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

• Published in: arXiv.org
• Main Authors: Grong, E, Vasil'ev, A
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 05.11.2011
• Subjects: Geodesy; Lie groups
• ISSN: 2331-8422

We study sub-Riemannian and sub-Lorentzian geometry on the Lie group \(\SU(1,1)\) and on its universal cover \(\CSU(1,1)\). In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both \(\SU(1,1)\) and \(\CSU(1,1)\), 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 \(\CSU(1,1)\), 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.