smooth function h,

dXdh - Y = 2 { grad h (WY)) - ([ grad h, X) JY) - (X1 [grad h, Y]) }. (9.81)

Hint: See Chapter 7 for the definition of grad, and use (9. 70).

24. (Continuation) A formula for the Laplacian for functions on a pseudo-Riemannianmanifold M is given by

Ah = V,,dh e;, (9.82)

where is any orthonormal frame field, and d is dual to the Levi-Civitaconnection.

(i) Show that this agrees with the usual formula Ia2h/ax? in the case of theEuclidean connection.

(ii) Show that the formula

A(uv) = uev+veu+2gradu grade (9.83)

holds on any pseudo-Riemannian manifold M; grad was defined in Chapter 7.

(iii) Show that Ah = div µ(grad h), where µ is the Riemannian volume form, and div

was defined in the exercises to Chapter 8.

25. A connection r on the tangent bundle of an m-dimensional manifold M will be calledpyramidie if, at each point p in M, there exists a coordinate system {x', ..., x"'} inwhich the Christoffel symbols satisfy:

r= 0 for k!5 max { i, j }, (9.84)11


rj = ri

. (xt,...,xk). (9.85)

For example, in three dimensions, the only nonzero Christoffel symbols of a pyramidicconnection are r 1 , r , r 2, r 1, r22, and is a function of xt and x2 only. Provethat, in three dimensions, a metric pyramidic connection is "flat," meaning that thecurvature forms are identically zero.

Hint: Write down expressions for Dig, where g is the 3 x 3 metric tensor, and use the fact thatD.Djg = for i, j E { 1, 2, 3}.


The theory of connections has many versions, most of which are summarized andcompared in Spivak [ 1979]. The parts we have studied here are mainly due to E.Christoffel (1829-1900), E. Cartan (1869-1951), T. Levi-Civita (1873-1941),J.L. Koszul (1921-), and S. Chern (1911-). For a much deeper treatment, consultSpivak [1979] and Kobayashi and Nomizu [1963]. For a stimulating and readablesurvey of some applications, see the article by Chern in Chern [1989]. Thesuggested sequel to this chapter is "Natural Operations in Differential Geometry" byKollar, Michor, and Slovak (Springer 1993).

10 Applications to Gauge Field Theory

The quantization of classical electromagnetic theory (see the appendix to Chapter 2)leads to the description of the photon, the quantum of electromagnetic radiation. One ofthe aims of quantum field theory is to explain all elementary particles as quanta ofappropriate classical field theories. In the search for such field theories, non-Abeliangauge theories have become leading candidates since their introduction in 1954 by Yangand Mills. Only in the 1970s did it become generally recognized that the mathematicaltheory of connections on vector bundles is the proper context for gauge theory.

In this chapter we shall start out by giving a geometric description of the Yang-MillsLagrangian, which must be minimized in order to construct the desired classical field.The minimization is actually carried out here using a special class of connections whichare called "self-dual." We reformulate electromagnetism as an Abelian gauge fieldtheory, and give a detailed account of a non-Abelian SU (2) gauge theory over S4,including the formula for "instantons," the connections which minimize the Yang-MillsLagrangian. Much of this chapter is based on Lawson [ 1985).

10.1 The Role of Connections in Field Theory

Imagine a structured particle, that is, a particle located at some point p in afour-dimensional manifold M ("space-time"), and with an internal structure, or set ofstates ("spin", etc.) labeled by elements of a complex Lie group G (e.g., SU (2) ). Inpractice we cannot observe this internal structure but only the action of G on somecomplex vector space V Thus the total space of all states of such a particle isrepresented by a vector bundle E over M with fibers isomorphic to V

An external field will cause the internal state of the particle to change as the particleflows along a vector field X in space-time. The internal state of the particle at position pin M is reflected by the value of a section a of Eat the point p. Therefore the external


performed in 1960 by Chambers, proved that the electromagnetic potential does play arole, even in the absence of a field. A coherent beam of electrons is reflected around aclosed path encircling a solenoid, which is considered as a perfectly insulated tube.Although the field outside the tube is zero, the phase shift caused by self-interaction ofthe beam is found to vary with the intensity of current in the solenoid.

The following table may be a useful reminder:

Differential Geometry

connection forms

curvature forms

Physics

gauge potential

field strength

Table 10.1 Comparison of mathematics and physics terminology

10.2 Geometric Formulation of Gauge Field Theory

The technical preliminaries involve some more material on complex Lie groups. andsome constructions of metrics on vector bundles.

10.2.1 Constructions with Complex Us Groups

Suppose G is a Lie subgroup of the complex general linear group GL°, (C) discussed inChapter 3; in other words G is a collection of invertible complex m x m matrices.closed under matrix multiplication, and with a smooth manifold structure. The tangentspace at the identity of G, under the operation of bracket of vector fields discussed inChapter 2, is known as the Lie algebra of G, which we will denote by L(G). In thischapter, we shall only be performing calculations with U (1) and SU (2), alreadymentioned in Chapter 3, whose Lie algebras are described below:

10.2.1.1 The Circle Group U ( 1)The unitary group U (1) is simply the set of complex numbers z with modulus 1, thatis, the circle in the complex plane, under ordinary multiplication of complex numbers.In electromagnetic theory, this group models the polarization of a photon. As amanifold, it may be identified with the circle S', and the tangent space at 1, which is ofcourse one-dimensional, is obtained by differentiating the curves t -+ efO' at t = 0 forall real numbers a, giving as the Lie algebra the complex line iR (here i is the complexnumber ); thus L (U (1)) = W.

10.2.1.2 The Special Unitary Group SU (2)This is a 3-dimensional manifold (over the reals) consisting of the 2 x 2 complexunitary matrices of determinant 1. Its importance in physics is that it is used to modelthe "isotopic spin" of particles related to the strong nuclear force. The reader may

226 Chapter 10 Applications to Gouge Field Theory

check using the hint in Exercise 6 in Section 10.5 that the Lie algebra L (SU (2) )consists of the span of the Pauli matrkest:

001,L0 a],Lioff.

In Section 10.3 we shall see how the algebra of quaternions provides concise andefficient machinery for performing calculations involving S U (2).

(10.1)

10.2.2 G-Bundles and G-Connections

Suppose G is represented as a set of m x m complex matrices, acting on a complexm-dimensions) vector space V; this means that, for every g e G, there is a linearmapping v -+ gv from V to V, with the property that 9, (920 = (9192) V'

A complex vector bundle n: E -4 M, whose fibers are isomorphic to V, is called aG-vector bundle if its transition functions (see Chapter 6) all belong to G. A Koszulconnection V on a G-vector bundle rt: E -4 M is called a G-connection if the associatedconnection matrix (with respect to any local frame field), when viewed as a 1-form onM with values in C"m, takes values in L (G). For example, the connection matrix ofan SU (m)-connection must satisfy o>k + wk = 0 (here the bar denotes complexconjugation) and c0 + ... + W. = 0, since the Lie algebra of SU (m) consists of theskew-Hermitian matrices whose trace is zero.

10.2.3 Action of G on the Fibers, Preserving a Metric

Given a local trivialization 0. for the vector bundle at r e M, G can be made to act onthe fiber E, = V in the obvious way, namely,

gnat (r, v) = 4;' (r, gv). (10.2)

Of course, this action depends on the trivialization; for example, in another localtrivialization gyp, with transition function gpa (r) as in Chapter 6, (10. 2) becomes

g4A' (r, w) _ P (r, gpa (r) ggnp (r) w). (10.3)

Nevertheless since {gp, (r) ggctp (r) g e G} = G, the resulting group G, of lineartransformations of E, does not depend on the trivialization. We are interested here in thecase where n: E -* M has a metric r -* such that, in every fiber, G, is identicalwith the set of isometries of E,, that is, the nonsingular linear transformations thatpreserve the metric; in other words G is the set of g e L (V -4 V) such that, for any r,

t These are IF- -I times the Pauli matrices as often given in the physics literature.


(gIg E = (II E. Vt E E,, (10.4)

where of course the meaning of depends on the trivialization.

102.4 The Gauge Group

Suppose n: E -4 M is a G-vector bundle with a metric r -4 The gauge group GEconsists of the set of sections of Hom (E,E) which are vector bundle isomorphisms(see Chapter 6), preserving the metric; the latter means that, for a E FE and 4) E GE,

($(yl$a)E = (a1a)E. (10.5)

In view of (10. 4), 0 locally takes the form of a smooth mapping from an open set in Mto G. 0 E GE takes a G-connection V to a G-connection VO, defined by

VXa = O (VX (O-'a) )- (10.6)

In Exercise 1 in Section 10.5 the reader may check that if V is a G-connection, then so isV9.

In terms of the covariant exterior derivatives dE and dE' for V and V, respectively,

dE.e(dE.oa) = dE.e(4)dE(4)-'a)) = 4)dE(dE(4-Ia))

= ctV Aa = $(9Vn0-1a). (10.7)

The elements of the gauge group are called gauge transformations; formula (10. 7)describes how the curvature of a G-connection changes under a gauge transformation.

10.2.5 Metrics on Various Vector Bundles

Suppose n: E -* M is a G-vector bundle with a metric r -* over ann-dimensional oriented Riemannian manifold (M, (4.)).

10.2.5.1 A Metric on the Vector Bundle Hom (AMTM,E)A metric on the vector bundle Horn (TM,E), whose sections are the E-valued I-forms,is given by the formula

(gla.) (1) = (µ eaI ? ea)E, (10.8)1SUSn

for g and. in S2' (M; E) , where { e1, ..., en} is an orthonormal frame field for TM in aneighborhood of r. The fact that this does not depend on the choice of { e , ..., en } is aspecial case of the exercise in Chapter 7 on constructing a metric on Horn (E,E').

228 Chapter 10 Applications to Gauge Field Theory

Next we seek a metric on the vector bundles Hom (AQTM,E) for q = 2, ..., n.Suppose that t and k are in S24 (M; E) for some 2:5 q:5 n ; for r E M, p (r) and A. (r)take values in L (AgTrM -4 E), which by a result in Chapter 1 can be viewed as the qthexterior power of L (TrM - Er). We learned in Chapter 1 how to extend an innerproduct on a vector space, in this case L (TAI -4 Er), to one on all of its exteriorpowers. Applying this construction here allows us to define a metric (WW q) for all .tand k in 1Zq (M;E). To be specific: If {s,} constitutes an orthonormal frame field forn: E -+ M in a neighborhood of r, and if {E1, ..., En} is an orthonormal coframe fieldfor the cotangent bundle in a neighborhood of r, then the set of E-valued q-forms

{SiA(Wa:i _ 1,aEIg}, (10.9)

where (,)a = F10) A ... A Ea (q), and Ig is the set of ascending multi-indices a with1 5 a (1) < ... < a (q) : n, constitutes an orthonormal frame field for the vector bundleHorn (A9TM,E) under this new metric.

Finally a "global" inner product of E-valued q-forms may be defined by the formula

(Lµ) = J(21Iµ)(g)p, (10.10)M

where p is the canonical volume form, at least when A and µ are zero outside a compactset.

10.2.6 The Norm of the Curvature

In the exercises to Chapter 7, we saw that a metric on Horn (E,E) - M is given by

(FlF')HO11 = Y(F(si)IF'(si))E, (10.11)i

where {s, } is any orthonormal frame field for n: E - M at r; here F and F' are any twosections of Horn (E,E) -4 M (this metric does not depend on the choice of { s1}).

With the construction given in 10.2.5.1, a metric on Horn (AqTM; Horn (EE) ),denoted r q, may be defined for all 1 S q 5 n. For example, the curvatureSR = 9tv of a connection V is a 2-form on M with values in Horn (E,E), and thisconstruction allows us to define

e

II `RV I12 =(gtVjgtv)H°m. 2

I r15a<05n i

(10.12)

The reader has the opportunity to check the last identity in Exercise 2 in Section 10.5.


10.2.6.1 The Norm of the Curvature Is Gauge InvariantFor any gauge transformation 0 (see 10.2.4) and any G-connection V,

II 9t°'II = II 9t°II;.

Proof: Since $ preserves the metric, { 4) s;} is an orthonormal frame field fortt: E -4 M, and for each a, 5

(10. 13)

AepII2 = VAsi ea^ebII2

However, 9Z°O n Si (9Z° n 4-'s;) by (10.7). it

10.2.7 The Yang-Mills Lagrangian

We want to find out which connections, and hence which field strengths, are consistentwith the Principle of Least Action. For this we will assume that, in the terminology ofChapter 7:

M is a compact, oriented, Riemannian manifold (for example, M = S4 with somemetric);

1t: E -4 M is a G-vector bundle with a metric r

The Yang-Mills Lagrangian, or Yang-Mills action, is the mapping from the set ofG-connections on it: E -+ M to R* given by

L(V) = 2 f jj9iVII2p,M

(10.14)

where p is the canonical volume form on the compact, Riemannian manifold M, and thenorm under the integral is as in (10. 12). It is important to note that, by 10.2.6.1,L (Vs) = L (V) for every gauge transformation 0 (see Section 10.2.4); in other words,the Lagrangian is gauge invariant. Do not be put off by the abstractness of this integral;more concrete versions will follow.

We are interested in finding connections that minimize the Yang-Mills action. To thisend, a G-connection V will be called a Yang-Mills connection if it is a stationary pointof L (V)(a stationary point means a local maximum or a local minimum). In that case

its curvature 9Z° will be called a Yang-Mills field.

10.2.8 Example: Maxwell's Equations as a Gauge Theory

Let us to try to rephrase Maxwell's equations (see the appendix to Chapter 2) in terms ofgauge theory. Recall that we encoded both the electric and the magnetic field in a2-form r = W E A dt + me on M = R4 with the Lorentz metric (so in this case, M ispseudo-Riemannian rather than Riemannian), and showed that solving Maxwell's


equations amounts to finding a I-form a, called the electromagnetic potential, such thatda = rl, * d (* T)) = 4ny, where y is the 4-current; of course if a solves the equation,then so does a + db, for any smooth function b.

Suppose n: E -a M is a complex line bundle, with transition functions in the groupU (I) (see Section 10.2.1.1); thus n: E - M is a U (1) bundle. Take the metric onn: E M defined by

(ak)P = Re (a (p) i (p) }, (10.15)

referring to the real part of a complex product (the right side does not depend on thetrivialization). The complex Lie group that preserves this metric is U (1); thus a gaugetransformation consists locally of multiplication in the fibers by a function of the formm (p) = exp (-ib (p)) e U (1), for some smooth real-valued function b on an openset in M.

Take a local trivialization Ou for the complex line bundle. Since every section is of theform a = sf, for some complex-valued function f, with respect to the frame fields (p) = 4)u) (p, i), we may define a connection V by

dE(sf) = sniaf+sndf. (10.16)

The reason for using is instead of a is that we want the "gauge potential" (i.e.,connection form) to take values in the Lie algebra iR of U (1); in other words, V is aU (1)-connection. Notice that under the gauge transformation induced by $ = e 'b,(10.6) gives

dE.4(sf) = e+'b(dE(e'bsf)) = sn (iaf+ifdb) +sndf. (10.17)

Thus the new gauge potential is ia4 = ia+ idb.

The formula for the curvature form ("field strength") is F = -a A a + ida fromChapter 9, but the a A a term is zero because a is an ordinary 1-form on M, and we areleft with a field strength of

iT) = ida = i (0E A di + QB). (10.18)

The Yang-Mills Lagrangian is now simply the action of the electromagnetic field,namely (in the absence of a current),

L (V) f11 dalI2p, (10.19)M

10.3 Special Unitary Groups and Quatemions 231

where p = dx A dy n dz A cdt (the minus sign can be cancelled out by suitable choiceof volume form). If a1 = a + to is a family of potentials, corresponding to connectionsVV then

II da,Il 2 = II da112 + 2r(daldp)H0 2 +:211dP112

= fL (V,) I 1 _ o = J (daadp)H0m. 2p f (* d (* da) IP)H°r".'P

M M

Here the inner product inside the integral turns out to be the Lorentz inner product of1-forms, and the last identity, which the reader is invited to check in Exercise 4 inSection 10.5, assumes that P is nonzero only inside a compact region. If Vo is aYang-Mills connection, then the last integral must be zero for every possible variationP; therefore in vacuo its field strength Tl satisfies

*d(*Tl) = 0. (10.20)

which is precisely Maxwell's equations in the absence of a current. To summarize:Minimizing the Yang-Mills Lagrangian leads to a set of differential equations for thefield, that is, for the curvature of the connection.

10.3 Special Unitary Groups and Quaternions

The purpose of this section is to introduce the formalism of quaternions, in order tocarry out calculations on SU(2)-bundles in a streamlined fashion, withoutmanipulating a lot of 2 x 2 complex matrices. The following account follows Atiyah[1979].

10.3.1 Auatemions and the Group Sp (1)

Just as the complex numbers C are formed from the real numbers R by adjoining asymbol i with i2 = -1, so the quaternions H are formed from R by adjoining threesymbols i, j, k satisfying the identities:

i2=j2=k2=-1ii = -ii = k, jk = -kj = i, ki = -ik = j.

Thus a general quaternion is of the form

z = zl+z2i+z3j+Z4k,

(10.21)

(10.22)

(10.23)

where z1, z2, z3, z4 are real numbers. Addition of quaternions is accomplished by simplyadding up the respective coefficients of 1, i, j. k. Multiplication, which is associative


(see Exercise 8 in Section 10.5) but not commutative, requires the use of (10. 21) and(10. 22). The conjugate quaternion z is defined by

zI -z2i-z3j-z4k, (10.24)

and by virtue of (10. 22), conjugation satisfies yz = (z) (y) . The identities (10. 21)and (10. 22) imply that

Isis4

and this quantity, denoted IzI2, is zero only for z = 0. It makes sense to define theinverse of a nonzero quaternion by

zt=z/IzI2. (10.26)

If two quaternions y and z have unit norm, that is, Iyl = Izl = 1, then so does theirproduct, because

IyzI2 = (yz) (Y z) = yzZy = ylzl2y = yy = 1.

Thus the set of quaternions of unit norms forms a multiplicative group, denoted Sp (1),in which z` 1 = z by (10. 26). Moreover Sp (1) may be identified with the differentia-ble manifold S3 by virtue of (10. 25).

10.3.2 Identification of Sp (1) and SU (2)

The groups Sp (1) and S U (2) are isomorphic; moreover the Lie algebra L (SU (2) )may be identified with the purely imaginary quaternions

Im (H) = {z1+z2i+z3j+z4k:z, =0}. (10.27)

Proof: Every quaternion has a unique expression of the form:

z = z,+z2i+z3j+z4k=u,+uaj,

where u, = z, + z2i, and u2 = z3 + z4i. This identifies each z e H with the pair ofcomplex numbers (u,, u2) a C2. Consider the quaternion multiplication z - zv wherev = v, +v2j, with (v,, v2) a C2:

zv = (u1 +u2j) (vl +v2j) = u,v, -u2v2+ (u,v2+u2v,)j

In other words, the multiplication z -+ zv amounts to multiplying the vector (u,, u2)on the right by the 2 x 2 complex matrix

10.4 Ouaternion Une Bundles 233

(10.28)

For v e SP MI V I i 1 + v2 v2 = 1, so this matrix has determinant I ; moreover matricesof this form make up the whole of SU (2) by Exercise 5 in Section 10.5. Thus the mapfrom H to C2,2 which sends (v 1, v2) to the matrix (10. 28) is one-to-one and onto fromSp (1) to SU (2), and preserves the group multiplication.

In particular, when v takes the values i, j, and k, respectively, then the correspondingforms of the matrix (10. 28) are precisely the Pauli matrices (10. 1). The reader mayverify that the three Pauli matrices behave under multiplication in exactly the same wayas the quaternions i. j. and k do in (10. 21) and (10. 22); this shows that we may identifythe Lie algebra of SU (2) with the purely imaginary quaternions. u

To summarize: We are now in the position where calculations involving SU (2), and itsLie algebra (spanned by the Pauli matrices), can be mimicked by calculations involvingunit quaternions, and the purely imaginary quaternions, respectively.

10.4 Quaternion Line Bundles

Recall from Chapter 6 that a complex line bundle over a differentiable manifold Mmeans a complex vector bundle whose fibers are copies of the complex line. Likewisewe may define tt: E -4 M to be a quaternion line bundle if the fibers are copies of H,the algebra of quaternions defined in Section 10.3.1. In checking the differentiabilityconditions in the definition of a vector bundle, as in Chapter 6, we may treat the fiberssimply as 4-dimensional real vector spaces. However, the transition functions { guu. }take values in the set of nonzero quaternions, acting on the right by quaternionmultiplication; in other words, for p e M,

guu' (p) (v) = vguu. (p), v e H. (10.29)

Evidently each transition function guu, is H-linear on the left, in the sense that

guu (p) (zv) = zvguu (p) z r= H. (10.30)

Note that the group Sp (1) of unit quaternions, defined in Section 10.3.1, acts on H bysimple quaternion multiplication. We shall call a quaternion line bundle anSp (1)-bundle if its transition functions are all unit quaternions.


10.4.1 Example: a Guatemion Line Bundle over the 4-Sphere

This comes from Lawson [ 1985]. Let M = S4 (we really have in mind afour-dimensional space-time which has been "compactified" by adding a "point atinfinity," which is more convenient to work with than R4 because integrals will now beover a compact manifold). Following along the lines of one of the exercises to Chapter5, we see that it is possible to pick an atlas consisting of the pair of charts (U, (p) and(V, W), where

(XI, ..., x4)U = S4- {(1,0,...,0)},ip(x0,...,x4) =

1-xo

V=S4-{(-1,0,...,0)},W(xo,...,X4) =(XV -X2, -X3, -X4)

1 + Xo

The change-of-chart map cp X V- : R4 - 101 -+ R4 - 10) has the property that if weidentify z e R4 with Z' + z2i + z3j + z4k e H as in (10. 23), then

'' -Z3, -z4) _ z

tp W (z'. z2, z3, z4) _ (Z_Z2,

Iz12Izl2,

(10.31)

where z is the conjugate quaternion as in (10. 24).

We may now construct a quaternion line bundle tt: E - M "of instanton number 1"(this is a topological classification defined in Section 10.7.3) using the vector bundleconstruction theorem given in Chapter 6. To apply this theorem, it suffices to specifythat the fiber of this vector bundle is H, the open cover of e is { U, V}, and thetransition function is

guv(W ' (z)) v = vz/Izl, (10.32)

where z e W (V) and v are both regarded as elements of H. In the other coordinatesystem {y', y2, y3, y4} induced by tp, where y = Z/Iz12 = z-1 by (10. 31), (10. 32)becomes

guv ((p-' (Y) ) v = vy/lyl .

Since z/Izl is a unit quaternion, this is an Sp (1)-bundle.

(10.33)

10.42 Connections on Ouaternion Line Bundles

Suppose it: E -4 M is an Sp (1)-bundle with the metric r -9 induced by quater-nion multiplication in the fibers, namely,

(alT)p = Re { a (p) T (p) } (10.34)

10.4 Quatemion Line Bundles 235

(the right side does not depend on the trivialization). The group of transformations thatpreserve the metric in each fiber is the group Sp (1) of unit quaternions (which wouldcorrespond to SU (2) if we viewed the bundle instead as having 2-dimensionalcomplex vector spaces as fibers). The gauge group consists of transformations whichtake a section a to 4)a, where 0 is locally a smooth mapping from M into the unitquaternions, and (4)a) (p) is simply the quaternion product 0 (p) a (p) computed inthe fiber over p, in some trivialization.

We are interested in Sp (1)-connections on rt: E -i M in the sense of Section 10.2.2; inother words, the connection form (see below) is a 1-form on M taking values in Im(H),the purely imaginary quaternions defined in (10. 27), since Im(H) corresponds to theLie algebra of SU (2) by 10.3.2. Moreover we require that the connection is linear withrespect to right multiplication by constant quaternions, meaning that the Leibniz rule forthe covariant exterior derivative, namely,

dE(p Aw) = dEµnw+ (-1)degµ tAdw, (10.35)

now holds for every E-valued p-form p., and every differential form won M with valuesin H (replacing R by H is the only extension of what was presented in Chapter 9).

Consider the case where µ is a nonvanishing section s of E, that is, an element ofi2° (M; E), which by itself constitutes a local frame field for E. If co is an H-valued0-form, that is, a function f = fi + if2 +jf3 + kf4 from M to H. then we obtain

dE(snf) = dE(sf) = (dES)f+sndf. (10.36)

Since the fibers of the bundle are only "1-dimensional" in the quaternion sense(although 4-dimensional in the real sense), the E-valued 1-form dEs is expressible asS A A for some Tm(H)-valued 1-form A on M, which corresponds to the "connectionmatrix" described in Chapter 9, and which will henceforward be called the gaugepotential with respect to the local frame field {s}.Thus the formula for differentiating ageneral section a = sf of a quaternion line bundle is

dE(sf) = sAAf+sndf. (10.37)

Note that if the gauge potential A is expressed in local coordinates, then the product Af isthe H-valued 1-form:

(Atdx'+A2dx2+A3dx3+A4dx4) (f1+if2+jf3+kf4). (10.38)

236 Chapter 10 Applications to Gouge Field Theory

10.4.3 Guidelines for Exterior Calculus with /valued Forms

The iteration rule and Leibniz rule for exterior differentiation of H-valued forms are thesame as for real-valued forms. However, since quaternion multiplication is notcommutative, we have

1.Aµ* (_1)pq(tAa.)

in general, for an H-valued p-form A, and an H-valued q-form p. For example, possiblyAf * fA in (10. 38). Ways to skirt around this difficulty will be apparent in Exercise 15in Section 10.5.

10.4.4 Curvature as "Field Strength"

Given an Sp (1)-connection on a quaternion line bundle it: E -> M, the curvature 9t,according to Chapter 9, is an H-valued 2-form characterized by

dE(dEa) = 91A0, ae n°(M;E). (10.39)

With respect to the local frame field (s}, the curvature form is the H-valued 2-form Fwhich satisfies:

dE (dES) = Fns. (10.40)

Actually it follows from (10.42) below that F is an Im(H)-valued 2-form. In terms of alocal coordinate system { x 1, ..., x') for M,

F = I Fap (dx(I A dxP).15 a< 5"

(10.41)

F, or the collection of Im(H)-valued functions { Fap }, is referred to as the field strengthassociated with the gauge potential AIdxl + ... +A"dx". The reasoning behind thisphysical interpretation of curvature was sketched in Section 10.1.

The reader may check in Exercise 10 in Section 10.5 that the formula in Chapter 9 forthe curvature forms in terms of the connection forms is now expressible as

F =AAA+dA. (10.42)

The fact that A is Im(H)-valued means that A A A is not zero in general, unlike the caseof real-valued 1-forms; see Exercise 18 in Section 10.9 for an example.

10.4 Guatemlon Line Bundles 237

10.4.5 Example: A Quatemlon Line Bundle over the 4-Sphere ContinuedThe notation continues from Section 10.4.1. The local trivializationl) :71' (U) -) U x H induces a local frame field { s} for E over U. wheres = Oul (p, 1). Thus an arbitrary section has a local expression

a = s(f,+if2+jf3+kf4), (10.43)

where the {fa} are real-valued. We would like to compare the local expressions for thegauge potential of an Sp (1)-connection in two different trivializations of the bundle.This will also tell us how A behaves under "gauge transformation." Suppose that over Uwe have

dE(a) = dE(sf) =sAAf+sndf, (10.44)

while over V. if t = 4V ,1 (p, 1), we have

dE(a) = dE(tg) = t ABg+t Adg. (10.45)

10.4.5.1 Transformation Rule for Gauge PotentialsThe formula for the gauge potential in the other trivialization is

B = u'Au + udu, (10.46)

where u = z/Izl = y/Iyl, and z = W (p). Y = cP (p)

10.4.5.2 Remarks

The formula (10.46) is a quaternion version of the identity 6 = g ' dg + g-' Qg,derived in the exercises to Chapter 9, relating two sets of connection forms under achange of local trivialization, where g denotes the transition function.

B is an Im(H)-valued 1-form, just as A is; check this in Exercise 11 in Section 10.5.

In terms of the respective coordinate systems on U and V, we could write the gaugepotentials as

A = A,dy'+A2dy2+A3dy3+A4dy4,B = B1dz'+B2dz2+B3dz3+B4dz4.

Here each A. is an Im(H)-valued function of {y', y2, y3, y4}, etc. In Exercise 14below, the reader can practice using the formula (10. 46) in going from y- toz-coordinates, where z = z' + z2i + z3j + z4k = y ' = (y' - y2i - y j - y4k) /Iy12.

Proof: By expressing a (p) in both trivializations, and using the definition of thetransition function, we see that

4'U(a(p)) (Ov(a(p)


(P,f (P)) = (P, g (P) u).

where u = z/izI = y/Iyl, by (10. 32). Thus a = tg = sf = sgu, and taking g a 1shows that t = su. Since S A Af + s A df = t A Bg + t A dg by (10.44) and (10. 45),we have on taking g : 1, f = u, that

SAAa+SAdu = sAUB

and therefore uB = Au + du; now the result follows on premultiplication by u, theelement of Sp (1) inverse to u. n

10.4.6 The Yang-Mills Lagranglan on a Quaternion Line Bundle

As the reader has guessed by now from our propaganda for quaternions, the case whereG = SU (2), acting on its Lie algebra, is going to be handled using a quaternion linebundle E, where the group is Sp (1) = SU (2). Forgetting about compactness for amoment, take M to be R4 with a metric

(. 1.) = ; gOdx® ® dxo. (10.47)1506 54

Let us take a local frame field { s } for E such that 1 s (p) 12 = 1; now the connection andits curvature are represented by equations (10. 37) and (10. 40). A special case ofExercise 2 in Section 10.5 shows that, in terms of the field strength defined in (10.41),and the coordinates { x', x2, x3, x4 },

11 vII2 =I

sai54elFaB; (10.48)

here the {F°O} are the Im(H)-valued functions given by

Fall= g"F 5ga0, (10.49)

r.

where (gY6) = (g0p)-1. Consequently

L (V) = 2 j ( F0llF0O) igi tie (dxl A dx2 Adx3 A dx4). (10.50)

M Sa< 54

10.5 Exercises

1. Verify that, in the setting of Section 10.2.4:

10.5 Exercises 239

(i) If V is a metric connection, then VO defined by (10. 6) is a metric connection forevery gauge transformation 4).

(ii) If V has connection matrix w with respect to some local frame field { st, ..., sm), andif the matrix g = (gJk) represents the gauge transformation tp with respect to this framefield, then Co. = gd (g-1) +gwg t is the connection matrix of VO.

Hint: Compare with one of the exercises for Chapter 9.

(iii) Deduce from (ii) that if w is an L (G)-valued 1-form, then so is w#; in other words,if V is a G-connection, then so is VO.

2. (i) Check the last identity in formula (10. 12), namely,

II91vII. _ AS eanepll2. (10.51)a< ;

(ii) Suppose that { D1, ..., D } is a local frame field for TM -> M, and { t, } is a localframe field for it: E -4 M, with

(DaIDp) = gap, (t,ItQE = h,j.

Prove that if (8T&) _ (gap)'' and (hkm) = (h,j)-t, then in terms of the curvaturetensor defined in Chapter 9,

II `II 2 = 2 8°p8ys hkmRka.RJmpbhrj.y.S i.J,k,m

(10.52)

Note: The comparison between (10.51) and (10. 52) may serve to remind the reader of theadvantages of coordinate-free notation!

3. Suppose the Riemannian metric (.1.) on TM is replaced by 4)2(.I.) for some smoothfunction 0. Show that the integrand in the Yang-Mills Lagrangian (10. 14) is multipliedby 0" "4, where n is the dimension of M, and hence is unchanged when M is4-dimensional.

Remark: The last property is called the conformal invariance of the Yang-Mills Lagrangian.

4. Suppose a and 0 are real-valued functions on R4 with the Lorentz inner product, wherethe canonical volume form is p = dx A dy A dz A cdt.

(i) Using a formula at the end of Chapter 1, and the rules for exterior differentiationgiven in Chapter 2, prove that

(dc4do)p = (*d(*da)I0)p-d(OA*da). (10.53)

(ii) Suppose that P is a compact 4-dimensional submanifold-with-boundary of R4, suchthat (3 vanishes on the boundary aP and on the complement of P Using Stokes'sTheorem, prove that


j(doddR)p = (10.54)

5. Prove that SU (2) consists of the complex matrices of the form

(10.55)

where u and v are complex numbers with I u12 + I v12 = 1.

Hint: A general matrix in C2 x 2 may be written in the form 1-Y uy = z = I when this matrix is in SU(2). v

Vzu

for y, z e C; prove that

6. Prove that the tangent space at the identity of the Lie group S U (2) (i.e., the Lie algebraof S U (2)) has as its basis the Pauli matrices listed in (10. 1).

Hint: In the light of Exercise 5, consider the following three curves in C2 X 2 and their derivativesat 0:

t- e,r 0 t 0

e 0 0(10.56)

Note that the derivatives of these curves at 0 are linearly independent, and so they must span aspace of dimension 3; also each tangent space to SU (2) is 3-dimensional.

7. Define the adjoint action of a Lie group G on its Lie algebra L (G) by the formula

Ad8(4) = G, (10.57)

where 4 e TeG (i.e., the tangent space at the identity of G), and tp (t) is a curve in Gwith tp (0) = e, 0 (0) = . Calculate the adjoint action of a general element (10. 55)of SU(2) on each of the Pauli matrices listed in (10. 1), using the curves described in(10.56).

8. Check that multiplication of quaternions is associative; that is, check using (10. 21) and(10. 22) that for any quaternions x, y, z, we have x (yz) = (xy) z.

9. Check the formula (10. 31) for the change-of-chart map for the 4-sphere(p*W-':R4- {0} -*R4- {0}.

10. Derive the formula F = A A A + dA that is, (10. 42), for the field strength in terms ofthe gauge potential on a quaternion line bundle, by using formula (10. 35) and the factthat dEs = s A A.

10.5 Exercises 241

11. Suppose that two H-valued 1-forms A and B are related by the formula (10. 46). Provethat if A is purely imaginary, that is, (1/2) (A -A) = A, then so is B.

12. Suppose z denotes the quaternion Z I + z2i + z3j + Z4k. Compute dz A dz and show thatit is not zero, where dz = dz' + dz2i +dz3j + dz4k. What is the field strengthassociated with the gauge potential A = (dz - di) /2?

13. Let z and dz be as in Exercise 12.

(i) Use the formula dl zl 2 = d (iz) to show that

dlzl =zdz+ (dz) z21 zI

(ii) Show that if u = z/Izl, then

(10.58)

[dzI - z (didu =

)(10.59)

IZI Iz13

(iii) Show that if y = z-1, for z * 0, then dy = -z 1 (dz) z-r .

14. Consider the formula (10. 46) in the case where the gauge potential A describes an"anti-instanton" (see Atiyah [ 1979], p. 21):

A = 1 [ydy - (dd) yl+lyl2

(10.60)

Using the results of Exercise 13, show that the corresponding gauge potential B is givenin terms of the quaternion variable z = y-1 by

B - 1 zdz - (di)[ I+IzI2

Z(10.61)

15. In the example described in Section 10.4.5, calculate the field strength of the gaugepotential C = ydj - (dy) Y, using the following steps:

(i) By exterior differentiation of 1y12, show that the real 1-form co = d (1y12) satisfies

(dy) Y = w-ydd.(ii) Using (i), show that

dC = MY AdY, (10.62)

CAC = 20oAC-4IyI2dyAdY. (10.63)


16. In 10.4.5.1, we compared the gauge potentials A and B computed on two different localtrivializations of a quaternion line bundle. Show that the field strengths F and P withrespect to these local trivializations are related by the formula

P = uFu. (10.64)

Hint: Compute B A B + dB using (10. 46); uu = I udu = - (du) u.

Remark: This is a quaternion version of the result = g-tRg" in the exercises to Chapter 9.concerning the way that curvature forms transform under a transition function g.

10.6 The Yang-Mills Equations

The Yang-Mills equations are the conditions on the curvature of a connection for it tobe a stationary point of the general Yang-Mills Lagrangian; thus in the special case ofthe Yang-Mills Lagrangian for the electromagnetic potential, discussed above, theYang-Mills equations are simply Maxwell's equations. To state these equations we needthe notion of codifferential.

10.6.1 The Codifferentlal

The codifferential 8E associated with a covariant exterior derivative dE on it: E -* Mis the mapping Qq+ 1 (M; E) -, f2g (M; E), for q = 0, 1, ..., n, defined by the formula

(A..SE t) (10.65)

for g e fg(M;E) and X E C29 (M;E), where the inner product is the one defined in(10. 10). As we shall see in Exercise 17 in Section 10.9, the codifferential stands insimilar relation to dE as * d* does to the exterior derivative d, where * is the Hodge staroperator.

10.6.1.1 The Codlfferential on Hom (E,E)Recall from the exercises of Chapter 9 that a connection V on n: E -+ M induces one onHom (E,E) - M, whose covariant exterior derivative dHom is characterized by

(dH'A) AA = dE(AAP) - (-1)degAAAdEt (10.66)

for every Hom (E,E) -valued form A and E-valued form µ. Its codifferential is denotedSH', and is calculated with respect to the inner product on Hom (EE)-valued q-formsgiven by

(A.B) = J(AIB)H-.gp, (10.67)M

10.6 The Yang-Mills Equations 243

in the notation of 10.2.6.

10.6.2 Formulation of the Yang-Mills Equations

Either of following conditions is necessary and sufficient for V to be a Yang-Millsconnection, that is, a stationary point of the Yang-Mills Lagrangian:

6HOm9ty = 0, (10.68)

A 9tv = 0, (10.69)

where A = dHom 6Hom + 6Hom dHom

Remarks: The condition 6H0m 9tv = 0 is called the Yang-Mills equations. As wenoted already, Maxwell's equations in vacuo, * d (* 1)) = 0, are a particular case. Thecondition A 9tv = 0 says that the curvature of a Yang-Mills connection is "harmonic,"with respect to the natural "Laplacian" on Hom (EE)-valued 2-forms.

Proof: Fix a connection V on a complex vector bundle it: E -> M (in the settingdescribed in Section 10.2.7), and select A e C (M; Hom (E,E) ). There exists a familyof connections V,, parametrized by t E R, whose covariant exterior derivatives aregiven by

dE. `p. = dEµ + to A µ. (10.70)

In the exercises to Chapter 9, it was proved that the Curvature of V, is given by theformula

9tv,r = 91"+tdH0mA+t2AAA, (10.71)

where dH0mA refers to (10. 66). Therefore, to first order in t, we have the formula

119ty.1112 = II 9tv 112 + 2t(dHomAI9tv)H0. 2 + 0 (12). (10.72)

Referring to (10. 14), and the definition of SHom on J12 (M; Hom (E,E)) given inSection 10.6.1.1, we see that

dtL(V') I7=0= j(dHomAl9tv)Hom.2p =M M

For V to be a Yang-Mills connection amounts to saying that this integral is zero for allchoices of A, which is precisely the condition 6Hom9t° = 0.

It only remains to check the equivalence of (10. 68) and (10. 69). Recall that in theexercises to Chapter 9, we formulated Bianchi's identity as

244 Chapter 10 Appllcatlons to Gauge Field Theory

dHom9ty = 0, (10.73)

which holds for any connection. Thus (10. 68) immediately implies (10. 69). To see theconverse, note

jWt°I9t°)Hom.2p

= f (dHom (8H-9tV) +SHom (dHom9t°)19 V)Hom.2P

M M

=f 11 dHom9ty112p+ f118H0m9tyII2P

M M

= f IISHom9tV112p,

M

and so A9ty = 0 implies 8Hom9{V = 0.

10.7 Self-duality

Since the curvature involves one order of differentiation of the connection forms (i.e.. ofthe gauge potential), the Yang-Mills equations SHom9ty = 0 are equations of secondorder in the gauge potential. However, there is a stronger, first-order system of equationswhose solutions automatically satisfy the Yang-Mills equations, and which is mucheasier to solve. The situation is analogous to replacing the study of Laplace's equation inthe plane by the study of the Cauchy-Riemann equations, which are much stronger.

10.7.1 Self-dual 2-Forms

We assume that it: E - M is a real or complex vector bundle with a metric rover a compact, oriented, 4-dimensional Riemannian manifold (M, Recall that ametric on Hom (AQTM,E) was constructed in Section 10.2.5.1, for q = 1, 2, 3, 4. Thediscussion of the Hodge star operator in Chapter 1 applies immediately to each vectorspace L (A9T,.M -. E,), for q = 1, 2, 3, 4, in the case where E is a line bundle;moreover it extends, with only minor notational changes, to the case of an arbitraryvector bundle E. If { s; } is an orthonormal frame field for n: E -4 M, and {Et, E2, e3, E°}is an orthonormal coframe field for T* M -> M, then the effect of the Hodge staroperator on E-valued 2-forms is given by:

* (s;(EaAEO)) = s1(EYAEs), (10.74)

where (a, 0, y, 8) is an even permutation of (1, 2, 3, 4); for reasons of space, we omitthe wedge symbol after Si. For example,

10.7 Self-duality 245

* (sj (£' A £3 + £4 A E2)) = sj (E4 A E2 + E' A E3). (10.75)

Define X e C12 (M; E) to be self-dual if * X = A,, and anti-self-dual if * X = -? anddenote the self-dual and the anti-self-dual E-valued 2-forms by Q2 (M;E) andL12 (M; E), respectively. For example, (10. 75) illustrates a self-dual E-valued 2-form.

It follows from (10. 74) that an orthonormal frame field for f12 (M; E) is given by

{s1(E' AE2+E3AE4),sj (E' AE3+£4AE2),Sk(E' AE4+E2AE3):i,j,kz 1},(10.76)

and an orthonormal frame field for f12 (M;E) is given by

{si(E' AE2-E3AE4),s) (E' AE3-£4AE2),sk(E' AE4-e2AE3):i,j,kz 1}.(10.77)

Moreover it is clear that the E-valued 2-forms listed in (10. 76) are orthogonal to thoselisted in (10. 77). We could summarize by saying that f12 (M;E) admits an orthogonaldecomposition

K12 (M; E) = !a2 (M; E) 9 02 (M; E) (10.78)

into the self-dual and the anti-self-dual E-valued 2-forms.

10.7.2 Self-dual Connections

Replace E by Hom (E,E) in the preceding discussion. For any connection V, theassociated curvature 9tv a C12 (M; Horn (E,E)) decomposes according to (10. 78) intoself-dual and anti-self-dual parts:

9tv = 9t°+9tv.

We say that Visa self-dual connection if * 9tv = 9tv, that is, if 9tv = 0.

(10.79)

10.7.2.1 Self-dual Connections Satisfy the Yang-Mills Equations

Suppose M has dimension four. If * 9tv = 9tv, then 8Hom 99V = 0.

Proof: Recall the following formula from Chapter 1, for the case of a four-dimensionalM, and an inner product of signature 0:

X A * µ = (;1µ)H0 . p, for 4, g e IV (M; Horn (E,E) ), (10.80)

where p is the canonical volume form. It follows that for ?. a S2' (M;Hom (E,E)) andg e C12 (M; Hom (E,E) )


f (dHom xiµ)Hom. 2p = JdHom X A -AM

JdHom ()LA *A)+ A. n dHom (* µ)

M M

by the Leibniz rule. Since the boundary of M is empty, the first integral is zero byStokes's Theorem. On the other hand * (* tp) _ -4) for 0 e j13. (M; Hom (E,E)) (seeChapter 1), so we obtain

J(dHom)dµ)Hom.2p = _JX n* (* (dHom*lt)) _ -f W* (dHom*it))Hom.Ip.

M M M

Now it follows from (10. 10) and (10.65) that SHom 9tv = -* (dHom * 9tv) If V isself-dual then dHom * 9tv = dHom9ty = 0, by the Bianchi identity (10. 73). ThereforeSHom 9 V = 0.

YI

10.7.2.2 Antl-self-dual ConnectionsIf the connection is anti-self-dual, that is, if * 9tv = -9tv, then simply reversing theorientation of M makes it self-dual, and so the preceding result applies.

10.7.3 Self-dual Connections Minimize the Yang-Mills Lagrangian

It follows from the Chern-Weyl theory of characteristic classes, which cannot becovered here (see Kobayashi and Nomizu [1963]), that

i(E) = 82J(II`RV1j2-II`RVjj2)pM

does not depend upon the choice of connection V; in other words, it is an invariant ofthe bundle. In the case where tt: E -. M is an Sp (1)-quaternion line bundle, asdescribed in Section 10.4, with the metric (10. 34), we call i (E) the instanton number.

This yields considerable insight into the minimization of the Yang-Mills Lagrangian.By the orthogonality aspect of decomposition (10. 78),

L(V) = 47[2i (E),M M

and the right side does not depend upon the connection; thus L (V) > 4tt2i (E), withequality if and only if V is self-dual. Thus, among the solutions to the Yang-Millsequations, self-dual connections are precisely the ones which absolutely minimize theYang-Mills Lagrangian.

10.8 Instantons 247

10.8 Instantons

Let us return to the study of the quaternion line bundle discussed in Section 10.4.1 andSection 10.4.5, and whose Yang-Mills Lagrangian was described in Section 10.4.6. Weshall take as the Riemannian metric on S4:

I 4 2 2dya ®dya = 1 4 2 2dza ®dza (10.82)15a54 (1+lyl ) ISa5.4 (1+IZI )

in terms of the two coordinate systems presented previously. Take a local frame field forE over U consisting of a nonvanishing section s of unit norm. Since the metric on S4 isjust a function multiplied by the Euclidean metric, it is clear from (10. 76) that anorthonormal frame field for 02 (M;E) over U is given by

{S (dy1 A dye + dy3 A dy4), s (dy' A dy3 +dy4 A dy2) , s (dy1 A dy4 +dy2 A dy3) }.

10.8.1 Self-dual Connections on a Quatemion Line Bundle

Self-dual solutions to the Yang-Mills equations over 54, with the metric (10. 82), arecalled instantons. The following instantons were reported by Belavin, Polyakov,Schwartz, and Tyupkin in 1975. Define a family of Sp (1)-connections V, for t > 0 bytaking their gauge potentials, with respect to this frame field {s} (see (10. 37)), to be

A' ydy - (dy) y2 C

t2 + 1y12 i. (10.83)

Comparing this expression to Exercise 15, we see that, for C = ydd - (dy) y andco = dlyI2, (10. 62) and (10.63) give

A'AA' =CAC WAC-21yl2dyAdy

4 (t2+ 1y12)2 2(t2+1y12)2

dA` = dy A dy W A C

t2+1)'12 - 2(r2+1y12)2.

According to (10. 42), the corresponding curvature 2-form is F' = dA' + As A A',which simplifies to

= t2dyAdy(10.84)

(12+1y12)2


which is self-dual, since by Exercise 18 in Section 10.9, dy n dy is a self-dual E-valued2-form. Changing to the other local trivialization as in Exercise 14, we find the gaugepotential transforms to

B"` = 2 rzdz - (dz) zLL

C2+1z12 1(10.85)

so clearly we have a family of self-dual connections defined over the whole vectorbundle. Using the metric (10. 82) and the formula (10. 48), we see that the norm of thecurvature is

11 `Rv"111 = ` I Z (1+ Iy12) 4 - 3t4 I + lyl') l4

(10.86)a 16 2 U2+1yl2)

using (10. 89) below.

Let us say that two G-connections are gauge equivalent if they are related by a gaugetransformation, as in (10. 6); this is indeed an equivalence relation. It follows from10.2.6.1 that two G-connections cannot be gauge equivalent if they lead to differentvalues for the normed curvature. In this light, formula (10. 86) tells us several thingsabout the equivalence classes of self-dual connections that we have constructed:

When t = I, the norm of the curvature is a constant, and this is the same for anychoice of the "pole" omitted from the chart U in 10.4.1.

t = 0 is not permitted, because in that case the gauge potential cannot be extendedto the other local trivialization, by (10. 85).

Changing from y to its antipode z = y u and from t to 1 /t would give the samevalues for the normed curvature; thus we should only consider t E (0, 1), if we aregoing to allow all choices of the pole.

For 0 < t < 1, (10. 86) achieves its unique maximum when y = (0, 0, 0, 0), that is,at the "pole" omitted from the chart U. Thus different choices of t e (0, 1) anddifferent choices of the "pole" p r= S4 omitted from the chart map lead to differentvalues for the normed curvature, and hence to different equivalence classes ofconnections, by the observation above.

In other words, we have constructed a family of distinct equivalence classes of self-dualconnections on a quaternion line bundle "of instanton number 1," parametrized by{ 1 } v (S4 x (0, 1) ). Using more advanced methods, Atiyah, Hitchin and Singer[ 1978] proved that these are the only classes of self-dual connections on this bundle,and gave this set the geometric structure of a 5-dimensional Riemannian manifold:

10.9 Exercises 249

10.8.2 Theorem

The set of connections described above represents all the equivalence classes ofself-dual connections on this quaternion line bundle; these equivalence classes can beidentified with 5-dimensional hyperbolic space.

To summarize: In view of 10.7.3, the connections defined by (10. 83), for t = I. and fort E (0, 1) with different choices of the "pole" p E S4 omitted from the chart map,comprise a complete set of those connections that minimize the Yang-Mills Lagrangianfor this Sp (1)-bundle over S4.

10.9 Exercises

17. Suppose x: E - M is a real or complex local U (1)-line bundle over M = R4 with theLorentz metric, and has the metric

(ah)P = Re { a (p) i (p) }.

The E-valued q-forms may be regarded as ordinary (real or complex) q-forms under theglobal trivialization, and an example of a covariant exterior derivative on this bundle isthe ordinary exterior derivative, dE = d. Prove that the codifferential SE defined inSection 10.6.1 is here given by the formula

SE9 = * (d *ll ), (10.87)

forp.E S29(M),q = 1,2,3,4.Hint: Use the methods of Exercise 4.

18. Suppose z is the quaternion z ' + z2i + z3j + z4k, and dz = dz ' + dz2i + dz3j + dz4k.

(i) Show that the H-valued 2-form dz A dz is self-dual, by showing that

-Z(dzAdz) = (dztndz2+dz3Adz4)i+(dzI Adz3+dz4Adz2)j (10.88)

+ (dz' A dz4 + dz2 A dz3) k.

(ii) Suppose Fup = (dz A dz) - Da A Dp, where Da = a/aza. Prove that

I FUDI 2 = 24 . (10.89)

19. In the situation described in Section 10.8, show that for any t > 0 the following gaugepotential represents an anti-self-dual connection ("anti-instanton"):


A' = 1 ydy - (dy) y11[ t2+Iy12 J

(10.90)

20. Check by direct integration over R4 that, for the curvatures of the self-dual connectionsdescribed in (10. 86),

fI19tv''II2P (10.91)

S,

is the same for all t > 0, and explain how this fact is related to the formula (10.81).


The unification of the electromagnetic and weak interactions by Weinberg and Salam,using Yang-Mills theory, received experimental confirmation in the discovery ofmassive particles called intermediate bosons, which were predicted by the theory. Inrecent years much work has been done on a U (1) x S U (2) x S U (3) Yang-Millstheory known as the "standard model," which seeks to provide the basic framework forunifying the electromagnetic, weak, and strong interactions; a complete description ofthe solution of the Yang-Mills equations for this case is available. Presently physicistsare postulating "grand unified theories" for the unification of the electromagnetic, weak,strong, and gravitational fields, but the sort of particles predicted by these theories seemto be beyond the range of present experimental capabilities.

Many physics books give accounts of gauge field theories. For a mathematician'saccount of the geometrization of physics in the twentieth century, see Bourguignon[ 1987]. The lecture series of Atiyah [ 1979] and of Lawson [ 19851 present gauge theoriesin greater depth.

Bibliography

Abraham, R.; Mardsen, J. E.; and Ratiu, T. (1988). Manifolds, Tensor Analysis, andApplications. Springer, New York.

Amarsi, S.-I.; Barndorff-Nielsen, 0. E.; Kass, R. E.; Lauritzen, S. L; and Rao, C. R.(1987). Differential Geometry in Statistical Inference. Institute of MathematicalStatistics Lecture Notes, Vol. 10. Institute of Mathematical Statistics, Hayward,CA.

Atiyah, M. F. (1979). Geometry of Yang-Mills fields. Lezione Fermiane, AcaademiaNazionale dei Lincei & Scuola Normale Superiore, Pisa.

Atiyah, M. F.; Hitchin, N. J.; Singer, I. M. (1978). Self-Duality in Four-DimensionalRiemannian Geometry. Proceedings of the Royal Society, London, A 362, 425-61.

Berger, M., and Gostiaux, B. (1988). Differential Geometry: Manifolds, Curves, andSurfaces. Springer, New York.

Bourguignon, J. P. (1987). Yang-Mills Theory: The Differential Geometric Side.Lecture Notes in Mathematics 1263, 13-54. Springer, New York.

Cartan, H. (1970). Differential Forms. Hermann, Paris.

Chern, S. S. (1989), ed. Global Differential Geometry. Mathematical Association ofAmerica, Studies in Mathematics, Vol. 27. Prentice-Hall, Englewood Cliffs, NJ.

Curtis, W. D. and Miller, F. R. (1985). Differential Manifolds and Theoretical Physics.Academic Press, New York..

do Carmo, M. P. (1992). Riemannian Geometry. Birkhauser, Boston.

Edelen, D. G. B. (1985). Applied Exterior Calculus. Wiley, New York.

252 Bibliography

Eisenhart, L. P. (1940). An Introduction to Differential Geometry with Use of the TensorCalculus. Princeton University Press, Princeton, NJ.

Flanders, H. (1989). Differential Forms with Applications to the Physical Sciences.Dover, New York.

Gallot, S.; Hulin, D.; and Lafontaine, J. (1990). Riemannian Geometry. Springer, Berlin.

Greub, W. (1978). MultilinearAlgebra. Springer, Berlin.

Helgason, S. (1978). Differential Geometry, Lie Groups, and Symmetric Spaces.Academic Press, New York.

Husemoller, D. (1975). Fiber Bundles. Springer, New York.

Klingenberg (1982). Riemannian Geometry. De Gruyter, Berlin.

Kobayashi, N., and Nomizu, K. (1963). Foundations of Differential Geometry.Wiley-Interscience, New York.

Lancaster, P., and Tismenetsky, M. (1985). The Theory of Matrices. Academic Press,New York.

Lang, S. (1972). Differential Manifolds. Addison-Wesley, Reading, MA.

Lawson, H. B. (1985). The Theory of Gauge Fields in Four Dimensions. CBMSRegional Conference Series 58, American Mathematical Society, Providence, RI.

Murray, M. K., and Rice, J. W. (1993). Differential Geometry and Statistics. Chapmanand Hall, London.

Okubo, T. (1987). Differential Geometry. Dekker, New York.

Sattinger, D. H., and Weaver, O. L. (1986). Lie Groups and Lie Algebras withApplications to Physics, Geometry, and Mechanics. Springer, New York.

Spivak, M. (1979). A Comprehensive Introduction to Differential Geometry, Vols. 1-VPublish or Perish, Berkeley.

Struik, D. (1961). Lectures on Classical Differential Geometry. Addison-Wesley,Reading, MA; republished by Dover, New York (1988).

Warner, F. W. (1983). Foundations of Differentiable Manifolds and Lie Groups.Springer, New York.

Index

adapted moving frame 82adjoint action 240alternating 5Ampere's Law SQanalytic function L63anti-instanton 241, 249antisymmetric 5area form 86Atiyah-Hitchin-Singer Theorem 249atlas 99

base manifold 120Bianchi's identity 20,6 211. 243bilinear 5binormal 80boundary L12

orientation of 169bracket of vector fields 22bump function 118bundle-valued forms 111

canonical parametrization 82canonical volume form 168. 182Cartan, the 49Cayley transform 24chain rule 42charge density 50chart 28

compatible 98positively oriented 1

Christoffel symbols 202. 216closed set

in a manifold 191Codazzi-Mainardi equations ,205codifferential 242codimension 105

compact setin a manifold 141

Compactness Lemma 109complex

general linear group 21line bundle 128special linear group 11vector bundle 128

component function 25connection 195

curvature 202dual 202Euclidean 195 196forms 78 201G- 226Levi-Civita 216matrix 201metric 216pyramidic 221self-dual 245Sp(1)- 235torsion of 212Yang-Mills 229

continuity equation 50contraction 42coordinate function; 25cotangent bundle L38

Euclidean 12.1cotangent space

Euclidean 28to manifold 13$

cotangent vector L38Euclidean 28

covariant derivative 195covariant exterior derivativecross product 7 19

199

254 Index

curl 37current 51curvature

forms 202

Gaussian 89. 205.220mean 82negative 92norm 228of a connection 202of a curve 80positive 92principal 91sectional 212tensor 203

curve in a manifold 103. 147cylinder 87 126

density form 36derivation 26. 142determinant 9diffeomorphism L02differentiable structure 99. 1114differential form

anti-self-dual 245bundle-valued 124Euclidean 30matrix-valued 28on a manifold 145. L481-form 145self-dual 245vector-bundle valued 122

differential manifold 99differential of a map 35., 41differential operator L42div 37, 188, 221Divergence Theorem 184. 189dot product 14dual

basis 28bundle 124frame 157space 11, 28

electric current density 54electric field 50electromagnetic potential 52electromagnetic theory 50 223ellipsoid 96 161embedded submanifold 108embedding 108equivalent atlas 92Euclidean metric 153exterior calculus L42, 23.6exterior derivative 36

bundle-valued form 129geometric meaning 186

exterior power 1of a linear transformation 1_Qvector bundle 125

exterior product 6bundle-valued forms 128differential forms 31

Faraday's Law 5Dfiber 120field strength 224, 236first fundamental form 95first structure equation 29flux form 36. 122frame field 144

positively oriented 181Frenet frame 80fundamental forms 95Fundamental Theorem of Calculus 184

gaugegroup 221potential 224, 237transformations 224, 227

Gauss's equation &1205general linear group 69geodesic 209grad 32

on a manifold 160Grassmann, Hermann 23Green's Theorem 184

hairy ball L29Hodge star operator 17.32homomorphism bundle 123hyperbolic paraboloid 97, L61hyperbolic plane L55hyperbolic space 155, 161hyperboloid 5.8.95, L65

immersion 5 , 102implicit function 63

parametrization 63theorem 64

inclusion map 1.03indefinite metric 15.2indefinite Riemannian metric L52induced metric 154inner product 13

on exterior powers 13instanton 242

number 234, 246integral of a form 174, 179interior LL2interior product 49intermediate bosons 254Inverse Function Theorem 64

Index

Iteration Rule 36

Jacobi identity 22

Koszul connection 195, L99

Laplacian 221Leibniz rule 195, 206level set, orientability of L65Levi-Civita connection 216Lie algebra 225Lie derivative 27, 48, 142

of differential forms 40using flows 48

Lie subgroup 69 1.62Lie, Sophus 62likelihood 221line bundle 121line integral 36, L26linear forms 11local connector 209local trivialization 126local vector bundle 120

isomorphism 123morphism 122section 120

locally finite LL2Lorentz group 24Lorentz inner product 14.50 L55.. L&2

magnetic field 59matrix groups 62Maxwell's equations 50 222metric 152

connection 216Riemannian 152

Mdbius strip 127, 169Moliere 125moving orthonormal frame 26multilinear 5

neighborhood 25non-degeneracy 13non-orientable manifold 165normal 80normal bundle 140

open setEuclidean 24in a manifold 191in a submanifold 109

orientable manifold 165orientation L65orientation of basis 12oriented 165orthogonal group 20

orthonormalbasis 14coframe 151frame L56

parametrization 61parametrized curve 61 126parametrized surface 61 159, L22partition of unity 111Pauli matrices 226permutations 3Poincarf. Henri 42projective plane 136pseudo-Riemannian manifold 152pullback 43 11 144

tangent bundle L24push-forward 44p-vectors 1

quadric surface 95quaternion 231

imaginary 232line bundle 233

regular mapping 60rep6re mobile 26Riemannian

manifold 152metric tensor 96, 153, 163volume 183

second fundamental form 96second structure equation 29.205section 134Serret-Frenet formulas 81sharp # 164signature

inner product space 15permutation 3

smooth map 24, 142special

linear group 211orthogonal group 20unitary group 71.225

sphere 57 154, L65spherical coordinates 45stereographic projection 144Stokes's Theorem 183strong nuclear force 225structure equations 18submanifold 195

Euclidean 56submanifold-with-boundary L12,169submersion 54 192surface integral 36. L21surface of revolution 54 87 94

255

256 Index

symmetry 13symplectic group 74

tangent bundle 136, 169atlas 131zEuclidean 121projective plane 137

tangent map 138,151Euclidean 122

tangent plane 81tangent space

Euclidean 24to manifold L36to submanifold 0

tangent vector 13-6

as class of curves 142Euclidean 25

tensor

(0,2)-tensor 33tensor product 32

of 1-forms 33vector bundles 124

Theorems Egregium E9 911torsion 212

of curve 84torsion-free 212torus 57, L65transition function 122transposition 3triple product 7 12trivial bundle 129trivializing cover 126

unitary group 71universal mapping property 5

vector 227

vector algebra 12vector bundle 125

base 126.construction 134, 141equivalent 129fiber 125metric 222morphism 129projection 125pullback 133section 134sub-bundle 122tensor product 124transition function 127

vector field 25 144volume

form 164Riemannian 183

work form 36, 12

Yang-Millsconformal invariance 232equations 243field 222Lagrangian 229, 238

zero section 120.

This book introduces the tools of modern differ-ential geometry-exterior calculus, manifolds, vectorbundles, connections-to advanced undergraduatesand beginning graduate students in mathematics,physics, and engineering. It covers both classicalsurface theory and the modern theory of connectionsand curvature, and includes a chapter on applicationsto theoretical physics. The only prerequisites aremultivariate calculus and linear algebra; no knowledge

of topology is assumed.

The powerful and concise calculus of differentialforms is used throughout. Through the use ofnumerous concrete examples, the author developscomputational skills in the familiar Euclidean context

before exposing the reader to the more abstractsetting of the manifolds. There are nearly 200exercises, making the b(x)k ideal for both class-room use and self-study.

COVER DESIGN BY JAMES F BRISSON

CAMBRIDGE www.cambridge.orgISBN 0-521-46800-0

UNIVERSITY PRESS11111 1IN111111111

9 780521 468008 0

darling - differential forms

Documents