Approximating coarse Ricci curvature on submanifolds of Euclidean space

Abstract

For an embedded submanifold Σ ⊂ 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 Σ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [2]. Specifically, we prove Proposition 3.2 in [2].