  The concept of the Zak phase lies at the core of the modern theory of
electric polarization. It is defined using the components of the Bloch wave
functions in a certain basis, which is not captured by the standard expression
for Berry potential. We provide a consistent geometric interpretation of the
Zak phase in terms of projected connections. In the context of Bloch states, we
relate the transformation law of projected Berry potential with classical
currents that contribute to the time derivative of the electric polarization.
This gives a new argument for the Zak phase formula for the electronic
contribution to the polarization. We demonstrate that the Wannier functions
play a key role in the description of an adiabatic current in a periodic
system.
