Equivalence problem for the orthogonal webs on the sphere

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.

32 pages

References in corpus (2)

• Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space
• Covariants,joint invariants and the problem of equivalence in the invariant theory of Killing tensors defined in pseudo-Riemannian spaces of constant curvature

Cited by in corpus (5)

• Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature
• Algebraic Integrability Conditions for Killing Tensors on Constant Sectional Curvature Manifolds
• Block-Separation of Variables: a Form of Partial Separation for Natural Hamiltonians
• Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional Hyperbolic and de Sitter spaces
• Classification of the Orthogonal Separable Webs for the Hamilton-Jacobi and Klein-Gordon Equations on 3-Dimensional Minkowski Space