Sagitta, Lenses, and Maximal Volume

Abstract

We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds M with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by Grove and Petersen by showing any such M that has volume sufficiently close to this upper bound is homeomorphic to the standard sphere $S^{n}$ or a standard lens space $S^n/{\mathbb {Z}}_m$ where $m\in \{2,3,\ldots \}$ is no larger than an a priori constant.