Upper bounds for the critical values of homology classes of loops

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of compact manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a compact and simply-connected n-dimensional manifold of positive Ricci curvature \(\text {Ric}\ge n-1\) $\text{Ric}\ge n-1$ has length \(\le n \pi .\) $\le n\pi .$ This improves the bound \(8\pi (n-1)\) $8\pi (n-1)$ given by Rotman (Positive Ricci curvature and the length of a shortest periodic geodesic. arXiv:2203.09492, 2022).