Nijenhuis Integrability for Killing Tensors

Abstract

Abstract. The fundamental tool in the classification of orthogonal coordinate systems in which the Hamilton-Jacobi and other prominent equations can be solved by a separation of variables are second order Killing tensors which satisfy the Nijenhuis integrability conditions. The latter are a system of three non-linear partial differential equations. We give a simple and completely algebraic proof that for a Killing tensor the third and most complicated of these equations is redundant. This considerably simplifies the classification of orthogonal separation coordinates on arbitrary (pseudo-)Riemannian manifolds. It is a natural problem to classify all coordinate systems in which a given partial differential equation can be solved by a separation of variables -the so called separation coordinates. For fundamental equations such as the Hamilton-Jacobi and the Schrödinger equation the theory of separation of variables is built on a characterisation of orthogonal separation coordinates by second order Killing tensors, i.e., solutions of the Killing equationwhich are integrable in the sense that they have simple eigenvalues and the orthogonal complements of each eigenvector field form an integrable distribution. The relation between orthogonal separation coordinates and integrable Killing tensors was observed in 1891 by Paul Stäckel in his Habilitation thesis [10] and then used by Luther P. Eisenhart in 1934 to classify orthogonal separation coordinates on 3-dimensional Euclidean space and on the 3-dimensional sphere [1]. His results were generalised to arbitrary spaces of constant curvature of any dimension by Kalnins and Miller in 1986 [3, 4]. The property of an endomorphism field to be integrable in the above sense has been cast into the form of a system of three non-linear partial differential equations by Nijenhuis in 1950 [5]. Explicitly, an endomorphism K with simple eigenvalues is integrable if and only if it satisfies the Nijenhuis integrability conditionswhere the square brackets stand for antisymmetrisation and N denotes the Nijenhuis torsion of K, defined by