Positive Ricci Curvature and the Length of a Shortest Periodic Geodesic

Let M^n be a closed Riemannian manifold of dimension n≥ 2 , with Ricci curvature Ric ≥ 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 π m . It follows that the length of a shortest periodic geodesic on M^n is bounded from above by 8 π (n-1) .