Metric lines in the jet space - Bravo-Doddoli Alejandro (Analysis and Geometry in Metric Spaces)
In the realm of sub-Riemannian manifolds, a relevant question is: what are the metric lines (isometric embedding of the real line)? The space of kk-jets of a real function of one real variable xx, denoted by Jk(R,R)J^{k}\left({\mathbb{R}},{\mathbb{R}}), admits the structure of a Carnot group. Every Carnot group is sub-Riemannian manifold, so is Jk(R,R)J^{k}\left({\mathbb{R}},{\mathbb{R}}). This study aims to present a partial result about the classification of the metric lines within Jk(R,R)J^{k}\left({\mathbb{R}},{\mathbb{R}}). The method is to use an intermediate three-dimensional sub-Riemannian space RF3{{\mathbb{R}}}_{F}^{3} lying between the group Jk(R,R)J^{k}\left({\mathbb{R}},{\mathbb{R}}) and the Euclidean space R2{{\mathbb{R}}}^{2}.