Before revision:  i. e., \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)  + O(h^{k-1}) .
After revision:  i. e., \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) + O(h^{k-1}) ) (i.e., the Manifold-MLS is nearly an isometry).