The Length of a Shortest Geodesic Loop

TL;DR

This paper establishes a lower bound for the length of non-trivial geodesic loops on certain compact Finsler manifolds, aiding in the geometric analysis of convexity properties.

Contribution

It provides a new lower bound estimate for geodesic loops on simply-connected, compact, even-dimensional Finsler manifolds with positive flag curvature.

Findings

Lower bound for geodesic loop length established

Application to characterizing dynamically convex Finsler metrics on 2-sphere

Supports geometric understanding of Finsler curvature properties

Abstract

We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper [HP] to give a geometric characterization of dynamically convex Finsler metrics on the 2-sphere.