Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature.
We study the Jones and Tod correspondence between selfdual conformal -manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl -manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kähler metrics and hypercomplex structures with symmetry. We also show that in the presence...
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