  A particle that is constrained to freely move on a hyperspherical surface in
an $N\left( \geq 2\right) $ dimensional flat space experiences a
curvature-induced gauge potential, whose form was given long ago (J. Math.
Phys. \textbf{34}(1993)2827). We demonstrate that the momentum for the particle
on the hypersphere is the geometric one including the gauge potential and its
components obey the commutation relations $\left[ p_{i},p_{j}\right] =-i\hbar
J_{ij}/r^{2}$, in which $\hbar $ is the Planck's constant, and $p_{i}$
($i,j=1,2,3,...N$) denotes the $i-$th component of the geometric momentum, and
$J_{ij}$ specifies the $ij-$th component of the generalized\textit{\ angular
momentum} containing both the orbital part and the coupling of the generators
of continuous rotational symmetry group $% SO(N-1)$ and curvature, and $r$
denotes the radius of the $N-1$ dimensional hypersphere.
