| 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 |
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