Sub-Riemannian geodesics on <i>SL</i>(2,ℝ)

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