Positive Ricci Curvature and the Length of a shortest periodic geodesic

Let $M^n$ be a closed Riemannian manifold of dimension $n\ge 2$, with Ricci curvature $Ric\ge n-1$. 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 $8\pi (n-1)$.