An introduction to differentiable manifolds and Riemannian geometry (2nd Ed)

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ISBN: 9780121160524 | 0121160521
Cover: Nonspecific Binding
Copyright: 4/1/1986

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The second edition of this text has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by "beginners" in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject.Line and surface integrals Divergence and curl of vector fields

Preface to the Revised Second Edition	p. xi
Preface to the Second Edition	p. xiii
Preface to the First Edition	p. xv
Introduction to Manifolds
Preliminary Comments on R[superscript n]	p. 1
R[superscript n] and Euclidean Space	p. 4
Topological Manifolds	p. 6
Further Examples of Manifolds. Cutting and Pasting	p. 11
Abstract Manifolds. Some Examples	p. 14
Functions of Several Variables and Mappings
Differentiability for Functions of Several Variables	p. 20
Differentiability of Mappings and Jacobians	p. 25
The Space of Tangent Vectors at a Point of R[superscript n]	p. 29
Another Definition of T[subscript a](R[superscript n])	p. 32
Vector Fields on Open Subsets of R[superscript n]	p. 36
The Inverse Function Theorem	p. 41
The Rank of a Mapping	p. 46
Differentiable Manifolds and Submanifolds
The Definition of a Differentiable Manifold	p. 52
Further Examples	p. 59
Differentiable Functions and Mappings	p. 65
Rank of a Mapping, Immersions	p. 68
Submanifolds	p. 74
Lie Groups	p. 80
The Action of a Lie Group on a Manifold. Transformation Groups	p. 87
The Action of a Discrete Group on a Manifold	p. 93
Covering Manifolds	p. 98
Vector Fields on a Manifold
The Tangent Space at a Point of a Manifold	p. 104
Vector Fields	p. 113
One-Parameter and Local One-Parameter Groups Acting on a Manifold	p. 119
The Existence Theorem for Ordinary Differential Equations	p. 127
Some Examples of One-Parameter Groups Acting on a Manifold	p. 135
One-Parameter Subgroups of Lie Groups	p. 142
The Lie Algebra of Vector Fields on a Manifold	p. 146
Frobenius's Theorem	p. 153
Homogeneous Spaces	p. 160
Tensors and Tensor Fields on Manifolds
Tangent Covectors	p. 171
Covectors on Manifolds	p. 172
Covector Fields and Mappings	p. 174
Bilinear Forms. The Riemannian Metric	p. 177
Riemannian Manifolds as Metric Spaces	p. 181
Partitions of Unity	p. 186
Some Applications of the Partition of Unity	p. 188
Tensor Fields	p. 192
Tensors on a Vector Space	p. 192
Tensor Fields	p. 194
Mappings and Covariant Tensors	p. 195
The Symmetrizing and Alternating Transformations	p. 196
Multiplication of Tensors	p. 199
Multiplication of Tensors on a Vector Space	p. 199
Multiplication of Tensor Fields	p. 201
Exterior Multiplication of Alternating Tensors	p. 202
The Exterior Algebra on Manifolds	p. 206
Orientation of Manifolds and the Volume Element	p. 207
Exterior Differentiation	p. 212
An Application to Frobenius's Theorem	p. 216
Integration on Manifolds
Integration in R[superscript n] Domains of Integration	p. 223
Basic Properties of the Riemann Integral	p. 224
A Generalization to Manifolds	p. 229
Integration on Riemannian Manifolds	p. 232
Integration on Lie Groups	p. 237
Manifolds with Boundary	p. 243
Stokes's Theorem for Manifolds	p. 251
Homotopy of Mappings. The Fundamental Group	p. 258
Homotopy of Paths and Loops. The Fundamental Group	p. 259
Some Applications of Differential Forms. The de Rham Groups	p. 265
The Homotopy Operator	p. 268
Some Further Applications of de Rham Groups	p. 272
The de Rham Groups of Lie Groups	p. 276
Covering Spaces and Fundamental Group	p. 280
Differentiation on Riemannian Manifolds
Differentiation of Vector Fields along Curves in R[superscript n]	p. 289
The Geometry of Space Curves	p. 292
Curvature of Plane Curves	p. 296
Differentiation of Vector Fields on Submanifolds of R[superscript n]	p. 298
Formulas for Covariant Derivatives	p. 303
[down triangle, open subscript x subscript p] Y and Differentiation of Vector Fields	p. 305
Differentiation on Riemannian Manifolds	p. 308
Constant Vector Fields and Parallel Displacement	p. 314
Addenda to the Theory of Differentiation on a Manifold	p. 316
The Curvature Tensor	p. 316
The Riemannian Connection and Exterior Differential Forms	p. 319
Geodesic Curves on Riemannian Manifolds	p. 321
The Tangent Bundle and Exponential Mapping. Normal Coordinates	p. 326
Some Further Properties of Geodesics	p. 332
Symmetric Riemannian Manifolds	p. 340
Some Examples	p. 346
Curvature
The Geometry of Surfaces in E[superscript 3]	p. 355
The Principal Curvatures at a Point of a Surface	p. 359
The Gaussian and Mean Curvatures of a Surface	p. 363
The Theorema Egregium of Gauss	p. 366
Basic Properties of the Riemann Curvature Tensor	p. 371
Curvature Forms and the Equations of Structure	p. 378
Differentiation of Covariant Tensor Fields	p. 384
Manifolds of Constant Curvature	p. 391
Spaces of Positive Curvature	p. 394
Spaces of Zero Curvature	p. 396
Spaces of Constant Negative Curvature	p. 397
References	p. 403
Index	p. 411
Table of Contents provided by Syndetics. All Rights Reserved.

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An introduction to differentiable manifolds and Riemannian geometry (2nd Ed)

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