A generalized maximal diameter sphere theorem

TL;DR

This paper proves a generalized maximal diameter sphere theorem for Riemannian manifolds with radial curvature bounds, extending previous results to a broad class of two-spheres of revolution and characterizing the equality case.

Contribution

It introduces a unifying theorem that encompasses earlier diameter sphere theorems and applies to a wide class of two-spheres of revolution.

Findings

Diameter of manifold bounded by that of a model sphere

Equality case implies isometry to the model

Includes ellipsoids of prolate type and constant curvature spheres

Abstract

We prove that if a complete connected $n$-dimensional Riemannian manifold $M$ has radial sectional curvature at a base point $p\in M$ bounded from below by the radial curvature function of a two-sphere of revolution $\widetilde M$ belonging to a certain class, then the diameter of $M$ does not exceed that of $\widetilde M.$ Moreover, we prove that if the diameter of $M$ equals that of $\widetilde M,$ then $M$ is isometric to the $n$-model of $\widetilde M.$ The class of a two-sphere of revolution employed in our main theorem is very wide. For example, this class contains both ellipsoids of prolate type and spheres of constant sectional curvature. Thus our theorem contains both the maximal diameter sphere theorem proved by Toponogov [9] and the radial curvature version by the present author [2] as a corollary.