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