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

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
Contributeur : Ludovic Sacchelli <>
Soumis le : mercredi 10 janvier 2018 - 15:24:25
Dernière modification le : jeudi 10 mai 2018 - 02:05:44

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  • HAL Id : hal-01573353, version 2
  • 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. 2017. 〈hal-01573353v2〉

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