Irreducible Killing Tensors From Third Rank Killing–yano Tensors

Abstract

We investigate higher rank Killing-Yano tensors showing that third rank Killing-Yano tensors are not always trivial objects being possible to construct irreducible Killing tensors from them. We give as an example the Kimura IIC metric were from two rank Killing-Yano tensors we obtain a reducible Killing tensor and from third rank Killing-Yano tensors we obtain three Killing tensors, one reducible and two irreducible. under some restrictions, pp-wave metrics and Siklos space-times admit non-generic supercharges [11]. On the other hand, there is a relation [12,13,14], called geometric duality, between spaces admitting irreducible Killing tensors of rank two and the spaces whose metrics are specified through those Killing tensors. Further generalizations of Killing tensors and their existence criteria were discussed [15,16,17,18]. Killing-Yano tensors and their corresponding Killing tensors have been studied extensively [19,20,21,22,23,24] in the related context of finding solutions of the Dirac-equation in non-trivial curved space-time. Moreover, in the context of generalized Dirac-type operators [25,26] the Killing-Yano tensors are indispensable tools.

Slice-reducible conformal Killing tensors, photon surfaces and shadows

“…In contrast to this, in spaces with totally umbilic foliation, such CKTs are always reducible. However, in the latter case slice-reducible CKTs with β α = 0 can also exist and it may turn out to be irreducible in the bulk [68]. We leave the analysis of this option as one of the directions for further researches.…”

Killing–Yano tensors of order n − 1

“…The simplest non-trivial case is = − p n 1. For instance, in [21] it was shown an example in which Killing-Yano tensors of order p = 3 in n = 4 dimensions lead to non-obvious conserved quantities. In spite of the simplicity, there are very few comments about Killing-Yano tensors of order − n 1 in the literature and the intent of the present article is to fill this gap.…”

Geometrical invariants, conserved currents and new symmetries in a covariant phase-space dynamics

“…are constants. The Kimura metric is the only one to imply in an irreducible second rank Killing tensor (nondegenerate) obtained by a contraction of third rank Killing Yano tensors [6]. This fact motivate us to consider this metric in our calculations.…”

“…Firstly, as a particular case, we consider the Kimura metric (for a recent use of this metric see [6]) and consider it in the equations (2.2) for n = 2 in order to find the Killing tensor compatible with the set of equations in (2.5).…”