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

David M J. Calderbank; Henrik Pedersen

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 3, page 921-963
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Calderbank, David M J., and Pedersen, Henrik. "Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics." Annales de l'institut Fourier 50.3 (2000): 921-963. <http://eudml.org/doc/75444>.

@article{Calderbank2000,
abstract = {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.},
author = {Calderbank, David M J., Pedersen, Henrik},
journal = {Annales de l'institut Fourier},
keywords = {selfdual manifold; Hermitian surface; conformal symmetry; Einstein-Weyl 3-manifold; geodesic congruence; twistor theory; abelian monopoles},
language = {eng},
number = {3},
pages = {921-963},
publisher = {Association des Annales de l'Institut Fourier},
title = {Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics},
url = {http://eudml.org/doc/75444},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Calderbank, David M J.
AU - Pedersen, Henrik
TI - Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 3
SP - 921
EP - 963
AB - 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.
LA - eng
KW - selfdual manifold; Hermitian surface; conformal symmetry; Einstein-Weyl 3-manifold; geodesic congruence; twistor theory; abelian monopoles
UR - http://eudml.org/doc/75444
ER -

References

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