Structural equations for Killing tensors of order two. II

Abstract

In a preceding paper, a new form of the structural equations for any Killing tensor of order two were derived; these equations constitute a system analogous to the Killing vector equations ∇α Kβ = ωαβ = −ωβα and ∇γ ωαβ = Rαβγδ Kδ. The first integrability condition for the Killing tensor structural equations is now derived. Our structural equations and the integrability condition have forms which can readily be expressed in terms of a null tetrad to furnish a Killing tensor parallel of the Newman–Penrose equations; this is briefly described. The integrability condition implies the new result, for any given space–time, that the dimension of the set of second order Killing tensors attains its maximum possible value of 50 only if the space–time is of constant curvature. Potential applications of the structural equations are discussed.