Stability of the Cut Locus and a Central Limit Theorem for Frechet Means of Riemannian Manifolds

We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lins Omnibus Central Limit Theorem for Frechet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the manifold is compact but may not be satisfied in the non-compact case.