Constant Angle Property and Canonical Principal Directions for Surfaces in $${\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}$$

Abstract

In this paper, we study surfaces in Lorentzian product spaces $${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$$ . We classify constant angle spacelike and timelike surfaces in $${{\mathbb{S}^{2} \times \mathbb{R}_1}}$$ and $${{\mathbb{H}^{2} \times \mathbb{R}_1}}$$ . Moreover, complete classifications of spacelike surfaces in $${{\mathbb{S}^{2} \times \mathbb{R}_1}}$$ and $${{\mathbb{H}^{2} \times \mathbb{R}_1}}$$ and timelike surfaces in $${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.