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

Calderbank, David M J.; Pedersen, Henrik

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

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.

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.

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