In the joint paper of the author with K. P. Tod [J. Reine Angew. Math. 491, 183-198 (1997; Zbl 0876.53029)] they showed all local solutions of the Einstein-Weyl equations in three dimensions, where the background metric is homogeneous with unimodular isometry group. In particular, they proved that there are no solutions with Nil or Sol as background metric. In this note, these two special cases are presented.
@article{701643,
title = {Impossible Einstein-Weyl geometries},
booktitle = {Proceedings of the 19th Winter School "Geometry and Physics"},
series = {GDML\_Books},
publisher = {Circolo Matematico di Palermo},
address = {Palermo},
year = {2000},
pages = {17-22},
mrnumber = {MR1758074},
zbl = {0984.53019},
url = {http://dml.mathdoc.fr/item/701643}
}
Eastwood, Michael. Impossible Einstein-Weyl geometries, dans Proceedings of the 19th Winter School "Geometry and Physics", GDML_Books, (2000), pp. 17-22. http://gdmltest.u-ga.fr/item/701643/