Local constraints on Einstein-Weyl geometries: The 3-dimensional case
An Einstein-Weyl structure on a manifold \(M\) of dimension \(n\geq 3\) consists of a conformal class \([g]\) and a torsion-free connection \(D\) (the Weyl connection) preserving \([g]\) such that the symmetric part of the Ricci tensor of \(D\) is proportional to \(g\in[g]\). Given a conformal metric a natural problem is to investigate whether there exists any Weyl connection which, together with the conformal metric, forms an Einstein-Weyl geometry. The Einstein-Weyl equations were considered by \textit{E. Cartan} [Ann. Sci. Éc. Norm. Supér., III. Ser. 60, 1-16 (1943; Zbl 0028.30802)] showing that when \(n=3\), Einstein-Weyl structures are determined by four functions of two variables. One expects the existence of conformal metrics in dimension 3 for which there are no solutions of the Einstein-Weyl equations. In general dimension this question was considered in [the authors, J. Reine Angew Math. 491, 183-198 (1997; Zbl 0876.53029)] where the authors found necessary conditions on a given conformal metric to form part of an Einstein-Weyl geometry. Moreover, they found explicit examples of conformal metrics in dimension \(n\geq 4\) which admit no solution of the Einstein-Weyl equations. In this paper the authors find a new technique to simplify the problem in dimension 3 and they classify all homogeneous 3-metrics (real and complex) where the isometry group is class A in the Bianchi classification of relativity theory (i.e., unimodular) according to whether they admit or not local solutions of the Einstein-Weyl equations. In particular, they find out that the general left-invariant metric on \(S^3\) and the metric 'Sol' (considered by Eastwood and Tod in the above cited paper) do not admit such solutions.
scientific article; zbMATH DE number 1422970