Killing–Yano tensors and multi-Hermitian structures

Abstract

We show that the Euclidean Kerr–NUT-(A)dS metric in 2 m dimensions locally admits 2 m 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 2 m almost complex structures that turn out to be integrable as a consequence of the conformal Killing–Yano equations. In the complexification, these lead to 2 m 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 generalises the shear-free condition from four 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 five-dimensional solution of Lü, 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 generalisation of the type D condition on a Weyl tensor from four dimensions.