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

TL;DR

This paper extends Whitney's classical theorem to the setting of $C^1$, horizontal curves in the Heisenberg group, providing necessary and sufficient conditions for such extensions from compact subsets of $ ^1$.

Contribution

It establishes a Whitney extension theorem for $C^1$, horizontal curves in the Heisenberg group, a novel result in geometric analysis.

Findings

Proves a Whitney extension theorem for $C^1$, horizontal curves in $H^n$

Provides necessary and sufficient conditions for extensions in the Heisenberg group

Advances understanding of regularity and extension problems in sub-Riemannian geometry

Abstract

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 $\mathbb{H}^n$.