Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds

Abstract

Abstract. Given a n-dimensional Riemannian manifold of arbitrary signature, we illustrate an algebraic method for constructing the coordinate webs separating the geodesic HamiltonJacobi equation by means of the eigenvalues of m ≤ n Killing two-tensors. Moreover, from the analysis of the eigenvalues, information about the possible symmetries of the web foliations arises. Three cases are examined: the orthogonal separation, the general separation, including non-orthogonal and isotropic coordinates, and the conformal separation, where Killing tensors are replaced by conformal Killing tensors. The method is illustrated by several examples and an application to the L-systems is provided.

Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations

“…Again, we shall not give any additional details on these topics at this stage since conformal deformations and R-separability will be studied in the remainder of this paper in the more general setting of Painlevé metrics. We refer to [2,3,4,8,9,25,26,27] for lucid accounts of the key results on the separability and R-separability properties of Stäckel metrics, their connection to Killing tensors, quadratic first integrals of the geodesic flow and symmetry operators for the Laplace-Beltrami operator. We also refer to [1,36] for recent surveys on separability on Riemannian manifolds and to [7] for a penetrating analysis of the relations between quadratic first integrals of the geodesic flow, symmetry operators and conserved currents, in the general setting of Riemannian or pseudo-Riemannian manifolds.…”

Hamilton-Jacobi Equation

“…In this paper we consider here separable systems without ignorable coordinates (asymmetric systems) only. It is easy to see from [7], [2] that the functions (8) behave in the same way also in the complex case, being the proof of this property based on the Killing-Eisenhart equations only (see the previous section). Therefore, the functions (8) depend on only one of the separable variables, real or complex, and can be used to define exactly this variable.…”

“…The papers [7], [2] show how the real eigenvalues of any basis of a Killing-Stäckel algebra can be combined together to build functions depending each on one only of the associated non-ignorable separable coordinated and that, in the case when all that functions are constants or undefined, the corresponding separable coordinate is ignorable. In the three-dimensional case, if (λ i 1 ) and (λ i 2 ) are the eigenvalues of a basis of a KS-algebra, the relevant functions depending on the single separable coordinate z h are…”

“…In the present paper we introduce complex separable variables on three-dimensional Minkowski space through a meaningful example and by making use of results exposed in [7] and [2] where separable coordinates are directly related to the eigenvalues of a basis of their KS algebra. These variables fit naturally with the standard real separable coordinate systems.…”

Complex variables for separation of the Hamilton-Jacobi equation on real pseudo-Riemannian manifolds

“…Remark 12 Given a characteristic Killing tensor K µν , or the associated Killing-Stäckel algebra, the functions z µ are given by integrating either the eigenforms or the eigenvectors of K µν or (as we will see) by the eigenvalues of a basis of the associated Killing-Stäckel algebra through the fundamental functions f bc a given in [5], generalized to the complex case.…”