Compact Einstein-Weyl four-dimensional manifolds

TL;DR

This paper classifies four-dimensional Einstein-Weyl manifolds with Bianchi metrics, identifying four unique families of conformally Kähler, positive scalar curvature solutions, and clarifies previous results with a simplified parametrization.

Contribution

It provides a complete classification of regular Einstein-Weyl structures with Bianchi metrics in four dimensions, correcting and simplifying earlier analyses.

Findings

Four 1-parameter families of regular Einstein-Weyl metrics found

All are of Bianchi IX type and conformally Kähler

Metrics have positive conformal scalar curvature

Abstract

We look for four dimensional Einstein-Weyl spaces equipped with a regular Bianchi metric. Using the explicit 4-parameters expression of the distance obtained in a previous work for non-conformally-Einstein Einstein-Weyl structures, we show that only four 1-parameter families of regular metrics exist on orientable manifolds : they are all of Bianchi $IX$ type and conformally K\"ahler ; moreover, in agreement with general results, they have a positive definite conformal scalar curvature. In a Gauduchon's gauge, they are compact and we obtain their topological invariants. Finally, we compare our results to the general analyses of Madsen, Pedersen, Poon and Swann : our simpler parametrisation allows us to correct some of their assertions.