Generally covariant geometric momentum and geometric potential for a Dirac fermion on a two-dimensional hypersurface

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.