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.

MSC codes

  1. 22E30
  2. 49J15
  3. 53C17

Keywords

  1. left-invariant sub-Riemannian geometry
  2. Carnot–Caratheodory distance
  3. global structure of the cut locus
  4. lens spaces-2pt

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Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 1851 - 1878
ISSN (online): 1095-7138

History

Submitted: 25 September 2007
Accepted: 17 February 2008
Published online: 25 June 2008

MSC codes

  1. 22E30
  2. 49J15
  3. 53C17

Keywords

  1. left-invariant sub-Riemannian geometry
  2. Carnot–Caratheodory distance
  3. global structure of the cut locus
  4. lens spaces-2pt

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