Pliability, or the whitney extension theorem for curves in carnot groups

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 numerous settings, among which the one of Carnot groups. However, the target space has generally been assumed to be equal to R^d for some d ≥ 1. We study here the case of maps from R to a general Carnot group and introduce for this purpose the notion of pliable Carnot group. We prove that this geometric condition, not far from the non-rigidity of curves as defined by Bryant and Hsu, is equivalent to the suitable statement for a horizontal Whitney extension theorem of class C^1. We provide examples of pliable and non-pliable Carnot groups and establish a relation with the Lusin approximation problem.