Introduction to Differentiable Manifolds and Riemannian Geometry (eBook)

Front Cover 1
An Introduction to Differentiable Manifolds and Riemannian Geometry 4
Copyright Page 5
Contents 8
Preface to the Second Edition 12
Preface to the First Edition 14
Chapter I. Introduction to Manifolds 18
1. Preliminary Comments on Rn 18
2. Rn and Euclidean Space 21
3. Topological Manifolds 23
4. Further Examples of Manifolds. Cutting and Pasting 28
5. Abstract Manifolds. Some Examples 31
Notes 35
Chapter II. Functions of Several Variables and Mappings 37
1. Differentiability for Functions of Several Variables 37
2. Differentiability of Mappings and Jacobians 42
3. The Space of Tangent Vectors at a Point of Rn 46
4. Another Definition of Ta(Rn) 49
5. Vector Fields on Open Subsets of Rn 54
6. The Inverse Function Theorem 58
7. The Rank of a Mapping 63
Notes 67
Chapter III. Differentiable Manifolds and Submanifolds 68
1. The Definition of a Differentiable Manifold 69
2. Further Examples 77
3. Differentiable Functions and Mappings 82
4. Rank of a Mapping. Immersions 86
5. Submanifolds 92
6. Lie Groups 98
7. The Action of a Lie Group on a Manifold. Transformation Groups 106
8. The Action of a Discrete Group on a Manifold 113
9. Covering Manifolds 118
Notes 121
Chapter IV. Vector Fields on a Manifold 123
1. The Tangent Space at a Point of a Manifold 124
2. Vector Fields 133
3. One-Parameter and Local One-Parameter Groups Acting on a Manifold 140
4. The Existence Theorem for Ordinary Differential Equations 148
5. Some Examples of One-Parameter Groups Acting on a Manifold 156
6. One-Parameter Subgroups of Lie Groups 164
7. The Lie Algebra of Vector Fields on a Manifold 168
8. Frobenius' Theorem 175
9. Homogeneous Spaces 182
Appendix Partial Proof of Theorem 4.1 189
Notes 191
Chapter V. Tensors and Tensor Fields on Manifolds 193
1. Tangent Covectors 194
2. Bilinear Forms. The Riemannian Metric 200
3. Riemannian Manifolds as Metric Spaces 204
4. Partitions of Unity 210
5. Tensor Fields 216
6. Multiplication of Tensors 223
7. Orientation of Manifolds and the Volume Element 232
8. Exterior Differentiation 236
Notes 244
Chapter VI. Integration on Manifolds 246
1. Integration in Rn. Domains of Integration 247
2. A Generalization to Manifolds 253
3. Integration on Lie Groups 261
4. Manifolds with Boundary 268
5. Stokes's Theorem for Manifolds with Boundary 276
6. Homotopy of Mappings. The Fundamental Group 283
7. Some Applications of Differential Forms. The de Rham Groups 291
8. Some Further Applications of de Rham Groups 298
9. Covering Spaces and the Fundamental Group 306
Notes 313
Chapter VII. Differentiation on Riemannian Manifolds 314
1. Differentiation of Vector Fields along Curves in Rn 315
2. Differentiation of Vector Fields on Submanifolds of Rn 324
3. Differentiation on Riemannian Manifolds 334
4. Addenda to the Theory of Differentiation on a Manifold 342
5. Geodesic Curves on Riemannian Manifolds 347
6. The Tangent Bundle and Exponential Mapping. Normal Coordinates 352
7. Some Further Properties of Geodesics 359
8. Symmetric Riemannian Manifolds 368
9. Some Examples 374
Notes 381
Chapter VIII. Curvature 382
1 . The Geometry of Surfaces in E3 383
2. The Gaussian and Mean Curvatures of a Surface 391
3. Basic Properties of the Riemann Curvature Tensor 399
4. The Curvature Forms and the Equations of Structure 406
5. Differentiation of Covariant Tensor Fields 413
6. Manifolds of Constant Curvature 420
Notes 431
References 434
Index 440