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

TL;DR

This paper explores sub-Riemannian and sub-Lorentzian geometries on the Lie group SU(1,1) and its universal cover, providing explicit descriptions of geodesics, distances, and the structure of reachable sets in both geometries.

Contribution

It offers a comprehensive analysis of geodesics and distance functions in sub-Riemannian and sub-Lorentzian settings on SU(1,1) and its cover, revealing their geometric relationships and differences.

Findings

Explicit formulas for sub-Riemannian geodesics and distances.

Connection between conjugate loci and geodesic multiplicity.

Comparison of reachable sets in Lorentzian and sub-Lorentzian geometries.

Abstract

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.