Sub-Riemannian distance on the Lie group $SO_0(2,1)$

TL;DR

This paper investigates the sub-Riemannian geometry of the Lorentz group SO_0(2,1), deriving explicit formulas for distances, cut loci, and conjugate sets under a specific invariant metric, advancing understanding of geometric structures on Lorentz groups.

Contribution

It provides explicit descriptions of the sub-Riemannian distance, cut locus, and conjugate set on SO_0(2,1) with a particular invariant metric, which was previously not well-understood.

Findings

Explicit distance formulas between elements

Description of the cut locus as a union of specific subgroups

Identification of the conjugate set for the unit element

Abstract

A left-invariant sub-Riemannian metric $d$ on the shortened Lorentz group $SO_0(2,1)$ under the condition that $d$ is right-invariant relative to the orthogonal Lie subgroup $1\otimes SO(2)$ is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup $1\otimes SO(2)$ with the antipodal set to the submanifold of symmetric matrices in the open solid torus $SO_0(2,1)$), and the conjugate set for the unit are found for $(SO_0(2,1),d)$.