paper

Killing-Yano tensors and multi-hermitian structures

arXiv:0805.3756 · doi:10.1016/j.geomphys.2010.02.008

Abstract

We show that the Euclidean Kerr-NUT-(A)dS metric in dimensions locally admits hermitian complex structures. These are derived from the existence of a non-degenerate closed conformal Killing-Yano tensor with distinct eigenvalues. More generally, a conformal Killing-Yano tensor, provided its exterior derivative satisfies a certain condition, algebraically determines almost complex structures that turn out to be integrable as a consequence of the conformal Killing-Yano equations. In the complexification, these lead to maximal isotropic foliations of the manifold and, in Lorentz signature, these lead to two congruences of null geodesics. These are not shear-free, but satisfy a weaker condition that also generalizes the shear-free condition from 4-dimensions to higher-dimensions. In odd dimensions, a conformal Killing-Yano tensor leads to similar integrable distributions in the complexification. We show that the recently discovered 5-dimensional solution of Lu, Mei and Pope also admits such integrable distributions, although this does not quite fit into the story as the obvious associated two-form is not conformal Killing-Yano. We give conditions on the Weyl curvature tensor imposed by the existence of a non-degenerate conformal Killing-Yano tensor; these give an appropriate generalization of the type D condition on a Weyl tensor from four dimensions.

28 pages - revised version. Typos and mistakes corrected. Statements of the results improved. Some remarks and references added

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