  We consider the response of an effective spin of a charge carrier to an
adiabatic rotation of its crystal momentum induced by electric field. This
rotation gives rise to Coriolis pseudo-force that is responsible for torque
acting on the orbital momentum of a particle. Mediated by a spin-orbit coupling
in the valence band this perturbation leads to a spin-electric coupling that
may affect the coherent transport properties of a charge carrier and cause a
spin precession in zero magnetic fields. In the static uniform electric field
the derived effective spin-Hamiltonians of the carriers in the conduction and
light hole bands are homologous to the Rashba Hamiltonian. These effects may be
also interpreted as a manifestation of, in general, a non-Abelian gauge
potential and can be described in purely geometric terms as a consequence of
the corresponding holonomy. We demonstrate that in the conduction band the
strength of the associated covariant gauge field is proportional to the
effective electron g-tensor and is controllable by gate fields or a strain
applied to the crystal.
