  Geometric momentum is the proper momentum for a moving particle constrained
on a curved surface, which depends on the outer curvature and has observable
effects. In the context of multi-component quantum states, geometric momentum
should be rewritten as generally covariant geometric momentum. For a Dirac
fermion constrained on a two-dimensional hypersurface, we give the generally
covariant geometric momentum, and show that on the pseudosphere and the helical
surface there exist no curvature-induced geometric potentials. These results
verify that the dynamical quantization conditions are effective in dealing with
constrained systems on hypersurfaces, and one could obtain the generally
convariant geometric momentum and the geometric potential of a spin particle
constrained on surfaces with definite parametric equations.
