  Starting with the first-order singular Lagrangian containing the redundant
variables, the noncommutative quantum mechanics on a curved space is
investigated by the constraint star-product quantization formalism of the
projection operator method. Imposing the additional constraints to eliminate
the reduntant degrees of freedom, the noncommutative quantum system with
noncommutativity among the coordinates on the curved space is exactly
constructed. Then, it is shown that the resultant Hamiltonian contains the
quantum corrections in the exact form. We further discuss the additional
constraints to realize the noncommutativities both of coordinates and momenta
on the curved space.
