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 $h_{ab}$, 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 $U(1)^n\rtimes S_n$ that stabilizes $h_{ab}$.