  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.
