Sweeping out sectional curvature

TL;DR

This paper introduces the concept of two-convexity for regions of constant curvature in Riemannian manifolds and uses it to establish new rigidity results in comparison geometry.

Contribution

It presents the novel notion of two-convexity and applies it to derive rigidity theorems for manifolds with curvature bounds.

Findings

Maximal open sets of constant curvature exhibit two-convexity.

Two-convexity leads to new rigidity statements in comparison geometry.

The approach generalizes previous convexity concepts in Riemannian geometry.

Abstract

We observe that the maximal open set of constant curvature k in a Riemannian manifold with curvature bounded below or above by k has a convexity type property, which we call "two-convexity". This statement is used to prove a number of rigidity statements in comparison geometry.