《MORSE THEORY》求取 ⇩

PART Ⅰ．NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD1

1．Introduction1

2．Definitions and Lemmas4

3．Homotopy Type in Terms of Critical Values12

4．Examples25

5．The Morse Inequalities28

6．Manifolds in Euclidean Space:The Existence of Non-degenerate Functions32

7．The Lefschetz Theorem on Hyperplane Sections39

PART Ⅱ．A RAPID COURSE IN RIEMANNIAN GEOMETRY43

8．Covariant Differentiation43

9．The Curvature Tensor51

10．Geodesics and Completeness55

PART Ⅲ．THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS67

11．The Path Space of a Smooth Manifold67

12．The Energy of a Path70

13．The Hessian of the Energy Function at a Critical Path74

14．Jacobi Fields:The Null-space of E＊＊77

15．The Index Theorem83

16．A Finite Dimensional Approximation to Ωc88

17．The Topology of the Full Path Space93

18．Existence of Non-conjugate Points98

19．Some Relations Between Topology and Curvature100

PART Ⅳ．APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES109

20．Symmetric Spaces109

21．Lie Groups as Symmetric Spaces112

22．Whole Manifolds of Minimal Geodesics118

23．The Bott Periodicity Theorem for the Unitary Group124

24．The Periodicity Theorem for the Orthogonal Group133

APPENDIX．THE HOMOTOPY TYPE OF A MONOTONE UNION149