On the geometry of non-degenerate surfaces in Lorentzian homogeneous $3$-manifolds

Abstract

In this paper we deal with non-degenerate surfaces Σ 2 immersed in the 3dimensional homogeneous space L 3 (κ, τ ) endowed with two different metrics, the one induced by the Riemannian metric of E 3 (κ, τ ) and the non-degenerate metric inherited by the Lorentzian one of L 3 (κ, τ ). Therefore, we have two different geometries on Σ 2 and we can compare them. In particular, we study the case where the mean curvature functions with respect to both metrics simultaneously vanish, and in this case we show that the surface is ruled. Furthermore, we consider the case where both mean curvature functions coincide but do not necessarily vanish and we also consider the situation where the extrinsic curvatures with respect to both metrics coincide.