Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover
Erlend Grong - 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.
Received: July 2010; Revised: June 2011; Available Online: July 2011.
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