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We review the theory of orthogonal separation of variables onpseudo-Riemannian manifolds of constant non-zero curvature via concirculartensors and warped products. We then apply this theory simultaneously to boththe three-dimensional Hyperbolic and de Sitter spaces, obtaining an invariantclassification of the thirty-four orthogonal separable webs on each space,modulo action of the respective isometry groups. The inequivalent coordinatecharts adapted to each web are also determined and listed. The results obtainedfor Hyperbolic 3-space agree with those in the literature, while the resultsfor de Sitter 3-space appear to be new.

journal of mathematical physics

Classification of the orthogonal separable webs for the Hamilton-Jacobi  and Laplace-Beltrami equations on 3-dimensional Hyperbolic and de Sitter  spaces