  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 Lin's Omnibus Central Limit Theorem for Fr\'echet 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.
