Approximating coarse Ricci curvature on submanifolds of Euclidean space

TL;DR

This paper develops a method to approximate the coarse Ricci curvature of submanifolds in Euclidean space using heat kernel-based Laplacian approximations, extending prior Laplacian approximation techniques.

Contribution

It derives asymptotics for approximating coarse Ricci curvature on submanifolds, specifically proving a key proposition from previous work.

Findings

Established asymptotic formulas for Ricci curvature approximation.

Validated the approximation method for submanifolds in Euclidean space.

Extended heat kernel Laplacian approximation techniques to curvature estimation.

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].