Whitney’s extension theorem and the finiteness principle for curves in the Heisenberg group

Consider the sub-Riemannian Heisenberg group $\mathbb{H}$. In this paper, we answer following question: given a compact set $K \subseteq \mathbb{R}$ and continuous map $f\colon K \to \mathbb{H}$, when is there horizontal $C^m$ curve $F\colon \mathbb{R} \mathbb{H}$ such that $F|\_K = f$? Whitney originally answered question for real valued mappings, Fefferman provided complete functions defined on subsets of $\mathbb{R}^n$. We also prove finiteness principle $C^{m,\sqrt{\omega}}$ curves in sense Brudnyi Shvartsman.