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The approximation of both geodesic distances and shortest paths on pointcloud sampled from an embedded submanifold $\mathcal{M}$ of Euclidean space hasbeen a long-standing challenge in computational geometry. Given a samplingresolution parameter $ h $, state-of-the-art discrete methods yield $ O(h) $provable approximations. In this paper, we investigate the convergence of suchapproximations made by Manifold Moving Least-Squares (Manifold-MLS), a methodthat constructs an approximating manifold $\mathcal{M}^h$ using informationfrom a given point cloud that was developed by Sober \& Levin in 2019. In thispaper, 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 thisresult, as well as the fact that $ \mathcal{M}^h $ can be sampled with anydesired resolution, to devise a naive algorithm that yields approximategeodesic distances with a rate of convergence $ O(h^{k-1}) $. We show thepotential and the robustness to noise of the proposed method on some numericalsimulations.

Approximating the Riemannian Metric from Point Clouds via Manifold Moving Least Squares