  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.
