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. 2025;11(2):24.
doi: 10.1007/s40879-025-00816-x. Epub 2025 Apr 10.

Coarse extrinsic curvature of Riemannian submanifolds

Marc Arnaudon  1 , Xue-Mei Li  2   3 , Benedikt Petko  2
Affiliations

Coarse extrinsic curvature of Riemannian submanifolds

Marc Arnaudon et al. Eur J Math. 2025.

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.

Keywords: Coarse curvature; Extrinsic curvature; Optimal transport.

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Conflict of interest statement

Competing interestsThe authors declare no competing interests.

Figures

Fig. 1
Fig. 1
Planar curve case: test measures in red with some transport pairs of T in blue (Color figure online)
Fig. 2
Fig. 2
Space curve case: test measures in red with some transport pairs of T in blue (Color figure online)
Fig. 3
Fig. 3
Fermi coordinates along γ adapted to the surface M embedded in R3
Fig. 4
Fig. 4
Test measures in red with some transport pairs of T in blue (Color figure online)
Fig. 5
Fig. 5
Top-down perspective for the transport map T
Fig. 6
Fig. 6
Cross-sectional perspective for the transport map T
See this image and copyright information in PMC

References

    1. Ambrosio, L., Gigli, N., Savaré, G.: Diffusion, optimal transport and Ricci curvature for metric measure spaces. Eur. Math. Soc. Newsl. 103, 19–28 (2017)
    1. Arnaudon, M., Li, X.M., Petko, B.: Coarse Ricci curvature of weighted Riemannian manifolds (2023). arXiv:2303.04228
    1. Boissonnat, J.-D., Guibas, L.J., Oudot, S.Y.: Manifold reconstruction in arbitrary dimensions using witness complexes. Discrete Comput. Geom. 42(1), 37–70 (2009) - PMC - PubMed
    1. Bonciocat, A.-I., Sturm, K.-T.: Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256(9), 2944–2966 (2009)
    1. Chazal, F., Cohen-Steiner, D., Lieutier, A., Mérigot, Q., Thibert, B.: Inference of curvature using tubular neighborhoods. In: Najman, L., Romon, P. (eds.) Modern Approaches to Discrete Curvature. Lecture Notes in Mathematics, vol. 2184, pp. 133–158. Springer, Cham (2017)

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