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

Ludovic Sacchelli (1, 2) , Mario Sigalotti (2, 1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 CaGE - Control And GEometry

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.

Mots clés: Whitney extension; nilpotent approximation; Lusin approximation; sub-Riemannian geometry

Domaines: Géométrie métrique [math.MG]; Optimisation et contrôle [math.OC]; Géométrie différentielle [math.DG]

Ludovic Sacchelli, Mario Sigalotti. On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds. 2017. ⟨hal-01573353v2⟩