Riemannian Geometry: Concepts, Examples and Techniques

314 Pages·2020·36.661 MB·English

by  S. Kumaresan

C oncepts, Examples and Techniques
S. Kumaresan
Formerly Professor of University of Hyderabad
Hyderabad
jlm M ,
: .
*4 TECVtt"0
TECHNO WORLD®
Kolkata
y;
r jL
Rlemannian Geometry
Concepts, Examples and Techniques
ByS.Kumaresan
Published by:
ffeCHNd World
90/6A, M.G.Road,
Kolkata - 7b0 007
©[phone],22571650
[email protected] k is sold sufc
that it shall not, by way of trade or otherwise,
Copyright be lent, resold, hired, hired out, or otherwise
© 2(^25, Reserved by the Author circulated without the publishers’ prior
written consent in any form of binding or
cover other than that in which it is published
First Edition: December, 2020 and without a similar condition including this
condition being imposed on the subsequent
purchase and without limiting the weights
under copyright reserved above, no part of
Printed by:
this publication may be reproduced, stored in
D.G. Offset, Kolkata
or introduced into retrieval system, or
transmitted in any form or by any means
Rinding by:
Standard Binding Works recording or otherwise), without the written
permission of both the copyright owner and
the author of this book.
ISBN: 978-93-88347-69-3 Information contained in this book work
believed to be reliable and are correct to the
Printed from the camera-ready
best of their knowledge. However, the
copy provided by the Author.
- and its author shall in no event be­
any errors, omissions or damages
information and specifically
X 295 implied warranties or
merchantability of fitness for any particular
' ” •
" m
~ > '.tt, % < t
I dedicate this book to
my wife, Kalai
my daughter Bharati and her husband Sivabalan
my son Sivaguru and his wife Snehal
and
my grandchildren Viayn and Ayansh.
Thank you for your love and support.

Preface
This book is based on a preliminary version of my manuscript "A Course in Riemannian
Geometry". It was distributed to the participants in an NBHM-sponsored Instructional
Conference on Riemannian Geometry, TIFR, October 22-November 10,1990. As a glance
at the table of contents would show, it was quite comprehensive and state of the art book
in 1990! My original intention was to publish it after the conference and after correcting
the typos, and inserting figures wherever needed and add a few more topics. But soon
after the conference, my life was on a different phase. It was a few years later, I again
turned my attention to the manuscript and did whatever I wanted to do. But it was utter
lack of professionalism that I did not send it for publication, as I was consumed by the
passion of my life. Mathematics Training and Talent Search (MTTS) Programme.
Meanwhile some excellent texts on Riemannin Geometry have appeared. Though
many of my friends who work in the subject convinced me that my book offers few things
which other books lack, I could not gather enough energy to send it for publication, as I
have more or less forgotten all technical aspects of the book. My dear friend Purnaprajna
Bangere of University of Kansas kept on persuading me to publish the manuscript. If
some readers find the book useful, they should thank him.
Now about the book. We assume that the reader has already learned basics of differ­
ential manifolds, tangent vectors, vector fields, diffeomorphisms, etc. We review them
quickly in Chapter 1. An enterprising reader may find this quite adequate to go ahead
with the rest of the material.
Chapter 2 deals with all the basic results and more. There are a few nonstandard
topics, with no readily available reference and a few novel proofs of some standard re­
sults. To make the reader develop intuition, we have delved into Gauss theory of surfaces
which will be frequently referred to elsewhere in the book to facilitate the understanding
and consolidate the ideas. Topics such as Musical isomorphisms, Holonomy, Algebraic
theory of curvature tensors, convex sets, Riemannian submersions are not usually done
in basic courses. In view of their increasing importance, we have included them. I believe
that I have brought out geometric significance of the primitive idea of parallel transport.
It is usually not given its due.
The theme of Chapter 3 is the variation of geodesics. This is the heart of Riemannian
Geometry. I took a lot of effort to emphasize the use of Jacobi fields, especially, in so-
called comparison theorems, a major theme of Chapter 4. Two unusual inclusions in
Chapter 3 are the discussion of focal points and Morse Index Theorem. We have closely
followed the papers of Osborne to give a simple proof of Morse index theorem.
Chapter 4 starts with various comparison theorems. At a preliminary level, (which
constitutes of Chapter 2), the results are obtained by comparing Riemannian manifolds
with Euclidean spaces. In Chapter 4, the comparison theorems deal with more general
results. It also introduces the reader with the standard techniques and they are illustrated
with some typical applications such as sphere theorems.
Variety of techniques are used to prove a result or to carry out curvature computa­
tions. To understand what we mean by this, we request you to look at the variety of
ways in which the geodesics on some of the spaces are found and the many ways in
which the curvature of complex projective spaces are computed.
Some of the noteworthy features of the book, even considered for a basic course in
Riemannian Geometry, are the following:
• Parallel transport and its geometric significance: Example 2.5.7. Theorem 2.5.10
and Theorem 2.5.12.
• Explicit determination of the holonomy group of the Poincare plane, at the end of
Section 2.5.
• The significant difference between Lie derivatives and covariant derivatives.
• Algebraic theory of curvature operators in Section 2.6, in particular, intrinsic deriva­
tion of the Weyl conformal tensor in Theorem 2.6.9, the fact that sectional curvatures
determine the curvature in Lemma 2.6.8.
• The geometry of Gausss lemma in Section 2.8.2.
• Hadamard's global result. Theorem 2.9.19.
• Kobayashi's version of Cartan-Hadamard, Theorem 2.12.2.
• A natural way of generating closed totally geodesic submanifolds. Theorem 2.9.22.
• Geometric meaning of sectional curvature. Lemmas 3.2.6-3.2.7.
Unlike my other books, this book has an appendix that deals with very specific col­
lection of examples. It is suggested that the reader whenever he meets a new concept,
he should visit Appendix A and understand the concepts through the examples. Rest as­
sured that in the main text itself, we do talk of examples as and when we introduce new
concepts. The reason why I kept them as separate Chapter is somewhat personal as well
as for pragmatic reasons. Many of the analysts are now working on analytical problems
which are set in the realm of Riemmanian Geometry. More often than not, they need
very concrete 'formulas' or expressions for the geometric quantities. This compendium
of examples in Appendix A may serve them well as a ready reference.
Appendix B deals with ever-increasing need for the notion of bundles and geometry
on them. I have strove hard to establish various ways of defining connections and their
equivalence. This appendix may be considered as laying down the basic technical terms
at the disposal of the reader. For a more thorough introduction, there could be no better
reference than [38].
I take this opportunity to thank those who were helpful during the preparation of the
preliminary version. I thank my friends C.S. Aravinda, D.S. Nagaraj and G. Santhanam
for having read through some portions of the preliminary notes and their suggestions to
enhance their readability. My special thanks are to Ravi Aithal who meticulously went
through a major portion of the text. Credits to clarity in exposition, consistency of nota­
tion and avoidance of some egregious errors etc. should go to him. Many of my nebulous
ideas took shape after extensive discussions with him. This book contains a lot more ma­
terial than the preliminary version.
I confess that I am poor at proof-reading. Almost all my books are plagued with typos
and some silly mistakes. My only excuse is that it is a confidence-boosting exercise for
any student to find such mistakes and convince himself that he is one up on the author.
Best wishes.
I would greatly appreciate any suggestions for further improvement. Feel free to send
typos and critical comments. Please mail them to kumaresaOgmail. com with Subject:
Comments on Riemannian Geometry. Flopefully, a young geometer may collaborate with
me to bring a future edition, if warranted.
S. Kumaresan
Suggestions for a Basic Course
I expect that the students have been exposed to basic theory of differential manifolds. If
so. Chapter 1 could serve as a review. Otherwise, the teacher may have to spend consid­
erable amount of time on Chapter 1. Even in such a scenario, the teacher may start the
course with sections 1.1-1.7 and return to other concepts as and when required.
In Chapter 2, the following is recommended.
• Section 2.1
• Subsections 2.2.1-2.2.4
• Section 2.3
• Section 2.4
• Subsections 2.5.1-2.5.2
• Subsections 2.6.1-2.6.3
• Subsections 2.7.1-2.7.4,
• Subsections 2.8.1-2.8.2
• Subsection 2.9.1
• Section 2.11
• Section 2.12
In Chapter 3, the following is recommended.
• Section 3.1
• Subsections 3.2.1-3.2.2
• Section 3.3
• Section 3.5, with a suggestion restrict to the simplest case.
In Appendix on Examples, A.l, A.2 , A.3 and A.6 are an absolute must.
Contents
Preface i
1 Foundations 1
1.1 Differential Calculus-A Review........................................ 1
1.1.1 Inverse Mapping Theorem ............................................................... 1
1.1.2 Basic Theorem in ODE.......................... 3
1.2 Differential Manifolds ........................................... 4
1.3 Smooth Maps & Diffeomorphisms............................................................ . 11
1.4 Tangent Spaces to a Manifold....................... 14
1.5 Derivatives of Smooth Maps........................................................................... 19
1.6 Immersions and Submersions................................................................... . 22
1.7 Submanifolds.......................................................................... 24
1.8 Tensor Fields..................................................................................................... 26
1.8.1 Vector Fields .............................................................................. 26
1.8.2 Left-Invariant Vector Fields on Lie Groups ...................................... 31
1.8.3 Tensor Fields ....................................................................................... 32
1.8.4 Lie Derivation....................................................................................... 34
1.9 Orientable Manifolds................................................... 35 2
2 Basic Riemannian Geometry 38
2.1 Covariant Differentiation................................................................................. 38
2.1.1 Connection on ]Rn and Hypersurfaces in Rn+1 ............................. . 38
2.1.2 Connections on Manifolds................. 44
2.2 Riemannian Metrics........................................... QI6 \
2.2.1 Riemannian M etrics........................... . 46
2.2.2 Local Isometry and Isometry.................................. 50
2.2.3 Volume Elements and Integration............................ 53
2.2.4 Inner Metric......................................................................................... 54
2.2.5 Musical Isomorphisms........................................................................ 54
2.3 The Levi-Civita Connection........................................................................... 56
2.4 Gauss Theory of Surfaces in R3 ..................................................................... 62
2.4.1 Gaussian Curvature........................................................................... 62
2.4.2 Gauss Theorema Egregium................................................................ 67
2.5 Curvature and Parallel Transport................................................................... 70
2.5.1 Curvature Operator ........................................................................... 70
2.5.2 Parallel Transport................................................................................. 71
2.5.3 Holonomy............................................................................................ 76
2.6 The Curvature Tensor....................................................................................... 79
2.6.1 The Riemannian Curvature................................................................ 79
2.6.2 Algebraic Theory of Curvature Operators......................................... 83
2.6.3 Definitions of Various Curvatures.................................................... 85
2.7 Geodesics ........................................................................................................ 86
2.7.1 Vector fields along Maps..................................................................... 86
2.7.2 Two Parameter Maps........................................................................... 88
2.7.3 Geodesics: Definition and Existence................................................. 90
2.7.4 Examples of Geodesics........................................................................ 92
2.7.5 Geodesics by Calculus of Variations................................................. 95
2.8 Exponential M ap............................................................................................ 97
2.8.1 Exponential Map & Normal Coordinates......................................... 97
2.8.2 Gauss Lemma....................................................................................... 100
2.8.3 Convex Sets.......................................................................................... 105
2.9 Riemannian Immersions................................................................................. 106
2.9.1 Riemannian Submanifolds .............................................. 106
2.9.2 Minimal submanifolds and convex hypersurfaces.......................... 108
2.9.3 Totally Geodesic Submanifolds.......................................................... 114
2.10 Riemannian Submersions.............................................................................. 116
2.11 Hopf-Rinow Theorem .................................................................................... 118
2.12 Cartan-Hadamard Theorem........................................................................... 123
3 Variations of Geodesics 126
3.1 Formulas of Arc-length Variation................................................................... 126
3.2 Jacobi Fields..................................................................................................... 129
3.2.1 Definition, Examples and Properties................................................. 129
3.2.2 Some Typical and Important Uses of Jacobi Fields.......................... 133
3.2.3 An Easy Comparison Theorem for Jacobi Fields............................. 138
3.3 Conjugate Points...................................................... 140
3.4 Focal Points..................................................................................................... 144
3.5 Index Form........................................................................................................ 147
3.6 The Morse Index Theorem............................... 153
3.7 Cut Locus ........................................................................................................ 157