Morse Theory and Closed Geodesics

Abstract

(This chapter includes an introduction to algebraic topology.)

In his famous habilitation address where he developed the foundations of Riemannian geometry, B. Riemann also expressed the idea that a manifold can be reconstructed from the level hypersurfaces of a continuous function. If the function is not only continuous, but also differentiable, all such hyper-surfaces that do not contain any critical points of the function are regular, i.e. embedded submanifolds, whereas the topological type of the level hyper-surfaces changes in critical points. In those critical points where the Hessian of the function has maximal rank, it is easy to describe this change of topological type. This is the content of Morse theory. The more difficult general case is the content of Conley theory which, however, we shall not be able to treat in this book.