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By considering the spin connection, we deduce the effective equation for aspin-1/2 particle confined to a curved surface with the non-relativistic limitand in the thin-layer quantization formalism. We obtain a pseudo-magnetic fieldand an effective spin-orbit interaction generated by the spin connection.Geometrically, the pseudo-magnetic field is proportional to the Gaussiancurvature and the effective spin-orbit interaction is determined by theWeingarten curvature tensor. Particularly, we find that the pseudo-magneticfield and the effective spin-orbit interaction can be employed to separate theelectrons with different spin orientations. All these results are demonstratedin two examples, a straight cylindrical surface and a bent one.

Pseudo-magnetic field and effective spin-orbit interaction for a spin-1/2 particle confined to a curved surface