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

@article{AIF_2000__50_3_921_0,
     author = {Calderbank, David M J. and Pedersen, Henrik},
     title = {Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics},
     journal = {Annales de l'Institut Fourier},
     pages = {921--963},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {3},
     year = {2000},
     doi = {10.5802/aif.1779},
     mrnumber = {2001h:53058},
     zbl = {0970.53027},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2000__50_3_921_0/}
}
Calderbank, David M J.; Pedersen, Henrik. Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics. Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 921-963. doi : 10.5802/aif.1779. https://aif.centre-mersenne.org/item/AIF_2000__50_3_921_0/

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