2008/05/31 by Lionel Mason, Arman Taghavi‐Chabert, Arman Taghavi-Chabert · 42 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebra over a field #Black Holes and Theoretical Physics #Hermitian matrix #Mathematical analysis #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Pure mathematics #gr-qc #hep-th #math.DG #msc:32L25 #msc:53B35 #msc:83C57
paper · pdf · doi:10.1016/j.geomphys.2010.02.008
published in Journal of Geometry and Physics 60(6-8), 907-923 (Elsevier BV) · 28 pages - revised version. Typos and mistakes corrected. Statements of the results improved. Some remarks and references added
arxiv created 2009/02/13 · openalex publication_date 2010/02/27 · arxiv updated 2011/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We show that the Euclidean Kerr-NUT-(A)dS metric in 2m dimensions locally admits 2m 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 2m almost complex structures that turn out to be integrable as a consequence of the conformal Killing-Yano equations. In the complexification, these lead to 2m 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.