An injectivity radius estimate in terms of metric sphere

TL;DR

This paper establishes a lower bound on the injectivity radius at a point in a Riemannian manifold based on the absence of cut points within a certain distance, answering a previously posed question.

Contribution

It provides a new estimate for the injectivity radius in terms of metric sphere properties, resolving a problem posed by Sun and Wan.

Findings

Injectivity radius exceeds a given radius if no cut points are within that distance.

Provides a geometric criterion for lower bounds on injectivity radius.

Answers an open problem in Riemannian geometry.

Abstract

In this paper we prove that if a point $p$ in a complete Riemannian manifold is not a cut point of any point whose distance to $p$ is $r$, then the injectivity radius of $p$ is strictly large than $r$. As a corollary we give a positive answer to a problem raised by Z. Sun and J. Wan.