Sagitta, Lenses, and Maximal Volume

TL;DR

This paper introduces a new metric invariant for Riemannian manifolds with curvature and radius bounds, establishing volume bounds and topological classifications near maximal volume cases.

Contribution

It defines a novel metric invariant and proves volume bounds and topological classification results for manifolds with curvature and radius constraints.

Findings

Established a uniform upper volume bound for manifolds with the invariant.

Proved manifolds close to the volume bound are diffeomorphic to spheres or lens spaces.

Generalized previous results on topological classification under curvature bounds.

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 and by Sill, Wilhelm, and the author by showing any such $M$ that has volume sufficiently close to this upper bound is diffeomorphic 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.