Short Simple Geodesic Loops on a 2-Sphere

TL;DR

This paper proves that on any Riemannian 2-sphere, at every point, there are at least two simple geodesic loops with lengths bounded by multiples of the sphere's diameter, extending classical geodesic existence results.

Contribution

It establishes the existence of two simple geodesic loops at any point on a 2-sphere with explicit length bounds, a new local geodesic result.

Findings

Existence of at least two simple geodesic loops at any point on a 2-sphere.

Lengths of these loops are bounded by 8d and 14d, where d is the sphere's diameter.

Extends classical global geodesic theorems to local loop existence.

Abstract

The classic Lusternik--Schnirelmann theorem states that there are three distinct simple periodic geodesics on any Riemannian 2-sphere $M$. It has been proven by Y. Liokumovich, A. Nabutovsky and R. Rotman that the shortest three such curves have lengths bounded in terms of the diameter $d$ of $M$. We show that at any point $p$ on $M$ there exist at least two distinct simple geodesic loops (geodesic segments that start and end at $p$) whose lengths are respectively bounded by $8d$ and $14d$.