Ricci Collineations of the Bianchi Type II, VIII, and IX Space-times

TL;DR

This paper classifies Ricci and contracted Ricci collineations in Bianchi type II, VIII, and IX space-times, identifying known solutions and conditions under which these models reduce to Robertson-Walker metrics.

Contribution

It provides a detailed analysis of Ricci collineations in specific Bianchi space-times, including new solutions and reduction conditions to Robertson-Walker metrics.

Findings

Identified known solutions for specific collineation vector fields.

Derived conditions under which Bianchi models reduce to Robertson-Walker metrics.

Classified Ricci collineations for Bianchi types II, VIII, and IX.

Abstract

Ricci and contracted Ricci collineations of the Bianchi type II, VIII, and IX space-times, associated with the vector fields of the form (i) one component of $\xi^a(x^b)$ is different from zero and (ii) two components of $\xi^a(x^b)$ are different from zero, for $a,b=1,2,3,4$, are presented. In subcase (i.b), which is $\xi^a= (0,\xi^2(x^a),0,0)$, some known solutions are found, and in subcase (i.d), which is $\xi^a =(0,0,0,\xi^4(x^a))$, choosing $S(t)=const.\times R(t)$, the Bianchi type II, VIII, and IX space-times is reduced to the Robertson-Walker metric.