Einstein–Weyl geometry explained

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$$\backslash nabla\; g\; =\; \backslash alpha\backslash otimes\; g$$where \alpha is a one-form.

The curvature tensor is defined in the usual manner by$$R(X,Y)Z\; =\; (\backslash nabla\_X\backslash nabla\_Y\; -\; \backslash nabla\_Y\backslash nabla\_X\; -\; \backslash nabla\_)Z,$$and the Ricci curvature is$$Rc(Y,Z)\; =\; \backslash operatorname(X\backslash mapsto\; R(X,Y)Z).$$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:$$Rc(X,Y)\; +\; Rc(Y,X)\; =\; f\backslash ,g(X,Y).$$

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.

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