Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 024, 4 pages      arXiv:1502.07516      https://doi.org/10.3842/SIGMA.2016.024
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

Nijenhuis Integrability for Killing Tensors

Konrad Schöbel
Mathematisches Institut, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany

Received October 30, 2015, in final form February 26, 2016; Published online March 07, 2016

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.

Key words: integrable systems; separation of variables; Killing tensors; Nijenhuis tensor; Haantjes tensor.

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