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

SIGMA 12 (2016), 041, 16 pages      arXiv:1510.09028      https://doi.org/10.3842/SIGMA.2016.041
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

Are Orthogonal Separable Coordinates Really Classified?

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

Received October 30, 2015, in final form March 15, 2016; Published online April 26, 2016

Abstract
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.

Key words: separation of variables; Stäckel systems; Deligne-Mumford moduli spaces; Stasheff polytopes; operads.

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