Lectures on Chern-Weil Theory and Witten Deformations

By: Weiping Zhang

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Hardcover | 1 December 2000

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132 Pages

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Based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics, this volume consists of two parts: the first part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and Andr Weil, as well as a proof of the Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction of Thom forms; the second part presents analytic proofs of the Poincar-Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten.

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Preface	p. vii
Chern-Weil Theory for Characteristic Classes	p. 1
Review of the de Rham Cohomology Theory	p. 1
Connections on Vector Bundles	p. 3
The Curvature of a Connection	p. 4
Chern-Weil Theorem	p. 6
Characteristic Forms, Classes and Numbers	p. 8
Some Examples	p. 10
Chern Forms and Classes	p. 10
Pontrjagin Classes for Real Vector Bundles	p. 11
Hirzebruch's L-class and A-class	p. 12
K-groups and the Chern Character	p. 14
The Chern-Simons Transgressed Form	p. 16
Bott Vanishing Theorem for Foliations	p. 17
Foliations and the Bott Vanishing Theorem	p. 18
Adiabatic Limit and the Bott Connection	p. 20
Chern-Weil Theory in Odd Dimension	p. 22
References	p. 26
Bott and Duistermaat-Heckman Formulas	p. 29
Berline-Vergne Localization Formula	p. 29
Bott Residue Formula	p. 35
Duistermaat-Heckman Formula	p. 37
Bott's Original Idea	p. 38
References	p. 39
Gauss-Bonnet-Chern Theorem	p. 41
A Toy Model and the Berezin Integral	p. 41
Mathai-Quillen's Thom Form	p. 43
A Transgression Formula	p. 46
Proof of the Gauss-Bonnet-Chern Theorem	p. 47
Some Remarks	p. 50
Chern's Original Proof	p. 51
References	p. 54
Poincare-Hopf Index Formula: an Analytic Proof	p. 57
Review of Hodge Theorem	p. 57
Poincare-Hopf Index Formula	p. 60
Clifford Actions and the Witten Deformation	p. 61
An Estimate Outside of U[subscript p[set membership]zero(V)]U[subscript p]	p. 63
Harmonic Oscillators on Euclidean Spaces	p. 64
A Proof of the Poincare-Hopf Index Formula	p. 67
Some Estimates for D[subscript T,i]'s, 2 [less than or equal] i [less than or equal] 4	p. 69
An Alternate Analytic Proof	p. 73
References	p. 74
Morse Inequalities: an Analytic Proof	p. 75
Review of Morse Inequalities	p. 75
Witten Deformation	p. 77
Hodge Theorem for ([Omega]* (M), d[subscript Tf])	p. 78
Behaviour of [square subscript Tf] Near the Critical Points of f	p. 79
Proof of Morse Inequalities	p. 81
Proof of Proposition 5.5	p. 83
Some Remarks and Comments	p. 88
References	p. 89
Thom-Smale and Witten Complexes	p. 93
The Thom-Smale Complex	p. 93
The de Rham Map for Thom-Smale Complexes	p. 95
Witten's Instanton Complex and the Map e[subscript T]	p. 97
The Map P[subscript [infinity],T]e[subscript T]	p. 100
An Analytic Proof of Theorem 6.4	p. 102
References	p. 102
Atiyah Theorem on Kervaire Semi-characteristic	p. 105
Kervaire Semi-characteristic	p. 106
Atiyah's Original Proof	p. 107
A proof via Witten Deformation	p. 108
A Generic Counting Formula for k(M)	p. 112
Non-multiplicativity of k(M)	p. 113
References	p. 115
Index	p. 117
Table of Contents provided by Syndetics. All Rights Reserved.

Lectures on Chern-Weil Theory and Witten Deformations

At a Glance

Hardcover

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