  In this paper, we consider the motion of a particle on a surface of
revolution under the influence of a central force field. We prove that there
are at most two analytic central potentials for which all the bounded,
nonsingular orbits are closed and that there are exactly two on some surfaces
with constant Gaussian curvature. The two potentials leading to closed orbits
are suitable generalizations of the gravitational and harmonic oscillator
potential. We also show that there could be surfaces admitting only one
potential that leads to closed orbits. In this case, the potential is a
generalized harmonic oscillator. In the special case of surfaces of revolution
with constant Gaussian curvature, we prove a generalization of the well-known
Bertrand theorem.
