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

Calderbank, David M J.; Pedersen, Henrik

Annales de l'Institut Fourier, Volume 50 (2000) no. 3, pp. 921-963.

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 Kähler 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.

Nous considérons la correspondance de Jones et Tod entre variétés conformes autoduales admettant un champ de vecteurs conforme et les monopoles abéliens sur les variétés de Weyl-Einstein de dimension 3, et nous montrons que les structures complexes invariantes correspondent aux congruences géodésiques sans distorsion. Comme les variétés de Weyl-Einstein tri-dimensionnelles admettent de nombreuses congruences de ce type, cette correspondance offre un mode de construction général de géométries autoduales, qui inclut les constructions bien connues des métriques kählériennes à courbure scalaire nulle et des structures hypercomplexes avec symétrie. Nous montrons également qu’en présence d’une telle congruence l’équation de Weyl-Einstein équivaut à une paire couplée d’équations de monopoles que nous résolvons dans un cas particulier. À partir de ces nouveaux exemples, appelés “espaces de Weyl-Einstein à symétrie géodésique”, nous construisons des structures hypercomplexes admettant deux champs de vecteurs tri-holomorphes commutant entre eux.

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