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 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.