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A particle that is constrained to freely move on a hyperspherical surface inan $N\left( \geq 2\right) $ dimensional flat space experiences acurvature-induced gauge potential, whose form was given long ago (J. Math.Phys. \textbf{34}(1993)2827). We demonstrate that the momentum for the particleon the hypersphere is the geometric one including the gauge potential and itscomponents obey the commutation relations $\left[ p_{i},p_{j}\right] =-i\hbarJ_{ij}/r^{2}$, 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{\ angularmomentum} containing both the orbital part and the coupling of the generatorsof continuous rotational symmetry group $% SO(N-1)$ and curvature, and $r$denotes the radius of the $N-1$ dimensional hypersphere.

The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere