  The full spectrum and eigenfunctions of the quantum version of a nonlinear
oscillator defined on an N-dimensional space with nonconstant curvature are
rigorously found. Since the underlying curved space generates a
position-dependent kinetic energy, three different quantization prescriptions
are worked out by imposing that the maximal superintegrability of the system
has to be preserved after quantization. The relationships among these three
Schroedinger problems are described in detail through appropriate similarity
transformations. These three approaches are used to illustrate different
features of the quantization problem on N-dimensional curved spaces or,
alternatively, of position-dependent mass quantum Hamiltonians. This quantum
oscillator is, to the best of our knowledge, the first example of a maximally
superintegrable quantum system on an N-dimensional space with nonconstant
curvature.
