Differential and Riemannian Manifolds

Front Cover
Springer Science & Business Media, Dec 6, 2012 - Mathematics - 364 pages
This is the third version of a book on differential manifolds. The first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. At the time, I found no satisfactory book for the foundations of the subject, for multiple reasons. I expanded the book in 1971, and I expand it still further today. Specifically, I have added three chapters on Riemannian and pseudo Riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the Hopf-Rinow and Hadamard-Cartan theorems, as well as some calculus of variations and applications to volume forms. I have rewritten the sections on sprays, and I have given more examples of the use of Stokes' theorem. I have also given many more references to the literature, all of this to broaden the perspective of the book, which I hope can be used among things for a general course leading into many directions. The present book still meets the old needs, but fulfills new ones. At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them (immersions, embeddings, isomorphisms, etc.).
 

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Contents

Preface
1
Manifolds
20
4 Manifolds with Boundary
36
3 Exact Sequences of Bundles
49
CHAPTER IV
64
2 Vector Fields Curves and Flows
86
3 Sprays
94
4 The Flow of a Spray and the Exponential Map
103
2 The Hilbert Group
173
3 Reduction to the Hilbert Group
176
4 Hilbertian Tubular Neighborhoods
179
5 The MorsePalais Lemma
182
6 The Riemannian Distance
184
7 The Canonical Spray
188
CHAPTER VIII
190
Covariant Derivatives and Geodesics
191

5 Existence of Tubular Neighborhoods
108
6 Uniqueness of Tubular Neighborhoods
110
CHAPTER V
114
2 Lie Derivative
120
3 Exterior Derivative
122
4 The Poincaré Lemma
135
5 Contractions and Lie Derivative
137
6 Vector Fields and 1Forms Under Self Duality
141
7 The Canonical 2Form
146
8 Darbouxs Theorem
148
CHAPTER VI
153
2 Differential Equations Depending on a Parameter
158
3 Proof of the Theorem
159
4 The Global Formulation
160
5 Lie Groups and Subgroups
163
CHAPTER VII
169
2 Sprays and Covariant Derivatives
194
3 Derivative Along a Curve and Parallelism
199
4 The Metric Derivative
203
5 More Local Results on the Exponential Map
209
6 Riemannian Geodesic Length and Completeness
216
CHAPTER IX
219
Curvature
225
2 Jacobi Lifts
233
3 Application of Jacobi Lifts to dexpx
241
5 Taylor Expansions
257
4 The Hodge Star on Forms
273
2 Change of Variables Formula
289
CHAPTER XII
307
CHAPTER XIII
321
5 Cauchys Theorem
335
Bibliography
355

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