  A strengthened canonical quantization scheme for the constrained motion on a
curved hypersurface is proposed with introduction of the second category of
fundamental commutation relations between Hamiltonian and positions/momenta,
whereas those between positions and moments are categorized into the first. As
an $N-1$ ($N\geq2$) dimensional hypersurface is embedded in an N dimensional
Euclidean space, we obtain the proper momentum that depends on the mean
curvature. For the surface is the spherical one, a long-standing problem on the
form of the geometric potential is resolved in a lucid and unambiguous manner,
which turns out to be identical to that given by the so-called confining
potential technique. In addition, a new dynamical group SO(N,1) symmetry for
the motion on the sphere is demonstrated.
