The Whitney Extension Theorem for $C^1$, horizontal curves in the Heisenberg group

For a real valued function defined on a compact set $K\subset {\mathbb{R}}^m$, the classical Whitney Extension Theorem from 1934 gives necessary and sufficient conditions for the existence of a $C^k$ extension to ${\mathbb{R}}^m$. In this paper, we prove a version of the Whitney Extension Theorem in the case of $C^1$, horizontal extensions for mappings defined on compact subsets of $\mathbb{R}$ taking values in the Heisenberg Group ${\mathbb{H}}^n$.