Invariant Carnot–Caratheodory Metrics on $S^3$, $SO(3)$, $SL(2)$, and Lens Spaces

Abstract

In this paper we study the Carnot–Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$, and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Besides 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.