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 [math]\displaystyle{ M }[/math] is a manifold with a conformal metric [math]\displaystyle{ [g] }[/math], then a Weyl connection is by definition a torsion-free affine connection [math]\displaystyle{ \nabla }[/math] such that [math]\displaystyle{ \nabla g = \alpha\otimes g }[/math] where [math]\displaystyle{ \alpha }[/math] is a one-form.

The curvature tensor is defined in the usual manner by [math]\displaystyle{ R(X,Y)Z = (\nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X,Y]})Z, }[/math] and the Ricci curvature is [math]\displaystyle{ Rc(Y,Z) = \operatorname{tr}(X\mapsto R(X,Y)Z). }[/math] The Ricci curvature for a Weyl connection may fail to be symmetric (its skew part is essentially the exterior derivative of [math]\displaystyle{ \alpha }[/math].)

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: [math]\displaystyle{ Rc(X,Y) + Rc(Y,X) = f\,g(X,Y). }[/math]

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 .