Separable components for three-dimensional complex riemannian spaces

TL;DR: In this paper, the geometric theory of separation of variables for time-independent Hamilton-Jacobi equation is extended to include the case of complex eigenvalues of a Killing tensor on pseudo-Riemannian manifolds.

TL;DR: Benenti et al. as discussed by the authors extended the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti.

TL;DR: In this paper, the authors develop the theory of R•separation for the Helmholtz equation on a pseudo-Riemannian manifold (including the possibility of null coordinates) and show that it, and not ordinary variable separation, is the natural analogy of additive separation for the Hamilton-Jacobi equation.

Abstract: We study concircular tensors in spaces of constant curvature and then apply the results obtained to the problem of the orthogonal separation of the Hamilton-Jacobi equation on these spaces. Any coordinates which separate the geodesic Hamilton-Jacobi equation are called separable. Specifically for spaces of constant curvature, we obtain canonical forms of concircular tensors modulo the action of the isometry group, we obtain the separable coordinates induced by irreducible concircular tensors, and we obtain warped products adapted to reducible concircular tensors. Using these results, we show how to enumerate the isometrically inequivalent orthogonal separable coordinates, construct the transformation from separable to Cartesian coordinates, and execute the Benenti-Eisenhart-Kalnins-Miller (BEKM) separation algorithm for separating natural Hamilton-Jacobi equations.

TL;DR: In this article, the authors investigated the relationship between the Weyl tensor and separable systems for the geodesic Hamilton-Jacobi equation in Riemannian and Lorentzian manifolds.

TL;DR: In this article, a detailed group theoretical study of the problem of separation of variables for the characteristic equation of the wave equation in one time and two space dimensions is presented, which includes several of the better known two-body problems, including the harmonic oscillator and Kepler problems.

TL;DR: In this paper, the authors classify and discuss the possible nonorthogonal coordinate systems which lead to R-separable solutions of the wave equation, each system is associated with a pair of commuting operators in the symmetry algebra so(3,2) of this equation, one operator first order and the other second order.