  We provide a general formulation of the spin-orbit coupling on a 2D curved
surface. Considering the wide applicability of spin-orbit effect in
spinor-based condensed matter physics, a general spin-orbit formulation could
aid the study of spintronics, Dirac graphene, topological systems, and quantum
information on curved surfaces. Particular attention is then devoted to the
development of an important spin-orbit quantity known as the spin-orbit torque.
As devices trend smaller in dimension, the physics of local geometries on
spin-orbit torque, hence spin and magnetic dynamics shall not be neglected. We
derived the general expression of a spin-orbit anisotropy field for the curved
surfaces and provided explicit solutions in the special contexts of the
spherical, cylindrical and flat coordinates. Our expressions allow spin-orbit
anisotropy fields and hence spin-orbit torque to be computed over the entire
surfaces of devices of any geometry.
