Lorentzian BCV Spaces: properties and geometry of their surfaces

We investigate the Lorentzian analogues of Riemannian Bianchi-Cartan-Vranceanu spaces. We provide their general description and emphasize their role in the classification of three-dimensional homogeneous Lorentzian manifolds with a four-dimensional isometry group. We then illustrate their geometric properties (with particular regard to curvature, Killing vector fields and their description as Lorentzian Lie groups) and we study several relevant classes of surfaces (parallel, totally umbilical, minimal, constant mean curvature) in these homogeneous Lorentzian three-manifolds.