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Abstract. In a recent comment Sneddon discussed the set of fourteen algebraic invariants of the Riemann curvature tensor in four dimensions. The focus was rectification of an error (in the form of lack of independence) in an earlier construction and the presentation of a corrected set suitable for application. Several authors who have worked on this problem were mentioned. The comment, however, did not mention the work of Narlikar and Karmarkar who presented a set of invariants well before the earliest work cited in the comment. The original publication by Narlikar and Karmarkar may not be readily available so we list and make a few comments on their set.Recently, Sneddon (1986) discussed briefly the form in which several authors have presented the fourteen independent algebraic invariants (Haskins 1902)$ of the Riemann curvature tensor in four dimensions. Among the authors mentioned were GChCniau and Debever (1956a, b, c) with implied priority of publication. This accords with common usage in which these scalars are almost universally referred to as the 'GChCniau-Debever' scalars. The fact is that Narlikar and Karmarkar (1948) published such a set substantially earlier.

Abstract. We review the theory of orthogonal separation of variables of the HamiltonJacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors. This results in the construction of a special class of separable coordinates known as Kalnins-Eisenhart-Miller coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins-Eisenhart-Miller coordinates which separate a given natural Hamilton-Jacobi equation. As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability of the three dimensional Calogero-Moser and Morosi-Tondo systems.

Abstract. The problem of the invariant classification of the orthogonal coordinate webs defined in Euclidean space is solved within the framework of Felix Klein's Erlangen Program. The results are applied to the problem of integrability of the Calogero-Moser model.

We study Killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the Hamilton-Jacobi equation. This work is motivated primarily by the case of spaces of constant curvature where warped products are abundant. We first characterize Killing tensors which have a natural algebraic decomposition in warped products. We then apply this result to show how one can obtain the Killing-Stäckel space (KS-space) for separable coordinate systems decomposable in warped products. This result in combination with Benenti's theory for constructing the KS-space of certain special separable coordinates can be used to obtain the KS-space for all orthogonal separable coordinates found by Kalnins and Miller in Riemannian spaces of constant curvature. Next we characterize when a natural Hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by Benenti can be reduced. Finally, we use this characterization and concircular tensors (a special type of torsionless conformal Killing tensor) to develop a general algorithm to determine when a natural Hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature. C 2014 AIP Publishing LLC. [http://dx.