  It is well-known that Dirac particles gain geometric phase, namely Berry
phase, while moving in an electromagnetic field. Researchers have already shown
covariant formalism for the Berry connection due to an electromagnetic field. A
similar effect is expected to happen due to the presence of Gravity. We use WKB
approximation to develop a covariant formalism of Berry-like connection in the
presence of Einstein gravity, which can be further used to describe the
Berry-like phase or simply Berry phase. We also extend this formalism for
massless Dirac particles (Weyl particles).Then we further show that this
connection can be split into two parts, one of which vanishes when the metric
is spherically symmetric and thus can be linked to the Aharonov-Bohm-like
effect in the 3 + 1 formalism. At the same time, the other term can be related
to the Pancharatnam-Berry like effect.
