A $C^m$ Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group

We characterize those mappings from a compact subset of $\mathbb{R}$ into the Heisenberg group ${\mathbb{H}}^n$ which can be extended to a $C^m$ horizontal curve in ${\mathbb{H}}^n$. The characterization combines the classical Whitney conditions with an estimate comparing changes in the vertical coordinate with those predicted by the Taylor series of the horizontal coordinates.