Geodesics and Shortest Arcs of Some Sub-Riemannian Metrics on the Lie Groups $ \operatorname{SU}(1,1)\times �� $ and $ \operatorname{SO}_{0}(2,1)\times �� $ with Three-Dimensional Generating Distributions

Abstract

We find geodesics, shortest arcs, cut loci, and first conjugate loci for some left-invariant sub-Riemannian metrics on the Lie groups \( \operatorname{SU}(1,1)\times 𝕉 \) and \( \operatorname{SO}_{0}(2,1)\times 𝕉 \).