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

In this paper, we consider space-like surfaces in Robertson–Walker spacetimes $$L^4_1(f,c)$$ 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 $$L^4_1(f,c)$$ 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 for 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 parameterizations 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.