Geodesics and shortest arcs of special sub-Riemannian metric on the Lie group $SL(2)$

TL;DR

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

Contribution

It explicitly computes geodesics and related structures for a left-invariant sub-Riemannian metric on SL(2), a novel analysis in the context of weakly symmetric spaces.

Findings

Explicit formulas for geodesics on SL(2)

Description of cut loci and conjugate sets

Insights into sub-Riemannian geometry on Lie groups

Abstract

The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)$).