Bertrand spacetimes as Kepler/oscillator potentials RID B-5702-2011 RID F-2453-2010

Perlick's classification of (3 + 1)-dimensional spherically symmetric and static spacetimes (M, eta = -1/v dt(2) + g) for which the classical Bertrand theorem holds (Perlick V 1992 Class. Quantum Grav. 9 1009) is revisited. For any Bertrand spacetime (M, eta) 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.