this is the third version of a book on differential manifolds. the first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. at the time, i found no satisfactory book for the foundations of the subject, for multiple reasons. i expanded the book in 1971, and i expand it still further today. specifically, i have added three chapters on riemannian and pseudo riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the hopf-rinow and hadamard-cartan theorems, as well as some calculus of variations and applications to volume forms. i have rewritten the sections on sprays, and i have given more examples of the use of stokes' theorem. i have also given many more references to the literature, all of this to broaden the perspective of the book, which i hope can be used among things for a general course leading into many directions. the present book still meets the old needs, but fulfills new ones.
目录
Contents Pretace CHAPTER Ⅰ Differentlal Calculus 1. Categories 2. Topological Vector Spaces 3. Derivatives and Composition of Maps 4. Integration and Taylor's Formula 5. The Inverse Mapping Theorem CHAPTER Ⅱ Manitolds 1. Atlases, Charts, Morphisms 2. Submanifolds, Immersions, Submersions 3. Partitions of Unity 4. Manifolds with Boundary CHAPTER Ⅲ Vector Bundles 1. Definition, Pull Backs 2. The Tangent Bundle 3. Exact Sequences of Bundles 4. Operations on Vector Bundles 5. Splitting of Vector Bundles CHAPTER Ⅳ Vector Fields and Ditterential Equatlons 1. Existence Theorem for Differential Equations 2. Vector Fields, Curves, and Flows 3. Sprays 4. The Flow of a Spray and the Exponential Map 5. Existence of Tubular Neighborhoods 6. Uniqueness of Tubular Neighborhoods CHAPTER Ⅴ Operations on Vector Flelds and Diffterential Forms 1. Vector Fields, Differential Operators, Brackets 2. Lie Derivative 3. Exterior Derivative 4. The Poincare Lemma 5. Contractions and Lie Derivative 6. Vector Fields and l-Forms Under Self Duality 7. The Canonical 2-Fonn 8. Darboux's Theorem CHAPTER Ⅵ The Theorem ot Frobenlus CHAPTER Ⅶ Metrlcs CHAPTER Ⅷ Covariant Derlvatlves and Geodesics CHAPTER Ⅸ Curvature CHAPTER Ⅹ Volume Forms CHAPTER Ⅺ Integratlon of Differentlal Forms CHAPTER Ⅻ Stokes' Theorem CHAPTER ⅩⅢ Appllcatlons of Stokes' Theorem APPENDIX The Spectral Theorem Bibliography Index
