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

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