  We have theoretically explored the spin-orbit interaction in carbon
nanotubes. We have found that, besides the dependence on chirality and
diameter, the effects of spin-orbit coupling are anisotropic: spin splitting is
larger for the higher valence or the lower electron band depending on the
specific tube. Different tube behaviors can be grouped in three families,
according to the so called chiral index. Curvature-induced changes in the
orbital hybridization have a crucial role, and they are shown to be
family-dependent. Our results explain recent experiments without invoking
external fields.
