  Structure of the spin-orbit coupling varies from material to material and
thus finding the correct spin-orbit coupling structure is an important step
towards advanced spintronic applications. We show theoretically that the
curvature in a carbon nanotube generates two types of the spin-orbit coupling,
one of which was not recognized before. In addition to the topological
phase-related contribution of the spin-orbit coupling, which appears in the
off-diagonal part of the effective Dirac Hamiltonian of carbon nanotubes, there
is another contribution that appears in the diagonal part. The existence of the
diagonal term can modify spin-orbit coupling effects qualitatively, an example
of which is the electron-hole asymmetric spin splitting observed recently, and
generate four qualitatively different behavior of energy level dependence on
parallel magnetic field. It is demonstrated that the diagonal term applies to a
curved graphene as well. This result should be valuable for spintronic
applications of graphitic materials.
