Nijenhuis Integrability for Killing Tensors

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 equation which 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 [Stä91] and then used by Luther P. Eisenhart in 1934 to classify orthogonal separation coordinates in 3-dimensional Euclidean space and on the 3-dimensional sphere [Eis34]. His results were generalised to arbitrary spaces of constant curvature of any dimension by Kalnins & Miller in 1986 [KM86, Kal86]. 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 [Nij51]. Explicitly, an endomorphism K with simple eigenvalues is integrable if and only if it satisfies the Nijenhuis integrability conditions where the square brackets stand for antisymmetrisation and N denotes the Nijenhuis torsion of K, defined by and given in local coordinates by Of course, these equations have not been known to Stäckel or Eisenhart. Neither did they play any role in the complete classification of Kalnins & Miller. However, they reveal that the classification of orthogonal separation coordinates is actually an algebraic geometric problem. Indeed, the Nijenhuis integrability conditions (2) are algebraic in K and its derivatives and hence constitute a set of homogeneous algebraic equations on the space of second order Killing tensors. Note that, as the space of solutions to the overdetermined linear equation (1), this space is a finite dimensional vector space. Consequently, the set of Killing tensors satisfying the Nijenhuis integrability conditions is a projective variety. Moreover, this variety comes along with a natural group action, namely the isometry group of the manifold.
To be more precise, orthogonal separation coordinates are in one-to-one correspondence with so-called Stäckel systems, i.e. n-dimensional spaces of integrable Killing tensors which mutually commute in the algebraic sense. This leads to the following remarkable observation: For any (pseudo-)Riemannian manifold the set of orthogonal separation coordinates carries a very rich structure: It is a projective variety isomorphic to a subvariety in the Grassmannian of ndimensional subspaces in the space of Killing tensors, equipped with a natural action of the isometry group. To the best of our knowledge, this point has never been made in the literature and the structure of these varieties had never been elucidated. The reason is probably that a general solution of the Equations (2) was deemed intractable [HMS05].
Recently it has now been possible to rewrite the Nijenhuis integrability conditions explicitly as algebraic equations for constant curvature manifolds [Sch12] and to solve them in the least non-trivial case, namely for the sphere of dimension three [Sch14]. The outcome, a detailed algebraic geometric description of the variety of integrable Killing tensors as well as the variety of Stäckel systems, has lead to a surprising connection between separation coordinates on spheres on one hand and algebraic curves on the other. More precisely, the set of orthogonal separation coordinates modulo isometries on the n-dimensional sphere is naturally parametrised by the real Deligne-Mumford moduli spaceM 0,n+2 (R) of stable algebraic curves of genus 0 with n + 2 marked points [SV15]. As a consequence, the well known classification of Kalnins & Miller can be reinterpreted in terms of the geometry and combinatorics of Stasheff polytopes. This also revealed a hitherto unknown operad structure on equivalence classes of separation coordinates on spheres.
To date, comparable results are unknown for manifolds other than spheres.
An important step in the explicit solution of the Nijenhuis equations has been the proof that the third of the Nijenhuis conditions is redundant when applied to a Killing tensor. According to a footnote in [HMS05], this had previously been proven by Steve Czapor for Euclidean space in dimension three using Gröbner bases. The author extended this to arbitrary constant curvature manifolds [Sch12].
The purpose of the present paper is to give a simple proof that this result holds in general. This will considerably simplify the classification of orthogonal separation coordinates on arbitrary manifolds.
Theorem. For a (diagonalisable) Killing tensor on an arbitrary (pseudo-)Riemannian manifold the third of the Nijenhuis equations (2) is redundant.
Remark. One can show that the first two Nijenhuis equations are independent [Sch14].
Remark. Instead of the Nijenhuis conditions (2), the vanishing of the Haantjes tensor is often used as a criterion for integrability. Being of order four in K, this condition is of the same complexity as the third of the Nijenhuis equations, while the first two are only of order two and three. Our result therefore shows that for Killing tensors the Nijenhuis conditions are better suited than the Haantjes tensor.

Proof.
A Killing tensor is symmetric by definition. Hence at any point of the manifold we can choose an orthonormal basis of the tangent space in which the Killing tensor is diagonal, i.e. K α β = λ α δ α β (no sum). In this basis, the Nijenhuis torsion (3) reads , where we have denoted the covariant derivative by a semicolon. Substituted into the Nijenhuis integrability conditions (2) we get where the Young operator α β γ denotes a complete antisymmetrisation in the indices α, β and γ. Using that K αβ;γ is symmetric in α, β and that a complete antisymmetrisation over α, β, γ can be split into an antisymmetrisation in α, β and a subsequent sum over the cyclic permutations of α, β, γ, we can rewrite the preceding equations as where "+cyclic" stands for a summation over the cyclic permutations of α, β, γ. These equations are one by one equivalent to the Nijenhuis integrability conditions (2). In the same way we can write the Killing equation (1) as 0 = K αβ;γ + cyclic.