Coarse extrinsic curvature of Riemannian submanifolds
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917982041538560 |
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| author | Arnaudon, Marc Li, Xue-Mei Petko, Benedikt |
| author_facet | Arnaudon, Marc Li, Xue-Mei Petko, Benedikt |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08031 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coarse extrinsic curvature of Riemannian submanifolds Arnaudon, Marc Li, Xue-Mei Petko, Benedikt Differential Geometry 53B25 (Primary) 49Q22 (Secondary) 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. |
| title | Coarse extrinsic curvature of Riemannian submanifolds |
| topic | Differential Geometry 53B25 (Primary) 49Q22 (Secondary) |
| url | https://arxiv.org/abs/2407.08031 |