UC Riverside

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}.