Invariant Carnot-Caratheodory metrics on $S^3$, $SO(3)$, $SL(2)$ and lens spaces

TL;DR

This paper explicitly computes invariant Carnot-Caratheodory metrics, geodesics, conjugate and cut loci on $SU(2)$, $SO(3)$, $SL(2)$, and lens spaces, providing the first such comprehensive analysis in sub-Riemannian geometry.

Contribution

It provides the first explicit computation of the entire cut locus in sub-Riemannian geometry for these spaces, including formulas for distances and geodesics.

Findings

Cut locus on $SU(2)$ is a maximal circle minus one point.

Cut loci on other spaces are stratified sets.

Distance expressed as inverse of an elementary function.

Abstract

In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric given on $SU(2)$ projects on the so called lens spaces $L(p,q)$. Also for lens spaces, we compute the cut loci (globally). For $SU(2)$ the cut locus is a maximal circle without one point. In all other cases the cut locus is a stratified set. To our knowledge, this is the first explicit computation of the whole cut locus in sub-Riemannian geometry, except for the trivial case of the Heisenberg group.