UC Riverside

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.