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 $$\mathbb {S}^3_1 (1)$$ .