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 is a manifold with a conformal metric, then a Weyl connection is by definition a torsion-free affine connection such that
where is a one-form.
The curvature tensor is defined in the usual manner by
and the Ricci curvature is
The Ricci curvature for a Weyl connection may fail to be symmetric
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:
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 is Einstein–Weyl.