Bounding Shortest Closed Geodesics with Diameter on compact 2-dimensional Orbifolds Homeomorphic to $S^2$

TL;DR

This paper extends length bounding techniques for shortest closed geodesics from manifolds to 2-dimensional orbifolds homeomorphic to $S^2$, establishing an inequality relating geodesic length to diameter.

Contribution

It generalizes length bounding methods to orbifolds with finite fundamental groups, providing new inequalities for shortest geodesics on these spaces.

Findings

Established a bound on shortest geodesic length in terms of diameter for orbifolds

Extended sweepout techniques from manifolds to orbifolds

Proved an inequality for orbifolds homeomorphic to $S^2$

Abstract

Length bounded sweepouts give a way to bound the length of the shortest closed geodesic of a closed manifold. In this paper, we generalized to the case of compact 2-dimensional orbifolds homeomorphic to $S^2$ as well as compact 2-dimensional orbifolds with finite orbifold fundamental groups. We proved an inequality for the length of the shortest closed orbifold geodesic in terms of the diameter.