陈省身-魏依理论及威腾变形讲义LECTURES ON CHERN-WEIL THEORY AND WITTEN DEFORMATIONS

This book is noteworthy for its combination of brevity, clarity, accessibility, and depth.

Preface
Chapter 1 Chern-Weil Theory for Chacteristic Closes
1.1 Review of the de Rham Cohomology Theory
1.3 The Curvature of a Connection
1.4 Chern Well Theorem
1.5 Characteristic Forms, Classes and Numnbers
1.6 Some Examples
1.6.1 Chern Forms and Classes
1.6.2 Pontrjagin Classes for Real Vector Bundles
1.6.3 Hirzebruch's L-class and A-class
1.6.4 K-groups and thc Chern Character
1.6.5 The Chern-Silnons Transgressed Form
1.7 Bott Vanishillg Theorem for Foliations
1.7.1 Foliations and the Bott Vanishing Theorem
1.8 Chcrn-Weil Theory in Odd Dimension
Chapter 2 Bott and Duistermaat-Heckman Formulas
2.1 Bcrline Vergne Loclization Formula
2.2 Bott Residue Formula
2.4 Bott's OrlginalIdea
2.5 References
Chapter 3 Gauss-Bonnet-Chern Theorem
3.1 A Toy Model and the Berezin Integral
3:2 Mathai-Quillen's Them Form
3.3 A Transgression Formula
3.4 Proof of the Gauss Bonnet-Chern Theorem
3.5 Some Remarks
3.6 Chern's Original Proof
3.7 References
Chapter 4 Poincare-HopfIndex Formula: an Analytic Proof
4.1 Review of Hedge Theorem
4.2 Poineare-Hopf Index Formula
4.4 An Estimate Outside of Upzero(v)Up
4.5 Harmonic Oscillators on Euclidean Spaces
4.6 A Proof of the Poincare-Hopf Index Formula
4.7 Some Estimates for DT'S, 2 < i < 4
4.8 An Alternate Analytic Proof
4.9 References
Chapter 5 Morse Inequalities: an Analytic Proof
5.1 Review of Morse Inequalitics
5.2 Witten Deformation
5.3 Hodge Theorem for (Ω*(M),dTf)
5.4 Bebaviour of Tf Near the Critical Points of f
5.5 Proof of Morse Inequalities
5.6 Proof of Proposition 5.5
5.7 Some Remards and Comments
5.8 References
Chapter 6 Thom-Smale and Witten Complexes
6.1 The Them Smalc Complex
6.2 The de Rham Map for Thom-Smale Complexes
6.3 Witten's Instanton Complex and the Map eT
6.4 The Map P∞.TeT
6.5 An Analytic Proof of Theorem 6.4
……
Chapter 7 Atyah Theorem on Kervaire Semi-characteristic
Index