A rigidity theorem in Alexandrov spaces with lower curvature bound

TL;DR

This paper investigates the extremal cases of metric inequalities in Alexandrov spaces with lower curvature bounds, establishing rigidity results that characterize these extremal configurations.

Contribution

It provides new rigidity theorems for Alexandrov spaces, extending understanding of metric inequalities in possibly infinite-dimensional, non-locally compact spaces.

Findings

Rigidity results for extremal metric inequalities

Characterization of spaces achieving equality in comparison inequalities

Extension of inequalities to non-locally compact Alexandrov spaces

Abstract

Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang-Schroeder-Sturm. The purpose of this paper is to study the extremal cases of these inequalities and to prove rigidity results. The spaces which we shall deal with here are Alexandrov spaces which possibly have infinite dimension and are not supposed to be locally compact.