As a nonrelativistic particle constrained to remain on an (N − 1)-dimensional ((N ≥ 2)) hypersurface embedded in an N-dimensional Euclidean space, two different components pi
and p
j
(i, j = 1, 2, 3,… N) of the Cartesian momentum of the particle are not mutually commutative, and explicitly commutation relations
[
p
ˆ
i
,
p
ˆ
j
]
≠
0
depend on products of positions and momenta in uncontrollable ways. The generalized Dupin indicatrix of the hypersurface, a local analysis technique, is utilized to explore the dependence of the noncommutativity on the curvatures around a local 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 tangential plane covering the local point, the noncommutativity leads to a rotation operator and the amount of the rotation is an angle anholonomy; and along each of the normal sectional curves centering the given 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.