Quantum Deformations and Superintegrable Motions on Spaces with Variable Curvature

An infinite family of quasi-maximally superintegrable Hamiltonians with a common set of $(2N-3)$ integrals of the motion is introduced. The integrability properties of all these Hamiltonians are shown to be a consequence of a hidden non-standard quantum $sl(2,\mathbb R)$ Poisson coalgebra symmetry. As a concrete application, one of this Hamiltonians is shown to generate the geodesic motion on certain manifolds with a non-constant curvature that turns out to be a function of the deformation parameter $z$. Moreover, another Hamiltonian in this family is shown to generate geodesic motions on Riemannian and relativistic spaces all of whose sectional curvatures are constant and equal to the deformation parameter $z$. This approach can be generalized to arbitrary dimension by making use of coalgebra symmetry.