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

Erlend Grong; Alexander Vasil’ev

doi:10.3934/jgm.2011.3.225

Article Contents

2011, Volume 3, Issue 2: 225-260. Doi: 10.3934/jgm.2011.3.225

This issue Previous Article Embedded geodesic problems and optimal control for matrix Lie groups Next Article Euler-Poincaré reduction for systems with configuration space isotropy

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

Erlend Grong^1, and
Alexander Vasil’ev^1,

1.
Department of Mathematics, University of Bergen, P.O. Box 7803, Bergen N-5020

Received: June 30, 2010

Revised: May 31, 2011

Published: May 2011

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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.

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