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

TL;DR

This paper provides a characterization of when functions from a compact subset of real numbers into the Heisenberg group can be extended to smooth horizontal curves, combining Whitney conditions with Taylor series estimates.

Contribution

It introduces a $C^m$ Whitney extension theorem specifically for horizontal curves in the Heisenberg group, blending classical Whitney conditions with new vertical coordinate estimates.

Findings

Characterization of extendable $C^m$ horizontal curves in $ ext{Heisenberg group$.

Extension criteria involve Whitney conditions and Taylor series estimates.

Provides a framework for smooth extensions in sub-Riemannian geometry.

Abstract

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.