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

2 CaGE - Control And GEometry
Inria de Paris, LJLL - 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.
Type de document :
Pré-publication, Document de travail
2017
Domaine :

https://hal.archives-ouvertes.fr/hal-01573353
Contributeur : Ludovic Sacchelli <>
Soumis le : mercredi 9 août 2017 - 11:30:35
Dernière modification le : vendredi 11 août 2017 - 01:08:09

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SR_Whitney_extension_theorem_f...
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### Identifiants

• HAL Id : hal-01573353, version 1
• ARXIV : 1708.02795

### Citation

Ludovic Sacchelli, Mario Sigalotti. On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds. 2017. 〈hal-01573353〉

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