Compact Einstein-Weyl four-dimensional manifolds

We look for complete four dimensional Einstein-Weyl spaces equipped with a 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 four 1-parameter families of compact metrics exist : they are all of Bianchi IX type and conformally Kähler ; moreover, in agreement with general results, they have a positive definite conformal scalar curvature. We obtain their topological invariants and check the correspondance with the general analysis of Madsen : our parametrisation allows us to prove some of his conjectures. PAR/LPTHE/98-25/gr-qc/9806037 May 1998 ∗Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS UMR 7589, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05. [email protected]