Sectional curvature-type conditions on metric spaces

TL;DR

This paper explores curvature-like conditions in metric spaces, establishing splitting, measure, and geometric properties under Busemann concavity and lower curvature bounds, and applies Banach space smoothness concepts to analyze Busemann functions.

Contribution

It introduces Busemann concavity as a non-negative curvature condition, proves a splitting theorem, and applies Banach space smoothness ideas to metric space geometry.

Findings

Busemann concavity implies doubling and Poincaré conditions.

A Bonnet-Myers type theorem is proven for spaces with positive lower curvature bounds.

Busemann functions are shown to be (quasi-)convex, leading to the existence of a weak soul.

Abstract

In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and satisfies a Poincar\'e condition and the measure contraction property. Using a comparison geometry variant for general lower curvature bounds $k\in\mathbb{R}$, a Bonnet-Myers theorem can be proven for spaces with lower curvature bound $k>0$. In the second part the notion of uniform smoothness known from the theory of Banach spaces is applied to metric spaces. It is shown that Busemann functions are (quasi-)convex. This implies the existence of a weak soul. In the end properties are developed to further dissect the soul.