Constant angle surfaces in the Lorentzian warped product manifold $I \times_{f} \mathbb E^2_1$

Abstract

Let $I \times_{f} \mathbb E^2_1$ be a 3-dimensional Lorentzian warped product manifold with the metric $\tilde g = dt^2 + f^2(t) (dx^2 - dy^2)$, where $I$ is an open interval, $f$ is a strictly positive smooth function on $I,$ and $\mathbb E^2_1$ is the Minkowski 2-plane. In this work, we give a classification of all space-like and time-like constant angle surfaces in $I \times_{f} \mathbb E^2_1$ with nonnull principal direction when the surface is time-like. In this classification, we obtain space-like and time-like surfaces with zero mean curvature, rotational surfaces, and surfaces with constant Gaussian curvature. Also, we have some results on constant angle surfaces of the anti-de Sitter space $ \mathbb H^3_1(-1)$.