Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

Details Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

1999-11-16

arxiv.org/abs/math/9911117v1

Ann. Inst. Fourier 50 (2000) 921-963

Mathematics

Differential Geometry

30 pages, 7 figures, to appear in Ann. Inst. Fourier. 50 (2000)

Scientific paper

We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-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 Kahler metrics and hypercomplex structures with symmetry. We also show that in the presence of such a congruence, the Einstein-Weyl equation is equivalent to a pair of coupled monopole equations, and we solve these equations in a special case. The new Einstein-Weyl spaces, which we call Einstein-Weyl ``with a geodesic symmetry'', give rise to hypercomplex structures with two commuting triholomorphic vector fields.

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