Sub-Riemannian geodesics on SL(2,ℝ)

Abstract

We explicitly describe the length minimizing geodesics for a sub-Riemannian structure of the elliptic type defined on SL(2, 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. We use classification results for three dimensional right invariant sub-Riemannian structures on Lie groups to identify exactly automorphic structures on which our results apply.