Coarse extrinsic curvature of Riemannian submanifolds

TL;DR

This paper introduces a new coarse extrinsic curvature concept for Riemannian submanifolds, using Wasserstein distances to infer geometric properties from data, with applications in manifold learning.

Contribution

It proposes a novel curvature measure based on Wasserstein distances, linking geometric analysis with statistical data and manifold learning.

Findings

Defines coarse extrinsic curvature using Wasserstein 1-distance

Provides a method to approximate mean curvature from point cloud data

Offers potential applications in manifold learning and metric embeddings

Abstract

We introduce a novel concept of coarse extrinsic curvature for Riemannian submanifolds, inspired by Ollivier's notion of coarse Ricci curvature. This curvature is derived from the Wasserstein 1-distance between probability measures supported in the tubular neighborhood of a submanifold, providing new insights into the extrinsic curvature of isometrically embedded manifolds in Euclidean spaces. The framework also offers a method to approximate the mean curvature from statistical data, such as point clouds generated by a Poisson point process. This approach has potential applications in manifold learning and the study of metric embeddings, enabling the inference of geometric information from empirical data.