arXiv:2108.06823v1 [math.DG] 15 Aug 2021 ON THE GEOMETRY OF NON-DEGENERATE SURFACES IN LORENTZIAN HOMOGENEOUS 3-MANIFOLDS ALMA L. ALBUJER * AND F ´ ABIO R. DOS SANTOS 1 Abstract. In this paper we deal with non-degenerate surfaces Σ 2 immersed in the 3- dimensional 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. Introduction In 1983 Kobayashi [9] proved that the only surfaces in the 3-dimensional Lorentz- Minkowski space L 3 which are simultaneously minimal and maximal are open pieces of spacelike planes or of the helicoid, in the region where it is spacelike. On the one hand, it is well known that a minimal surface in R 3 is a surface with zero mean curvature. On the other hand, let us recall that a spacelike surface in the Lorentz-Minkowski space is an immersed surface such that the metric induced from L 3 is a Riemannian one. Fur- thermore, a maximal surface is a spacelike surface with identically zero mean curvature. The terminology minimal (maximal, respectively) comes from the fact that these surfaces locally minimize (maximize, respectively) area among close enough surfaces. Since any spacelike surface in L 3 can also be endowed with a second Riemannian metric, the one induced from the Euclidean space R 3 , the problem studied by Kobayashi makes sense. During the last years several authors have generalized the above result both to general dimension and to surfaces in different ambient spaces. As a first extension, in 2009 Kim, Koh, Shin and Yang [7] considered the situation of a simultaneously minimal and maximal surface in a Lorentzian product space of the type M 2 × R 1 , where M 2 is a Riemannian surface, and they proved that those surfaces are horizontally ruled by geodesics. Besides, Date : January 26, 2022. 1 2020 Mathematics Subject Classification. Primary 53C42; Secondary 53A10, 53C20, 53C50. Key words and phrases. Non-degenerate surfaces, Homogeneous manifolds, Minimal and maximal sur- faces, Extrinsic curvature, Hopf cylinders. * Corresponding author. 1