多様体入門（テキスト） An Introduction to Manifolds - Universitext

Tu, Loring W.

Springer（2007/11発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 360 p.／サイズ 104 illus.
• 言語 ENG
• 商品コード 9780387480985

基本説明

Features - Assumes only one semester of algebra and two semesters of undergraduate analysis, so is accessible to undegraduates and graduate students; Provides hints and solutions to about half of the exercises, so may be used in the classroom or for independent study.

Full Description

In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages, while other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems.

Contents

A Brief Introduction.- Part I. The Euclidean Space.- Smooth Functions on R(N).- Tangent Vectors In R(N) as Derivations.- Alternating K-Linear Functions.- Differential Forms on R(N).- Part II. Manifolds.- Manifolds.- Smooth Maps on A Manifold.- Quotient.- Part III. The Tangent Space.- The Tangent Space.- Submanifolds.- Categories And Functors.- The Image of A Smooth Map.- The Tangent Bundle.- Bump Functions and Partitions of Unity.- Vector Fields.- Part IV. Lie Groups and Lie Algebras.- Lie Groups.- Lie Algebras.- Part V. Differential Forms.- Differential 1-Forms.- Differential K-Forms.- The Exterior Derivative.- Part VI. Integration.- Orientations.- Manifolds With Boundary.- Integration on A Manifold.- Part VII. De Rham Theory.- De Rham Cohomology.- The Long Exact Sequence in Cohomology.- The Mayer-Vietoris Sequence.- Homotopy Invariance.- Computation of De Rham Cohomology.- Proof of Homotopy Invariance.- Appendix A. Point-Set Topology.- Appendix B. Inverse Function Theorem of R(N) And Related Results.- Appendix C. Existence of A Partition of Unity in General.- Appendix D. Solutions to Selected Exercises.- Bibliography.- Index.