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Coarse extrinsic curvature of Riemannian submanifolds

Marc Arnaudon, Xue-Mei Li, Benedikt Petko

TL;DR

A novel concept of coarse extrinsic curvature for Riemannian submanifolds is introduced, inspired by Ollivier’s notion of coarse Ricci curvature, derived from the Wasserstein 1-distance between probability measures supported in the tubular neighborhood of a submanifold.

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.

Coarse extrinsic curvature of Riemannian submanifolds

TL;DR

A novel concept of coarse extrinsic curvature for Riemannian submanifolds is introduced, inspired by Ollivier’s notion of coarse Ricci curvature, derived from the Wasserstein 1-distance between probability measures supported in the tubular neighborhood of a submanifold.

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.
Paper Structure (17 sections, 29 theorems, 257 equations, 6 figures)

This paper contains 17 sections, 29 theorems, 257 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ambient volume disintegration
  4. Approximate transport maps
  5. Test measures in Fermi coordinates
  6. Proposed transport map
  7. Curves and surfaces
  8. The circle example
  9. Planar curve
  10. Space curve
  11. Surface
  12. General Riemannian submanifolds
  13. Frame extension
  14. Main theorem
  15. Applications
  16. ...and 2 more sections

Key Result

Proposition $\mathbf{}$

Let $\gamma$ be a smooth unit speed curve in $\mathbb{R}^2$ such that $\gamma(0) = x_0$ and $\gamma(\delta)=y$. For all $\delta, \varepsilon, \sigma >0$ sufficiently small with $\sigma \vee \varepsilon \leqslant \frac{\delta}{4}$, it holds that where $R$ is the radius of the osculating circle of the curve at $x_0$.

Figures (6)

  • Figure 1: Planar curve case: test measures in red with some transport pairs of $T$ in blue.
  • Figure 2: Space curve case: test measures in red with some transport pairs of $T$ in blue.
  • Figure 3: Fermi coordinates along $\gamma$ adapted to the surface $M$ embedded in $\mathbb{R}^3$.
  • Figure 4: Test measures in red with some transport pairs of $T$ in blue.
  • Figure 5: Top-down perspective for the transport map $T$.
  • ...and 1 more figures

Theorems & Definitions (71)

  • Proposition $\mathbf{}$
  • Theorem $\mathbf{}$
  • Corollary $\mathbf{}$
  • Lemma $\mathbf{}$
  • proof
  • Definition $\mathbf{}$
  • Lemma $\mathbf{}$
  • proof
  • Proposition $\mathbf{}$: Change of volume
  • proof
  • ...and 61 more