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.
Nonparame tric inference 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.