  We study the electronic properties of electrons in flat and curved zigzag
graphene ribbons using a tight-binding model within the Slater Koster
approximation.
  We find that curvature dramatically enhances the action of spin orbit effects
in graphene ribbons and has a strong effect on the spin orientation of the edge
states: whereas spins are normal to the surface in the case of flat ribbons,
this is no longer the case in the case of curved ribbons. We find that for the
edge states, the spin density lies always in the plane perpendicular to the
ribbon axis, and deviate strongly from the normal to the ribbon, even for very
small curvature and the small spin orbit coupling of carbon. We find that
curvature results also in an effective second neighbor hopping that modifies
the electronic properties of zigzag graphene ribbons. We discuss the
implications of our finding in the spin Hall phase of curved graphene Ribbons.
