Local constraints on Einstein-Weyl geometries: The 3-dimensional case
type: Item
instance of: scholarly article
title: Local constraints on Einstein-Weyl geometries: The 3-dimensional case
zbMATH Open document ID: 0957.53015
DOI: 10.1023/A:1006621831435
author: M. Eastwood
author: K. P. Tod
published in: Annals of Global Analysis and Geometry
publication date: 2000-06-04T00:00:00Z
review text: 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.
reviewed by: Anna Fino
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 1422970
zbMATH Keywords: conformal geometry
zbMATH Keywords: Einstein-Weyl geometry
zbMATH Keywords: homogeneous manifold
zbMATH Keywords: conformal class
zbMATH Keywords: Weyl connection
MaRDI profile type: Publication
full work available at URL: a:1006621831435
OpenAlex ID: W20680073
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description: scientific article; zbMATH DE number 1422970