Mathematics - Differential Geometry - Analysis and Physics

Mathematics - Differential Geometry - Analysis and Physics...

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Unformatted text preview: Differential Geometry, Analysis and Physics Jeffrey M. Lee c 2000 Jeffrey Marc lee ii Contents 0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Preliminaries and Local Theory 1 1.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Chain Rule, Product rule and Taylor’s Theorem . . . . . . . . . 11 1.3 Local theory of maps . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Differentiable Manifolds 15 2.1 Rough Ideas I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Topological Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Differentiable Manifolds and Differentiable Maps . . . . . . . . . 17 2.4 Pseudo-Groups and Models Spaces . . . . . . . . . . . . . . . . . 22 2.5 Smooth Maps and Diffeomorphisms . . . . . . . . . . . . . . . . 27 2.6 Coverings and Discrete groups . . . . . . . . . . . . . . . . . . . 30 2.6.1 Covering spaces and the fundamental group . . . . . . . . 30 2.6.2 Discrete Group Actions . . . . . . . . . . . . . . . . . . . 36 2.7 Grassmannian manifolds . . . . . . . . . . . . . . . . . . . . . . . 39 2.8 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.9 Manifolds with boundary. . . . . . . . . . . . . . . . . . . . . . . 43 3 The Tangent Structure 47 3.1 Rough Ideas II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 The Tangent Map . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 The Tangent and Cotangent Bundles . . . . . . . . . . . . . . . . 55 3.5.1 Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5.2 The Cotangent Bundle . . . . . . . . . . . . . . . . . . . . 57 3.6 Important Special Situations. . . . . . . . . . . . . . . . . . . . . 59 4 Submanifold, Immersion and Submersion. 63 4.1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Submanifolds of R n . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Regular and Critical Points and Values . . . . . . . . . . . . . . . 66 4.4 Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 iii iv CONTENTS 4.5 Immersed Submanifolds and Initial Submanifolds . . . . . . . . . 71 4.6 Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.7 Morse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.8 Problem set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5 Lie Groups I 81 5.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 81 5.2 Lie Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . 84 6 Fiber Bundles and Vector Bundles I 87 6.1 Transitions Maps and Structure . . . . . . . . . . . . . . . . . . . 94 6.2 Useful ways to think about vector bundles . . . . . . . . . . . . . 94 6.3 Sections of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . ....
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This note was uploaded on 10/22/2010 for the course MATH 9999 taught by Professor Dr.abc during the Spring '10 term at CUHK.

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Mathematics - Differential Geometry - Analysis and Physics...

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