Quadrics on Complex Riemannian Spaces of Constant Curvature, Separation of Variables and the Gaudin Magnet

TL;DR

This paper classifies orthogonal separation of variables in complex Riemannian spaces of constant curvature, linking them to the hyperbolic Gaudin magnet and using $L$-matrices for a comprehensive understanding.

Contribution

It extends the isomorphism between integrable systems and the hyperbolic Gaudin magnet to complex spaces and provides a complete classification of separable coordinate systems.

Findings

Classification of all orthogonal separable coordinate systems in complex Riemannian spaces.

Extension of the isomorphism to include complex spaces of constant curvature.

Use of limiting procedures to relate degenerate coordinate systems.

Abstract

We consider integrable systems that are connected with orthogonal separation of variables in complex Riemannian spaces of constant curvature. An isomorphism with the hyperbolic Gaudin magnet, previously pointed out by one of us, extends to coordinates of this type. The complete classification of these separable coordinate systems is provided by means of the corresponding $L$-matrices for the Gaudin magnet. The limiting procedures (or $\epsilon $ calculus) which relate various degenerate orthogonal coordinate systems play a crucial result in the classification of all such systems.