Einstein–Weyl geometry

An Einstein–Weyl geometry is a smooth conformal manifold, together with a compatible Weyl connection that satisfies an appropriate version of the Einstein vacuum equations, first considered by Cartan (1943) and named after Albert Einstein and Hermann Weyl. Specifically, if ${\displaystyle M}$ is a manifold with a conformal metric ${\displaystyle [g]}$, then a Weyl connection is by definition a torsion-free affine connection ${\displaystyle \nabla }$ such that

$${\displaystyle \nabla g=\alpha \otimes g}$$

where ${\displaystyle \alpha }$ is a one-form.

The curvature tensor is defined in the usual manner by

$${\displaystyle R(X,Y)Z=(\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-\nabla _{[X,Y]})Z,}$$

and the Ricci curvature is

$${\displaystyle Rc(Y,Z)=\operatorname {tr} (X\mapsto R(X,Y)Z).}$$

The Ricci curvature for a Weyl connection may fail to be symmetric (its skew part is essentially the exterior derivative of ${\displaystyle \alpha }$.)

An Einstein–Weyl geometry is then one for which the symmetric part of the Ricci curvature is a multiple of the metric, by an arbitrary smooth function:[1]

$${\displaystyle Rc(X,Y)+Rc(Y,X)=f\,g(X,Y).}$$

The global analysis of Einstein–Weyl geometries is generally more subtle than that of conformal geometry. For example, the Einstein cylinder is a global static conformal structure, but only one period of the cylinder (with the conformal structure of the de Sitter metric) is Einstein–Weyl.