Abstract
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
k
k
-jets of a real function of one real variable
x
x
, denoted by
J
k
(
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
J
k
(
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
J
k
(
R
,
R
)
{J}^{k}\left({\mathbb{R}},{\mathbb{R}})
. The method is to use an intermediate three-dimensional sub-Riemannian space
R
F
3
{{\mathbb{R}}}_{F}^{3}
lying between the group
J
k
(
R
,
R
)
{J}^{k}\left({\mathbb{R}},{\mathbb{R}})
and the Euclidean space
R
2
{{\mathbb{R}}}^{2}
.