Einstein metrics, hypercomplex structures and the Toda field equation

We obtain explicitly all solutions of the SU(∞) Toda field equation with the property that the associated Einstein–Weyl space admits a 2-sphere of divergence-free shear-free geodesic congruences. The solutions depend on an arbitrary holomorphic function and give rise to new hyperKähler and selfdual...

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Bibliographic Details
Published in:Differential geometry and its applications 2001-03, Vol.14 (2), p.199-208
Main Authors: Calderbank, David M.J., Tod, Paul
Format: Article
Language:English
Subjects:
Einstein metric
Einstein–Weyl geometry
hypercomplex structure
Toda equation
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Summary:We obtain explicitly all solutions of the SU(∞) Toda field equation with the property that the associated Einstein–Weyl space admits a 2-sphere of divergence-free shear-free geodesic congruences. The solutions depend on an arbitrary holomorphic function and give rise to new hyperKähler and selfdual Einstein metrics with one-dimensional isometry group. These metrics each admit a compatible hypercomplex structure with respect to which the symmetries are triholomorphic
ISSN:0926-2245
1872-6984
DOI:10.1016/S0926-2245(01)00037-7