Length of closed geodesics on Riemannian manifolds with good covers

TL;DR

This paper establishes a bound on the length of the shortest closed geodesic on certain Riemannian manifolds, depending only on volume, diameter, and the number of elements in a good cover.

Contribution

It generalizes previous results by providing a bound based on volume, diameter, and good cover properties for simply connected Riemannian manifolds.

Findings

Bound on shortest closed geodesic length depending on V, D, N

Applicable to manifolds with good covers

Extends previous geodesic length estimates

Abstract

In this article, we prove a generalization of our previous result in [12]. In particular, we show that for an $n$-dimensional, simply connected Riemannian manifold with diameter $D$ and volume $V$. Suppose that $M$ admits a good cover consisting of $N$ elements. Then, the length of a shortest closed geodesic on $M$ is bounded by some function that only depends on $V, D$, and $N$.