  We consider momentum operators on intrinsically curved manifolds. Given that
the momentum operators are Killing vector fields whose integral curves are
geodesics, it is shown that the corresponding manifold is either flat, or
otherwise of compact type with positive constant sectional curvature and
dimension equal to 1, 3 or 7. Explicit representations of momentum operators
and the associated Casimir element will be discussed for the 3-sphere $S^3$. It
will be verified that the structure constants of the underlying Lie algebra are
proportional to $2\hbar/R$, where $R$ is the curvature radius of $S^3$. This
results in a countable energy and momentum spectrum of freely moving particles
in $S^3$. It is shown that the maximum resolution of the possible momenta is
given by the de-Broglie wave length $\lambda_R=\pi R$, which is identical to
the diameter of the manifold. The corresponding covariant position operators
are defined in terms of geodesic normal coordinates and the associated
commutator relations of position and momentum are established.
