A Course in Differential Geometry

By: Thierry Aubin

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Hardcover | 1 January 2001

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This textbook for second-year graduate students is an introduction to differential geometry with principal emphasis on Riemannian geometry. The author is well-known for his significant contributions to the field of geometry and PDEs - particularly for his work on the Yamabe problem - and for his expository accounts on the subject. The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory.

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Non-Fiction Mathematics Geometry Differential & Riemannian Geometry Calculus & Mathematical Analysis

Preface	p. ix
Background Material	p. 1
Topology	p. 1
Tensors	p. 3
Differential Calculus	p. 7
Exercises and Problems	p. 13
Differentiable Manifolds	p. 19
Basic Definitions	p. 19
Partition of Unity	p. 25
Differentiable Mappings	p. 27
Submanifolds	p. 29
The Whitney Theorem	p. 30
The Sard Theorem	p. 34
Exercises and Problems	p. 38
Solutions to Exercises	p. 40
Tangent Space	p. 43
Tangent Vector	p. 44
Linear Tangent Mapping	p. 46
Vector Bundles	p. 48
The Bracket [X, Y]	p. 49
Exterior Differential	p. 52
Orientable Manifolds	p. 55
Manifolds with Boundary	p. 58
Exercises and Problems	p. 60
Solutions to Exercises and Problems	p. 66
Integration of Vector Fields and Differential Forms	p. 77
Integration of Vector Fields	p. 77
Lie Derivative	p. 79
The Frobenius Theorem	p. 81
Integrability Criteria	p. 85
Exercises and Problems	p. 87
Solutions to Exercises and Problems	p. 93
Linear Connections	p. 99
First Definitions	p. 99
Christoffel Symbols	p. 100
Torsion and Curvature	p. 101
Parallel Transport. Geodesics	p. 103
Covariant Derivative	p. 105
Exercises and Problems	p. 107
Solutions to Exercises	p. 108
Riemannian Manifolds	p. 111
Some Definitions	p. 111
Riemannian Connection	p. 114
Exponential Mapping	p. 117
Some Operators on Differential Forms	p. 121
Spectrum of a Manifold	p. 125
Exercises and Problems	p. 128
Solutions to Exercises and Problems	p. 145
The Yamabe Problem: An Introduction to Research	p. 169
Bibliography	p. 177
Subject Index	p. 179
Notation	p. 183
Table of Contents provided by Syndetics. All Rights Reserved.

A Course in Differential Geometry

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