Positive Ricci Curvature and the Length of a Shortest Periodic Geodesic

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