Schrödinger equation for the nonrelativistic particle constrained on a hypersurface in a curved space

Dirac-bracket quantization of the nonrelativistic particle whose motion is constrained on the hypersurface f(x)=const embedded in a general curved space is discussed. The noncanonical nature of commutation relations makes it difficult to obtain the coordinate representation. Then the system with the derivative-type constraint df(x)/dt=0 is alternatively quantized, treating carefully the operator-ordering problem. In this system it is shown that there exist no constraints on coordinates and momenta and that one can thus have a straightforward representation, which leads, in turn, to the representation and the Schr\"odinger equation in the former system.