Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups

We show that, in arbitrary Carnot groups, pliability in a subset of directions is sufficient to guarantee the existence of a Whitney-type extension and a Lusin approximation for curves with tangent vectors in the same set of directions. We apply this to show that every horizontal curve in the Engel group must intersect a $C^1$ horizontal curve in a set of positive measure.