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

• Main Authors: Grong, E, Vasil'ev, A
• Format: Journal Article
• Language: English
• Published: 06.10.2009
• Subjects: Mathematics - Differential Geometry; Mathematics - Optimization and Control
• DOI: 10.48550/arxiv.0910.0945

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.