Critical values of homology classes of loops and positive curvature

Abstract

We study compact and simply-connected Riemannian manifolds $(M,g)$ with positive sectional curvature $K \geq 1$. For a nontrivial homology class of lowest positive dimension in the space of loops based at a point $p \in M$ or in the free loop space one can define a critical length $\mathsf{crl}_p (M,g)$ resp. $\mathsf{crl} (M,g)$. Then $\mathsf{crl}_p (M,g)$ equals the length of a geodesic loop with base point $p$ and $\mathsf{crl} (M,g)$ 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 & Fet in the general case. It is the main result of the paper that the numbers $\mathsf{crl}_p (M,g)$ resp. $\mathsf{crl} (M,g)$ attain its maximal value $2\pi$ only for the round metric on the $n$-sphere.

Under the additional assumption $K \leq 4$ this result for $\mathsf{crl} (M,g)$ follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson & Ziller in odd dimensions.