A. V. Podobryaev, Yu. L. Sachkov, “Symmetric riemannian problem on the group of proper isometries of hyperbolic plane”, Journal of Dynamical and Control Systems, 24:3 (2018), 391–423
We consider the Lie group PSL_2(R) (the group of orientation preserving isometries of the hyperbolic plane) and a left-invariant Riemannian metric on this group with two equal eigenvalues that correspond to space-like eigenvectors (with respect to the Killing form). For such metrics we find a parametrization of geodesics, the conjugate time, the cut time and the cut locus. The injectivity radius is computed. We show that the cut time and the cut locus in such Riemannian problem converge to the cut time and the cut locus in the corresponding sub-Riemannian problem when the third eigenvalue of the metric tends to infinity. Also similar results are obtained for SL_2(R).