A note on proper affine symmetry in Bianchi types VIII and IX space-times

TL;DR

This paper investigates proper affine symmetries in Bianchi types VIII and IX space-times, revealing that such symmetries occur only under a unique condition, using holonomy, decomposability, and Riemann matrix analysis.

Contribution

It provides a comprehensive analysis of proper affine symmetries in Bianchi types VIII and IX space-times, identifying the specific condition under which these symmetries exist.

Findings

Proper affine symmetry exists only under a unique condition in these space-times.

Holonomy and Riemann matrix analysis are effective tools for studying symmetries.

The study narrows down the circumstances for affine symmetries in complex cosmological models.

Abstract

We consider the most general form of Bianchi types VIII and IX space-times for studying proper affine symmetry by using holonomy and decomposability, the rank of the 6X6 Riemann matrix and direct integration techniques. Studying proper affine symmetry in the above space-times it is shown that there exists only one possibilty when the above space-times admit proper affine vector field.