Compact Einstein-Weyl four-dimensional manifolds

Details Compact Einstein-Weyl four-dimensional manifolds Compact Einstein-Weyl four-dimensional manifolds

1998-06-08

arxiv.org/abs/gr-qc/9806037v2

Class.Quant.Grav. 16 (1999) 1057-1068

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

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

Scientific paper

10.1088/0264-9381/16/3/031

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 K\"ahler ; 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.

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