  The Virial Theorem in the one- and two-dimensional spherical geometry are
presented, in both classical and quantum mechanics. Choosing a special class of
Hypervirial operators, the quantum Hypervirial relations in the spherical
spaces are obtained. With the aid of the Hellmann-Feynman Theorem, these
relations can be used to formulate a \emph{perturbation theorem without wave
functions}, corresponding to the Hypervirial-Hellmann-Feynman Theorem
perturbation theorem of Euclidean geometry. The one-dimensional harmonic
oscillator and two-dimensional Coulomb system in the spherical spaces are given
as two sample examples to illustrate the perturbation method.
