Overview
This research investigates time-like surfaces situated within a static spacetime defined by a warped product structure. Specifically, the focus is on $\mathbb L^3_1(c)\, _f\times (I,dz^2)$, where $\mathbb L^3_1(c)$ denotes a Lorentzian space form characterized by a constant sectional curvature $c$, which can take values of $-1, 0, \text{or } 1$. The study concentrates on surfaces where the vector field $\left(\frac{\partial}{\partial z}\right)^T$ is light-like.
Approach
The methodology employed in this study involved several sequential steps. Initially, a globally defined pseudo-orthonormal frame field was constructed on surfaces that satisfy the condition of having a light-like $\left(\frac{\partial}{\partial z}\right)^T$. This frame field serves as a foundational element for subsequent analysis.
Following the construction of the frame field, the research proceeded to analyze the invariants associated with it. The identification and characterization of these invariants were central to understanding the geometric properties of the surfaces under consideration.
The core of the research approach culminated in the derivation of a complete classification theorem. This theorem specifically addresses class $\mathcal A$ surfaces within the defined warped spacetime. The classification is predicated on the properties and invariants previously established through the frame field analysis.
Findings
- A globally defined pseudo-orthonormal frame field was constructed for time-like surfaces in the static warped product $\mathbb L^3_1(c)\, _f\times (I,dz^2)$, provided the condition that $\left(\frac{\partial}{\partial z}\right)^T$ is light-like.
- Invariants associated with this constructed frame field were identified and dealt with.
- A complete classification theorem for class $\mathcal A$ surfaces within this specific warped spacetime characterized by a Lorentzian fiber was obtained.
- Applications of this classification theorem were considered.