Introduction to Manifolds

Loring W. Tu0387480986, 9780387480985

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory.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; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems.This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, Introduction to Manifolds is also an excellent foundation for Springer GTM 82, Differential Forms in Algebraic Topology.

---

Table of contents :
Preface……Page 5
Contents……Page 7
A Brief Introduction……Page 14
Smooth Functions on a Euclidean Space……Page 16
Tangent Vectors in Rn as Derivations……Page 22
Alternating k-Linear Functions……Page 30
Differential Forms on Rn……Page 44
Manifolds……Page 56
Smooth Maps on a Manifold……Page 65
Quotients……Page 71
The Tangent Space……Page 83
Submanifolds……Page 96
Categories and Functors……Page 106
The Rank of a Smooth Map……Page 110
The Tangent Bundle……Page 123
Bump Functions and Partitions of Unity……Page 131
Vector Fields……Page 139
Lie Groups……Page 151
Lie Algebras……Page 163
Differential 1-Forms……Page 174
Differential k-Forms……Page 179
The Exterior Derivative……Page 187
Orientations……Page 197
Manifolds with Boundary……Page 206
Integration on a Manifold……Page 216
De Rham Cohomology……Page 227
The Long Exact Sequence in Cohomology……Page 235
The Mayer–Vietoris Sequence……Page 241
Homotopy Invariance……Page 248
Computation of de Rham Cohomology……Page 254
Proof of Homotopy Invariance……Page 263
Appendices……Page 268
Point-Set Topology……Page 269
The Inverse Function Theorem on Rn and Related Results……Page 287
Existence of a Partition of Unity in General……Page 294
Linear Algebra……Page 298
Solutions to Selected ExercisesWithin the Text……Page 302
List of Symbols……Page 326
References……Page 334
Index……Page 335