Metric lines in metabelian Carnot groups.

A Carnot group is a simple connected Lie group whose Lie algebra is stratified, graded, and nilpotent. Every Carnot group has the structure of a left-invariant sub-Riemannian manifold. The sub-Riemannian geodesic flow defines locally minimizing curves. A natural question is: Under which condition is a sub-Riemannian geodesic globally minimizing? A metric line is globally minimizing geodesic; an alternative term for "metric line" is "an isometric embedding of the real line". We talk about some results in metric lines in metabelian Carnot groups (a group is metabelian if its commutator subgroup is abelian).