Characterizations of the round two-dimensional sphere in terms of closed geodesics

TL;DR

This paper investigates the properties of closed geodesics on two-dimensional spheres, providing characterizations of the round sphere by analyzing geodesic counts and refining existing results for surfaces of revolution.

Contribution

It refines previous results by quantifying geodesic counts on spheres of revolution and offers new characterizations of the round two-sphere based on closed geodesic properties.

Findings

Quantified the number of closed geodesics of bounded length on spheres of revolution.

Provided new characterizations of the round two-sphere.

Reframed existing results to better understand the geometry of the sphere.

Abstract

The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere.