Sub-Riemannian Geodesics on $SL(2, \mathbb{R})$

TL;DR

This paper explicitly characterizes sub-Riemannian geodesics on SL(2, R) using symmetry reduction, providing visualizations and an alternative derivation of the cut-locus, advancing understanding of geometric structures on Lie groups.

Contribution

It introduces a symmetry reduction method to explicitly describe geodesics on SL(2, R), linking sub-Riemannian geometry with Riemannian problems on quotient spaces.

Findings

Explicit description of length minimizing geodesics on SL(2, R)

Visualization of geodesic projections on a 2D quotient space

Alternative derivation of the cut-locus characterization

Abstract

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on $SL(2, \mathbb{R})$. Our method uses a symmetry reduction which translates the problem into a Riemannian problem on a two dimensional quotient space, on which projections of geodesics can be easily visualized. As a byproduct, we obtain an alternative derivation of the characterization of the cut-locus obtained in \cite{BoscaRossi}. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups \cite{AGBD}, \cite{Biggs}, \cite{HB2} to identify exactly automorphic structures on which our results apply.