  The Hermitian Hamiltonian of a spin one-half particle with spin-orbit
coupling (SOC) confined to a surface that is embedded in a three-dimensional
space spanned by a general Orthogonal Curvilinear Coordinate (OCC) is
constructed. A gauge field formalism, where the SOC is expressed as a
non-Abelian $SU(2)$ gauge field is used. A new practical approach, based on the
physical argument that upon confining the particle to the surface by a
potential, then it is the physical Hermitian momentum operator transverse to
the surface, rather than just the derivative with respect to the transverse
coordinate that should be dropped from the Hamiltonian.Doing so, it is shown
that the Hermitian Hamiltonian for SOC is obtained with the geometric potential
and the geometric kinetic energy terms emerging naturally. The geometric
potential is shown to represent a coupling between the transverse component of
the gauge field and the mean curvature of the surface that replaces the
coupling between the transverse momentum and the gauge field. The most general
Hermitian Hamiltonian with linear SOC on a general surface embedded in any 3D
OCC system is reported. Explicit plug-and-play formulae for this Hamiltonian on
the surfaces of a cylinder, a sphere and a torus are given. The formalism is
applied to the Rashba SOC in three dimensions (3D RSOC) and the explicit
expressions for the surface Hamiltonians on these three geometries are worked
out.
