Remarks on manifolds with two sided curvature bounds

TL;DR

This paper examines well-known but informal claims regarding the properties of distance functions in manifolds with bounded curvature, focusing on regularity, reach, and cut locus.

Contribution

It provides a detailed analysis and clarification of folklore statements about distance functions in manifolds with two-sided curvature bounds.

Findings

Clarifies regularity properties of distance functions.

Analyzes subsets of positive reach in curved manifolds.

Discusses the structure of the cut locus.

Abstract

We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.