A particle that is constrained to freely move on a hyperspherical surface in an $N\left(\ge 2\right)$ dimensional flat space experiences a curvature-induced gauge potential, whose form was given long ago (Ohnuki and Kitakado, 1993). 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 $\left[p_i,p_j\right]=-iħJ_{ij}∕r^2$, in which $ħ$ is the Planck’s constant, and $p_i$ ($i,j=1,2,3,\dots N$) denotes the $i$-th component of the geometric momentum, and $J_{ij}$ specifies the $ij$-th component of the generalized 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.

被约束在超球面上自由运动的粒子 $N\left(\ge 2\right)$维平坦空间经历曲率引起的规范势，其形式很久以前就已给出（Ohnuki 和 Kitakado，1993）。我们证明了超球面上粒子的动量是几何动量，包括规范势及其分量服从对易关系$\left[{磷}_{\mathrm{一世}},{磷}_j\right]=-\mathrm{一世}H{J}_{\mathrm{一世}j}∕r^2$，其中 $H$ 是普朗克常数，并且 ${磷}_{\mathrm{一世}}$ ($\mathrm{一世},j=1,2,3,\dots N$) 表示 $\mathrm{一世}$- 几何动量的第一个分量，和 $J_{\mathrm{一世}j}$ 指定 $\mathrm{一世}j$-包含轨道部分和连续旋转对称群发生器的耦合的广义角动量的第-分量 $秒哦(N-1)$ 和曲率，和 $r$ 表示半径 $N-1$ 维超球面。