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

TL;DR

This paper investigates the Whitney $C^1$ extension property for horizontal curves in sub-Riemannian manifolds, establishing conditions under which the property holds in equiregular and step-2 cases, and linking it to approximation theorems.

Contribution

It provides new criteria for the Whitney $C^1$ extension property in sub-Riemannian manifolds, especially in singular and step-2 cases, and connects it to Lusin-type approximation results.

Findings

Extension property holds in equiregular cases with non-singularity conditions.

All step-2 sub-Riemannian manifolds satisfy the $C^1$ extension property.

The $C^1$ extension property implies a Lusin-like approximation theorem.

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.