Supersymmetric quantum mechanics in dynamical systems on a curved space

The quantum mechanics of a dynamical system constrained to a curved space embedded in the higher-dimensional Euclidean space is investigated by using the projection operator method (POM) for the constrained systems in case the system has supersymmetry. It is shown that the projected Hamiltonian contains additional terms interpreted as the quantum corrections caused by the non-commutativity of the constraints, which are completely missed in the usual approach like the Dirac formulation. These quantum corrections break the supersymmetry at the operator level, which holds at the classical level, and present the geometrical interpretation of them. The ordering problem for the process of the projections of the system is also discussed