Cohomogeneity - one Einstein - Weyl structures : a local approach

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 space is a homogeneous reductive Riemannian space with a 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 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 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 G compact, a one-dimensional subgroup H’ and the (n-2)-dimensional space G/(H×H’) being an arbitrary compact symmetric Kähler coset space, corresponds to n-dimensional Riemannian locally conformally Kähler metrics. 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 most of his conjectures. PAR/LPTHE/99-36/gr-qc/9912067 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]