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.

A fake real projective space is a manifold homotopy equivalent to real projective space, but not diffeomorphic to it. Equivalently, it is the orbit space of a free involu- tion on a (homotopy) sphere.

In this thesis, we show that some of the fake RP6s , constructed by Hirsch and Milnor in 1963, and the analogous fake RP14s admit metrics that simutaneously have almost nonnegative sectional curvature and positive Ricci curvature. These spaces are obtained by taking the Z2 quotients of the embedded images of the standard spheres of codimension one in some of Milnor’ s exotic 7−spheres and the analogous Shimada’s exotic 15−spheres. This part of my thesis is joint work with F. Wilhelm.

Hirsch and Milnor also constructed fake RP5s using invariant subspheres of codi- mension two. Octonionically, this construction yields closed 13−manifolds, that are homotopy equivalent to RP13s. The analog to their proof that fake RP5s are not diffeomorphic to standard RP5 breaks down; since in contrast to dimension 6, there is an exotic 14−sphere. We show that some of the Hirsch-Milnor RP13s are not diffeomorphic to standard RP13s. Here we obtain a complete diffeomorphism clas- sification of the Hirsch-Milnor RP13s. This part of my thesis is joint work with C. He.

Using techniques of group diagrams we provide an elementary proof that certain 3-dimensional Brieskorn varieties are SO(2) × SO(2)−equivariantly diffeomorphic to certain Lens Spaces.

If we start with a generic nonnegatively curved metric on a Riemannian manifold and apply a deformation to the metric, then nonnegative curvature is usually not preserved. From the perspective of systems of equations, the problem of finding a metric with a prescribed curvature tensor is a highly overdetermined system. However, if every zero curvature plane is tangent to a totally geodesic flat that is fixed by a deformation, then the problem of finding the prescribed metric becomes more manageable. In 2008 Petersen and Wilhelm exploited this idea and used it to develop a deformation called Orthogonal Partial Conformal Change (OPCC). Then they successfully combined it with other deformations to put a positively curved metric on the Gromoll-Meyer sphere.

Here we introduce new examples of manifolds that admit an OPCC deformation. In particular, we generalize a result obtained by Walschap on the canonical line bundle over the complex projective space. One peculiar characteristic of one of our OPCC deformations is that they are not verifiable via the machinery developed by Petersen and Wilhelm.