  We show Zeeman-like splitting in the energy of spinors propagating in a
background gravitational field, analogous to the spinors in an electromagnetic
field, otherwise termed the Gravitational Zeeman Effect. These spinors are also
found to acquire a geometric phase, in a similar way as they do in the presence
of magnetic fields. However, in a gravitational background, the Aharonov-Bohm
type effect, in addition to Berry-like phase, arises. Based on this result, we
investigate geometric phases acquired by neutrinos propagating in a strong
gravitational field. We also explore entanglement of neutrino states due to
gravity, which could induce neutrino-antineutrino oscillation in the first
place. We show that entangled states also acquire geometric phases which are
determined by the relative strength between gravitational field and neutrino
masses.
