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arXiv · 1708.02795

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

Ludovic Sacchelli·Mario Sigalotti
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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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Ludovic Sacchelli, Mario Sigalotti. 2017-08-09. On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds. https://arxiv.org/abs/1708.02795

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