  A definition of quantum mechanics on a manifold $ M $ is proposed and a
method to realize the definition is presented. This scheme is applicable to a
homogeneous space $ M = G / H $. The realization is a unitary representation of
the transformation group $ G $ on the space of vector bundle-valued functions.
When $ H \ne \{ e \} $, there exist a number of inequivalent realizations. As
examples, quantum mechanics on a sphere $ S^n $, a torus $ T^n $ and a
projective space $ \RP $ are studied. In any case, it is shown that there are
an infinite number of inequivalent realizations.
