Positive Ricci Curvature and the Length of a shortest periodic geodesic

TL;DR

This paper proves that on a closed Riemannian manifold with Ricci curvature at least n-1, the shortest periodic geodesic length is bounded above by 8π(n-1), linking curvature conditions to geodesic length bounds.

Contribution

It establishes an explicit upper bound for the length of the shortest periodic geodesic based on Ricci curvature and dimension, connecting geometric curvature constraints to geodesic properties.

Findings

Shortest periodic geodesic length ≤ 8π(n-1)

Sphere of dimension m in loop space homotopic to a sphere of controlled length

Bound applies to manifolds with Ricci curvature ≥ n-1

Abstract

Let $M^n$ be a closed Riemannian manifold of dimension $n\geq 2$, with Ricci curvature $Ric \geq n-1$. We will show that any sphere of dimension $m$ in the space of closed loops on $M^n$ is homotopic to the sphere in the space of closed loops of length at most $8 \pi m$. It follows that the length of a shortest periodic geodesic on $M^n$ is bounded from above by $8 \pi (n-1)$.