The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold $mathcal{M}$ of Euclidean space has been a long-standing challenge in computational geometry. Given a sampling resolution parameter $ h $, state-of-the-art discrete methods yield $ O(h) $ provable approximations. In this paper, we investigate the convergence of such approximations made by Manifold Moving Least-Squares (Manifold-MLS), a method that constructs an approximating manifold $mathcal{M}^h$ using information from a given point cloud that was developed by Sober & Levin in 2019. In this paper, we show that provided that $mathcal{M}in C^{k}$ and closed (i.e. $mathcal{M}$ is a compact manifold without boundary) the Riemannian metric of $ mathcal{M}^h $ approximates the Riemannian metric of $ mathcal{M}, $. Explicitly, given points $ p_1, p_2 in mathcal{M}$ with geodesic distance $ rho_{mathcal{M}}(p_1, p_2) $, we show that their corresponding points $ p_1^h, p_2^h in mathcal{M}^h$ have a geodesic distance of $ rho_{mathcal{M}^h}(p_1^h,p_2^h) = rho_{mathcal{M}}(p_1, p_2)(1 + O(h^{k-1})) $ (i.e., the Manifold-MLS is nearly an isometry). We then use this result, as well as the fact that $ mathcal{M}^h $ can be sampled with any desired resolution, to devise a naive algorithm that yields approximate geodesic distances with a rate of convergence $ O(h^{k-1}) $. We show the potential and the robustness to noise of the proposed method on some numerical simulations.
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