Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 021, 21 pages      nlin.SI/0612042      https://doi.org/10.3842/SIGMA.2007.021
Contribution to the Vadim Kuznetsov Memorial Issue

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

Claudia Chanu and Giovanni Rastelli
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Received November 02, 2006, in final form January 16, 2007; Published online February 06, 2007

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 ≤ 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.

Key words: variable separation; Hamilton-Jacobi equation; Killing tensors; (pseudo-)Riemannian manifolds.

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