PART Ⅰ．NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD 1
1．Introduction 1
2．Definitions and Lemmas 4
3．Homotopy Type in Terms of Critical Values 12
4．Examples 25
5．The Morse Inequalities 28
6．Manifolds in Euclidean Space:The Existence of Non-degenerate Functions 32
7．The Lefschetz Theorem on Hyperplane Sections 39
PART Ⅱ．A RAPID CORSE IN RIEMANNIAN GEOMETRY 43
8．Covariant Differentiation 43
9．The Curvature Tensor 51
10．Geodesics and Completeness 55
PART Ⅲ．THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS 67
11．The Path Space of a Smooth Manifold 67
12．The Energy of a Path 70
13．The Hessian of the Energy Function at a Critical Path 74
14．Jacobi Fields:The Null-space of E＊＊ 77
15．The Index Theorem 83
16．A Finite Dimensional Approximation to Ωc 88
17．The Topology of the Full Path Space 93
18．Existence of Non-conjugate Points 98
19．Some Relations Between Topology and Curvature 100
PART Ⅳ．APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES 109
20．Symmetric Spaces 109
21．Lie Groups as Symmetric Spaces 112
22．Whole Manifolds of Minimal Geodesics 118
23．The Bott Periodicity Theorem for the Unitayy Group 124
24．The Periodicity Theorem for the Orthogonal Group 133
APPENDIX．THE HOMOTOPY TYPE OF A MONOTONE UNION 149