Compact Einstein-Weyl four-dimensional manifolds

Abstract

We look for four dimensional Einstein-Weyl spaces equipped with a regular Bianchi metric. Using the explicit 4-parameters expression of the distance obtained in a previous work for non-conformally-Einstein Einstein-Weyl structures, we show that only four 1-parameter families of regular metrics exist on orientable manifolds : they are all of Bianchi $IX$ type and conformally Kähler ; moreover, in agreement with general results, they have a positive definite conformal scalar curvature. In a Gauduchon's gauge, they are compact and we obtain their topological invariants. Finally, we compare our results to the general analyses of Madsen, Pedersen, Poon and Swann : our simpler parametrisation allows us to correct some of their assertions.

Latex file, 13 pages, an important reference added and a critical discussion of its claims offered, others minor modifications

References in corpus (1)

• Einstein-Weyl structures and Bianchi metrics

Cited by in corpus (4)

• Einstein four-manifolds with skew torsion
• Flows and particles with shear-free and expansion-free velocities in (L^-_n,g)- and Weyl's spaces
• Cohomogeneity-one Einstein-Weyl structures : a local approach
• Einstein Structure of Four-Manifolds