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

Positive Ricci Curvature and the Length of a shortest periodic geodesic