Surfaces in Robertson-Walker Space-Times with Positive Relative Nullity

Abstract

In this article, we study space-like and time-like surfaces in a Robertson-Walker space-time,, denoted by $L_1^4(f,c)$, having positive relative nullity. First, we give the necessary and sufficient conditions for such space-like and time-like surfaces in $L_1^4(f,c)$. Then, we obtain the local classification theorems for space-like and time-like surfaces in $L_1^4(f,0)$ with positive relative nullity. Finally, we consider the space-like and time-like surfaces in ${\mathbb{E}}_1^1\times {\mathbb{S}}^3$ and ${\mathbb{E}}_1^1\times {\mathbb{H}}^3$ with positive relative nullity. These are the special spaces of $L_1^4(f,c)$ when the warping function $f$ is a constant function, with $c=1$ for ${\mathbb{E}}_1^1\times {\mathbb{S}}^3$ and $c=-1$ for ${\mathbb{E}}_1^1\times {\mathbb{H}}^3$.