Differential Geometric Structures

Name: Differential Geometric Structures
Price: 81.6 USD
Availability: InStock
ISBN: 9780070504356

ISBN-10: 0070504350

ISBN-13: 9780070504356

Edition: 1981

Authors: Walter A. Poor

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Description:

Useful for independent study and as a reference work, this introduction to differential geometry features many examples and exercises. It defines geometric structure by specifying the parallel transport in an appropriate fiber bundle, focusing on the simplest cases of linear parallel transport in a vector bundle. The treatment opens with an introductory chapter on fiber bundles that proceeds to examinations of connection theory for vector bundles and Riemannian vector bundles. Additional topics include the role of harmonic theory, geometric vector fields on Riemannian manifolds, Lie groups, symmetric spaces, and symplectic and Hermitian vector bundles. A consideration of other differential…

Book details

List price: $59.95
Copyright year: 1981
Publisher: McGraw-Hill Companies, The
Binding: Cloth Text
Pages: 320
Language: English

Preface	p. ix
An Introduction to Fiber Bundles	p. 1
The Definition of a Fiber Bundle	p. 1
Vector Bundles	p. 12
The Vertical Bundle	p. 17
Operations on Vector Bundles	p. 21
Principal and Associated Bundles	p. 25
Sections of Fiber Bundles	p. 31
Connection Theory for Vector Bundles	p. 40
Parallelism Structures in Vector Bundles	p. 41
Holonomy in Vector Bundles	p. 51
Connections on Vector Bundles	p. 54
The Curvature Tensor	p. 67
Covariant Derivative Operators	p. 71
The Structure Equation	p. 80
The Space of Connections on a Vector Bundle	p. 85
Characteristic Classes	p. 90
The Tangent Bundle: Linear Connections	p. 93
The Tangent Bundle: Affine Connections	p. 103
Affine Transformations	p. 107
Riemannian Vector Bundles	p. 112
Riemannian Metrics	p. 112
Riemannian Connections	p. 119
The Levi-Civita Connection	p. 122
The Metric Structure of a Riemannian Manifold	p. 131
The Gauss-Bonnet Theorem	p. 138
Harmonic Theory	p. 150
The Basic Differential Operators	p. 150
Green's Theorem and Some Applications	p. 157
Weitzenbock's Formula for the Laplacian	p. 158
Chern's Formula for the Laplacian	p. 161
Geometric Vector Fields on Riemannian Manifolds	p. 166
Harmonic Fields	p. 167
Killing Fields	p. 169
Conformal Fields	p. 171
Affine Fields	p. 181
Projective Fields	p. 181
Lie Groups	p. 185
A Negative Curvature Example	p. 185
Bi-invariant Metrics	p. 188
Some Simple Examples	p. 198
Homogeneous Spaces	p. 210
Symmetric Spaces	p. 221
Affine Symmetric Spaces	p. 221
Locally Affine Symmetric Spaces	p. 228
Symmetric Lie Algebras	p. 234
Riemannian Symmetric Spaces	p. 236
Symplectic and Hermitian Vector Bundles	p. 243
Symplectic Vector Bundles	p. 243
Hermitian Vector Bundles	p. 251
Complex Manifolds	p. 254
The Curvature of Kahler Manifolds	p. 270
Other Differential Geometric Structures	p. 274
Parallelism in Principal Fiber Bundles	p. 275
Holonomy and Curvature in Principal Fiber Bundles	p. 280
Characteristic Classes of Principal Bundles	p. 288
Parallel Transport in Fiber Bundles	p. 290
Cartan Connections	p. 293
Spin Structures	p. 302
Bibliography	p. 317
Index of Notation	p. 324
Index	p. 331
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