The Injectivity Radius of Souls of Alexandrov Spaces

TL;DR

This paper extends a known lower bound on the injectivity radius from noncompact nonnegatively curved Riemannian manifolds to Alexandrov spaces, relating it to the space's soul and curvature bounds.

Contribution

It generalizes the injectivity radius bound to Alexandrov spaces and introduces the concept of the soul in this setting, comparing different notions of injectivity radius.

Findings

Injectivity radius bound of at least π K^{-1/2} when it differs from the soul's radius.

Comparison of two notions of injectivity radius in Alexandrov spaces.

Extension of classical Riemannian results to Alexandrov spaces.

Abstract

A sharp lower bound for the injectivity radius in noncompact nonnegatively curved Riemannian manifolds involving their soul goes back to \v{S}arafutdinov. We generalize this bound to the setting of Alexandrov spaces. Our main theorem reads as follows. If the injectivity radius of an Alexandrov space of nonnegative curvature does not coincide with the one of its souls, then it is at least $ \pi K^{-1/2} $, where $ K $ is an upper curvature bound. We introduce the soul of Alexandrov spaces in some detail and compare two notions of injectivity radii.