(Locally) shortest arcs of special sub-Riemannian metric on the Lie group $SO_0(2,1)$

TL;DR

This paper characterizes geodesics, shortest paths, cut loci, and conjugate points for a specific sub-Riemannian metric on the Lie group SO_0(2,1), revealing geometric properties relevant to control theory and differential geometry.

Contribution

It provides explicit descriptions of geodesics and cut loci for a particular sub-Riemannian structure on SO_0(2,1), advancing understanding of geometric control on this Lie group.

Findings

Explicit formulas for geodesics on SO_0(2,1)

Identification of cut locus and conjugate sets

Analysis of shortest arcs under the given metric

Abstract

The author finds geodesics, shortest arcs, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group $SO_0(2,1)$ under the condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO_0(2,1)$.