On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane H2

Two Harmonic Oscillators (isotropic and nonisotropic 2:1) are studied on the two-dimensional sphere S2 and the hyperbolic plane H2. Both systems are integrable and super-integrable with constants of motion quadratic in the momenta. These properties are shown to derive from a complex factorization for the constants of motion, which holds for arbitrary values of the curvature κ, and the dynamics of the Euclidean harmonic 1:1 and 2:1 oscillators is directly recovered for κ=0. The harmonic oscillators on either the standard unit sphere (radius R=1) or the unit Lobachewski plane ('radius' R=1) appear as the particular values of the κ-dependent potentials for the values κ=1 and κ=-1. Finally a particular potential is proposed for representing the general spherical (hyperbolic) n:1 anisotropic harmonic oscillator on a two-dimensional manifold of constant curvature