  A family of geometric models of quantum relativistic rotating oscillator is
defined by using a set of one-parameter deformations of the static (3+1) de
Sitter or anti-de Sitter metrics. It is shown that all these models lead to the
usual isotropic harmonic oscillator in the non-relativistic limit, even though
their relativistic behavior is different. As in the case of the (1+1) models,
these will have even countable energy spectra or mixed ones, with a finite
discrete sequence and a continuous part. In addition, all these spectra, except
that of the pure anti-de Sitter model, will have a fine-structure, given by a
rotator-like term.
