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arXiv · 2407.08031

Coarse extrinsic curvature of Riemannian submanifolds

Marc Arnaudon·Xue-Mei Li·Benedikt Petko
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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.

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Marc Arnaudon, Xue-Mei Li, Benedikt Petko. 2024-07-10. Coarse extrinsic curvature of Riemannian submanifolds. https://doi.org/10.1007/s40879-025-00816-x

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