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

By Ugo V. Boscain, Francesco Rossi. 2007

153 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/14489d5e3f5b31b129d177831f49cbf8e692f509

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups (2008, background): “Indeed, in [5] the complete description of the cut and conjugate loci for these groups was obtained.”
Sub-Riemannian structures on 3D lie groups (2010, background): “In particular, thanks to the group structure, in some of these cases it is also possible to compute explicitly the sub-Riemannian distance and geodesics (see in particular [13] for the Heisenberg group, [8] for semisimple Lie groups with Killing form and [16, 20] for a detailed study of the sub-Riem”
Small time heat kernel asymptotics at the sub-Riemannian cut locus (2012, background): “on sub-Riemannian manifolds has been an object of attention starting from the late 70s [13, 15, 16, 18, 20, 25, 31, 37, 40, 53, 54], as have the geodesics and cut and conjugate loci of such manifolds [2, 3, 5, 8, 9, 24, 28, 42, 51]. In this paper, we provide a general approach to relate the sub-Riem”