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 $M$ is a manifold with a conformal metric $[g]$, then a Weyl connection is by definition a torsion-free affine connection $\nabla$ such that

$$\nabla g=\alpha \otimes g$$

where $\alpha$ is a one-form.

The curvature tensor is defined in the usual manner by

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

and the Ricci curvature is

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

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

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:

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

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.

References

• Cartan, Élie (1943), "Sur une classe d'espaces de Weyl", Ann Sci École Norm Sup, 60 (3).
• Mason, Lionel; LeBrun, Claude (2009), "The Einstein–Weyl equations, scattering maps, and holomorphic disks", Math Res Lett, 16: 291–301.