Einstein-Weyl geometry (Q2771301)

A Weyl manifold is a conformal manifold equipped with a torsion free connection preserving the conformal structure, called a Weyl connection. It is said to be Einstein-Weyl if the symmetric trace-free part of the Ricci tensor of this connection vanishes. The implications in Einstein-Weyl geometry are more subtle as shown in the following theorem: closed Einstein-Weyl structures have parallel scalar curvature, and the converse holds in the compact case or when the dimension is not four. NEWLINENEWLINENEWLINEMany of the general theorems about compact Einstein-Weyl manifolds follow from the existence of a distinguished compatible metric, the Gauduchon metric. These results are given in this survey and imply that, apart from the Einstein case, the isometry group of the Gauduchon metric on a compact Einstein-Weyl manifold is at least one dimensional. The sign of the scalar curvature is constant in four or more dimensions, contrary to some previous claims. It is shown that this need not hold in dimensions two and three. An extensive supply of examples of Einstein-Weyl manifolds is given. These examples are often obtained from Riemannian submersions, which are discussed within the more general framework of conformal submersions. In particular, the authors give an ansatz aimed at studying of submersions between Einstein-Weyl manifolds, which includes as special cases both circle bundles over Kähler-Einstein manifolds and hyper-complex 4-manifolds (which are Einstein-Weyl) over Einstein-Weyl 3-manifolds. Using submersions, a direct proof of the Jones-Tod result concerning the construction of three dimensional Einstein-Weyl spaces from self-dual 4-manifolds with a conformal vector field is given. NEWLINENEWLINENEWLINEEinstein-Weyl moduli spaces near Einstein metrics are studied. The authors illustrate how additional conditions on Einstein-Weyl manifolds often lead to closed structures, and highlight the role of Weyl structures in complex and quaternionic geometry. This is further amplified on four dimensions, where Weyl geometry and complex geometry are intimately linked. NEWLINENEWLINENEWLINEThe interactions between four dimensional Weyl geometry and twistor theory are discussed, and a local formula for the Bach tensor on an Einstein-Weyl manifold is given. After discussing topological constraints given by an analogue of the Hitchin-Thorpe inequality, the classification of four dimensional Einstein-Weyl manifolds with symmetry group of dimension at least four is presented. After discussing the twistor theory of Einstein-Weyl 3-manifolds and the Jones-Tod construction, some special classes of three dimensional Einstein-Weyl geometries are presented and placed in an unified framework. The analogous classification result in dimension two is given.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00021].