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Perlick's classification of (3+1)-dimensional spherically symmetric andstatic spacetimes (\cal M,\eta=-1/V dt^2+g) for which the classical Bertrandtheorem holds [Perlick V Class. Quantum Grav. 9 (1992) 1009] is revisited. Forany Bertrand spacetime (\cal M,\eta) the term V(r) is proven to be either theintrinsic Kepler-Coulomb or the harmonic oscillator potential on its associatedRiemannian 3-manifold (M,g). Among the latter 3-spaces (M,g) we explicitlyidentify the three classical Riemannian spaces of constant curvature, ageneralization of a Darboux space and the Iwai-Katayama spaces generalizing theMIC-Kepler and Taub-NUT problems. The key dynamical role played by the Keplerand oscillator potentials in Euclidean space is thus extended to a wide classof 3-dimensional curved spaces.

Bertrand spacetimes as Kepler/oscillator potentials