This thesis consists of work that was carried out in three separate papers that were written during my time at UC, Riverside.

Abstract of chapter II: If $\pi:M\rightarrow B$ is a Riemannian Submersion and $M$ has non-negative sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have non-negative curvature. We find constraints on the extent to which O'Neill's horizontal curvature equation can be used to create positive curvature on the base space of a Riemannian submersion. In particular, we study when K. Tapp's theorem on Riemannian submersions of compact Lie groups with bi-invariant metrics generalizes to arbitrary manifolds of non-negative curvature.

Abstract of Chapter III: Though Riemannian submersions preserve non-negative sectional curvature this does not generalize to Riemannian submersions from manifolds with non-negative Ricci curvature. We give here an example of a Riemannian submersion $\pi: M\rightarrow B$ for which $\textrm{Ricci}_p(M)>0$ and at some point $p\in B$, $\text{Ricci}_p(B)<0$.

Abstract of Chapter IV: The smallest $r$ so that a metric $r$--ball covers a metric space $M$ is called the radius of $M.$ The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.

The Guest Editors of the special issue offer thoughts on Indigenous Peoples' leadership in the responsibility of all people to protect the oecans and waters of this planet.

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.

In this work, we describe a method to construct new examples of collapse with a lower curvature bound inspired by Cheeger and Gromov. Unlike with collapse with an upper and lower curvature bound, which is now completely understood, the structure of collapse with a lower curvature bound is still a mystery.

In [4], Grove and Petersen showed that if a sequence of Riemannian manifolds (M i , g i ) has uniform lower curvature bound, k, and M i → X, then X is an Alexandrov space with lower curvature bound k. Petersen, Wilhelm, and Zhu then showed that the converse is false [7]. Perelman showed that given a sequence of n-dimensional Alexandrov spaces with a uniform lower curvature bound, with limit space X such that dim X = n, all but finitely many of the prelimits are homeomorphic to X.

In 1985, Cheeger and Gromov introduced the concept of an F-structure, which can be thought as a generalized torus action on a manifold [1]. They showed that a manifold collapses with bounded curvature if and only if it admits an F-structure [2]. An F-structure is an example of a more general construct known as a g-structure, which is a sheaf of Lie groups actions, and is one of the main tools used in our approach.

The second main tool we use was also defined by Cheeger and is known as a Cheeger deformation. Cheeger generalized the method used by Berger in his classic example now known as the Berger spheres, which helped create the study of collapse in Riemannian geometry. Berger showed that scaling the metric of S 3 along the Hopf circles collapses S 3 to the 2-sphere of radius 1/2.

In this work, using g-structures and Cheeger deformations, we construct new examples of collapse with a lower curvature bound.

This dissertation will present two new rigidity theorems for manifolds with sectional curvature bounded below. The main new result is a new splitting theorem for Jacobi fields on manifolds with positive sectional curvature.