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Triangulating stratified manifolds I: a reach comparison theorem

Jean-Daniel Boissonnat 1 Mathijs Wintraecken 1
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : In this paper, we define the reach for submanifolds of Riemannian manifolds, in a way that is similar to the Euclidean case. Given a d-dimensional submanifold S of a smooth Riemannian manifold M and a point p ∈ M that is not too far from S we want to give bounds on local feature size of exp −1 p (S). Here exp −1 p is the inverse exponential map, a canonical map from the manifold to the tangent space. Bounds on the local feature size of exp −1 p (S) can be reduced to giving bounds on the reach of exp −1 p (B), where B is a geodesic ball, centred at c with radius equal to the reach of S. Equivalently we can give bounds on the reach of exp −1 p • exp c (B c), where now B c is a ball in the tangent space T c M, with the same radius. To establish bounds on the reach of exp −1 p • exp c (B c) we use bounds on the metric and on its derivative in Riemann normal coordinates. This result is a first step towards answering the important question of how to triangulate stratified manifolds.
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https://hal.inria.fr/hal-01661233
Contributor : Jean-Daniel Boissonnat <>
Submitted on : Monday, December 11, 2017 - 6:45:39 PM
Last modification on : Wednesday, October 10, 2018 - 10:09:42 AM

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Jean-Daniel Boissonnat, Mathijs Wintraecken. Triangulating stratified manifolds I: a reach comparison theorem. 2017. ⟨hal-01661233⟩

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