In Ref [1], the authors show how the Berry connection is a geometric quantity using the fiber bundle approach. My question is about the idea of taking a local section of a fiber bundle (corresponding to a closed loop trajectory over the base space, for example). Since the section is horizontal, its projection to the total space would mean that the loop 'intersects' (or 'passes through') with fibers different from the initial/final fiber. I understand how, by returning to the same final point, the initial and final states are the same (up to some U(1) phase). This is consistent with the adiabatic theorem in the sense that the system remained in its initial state AFTER the adiabatic perturbation that took it around a loop. But my issue is that this loop touched other fibers/states along the way. Does this mean that the adiabatic theorem is violated DURING the trajectory (because the initial state became several other states before returning to the initial state)? This doesn't seem right.
So, maybe my confusion is that I did not understand something about local sections potentially ignoring the other fibers along the trajectory. Any clarification would be appreciated.
[1] Bohm, A., Boya, L. J., & Kendrick, B. (1991). Derivation of the geometrical phase. Physical Review A, 43(3), 1206–1210. doi:10.1103/physreva.43.1206
