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PREFACEPART 1 NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLDIntroductionDefinitions and LenmmsHomotopy Type in TeRns of Critical ValuesExamplesThe Morse InequalitiesManifolds in Euclidean Space: The Existence of Non-degenerate FunctionsThe Lefschetz Theorem on Hyperplane SectionsPART 2 A RAPID COURSE IN RIEMANNIAN GEOMETRYCovariant DifferentiationThe Curvature TensorGeodesics and CompletenessPART 3 THE CALCULUS OF VARIATIONS APPLIED TO GEODESICSThe Path Space of a Smooth ManifoldThe Fnergy of a PathThe Hessian of the Energy Function at a Critical Path Jacobi Fields: The -space of E**The Index TheoremA Finite Dimensional Approximation to ncThe Topology of the Full Path SpaceExistence of Non-conjugate PointsSome Between Topology and CurvaturePART 4 APPLICATIONS TO LIE GROUPS AND SYMMEIRIC SPACESSymmetric SpacesLie Groups as Symmetric SpacesWhole Manifolds of Minimal GeodesicsThe Bott Periodicity Theorem for the Unitary GroupThe Periodicity Theorem for the Orthogonal GroupAPPENDIX THE HOMOTOPY TYPE OF A MONOTONE UNION