Equivalence problem for the orthogonal webs on the 3-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 threesphere.

Classification of the orthogonal separable webs for the Hamilton-Jacobi and Laplace-Beltrami equations on 3-dimensional hyperbolic and de Sitter spaces

“…We find that this new approach allows us en passant to simultaneously determine and classify the thirty-four orthogonal separable webs on 3-dimensional hyperbolic space H 3 , the Riemannian space of constant negative curvature, in agreement with the results obtained by Olevsky [10]. We therefore get a complete classification of separable webs on both dS 3 and H 3 , ultimately due to the fact that each space can be isometrically embedded in 4-dimensional Minkowski space E 4 1 . It will become evident that the procedure presented here generalizes and allows one to obtain a complete classification of the separable webs on dS n and H n simultaneously.…”

“…The problem for H 3 has been studied by other methods by different authors including Olevsky [10], Kalnins, Miller and Reid [8], Kalnins [7] and Adlam, McLenaghan and Smirnov [4]. For a review and comparison of these methods with those used in the present paper, see [11].…”

First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator

“…The surfaces η = const are flat tori spanned by the rotations ∂ ξ i which are Killing vectors of the manifold. These coordinates correspond to a cylindrical rotational separable system, they are associated with a Killing 2-tensor K which is described in appendix in [6] and which provides a quadratic first integral of the geodesics H 1 = 1 2 K ij p i p j . Therefore, the geodesic Hamiltonian G = 1 2 g ii p 2 i of S 3 admits the following four independent quadratic in the momenta first integrals G, H 1 , p 2 2 , p 2 3 and is Liouville integrable.…”

Block-Separation of Variables: a Form of Partial Separation for Natural Hamiltonians

“…An analogue result about characteristic Killing tensors of Stäckel systems is given in [10] making use of theorems due to Tonolo, Schouten and Nijenhuis (see Section 2). The main difference is due to the fact that, in that case, the eigenvalues of the tensors are simple.…”