Curvature-induced noncommutativity of two different components of momentum for a particle on a hypersurface

As a nonrelativistic particle constrained to remain on an $N-1$ ($N\ge 2$) dimensional hypersurface embedded in an $N$ dimensional Euclidean space, two different components $p_i$ and $p_j$ ($i,j=1,2,3,\mathrm{.}\mathrm{.}\mathrm{.}N$) of the Cartesian momentum of the particle are not mutually commutative, and explicitly commutation relations $[p_i,p_j](\mathrm{\ne }0)$ depend on products of positions and momenta in uncontrollable ways. The \textit{% generalized} Dupin indicatrix of the hypersurface, a local analysis technique, is utilized to explore the dependence of the noncommutativity on the curvatures on a $\mathit{\text{local}}\mathit{\text{point}}$of the hypersurface. The first finding is that the noncommutativity can be grouped into two categories; one is the product of a sectional curvature and the angular momentum, and another is the product of a principal curvature and the momentum. The second finding is that, for a small circle lying a $\mathit{\text{tangential}}\mathit{\text{plane}}$ covering the $\mathit{\text{local}}\mathit{\text{point}}$, the noncommutativity leads to a rotation operator and the amount of the rotation is an angle anholonomy; and along each of the $\mathit{\text{normal}}\mathit{\text{sectional}}\mathit{\text{curves}}$ centering the $\mathit{\text{given}}\mathit{\text{point}}$ the noncommutativity leads to a translation plus an additional rotation and the amount of the rotation is one half of the tangential angle change of the arc.