Pliability, or the Whitney extension theorem for curves in Carnot groups

Abstract

The Whitney extension theorem is a classical result in analysis giving a necessary and sufficient condition for a function defined on a closed set to be extendable to the whole space with a given class of regularity. It has been adapted to several settings, including the one of Carnot groups. However, the target space has generally been assumed to be equal to ${ℝ}^d$ for some $d\ge 1$.

We focus here on the extendability problem for general ordered pairs $\left(G_1,G_2\right)$ (with $G_2$ nonabelian). We analyse in particular the case $G_1=ℝ$ and characterize the groups $G_2$ for which the Whitney extension property holds, in terms of a newly introduced notion that we call pliability. Pliability happens to be related to rigidity as defined by Bryant and Hsu. We exploit this relation in order to provide examples of nonpliable Carnot groups, that is, Carnot groups such that the Whitney extension property does not hold. We use geometric control theory results on the accessibility of control affine systems in order to test the pliability of a Carnot group. In particular, we recover some recent results by Le Donne, Speight and Zimmerman about Lusin approximation in Carnot groups of step 2 and Whitney extension in Heisenberg groups. We extend such results to all pliable Carnot groups, and we show that the latter may be of arbitrarily large step.