Equivalence problem for the orthogonal webs on the sphere

TL;DR

This paper addresses the classification of orthogonally separable coordinate systems on the three-sphere by solving the equivalence problem under isometries, completing a classical project on separation of variables for the Hamilton-Jacobi equation.

Contribution

It provides a complete solution to the equivalence problem for orthogonal webs on the three-sphere, extending Olevsky's canonical forms to an equivalence classification.

Findings

Classification of orthogonal webs on the three-sphere achieved

Invariant properties of Killing tensors used for classification

Application to a natural Hamiltonian on the three-sphere demonstrated

Abstract

We solve the equivalence problem for the orthogonally separable webs on the three-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved the corresponding canonical forms problem. The solution to the equivalence problem together with the results by Olevsky forms a complete solution to the problem of orthogonal separation of variables to the Hamilton-Jacobi equation defined on the three-sphere via orthogonal separation of variables. It is based on invariant properties of the characteristic Killing two-tensors in addition to properties of the corresponding algebraic curvature tensor and the associated Ricci tensor. The result is illustrated by a non-trivial application to a natural Hamiltonian defined on the three-sphere.