On Time-Like Class $\mathcal A$ Surfaces in a Static Space Time

In this paper, we consider time-like surfaces in the static space-time given by the warped product $\mathbb L^3_1(c)\, _f\times (I,dz^2)$, where $\mathbb L^3_1(c)$ denotes the Lorentzian space form with the constant sectional curvature $c\in\{-1,0,1\}$. In particular, we study the surfaces with light-like $\left(\frac{\partial}{\partial z}\right)^T$. 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.