  A quantum sl(2,R) coalgebra is shown to underly the construction of a large
class of superintegrable potentials on 3D curved spaces, that include the
non-constant curvature analogues of the spherical, hyperbolic and (anti-)de
Sitter spaces. The connection and curvature tensors for these "deformed" spaces
are fully studied by working on two different phase spaces. The former directly
comes from a 3D symplectic realization of the deformed coalgebra, while the
latter is obtained through a map leading to a spherical-type phase space. In
this framework, the non-deformed limit is identified with the flat contraction
leading to the Euclidean and Minkowskian spaces/potentials. The resulting
Hamiltonians always admit, at least, three functionally independent constants
of motion coming from the coalgebra structure. Furthermore, the intrinsic
oscillator and Kepler potentials on such Riemannian and Lorentzian spaces of
non-constant curvature are identified, and several examples of them are
explicitly presented.
