Title: Symmetric Riemannian problem on the group of proper isometries of hyperbolic plane,
Abstract: We consider the Lie group PSL(2) (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 as the third eigenvalue of the
metric tends to infinity. Similar results are also obtained for SL(2).