【珍藏书籍】(eBook-pdf) Mathematics - Differential Geometry, Analysis, and Physics.pdf

Differential Geometry, Analysis and Physics
Jeffrey M. Lee
c 2000 Jeffrey Marc lee
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Preliminaries and Local Theory 1
Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 11
Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Differentiable Manifolds 15
Rough Ideas I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 16
Differentiable Manifolds and Differentiable Maps . . . . . . . . . 17
Pseudo-Groups and Models Spaces . . . . . . . . . . . . . . . . . 22
Smooth Maps and Diffeomorphisms . . . . . . . . . . . . . . . . 27
Coverings and Discrete groups . . . . . . . . . . . . . . . . . . . 30
Covering spaces and the fundamental group . . . . . . . . 30
Discrete Group Actions . . . . . . . . . . . . . . . . . . . 36
Grassmannian manifolds . . . . . . . . . . . . . . . . . . . . . . . 39
Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Manifolds with boundary. . . . . . . . . . . . . . . . . . . . . . . 43
3 The Tangent Structure 47
Rough Ideas II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
The Tangent Map . . . . . . . . . . . . . . . . . . . . . . . . . . 54
The Tangent and Cotangent Bundles . . . . . . . . . . . . . . . . 55
Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . 55
The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . 57
Important Special Situations. . . . . . . . . . . . . . . . . . . . . 59
4 Submanifold, Immersion and Subm