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The Whitney extension theorem is a classical result in analysis giving anecessary and sufficient condition for a function defined on a closed set to beextendable to the whole space with a given class of regularity. It has beenadapted to several settings, among which the one of Carnot groups. However, thetarget 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(G\_1,G\_2) (with G\_2 non-Abelian). We analyze in particular the case G\_1 = Rand characterize the groups G\_2 for which the Whitney extension propertyholds, in terms of a newly introduced notion that we call pliability.Pliability happens to be related to rigidity as defined by Bryant an Hsu. Weexploit this relation in order to provide examples of non-pliable Carnotgroups, that is, Carnot groups so that the Whitney extension property does nothold. We use geometric control theory results on the accessibility of controlaffine systems in order to test the pliability of a Carnot group. Inparticular, we recover some recent results by Le Donne, Speight and Zimmermannabout Lusin approximation in Carnot groups of step 2 and Whitney extension inHeisenberg groups. We extend such results to all pliable Carnot groups, and weshow that the latter may be of arbitrarily large step.

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