Pauli equation for a charged spin particle on a curved surface in an electric and magnetic field

We derive the Pauli equation for a charged spin particle confined to move on a spatially curved surface $\mathcal{S}$ in an electromagnetic field. Using the thin-layer quantization scheme to constrain the particle on $\mathcal{S}$, and in the transformed spinor representations, we obtain the well-known geometric potential $V_g$ and the presence of $e^{-i\varphi}$, which can generate additive spin connection geometric potentials by the curvilinear coordinate derivatives, and we find that the two fundamental evidences in the literature [Giulio Ferrari and Giampaolo Cuoghi, Phys. Rev. Lett. 100, 230403 (2008).] are still valid in the present system without source current perpendicular to $\mathcal{S}$. Finally, we apply the surface Pauli equation to spherical, cylindrical, and toroidal surfaces, in which we obtain expectantly the geometric potentials and new spin connection geometric potentials, and find that only the normal Pauli matrix appears in these equations.