We explicitly calculate the Riemannian curvature of D-dimensional metrics recently discussed by Chen, Lü and Pope. We find that they can be concisely written by using a single function. The Einstein condition which corresponds to the Kerr-NUT-de Sitter metric is clarified for all dimensions. It is shown that the metrics are of type D.
In this paper, we consider the symmetries of the Dirac operator derived from a connection with skew-symmetric torsion, ∇ T . We find that the generalized conformal Killing-Yano tensors give rise to symmetry operators of the massless Dirac equation, provided an explicitly given anomaly vanishes. We show that this gives rise to symmetries of the Dirac operator in the case of strong Kähler with torsion (KT) and strong hyper-Kähler with torsion (HKT) manifolds.
We elaborate on basic properties of generalized Killing-Yano tensors which
naturally extend Killing-Yano symmetry in the presence of skew-symmetric
torsion. In particular, we discuss their relationship to Killing tensors and
the separability of various field equations. We further demonstrate that the
Kerr-Sen black hole spacetime of heterotic string theory, as well as its
generalization to all dimensions, possesses a generalized closed conformal
Killing-Yano 2-form with respect to a torsion identified with the 3-form
occuring naturally in the theory. Such a 2-form is responsible for complete
integrability of geodesic motion as well as for separability of the scalar and
Dirac equations in these spacetimes.Comment: 33 pages, no figure
Assuming the existence of a single rank-2 closed conformal Killing-Yano tensor with a certain symmetry we show that there exist mutually commuting rank-2 Killing tensors and Killing vectors. We also discuss the condition of separation of variables for the geodesic Hamilton-Jacobi equations.Recently, it has been shown that geodesic motion in the Kerr-NUT de Sitter spacetime is integrable for all dimensions [1,2,3,4,5,6]. Indeed, the constants of motion that are in involution can be explicitly constructed from a rank-2 closed conformal Killing-Yano (CKY) tensor. In this paper we consider the problem of integrability of the geodesic equation in a more general situation. We assume the existence of a single rank-2 closed CKY tensor with a certain symmetry for D-dimensional spacetime M with a metric g. It turns out that such a spacetime admits mutually commuting k rank-2 Killing tensors and k Killing vectors. Here we put D = 2k for even D , and D = 2k − 1 for odd D. Although the existence of the commuting Killing tensors was shown in [5,6], we reproduce it more directly. We also discuss the condition of separation of variables for the geodesic Hamilton-Jacobi equations using the result given by and Kalnins-Miller [8] (see also [9]).
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