Lectures on Chern-Weil Theory and... | Buy

Based on the notes of a graduate course on differential geometry which the author gave at the Nankai Institute of Mathematics in Tianjin, P R China.

Preface

vii

Chern-Weil Theory for Characteristic Classes

1

(28)

Review of the de Rham Cohomology Theory

1

(2)

Connections on Vector Bundles

3

(1)

The Curvature of a Connection

4

(2)

Chern-Weil Theorem

6

(2)

Characteristic Forms, Classes and Numbers

8

(2)

Some Examples

10

(7)

Chern Forms and Classes

10

(1)

Pontrjagin Classes for Real Vector Bundles

11

(1)

Hirzebruch's L-class and A-class

12

(2)

K-groups and the Chern Character

14

(2)

The Chern-Simons Transgressed Form

16

(1)

Bott Vanishing Theorem for Foliations

17

(5)

Foliations and the Bott Vanishing Theorem

18

(2)

Adiabatic Limit and the Bott Connection

20

(2)

Chern-Weil Theory in Odd Dimension

22

(4)

References

26

(3)

Bott and Duistermaat-Heckman Formulas

29

(12)

Berline-Vergne Localization Formula

29

(6)

Bott Residue Formula

35

(2)

Duistermaat-Heckman Formula

37

(1)

Bott's Original Idea

38

(1)

References

39

(2)

Gauss-Bonnet-Chern Theorem

41

(16)

A Toy Model and the Berezin Integral

41

(2)

Mathai-Quillen's Thom Form

43

(3)

A Transgression Formula

46

(1)

Proof of the Gauss-Bonnet-Chern Theorem

47

(3)

Some Remarks

50

(1)

Chern's Original Proof

51

(3)

References

54

(3)

Poincare-Hopf Index Formula: an Analytic Proof

57

(18)

Review of Hodge Theorem

57

(3)

Poincare-Hopf Index Formula

60

(1)

Clifford Actions and the Witten Deformation

61

(2)

An Estimate Outside of Up∈zero(V)Up

63

(1)

Harmonic Oscillators on Euclidean Spaces

64

(3)

A Proof of the Poincare-Hopf Index Formula

67

(2)

Some Estimates for DT, is, 2 ≤ i ≤ 4

69

(4)

An Alternate Analytic Proof

73

(1)

References

74

(1)

Morse Inequalities: an Analytic Proof

75

(18)

Review of Morse Inequalities

75

(2)

Witten Deformation

77

(1)

Hodge Theorem for (Ω(M), dTf)

78

(1)

Behaviour of Tf Near the Critical Points of f

79

(2)

Proof of Morse Inequalities

81

(2)

Proof of Proposition 5.5

83

(5)

Some Remarks and Comments

88

(1)

References

89

(4)

Thom-Smale and Witten Complexes

93

(12)

The Thom-Smale Complex

93

(2)

The de Rham Map for Thom-Smale Complexes

95

(2)

Witten's Instanton Complex and the Map eT

97

(3)

The Map P∞, TeT

100

(2)

An Analytic Proof of Theorem 6.4

102

(1)

References

102

(3)

Atiyah Theorem on Kervaire Semi-characteristic

105

(12)

Kervaire Semi-characteristic

106

(1)

Atiyah's Original Proof

107

(1)

A Proof via Witten Deformation

108

(4)

A Generic Counting Formula for k(M)

112

(1)

Non-multiplicativity of k(M)

113

(2)

References

115

(2)

Index

117

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