  Bertrand's theorem asserts that any spherically symmetric natural Hamiltonian
system in Euclidean 3-space which possesses stable circular orbits and whose
bounded trajectories are all periodic is either a harmonic oscillator or a
Kepler system. In this paper we extend this classical result to curved spaces
by proving that any Hamiltonian on a spherically symmetric Riemannian
3-manifold which satisfies the same conditions as in Bertrand's theorem is
superintegrable and given by an intrinsic oscillator or Kepler system. As a
byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems.
The demonstration relies on Perlick's classification of Bertrand spacetimes and
on the construction of a suitable, globally defined generalization of the
Runge-Lenz vector.
