Metric lines in Jet Space

Given a sub-Riemannian manifold, a relevant question is: what are the metric lines (isometric embedding of the real line)? The space of $k$-jets of a real function of one real variable $x$, denoted by $J^k(\mathbb{R},\mathbb{R})$, admits the structure of a Carnot group, as every Carnot group $J^k(\mathbb{R},\mathbb{R})$ is a sub-Riemannian Manifold. This work is devoted to provide a partial result about the classification of the metric lines in $J^k(\mathbb{R},\mathbb{R})$. The method to prove the main Theorems is to use an intermediate $3$-dimensional sub-Riemannian space $\mathbb{R}^{3}_F$ lying between the group $J^k(\mathbb{R},\mathbb{R})$ and the Euclidean space $\mathbb{R}^{2} \simeq J^k(\mathbb{R},\mathbb{R}) / [J^k(\mathbb{R},\mathbb{R}),J^k(\mathbb{R},\mathbb{R})]$.