  We study the influence of geometry of quantum systems underlying space of
states on its quantum many-body dynamics. We observe an interplay between
dynamical and topological ingredients of quantum non-equilibrium dynamics
revealed by the geometrical structure of the quantum space of states. As a
primary example we use the anisotropic XY ring in a transverse magnetic field
with an additional time-dependent flux. In particular, if the flux insertion is
slow, non-adiabatic transitions in the dynamics are dominated by the dynamical
phase. In the opposite limit geometric phase strongly affects transition
probabilities. We show that this interplay can lead to a non-equilibrium phase
transition between these two regimes. We also analyze the effect of geometric
phase on defect generation during crossing a quantum critical point.
