Differentiable manifolds

Contents......Page 5
Preface......Page 3
§1. Topological Spaces......Page 9
§2. Vector Spaces......Page 14
§3. The n-Dimensional Real Space R^n and C Functions......Page 21
§4. The Inverse Function Theorem......Page 24
§1. The Definition of a Manifold......Page 33
§2. Examples of Differentiable Manifolds......Page 37
§3. Differentiable Functions and Local Coordinate Systems......Page 41
§4. Differentiable Mappings......Page 46
§5. Tangent Vectors and Tangent Spaces, the Riemannian Metric......Page 48
§6. The Differential of a Function and Critical Points......Page 54
§7. The Differential of a Map......Page 61
§8. Sard\'s Theorem......Page 64
§9. The Motion in a Riemannian Manifold......Page 67
§10. Immersion and Imbedding of Manifolds; Submanifolds......Page 70
§11. Vector Field and Derivations......Page 78
§12. Vector Fields and One-Parameter Transformation Groups......Page 86
§13. The Infinitesimal Motion of a Riemannian Manifold......Page 95
§14. Paracompact Manifolds and the Partition of Unity......Page 99
§15. Some Remarks on the Topology of Manifolds......Page 103
§16. Complex Manifolds......Page 108
§17. Almost Complex Structures......Page 121
§1. p-Linear Functions......Page 131
§2. Symmetric Tensors and Alternating Tensors; the Exterior Product......Page 133
§3. Covariant Tensor Fields on a Manifold and Differential Forms......Page 141
§4. The Lie Differentiation of Tensor Fields and the Exterior Differentiation of Differential Forms......Page 147
§5. Transformations of Covariant Tensor Fields by Maps......Page 153
§6. The Cohomology Algebra of a Manifold......Page 156
§7. Complex Differential Forms on a Complex Manifold......Page 160
§8. Differential Systems and Integral Manifolds......Page 166
§9. An Application to Integrable Almost Complex Structures......Page 176
§10. Maximal Connected Integral Manifolds......Page 180
§1. Topological Groups......Page 184
§2. Subgroups and Quotient Spaces of Topological Groups......Page 186
§3. Isomorphisms and Homomorphisms of Topological Groups......Page 188
§4. The Connected Component of a Topological Group......Page 189
§5. Homogeneous Spaces of Topological Groups, Locally Compact Groups......Page 192
§6. Lie Groups and Lie Algebras......Page 196
§7. Invariant Differential Forms on Lie Groups......Page 200
§8 One-Parameter Subgroups and the Exponential Map......Page 203
§9. Examples of Lie Groups......Page 208
§10. The Canonical Coordinate Systems of Lie Groups......Page 211
§11. Complex Lie Groups and Complex Lie Algebras......Page 218
§12. Lie Subgroups of a Lie Group (I)......Page 225
§13. Linear Lie Groups......Page 229
§14. Quotient Spaces and Quotient Groups of Lie Groups......Page 233
§15. Isomorphisms and Homomorphisms of Lie Groups; Representations of Lie Groups......Page 236
§16. The Structure of Connected Commutative Lie Groups......Page 241
§17. Lie Transformation Groups and Homogeneous Spaces of Lie Groups......Page 242
§18. Examples of Homogeneous Spaces......Page 247
§19. The Differentiability of One-Parameter Subgroups......Page 252
§20. Lie Subgroups of a Lie Group (II)......Page 254
§1. Orientation of Manifolds......Page 261
§2. Integration of Differential Forms......Page 272
§3. Invariant Integration of Lie Groups......Page 280
§4. Applications of Invariant Integration......Page 283
§5. Stokes\' Theorem......Page 288
§6. Degree of Mappings......Page 297
§7. Divergences of Vector Fields; Laplacians......Page 300
Bibliography......Page 308
Index......Page 309