Structure of the Riemann tensor in higher-dimensional Kerr-NUT-(A)dS spaces
Abstract
We study the algebraic and differential structure of the Riemann curvature tensor of the higher-dimensional Kerr-NUT-(A)dS spaces, motivated by the eventual goal of finding an IDEAL (Intrinsic, Deductive, Explicit and ALgorithmic) characterization of this family of metrics. The special geometry of these spaces is governed by the principal tensor hab, a non-degenerate closed conformal Killing-Yano $2$-form, whose existence singles out the Kerr-NUT-(A)dS family; we therefore focus on the algebraic relationship between the Riemann curvature Rabcd and hab. In our investigations we encountered an obstruction, which results in a no-go theorem: no non-trivial $2$-form, including hab itself, can be constructed covariantly from the undifferentiated Riemann tensor alone. Despite that, we do characterize the family of Riemann-symmetric tensors that could be the curvature of a Kerr-NUT-(A)dS metric by an algebraic and a differential condition, stemming from the integrability of the defining equation of hab and the second Bianchi identity. Our calculations apply in all higher dimensions, which becomes feasible by a judicious application of representation theoretic techniques related to a semi-direct product group U(1)nrtimes Sn that stabilizes hab.
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BibTeX
@misc{khavkine2026structure,
title = {{Structure of the Riemann tensor in higher-dimensional Kerr-NUT-(A)dS spaces}},
author = {Igor Khavkine and David Matejov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30448},
eprint = {2608.30448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30448},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Structure of the Riemann tensor in higher-dimensional Kerr-NUT-(A)dS spaces
AU - Igor Khavkine
AU - David Matejov
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30448
UR - https://arxiv.org/abs/2608.30448
AB - We study the algebraic and differential structure of the Riemann curvature tensor of the higher-dimensional Kerr-NUT-(A)dS spaces, motivated by the eventual goal of finding an IDEAL (Intrinsic, Deductive, Explicit and ALgorithmic) characterization of this family of metrics. The special geometry of these spaces is governed by the principal tensor h_ab, a non-degenerate closed conformal Killing-Yano $2$-form, whose existence singles out the Kerr-NUT-(A)dS family; we therefore focus on the algebraic relationship between the Riemann curvature R_abcd and h_ab. In our investigations we encountered an obstruction, which results in a no-go theorem: no non-trivial $2$-form, including h_ab itself, can be constructed covariantly from the undifferentiated Riemann tensor alone. Despite that, we do characterize the family of Riemann-symmetric tensors that could be the curvature of a Kerr-NUT-(A)dS metric by an algebraic and a differential condition, stemming from the integrability of the defining equation of h_ab and the second Bianchi identity. Our calculations apply in all higher dimensions, which becomes feasible by a judicious application of representation theoretic techniques related to a semi-direct product group U(1)^nrtimes S_n that stabilizes h_ab.
ER -