Left-invariant Riemannian problems on the groups of proper motions of hyperbolic plane and sphere

Abstract

On the Lie groups PSL₂(ℝ) and SO₃ we consider left-invariant Riemannian metrics with two equal eigenvalues. The global optimality of geodesics is investigated. We find the parametrization of geodesics, the cut locus and the equations for the cut time. When the third eigenvalue of a metric tends to the infinity the cut locus and the cut time converge to the cut locus and the cut time of the sub-Riemannian problem.