On orthogonal curvilinear coordinate systems in constant curvature spaces

TL;DR

This paper introduces a modified method for constructing orthogonal curvilinear coordinate systems in constant curvature spaces, extending Krichever's approach to non-Euclidean geometries like the sphere and hyperbolic plane.

Contribution

It presents a novel modification of Krichever's method tailored for constant curvature spaces, with explicit examples on the sphere and hyperbolic plane.

Findings

Constructed orthogonal coordinate systems on the sphere and hyperbolic plane.

Demonstrated the method's applicability when the spectral curve is reducible.

Extended the theory of orthogonal coordinate systems to non-Euclidean geometries.

Abstract

We describe a method for constructing $n$-orthogonal coordinate systems in constant curvature spaces. The construction proposed is a modification of Krichever's method for producing orthogonal curvilinear coordinate systems in the $n$-dimensional Euclidean space. To demonstrate how this method works, we construct examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.