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 12
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 112
9. Covering Manifolds 117
Notes 121
Chapter IV. Vector Fields on a Manifold 122
1. The Tangent Space at a Point of a Manifold 123
2. Vector Fields 132
3. One-Parameter and Local One-Parameter Groups Acting on a Manifold 139
4. The Existence Theorem for Ordinary Differential Equations 147
5. Some Examples of One-Parameter Groups Acting on a Manifold 155
6. One-Parameter Subgroups of Lie Groups 162
7. The Lie Algebra of Vector Fields on a Manifold 166
8. Frobenius' Theorem 173
9. Homogeneous Spaces 181
Notes 188
Appendix: Partial Proof of Theorem 4.1 189
Chapter V. Tensors and Tensor Fields on Manifolds 191
1. Tangent Covectors 192
2. Bilinear Forms. The Riemannian Metric 198
3. Riemannian Manifolds as Metric Spaces 202
4. Partitions of Unity 208
5. Tensor Fields 214
6. Multiplication of Tensors 221
7. Orientation of Manifolds and the Volume Element 230
8. Exterior Differentiation 234
Notes 242
Chapter Vl. Integration on Manifolds 243
1. Integration in Rn Domains of Integration 244
2. A Generalization to Manifolds 250
3. Integration on Lie Groups 258
4. Manifolds with Boundary 265
5. Stokes's Theorem for Manifolds with Boundary 273
6. Homotopy or Mappings. The Fundamental Group 280
7. Some Applications of Differential Forms. The de Rham Groups 288
8. Some Further Applications of de Rham Groups 295
9. Covering Spaces and the Fundamental Group 303
Notes 309
Chapter VII. Differentiation on Riemannian Manifolds 310
1. Differentiation of Vector Fields along Curves in Rn 311
2. Differentiation of Vector Fields on Submanifolds of Rn 320
3. Differentiation on Riemannian Manifolds 330
4. Addenda to the Theory of Differentiation on a Manifold 338
5. Geodesic Curves on Riemannian Manifolds 343
6. The Tangent Bundle and Exponential Mapping. Normal Coordinates 348
7. Some Further Properties of Geodesics 355
8. Symmetric Riemannian Manifolds 364
9. Some Examples 370
Notes 377
Chapter VIII. Curvature 378
1. The Geometry of Surfaces in E3 379
2. The Gaussian and Mean Curvatures of a Surface 387
3. Basic Properties of the Riemann Curvature Tensor 395
4. The Curvature Forms and the Equations of Structure 402
5. Differentiation of Covariant Tensor Fields 408
6. Manifolds of Constant Curvature 416
Notes 427
References 430
Index 434