Are Orthogonal Separable Coordinates Really Classified?

TL;DR

This paper introduces an algebraic geometric framework for classifying orthogonal separable coordinates on (pseudo-)Riemannian manifolds, revealing new structural insights and posing open problems.

Contribution

It establishes that the set of orthogonal separable coordinates forms a projective variety with a group action, enabling a novel algebraic geometric classification approach.

Findings

The set of orthogonal separable coordinates has a natural projective variety structure.

This approach uncovers unexpected structures in known classifications.

The paper proposes new problems in the algebraic geometric study of coordinate systems.

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.