Stability of the Cut Locus and a Central Limit Theorem for Fr\'echet Means of Riemannian Manifolds

TL;DR

This paper establishes a Central Limit Theorem for Fréchet means on closed Riemannian manifolds, clarifying geometric conditions related to the cut locus, with implications for both compact and non-compact cases.

Contribution

It provides a CLT for Fréchet means on Riemannian manifolds and clarifies the geometric stability conditions necessary for the theorem.

Findings

CLT proven for closed Riemannian manifolds under stability conditions

Clarification of geometric meaning of hypotheses in previous CLT work

Stability hypothesis holds for compact manifolds but not necessarily for non-compact ones

Abstract

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.