A CLT for Fréchet means on closed Riemannian manifolds follows from Bhattacharya-Lin's omnibus theorem once the cut locus is topologically stable and avoids the measure, and the stability condition is essential.
Large sample the ory of intrinsic and extrinsic sample means on manifolds
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Stability of the Cut Locus and a Central Limit Theorem for Fr\'echet Means of Riemannian Manifolds
A CLT for Fréchet means on closed Riemannian manifolds follows from Bhattacharya-Lin's omnibus theorem once the cut locus is topologically stable and avoids the measure, and the stability condition is essential.