A Course in Differential Geometry and Lie Groups

306 Pages·2002·20.474 MB·English

by  S. Kumaresan

TEXTS AND READINGS
IN MATHEMATICS 22
A Course in
Differential Geometry
and Lie Groups
Texts and Readings in Mathematics
Advisory Editor
C. S. Seshadri, Chennai Mathematical Institute, Chennai.
Managing Editor
Rajendra Bhatia, Indian Statistical Institute, New Delhi.
Editors
V. S. Borkar, Tata Institute of Fundamental Research, Mumbai.
R. L. Karandikar, Indian Statistical Institute, New Delhi.
C. Musili, University of Hyderabad, Hyderabad.
K. H. Paranjape, Institute of Mathematical Sciences, Chennai.
T. R. Ramadas, Tata Institute of Fundamental Research, Mumbai.
V. S. Sunder, Institute of Mathematical Sciences, Chennai.
Already Published Volumes
R. B. Bapat: Linear Algebra and Linear Models (Second Edition)
R. Bhatia: Fourier Series
C. Musili: Representations of Finite Groups
H. Helson: Linear Algebra ( Second Edition)
D. Sarason: Notes on Complex Function Theory
M. G. Nadkarni: Basic Ergodic Theory (Second Edition)
H. Helson: Harmonic Analysis ( Second Edition)
K. Chandrasekharan: A Course on Integration Theory
K. Chandrasekharan: A Course on Topological Groups
R. Bhatia (ed.): Analysis, Geometry and Probability
K. R. Davidson: C· - Algebras by Example
M. Bhattacharjee et af.: Notes on Infinite Permutation Groups
V. S. Sunder: Functional Analysis - Spectral Theory
V. S. Varadarajan: Algebra in Ancient and Modern Times
M. G. Nadkarni: Spectral Theory of Dynamical Systems
A. Borel: Semisimple Groups and Riemannian Symmetric Spaces
M. Marcolli: Seiberg-Witten Gauge Theory
A. Bottcher and S. M. Grudsky: Toeplitz Matrices, Asymptotic Linear
Algebra and Functional Analysis
A. R. Rao and P. Bhimasankaram: Linear Algebra ( Second Edition)
C. Musili: Algebraic Geometry for Beginners
A. R. Rajwade: Convex Polyhedra with Regularity Conditions and
Hilbert's Third Problem
A Course in
Differential Geometry
and Lie Groups
s.
Kumaresan
University of Mumbai
~HINDUSTAN
U LQj UBOOKAGENCY
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Copyright © 2002 by Hindustan Book Agency ( India)
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All export rights for this edition vest exclusively with Hindustan
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right Law and is subject to legal action.
ISBN 978-81-85931-67-8 ISBN 978-93-86279-08-8 (eBook)
DOI 10.1007/978-93-86279-08-8
Dedicated to the memory of my mother
S. Susila
Contents
Preface ix
1 Differential Calculus 1
1.1 Definitions and examples ............. . 1
1.2 Chain rule, mean value theorem and applications 16
1.3 Directional derivatives . . . . . . . . . . . . 21
1.4 Inverse mapping theorem ......... . 32
1.5 Local study of immersions and submersions 44
1.6 Fundamental theorem of calculus .... 46
1. 7 Higher derivatives and Taylor's theorem 48
1.8 Smooth functions with compact support 55
1.9 Existence of solutions of ODE . . . . . . 58
2 Manifolds and Lie Groups 64
2.1 Differential manifolds ... 64
2.2 Smooth maps and diffeomorphisms 75
2.3 Tangent spaces to a manifold 81
2.4 Derivatives of smooth maps 90
2.5 Immersions and submersions 96
2.6 Submanifolds ....... . 100
2.7 Vector fields ........ . 106
2.8 Flows and exponential map 125
2.9 Frobenius theorem ..... 136
2.10 Lie groups and Lie algebras 144
2.11 Homogeneous spaces · 155
3 Tensor Analysis 165
3.1 Multilinear algebra · 165
3.2 Exterior algebra 172
3.3 Tensor fields. . . . · 183
viii CONTENTS
3.4 The exterior derivative , . 190
3.5 Lie derivatives . 199
4 Integration 207
4.1 Orient able manifolds .207
4.2 Integration on manifolds . .214
4.3 Stokes' theorem ... .223
5 Riemannian Geometry 232
5.1 Covariant differentiation .232
5.2 Riemannian metrics .. .238
5.3 The Levi-Civita connection .249
5.4 Gauss theory of surfaces in 1R3 .253
5.5 Curvature and parallel transport .264
5.6 Cartan structural equations . .272
5.7 Spaces of constant curvature .. .278
A Tangent Bundles and Vector Bundles 281
B Partitions of Unity 286
Bibliography 288
List of Symbols 290
Index 292
Preface
This book arose out of the courses offered by me at T.I.F.R. Centre,
Bangalore, T.I.F.R., Bombay (twice), Indian Institute of Technology,
Kanpur and Ramanujan Institute of Mathematics, University of Madras.
I plunged into writing this book thanks to the encouragement and per
suasions of the audience of my courses.
The book covers the traditional topics of differential manifolds,
tensor fields, Lie groups, integration on manifolds and a short but moti
vated introduction to basic differential and Riemannian geometry. This
book will be suitable for a course for students of Physics and Mathe
matics at the graduate level of western universities or at M.Phil. level
of Indian universities. While the topics are traditional, the discern
ing reader will find the approach to the topics and the proofs at many
places quite novel. Our main emphasis is on the geometric meaning of
the concepts, so that the reader will feel confident and acquire a working
knowledge. For this reason, motivations are given, many simple exer
cises are included and illuminating nontrivial examples are discussed in
detail.
Some of the salient features of the book are the following:
1. Geometric and conceptual treatment of Differential Calculus with
a wealth of nontrivial examples.
2. A thorough discussion of the much-used result on the existence,
uniqueness and smooth dependence of solutions of ODE.
3. Special care in introducing the concept of tangent space to a
manifold.
4. An early and simultaneous treatment of Lie Groups and related
concepts as we develop the basic topics in differential manifolds.
5. An early and elementary proof of the fact that all classical groups
are. Lie groups.
6. A motivated and highly geometric proof of the Frobenius theorem.
7. A constant reconciliation with the classical (such as tensor calcu
Ius) treatment or notation and the modern approach.
x Preface
8. Simple proofs of the Hairy-Ball theorem and Brouwer's fixed point
theorem.
9. Construction of manifolds of constant curvature a la Chern.
10 . Merits and comparisons of different view points, whenever pos
sible.
Major portion of this book was typeset when I was at the Tata Insti
tute and I thank the Tata Institute for it. A substantial portion of the
preliminary version was typed by G. Santhanam and was proof-read by
C.S. Aravinda. Kirti Joshi and Kapil Paranjape introduced me to the
world of 'lEX and Jb.'lEX. V. Muruganandam went through the prelim
inary version ~nd made suggestions for improvement. V. Nandagopal
formatted the book for the TRIM series. Ajit Kumar drew the figures
on computers and helped me a lot in the final version. It gives me great
pleasure to record my sincere thanks to all these friends.
The book was in hibernation for about ten years and the sustained
efforts of many of my friends, especially S. Ilangovan, led to its seeing
the light of day.
I take this opportunity to record my sense of gratitude to one of
my relatives Mr. G.Gnanasambandam, B.E., who honed my scientific
thinking by getting me into serious discussions on scientific matters dur
ing my high-school days. I also record my deep sense of appreciation
for the numerous hours of pleasant discussions on a variety of topics in
Mathematics which I had with Adimurthi, Akhil Ranjan, K. Okamoto,
M.S. Raghunathan and R. Parthasarathy during my days at the Tata
Institute. My ideas, knowledge and appreciation of Mathematics owe a
lot to these people.
I thank many of my colleagues, especially juniors, at the Tata Insti
tute who asked me to give many seminars and discussed their difficulties
and problems, mathematical as well as personal. These helped me per
r.eive the difficulties of a beginner and also made me a better human
being.
I also record my appreciation for Rajendra Bhatia, Managing Editor
of the series and the referees. Their corrections and persuasive sugges
tions made me realize that my enthusiasm for dissemination of knowl
edge cannot be an excuse for being sloppy. I am sure that paying heed
to their suggestions has enhanced the value of this book and eliminated
some of the egregious errors and tactless remarks.
I thank my wife Kalai and my children Sivaguru and Bharathi who
bore with me while I was busy with the preparation of the book.
Books and expositions at this level, as a rule, are written in a formal
and precise way. I broke away from this and wrote the book in a con-