Shortest paths on 3-D simple Lie groups with nonholonomic constraint

— In this paper we study the Carnot-Caratheodory metrics on SU(2) S3 , 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 CarnotCaratheodory distance as the inverse of an elementary function. 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.

• Type: Conference
• Year: 2008
• Where: CDC
• Authors: Ugo V. Boscain, Francesco Rossi
• Keywords: Cartan Decomposition, CDC 2008, Control Systems, Cut Locus, first Explicit Computation