Differential Geometry 9780824777005

Citation preview

PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks

DIFFERENTIAL GEOMETRY

Tanjiro Okubo

DIFFERENTIAL GEOMETRY

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS

Earl J. Taft

Zuhair Nashed

Rutgers University New Brunswick, New Jersey

University of Delaware Newark, Delaware

CHAIRMEN OF THE EDITORIAL BOARD

S. Kobayashi

Edwin Hewitt

University of California, Berkeley Berkeley, California

University of Washington Seattle, Washington

EDITORIAL BOARD

M. S. Baouendi Purdue University

Donald Passman University of Wisconsin-Madison

Jack K. Hale Brown University

Fred S. Roberts Rutgers University

Marvin Marcus University of California, Santa Barbara

Gian-Carlo Rota Massachusetts Institute of Technology

W. S. Massey Yale University

Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas and University of Rochester Anil Nerode Cornell University

David Russell University of Wisconsin-Madison

Jane Cronin Scanlon Rutgers University Walter Schempp Universitat Siegen

Mark Teply University of Wisconsin-Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K. Yano, Integral Formulas in Riemannian Geometry (\970)(out of print) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) (out of print) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, editor; A. Littlewood, translator) (1970) (out of print) 4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation editor; K. Makowski, translator) (1971) 5. L. Narici, E. Beckenstein, and G. Bachman, Functional Analysis and Valuation Theory (1971) 6. D. S. Passman, Infinite Group Rings (1971) 7. L. Dornhoff Group Representation Theory (in two parts). Part A: Ordinary Representation Theory. Part B: Modular Representation Theory

(1971,1972) 8. W. Booth by and G. L. Weiss (eds.), Symmetric Spaces: Short Courses Presented at Washington University (1972) 9. Y. Matsushima, Differentiate Manifolds (E. T. Kobayashi, translator) (1972) 10. L. E. Ward, Jr., Topology: An Outline for a First Course (1972) (out of prin t)

11. A. Babakhanian, Cohomological Methods in Group Theory (1972) 12. R. Gilmer, Multiplicative Ideal Theory (1972) 13. /. Yeh, Stochastic Processes and the Wiener Integral (1973) (out of print) 14. /. Barros-Neto, Introduction to the Theory of Distributions (1973) (out of print) 15. R. Larsen, Functional Analysis: An Introduction (1973) (out of print) 16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry (1973) (out of print) 17. C. Procesi, Rings with Polynomial Identities (1973) 18. R. Hermann, Geometry, Physics, and Systems (1973) 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) (out of print) 20. /. Dieudonne, Introduction to the Theory of Formal Groups (1973) 21. /. Vaisman, Cohomology and Differential Forms (1973) 22. B. -Y. Chen, Geometry of Submanifolds (1973) 23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975) 24. R. Larsen, Banach Algebras: An Introduction (1973) 25. R. O. Kujala and A. L. Vitter (eds.), Value Distribution Theory: Part A; Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973) 26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) 28. B. R. McDonald, Finite Rings with Identity (1974) 29. /. Satake, Linear Algebra (S. Koh, T. A. Akiba, and S. Ihara, translators) (1975)

30. J. S. Golan, Localization of Noncommutative Rings (1975) 31. G. Klambauer, Mathematical Analysis (1975) 32. M. K. Agoston, Algebraic Topology: A First Course (1976) 33. K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules (1976) 34. L. E. Mansfield, Linear Algebra with Geometric Applications: Selected Topics (1976) 35. N. J. Pullman, Matrix Theory and Its Applications (1976) 36. B. R. McDonald, Geometric Algebra Over Local Rings (1976) 37. C. W. Groetsch, Generalized Inverses of Linear Operators: Representation and Approximation (1977) 38. J. E. KuczkowskiandJ. L. Gersting, Abstract Algebra: A First Look (1977) 39. C. O. Christ ens on and W. L. Voxman, Aspects of Topology (1977) 40. M. Nagata, Field Theory (1977) 41. R. L. Long, Algebraic Number Theory (1977) 42. W. F. Pfeffer, Integrals and Measures (1977) 43. R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis (1977) 44. J. H. Curtiss, Introduction to Functions of a Complex Variable (1978) 45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978) 46. W. S. Massey, Homology and Cohomology Theory (1978) 47. M. Marcus, Introduction to Modern Algebra (1978) 48. E. C. Young, Vector and Tensor Analysis (1978) 49. S. B. Nadler, Jr., Hyperspaces of Sets (1978) 50. S. K. Segal, Topics in Group Rings (1978) 51. A. C. M. van Rooij, Non-Archimedean Functional Analysis (1978) 54. L. Corwin and R. Szczarba, Calculus in Vector Spaces (1979) 53. C. Sadosky, Interpolation of Operators and Singular Integrals: An Introduction to Harmonic Analysis (1979) 54. J. Cronin, Differential Equations: Introduction and Quantitative Theory (1980) 55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980) 56. /. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980) 57. H. I. Freedman, Deterministic Mathematical Models in Population Ecology (1980) 58. S. B. Chae, Lebesgue Integration (1980) 59. C. S. Rees, S. M. Shah, and C. V. Stanojevic, Theory and Applications of Fourier Analysis (1981) 60. L. Nachbin, Introduction to Functional Analysis: Banach Spaces and Differential Calculus (R. M. Aron, translator) (1981) 61. G. Orzech and M. Orzech, Plane Algebraic Curves: An Introduction Via Valuations (1981) 62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis (1981) 63. W. L. Voxman and R. H. Goetschel, Advanced Calculus: An Introduction to Modern Analysis (1981) 64. L. J. Corwin and R. H. Szcarba, Multivariate Calculus (1982) 65. V. I. Istratescu, Introduction to Linear Operator Theory (1981) 66. R. D. Jarvinen, Finite and Infinite Dimensional Linear Spaces: A Comparative Study in Algebraic and Analytic Settings (1981)

67. /. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry (1981) 68. D. L. Armacost, The Structure of Locally Compact Abelian Groups (1981) 69. /. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute to Her Life and Work (1981) 70. K. H. Kim, Boolean Matrix Theory and Applications (1982) 71. T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982) 72. D. B. Gauld, Differential Topology: An Introduction (1982) 73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) 74. M. Carmeli, Statistical Theory and Random Matrices (1983) 75. /. H. Carruth, J. A. Hildebrant, and R. J. Koch, The Theory of Topological Semigroups (1983) 76. R. L. Faber, Differential Geometry and Relativity Theory: An Introduction (1983) 77. S. Barnett, Polynomials and Linear Control Systems (1983) 78. G. Karpilovsky, Commutative Group Algebras (1983) 79. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings: The Commutative Theory (1983) 80. /. Vaisman, A First Course in Differential Geometry (1984) 81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) 82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) 83. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) 84. T. Albu and C. Nastasescu, Relative Finiteness in Module Theory (1984) 85. K. Hrbacek and T. Jech, Introduction to Set Theory, Second Edition, Revised and Expanded (1984) 86. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings: The Noncommutative Theory (1984) 87. B. R. McDonald, Linear Algebra Over Commutative Rings (1984) 88. M. Namba, Geometry of Projective Algebraic Curves (1984) 89. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) 90. M. R. Bremner, R. V. Moody, and J. Patera, Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) 91. A. E. Fekete, Real Linear Algebra (1985) 92. S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985) 93. A. J. Jerri, Introduction to Integral Equations with Applications (1985) 94. G. Karpilovsky, Projective Representations of Finite Groups (1985) 95. L. Narici and E. Beckenstein, Topological Vector Spaces (1985) 96. /. Weeks, The Shape of Space: How to Visualize Surfaces and Three­ Dimensional Manifolds (1985) 97. P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research (1985) 98. J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis (1986) 99. G. D. Crown, M. H. Fenrick, and R. J. Valenza, Abstract Algebra (1986)

100. J. H. Carruth, J. A. Hildebrant, and R. J. Koch, The Theory of Topological Semigroups, Volume 2 (1986)

101. R. S. Doran and V. A. Belfi, Characterizations of C*-Algebras: The Gelfand-Naimark Theorems (1986) 102. M. W. Jeter, Mathematical Programming: An Introduction to Optimization (1986)

103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications: A New Approach to Nonlinear Partial Differential

Equations (1986) 104. A. Verschoren, Relative Invariants of Sheaves (1987) 105. R. A. Usmani, Applied Linear Algebra (1987) 106. P. BlassandJ. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0 (1987) 107. J. A. Reneke, R. E. Fennell, and R. B. Minton. Structured Hereditary Systems (1987) 108. H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesies (1987) 109. R. Harte, Invertibility and Singularity for Bounded Linear Operators (1987) 110. G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments (1987) 111. L. Dudkin, I. Rabinovich, and I. Vakhutinsky, Iterative Aggregation Theory: Mathematical Methods of Coordinating Detailed and Aggregate Problems in Large Control Systems (1987) 112. T. Okubo, Differential Geometry (1987) 113. D. L. Stand and M. L. Stand, Real Analysis with Point-Set Topology (1987) 114. H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations (1987) 1 15. T. C. Gard, Stochastic Differential Equations (1987) Other Volumes in Preparation

DIFFERENTIAL GEOMETRY TANJIRO OKUBO McGill University Montreal, Quebec Canada

MARCEL DEKKER, INC.

New York and Basel

Library of Congress Cataloging-in-Publication Data Okubo, Tanjiro

Differential geometry.

(Monographs and textbooks in pure and applied mathematics ; v. 112) Includes index.

1. Geometry, Differential. I. Title. II. Series. QA6U.0lj-8 1987 516.3!6 87-13530 ISBN 0-82^7-7700-X

COPYRIGHT © 1987 by MARCEL DEKKER, INC. ALL RIGHTS RESERVED

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. MARCEL DEKKER, INC.

270 Madison Avenue, New York, New York 10016

Current printing (last digit)

10 98765^-321

PRINTED IN THE UNITED STATES OF AMERICA

Preface

I have written this book with the upper-level undergraduate and graduate students in mind. The purpose of the text is to provide a background for the study of local and global differential geometry. In particular, we are interested in tensor analysis developed on differentiable manifolds. The prerequisites are kept at a minimum, avoiding complete proofs of propositions and theorems that might require elaborate machinery. The material is arranged so as to proceed from the general to the special, and is accompanied in almost every section by exercises designed to enhance understanding

of the definitions and theorems. Each topic is explained in a manner relating notation expressed in coordinates with coordinate-free

expressions. This will help students explore the classical references and the abstractions currently adopted in most recent research papers.

Chapter 1 gives a survey of differentiable manifolds and Lie groups whose foundation may be found in the texts by Matsushima [^], Pontrjagin [1], Chevalley [1], Bishop and Goldberg [1], Warner [2], and Poor [1]. The last three books, in particular, are recommended for students wishing to fill the gap between the foundations of manifolds and diverse topics in differential geometry. In 1-1,

Section 1, we topologize an abstract point set by means of a fundamental neighborhood system. Assigning a coordinate map to each

element of the system, we construct a manifold structure in 1.2,

and then introduce in it the differentiability class to obtain a differentiable manifold. Since a local chart of any differentiable

in

IV

Preface

manifold M is defined by a coordinate map cp of a neighborhood of M

into the number space of a certain dimension, a function is defined by f o cp"1, where f denotes a map from an open subset U of M into the number space R. To carry out the global analysis of functions on a paracompact manifold M, 1.3 of Section 1 describes a device known as the partition of unity subordinate to the given open cover of M.

Section 2 of Chapter 1 treats the notion of tangent vectors and vector fields. We define the tangent space at a point x of M as a linear derivation on germs of functions. Calling a field of tangent

vector on U ( R defined by

where a = (a ,...,a ) € Rn, is called the ith standard coordinate function on R. If f is a map from a point set X to R, we let

f1 = r1 o f (2.2)

and call f the ith component function of f.

1 Differentiable Manifolds

7

If f: g -» g and t € g, the derivative of f at t is denoted by

(£Vf = (S) =h-»0 ii->(t + h)-f(t)] If f: gn -» g and t = (t1,.. . ,tn) € gn, the partial derivative of f with respect to t (i = 1, 2, ..., n) is denoted by

\X'f t= 6r (^ = tlim t1 +h\ ti+1, 6r ^ i[f(t, h .,...,tn)t1"1, - f(t)] For a point a of R we will denote by B (a) the open ball of radius r about a. By C(r) we denote the open cube with sides of

length 2r about the origin 0 = (0,...,0) in g . We shall use C to denote the field of complex numbers and C to denote the complex n-space

k'k

0 t * Exercise 3-6. Let X,' Y,' $u' , andv $ be as in Theorem 3.1. Then

[X,Y] = 0 if and only if $ and $ commute for every u and v. Exercise 3-7• Let A and B be Lie algebras over K. If a linear map oi from A to B satisfies or([X,Y]) = [or(X) ,or(Y) ] for X, Y € A, then oi

is called a homomorphism from A into B. If a is one-to-one, it is an isomorphism from A into B. [If a{k) = B, then a is an isomorphism from A onto B. If there is an isomorphism from A onto B, then A and B are said to be isomorphic, and we write A ss B. An isomorphism of A onto itself is called an automorphism of A.] Let M and N be connected, paracompact C manifolds, and 3E(M) and X(N) the Lie alge­ 00

bras of all C vector fields with compact supports on M and N, respectively. Prove that there exists an isomorphism $ of X(N) if and only if there exists a C diffeomorphism i|r of M onto N such CO

that $ = d\|r [L. E. Pursell and M. E Shanks, The Lie algebra of vector fields of a smooth manifold, Proc. Amer. Math- Soc. 5(195*0* ^68­ Vf2].

2.X Submanifolds CO

Let cp be a C mapping from a manifold M into a manifold N. If at each point p of M, (dcp) is one-to-one, then cp is called an JTmnersion

Chap. 1. Differentiable Manifolds

32

of M into N. If, moreover, cp is one-to-one, then cp is called an embedding of M into N.

Definition. If M and N satisfy the following conditions, M is called a submanifold of N:

(1) The point set M is a subset of N. (2) The inclusion map 1 from M into N is an embedding of M into N.

If M is a closed subset of N, we call M a closed submanifold of N. It should be noted that the intrinsic topology of a submanifold

M does not necessarily coincide with the relative topology of M induced from N by 1 : M -» N. In the case that the intrinsic topology of M coincides with the relative topology, M is said to be a regular submanifold of N.

h °° °°

Example 1. Let M be a regular submanifold of N, and let dim M = n and dim N = n + k. Then for local charts {Ujx1} of N and {V;y^} of M

such that 1(v) c u, we see that the x n (h = 1, ..., n + k) are C functions on V such that x are C functions on 1 and 1 is a C

mapping. Hence there exist C functions f (t ,...,t ) in a certain domain of g such that x u = f (y ,...,y ) on V. Classically, one denotes this by x = x (y ,...,y ) and calls it a local expression of the (regular) submanifold M of N. 2

Example 2. In g the limagon M given by

r = a cos 0 - b is not a regular submanifold, for at the origin (0,0) the tangent space (now one-dimensional) of M is not uniquely determined, By

setting a = 1, b = cos a, 6 = t + 01 (a = constant), as an example

00 00

of a limagon, we consider M with the parametric expression

x = cos(t + a)(cos(t + a) - cos a) y = sin(t + aO(cos(t + a) - cos a)

and take t as the local coordinate on M. Then M is endowed with a one-dimensional C structure; namely, it is the image of a C map­ 0

ping of (0,2tt) or of (-2a, 2rr - 2a) into R .

2 Vector Fields

12 3

33

Example 3» The Veronese surface in g is given by

x=w x=wu x=uv 2 25=:u 2+ .,v -22w o2 4), u-v X =—"2— X "7 2

2 *"t 2 2 i i

where u, v, w satisfy u p v + w = 1. Since x (u,v,w) = x (-u,

1 2h 35

-v,-w) for i = 1, 2, 3> ^> 5> the surface is a submanifold diffeo­ morphic to the projective space P (g). Example k. Consider the subset N of g given by

x = cos u x = sin u cos v yz = sin u sin v

x = cos 2u x = sin 2v

Then we have x (u',v') = x (u,v) for i = 1, 2, 3, ^, 5 if and only if u' = (-l)n + 2mn , v' = v + nn (m, n = 0, 1, 2, ...). N with the relative topology induced from the natural topology of g is dif­ feomorphic to the Klein bottle.

Exercise k.l. Show that S cannot be immersed in g . Exercise k.2. Let M be a submanifold of N, and let dim M = n, dim

N = n + k. Assume that at each point p of M, there are a neighborhood U of p in N and C^ functions f ,. . . ,r such that (i) U fl M =

{q € U : f1^) = 0 (i = 1, ..., k)} and (ii) (df1)^ .. ., (dfk)p are

linearly independent. Prove that M is a regular submanifold of N. Exercise k.J>. Let {tl ;x/ \} be an open covering of M by local

charts and H be an open set in M. If the restriction of the coordinate function X/ \ to U fl a is denoted by 'x/. \, show that UL {II fl H; ,yi(\ \3 is an open covering of H, providing H with a C manifold structure. Show also that H with this structure satisfies conditions (i) and (ii) and, moreover, H is regularly embedded in M. (Such a manifold is called an open submanifold of M. Differential

geometry that treats only open submanifolds is called local differential geometry.)

Chap. 1. Differentiable Manifolds

3h

3. TENSOR FIELDS

3.1 Tensor Algebra Let A and B be finite-dimensional vector spaces over the field K where in the applications K is either the real number system R or the complex number system C. We denote by L(A x B) the vector space spanned by all ordered pairs (a,b), where a € A and b € B, and denote by R(A x B) the subspace of L(A x B) consisting of the forms

(ka,b) - k(a,b) (a,kb) - k(a,b) (k € K) (ax + a2, b) - (a-^b) - (a2,b), (a, bj_ + bg) - (a,b1) - (a,bg) We define A ® B and p,: AxB->A®Bby

n(a,b) = a ® b (1.2) where a®b€A®Bis the coset of (a,b) by R(A x B). Let 0 be a bilinear mapping of A x B into a vector space D. We say that the couple (0,D) has the universal mapping property of A x B if for any vector space C and every bilinear mapping a : A x B

-» C there exists a unique linear mapping a such that a = a o 0; that is, the following diagram is commutative.

j(: A x B-*D is said to be a bilinear mapping if the following conditions are satisfied:

0(ka,b) = k0(a,b) 0(a,kb) = k#(a,b) (k € k) Jf(a + a2, b) = 0(ax,b) + 0(a2,b) 0(a, bi + b2) = 0(a,bi) + 0(a,b2) (We can generalize to a trilinear or more general r-linear mapping in the natural way.)

3 Tensor Fields

35

Proposition 1.1. The couple (p,, A ® B) has the universal mapping property and the vector space A ® B as defined above is unique up to an isomorphism.

Proof: From the construction of A ® B, we have (ka) ® b = a ® (Kb) = ka ® b

(ai + a2) ®b = ax®b = a]_ ® b + a2 ® b a ® (bi + b2) = a ® bi + a ® b2

Hence p,:AxB-»A®Bisa bilinear mapping, and A ® B is spanned by p,(A x B). Let C be a vector space and \ : A x B -» C a bilinear mapping. Then \ can be uniquely extended to a homomorphism \':

L(A x B) -> C such that \': R(a x b) -» 0, and this naturally induces a mapping X : A ® B -> C, where X (a ® b) = \(a,b). This implies

that \ = \ op,. To show the uniqueness of A ® B, assume that there is another vector space A ®' B and another bilinear mapping p/: A x B -» A ®' B satisfying the universal mapping property. Then considering A ®' B as C and p/ as \, we have a homomorphism p, :A®B-»A®/Bas\ and p/ «= p, op,. Similarly, we have a homomorphism p,' : A ®7 B ->

A ® B and p, = p,' op,'. From p,' = p, op, and p, = p,' o p,', we find

that p, op,7 and p/ op, are identities when restricted to p/(A X B) and p,(A x B), respectively. As A ® B and A ®7 B are generated by the elements of p,(A x B) and p,/(A x B), respectively, it follows

that p, op,7 and p/ op, are identities when considered as operators acting on A ®' B and A ® B, respectively. Hence p,' is an isomorphism of A ®' B onto A ® B.

A ® B is called the tensor product of A and B. The proof of the Proposition may be said to solve the universal mapping problem on the couple (p,, A ® B) for A x B. By solving the problem on various couples in a similar way, we have the following isomorphisms. Proposition 1.2. (l)° A ® B ^ B ® A. (2)° (A ® B) ® C s- A ® (B ® C). (3)° K A ^ A.

Chap. 1. Differentiate Manifolds

56

(2)° in Proposition 1.2 makes it meaningful to write A ® B ® C. Since L(A x B)/r(A x B) is a vector space, we have for any element (a]_ + a2) ® ("bi + b2) of A ® B, the distributive law:

(ax + a2) ® (bi + b2) = ax ® bi + a2 ® bi + ax ® b2 + a2 ® b2 and from this law we have

Proposition 1.3- Let (e1,...,e ) and (f1,...,f ) be bases of the vector spaces A and B, respectively. Then (e. ® f , i = l,...,m; v = 1,. . . ,n) is a basis of A ® B and dim A ® B = dim A • dim B.

Thus any element of A ® B is expressed in the form e e. ® f , where we have adopted the Einstein summation convention for the sum

2?1=1 , 2? ,V=l elve. 1 ®V f . This convention will be followed in the sequel.

Let Ai and A2 be vector subspaces of A. If for an element a of A, there are vectors ax in A]_ and a2 in A2 such that a = a.± + a2, then A is said to be the sum of A]_ and A2. Furthermore, if a is expressed uniquely in this form, that is, if a = a: + a2 = a.± + a2 (a{ € Ax, a2 € A2) means that ax = a£ and a2 = a2, then A is said to be the direct sum of Ai and A2 and we write

A = Ax + A2 (direct sum) By the distributive law for the tensor product, we then have Proposition l.k. For the direct sum Ai + A2 of vector spaces A]_ and

-ft -X­

(Ai + A2) ®Bs-Ai®B+A2®B (direct sum) Similarly, we have for the direct sum Bi + B2,

Proposition 1.5. A ® (B± + B2) = A ® Bi + A ® B2 (direct sum).

Let A be the dual vector space of A. An element a of A is a

a: a -» (a,a) (1-3)

linear mapping a: A -» g, which we write, as before,

for each a € X. Thus (a,a) is the value of a taken on a.

3 Tensor Fields

37

Let H(A ,B) [= Hom(A,B)] be the space of linear mappings from

A to B defined by (a,b)(a) = A = 0

If AT = T, T is called a skew-symmetric covariant tensor field of

type (0,r) or a differential form of degree r (or an r-form). If ou is an r-f orm and u/ is an s-form, then A(ou ® u/) is a skew­

symmetric covariant tensor field of degree r + s. It is written by adopting the symbol wedge:

A(u> 8 u/) = u> A u/ (2.3)

and is called the exterior product of ou and u/. Applying (2.2), we then have the formula for

Chap. 1. Differentiable Manifolds

k2

1 r+s [T + S)- tt(1) n(r)

(w A u>')(x,..., X ) =7——77 sgn(TT)u>( X ,..., X )

X n(r+l) u/( X ,..., X ) (2A) n(r+s)

where tt is an arbitrary permutation of (l, ..., r + s). Example 1. A Riemann metric on M is a covariant symmetric tensor

field g of degree 2 which is positive definite: g(X,X) ^ 0 for all X € 3E(M). g(X,X) = 0 if and only if X = 0, and g(X,Y) = g(Y,X) for all X,Y € X(M). We shall show that a Riemann metric exists on any

paracompact C manifold. Let ({II : x, \} ). ,. be a locally finite covering of M with compact II and {f } (\ € A) aC partition of unity subordinate to the covering. Denote by ( , ) the Euclidean A.

inner product on II , namely

c T^j(M) (c) 0). We define the

1r1r

interior product by X to be the (r - l)-form i(x)ou given by

i(X)u>(X,... ,X) = ru>(X,X,. .. ,x) (3-6) For 0-form f we define

i(x)f = 0 for every f € 3(M) (3-7) By (2A), 2.2, Section 2, we have for an r-form ou and an s-form u/

i(X)(u> A u/) = i(x)«) A «/ + (-l)rcu A i(x)u/ (3-8) Let ou be an r-form. We define the exterior derivative dou by

1 r+1

1 r+1 r "" i=l i 1 i r+1 (5>9)

da>(X,..., X ) =-—T L (-1)1 ^(X,...,^,..., X )

i = f, then (df)X = Xf. oo Let i|r: M -* N be a C mapping, and let K be a covariant tensor

field on N of degree r (r > 0). For each point p of M and for any

vectors v,,...,v € T (M), we set

p r yyyj i r

(*tK) (v,...,v) = Ky, x(d\|fv,...3d\|r.v) This defines an element (6i|f'K) of T/ \(M), which we call the pullback to M of the covariant tensor field K on N by the map \|r

(see 2.3, Section 2). If K is of degree 0, i.e., if K is a function f, we define a function 6 Section 2, and a equals L ($, -e). If we denote by U a neighborhood of

a a ■€> a "C a

■e, and set U = L (U ) and $v ; = L (* -e) L (|t| < e), we see (a)

that {U ,$ } ^-r with t < e is a local one-parameter group of transformations of G generated by X, and as e is a constant independent of a, X is complete (see Exercise 2.k9 2.2, Section 2). If

we put a, = $,-e, then a. = a, a . We call a, the one-parameter subgroup of G generated by X. Since the tangent vector of a, is given by a, = (L ) X = X • L and X commutes with a, , we see

"C a, ■*■ ■€> -6- a, -6- w

that a, is a solution of the differential equation a, a = X with the initial condition a0 = -e, and the solution is unique. We denote al = ^i6 by exp X* Hence exp tX = a, for all t. The mapping X -» exp X of g into G is called the exponential mapping. It is one-to­

one near the origin of g, that is, there exists an open

CO ^

neighborhood N of 0 in g such that exp is a diffeomorphism of N onto an

open neighborhood U of -e in G (L. Pontrjagin [1]). Considering cp : R (u) = Ua -» N to be the inverse mapping of R exp: N -» Ua (a 6 G), and regarding N as an open set of R (by identifying g with R ), we obtain for G local chart (Ua,cp ), which shows that the C -structure of a Lie group G becomes a C^-structure, and the group

aaaa

k Lie Groups and Lie Algebras

55

operations (a,b) -» a»b and a -» a" are real analytic (Pontrjagin [1], p. 257). A differentiable 1-form oo on G is called left invariant if (L ) oo = oo o L for every a € G, where (L ) is the pullback of L . The vector space g formed by all left-invariant 1-forms is the dual of g; namely, if X € g and oo € g , then oo(x) = constant. For a left-invariant 1-form oo, its exterior differential doo is left invariant, too, because exterior differentiation commutes with (L ). From the definition of exterior differentiation given in (3.11), 3.3 > Section 3> we obtain for 00,

dco(x,Y) = - § d)([X,Y]) (X,Y € g) (1.6) For a basis (X-., — ,X ) we take the set (6 , ..., 6 ) of 1­ forms in g satisfying 6 (X.) = 6 . and call it the canonical 1-form on G or the left-invariant g-valued 1-form on G. Expressing this as 6 = 6 X., we have by (l.k) and (1.6),

deh = - i c.h.eJ a e1 (= - s c^ag1) (1.7) which is called the Maurer-Cartan equation of a Lie group G expressed in terms of the left invariant g-valued 1-form.

Remark. We can also consider a complex Lie group G where the manifold G is a complex manifold and the maps (a,b) -* a*b from G X G to

G and a -> a" from G to G are holomorphic.

Examples of Lie groups (l) Let X = (x ,... ,x ) and y = (y ,...,y ) be points in E . By defining the sum x+y=(x + y , —, x +y ), R becomes a commutative group, and with the differentiable

' a a' a

structure as an affine space, R is a Lie group. As R is commutative, L = R , where L : x -» x + a. The left-invariant vector fields

on Rn are Z? a1(b/bx1) (a1 € g). (2) The set of all complex numbers with the absolute value 1 forms a commutative group T with respect to multiplication. Transporting

Chap. 1. Differentiable Manifolds

56

the differentiable structure of S onto T in the natural manner, T becomes a Lie group.

(3) Let G and G' be two Lie groups and define the direct product

r r 111r 1

on G x G' by (a,b)*(a',b') = (a*a', b-b') for (a,b) € G, (a',b') € G'. Then G x G' is a group. It is a manifold as a direct product manifold of G and G', and is a Lie group by the group structure defined

above. The Lie group is called the direct product of G and G'. (We can have the direct product G x ••• X G of r Lie groups G , • • •,

G , too. For instance, T = T x ••« X T is a product Lie group, called the r-dimensional torus. It is compact, connected, and commutative. ) 00

In a Lie group G, if there is a subgroup H of G which has a C manifold structure with respect to which H is a submanifold of G and at the same time is a Lie group, H is called a Lie subgroup of G. We then have

—1 °°

Proposition l.k. If H is a subgroup of G and is a regular submanifold, it is a Lie subgroup of G.

Proof: It suffices to prove that the mappings (h1,hp) -» h-.hu from H*H to H and h -» h from H to H are of class C . The composite mapping (h^t^) € H x H -> (h-^fc^) € G x G -* h^ ^HCGisaC mapping of H x H into G. H being a regular submanifold of G, this mapping has values in H, so it is of class C even regarded as the mapping H x H -> H. A similar situation holds for h -> h CD

We note that the Lie subgroup H arising from Proposition l.k

is a closed subset of G. In fact, if a sequence of elements {h, } (X € A) in H converges to an element a of G, then {ii } is a Cauchy

sequence in H. On the other hand, each element of H has a neighborhood whose closure is compact. Hence {ii } converges to an element h of H.

Remark. If for a Lie subgroup H of G any neighborhood of -e in G taken small enough has no element of H in common except -e, H is

said to be a discrete subgroup of G and is counted as a zero-dimensional Lie subgroup.

k Lie Groups and Lie Algebras

57

In general, by a discrete group we mean a group II with a countable number of elements, every element of which is an open set. A

discrete group II is said to act properly discontinuously on a manifold M on the right if the action satisfies the following conditions:

(1) If two points x and x' of M are not in the same orbit, that is,

xa

R x ^ x' for every a 6 II, there are neighborhoods U of x and u' of x' such that R U D u' = $.

(2) For each x 6 M, the isotropy group II = {a€II:Rx = x}is finite.

xa

(3) Each x € M has a neighborhood U such that RUr)U = 0ifa€II and stable by II (i.e., R U = U). [Condition (l) implies that M = M/lI is Hausdorff.]

Let g be the Lie algebra of a Lie group G. A subalgebra ^ of 9 is a vector subspace of g whose Poisson brackets are closed in fy. An ideal a of g is a subalgebra such that for any X € g, X' € a

satisfies [x',X] € Q. A homomorphism of one Lie group into another is a map that is a homomorphism of the underlying abstract groups and a C mapping of the underlying manifolds. If p: G-. -> Gp is a homomorphism, the left-invariant vector fields corresponding to v■6-1 € T-61 (G-.) -L and CD

(p)v 6 T (G0) are p -related, where -en and «eQ are the identity elements of G-, and Gp, respectively. Thus p gives rise to the Lie algebra homomorphism (p) : gn -» g , where g, and gp are the Lie algebras of G-. and G , respectively. Here a Lie algebra homomorphism

of one Lie algebra into another is a linear transformation that preserves the Poisson brackets: [(p)*X]_> (p)*xp-' = (p)*txi>xp-' for Xl>

XP ^ ^1 (see Exercise 3-6, 2.3, Section 2).

Exercise 1.1. Let G be a Lie group. Show that G is a closed normal subgroup of G.

Exercise 1.2. Prove that G satisfies the second axiom of countabil­ ity if and only if the factor group G/G consists of finitely or countably many elements.

Chap. 1. Differentiable Manifolds

58

Exercise 1.3• Let g' be the set of all right-invariant vector fields on G. Show that g' is a Lie algebra and is isomorphic with g, the Lie algebra of G.

Exercise l.k. Prove that if X is left invariant and Y is right invariant on g, then [X,Y] =0.

Exercise 1.5- If X, Y are elements of the Lie algebra of a Lie group G such that [X,Y] = 0, show that t -* exp tX«exp tY is a one­ parameter subgroup of G.

Exercise 1.6. Prove that if H is an invariant Lie subgroup of G, the Lie algebra t) of H is an ideal of the Lie algebra g of G, and conversely, the connected Lie subgroup H generated by an ideal t) of g is an invariant subgroup of G. Exercise 1.7- Let H be a Lie subgroup of a Lie group G and H the connected component of H containing the identity element -e. Show that H is the maximal integral submanifold through -e of the involu­

tive distribution determined by the Lie algebra of H. Another connected component is also a maximal integral submanifold of the

distribution obtained by the left translation of H.

Exercise 1.8. Let H be as above and suppose that cp is a differentiable mapping of a differentiable manifold N into G. If cp(N) ^ h,

show that cp is a differentiable map of N into H (C. Chevalley [1], P. hk). k.2 GL(n:R) and GL(n:C)

i^

The set GL(n:g) of all nonsingular n x n square matrices A = [a.], [the matrix whose ith row and jth column entry is a.] is an open submanifold of R 2 and is a group under the matrix multiplication given by (AB) = ajb^ for A = [a3:] and B = [b^]. Since the entries of any nonsingular matrix are the local coordinates of this matrix, when we regard the latter as a point of E 2, the mapping A x B -> A«B is of class C , hence C , and thus GL(n;R) is a Lie group, called

(JL) °° / \

the real general linear Lie group of degree n. Similarly, the set of nonsingular n x n square matrices with entries consisting of complex numbers can be regarded as an open submanifold of Rp 2 if

k Lie Groups and Lie Algebras

59

the entries are decomposed into the sum of real and imaginary parts. The set GL(n;C) of all nonsingular complex n x n square matrices is a Lie group and is called the complex general linear Lie group of degree n. For a matrix group there exists another exponential mapping

which actually is the same as the one defined in the preceding section for general Lie groups. Let A(t) be a nonsingular matrix of degree n whose entries are continuous functions of the parameter t satisfying

A(s + t) = A(s) • A(t) (s, t € R) (2.1)

-1 ^ t

for every s and t. Then it is commutative and A(0) = A(o)*A(0). Since A(t) is nonsingular, A (0) exists and multiplication of A"1(0) gives A(0) = I (the identity matrix [61:]). From A(t)*A(-t)

= A(0), we have A~ (t) = A(-t). Differentiating (2.1) with respect to s at s = 0 and setting A(0) = X, we have

*t

A(t) = A(0) • A(t) = X*A(t) or by commutativity,

A-1(t) • A(t) = X (2.2)

Thus A(t) is the solution of the ordinary linear differential equation (2.2) with the initial condition A(0) = I, which implies that A(t) generates a local one-parameter group of transformations. To show that the solution is a global one analytically, we first define the norm in GL(n;C).

The inner product of the two matrices A and B on C is defined by

(A,B) = Tr A*B

where B denotes the complex conjugate of the transpose B, that is, *Bt~~ = B. The norm of A is then given by For the proof of differentiability, see, for example, T. Yamanouchi and M. Sugiura [1], p. 2k.

Chap. 1. Differentiate Manifolds

60

[] A« =^AAT We define for a matrix A its exponential function e = exp A by

2 » n exPA = I+A+|r+..-= E ^r (2.3) n=0

The infinite series here means the limit of the partial sum,

lirn^ £k=0 (A /k!). A sequence of matrices {A } is said to converge to a matrix A if each sequence {a.(m)} of the (i,j)-entry of

A converges to the (i, j)-entry of A. This statement makes sense, since the set of n-square matrices is homeomorphic to C 2 and there the distance between A = [a.] and B = [b^] is given by || A - B ]| (= the norm of the matrix [a. - b.]), so that the convergence of [A } can be determined by this metric. Now for exp tX let the mth impartial sum of (2.3) and of the positive term S_0 (||a|| /k!) be denoted by s and S , respectively; then we see from the relation 1M| < M ' llBH *ha* IIsn - sj < flsn - Sj| and since Sn converges, s satisfies the Cauchy condition and hence it converges. Thus each entry of the function exp tX (t € g) can be expanded in powers of t which converges for all t. Hence A(t) = exp tX is an analytic function of t and allows termwise differentiation

Ji IJ

oo

^j^- = X + 37- + ••• = X-exp tX = X-A(t)

which proves that the unique solution of (2.2) satisfying the initial condition A(0) = I is A(t) = exp tX (-00 < t < »). exp satisfies the following well-known properties: (1) If AB = BA, then exp (A + B) = exp A exp B. (2) B(exp A)B"1 = exp(BAB"1).

(3) exp A = exp A, exp *A = *exp A. (k) det(exp A) = exp Tr A. A linear Lie group is by definition a subgroup G of GL(n;C).

It is closed in the latter, and consequently G is a regular sub­ manifold of GL(n;C). The Lie algebra gI(n;C) of GL(n;C) is defined by the set of all matrices X of degree n such that exp tX 6 GL(n;C)

k Lie Groups and Lie Algebras

61

for all t, and the Lie algebra g of a linear Lie group G is the set of those X such that exp tX 6 G for all t. Then g is seen to be a vector space over C and the Poisson brackets are closed in g.

The following Lie groups are examples of the so-called classical linear Lie groups.

i' ?'i K(x,y) = *xKy = K..xV (2.k)

An element A = [a.] of GL(n;C) acts on C as the coordinate

transformation x = a.xJ, subject to which a bilinear form

is transformed to

K(x',y') = ScLy = J^xV (2.5) where

L = tAKA (je v ji.. =3 aV Jimam) iy The set of all elements A which keeps K invariant, that is, K = AKA, is a Lie subgroup of GL(n;C), which we denote by G(k): G(K) = {A € GL(n;C): tAKA = K}

Its Lie algebra is characterized by t(exp tA)K(exp tA) = K (-00 < t < «>) By differentiation, we have *AK + KA = 0

Similarly, GK J (K) is given by G^(K) = {A 6 GL(n;R); tAKA = K}

If the bilinear form (2.k) is replaced by a complex bilinear form

K(x,y) = k..xJy . (y = complex conjugate of y1) the group G (K) leaving K invariant is a Lie subgroup of GL(n;C).

For the particular case that K is the identity matrix I, we denote

Chap. 1. Differentiable Manifolds

62

the corresponding Lie groups G(l), G^ (i), and G (i) by 0(n;C), 0(n), and u(n), respectively. 0(n;C) = {A € GL(n;C); *AA = l}

U(n) = {A € GL(n;C); *AA = i}

0(n) = {A 6 GL(n;g); *AA = i}

are called the complex orthogonal group, the unitary group, and the real orthogonal group of dimension n, respectively. Their identity components, consisting of elements of determinant 1, are denoted by SO(n;C), SU(n), and SO(n), whose Lie algebras are given, respectively, by

AX + v of R .

Example 2. Let P (g) be the real projective space. For the projection tt: R - {0} -> P (R) and a matrix a in GL(n + 1; g), we n 4-1

set a • tt(x) = rr(ax) (x € g - {0}). Then we have a differentiable map (a,TT(x)) -» a • tt(x) from GL(n + 1; R) x Pn(g) to Pn(g), and GL(n + 1; g) becomes a Lie transformation group acting transitively on P^(g)• The isotropy subgroup of GL(n + 1; g) fixing a point

tt(xq) [xq = (1,0,...,0)] of Pn(g) is a closed subgroup H of GL(n + 1; g) consisting of matrices of the form

a* 0A

^ 0 A € GL(n; g)

Similarly, S0(n + l) and the special linear group SL(n + 1; g) [= B 6 GL(n + 1; g); det B = l] act transitively on Pn(g). The isotropy subgroup of S0(n + l) at tt(x0) is the closed subgroup H of S0(n + 1) consisting of all the orthogonal matrices.

a0

a=1

A € 0(n)

/\N 0A

a det A = 1

Hence SL(n + 1; R)/S and S0(n + l)/S are both diffeomorphic to Pn(R).

Example 3- Let V be an N (= n + p)-dimensional vector space and let G(n,p) be the set of all n-dimensional subspaces of V. Fix a basis

(e,,...,e ) and identify V with g . Then the group of all nonsingu­ lar linear transformations of V is identified with GL(N;g). Let W be an n-dimensional subspace of V. Then for a 6 GL(N;R), aW = fax;x 6 W} is an n-dimensional subspace of V. If W and w' are two n­ dimensional subspaces of V, there is an element a 6 GL(N;g) such that

k Lie Groups and Lie Algebras

67

w' = aW. Thus GL(N;R) acts on G(n,p) transitively. If Wn 0 is a particular n-dimensional subspace spanned by e,, ..., e , then b € GL(N;)g) satisfying bWQ = WQ is a matrix of the form A € GL(n;R)

B € GL(p;R) (n + p = N) These matrices form a closed Lie subgroup H of GL(n;R), and since GL(N;g)/H is a differentiable manifold, we have the diffeomorphism a: aH -> aW and thus G(n,p) becomes a homogeneous space of GL(n;R).

[If p = 1, it is the projective space P (R)-] By taking e.., ..., e as an orthonormal basis, we define a positive inner product in V. If W and W are n-dimensional subspaces of V with orthonormal bases a and b,, ., b , respectively, then by renumbering the an subscripts of b. b if necessary, we can find an a 6 SO(n)

n 1'

such that a. =b. (i=l, ...,n). Thus we have aW = W', so that SO(n) acts transitively on G(n,p) = GL(N;R)/5. The isotropy subgroup H of SO(N) at WQ is the closed subgroup of SO(N) consisting

of the orthogonal matrices of the form

ta

A € 0(n) B € 0(p)

(n + p = N)

det A • det B

Thus G(n,p) = 0(n + p)/o(n) x 0(p) is also diffeomorphic to SO(N)/H and hence is compact. G(n,p) is called the Grassmann manifold of n

N/N

planes in R (N = n + p). An n-dimensional orthonormal frame in R is an ordered set of n orthonormal vectors in V (as identified with R ). The group 0(N) acts transitively on the set S(n,p) of n-dimensional orthonormal frames in Rn, and the elements of 0(N) that fix the n-dimensional orthonormal frame (e.,. ,e 9 n7) are the matrices of the form I=nxn identity matrix H

B £ 0(p) Thus we have

S(n,p) =

0(n + P) 0(P)

(n + p = N)

68 Chap. 1. Differentiate Manifolds The manifold s(n,p) is called the Stiefel manifold of n frames in RN (N = n + p).

Exercise 3.1. If G acts effectively on M on the right, show that the map a in Proposition 3-1 is a Lie algebra isomorphism of g into 3E(M).

2 Theory of Connections

00 00

1. FIBRE BUNDLES

1.1 Principal Fibre Bundles and Associated Fibre Bundles

Definition. Let M be a C manifold and G a Lie group. A C principal fibre bundle P(M,G) over M with group G consists of a manifold P

and an action of G on P satisfying the following conditions:

(l)° G acts freely on P on the right: P -» P x G defined by (u,a) -» ua (= Ra u) for u 6 P and a 6 G.

(2)° M is isomorphic with the quotient space of P by the equivalence relation under G, M ^ P/G, and the projection tt: P -» M

is of class C . (3)° P is locally trivial; that is, for each point x of M there 00

exists a neighborhood U in M such that tt"1(u) is isomorphic with the product manifold U x G. This isomorphism is given by a C diffeomorphism i|r: tt"1(u) -» U x G such that

t(u) = (tt(u), 0(h)) (tt(u) =x) (1.1)

0(ua) = 0(u)a (1.2)

where 0 is a mapping of tt~1(u) into G satisfying

A principal fibre bundle P(M,G) will often be denoted simply by P. We call M the base manifold, G the structure group, tt the

projection, and P the bundle space or the total space. For each point x of M, tt"1(x) is a closed submanifold of P, called the fibre

over x. If u is a point of tt"1(x) in P, tt~1(x) is the set of all 69

Chap. 2. Theory of Connections

TO

points of the form ua (a € G) and is called the fibre through u. Each fibre is diffeomorphic to G. A C map \ : M -» P is called a cross section of P if tt «\ is the identity transformation of M. Example 1: Trivial (product) bundle. A product manifold M x G is turned into a principal fibre bundle when it is provided with the

right action of G on itself in such a way that (x,a) -> (x,ab) for x 6 M and a, b 6 G. It is called a trivial (product) bundle. A principal fibre bundle P(M,G) is isomorphic with a trivial bundle if and only if there is a C cross section on all of M (N. Steenrod

[l], P. 36). Example 2: 0(n + l)(sn,0(n)). The isotropy subgroup of 0(n + l) which fixes the point O = (1,0,...,0) of the unit sphere S is 0(n), and hence the quotient space 0(n + l)/o(n) is diffeomorphic to S . 0(n + l)(S ,0(n)) is a principal bundle over S with structure group Q(n). The projection tt: 0(n + l) -> S is given by n-(r) =0 «r [r € 0(n + 1)]. More generally, let G be a Lie group and K a closed subgroup

of G. Every a 6 K maps u 6 G to ua. With this right action on G, G(g/K,K) is a principal fibre bundle. [For the proof of local triviality of g(g/K,K), see C. Chevalley [1], pp. 109-110.] Example 3: SO(*0(S3,SO(3) )• Let s^ be realized by the set of quaternions

S3 = {q = x1 + ix2 + Jx3 + kx\ |]q|| = 1}

For each point q of S the transformation \(q)(€ S0(^)): q' -> q«q' preserves the norm. Denoting by -e 6 S the unit quaternion, we

00 .

define tt : SO(^) -» S3 by TT-r = -er [r 6 S0(^)]. Then since TT-\(q)

= -e «X(q) = -eq = q, X is a cross section. Hence SO(^) is a trivial bundle, that is, SOOO is diffeomorphic to S3 X SO(3). Example k: Bundle of linear frames L(M). A linear frame u at a point x of a C manifold M is an ordered basis (X-,,Xp,...,X )

1 Fibre Bundles

71

(n = dim M) of the tangent space T (m). Let L(m) be the set of all linear frames u at all points x of M, and define tt : l(m) -> M by tt-q = x. The general linear Lie group GL(n;g) acts on L(m) on the right as follows. For u = (X-, ,Xp,... ,X ) at x and a = [a.] 6 GL(n;g), ua

is a linear frame at x given by ua = (a!:X .,.. . ,aJX .) (i,j = 1, 2, .. ., n). It is clear that GL(n;g) acts freely on L(m) and tt-u = tt«v if and only if v = ua for some a € G. Let {U;x } be a local chart about x in M. Every frame u at x is uniquely expressed in the form u = (X,,Xp,... ,X ), where

bx x and [X.] is a nonsingular matrix of degree n. This implies that TT"1(u) is in one-to-one correspondence with U x GL(n;g), and hence

we can take (x ) and as the local coordinate system in the product manifold tt"1(u). Thus L(m) is made into an (n + n ^dimensional C manifold. It is a trivial task to ascertain that L(m)(M,

GL(n;g)) is a principal fibre bundle. L(m) is called the bundle of linear frames or the frame bundle over M. We note that a linear frame u is regarded as a nonsingular linear mapping of g onto

Tx(M). For if we let 71 = U-^^,... ,-en), where ^ (i = 1, 2, ..., n) is the basis of R given by

l.

•ft.

= (0,...,0,1,0,...,0) (i = 1, 2, ..., n) (lA)

then a linear frame at x is obtained by a linear map u: g -» T (m) such that u-Tl = (X-^Xp,.. . ,X ), i.e., u--e. = X± (i = 1, 2, . .., n). The action of a = [a^] € GL(n;g) on L(m) at x is given by (ua) =

1 ,V 2 jJ n j' x 1 y n y '

(a^X .,X^X .,. .. ,aJX .) = (a^u«-e .,. .. ,aJu«-e .), which by the linearity

of u equals (ua^-e .,. .. ,ua -e .) =u(aTl). Hence

(ua).Tl = u(aTl) (1.5)

and this implies that ua: g -» T (M) is the composition of the map a: gn -> gn with the map u: gn -> T (m) .

Chap. 2. Theory of Connections

72

Let P(M,G) be a principal fibre bundle and F be a C manifold on which G acts on the left: (a, %) -* a? for a 6 G and 5 6 F. We let G act on P x G on the right: (u,5)a = (ua^a"1?). The quotient space of P x G defined by the equivalence relation under G is denoted by E = P x„ F. We define the projection tt : E -> M by tt ((u,?)g) =

G iL iij

tt(u). If x € M, the set tT1(x) is called the fibre over x in E. Since every point x of M has a neighborhood U such that tt"1(u) ^ U X G (3 (x,a)), G acts on tt"1(u) x F on the right by

(x,a,5) -» (x^ab^b*1?) (b € G) for (x,a,|) € U x G x F. Hence the isomorphism of tt*1(u) with U x G induces an isomorphism of tt"1 with U x F. tt"1(u) is an open submani­ fold of E and is C diffeomorphic to U x F under the isomorphism tt,-, (U) ^U xF, and consequently rr„ is a C mapping of E ontoM. E = CO

E(M,F,G,P) is called the fibre bundle over Mwith fibre F associated

with the principal fibre bundle P(M,G). It is an easy matter to prove.

Proposition 1.1. Let P(M,G) be a principal fibre bundle and E an associated fibre bundle with fibre F. The quotient projection P x F -» E [= (P X F)/G] defines for every u 6 P a C map u: F -> E, given

(ua)5 =u(a?) (1.6)

by u(?) = (u,5)G. It satisfies the relation similar to (1.5):

Example 5*. Tangent bundle T(m). Let GL(n;g) act on the vector space

g . The tangent bundle T(m) over M is a fibre bundle over M associated with the bundle of frames L(m) with fibre gn. The fibre of T(m)

over x 6 M can be considered as the tangent space T (M), and T(m) itself is the set of all tangent planes of M: T(m) = U T (m). Let

./x

{U;x } be a local chart about x in M such that tt"" ^-(tj) is diffeomorphic to U x g • Denoting by tt again the projection T(m) -» M, the

coordinates of a point u € tt"1(x) are given by (x1,y ), where the y are the components in U of a vector X at x: X = y (x)(b/bx1) (i =

1, 2, ..., n). If {u';x } is another chart at x, the coordinate transformation in tt~1(U H u') is

x1'-*1'^) yi'=Afyi (a£'-^) (1.7) bx

1 Fibre Bundles

73

If we adopt the convention of writing i = n + i for any index i, we can describe the 2n coordinates of a point u in T(M) as x

(A, B, C, ... = 1, 2, ..., 2n), with x1 = x1

= y1, and (1.7)

can be written

x xA'= A', x ^xj The Jacobian of the transformation is bx

A' bx

bx^

bx^

tf*

bx^

bxJbx

>0

bx

bx^

and hence T(m) is orientable. The coordinate system (x ) was assigned to tt"1(u), which is diffeomorphic to U x G. Thus if M is covered by a single coordinate system, T(m) is diffeomorphic to the product manifold M x g .

Example 6: Tensor bundles.

(r)( be the tensor space of tensors Let T>

of type (r,s) on the vector space Rn as defined in 3.1, Section 3, Chapter 1. GL(n;g) acts on T^n as a linear transformation. With T as fibre, we obtain the tensor bundle T/ \(M) of type (r,s) over M associated with L(m) .

Example J: The principal fibre bundle P(M,H). Let H be a closed subgroup of the structure group G of P(M,G). Since H acts on P on the right as a subgroup of G, we have the quotient space P/^\ of P. On the other hand, G acts on the homogeneous space G/H (with projection p: G -» G/h) on the left, and by this action we obtain the fibre bundle E(M,G/H,G,P) [= P xG (g/h)] associated with P(M,G). We can identify P/H with E by the correspondence (u,aH) € P xG (G/H) -> ua 6 P/H

where u 6 P and a € G. P(E,H) is a principal fibre bundle over E = P/H with structure group H and projection a: P(E,H) -> E defined by the map u € EH of P(E,H) onto E. The local triviality of P(E,H) is proved as follows. Let U be an open set of M such that tt~1(u)

Chap. 2. Theory of Connections

7^

U x G/H and V be an open set of G such that p~1(v) ^ V x H. Then e can we cantake takean anoper: open set A of tt~1(u) corresponding to U x V, obtain­ ing oT1(A) ^ A x H.

A manifold M is said to be parallelizable if there are C vector fields X.,, Xp, ..., X (n = dim M) defined on all M such that for every point x of M, the set {X.,(x), Xp(x),...,X (x)} is a basis of T (M). In this case the set of vector fields Xn, — ,X is called a parallelization of M. We formulate this property in the form of a proposition as follows (whose proof may be found, for instance, in R. L. Bishop and S. I. Goldberg [1], pp. l60-l6l).

xv ' 1* 9 n

Proposition 1.2. M is parallelizable if and only if there exists a diffeomorphism |jl: T(m) -» M x R (n = dim M) such that the first factor of (j, is tt: T(m) -» M and the second factor of (j, is a linear function T (m) -» R for each point x of M.

12 7

Example 8. The spheres S , S , and S are parallelizable (R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. ft-(1959), 86-89).

Exercise 1.1. Use (1.2) to prove that the right action of G on the

total space of P(M,G) is a right translation. [Hint: For (l.l), i.e., i|r : u -> (tt(u) ,0(u) ), we have ua -» (tt(u) ,^(ua)) by virtue of

(1.2).] Exercise 1.2. S2n+1(Pn(c),S1). In the set £n+1 - {0} obtained by removing the origin from C we define an equivalence relation as follows: Two points z = (z ) and w = (w ) are equivalent if there is a nonzero complex number k such that z = kw1 (i = 1, 2, ..., n + l). The set of equivalence classes obtained by this equivalence relation is the n-dimensional complex projective space and is denoted by Pn(£). Show that S2n+1(Pn(c),S1) is a principal fibre bundle.

Using quaternions, define similarly the n-dimensional quaternion projective space Pn(Q), and show that S n (Pn(Q),S ) is a

principal fibre bundle.

1 Fibre Bundles

75

1.2 Reduction of Structure Groups

We first relate our definition of a principal fibre bundle to the definition and construction by means of an open cover by coordinate neighborhoods. For a principal fibre bundle P(M,G), let {U } ­ be an open cover of M by local charts such that each tt-1(U ) is

pd dp dp

represented as a trivial bundle by 0 : tt-1(U ) -» G [see (3)°, 1.1, Section 1]. We define for each a, P for which U 0 U^ $> a map ♦ a : u fl U0 -> G in the following way. If x 6 U D UQ, let u € tt_1(x) and set

♦ pa(x) =^(u) • (^(u))"1 (2.1)

pd

Then it can be shown that the definition of +.pd(x) is independent

of the choice of u in tt_1(x). In fact, if u' is also in tt_1(x), then u' = ua (a € G). We then have, by (1.2), 1.1, Section 1,

0p(u') • (^(u'))"1 =^(ua) • (^(ua))"1

= (^(u)a)(a-1(0a(u))-1 = 0p(u) • (0JU))"1

as required. The functions i|fQ are called the transition functions corre­ sponding to the open cover {U } ,-. . They satisfy further properties

*Ya(x) = Vx)Wx) x 6 Ua n Up n UY (2'2)

Old dp pd

[The relations t (x) = -e and i|f . (x) = (tft (x))_1 are an immediate consequence of (1.2).] We then have the theorem due to N. E. Steen­

rod [1], p. Ik: Theorem 2.1. Let [U -. be an open cover of a C manifold M and L djd€h G a Lie group. If a mapping +R : U H U -> G is assigned for every nonempty intersection U PI IL in such a way that (2.2) is satisfied,

then we can construct a principal fibre bundle P(M,G) whose transition functions corresponding to {u} - are the #fi .

76

/ °° Gr

Chap. 2. Theory of Connections

Let P(M,G) and P'(m',G') be two principal fibre bundles. A map f: P -» P is said to be a homomorphism if f consists of a C map f-,: P' -* P and an injection (monomorphism) f • G' -> G such that

r—rG

fD(u'a') = f-nCu') • f„(a') for all u' € P' and a' € G'. Every

homomorphism f: P' -» P sends a fibre of P' into a fibre of P and induces a mapping f : M' -» M such that f • tt' = tt • fp, where tt and tt' are the projections tt: P -» M and tt': P' -> M', respectively. A homomorphism f: P' -> P is called an embedding if f is an embedding and f_ is an injection. When P' is identified with f^P'), G with t (G')9 and M with fAlA'), P'(m',G') is called a subbundle of P(M,G). Furthermore, f: P' -» P is called the reduction of the structure group G of P(M,G) to G' if M' = M and the induced map f : M' -> M is the identity transformation. The subbundle P'Cm'^G') thus obtained is called a reduced bundle. Given a principal fibre bundle P(M,G) and Lie subgroup G' of G, we say that the structure group G is reducible to G' if there exists a reduced subbundle P'(M',G').

Theorem 2.2. In order that the structure group G of a principal fibre bundle P(M,G) may be reduced to a Lie subgroup G' of G, it is necessary and sufficient that there exists an open cover {UJ ,-. of M with a set of transition functions t« pawhich take their values in G'.

Proof: Assume that G is reducible to G'. Let P'(M,g') be the sub­ bundle obtained by this reduction and tt': P' -» M be the projection. Then we can take an open cover {U ] .. of M such that tt/~1(tJ) is provided with an isomorphism i|f': u -> (tt'(u),0 (u)) of XT'"1 (U ) onto

U x G'. The corresponding transition functions i|r_ on U PI IL

take values in G'. Using this isomorphism i|r' we can obtain an isomorphism t of tt~1(u ) onto U x G in the following way. Since every

point v of tt_1(tj ) is expressed in the form v = ua (a € G), we have

by setting 0 (v) = 0'(u)a an extension of t': u -» (tt'(u),0 (u)) in

P' to t : v -> (tt(v),0 (v)) in P. Clearly, this extension does not depend on the expression of v (= ua). Hence i|f (u) defines an isomorphism

1 Fibre Bundles

77

of tt"1(u ) onto U x G, and the corresponding transition functions take values in G', since

= ^'(u)(^(u))-1 6 G'

pet pot

where tt(v) = tt(u) = x € U flU.. Assume, conversely, that there is an open cover {U ] ,. of M for which the transition functions i|fQ all take values in G'. to is a different iable mapping of U fl U. (/ f() into G such that tfl (U fl Uft) CG. Being a Lie subgroup of G, G' is second countable by definition (see 4.1, Section 4, Chapter l) and the connected component GL of G containing the identity element -e is the maximal

integral submanifold through -e of the involutive distribution corresponding to the Lie algebra of G'. Hencepeti|f is a different iable mapping of ct U flpUa into G' with respect to the different iable struc­ ture of G' (see Proposition 4.1, 3-4, Section 3> Chapter l). By Proposition 2.1, we construct a principal fibre bundle P'^G') by {U } £. and i|f . Also, we have by the monomorphism f : G' -» G a mapping ctf of P into P defined by

ct p ct p 9 ct ct ct ct ct

f : tt,"1(U In'xG'n x G ^ tt"1(U )

and f coincides with fQ on U fl UQ . Thus f: P' -» P is an injection, namely, P'(M,G') is a reduced bundle of P(M,G).

Let E(M,F,G,P) be a fibre bundle associated with a principal fibre bundle P(M,G). A cross section of E is a mapping \ : M -» E such that tt «X is the identity transformation of M. For the cri­ terion of reducibility of P(M,G), we have the following: Lemma 2.3. Let H be a closed subgroup of the structure group G of P(M,G). P is reducible to the subbundle Q(M,H) if and only if the associated fibre bundle E(M,G/H,G,P) admits a cross section M -> E (=P/h). (For the proof, see S. Kobayashi and K. Nomizu [2], Vol. I,

p. 57.)

Chap. 2. Theory of Connections

78

As for the property of cross section in E(M,F,G,P) we know the following fact (R. Godement [1], p. 150). Lemma 2.k. Let E(M,F,G,P) be a fibre bundle over a paracompact manifold M such that the fibre F is diffeomorphic with Euclidean

space R . Then every cross section X of E defined on a closed subset A of M can be extended to all of M.

We have hitherto assumed that the structure group G of a

principal fibre bundle or of its associated fibre bundle is a Lie group. We recall the Iwasawa decomposition of a Lie group (K. Iwa­ sawa [1]): Theorem 2.5. A connected Lie group G is diffeomorphic with the direct product of any of its maximal compact subgroup H and a Euclidean space. Combining Lemma 2.k and Theorem 2.5, we have by Lemma 2.3

Theorem 2.6. Let P(M,G) be a principal fibre bundle over a C manifold M with connected structure Lie group G, and let H be a

maximal compact subgroup of G. G is reducible to H as the structure group of P if M is paracompact.

Example. Let L(m) be the bundle of linear frames over a paracompact manifold M. The coset space E = L(M)/o(n) (n = dim M) is a fibre bundle associated with L(m) whose fibre is GL(n;g)/o(n). Since any nonsingular matrix of degree n is represented as the product of an element of 0(n) and a positive definite symmetric matrix of degree

n, called the polar representation of a nonsingular matrix (see, e.g., T. Yamanouchi and M. Sugiura [1], p. 1^2), the fibre GL(n;R)/ 0(n) is diffeomorphic with R ^ ' , and hence the structure group of L(M) is reducible to 0(n). 1.3 Covering Manifolds Let X be a connected and locally arcwise connected topological space. A connected space B is called a covering space of X with projection

1 Fibre Bundles

79

p: B -» X if every point x of g has a neighborhood U such that each connected component W (c B) of p~1(u) is open in B, and p: W ■* U is a homeomorphism. A covering B of X is called a universal covering

space if it is simply connected. It is known that the universal covering space B of X is a principal fibre bundle over X with structure group ni(x), where ^iCx) is the fundamental group of X. Moreover

(N. E. Steenrod [l]. pp. 6I-67), Theorem 3.1. Let H be a subgroup of tt1(x). The covering space B = B/H of X is a fibre bundle over g "with fibre tt1(x)/h associated with the principal fibre bundle bQ^tt^X)). If H is a normal subgroup of TTiQc), then B = B/H is a principal fibre bundle with group II = tt1(x)/h, called a regular covering space of X. The right translations of B by the elements of this group map each fibre onto itself and their action (on B) is properly discontinuous. These right translations are referred to as the covering home omorph isms or, classically, the deck transformations of B. 00

Let M be a C manifold and B its covering space with00 projection 00 p: B -» M. We call p a differentiable covering of class C or C 00 covering if p is a C map of maximal rank. In view of the inverse function theorem, this condition is equivalent to the requirement that if U is a neighborhood in M and V is a connected component of P_1(u), the projection p: V -» U is a C diffeomorphism.

00 00

Let II be the group of deck transformations of a covering p: B-* M. CD

Theorem 3»2. B admits a unique C manifold structure for which p: B -» M is a C covering. Suppose that the covering is of class C ; then every element of II is a C diffeomorphism of B. CD

Proof: Let {U;x } and {u';x } be local charts in M such that UflU' ^ 0. If V is a connected component of p_1(u), the local coordinates

y of a point v of V are given by y (v) = x (p(v)). Similarly, if V' is a connected component of p~1(U/) containing v, the local coordinates y of v on V' are y (v) = x (p(v)). Then, on V H V', we get by /by = bx /bx . Hence B is a C manifold and p: B -» M CO

is a C diffeomorphism, which completes the proof of the first

Chap. 1. Theory of Connections

80

statement. For the proof of the second statement, let y € II and V be a connected component of p_1(u). Then the transformation of V

-.00 00

by y: v -» y(v) is given by v -> (p| / 0-1p(v), where v is an arbitrary point of V. Since this mapping is C , so is y. Similarly, Y is C , and hence y is a C diffeomorphism.

Example 1. R is the universal covering of S .

Example 2. S (n ^ 2) is the universal covering manifold of the projective space P (g).

Exercise 3-1: Lens space. Let k and 4.,,..., & be relatively prime integers and consider the map

/ 0 1 n\ / 0 ^rri 1 1 n n >

(z ,z ,... ,z ) -» (z exp —, z exp k ,..., z exp ——) n c2n+l / 0,... 1 n\ Sn S = | (zf ,z ,z c; 6pn+l C ; "Si=0 ||zin

Then this map defines the cyclic group Z, . The orbit space M =

S /z, is called a lens space. Show that this space is a different iable manifold and that ~kZ, can be used as the deck trans­ formations of M. 2. COMIECTIONS IN PRINCIPAL FIBRE BUNDLES

2.1 Connection in P(M,G) Let M be a C manifold and P(M,G) a principal fibre bundle over M with structure group G. The action of G on P induces a homomor­

phism a of G into the Lie algebra 3E(P) of vector fields on P (see Proposition 3.1, k.J, Section k9 Chapter l). For each element A of the Lie algebra g of G, A 6 a (A) is called the fundamental vector field on P corresponding to A. Since G sends each fibre to

itself, the vector A is tangent to the fibre at each point u of P.

*u

2 Connections in Principal Fibre Bundles

81

Proposition 1.1. The map a: A -» A of g into the tangent space T (P) of P at u is a linear isomorphism of g onto the tangent space at u of the fibre through u. Proof: We only have to show that in the correspondence a: A -» A , if A is not the O-vector in a, then A never vanishes. Assume that A =0; then since the action of G on P is free, the one-parameter group of transformations Ra (a = exp tA) of the fibre through u t "t is trivial, and hence A = 0.

u&u

We denote by G the tangent space at u of the fibre through u. G is then the set of all tangent vectors of the form A . Let a be an element of G. By Theorem 3.2, 2.3, Section 2,

a a, a a a, a a « «

Chapter 1, (R ) A is induced by the one-parameter group of transformations R-i (=RR R-i), and since A -i(a, ) = a_1a,a = exp(t ad(a~1)A), where ad(a~1)A = (A _i) A (see 4.3, Section 4, Chapter l), we have

Proposition 1.2. (R ) A is the fundamental vector field corresponding to ad(a_1)A (€ g).

We shall show that the fundamental vector field can always be obtained if we utilize the coordinate bundle structure of P(M,G). Let {U } £A be an open cover of M by coordinate neighborhoods such

that each if^U ) is a trivial bundle, i.e., there is an

isomorphism \|f: u -> (tt(u), 0(u)) € U x G, satisfying the condition (1.2),

i • R = R 0 (1.1)

1.1, Section 1:

We have the transition function on U PI UQ / 0:

Wx) = V^ • ^1M)€ G [x = tt(u) € u*n V and since 0~1 induces a diffeomorphism of a 6 G with u € tt"1(x),

u = ^(a) = SZfp-^ba) (1.2)

Chap. 2. Theory of Connections

82

% Per a

where we have put b = +_ (x). In each tt"1(u ) the fundamental vector field A corresponding to A € g is defined by

A*f = tlim U t->0 at i [f(R u) - f(u)] (1.3) where a = exp tA, and f is any differentiable function defined on tt'^U ). On the other hand, a vector A of g induces a differenti­ able vector field A on the fibre n-1(x) by the differential of jZf-i: a = (0"1) A. x>

Proposition 1.3- A is identified with the fundamental vector field A*.

Proof: Let f be as above. Since 0 _1 induces a diffeomorphism a € G -» u = tt_1(x),

u ra. * u rot

A f = ((Qf"1) A) f = A • (f • jfC1)(a)

= lim \ [f • jf( "1(Rn a) - f • 0 ■1(a)] (a = exp tA) and by (l.l) the last term equals

limi [f-R at 0_1(a)"- f.^"1(a)] ^0t " which is equal to (1.3) by virtue of the relation (1.2). Definition. A connection T in P(M,G) is an assignment of a sub­

space Q to the tangent space T (P) at u of P satisfying the following conditions:

(1)° Tu(P) = G^ + Q^ (direct sum).

W° *R au = (Ra} A f0r a ^ G­ (3)° Q depends differentiably on u. Condition (l)° means that G and Q are complementary to each other

in T (P), and (2)° implies that the distribution u -» 0 is right invariant under the action of G. We call G the vertical subspace and Q the horizontal subspace of T (P). A vector X 6 T (P) is

2 Connections in Principal Fibre Bundles

83

called the vertical (resp. horizontal) component of X if X lies in G (resp. Q ). Every X 6 T (P) is uniquely expressed in the form X =«X +AX

where ** (resp.A ) denotes the vertical (resp. horizontal) projection of T (P) onto G (resp. Q ). Given a connection F in P(M,G) we define the connection form go

of r as follows, oo is a g-valued 1-form on P such that

cu(A*) = A co(Zu) = 0 (1A)

where A 6 G is the fundamental vector at u corresponding to A € g and Z is any element of Q . Clearly, uo(X) =0 if and only if X

Proposition l.k. The connection form has the property

(Ra)*oj = ad(a_1)oj (1.5)

that is, co((Ra)^X) = ad(a-1)co(x)

[An r-form on P possessing property (1-5) is said to be equivariant.] Proof: It suffices to prove the statement for the following two cases.

(1) X is horizontal. In this case, (E ) X is horizontal by condition (2)° , and hence u>( (R ) X) =0. On the other hand, cw(x) = 0 gives ad(a_1)oo(x) = 0.

(2) X is vertical. For this case, we can assume that X is some fundamental vector field A* corresponding to A € g. Then by

Proposition 1.2, (R ) A is the fundamental vector field corresponding to ad(a_1)A € g. Recalling the definition of the pullback of 1-forms (see 2.3, Section 2, Chapter l), we have ((Ra)*uj)u(x) = %a((\)*X) = coua(ad(a-1)x) = ad(a" 1)(a\i(x))

as required.

Chap. 2. Theory of Connections

8k

Proposition 1.5. Given a connection r in P(M,G) and a vector field X € 3E(m), there exists a unique vector field X in 3f(P) such that

tt (X ) = X. X* is right invariant by G, that is,

v a7* a v

(R ) X* = X* • R (a e G) We call X the horizontal lift (or simply the lift) of a vector field X.

Proof: The projection tt: P-» G being differentiable, it induces a linear map tt : T (P) -* T (m) at each point u of P, where x = tt(u). In the decomposition T(P)=G + Q , tt G =0, since G is spanned by the fundamental vectors A which are tangent to the fibre tt~1(x). Hence tt maps Q isomorphic ally onto T (m) . From this isomorphism

the existence and uniqueness of the lift of a vector field X on M follow. Let U be a neighborhood of M containing x such that tt~1(u) ^ U x G. There exists a vector field X in tt~1(tj) such that tt X = X. The horizontal component AX of X is a lift of X and is invariant by R . Now, by the uniqueness we have/X = X, proving the

ct ct °

statement.

rct rct ct ct

Let {U } ,. be an open cover of M by coordinate neighborhoods

such that each tt~1(U ) is a trivial bundle, that is, t : u -» (tt(u),

(z),f] Theorem 3A (Bianchi identity). W. = 0. Proof: Let [E ] be a basis of g, and set 00 = u/*E , CI = Q^E . Then we have from (3»2) the equations

to" = - § cYapoY A^+(f (3.6) where c are the structure constants of G. Now, for proving Theorem 3A, we have only to show that dH(X,Y,z) = 0, where X, Y,

and Z are horizontal vectors. Applying exterior differentiation to both sides of (3-6), we have

2 y6 2 yP

0 = d do/* = - i c*dcuY A^+icVA duf + 6Cf Since oo(x) = 0 whenever X is horizontal, we obtain dTi(X,Y,z) = 0.

Note. In P(M,G) let 0 be a 2-form with value in g. To inquire into the condition that 0 may be the curvature form of a connection F in P is called Novikov's problem. For dim M = 3 or ^ and G = SU(2), the complete solution was found by S. P. Tsarev [1] in 1982.

2 Connections in Principal Fibre Bundles

95

2.k Homomorphisms of Connections

P P b oo

Let P(M,G) and P'(m',G') be two principal fibre bundles. We recall the definition of a homomorphism f of p'(m',G') into P(M,G): f con­ sists of a C map f : P -» P and a monomorphism f : G -» G such that

f (u'a') = f (u')-f-(a') for all u' € P' and a'^G'. f then sends a fibre of P into a fibre of P and induces a C map f„: MM' -» M such that -►P

V17' =TT,fP *'Jf P J* ^^ m' y—^M

that is, the diagram commutes. Theorem Ij-.l. Let f: P'(m',g') -* P(M,G) be a homomorphism such that the induced map f : M' -» M is a diffeomorphism. Given a connection

Tx in P', there exists in P a unique connection r such that the horizontal subspaces of F are the images of horizontal subspaces of r' by the differential f * of the map f : P' -* P.

Proof: Let u € P, u' € P', and a € G such that u = R f(u'). Denote by Q / the horizontal subspace of T /(p') with respect to r'. We define the subspace 0 of T (p) by

Si - Vp^V) ik-2) a s. d a ir

Then the distribution u -» Q is seen to be a connection in P as follows. For u = R f *(u'), we have ub = R.R f *(u'), so that by

and hence Q is right invariant. Our next task is to show that T (P) is the direct sum of Q and the tangent space G to the fibre, so that Q is a horizontal subspace of T (p). By the right invariance of Q we may assume for simplicity that u = f(u'). Since f : M' -» M is by hypothesis a diffeomorphism, f is a linear isomorphism of T / (M')

onto T (M) where x = tt(u) = TT(fp(u')) and x' = (u'). By taking

Chap. 2. Theory of Connections

the differentials of the maps in the commutative diagram (^.l),

M**P

we have for fp*(Qv,') = Qy,>

V 'n* ' V = TT* • V(V> = TTA

"" U X kX U

which implies that tt maps Q onto T (M). Assume that Q PI G ^ 0, and let X = fp*(x') (x' € 0 /) be a nonzero vector in P at u such

that X^flG . Then X 6 G implies that tt^X = 0. Applying (k.3) to X', we have

M**P*

f * • tt'x' = tt • fL*(x') = tt x = 0 and hence f^-arr'x' = 0. But since f__* and tt are both isomorphisms, f-.-KTT'x' = 0 implies that X = 0, which is a contradiction. In the case where M' = M and f is an embedding of P'(m',G') into P(M,G) (see 1.2, Section l), we say that a given connection F

in P is reducible to r' in P' if f maps r' into T. In the case where f is an automorphism of the bundle P onto itself which maps a given connection r into itself, we call f an automorphism of the connection of P.

Theorem k.2 (S. Kobayashi and K. Nomizu [2], Vol. I, p. 80). Let oo and O (resp. uo' and Ox) be the connection form and the curvature form of r (resp. r'), respectively. Then f*oo = foo' and f*Q = fQ', where foo' and fQ' are the g-valued forms on P' defined by (foo')(x')

000

= f (u/(x')), (fO,)(X,,Y/) = f (Q,(X,,Y,))> and f denotes the homomorphism g -> g induced by f : G -» G.

Theorem k.J. If u € P and u € f^u') € P, then f. maps h(u') onto h(u) [resp. h (u') onto h (u)], where h(u) and h(u') [resp. h (u) and h (u')] are the holonomy group and the restricted holonomy group

of r (resp. r') with reference point u (resp. u').

2 Connections in Principal Fibre Bundles

97

2.5 Holonomy Theorem 00

Let P(M,G) be a principal fibre bundle over a connected C manifold

M and u a point of P. We denote by P(u) the set of all points in P each of which can be joined to u by a piecewise differentiable horizontal curve. Then P(u) is connected and P(u) = P(v) for any point

v € P(u). Also, h(u) = h(v) [resp. h (u) = h (v)] by Proposition 2.5, 2.2, Section 2. Lemma 5.1. Let u € p(u0) and- v = ua (a € G). v € P(u0) if and only if a € h(uQ). Proof: If v € P(u0), u and v (= ua) are joined by a horizontal curve. Hence a is an element of h(u). Since u is joined to uQ by a horizontal curve, h(u) = h(un) by Proposition 2.5, 2.2, Section 2. Hence a € h(uQ). Conversely, if a € h(uQ), then v = ua € P(uQ). Lemma 5»2. Every coordinate neighborhood U of M has a differenti­ able cross section \ : U -> P such that \ (u) € P(uQ).

Proof: Let x be a point of U. By a suitable coordinate

transformation we can take a coordinate system (x ,...,x ) in U such that x

is the origin. U is then covered by rays of the form x = c t (i = 1, 2, ..., n), where c = const for each ray. For a fixed point u € P(uq) H tt-1(x), the lifts passing through u of all rays determine a cross section X(u) as indicated above. X(x) is differentiable.

For the lift t (ct) of a ray x = ct is a solution of the differential equation as given in (2.1), 2.2, Section 2, and consequently, different iable in c and t, that is, differentiable on U. 00

Theorem 5.3. Let M be a connected, paracompact, and C manifold and P(M,G) its principal fibre bundle with a connection F. Then:

(1) P(uQ) (uQ € P) is a reduced bundle of P(M,G) (see 1.2, Section l) with structure group h(uQ). (2) The connection T is reducible to a connection r' in P(un) (see Theorem k.l) whose holonomy group is h(un).

Proof: (l) For a point u = tt~1(x) Crr(u), we can determine an element a of G by u = \(x)a. Since an isomorphism i|r : TT"1(Uof) ^ U01 x G

Chap. 2. Theory of Connections

98

is given by +(u) = (x, a = (ZL (x)), the restriction of t to tt~1(U ) H P(u0) gives an isomorphism of tt_1(u ) with U x h(uQ) by virtue

of Lemma 5.2. The transition function + (x) = 0R(u)(0 (u))"1 restricted again to P(iO is an element of h(uQ), and it is differ­ entiable by Proposition If. 2, 3. If, Section 3, Chapter 1. Thus P(uQ) is a principal fibre bundle over M with structure group h(uQ), which is a reduced bundle of P(M,G). (2) For any point u € P(u0) the horizontal subspace Q of

T (P) is contained in T (P(u0)), since 0 is tangent to all horizontal curves starting at u which are necessarily contained in P(u~). By using Q in each tangent space at u 6 P(uQ), we can define a connection r' in P(uQ) whose holonomy group is just h(uQ). P(uQ) in Theorem 5.3 is called the holonomy bundle through uQ € P(M,G).

Theorem 5.if Holonomy theorem (W. Ambrose and I. M. Singer [l]). Let P(M,G) be as above with a connection F, and t) (un) be the Lie algebra of the restricted holonomy group h (u0) [u0 € P(M,G)]. Then

t) (u ) is a subalgebra of the Lie algebra g of the structure group G generated by all elements of the form Q (X,Y), where u runs over the points of P each of which can be joined to u0 by a horizontal curve, and X, Y are arbitrary horizontal vectors at u in P(M,G). Proof: Let T* be the connection in the holonomy bundle P(u0) and

denote by u/ its connection form. Then u/ is the restriction to P(u0) of the connection form 00 of the connection T in P(M,G), and

hence its exterior differential du>' is also the restriction to P(u~) of the exterior differential doo. If 0 and Q' are the curvature forms of r and r', respectively, then we have at u 6 P(uQ),

nu(x,Y) =^(x,y) (5.1)

where X and Y are horizontal vectors at u. By Theorem 5.k it suffices to prove the statement for the holonomy bundle P(un); that is, we may assume that P is P(uQ) so that every point of P can be joined to u0 by a horizontal curve and Q = ^ (un)* ^e^ &' 1°e ^e subalgebra

2 Connections in Principal Fibre Bundles 99 of g generated by all elements of the form Q (X,Y), where u 6 P and

X, Y are arbitrary horizontal vector fields on P. Let T'(P) be the direct sum of the horizontal plane Q and the set of fundamental

vectors at u corresponding to cj'. T'(p) determines on P a distribution T'(p). If we denote by p' the (natural) projection T'(p) ->

T (P), p' is differentiable and /i =>S«p/ where /i is the horizontal projection.

Let {E } (a = 1, 2, ..., r: r = dim G) be a basis of g. Then oo = u/*E a , where each u/* is a differentiable 1-form on P. We define

3"(x) = u/*(x) - of(p'.X) [X € 3E(P)] Then we have, from (3-6),

doo = - -2cywT 3 A go by virtue of (5-1), that is, Q°^(x,Y) = 0°^(/X/*Y) = Q^p' viX,p'viY). Hence each go = 0 is completely integrable by the Frobenius theorem,

and the resulting maximal integral manifold P contains all the hori­ zontal curves starting at u0 because u/*(x) = 0 for every horizontal vector field X on P. We have P = P by assumption, and this means

that for X € £(P), oj(x) = u/*(x)E = u/*(p'-X)E is g-valued, that is,

S' =9­ 2.6 Local Holonomy Groups

Let t and t ' be two curves in M. t is said to be equivalent to t ' if t is obtained from t ' by identifying the finite parts in t' of the form y"1Y with single points. If t is a piecewise differentiable closed curve starting and ending at a point x 6 M and p, is a piece­ wise differentiable curve joining a point xn 6 M to x, we call the curve mT1T|j) a lasso at xn.

The notions of local and infinitesimal holonomy groups on P(M,G)

are due to H. Ozeki [1] as the generalization of the cases for linear connections considered by A. Nijenhuis [3]­

100

Chap. 2. Theory of Connections

Lemma 6.1. Factorization lemma (A. Borel and A. Lichnerowicz) Let {U } .. be an open cover of M by neighborhoods. A piecewise differentiable 0-homotopic curve x(t) at xQ [t € [0,1]j xQ = x(0)]

is equivalent to the product of finitely many piecewise differenti­ able lassos at x, each of which is small in the sense that its loop

part is contained in a single member of {U } -. . Hence every element of the restricted holonomy group h at u 6 P(M,G) is decomposed

into the product of finitely many elements each of which is induced by the parallel displacement of the fibre along a small lasso at tt(u) = x 6 M.

Proof: We recall Lemma 2.7, 2.2, Section 2, which says that a homotopy deformation f(t,s) (t, s 6 I = [0,1]) of a curve x, : f(t,0) = xQ, f(t,l) = x.^ can be covered by a finite number of the members in {U } ^., which we still denote by {U } . f(t,s) is also covered by a set of coordinate neighborhoods {V.}. By taking its refinement, which is still denoted by {V.}, and also by dividing the interval I of s and t, we assume that x, covers a net consisting of a finite number of meshes over f(t,s) each of which is contained in a single coordinate neighborhood V. (c Ua) of {V.}. In V. we replace the threads of a mesh by differentiable curves which still

lpii

lie in UP. Then, for a net v in M corresponding to the lattice points a, b, c, d, e, f of the subdivision of I x I, we see that v is equivalent to the product of the two lassos abefa and afebcdefa. For the general net we proceed with the similar proof by induction

in which all lassos are assumed small enough to satisfy the condition of the lemma.

In the remainder of this section we assume that M is a con­ 00 nected and paracompact C manifold. For defining the holonomy group at u 6 Gx [tt(u) = x 6 M], we restrict a closed curve t as contained

2 Connections in Principal Fibre Bundles

101

in a single neighborhood U c M (see 2.2, Section 2). Then we obtain a (connected) Lie subgroup of the restricted holonomy group h (u) at u. Denoting such a group by h (u,U), let LL, LU, ... be a sequence of neighborhoods of x 6 M such that

Ui3 UI+1 i=l n Ui = X (i = 1, 2, ...) Then it is obvious that

^(u,!^) =Dh°)u,U2) 3 ... We put

i=l x

h*(u) = n h°(u,u.)

•# 0

and call h (u) the local holonomy group at u 6 P(M,G). It follows from this definition that there exists an integer k such that dim

h(u,U, ) = const for m = 0, 1,

Proposition 6.2. h (u) is a Lie subgroup of h (u). Proof: The statement follows from the fact that h (u) is a Lie subgroup of G and from Yamabe's theorem stated in 2.2, Section 2.

Let U run over the neighborhoods ofrr(u) = x € M. Then h*(u) = n h°(u,U) Hence

*0

Proposition 6.3- Each point x = ^(u) € M has a neighborhood U such

that h (u) = h (u,v) for any neighborhood V c: u. In particular, h(u) = h°(u,U).

Proposition 6.k. Let U be as above. If v is a point in the fibre tt_1(x) which can be Joined to u by a horizontal curve in tt-1(u), then h*(u) => h*(v).

Proof: By Proposition 6.3 there is a neighborhood V of y = tt(v) c u such that V c u and h (v) = h(v,v). h (v) is then contained in h (v,u). Since v is joined to u by a horizontal curve in tt""1(u),

Chap. 2. Theory of Connections

102

we have h°(v,u) = h(u,u) = h°(u) by Proposition 2.5, 2.2, Section 2. Hence h*(v) c h*(u).

Proposition 6.5. For every integer m (^ 0), the set of points x = tt(u) € M such that dim h (u) ^ m is open. Proof: Since h (u) and h (u') are conjugate in G, dim h (u) = dim

h (u') for all u,u' € tt'^x). Hence the statement follows from Propo s it ion 6. k.

Definition. The Lie algebra of a local holonomy group h (u) is called the local holonomy algebra at u and is denoted by ^ (u). Proposition 6.6. Let uQ be a point of P(M,G) and P(u0) be the holonomy bundle at uQ. The restricted holonomy group h (uQ) at uQ contains every local holonomy group h (u) where u runs over the points in P(uQ). Any element of h (uQ) is expressed in the form

a = an«a0...a ± d m (m: finite) where a. (i = 1, 2, ..., m) is an element of h (u.) at u. € P(uQ). Proof: Let u be apoint of P(uQ). u is joined with uQ by a horizontal curve |j, . Then tt(u0) = x 6 M can be joined to tt(u) =x€M by the curve tt(|J. ) = M-. There is a neighborhoodU of x such that h*(u) =h (u,U) (see

Exercise 6.1). Let t be an arbitrary curve at x contained in U. The parallel displacement along t sends u to ua, where a 6 h (u). The parallel

displacement along mT1 sends ua to u0a € tt_1(x0), and the lift of IjT1 passing through ua is R [j, . Hence the element induced by the parallel displacement along fjT1«T«|j, is a, and this implies that h (u) c: h (uQ) for every point u of P(uQ). To prove the latter statement of the proposition we use the factorization lemma and Proposition 6.3. Namely, let {U } be an open cover of M such that each member U is a neighborhood of a point y 6 M such that h (u) = h (u,U) where u 6 p(uq) ^ ^""""(y)­ Then each 0-homotopic curve at x0 = tt(u0) is decomposed into the product of finite lassos mT1t[ji where |j, denotes a curve joining x0

* 0 °^

to x and t is a closed curve at x contained in U. An element a' of

2 Connections in Principal Fibre Bundles

103

h (un) induced by the parallel displacement along the lasso |j.~1T|jl coincides with the element of h (u), and since an element a of

h (u0) is the product of finite elements such as a', the latter part of the proposition follows immediately.

Proposition 6.7- Suppose that dim h (u) is constant for all points of P(uQ); then h*(u) = h*(uQ).

Proof: u € p(uq) is joined with xQ by the (unique) lift t of t 6 M. Let v be a point of t and put tt(v) = y 6 t. By Proposition 6.k there is a neighborhood U such that h (v) zd h (v') for any v' € tt"1^ ) H t . Since dim h (v) = dim h (v') by hypothesis, h (v) equals h (v'), and since the set {U } (y € t) covers t, we can select finite subsets U , U , ..., U [yQ = TT(u0), y^ ..., ym = tt(u) € t], which cover tt (see Lemma 2.7, 2.2, Section 2) and U PI U H t

yi yi+l

y0 yl

yl yi yi+l

^ (jl. Then there exists a point v' in tt~1(u ) D tt_1(u JOt* for which h (u0) = h (v') = h (v) holds, where v is an arbitrary point

of if^U ) PI t . Continuing this process for U , U , .. ., we

get h*(uQ) = ••• = h*(u). Combining Proposition 6.7 with Proposition 6.6, we have

Theorem 6.8. If dim h (u) is constant for all u 6 P, then h (u) coincides with h (u).

Proposition 6.9- Let f be a function on P(M,G) such that for every point u of P, f(u) is an element of the local holonomy algebra ^ (u). If X is an element of Q , then Xf(u) is also an element of h (u). Proof: Let u0 be a point of P and U be a neighborhood of tt(u0) € M. If u is an arbitrary horizontal curve in tt"1^) starting at u0, we see by Proposition 6.k that h (u0) z> h (u,). Since f(u,) is an element of ^ (u.) by hypothesis, it is also an element of ^*(uQ), that is, f(ut) = f*(ut) E^, where {E^} [a = 1, 2, ..., r : r = dim $*(uQ)] is

a basis of fy*(u0), and r (u,) are C functions of t. Therefore, df(u, )/dt is also an element of *) (uQ), and, in particular,

Chap. 2. Theory of Connections

104

df(u, )/dt) . Since u, is a horizontal curve, it must be the orbit of u0 generated by a horizontal vector X; that is, we can set {Xf(u)} = (dt(u)/dt) , which is indeed an element of ^ (un). Remark. In the statement of Proposition 6.9, h (u) can be replaced by h (u) or by h(u).

Exercise 6.1. Show that if U has the property described in Proposition 6.3, then the same proposition holds for any point u' €

Exercise 6.2. Let U be as in Proposition 6.3- Prove that the holonomy algebra ^ (u0) at u0 € P(M,G) is generated by all elements of the form Q (X,Y), where u runs over the points in tt""1^), and

X, Y are arbitrary horizontal vectors at u in tt_1(u). 2.7 Infinitesimal Holonomy Groups

Let 0 be the curvature form of a connection r in P(M,G). We consider the set of all g-valued functions on P defined by

ZrZr-l "" ziQ(X>Y) (r = °> ^ •••) (7.1) where X, Y, Z,,...,Z run over horizontal vector fields on P. Let

*)'(u) denote the set of all values at u 6 P of (7-l)- In a neighborhood tt_1(u) of u in P, §'(u) is determined by all values of

ZrZr-l "■ ZiQ(X^Y) (r = 0, 1, ...) (7.2) where X, Y, Z, ..., Z are the lifts of vector fields defined on U, which is a coordinate neighborhood of M containing x = tt(u).

Theorem 7.1. Jj'(u) forms a Lie subalgebra of g and is contained in the local holonomy algebra ^ (u) at u 6 P(M,G). §'(u) is then a connected Lie subgroup of h (u). Hence

t>'(u) = $*(u) C$(u) Cg and consequently,

h'(u) c h*(u) c h°(u) c h(u) c G

2 Connections in Principal Fibre Bundles

105

where h'(u) denotes a connected Lie subgroup of G generated by

h'(u) and fy' (u) are called the infinitesimal holonomy group and the infinitesimal holonomy algebra at u, respectively. Proof: For each r we denote the set of all g-valued functions (7-2)

on tt-1(u) by Cl^ [u 6 tt~1(u)]. If fQ(u) € 0^ and fp(u) € Q^\ fQ(u) = nu(x,Y) and fp(u) = (Zp ••• Z-LQ(X,Y,))u. Since Zg (s = 0,

1, ...) are right invariant by G: (R ) Z = Z a (a € G), and CI is equivariant, f is also equivariant. We have, by Proposition 3-3> 2.3, Section 2,

[f0,fr] = [G(X,Y),fr] =|*[X,Y]fr = | ([X,Y] -/,[X,Y])fr = \ (x(Yfr) - Y(Xfr) - ^[X,Y]fr)

Since X (Yf ), Y(Xf ) € o'r+2\ and^[X,Y]f € ^r+1', the equality above yields

L 0' r+1 u u

[fn, f +1] €Q dim h'(u) for all v € tt_1(u). Proof: As was noted above, f (u) [and hence fy'(u)] is equivariant,

so that h'(u) is also equivariant, i.e., h'(ua) = A -i*h'(u) (a € G). Hence dim h'(u) is constant on each fibre over x = tt(u) € M. If dim h'(u) = m, there exist m linearly independent elements on h'(u) of the form (T-l)- Since these are continuous functions on P, there

is in P a neighborhood U where they are linearly independent. Setting U = tt(u), we see that dim h'(v) ^ dim h'(u) for all v € tt~1(u).

Corollary J.k. The set of all points tt(u) = x 6 M such that dim h'(u) > m, where m is an arbitrary integer, is open in M.

2 Connections in Principal Fibre Bundles

107

Lemma 7.5. Assume that h'(u) = h (u) holds at a point u 6 P. Then there exists a neighborhood U of tt(u) = x € M such that

h*(v) =h'(v) =h'(u) (7A)

for all v € tt_1(u), which can be joined with u by a horizontal curve in tt~1(u).

Proof: By Proposition 6.k in the preceding section, there is a neighborhood V of x = tt(u) such that dim h (v) ^ dim h (u) for all v € tt~1(v) which can be joined to u by a horizontal curve. By Lemma 7.3 there is a neighborhood W of x = tt(u) such that dim h'(u) G is the natural projection, then oo = 0 9 is the connection form of the flat canonical connection in P. The Maurer-Cartan equation [see (1.6) or (1.7), **■•!* Section k9 Chapter 1] gives dco = d(0*9 ) = 0*d9 = 0* ( - \ [9 ,9 ] )

= - \ [0*9,0*9] = - \ [«,,«,] and by comparing this with (3.2), 2.3, Section 2, we have Q = 0. A connection in a principal fibre bundle P(M,G) is called locally flat if every point x € M has a neighborhood U such that the induced connection in tt_1(u) is isomorphic with the canonical flat connection in U x G.

Proposition 8.1. A necessary and sufficient condition that P(M,G)

be locally flat is that the curvature form be equal to 0. Proof: The necessity is obvious. For proving the sufficiency, assume that the curvature form is zero. We take a simply connected neighborhood U of any point x 6 M and consider the induced connection

in tt_1(u) c P. The holonomy group in tt_1(u) consists of the identity element only by the holonomy theorem in 2.k and by Theorem k.J, 2A,

Section 2, the connection in tt_1(u) is trivial. Corollary 8.2. Let P(M,G) be a principal fibre bundle over a para­ 00 compact, simply connected C manifold M. If P admits a locally flat connection, then P(M,G) is a trivial fibre bundle.

2 Connections in Principal Fibre Bundles

109

Exercise 8.1. Assume that M is not simply connected and P(M,G) is locally flat. Show that the holonomy bundle P(uQ) is a covering space of M.

Exercise 8.2. Let P(M,G) be as above. Let N be a normal subgroup of h(u0) and set M' = P(uQ)/N. Show that M' is a principal fibre bundle over M with structure group h(uQ)/N. 2.9 Connections in Associated Bundles Let P(M,G) be a C principal fibre bundle and E(M,F,G,P) be its associated fibre bundle in which G acts differentiably on F from

the left (see 1.1, Section l). A point u of P can be regarded as a mapping of G onto the fibre F (x € M) such that (ua)T] = u(aT]) for a 6 G and 11 6 F. A connection in E is defined as an assignment of a distribution Q of dimension n (= dim M) at each point e of E satisfying the following conditions:

eeeeee

"C X "C "C Xo Xi

(l)° T (E) = T .(F ) + Q (direct sum), where T (F ) is the subspace of T (E) at e and is tangent to the fibre F through e. (2)° For a curve x, (t € [0,1]) from xQ to x1 in M, there is an integral curve e, which starts at any given point of F and covers x, . e, defines an isomorphism of F onto F and the isomorphism depends differentiably on t 6 [0,1].

(3)° e -> Q is differentiable.

Proposition 9-1- Given a connection F in P(M,G), there is a

u x u a * u nj.a

connection in E(M,F,G,P) which is induced by T.

Proof: Let u and e be points in P and E, respectively. There exists a point ? of F such that u«£ = e. For £ fixed, we have a dif f erentiable mapping of P into E given byu € P -» u«£ =e 6 E,

and it induces the map of the fibre G of P into the fibre F over x =tt(u) 6 M. We define the tangent subspace Q to be the image of Q by this mapping. Since (R ) Q = Q and (ua)(a"1?) = e, this map does not depend on the choice of u in n-1(x); namely, Q is the

subspace of E (= P x GF). It is easy to see that T (E ) is the direct sum of T (F ) and Q , and that the distribution Q at e is

Chap. 2. Theory of Connections

110

differentiable. Let x (t € [0,1]) be a curve in M starting at xQ and ending at x... There exists a unique horizontal curve u in P which starts at uQ and covers x, . u. •§ = e, is then an integral curve of the distribution e -> Q , which covers u,. Since e, is the map u •£ € F -> un •§ € F where u, = t(u~) and t is the parallel displacement of the fibre in P along t = x. (t € [0,1]) in M, it gives an isomorphism of F onto F . Hence e -» Q defines a con­

^ 0 * x0 1 xi 1 0'

& u0 ui ^e nection in E.

Exercise 9-1- Prove the statement converse to Proposition 9-1* and then verify Proposition 9-2. There is a one-to-one correspondence between the set of connections in P and the set of connections in E. 3- LINEAB COIWIECTIONS

3-1 Canonical 1-Forms and Connection Forms

Let L(M) be the bundle of linear frames over an n-dimensional C manifold. L(m) is a principal fibre bundle with structure group G = GL(n;g). We define the canonical 1-form 9 on L(M) by

9U(X) =u"1(tt^(x)) [X € Tu(L(M))] where u is a point of L(m) and is considered to be a linear mapping of gn onto T (M) [x = tt(u)] (see Example k9 1.1, Section l).

Proposition 1.1. For any element a of GL(n;g), 9 satisfies the

R*9 = a-19 (1.1)

property

Proof:

= WX«Ra># = a'^rT^T (X) = a-19 (X)

A connection in the principal fibre bundle L(M) is called a linear connection of M. Given a linear connection r on M, we asso­

3 Linear Connections

111

ciate with each element ? of R a horizontal vector field B(?) on L(m) in the following way. At each point u in L(m) the map u -*

(b(?)) is a linear isomorphism of R onto the horizontal subspace Q of T (L(M)) such that

n#(B(5))u = u • 5 [€ Tx(M), x = tt(u)] We call B(5) the basic vector field on L(M) corresponding to ?.

Proposition 1.2. B(?) =0 if and only if 5 = 0. Proof: If B(?) = 0 at some point u € L(M), we have tt (b(?)) = u-5 = 0. Since u is a linear isomorphism of R onto ^u^^ (M), § is the zero vector.

Proposition 1.3- 6(b(?)) = ?. Proof: eu(B(5)u) = u"1(n#(B(5))u) = u^-uS = 5­

Proposition l.k. (R ) B(§) = BCa"1?). Proof: By Propositions 1.1 and 1.3 we have

(Ra) e(B(5)) = a-MBCS)) = a"1? = e^a"1?))

Since the left-hand side equals 9((R ) B(?)), we get 9((Ra)#B(5) - BCa"1?)) = 0

and by Proposition 1.2, the statement follows.

Proposition 1.5. Let B(?) be the basic vector field corresponding to ? € R and A the fundamental vector field corresponding to A 6 gI(n;R). Then [A*,B(5)] =B(A?)

Proof: Let a, be the one-parameter subgroup of GL(n;R) generated by A: a, = exp tA. Then

t*0 t at

[A*,B(5)] = limj [B(5) - R B(5)]

t^O t t

= B(limk5 - aT1.?]) = B(A?)

Chap. 2. Theory of Connections

112

J*i J ^*i^i

■*i i1

Let {-e, ,...,-e } be a basis of R and {E .} (i, j = 1, 2 ..., n) be a basis of gl(n;g). Then it can be shown that the basic vector fields B. corresponding to -e. and the fundamental vector fields E . corresponding to E. are linearly independent at each point u of L(M). First, the B. are linearly independent everywhere in L(m) by Proposition 1.1. The E. are linearly independent. For E. € gl(n; g) -> {E .} is a linear isomorphism of gl(n;g) onto the vertical subspace G of T (L(m)). Since the E. are vertical and the B. are horizontal, they are linearly independent at every point u € L(m) .

j u ^.. ^ u uv v JJ j l Let oo be the connection form of r. We write 9 = e1^. oo = oo2.E?

Then the sets fB.,E.} and [9 ,oo.} are dual to each other, since

eJ(B.) = 0 t t

lim^ta^fCu) - f(u)] = -Af(u) 1>^0 t t by (2.3).

Chap. 2. Theory of Connections

116

By similar arguments we can extend the correspondence to that of forms on L(m) and tensor fields on M. First, we have

Theorem 2.3- There is a one-to-one correspondence between the set of Rn-valued horizontal 1-forms a on L(m) satisfying R a = a"1^

[a € GL(n;R)] and the set of tensor fields K of type (1^1). The correspondence is given by %(X*)=u"1K(n 1-1* Section 1]. For

i' i'/ n

another chart {V^1} such that U PI V 1 $ and tt"1(v) ^ V x GL(n;R),

x = x (x) are C functions on U PI V. Putting

. / v i' . v i

A^ = ^ A1, = ^TT

bxJ J bxJ

we have

*--£'bx -Kr bx and each frame u € tt"1(v) is expressible in the form X' = X. (b/bx ) 9

. / . /1, 1• / 1. / K.

where detfX1!" } ^ 0. Then we have for u € tt_1(u D v), X?(b/bx^) =

JJJ

X. (b/bx. ), where Xt = A^ XV, or in the matrix representation, X7 = AX

where X = [X.1], X7 = [X^ ], and A = [A3: ]. The transition function i|r of L(M) is given by *TO = 0v(u)(0u(u))"1 = X7 • T1 = A € GL(n;R)

"i Vi ^

Let i|r be the isomorphism u € tt_1(u) -> (X,jZL(u)) € U x GL(n;g),

where 0TT(u) = [X.]. Then the cross section UrT(x) = i|r"1(x,-e) corresponding to the identity element -e (= I ) of GL(n;g) has the local

expression (x ,6 .) in tt_1(u). As in 2.1, Section 2, we put ul = oo-dy, where d^ is the differential of the map x -» ^tT(x) given by (dx1) -> (dx1^). Then w is a 9I (n;g)-valued 1-form on tt"1(u):

where the 00. are ordinary 1-forms onU. Let{V;x}be another local J chart. Then we have on V,

*i' j

u^Cx) = fv(x, -e) cuy = cu-du^ = wyE±

3 Linear Connections

119

Since # = i|r^i = A"1 on U PI V, we have by (1.7), 2.1, Section 2, the transformation law (3-D

5' = AuoA."1 + A dA"1

that is, *i' fli'*kflh , ni' JAh

(3-2)

We put

*i _i , k *i' _i' , k'

(3-3)

Then (3-2) gives

kj h^jmp hkj

A'., =A^Am/A?/rh +A?;' bv,Ah,

d9a ,. dx1) . , =, (-4) dx° A dx1 = d(Y* bTa N z, 6-YT bxJ

= (Y^rh. - ycYbf\) dxd a dx1

-«J a e* + I Y^.x^ec a xje where

Th. =rh. -rh. (3.10)

Putting ~a

" i^i^0 A e* (5-11}

3 Linear Connections

121

we have

ae1 = -co1 a eJ + e1 (3.12) J

which was obtained intrinsically in the preceding section [see (l.*0, 3-1, Section 3]. Similarly, applying exterior differentiation to (3-5)>: we get the second structure equations [also given in (lA), 3-l> Section 3]>

that is,

duK = -ai1 A uk + Q3: (3.13) where

and

Again applying exterior differentiation to (3-12) and (3.13)> respectively, we obtain the first and second Bianchi identities

JJ

d©1 + ©^ A co1 = 9^ A Q1 (3.16) dD^ + Qk A co^ + Qk A co^ = 0 (3-17) [see (1.5) and (1.6), 3.1, Section 3]. The torsion form (3-H) and the curvature form (3-12) can be derived in the following way (H. Wakakuwa [3])- We define the horizontal part wA of the connection form by («>*4)(X) = w(/iX) for all X € X(L(M))

Since cu(/$«x) = 0 for all vector fields on L(m), wA = 0

which in terms of (x' may be written as

j fcr j km j

o/S/s = Y?;(dxh*/e+ i\h x^ dxk-^) = 0

122

Ji k ^ k k

Chap. 2. Theory of Connections

by virtue of (3.5)> where OX* •A and dx %A are the horizontal pro­

jection of dX". and dx , respectively. We remark that dx •A = dx , J because by the definition of the projection tt: l(m) -> M,

TT(dx)=dx«TT=dx

while the left-hand side is dx •/*, since tt (a) = a«/for any 1-form a on L(M). Hence go. •/Sis given by the condition J

ax5,'t= 'Tih^i ^ (5-18)

[The condition (3-l8) is equivalent to

dY^/e^.^dx* J kj m since A = [X.] and Y = [Y.] are inverses.] For the torsion form ®(X,Y) = d6 (/X,/Y), we have by applying

exterior differentiation to (3-6), 01 = ae1/ = dCY1J dxJ)/*

= dY1-/ A dx^A J

Since dx / = dx and dY.«/S is given by (3*19) > we get J

©a = Y^. h kjdxk A dxJ

= 2iY^. h kjdxkA dxJ which is identical to (3.11), since dx = X^

a a i 2 L^jJ ' 9 cb 9 cb deb

A similar procedure applied to (3»5) yields the curvature form (3.14).

Exercise 3-1- Derive (3.IM from (3-5) in the way indicated above. Exercise 3»2. Let 9 = Q. dx be n 1-forms of class C defined on M such that detfe1] 4 0, and set d9a = T\eC A 9b, dTa, = T. a9d.

Show that T, , satisfies the equations

3 Linear Connections

123

T[dcb] " ^[dc Tb]e " ° where [deb] denotes the skew-symmetric part. For example,

A[dcb] = 5^Adcb + Acbd + \dc " Adbc " ^cd " Acdb^

Remark. For the Novikov problem (see 2.4, Section 2) with regard to linear connections, see G. Caviglia [1]. 3-4 Expression of Covariant Derivatives in Local Coordinates

Let t = x, (t € [0,1]) be a piecewise differentiable curve in M and t be a piecewise different iable curve in L(m) such that tt(t ) = t. Let [U;x } be a local chart of M such that the initial point xQ of t is contained in U. Then t has the expression u, = (x (t), xT(t)) in tt 1(u) and the tangent vector u, = (d/dt) is given by

Z Ut

, i x d^ x

bxdtb: t dt __i ^h Since t is the lift of t if and only if oo(u, ) = 0, we have by making use of (3-5) in the preceding section, , q.

I h mk j h jj \ dt pr dt q./ v

hhh j£ dxm + y£ «*}(# -^- + -£ -M = 0 (*.i) Hence

dt

+ 44(^)=° ^

since det[Y.] ^ 0. This condition is equivalent to J

_i rh hV ^{^-^dt=0/ (k 3) dt-kj ^ o; dY3: u . ,A kv

Let t be the integral curve of a vector field Y = 11 (x)(b/bx ) Taking account of (k.l) and (4.2), the lift Y* of Y is seen to be

Chap. 2. Theory of Connections

12k

JJ

f-Ti1(x(t))(^) -r^(x(t))(-|) Since t is an arbitrary curve in M, we have

Proposition k.l. Let X. be the horizontal lift of b/bx . In terms of the local coordinate system (x1,*?) in tt"1(u) c L(m), X* has the expression

We recall Theorem 2.1, 3.2, Section 5. Let f(u) be the E ­ valued function on n_1(u) which corresponds to X.. Then f is of the form A.-e , [A.] € gl(n;g), and we have by (^.5),

jnj

*,. h n „k Ah

0 0/bxJ

y 1 hy jikh

X.fA.-e. ) = r ..A,^.

By the same theorem, X.,f is the function corresponding to V _. x (b/bx ), and hence

b/bx° bx ° bx

. . xt

t j xt ^ X0 - i

The local expression of V„X can also be obtained from

definition (2.1), 3-2, Section 3> as follows. Let t be a curve in M as above. The inverse parallel displacement tT1 maps the frame {(X.) }

onto the frame {(X.) }. Let X be an arbitrary vector field on M with the local expression X = ?(x)(b/bx ) in U. Then the vector X

xt J J xt t

= 51 (x(t))(b/bx1) = 51-Y^(t)(x,)v € T (M) is mapped to

t^(xx ) = 5i(x(t))yJ(t)(x.)x0

3 Linear Connections

125

In particular,

r-(-\) (k.7) bx"' x -*i(t)(X,)_ ° ° x0

Making use of (k-.j), we have

dT~1[(x) ]

dt Vbxj 0k* Adt ^ 1% As before, let x, be the integral curve through xQ of a vector field Y = 11 (x)(b/bx ), which starts at x~. In the definition of V„X given in (2.1), 3.2, Section 3> we set

dt V bx1 /X0 1 J X0 X0

t xt t_u xo

and noting that (t~1(X ))+ ~ = (x) , we have Proposition k.2. The covariant derivative of a vector field X = 5 (b/b^. ) in the direction of a vector field Y = 11 (b/bx ) at x0 is given by

J ^0'bx^'x0 where we have put

In particular, we have (^.6), i.e.,

0 bx1 b/b^ bx1 J1bxk We define for a differentiable function f(x) on M its covariant derivative at x0 by

(VYf) - f(x(0))] Y X0 =limkf(x(t)) t-K) *

126 Chap. 2. Theory of Connections where x, is the integral curve of Y. Hence Vyf (x) = Yf (x) = TlVf (x) where

J' bxJ' bx1

V f = ^0 Y = -ni _A_ Let (91) be the coframe field on M dual to (X.). Then the 91 d

(i = 1, ..., n) are given in U by

xt J t

(e1) =Y5(t)(dx\

If w is a covector field on M, it has the local expression

xt j xt j l xt

w^ = cp.(x(t))(dx\ =cpxj(t)(9i)x along x, . Defining the dual map t by

= Vi-eixt((Vxt)=cpoxi6k

0 l xQ X. xQ

= cp.(x(t)).x^(t).ei ((x.) ) we have

rt(wx (X(t))xj:(t).ei (t.io) xt j) =cp i xQ In particular,

T*(dx\ z xt i xQ=x}(t).eiv (k.n)

From (^.T),

d{r*(w )}

dt ' =k^(vvcpjx^(t).e\ y iv xQ

3 Linear Connections

127

where we have put

W^-r*A (^12) bcp.

Letting x, be the integral curve through x of a vector field Y on M with the local expression 11 (b/bx ), we define the covariant

derivative of w in the direction of Y at x by (V__w) = lim t-[t, (w ) - w ]

Y x t-»0 t * Xt X0

Then we have in a similar way as in the case of a vector field X,

(Vyw) = (71 V ) • (dx1) (1*.13) In particular,

V. dx1 = V . dx1 = -ri- dxk (^-1^) (r)\ be a tensor field of type (r,s) on M with the local Let K/ expression

bx bx

K.1 .r(-V) ® ... ® (-^-} ® (dx 1) ® .-^ ® (dx S) (r)

on U. The map t of K) < at x, onto that at x0 is defined by

tu (s)'x VV * *bx xl * bx x Xr

~+((Kjr)0 ) =K^1"^r(xJ.r;1 -^-® -.. OrT1-^­ t (dx ) (x) [€ t)(uQ)] are constant functions. Show that g forms a Lie algebra on P(u0) with dim g = dim P(uQ).

Exercise 6.3. Prove that the Lie algebra g of a linear connection

h h J1 J 1 1 J 1

whose components in {U;x } are of the form r.. = p .&. + p.6 . (p. = const) is spanned by (6VPV - 6 .p, )p.. 3-T Affine Connections

We regard g as an affine space A (see 1.2, Section 1, Chapter l). Let o be the origin of R and (-e ,. .. ,-e ) be a basis for gn. We call the set (0; 0 bx° x, bx xQ

bx (8.6)

Since the point pQ = x0 and the curve x, can be taken arbitrarily, we express these equations in the form of differentials:

ap = dxi(-*I)

a(JL) = r*. ^(-M V5xW ^ V6xi; which characterizes a natural affine connection under the notion of development.

If in (8.6) the natural frame (b/bx ) is replaced by a gen­ x0 eral frame (X.) [with X. = X?(b/bxJ)], then by writing (X.) = e.,

1 XQ 1 1 1 XQ 1

we have the equations

dp = eV

1iJ

(8.7)

de. = o)?e .

which were used by E. Cartan for the indication of a (natural) affine connection on M, where 01 = Y"^ dxJ and [Y"^] = [X.]"1.

Hereafter we consider the natural affine connection in the above sense, and call it an affine connection on M. M as such is called an affinely connected manifold. By taking Proposition 7.1 of the preceding section and Proposition 8.1, 2.8, Section 2, into account, we have

Theorem 8.1. An affinely connected manifold is locally flat if the torsion and curvature tensors of the corresponding linear connection vanish identically.

3 Linear Connections

1U

The holonomy group H(x) [resp. restricted holonomy group H (x)]

at x 6 M of an affinely connected manifold M is defined as in a linear holonomy group (resp. restricted linear holonomy group). If M is connected and paracompact, H(x) [resp. H (x)] is a Lie subgroup of A(n;g) [resp. connected component of the identity element of A(n;g)']. The homogeneous part h(x) [resp. h (x)] is called the homogeneous holonomy group (resp. restricted homogeneous holonomy

group) of the affine connection on M. It is obvious that h(x) [resp. h (x)] coincides with the linear holonomy group (resp. restricted linear holonomy group) of the linear connection in L(M) which determines the affine connection.

Theorem 8.2. If M is locally flat, the affine holonomy group is discrete, that is, the group is a zero-dimensional Lie group (see Exercise 8.1, 2.8, Section 2).

Exercise 8.1. Derive (8.7) from (8.6). [Hint: In order to obtain the first of equations (8.7) we need only substitute e. = X*?(b/bxJ) into the first of equations (8.6). For the second we have

bx , bx Now use (j.k), 3-3> Section 3 to complete the proof.]

Exercise 8.2. Let X be the vector field tangent to a curve t = x, (t € [0,1]) in M. Show that if 7x\xt = 0, the development of t is a parabola.

3-9 Geodesies Let M be an n-dimensional affinely connected manifold of class C .

A curve y = x, (t 6 [0,1]) of class C in M is called a geodesic if the development of y is a straight line. Let (x^b/bx1),... ,(b/bxn))

Ik2

Chap. 2. Theory of Connections

be the affine frame in the affine tangent space A (m) at x 6 M. If Y is a geodesic of M, then it is of the form p(t) = orb + p where a and 3 are constant vectors. Differentiating p(t) twice, we have by equations (8.5) in the preceding section,

,2 h , , j . i

dt °

which are the equations of a geodesic y expressed in a local coor­ 00 dinate system. Thus a geodesic is of class C provided that the 00 given affine connection on M is of class C . The tangent vector X of y is expressed as X = (dx /dt)(b/bx ), so equations (9-1) imply

that V Y = 0 along y; that is, X is parallel along y if y is a geodesic. In this definition of geodesies, the parametrization of the curve in question is important.

Theorem 9-1. A necessary and sufficient condition for a curve t;

at dt

x, = x (t) (t 6 [0,1]) to be a geodesic is that t satisfy the equations

dxk - ' d\ /J4_2 xd2xh + „h ]hdx° dxdx^\ \ _ /dx /V d/ xd2xk dt dt ji dtdxH dt /,Q ^ }* .,,2 ji dtjidx^ dt /dt dt dxh \dt_. 2ja_2k]dx^, Proof: For a solution x1(t) of equations (9-2) there exists a function n(t) which satisfies

d x , _h dx° dx /. n dx

dt °

On putting t = t(s), we have by the chain rule

ds ° dt ds °

d2xh h dxj dx1 _ d2xh / dt \2 dxh d2t h dxj dx1 / dt \c

. 2 ] ji ds ds J4_2 \ ds / dt _ 2 ji dt dt \ ds /

Ikj

3 Linear Connections

/d2s r+\(te\\ fat\3dxh dt

If we denote by s = s(t) the solution of the ordinary differential equation

at and use it as the parameter, we have

nh /Ji / \\dxJ -t;— dsdx1 ds v ,s d,2xh+±r..(x(s))-=—■=— =no fr. (9-3;

Conversely, if (9-3) holds for a parameter s, then (9-2) holds. A parameter t for which equations (9-1) hold is called the affine parameter of a geodesic, and by (9-3)> s = at + b is an affine parameter if t is such, where a and b are constants. We now consider a solution of the differential equations (9-1)

which satisfies the initial conditions x (0) = (x ) , (dx /dt) = (X )Q. Since r.. involves only x (t),..., x (t) as the unknown variables, the equations admit unique solutions x (t) in an interval

0 £ t ^ a if a is sufficiently small (see K. S. Miller and F. I. Murray [1]). For a point p in {U;x } we consider x = x0 and (dx /dt) = X0 6 T (M). We then obtain a unique solution curve x (t); 0 ^ t ^ a, which is contained in the coordinate neighborhood

U. Let the point x (a) 6 U be denoted by p. Then x = x (t) represents in U a geodesic ray y0: [0,a] -> U such that Yo(0) = P>

Y0(a) = p'. It is obvious that y0 is extendable to a geodesic ray y: (-e, a + e) -» u. If a geodesic y("t) is extendable for infinitely large values of the affine parameter t, the manifold M is said to be

2 11

complete (with respect to the affine connection of M).

Example 1 (Y. Tashiro). Let T be the torus S x S with the coordinate system (x,y) (mod 2rr) and consider the geodesic defined by

ikh

Chap. 2. Theory of Connections

d2x ^ .2 /dx\2 2 /dy\2

~2 + sinxldtJ " COSXldt) dt

. .2 at

V dt / V dt J

The solution curve passing through (x,y) = (0,0) is given by x = t and y = log cos t (| t | < tt/2). The geodesic is not determined at

t = tt/2. Hence T is not complete with respect to the given connection.

Exercise 9-1- Let L(M) be the bundle of linear frames over M and r a linear connection. Prove that the projection onto M of any integral curve of a horizontal curve is a geodesic and, conversely, every geodesic is obtained in this manner. Exercise 9-2 (Y. Muto). Let M be an affinely connected manifold covered by local charts for which the components of the corresponding

Ji

J1 J \ J i h

linear connection r are of the form r.. = p.6. (p. = constants). Show that the geodesic y("t) of M is given by x = (p.c )c log(l +

p.c t), where the c are constants, or x = c t, where the c are constants such that p.c = 0. 3 Exercise 9.3* Show that the equations of a geodesic are given in

dt dt dt Ky J

terms of a frame field (e.) [= (X.) : X. = X