Symmetric Riemannian problem on the group of proper isometries of hyperbolic plane

TL;DR

This paper analyzes a specific Riemannian metric on the group of proper isometries of the hyperbolic plane, deriving geodesic properties, conjugate and cut loci, and examining their limits compared to sub-Riemannian problems.

Contribution

It provides explicit parametrizations of geodesics, conjugate points, and cut loci for a class of left-invariant metrics on PSL(2) and SL(2), including convergence results to sub-Riemannian cases.

Findings

Computed the injectivity radius for the metrics.

Derived the parametrization of geodesics and cut loci.

Showed convergence of cut time and locus to sub-Riemannian cases.

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).