Upper bounds for the critical values of homology classes of loops

TL;DR

This paper establishes upper bounds for the critical values of homology classes in loop spaces of manifolds with positive Ricci curvature, leading to bounds on the length of shortest closed geodesics.

Contribution

It provides new upper bounds for critical values of homology classes in loop spaces of positively curved manifolds, connecting geometric curvature conditions to geodesic length bounds.

Findings

Shortest closed geodesic length on simply-connected manifolds with Ric ≥ n-1 is ≤ nπ.

Upper bounds for critical values depend on the manifold's curvature properties.

Results apply to both Riemannian and Finsler metrics.

Abstract

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a simply-connected $n$-dimensional manifold of positive Ricci curvature $\textrm{Ric} \ge n-1$ has length $\le n \pi.$