  We give an algebraic derivation of the eigenvalues of energy of a quantum
harmonic oscillator on the surface of constant curvature, i.e. on the sphere or
on the hyperbolic plane. We use the method proposed by Daskaloyannis for fixing
the energy eigenvalues of two-dimensional (2D) quadratically superintegrable
systems by assuming that they are determined by the existence of
finite-dimensional representation of the polynomial algebra of the motion
integral operators. The tool for realizing representations is the deformed
parafermionic oscillator. The eigenvalues of energy are calculated and the
result derived by us algebraically agrees with the known energy eigenvalues
calculated by classical analytical methods. This assertion which is the main
result of this article is demonstrated by a detailed presentation. We also
discuss the qualitative difference of the energy spectra on the sphere and on
the hyperbolic plane.
