微分流形和黎曼流形

出版时间：1998-3  出版社：世界图书出版公司  作者：S.lang  页数：364  
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内容概要

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.

书籍目录

ContentsPretaceCHAPTER 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 TheoremCHAPTER Ⅱ Manitolds  1. Atlases, Charts, Morphisms  2. Submanifolds, Immersions, Submersions  3. Partitions of Unity  4. Manifolds with BoundaryCHAPTER 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 BundlesCHAPTER 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 NeighborhoodsCHAPTER 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 TheoremCHAPTER Ⅵ The Theorem ot FrobenlusCHAPTER Ⅶ MetrlcsCHAPTER Ⅷ Covariant Derlvatlves and GeodesicsCHAPTER Ⅸ CurvatureCHAPTER Ⅹ Volume FormsCHAPTER Ⅺ Integratlon of Differentlal FormsCHAPTER Ⅻ Stokes' TheoremCHAPTER ⅩⅢ Appllcatlons of Stokes' TheoremAPPENDIX The Spectral TheoremBibliographyIndex

用户评论 (总计1条)

•   S.Lang 的书不错，起点比较高。