Nijenhuis Integrability for Killing Tensors

TL;DR

This paper simplifies the classification of orthogonal coordinate systems in (pseudo-)Riemannian manifolds by proving that one of the Nijenhuis integrability conditions for Killing tensors is redundant, streamlining the separation of variables approach.

Contribution

It provides an algebraic proof showing the redundancy of the most complex Nijenhuis integrability condition for Killing tensors, simplifying their classification.

Findings

Proves the third Nijenhuis integrability condition is redundant for Killing tensors.

Simplifies the process of classifying orthogonal separation coordinates.

Facilitates solving Hamilton-Jacobi equations on (pseudo-)Riemannian manifolds.

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.