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.

A closed expression for the chemical potential of Bose gases in an external one-dimensional harmonic trap, reported recently in this journal (Mungan 2009 Chemical potential of one-dimensional simple harmonic oscillators Eur. J. Phys. 30 1131–6), is approximate and not applicable for temperatures lower than a characteristic value below which the ground state becomes occupied by a macroscopic number of particles. In this letter, the correct behaviour of the chemical potential at low temperature is addressed.