Killing-Yano tensors and multi-hermitian structures

TL;DR

This paper demonstrates that certain higher-dimensional metrics admit multiple complex structures derived from conformal Killing-Yano tensors, revealing new geometric properties and integrable distributions in complexified and Lorentzian manifolds.

Contribution

It generalizes the existence of complex structures and integrable distributions from four dimensions to higher dimensions using conformal Killing-Yano tensors.

Findings

Euclidean Kerr-NUT-(A)dS metrics admit 2^m hermitian complex structures.

Conformal Killing-Yano tensors determine integrable almost complex structures.

Conditions on Weyl curvature generalize type D classification to higher dimensions.

Abstract

We show that the Euclidean Kerr-NUT-(A)dS metric in $2m$ 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 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.