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

Abstract

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.