Constant Angle Surfaces in the Lorentzian Warped Product Manifold $-I \times _{f} \mathbb {E}^2$

Abstract

In this work, we study constant angle space-like and time-like surfaces in the 3-dimensional Lorentzian warped product manifold $-I \times _{f} \mathbb {E}^2$ with the metric ${\tilde{g}} = - \mathrm{d}t^2 + f^2(t) (\mathrm{d}x^2 + \mathrm{d}y^2)$, where I is an open interval, f is a strictly positive function on I, and $\mathbb {E}^2$ is the Euclidean plane. We obtain a classification of all constant angle space-like and time-like surfaces in $-I \times _{f} \mathbb {E}^2$. In this classification, we determine space-like and time-like surfaces with zero mean curvature, rotational surfaces, and surfaces with constant Gaussian curvature. Also, we obtain some results on constant angle space-like and time-like surfaces of the de Sitter space $\mathbb {S}^3_1 (1)$.