On space-like class A $\mathcal {A}$ surfaces in Robertson–Walker spacetimes

In this paper, we consider space-like surfaces in Robertson–Walker spacetimes $L_1^4(f,c)$ with the comoving observer field $\frac{\partial }{\partial t}$. We study some problems related to such surfaces satisfying the geometric conditions imposed on the tangential and normal parts of the unit vector field $\frac{\partial }{\partial t}$, as naturally defined. First, we investigate space-like surfaces in $L_1^4(f,c)$ satisfying that the tangent component of $\frac{\partial }{\partial t}$ is an eigenvector of all shape operators, called class $A$ surfaces. Then, we get a classification theorem for space-like class $A$ surfaces in $L_1^4(f,0)$. Also, we examine minimal space-like class $A$ surfaces in $L_1^4(f,0)$. Finally, we give the parameterizations of space-like surfaces in $L_1^4(f,0)$ when the normal part of the unit vector field $\frac{\partial }{\partial t}$ is parallel.