UC Riverside

Morse theory is based on the idea that a smooth function on a manifold yields data about

the topology of the manifold. In this way it provides a tool for visualizing the shape of a space. Specifically, Morse's Isotopy Lemma tells us that the homotopy type of a manifold does not change in regions without critical points. The topology only changes in the presence of a critical point. Morse's Theorem states that the specific topological change is determined by the index of the Hessian at each critical point. In Morse Theory a smooth function is essential so that the differential and Hessian exist.

In Riemannian geometry, the distance function is not smooth everywhere. This means the differential as well the Hessian do not exist and Morse Theory cannot be applied. In order to generalize Morse Theory to this non-smooth function, an alternate definition of critical point and index are required. Grove and Shiohama developed a definition of critical point for the Riemannian distance function and used it to generalize Morse's Isotopy Lemma. Their generalization had a profound impact on the study of Riemannian geometry. Since no definition of index currently exists, Morse's Theorem has not been generalized.

The purpose of this dissertation is to define a new notion, called sub-index, for critical points of Riemannian distance functions. We show that Morse's connectedness corollary holds for the distance function when index is replaced by sub-index.