Structural equations for Killing tensors of order two. I

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Abstract
For any Killing tensor Kαβ of order two, a system of linear homogeneous first−order differential equations of the form (FA);α = JB(ΓαABFB) is derived. F1, F2, ... are the components of the tensors Kαβ, Lαβγ = 2Kγ[β;α], and Mαβγδ = (1/2)(Lαβ[γ;δ] + Lγδ[α;β]). The coefficients ΓαAB are linear expressions in the Riemann tensor and its covariant derivative. These equations are analogous to those satisfied by a Killing vector Kα and the Killing bivector ωαβ = Kβ;α, with Lαβγ and Mαβγδ playing roles analogous to ωαβ. The tensor Lαβγ has the symmetries Lαβγ = − Lβαγ and L[αβγ] = 0, and Mαβγδ has the symmetries of the Riemann tensor. Several relations similar to those satisfied by covariant derivatives of Killing vectors are derived. Perspectives for further work are briefly discussed with the idea of using the equations to investigate space−times which admit Killing tensors of order two.

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