D ec 1 99 9 Cohomogeneity - one Einstein - Weyl structures : a local approach Guy Bonneau

We analyse in a systematic way the (non-)compact n-dimensional Einstein-Weyl spaces equipped with a cohomogeneity-one metric. In that context, with no compactness hypothesis for the manifold on which lives the Einstein-Weyl structure, we prove that, as soon as the (n-1)-dimensional basis space is an homogeneous reductive Riemannian space with an unimodular group of left-acting isometries G : • there exists a Gauduchon gauge such that the Weyl form is co-closed and its dual is a Killing vector for the metric, • in that gauge, a simple constraint on the conformal scalar curvature holds, • a non-exact Einstein-Weyl stucture may exist only if the (n-1)-dimensional basis homogeneous space G/H contains a non trivial subgroup H’ that commutes with the isotropy subgroup H, • the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H’, the other generators relating the different solutions that correspond to the chosen generator. The first two results are well known when the Einstein-Weyl structure lives on a compact manifold, but our analysis gives the first hints on the enlargement of the symmetry due to the Einstein-Weyl constraint. We also prove that the subclass with a one-dimensional subgroup H’ corresponds to n-dimensional Riemannian locally conformally Kähler metrics, the (n-2)-dimensional basis being an arbitrary compact symmetric Kähler coset space. The explicit family of structures of cohomogeneity-one under SU(n/2) being, thanks to our results, invariant under U(1) ⊗ SU(n/2), it coincides with the one first studied by Madsen ; our analysis allows us to prove some of his conjectures. PAR/LPTHE/99-36 November 1999 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]