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Bertrand's theorem asserts that any spherically symmetric natural Hamiltoniansystem in Euclidean 3-space which possesses stable circular orbits and whosebounded trajectories are all periodic is either a harmonic oscillator or aKepler system. In this paper we extend this classical result to curved spacesby proving that any Hamiltonian on a spherically symmetric Riemannian3-manifold which satisfies the same conditions as in Bertrand's theorem issuperintegrable and given by an intrinsic oscillator or Kepler system. As abyproduct we obtain a wide panoply of new superintegrable Hamiltonian systems.The demonstration relies on Perlick's classification of Bertrand spacetimes andon the construction of a suitable, globally defined generalization of theRunge-Lenz vector.

Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds