Orthogonal separation of variables on manifolds with constant curvature

Abstract

Coordinates which allow the integration by separation of variables of the geodesic Hamilton-Jacobi equation are called separable. Particular interest is placed on orthogonal separable coordinates. In this paper it is proved that on a Riemannian manifold with constant curvature and on a Lorentzian manifold with constant positive curvature every system of separable coordinates has an orthogonal equivalent, i.e. that in these manifolds the integration by separation of variables of the geodesic Hamilton-Jacobi equation always occurs in orthogonal coordinates. Proofs of this property concerning strictly-Riemannian manifolds of positive, negative and zero constant curvature (and also for conformally flat manifolds) were firstly given by Kalnins and Miller (1982-1986). The proof presented here is based on elementary properties of Killing vectors of an affine space and on a geometrical characterization of the equivalence classes of separable coordinates. © 1992.