Bertrand spacetimes as Kepler/oscillator potentials

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Perlick's classification of (3 + 1)-dimensional spherically symmetric and static spacetimes for which the classical Bertrand theorem holds (Perlick V 1992 Class. Quantum Grav. 9 1009) is revisited. For any Bertrand spacetime the term V(r) is proven to be either the intrinsic Kepler–Coulomb or the harmonic oscillator potential on its associated Riemannian 3-manifold (M, g). Among the latter 3-spaces (M, g) we explicitly identify the three classical Riemannian spaces of constant curvature, a generalization of a Darboux space and the Iwai–Katayama spaces generalizing the MIC–Kepler and Taub–NUT problems. The key dynamical role played by the Kepler and oscillator potentials in Euclidean space is thus extended to a wide class of three-dimensional curved spaces.