The effects of the geometric phase on vibronic states associated with the lower potential surface of two electronic level Hamiltonians are examined. We obtain the general formula for the gauge potential arising from the vibronic interaction. It is shown that this gauge potential is split into a topological part and a magnetic part, where the topological part gives rise to the phase factor of +1 or -1 when it is integrated along a closed trajectory in the nuclear coordinate space, and the magnetic part gives rise to a contribution depending on the local character of the trajectory and exists only in systems without time-reversal symmetry. For particular examples, we consider the E circle times e and E circle times(b(1) + b(2)) Jahn-Teller systems with strong vibronic interactions. It is demonstrated that the ground states have vibronic standing wave states whose nuclear probability density distributions are localized in one of the equivalent minima on the lower potential surface. We also consider Zeeman splittings of degenerate vibronic states, where the degeneracy arises from time-reversal symmetry.