A characterization of Zoll Riemannian metrics on the 2-sphere

TL;DR

This paper proves that on a Riemannian 2-sphere with a simple length spectrum of a single value, all geodesics are simple closed with that length, confirming a conjecture by Lusternik.

Contribution

It establishes a characterization of Zoll Riemannian metrics on the 2-sphere based on the simple length spectrum.

Findings

All geodesics are simple closed with the same length L.

The simple length spectrum consisting of one element implies Zoll metrics.

Confirms Lusternik's conjecture for 2-spheres.

Abstract

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any geodesic is simple closed with length L.