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

Abstract

Given a $n$-dimensional Riemannian manifold of arbitrary signature, we illustrate an algebraic method for constructing the coordinate webs separating the geodesic Hamilton-Jacobi equation by means of the eigenvalues of $m\le n$ 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.

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (3)

• Invariant classification of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space
• Separable bi-Hamiltonian systems with quadratic in momenta first integrals
• Complex variables for separation of Hamilton-Jacobi equation on three-dimensional Minkowski space

Cited by in corpus (5)

• Complex variables for separation of Hamilton-Jacobi equation on real pseudo-Riemannian manifolds
• Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature
• Invariant classification of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space
• Complex variables for separation of Hamilton-Jacobi equation on three-dimensional Minkowski space
• Separation of Variables and Superintegrability on Riemannian Coverings