On Space-like Class \(\mathcal A\) Surfaces in Robertson-Walker Space Times

In this article, we consider space-like surfaces in Robertson-Walker Space times \(L^4_1(f,c)\) with comoving observer field \(\frac{\partial}{\partial t}\). We study some problems related to such surfaces satisfying the geometric conditions imposed on the tangential part and normal part of the unit vector field \(\frac{\partial}{\partial t}\) naturally defined. First, we investigate space-like surfaces in \(L^4_1(f,c)\) satisfying that the tangent component of \(\frac{\partial}{\partial t}\) is an eigenvector of all shape operators, called class \(\mathcal A\) surfaces. Then, we get a classification theorem of space-like class \(\mathcal A\) surfaces in \(L^4_1(f,0)\). Also, we examine minimal space-like class \(\mathcal A\) surfaces in \(L^4_1(f,0)\). Finally, we give the parametrizations of space-like surfaces in \(L^4_1(f,0)\) when the normal part of the unit vector field \(\frac{\partial}{\partial t}\) is parallel.