Einstein-Weyl structures and Bianchi metrics

TL;DR

This paper systematically studies four-dimensional Einstein-Weyl spaces with Bianchi metrics, revealing conditions for conformal scalar curvature, conformally Einstein or Kähler structures, and explicit metric forms for certain Bianchi types.

Contribution

It extends previous work by analyzing non-diagonal Einstein-Weyl Bianchi metrics and providing explicit metric expressions for specific Bianchi types.

Findings

Einstein-Weyl structures with Class A Bianchi metrics have constant sign conformal scalar curvature.

Most such structures are conformally Einstein or Kähler.

Explicit 4-parameter diagonal metric forms are derived for non-exact Einstein-Weyl cases of types VII_0, VIII, IX.

Abstract

We analyse in a systematic way the (non-)compact four dimensional Einstein-Weyl spaces equipped with a Bianchi metric. We show that Einstein-Weyl structures with a Class A Bianchi metric have a conformal scalar curvature of constant sign on the manifold. Moreover, we prove that most of them are conformally Einstein or conformally K\"ahler ; in the non-exact Einstein-Weyl case with a Bianchi metric of the type $VII_0, VIII$ or $IX$, we show that the distance may be taken in a diagonal form and we obtain its explicit 4-parameters expression. This extends our previous analysis, limited to the diagonal, K\"ahler Bianchi $IX$ case.