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Local Constraints on Einstein–Weyl Geometries: The 3-Dimensional Case

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Abstract

Following an earlier study [3], we consider the Einstein–Weyl equations on a fixed (complex) background metric as an equation for a 1-form and its first few derivatives. If the background is flat then we conclude that the only solutions are conformal rescalings of constant curvature metrics. If the background is a homogeneous 3-geometry in Bianchi class A (i.e., with unimodular isometry group), we find necessary and sufficient conditions on the 3-geometry for solutions of the Einstein–Weyl equations to exist. The solutions we find are complexifications of known ones. In particular, we find that the general left-invariant metric on S3 and the metric 'Sol' admit no local solutions of the Einstein–Weyl equations.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Adelaide, Adelaide, SA 5005, Australia. e-mail

    M. G. Eastwood

  2. Mathematical Institute, 24–29 Saint Giles, Oxford, OX1 3LB, England. E-mail

    K. P. Tod

Authors
  1. M. G. Eastwood
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  2. K. P. Tod
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Eastwood, M.G., Tod, K.P. Local Constraints on Einstein–Weyl Geometries: The 3-Dimensional Case. Annals of Global Analysis and Geometry 18, 1–27 (2000). https://doi.org/10.1023/A:1006621831435

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