  We construct a general theoretical framework for describing curvature-induced
spin-orbit interactions on the basis of group theory. Our theory can
systematically determine the emergence of spin splitting in the band structure
according to symmetry in the wavenumber space and the bending direction of the
material. As illustrative examples, we derive the curvature-induced spin-orbit
coupling for carbon and silicon nanotubes. Our theory offers a strategy for
designing valley-contrasting spin-orbit coupled materials by tuning their
curvatures.
