Tensor Analysis on Manifolds

290 Pages·02.741 MB·English

by  Richard L. Bishop, Samuel I. Goldberg

Tensor Analysis
Manifolds
on
Richard L. Bishop
University of Illinois
Samuel I. Goldberg
University of Illinois
Big"' _ cmtrale
Facjlia Inyagneria
0G0312
Dover Publications, Inc.
New York
Copyright © 1968, 1980 by Richard L Bishop and Samuel
I Goldberg
All rights reserved under Pan American and International
Copyright Conventions
This Dover edition, first published in 1980, is an unabridg(
and corrected republication of the work originally publishi
by The Macmillan Company in 1968
International Standard Book Number 0-486-64039-6
Library of Congress Catalog Card Number 80-66959
Manufactured in the United States of America
Dover Publications, Inc
31 East 2nd Street, Mineola, N.Y. 11501
Preface
"Sie bedeutet einen wahren Triumph der durch Gauss,
Riemann, Christoffel, Ricci ... begrundeten Methoden des
allgemeinen Differentialcalculus." ALBERT EINSTEIN, 1915
SINCE ITS DEVELOPMENT BY Ricci between 1887 and 1896, tensor analysis has
had a rather restricted outlook despite its striking success as a mathematical
tool in the general theory of relativity and its adaptability to a wide range of
problems in differential equations, geometry, and physics. The emphasis has
been on notation and manipulation of indices. This book is an attempt to
broaden this point of view at the stage where the student first encounters the
subject. We have treated tensor analysis as a continuation of advanced calcu-
lus, and our standards of rigor and logical completeness compare favorably
with parallel courses in the curriculum such as complex variable theory and
linear algebra.
For students in the physical sciences, who acquire mathematical knowledge
on a "need-to-know" basis, this book provides organization. On the other
hand, it can be used by mathematics students as a meaningful introduction to
differential geometry.
A broad range of notations is explained and interrelated, so the student will
be able to continue his studies among either the classical references, those in
the style of E. Cartan, or the current abstractions.
The material has been organized according to the dictates of mathematical
structure, proceeding from the general to the special. The initial chapter has
been numbered 0 because it logically precedes the main topics. Thus Chapter 0
establishes notation and gives an outline of a body of theory required to put
the remaining chapters on a sound and logical footing. It is intended to be a
handy reference but not for systematic study in a course. Chapters I and 2 are
independent of each other, representing a division of tensor analysis into its
function-theoretical and algebraic aspects, respectively. This material is com-
bined and developed in several ways in Chapters 3 and 4, without specializa-
tion of mathematical structure. In the last two chapters (5 and 6) several
important special structures are studied, those in Chapter 6 illustrating how
the previous material can be adapted to clarify the ideas of classical mechanics.
Advanced calculus and elementary differential equations are the minimum
background necessary for the study of this book. The topics in advanced calculus
which,are essential are the theory of functions of several variables, the implicit
function theorem, and (for Chapter 4) multiple integrals. An understanding
N/
iv Preface
of what it means for solutions of systems of differential equations to
exist and be unique is more important than an ability to crank out general
solutions. Thus we would not expect that a student in the physical sciences
would be ready for a course based on this book until his senior year. Mathe-
matics students intent on graduate study might use this material as early as
their junior year, but we suggest that they would find it more fruitful and make
faster progress if they wait until they have had a course in linear algebra and
matrix theory. Other courses helpful in speeding the digestion of this material
are those in real variable theory and topology.
The problems are frequently important to the development of the text.
Other problems are devices to enforce the understanding of a definition or a
theorem. They also have been used to insert additional topics not discussed in
the text.
We advocate eliminating many of the parentheses customarily used in
denoting function values. That is, we often writefx instead off(x).
The end of a proof will be denoted by the symbol 1.
We wish to thank Professor Louis N. Howard of MIT for his critical
reading and many helpful suggestions; W. C. Weber for critical reading, useful
suggestions, and other editorial assistance; E. M. Moskal and D. E. Blair for
proofreading parts of the manuscript; and the editors of The Macmillan
Company for their cooperation and patience.
Suggestions for the Reader
The bulk of this material can be covered in a two-semester (or three-
quarter) course. Thus one could omit Chapter 0 and several sections of the
later chapters, as follows: 2.14, 2.22, 2.23, 3.8, 3.10, 3.11, 3.12, the Appendix
in Chapter 3, 4.4, 4.5, 4.10, 5.6, and all of Chapter 6. If it is desired to cover
Chapter 6, Sections 2.23 and 4.4 and Appendix 3A should be studied. For a
one-semester course one should try to get through most of Chapters 1 and 2
and half of Chapter 3. A thorough study of Chapter 2 would make a reason-
able course in linear algebra, so that for students who have had linear algebra
the time on Chapter 2 could be considerably shortened. In a slightly longer
course, say two quarters, it is desirable to cover Chapter 3, Sections 4.1, 4.2,
and 4.3, and most of the rest of Chapter 4 or all of Chapter 5. The choice of
either is possible because Chapter 5 does not depend on Sections 4.4 through
4.10. The parts in smaller print are more difficult or tangential, so they may
be considered as supplemental reading.
R. L. B.
S.LG.
Contents
Chapter 0/Set Theory and Topology
1
0.1. SET THEORY 1
0.1.1. Sets 1
0.1.2. Set Operations 2
0.1.3. Cartesian Products 3
0.1.4. Functions 4
0.1.5. Functions and Set Operations 6
0.1.6. Equivalence Relations 7
0.2. TOPOLOGY 8
0.2.1. Topologies 8
0 2.2. Metric Spaces 10
0.2.3. Subspaces 11
0.2.4. Product Topologies 11
0.2.5. Hausdorff Spaces 12
0.2.6. Continuity 12
0.2.7. Connectedness 13
0 2.8. Compactness 15
0.2.9. Local Compactness 17
0.2.10. Separability 17
0.2.11. Paracompactness 17
Chapter 1 /Manifolds 19
1.1. Definition of a Manifold 19
1.2. Examples of Manifolds 22
1.3. Differentiable Maps 35
1.4. Submanifolds 40
1.5. Differentiable Curves 43
1.6. Tangents 47
1.7. Coordinate Vector Fields 50
1.8. Differential of a Map 55
vi Contents
Chapter 2/Tensor Algebra 59
2.1. Vector Spaces 59
2.2. Linear Independence 61
2.3. Summation Convention 65
2.4. Subspaces 67
2.5. Linear Functions 69
2.6. Spaces of Linear Functions 71
2.7. Dual Space 75
2.8. Multilinear Functions 76
2.9. Natural Pairing 77
2.10. Tensor Spaces 78
2.11. Algebra of Tensors 79
2.12. Reinterpretations 79
2.13. Transformation Laws 83
2.14. Invariants 85
2.15. Symmetric Tensors 87
2.16. Symmetric Algebra 88
2.17. Skew-Symmetric Tensors 91
2.18. Exterior Algebra 92
2.19. Determinants 97
2.20. Bilinear Forms 100
2.21. Quadratic Forms 101
2.22. Hodge Duality 107
2.23. Symplectic Forms 111
Chapter 3/Vector Analysis on Manifolds 116
3.1. Vector Fields 116
3.2. Tensor Fields 118
3.3. Riemannian Metrics 120
3.4. Integral Curves 121
3.5. Flows 124
3.6. Lie Derivatives 128
3.7. Bracket 133
3.8. Geometric Interpretation of Brackets 135
3.9. Action of Maps 138
3.10. Critical Point Theory 142
3.11. First Order Partial Differential Equations 149
3.12. Frobenius' Theorem 155
Contents Vii
Appendix to Chapter 3 158
3A Tensor Bundles 158
3B. Parallelizable Manifolds 160
3C. Orientability 162
Chapter 4/Integration Theory 65
4.1. Introduction 165
4.2. Differential Forms 166
4.3. Exterior Derivatives 167
4 4. Interior Products 170
4.5. Converse of the Poincare Lemma 173
4.6. Cubical Chains 778
4.7. Integration on Euclidean Spaces 187
4.8. Integration of Forms 190
4.9. Stokes' Theorem 195
4.10. Differential Systems 199
Chapter 5/Riemannian and Semi-riemannian Manifolds 206
5.1. Introduction 206
5.2. Riemannian and Semi-riemannian Metrics 207
5.3. Length, Angle, Distance, and Energy 208
5.4. Euclidean Space 212
5.5. Variations and Rectangles 213
5.6. Flat Spaces 216
5.7. Affine Connexions 219
5.8. Parallel Translation 224
5.9. Covariant Differentiation of Tensor Fields 228
5.10. Curvature and Torsion Tensors 231
5.11. Connexion of a Semi-riemannian Structure 238
5.12. Geodesics 244
5.13. Minimizing Properties of Geodesics 247
5.14. Sectional Curvature 250
Chapter 6/Physical Application 255
6.1. Introduction 255
6.2. Hamiltonian Manifolds 256
6.3. Canonical Hamiltonian Structure on the Cotangent Bundle 259
viii Contents
6.4. Geodesic Spray of a Semi-riemannian Manifold 262
6.5. Phase Space 264
6.6. State Space 269
6.7. Contact Coordinates 269
6.8. Contact Manifolds 271
Bibliography 273
Index 275
o
CHAPTER
Set Theory and Topology
0.1. SET THEORY
Since we cannot hope to convey the significance of set theory, it is mostly for
the sake of logical completeness and to fix our notation that we give the
definitions and deduce the facts that follow.
0.1.1. Sets
Set theory is concerned with abstract objects and their relation to various
collections which contain them. We do not define what a set is but accept it as
a primitive notion. We gain an intuitive feeling for the meaning of sets and,
consequently, an idea of their usage from merely listing some of the synonyms:
class, collection, conglomeration, bunch, aggregate. Similarly, the notion of an
object is primitive, with synonyms element and point. Finally, the relation
between elements and sets, the idea of an element being in a set, is primitive.
We use a special symbol to indicate this relation, c, which is read "is an
element of." The negation is written 0, read "is not an element of."
As with all modern mathematics, once the primitive terms have been
specified, axioms regarding their usage can be specified, so the set theory can
be developed as a sequence of theorems and definitions. (For example, this is
done in an appendix to J. Kelly, General Topology, Van Nostrand, Princeton,
N.J., 1955.) However, the axioms are either very transparent intuitively or
highly technical, so we shall use the naive approach of dependence on intui-
tion, since it is quite natural (deceptively so) and customary.
We do not exclude the possibility that sets are elements of other sets. Thus
we may have x e A and A E T, which we interpret as saying that A and T are
sets, x and A are elements, and that x belongs to A and A belongs to T. It may
also be that x belongs to the set B, that x is itself a set, and that T is an element
I