On Time-Like Class A Surfaces of a Warped Space-time with Lorentzian Fiber

In this paper, we consider time-like surfaces in the static space-time given by the warped product ${\mathbb{L}}_1^3(c){}_f\times (I,d{z}^2)$, where ${\mathbb{L}}_1^3(c)$ denotes the Lorentzian space form with the constant sectional curvature $c\in \{-1,0,1\}$. In particular, we study the surfaces with light-like ${\left(\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right)}^T$. First, we construct a globally defined pseudo-orthonormal frame field on a surface satisfying this condition and deal with the invariants associated with this frame field. Then, we obtain a complete classification theorem for class~$\mathcal{A}$ surfaces. Finally, we consider some applications of this theorem.