M ar 1 99 8 Einstein - Weyl structures and Bianchi metrics

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 Kähler ; in the non-exact Einstein-Weyl case with a Bianchi metric of the type V II0 , V III 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, Kähler Bianchi IX case. PAR/LPTHE/98-06/hep-gr-qc/9803009 February 1998 ∗Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS URA 280, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05. [email protected]