Approximating coarse Ricci curvature on submanifolds of Euclidean space

Abstract

For an embedded submanifold $\Sigma \subset {\mathbb{R}}^N$, Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds $\Sigma$ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [AW19]. Specifically, we prove Proposition 3.2 in [AW19].

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