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