  For a particle that is constrained to freely move on a hypersurface, the
curvature of the surface can induce a gauge potential; and for a particle on
the hypersphere, the gauge potential derived from the \textit{generalized
angular momentum algebra }on it has been known long before (J. Math. Phys.
\textbf{34}(1993)2827). We demonstrate that the momentum for the particle on
the hyperspherecan be the geometric one which obey commutation relations $%
\left[ p_{i},p_{j}\right] =-i\hbar J_{ij}/r^{2}$, in which $\hbar $ is the
Planck's constant, and $p_{i}$ ($i=1,2,3,...N$) symbolizes the $i-$th component
of the geometric momentum, and $J_{ij}$ specifies the $ij-$th component of the
angular momentum containing the spin-curvature coupling, and $r$ denotes the
radius of the $N-1$ dimensional hypersphere.
