Lectures On Chern-weil Theory And Witten Deformations

by Weiping Zhang

Overview

This invaluable book is based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics. It consists of two parts: the first part contains an introduction to the geometric theory of characteristic classes due to Shiing-shen Chern and André 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 Poincaré-Hopf index formula, as well as the Morse inequalities based on deformations introduced by Edward Witten.

Product Details

ISBN-13:	9789810246860
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/25/2001
Series:	Nankai Tracts In Mathematics , #4
Pages:	132
Product dimensions:	10.00(w) x 9.90(h) x 0.10(d)

Table of Contents

	Preface	vii
Chapter 1	Chern-Weil Theory for Characteristic Classes	1
1.1	Review of the de Rham Cohomology Theory	1
1.2	Connections on Vector Bundles	3
1.3	The Curvature of a Connection	4
1.4	Chern-Weil Theorem	6
1.5	Characteristic Forms, Classes and Numbers	8
1.6	Some Examples	10
1.6.1	Chern Forms and Classes	10
1.6.2	Pontrjagin Classes for Real Vector Bundles	11
1.6.3	Hirzebruch's L-class and A-class	12
1.6.4	K-groups and the Chern Character	14
1.6.5	The Chern-Simons Transgressed Form	16
1.7	Bott Vanishing Theorem for Foliations	17
1.7.1	Foliations and the Bott Vanishing Theorem	18
1.7.2	Adiabatic Limit and the Bott Connection	20
1.8	Chern-Weil Theory in Odd Dimension	22
1.9	References	26
Chapter 2	Bott and Duistermaat-Heckman Formulas	29
2.1	Berline-Vergne Localization Formula	29
2.2	Bott Residue Formula	35
2.3	Duistermaat-Heckman Formula	37
2.4	Bott's Original Idea	38
2.5	References	39
Chapter 3	Gauss-Bonnet-Chern Theorem	41
3.1	A Toy Model and the Berezin Integral	41
3.2	Mathai-Quillen's Thom Form	43
3.3	A Transgression Formula	46
3.4	Proof of the Gauss-Bonnet-Chern Theorem	47
3.5	Some Remarks	50
3.6	Chern's Original Proof	51
3.7	References	54
Chapter 4	Poincare-Hopf Index Formula: an Analytic Proof	57
4.1	Review of Hodge Theorem	57
4.2	Poincare-Hopf Index Formula	60
4.3	Clifford Actions and the Witten Deformation	61
4.4	An Estimate Outside of U[subscript p[set membership]zero(V)]U[subscript p]	63
4.5	Harmonic Oscillators on Euclidean Spaces	64
4.6	A Proof of the Poincare-Hopf Index Formula	67
4.7	Some Estimates for D[subscript T,i]'s, 2 [less than or equal] i [less than or equal] 4	69
4.8	An Alternate Analytic Proof	73
4.9	References	74
Chapter 5	Morse Inequalities: an Analytic Proof	75
5.1	Review of Morse Inequalities	75
5.2	Witten Deformation	77
5.3	Hodge Theorem for ([Omega]* (M), d[subscript Tf])	78
5.4	Behaviour of [square subscript Tf] Near the Critical Points of f	79
5.5	Proof of Morse Inequalities	81
5.6	Proof of Proposition 5.5	83
5.7	Some Remarks and Comments	88
5.8	References	89
Chapter 6	Thom-Smale and Witten Complexes	93
6.1	The Thom-Smale Complex	93
6.2	The de Rham Map for Thom-Smale Complexes	95
6.3	Witten's Instanton Complex and the Map e[subscript T]	97
6.4	The Map P[subscript [infinity],T]e[subscript T]	100
6.5	An Analytic Proof of Theorem 6.4	102
6.6	References	102
Chapter 7	Atiyah Theorem on Kervaire Semi-characteristic	105
7.1	Kervaire Semi-characteristic	106
7.2	Atiyah's Original Proof	107
7.3	A proof via Witten Deformation	108
7.4	A Generic Counting Formula for k(M)	112
7.5	Non-multiplicativity of k(M)	113
7.6	References	115
	Index	117