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.

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.

\indent In 1997, Wilhelm \cite{OnIntR} proved the following generalization of Synge's Theorem: let $(M, g_M)$ be a compact Riemannian $n$-manifold with $\textnormal{Ric}_k (M, g_M)\geq k$ and $\textnormal{sys}_1(M, g_M)> \pi\sqrt{\frac{k-1}{k}}$; if $n$ is even and $M$ is orientable, then $M$ is simply connected; if $n$ is odd, then $M$ is orientable (Theorem \ref{Fredsmainresult}). Furthermore, he proved that this lower bound on $\textnormal{sys}_1$ is optimal when $k=n-1$. In 2020, Mouillé \cite{LawrenceThesis} proved that $S^3\times S^3$ admits a metric $g_\ell$ with $\textnormal{Ric}_2(S^3\times S^3, g_\ell)>0$ (Theorem \ref{LawrenceMain}). \vskip 1em

In this dissertation, we first show that the metric $g_\ell$ (which is a Cheeger deformation) is canonical variation. This follows from a more general result we prove (Theorem \ref{nhmetric}), which is that if $(M, g_\ell)$ is a Cheeger deformation by $(G, g_\textnormal{bi})$ that satisfies what we call the generalized Petersen-Wilhelm hypothesis (Definition \ref{pwhypothesisgen}), then for all $p\in M$, the orbit $G(p)$ is normal homogeneous and $g_\ell|_p$ is canonical variation with respect to the Riemannian submersion $\pi:(M, g_M)\stackrel{\ref{RQT}}{\longrightarrow}(M/G, \overline{g})$. Moreover, if $G(p)$ is totally geodesic for all $p\in M$, then $g_\ell$ is canonical variation (Theorem \ref{cheegtocangen}). \vskip 1em

\indent We then develop a technique for finding an optimal lower bound on $\textnormal{Ric}_k$ for any Riemannian manifold $(M, g_M)$ with dimension $n\geq 4$. Specifically, for any $p\in M$, unit vector $x\in T_pM$, and $k\in\mathbb{N}$ such that $2 \leq k \leq n-2$, we prove that $\min \textnormal{Ric}_{k}(x; \bullet)= \textnormal{Ric}(x)-\max \textnormal{Ric}_{n-1-k}(x; \bullet)$ (Theorem \ref{Ricklb}). \vskip 1em

\indent From there, letting $\mathbb{Z}_2$ act on $S^3\times S^3$ in two ways---as the antipodal map $a:(N_1, N_2)\mapsto (-N_1, -N_2)$ and as $f:(N_1, N_2)\mapsto (-N_1, N_2)$---we study for each value of $t\in (0,1)$ the manifold $\frac{S^3\times S^3}{\mathbb{Z}_2}$ paired with the unique metric $\overline{g_t}$ that makes the quotient map $ (S^3\times S^3, g_t)\longrightarrow \left(\frac{S^3\times S^3}{\mathbb{Z}_2}, \overline{g_t}\right)$ a local isometry. In particular, we establish $t$-independent and $t$-dependent upper bounds on the product $\min\textnormal{Ric}_k\left(\frac{S^3\times S^3}{\mathbb{Z}_2}, \overline{g_t}\right) \cdot \left(\textnormal{sys}_1\left(\frac{S^3\times S^3}{\mathbb{Z}_2}, \overline{g_t}\right)\right)^2$ when $k=2, 3, 4$ (Theorems \ref{mainresultspecific} and \ref{mainresult}). \vskip 1em

\indent Finally, we notice that our upper bounds are smaller than Wilhelm's upper bounds. We conclude that when restricted to the family $\left\{\left(\frac{S^3\times S^3}{\mathbb{Z}_2}, \overline{g_t}\right) \ | \ 0

In this dissertation, we study manifolds that have positive kth-intermediate Ricci curvature, which we abbreviate as Ric_k>0. This condition interpolates between having positive sectional curvature and having positive Ricci curvature. Specifically, we study this curvature condition in the presence of isometric group actions.

First, we show that if M is a positively curved homogeneous space, then M x M admits a metric with Ric_2>0. We also construct metrics with Ric_2>0 on several other products of homogeneous spaces. It follows from these examples that the Hopf Conjectures, Petersen-Wilhelm Conjecture, Berger Fixed Point Theorem, and Hsiang-Kleiner Theorem for positive sectional curvature do not hold in the setting of Ric_2 > 0.

Second, we establish the following: In a manifold with Ric_k>0, if a submanifold N has a tangential subspace on which the intrinsic kth-intermediate Ricci curvatures are non-positive, then the dimension of that subspace is bounded above by codim(N) + k. As a consequence, we obtain a local symmetry rank bound for manifolds with Ric_k>0.

Finally, we establish that if there are k+1 commuting Killing fields on a compact manifold with Ric_k>0, then there exists a point at which the Killing fields are linearly dependent. Using this, we establish a cohomogeneity obstruction and a symmetry rank bound for compact manifolds with Ric_k>0.

In this paper, we seek to provide counter examples to two volume comparison lemmas found in \cite{GP2} if we generalize their assumptions to a lower Ricci curvature bound. Second we seek to further understand Riemannian manifolds which contain embedded submanifolds of certain focal radius. First, in papers \cite{GP1}, \cite{GP2} Grove and Petersen discussed the relationship between bounds on sectional curvature, radius, and volume and its effects on the topology of a closed Riemannian manifolds. They prove that for manifolds with a lower sectional curvature bound, an upper radius bound and almost maximal volume, one can give topological equivalence to either $S^n$ or $\mathbb{R}P^n$. It was later proved to be diffeomorphic \cite{PSW1}. Also, Grove and Petersen showed that the limit space of a convergent sequence of manifolds with maximal volume has certain geometric properties.

In the proofs of these theorems they use two powerful volume comparison lemmas. We discuss why the methods used in \cite{GP2} cannot be extended in general to manifolds with a lower Ricci curvature bound by looking at some interesting counter examples. Second, in a paper by Guiljaro and Wilhelm \cite{GW1} we see a relationship between closed embedded submanifolds of maximal focal radius and topological type, and seek to understand the necessity of the submanifold being closed by providing counterexamples in which we consider Manifolds with embedded open submanifolds of certain focal radius which do not satisfy the conclusions of the results in \cite{GW1}.

This thesis explores families of metric spaces. It has two parts. First, the Kuratowski embedding is an isometric embedding of a metric space, M, into the space of bound, Borel functions over M, equipped with the sup norm. We extend this map to a family of maps by averaging over metric r--balls. The image of M under this map can be regarded as a deformation. After restricting our metric spaces to Riemannian manifolds, we explore how curvature affects this deformation. Furthermore, we give a complete description of the deformation of the n-sphere. Second, we prove a diffeomorphsim stability theorem. 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 ≥ k and radius ≤ 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.