Konrad Schöbel

Algebraic integrability conditions for Killing tensors on constant sectional curvature manifolds

We use an isomorphism between the space of valence two Killing tensors on an n-dimensional constant sectional curvature manifold and the irreducible GL(n + 1)-representation space of algebraic curvature tensors [MMS04] in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integrability conditions for the corresponding algebraic curvature tensor, resulting in two simple algebraic equations of degree two and three. As a first application of this we construct a new family of integrable Killing tensors.

The Variety of Integrable Killing Tensors on the 3-Sphere

Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere S3 and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all Stäckel systems in these varieties. This allows us to recover the known list of separation coordinates on S3 in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron K4.

ARE ORTHOGONAL SEPARABLE COORDINATES REALLY CLASSIFIED

We prove that the set of orthogonal separable coordinates on an arbitrary (pseudo-)Riemannian manifold carries a natural structure of a pro- jective variety, equipped with an action of the isometry group. This leads us to propose a new, algebraic geometric approach to the classification of orthogo- nal separable coordinates by studying the structure of this variety. We give an example where this approach reveals unexpected structure in the well known classification and pose a number of problems arising naturally in this context.We prove that the set of orthogonal separable coordinates on an arbitrary (pseudo-)Riemannian manifold carries a natural structure of a projective variety, equipped with an action of the isometry group. This leads us to propose a new, algebraic geometric approach to the classification of orthogonal separable coordinates by studying the structure of this variety. We give an example where this approach reveals unexpected structure in the well known classification and pose a number of problems arising naturally in this context.

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.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.

An algebraic geometric approach to separation of variables

The Foundation: The Algebraic Integrability Conditions.- The Proof of Concept: A Complete Solution for the 3-Sphere.- The Generalisation: A Solution for Spheres of Arbitrary Dimension.- The Perspectives: Applications and Generalisations.

The perspectives: applications and generalisations

We have proposed a new, purely algebraic geometric approach to the problem of separation of variables and we have demonstrated that this approach is viable by successfully carrying it out for the simplest non-trivial family of examples – that of spheres. In particular, we elucidated the natural algebro-geometric structure of the parameter space classifying equivalence classes of separation coordinates, which for a long time had only been known as a mere set, and gave a precise description of its topology. In this way we discovered that the theory of Deligne-Mumford-Knudsen moduli spaces and Stasheff polytopes provides the right framework for the classification and construction of all orthogonal separation coordinates on spheres.

The foundation: the algebraic integrability conditions

In this chapter we translate the Nijenhuis integrability conditions for a Killing tensor on a constant curvature manifold into algebraic conditions on the corresponding algebraic curvature tensors. To this end, we substitute (0.7) into (0.2) and both into (0.3) and then use the representation theory for general linear groups to get rid of the dependence on the base point in the manifold.

The generalisation: a solution for spheres of arbitrary dimension

In Definition 0.1, a Killing tensor is a symmetric bilinear form K αβ on the manifold M. In what follows we will interpret it in two other ways, each of which gives rise to a Lie bracket and hence to a Lie algebra generated by Killing tensors. On one hand, we can use the metric to identify the symmetric bilinear form KK αβ with a symmetric endomorphism \({{K}^{\alpha }}_{\beta}\).

The proof of concept: a complete solution for the 3-dimensional sphere

Given a scalar product g on V, we can raise and lower indices. The symmetries (0.6a) and (0.6b) then allow us to regard an algebraic curvature tensor \({{R}_{a\text{1}\ a\text{2}\ b\text{1}\ b\text{2}}}\) on V as a symmetric endomorphism \({{R}^{a\text{1}\ b\text{1}}}_{a\text{2}\ b\text{2}}\) on the space ʌ2V of 2-forms on V . Since we will frequently change between both interpretations, we denote endomorphisms by the same letter in boldface.