Einstein-Weyl structures from Hyper-Kähler metrics with conformal Killing vectors

Details Einstein-Weyl structures from Hyper-Kähler metrics with conformal Killing vectors Einstein-Weyl structures from Hyper-Kähler metrics with conformal Killing vectors

1999-07-23

arxiv.org/abs/math/9907146v1

Differ.Geom.Appl. 14 (2001) 39-55

Mathematics

Differential Geometry

14 pages

Scientific paper

We consider four (real or complex) dimensional hyper-K\"ahler metrics with a conformal symmetry K. The three-dimensional space of orbits of K is shown to have an Einstein-Weyl structure which admits a shear-free geodesics congruence for which the twist is a constant multiple of the divergence. In this case the Einstein-Weyl equations reduce down to a single second order PDE for one function. The Lax representation, Lie point symmetries, hidden symmetries and the recursion operator associated with this PDE are found, and some group invariant solutions are considered.

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