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On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds

Ludovic Sacchelli 1, 2 Mario Sigalotti 1, 2
2 CaGE - Control And GEometry
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : In this article we study the validity of the Whitney $C^1$ extension property for horizontal curves in sub-Riemannian manifolds endowed with 1-jets that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the input-output maps on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the $C^1$ extension property. We conclude by showing that the $C^1$ extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.
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https://hal.archives-ouvertes.fr/hal-01573353
Contributor : Mario Sigalotti <>
Submitted on : Thursday, December 20, 2018 - 1:25:41 PM
Last modification on : Monday, March 29, 2021 - 2:47:34 PM

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  • HAL Id : hal-01573353, version 3
  • ARXIV : 1708.02795

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Ludovic Sacchelli, Mario Sigalotti. On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds. Calculus of Variations and Partial Differential Equations, Springer Verlag, 2018. ⟨hal-01573353v3⟩

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