  We develop an effective tight-binding Hamiltonian for spin-orbit (SO)
interaction in bent carbon nanotubes (CNT) for the electrons forming the $\pi$
bonds between the nearest neighbor atoms. We account for the bend of the CNT
and the intrinsic spin-orbit interaction which introduce mixing of $\pi$ and
$\sigma$ bonds between the $p_z$ orbitals along the CNT. The effect contributes
to the main origin of the SO coupling--the folding of the graphene plane into
the nanotube. We discuss the bend-related contribution of the SO coupling for
resonant single-electron spin and charge transitions in a double quantum dot.
We report that although the effect of the bend-related SO coupling is weak for
the energy spectra, it produces a pronounced increase of the spin transition
rates driven by an external electric field. We find that spin-flipping
transitions driven by alternate electric fields have usually larger rates when
accompanied by charge shift from one dot to the other. Spin-flipping transition
rates are non-monotonic functions of the driving amplitude since they are
masked by stronger spin-conserving charge transitions. We demonstrate that the
fractional resonances--counterparts of multiphoton transitions for atoms in
strong laser fields--occurring in electrically controlled nanodevices already
at moderate ac amplitudes--can be used to maintain the spin-flip transitions.
