  We derive the Schroedinger equation for a spinless charged particle
constrained to a curved surface with electric and magnetics fields applied. The
particle is confined on the surface using a thin-layer procedure, giving rise
to the well-known geometric potential. The electric and magnetic fields are
included via the four-potential. We find that there is no coupling between the
fields and the surface curvature and that, with a proper choice of the gauge,
the surface and transverse dynamics are exactly separable. Finally, the
Hamiltonian for the cylinder, sphere and torus are analytically derived.
