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

TL;DR

This paper classifies space-like and time-like surfaces with positive relative nullity in Robertson-Walker space-times, providing necessary conditions, local classifications, and examining special cases involving product spaces with constant warping functions.

Contribution

It offers new classification results for surfaces with positive relative nullity in Robertson-Walker space-times, extending understanding of their geometric properties.

Findings

Necessary and sufficient conditions for such surfaces in $L^4_1(f,c)$.

Local classification theorems for surfaces in $L^4_1(f,0)$.

Analysis of surfaces in product spaces $ ext{E}^1_1 imes ext{S}^3$ and $ ext{E}^1_1 imes ext{H}^3$.

Abstract

In this article, we study space-like and time-like surfaces in a Robertson-Walker space-time,, denoted by $L^4_1(f,c)$, having positive relative nullity. First, we give the necessary and sufficient conditions for such space-like and time-like surfaces in $L^4_1(f,c)$. Then, we obtain the local classification theorems for space-like and time-like surfaces in $L^4_1(f,0)$ 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 $L^4_1(f,c)$ 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 $\mathbb{E}^1_1\times\mathbb{H}^3$.