Critical values of homology classes of loops and positive curvature

TL;DR

This paper investigates the critical lengths of homology classes of loops in positively curved manifolds, showing these lengths reach their maximum only for the round sphere, thus characterizing the sphere among such manifolds.

Contribution

It proves that the critical lengths of homology classes of loops attain their maximum only for the round sphere, providing a new characterization of the sphere in positive curvature.

Findings

Critical lengths equal to 2π only for the round sphere.

Maximum critical length characterizes the round sphere among positively curved manifolds.

Results extend previous work under additional curvature bounds.

Abstract

We study compact and simply-connected Riemannian manifolds with positive sectional curvature $K\ge 1.$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define a critical length ${\sf crl}_p\left(M,g\right)$ resp. ${\sf crl}\left(M,g\right).$ Then ${\sf crl}_p\left(M,g\right)$ equals the length of a geodesic loop and ${\sf crl}\left(M,g\right)$ equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and by Lusternik and Fet in the general case. It is the main result of the paper that the numbers ${\sf crl}_p\left(M,g\right)$ resp. ${\sf crl}\left(M,g\right)$ attain its maximal value $2\pi$ only for the round metric on the $n$-sphere. Under the additional assumption $K \le 4$ this result for ${\sf crl}\left(M,g\right)$ follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson and Ziller in odd dimensions.