A min-max characterization of Zoll Riemannian metrics

TL;DR

This paper provides a new characterization of Zoll Riemannian metrics on simply connected spin closed manifolds using min-max values in loop spaces, and explores implications for closed geodesics on odd-dimensional spheres.

Contribution

It introduces a min-max characterization of Zoll metrics and links min-max value coincidences to the existence of closed geodesics on spheres.

Findings

Zoll metrics are characterized by coinciding min-max values.

On odd-dimensional spheres, min-max value coincidences imply every point lies on a closed geodesic.

Provides a new variational approach to study Zoll metrics and geodesics.

Abstract

We characterize the Zoll Riemannian metrics on a given simply connected spin closed manifold as those Riemannian metrics for which two suitable min-max values in a finite dimensional loop space coincide. We also show that on odd dimensional Riemannian spheres, when certain pairs of min-max values in the loop space coincide, every point lies on a closed geodesic.