Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover

Pages: 225 - 260, Volume 3, Issue 2, June 2011

doi:10.3934/jgm.2011.3.225       Abstract        References        Full Text (741.6K)       Related Articles

Erlend Grong - Department of Mathematics, University of Bergen, P.O. Box 7803, Bergen N-5020, Norway (email)
Alexander Vasil’ev - Department of Mathematics, University of Bergen, P.O. Box 7803, Bergen N-5020, Norway (email)

Abstract: We study sub-Riemannian and sub-Lorentzian geometry on the Lie group $SU(1,1)$ and on its universal cover SU^~(1,1). In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both $SU(1,1)$ and SU^~(1,1), connecting two fixed points. In particular, we prove that there is a strong connection between the conjugate loci and the number of geodesics. In the sub-Lorentzian case, we describe the geodesics connecting two points on SU^~(1,1), and compare them with Lorentzian ones. It turns out that the reachable sets for Lorentzian and sub-Lorentzian normal geodesics intersect but are not included one to the other. A description of the timelike future is obtained and compared in the Lorentzian and sub-Lorentzain cases.

Keywords:  SU(1,1), universal cover, optimal control, sub-Riemannian and sub-Lorentzian manifolds, Carnot-Carathéodory metric, geodesic.
Mathematics Subject Classification:  Primary: 53C17, 53B30, 22E30; Secondary: 53C50, 83C65.

Received: July 2010;      Revised: June 2011;      Published: July 2011.

 References