The Faraday 2-form in Einstein-Weyl geometry

Abstract

On a conformal manifold, a compatible torsion free connection $D$ need not be the Levi-Civita connection of a compatible Riemannian metric. The local obstruction is a real $2$-form $F^D$, the Faraday curvature. It is shown that, except in four dimensions, $F^D$ necessarily vanishes if it is divergence free. In four dimensions another differential operator may be applied to $F^D$ to show that an Einstein-Weyl $4$-manifold with selfdual Weyl curvature also has selfdual Faraday curvature and so is either Einstein or locally hypercomplex. More generally, the Bach tensor and the scalar curvature are shown to control the selfduality of $F^D$. Finally, the constancy of the sign of the scalar curvature on compact Einstein-Weyl $4$-manifolds [24] is generalised to higher dimensions. The scalar curvature need not have constant sign in dimensions two and three.