  We study the flavor evolution of a dense gas initially consisting of pure
mono-energetic $\nu_e$ and $\bar\nu_e$. Using adiabatic invariants and the
special symmetry in such a system we are able to calculate the flavor evolution
of the neutrino gas for the cases with slowly decreasing neutrino number
densities. These calculations give new insights into the results of recent
large-scale numerical simulations of neutrino flavor transformation in
supernovae. For example, our calculations reveal the existence of what we term
the ``collective precession mode''. Our analyses suggest that neutrinos which
travel on intersecting trajectories subject to destructive quantum interference
nevertheless can be in this mode. This mode can result in sharp transitions in
the final energy-dependent neutrino survival probabilities across all
trajectories, a feature seen in the numerical simulations. Moreover, this
transition is qualitatively different for the normal and inverted neutrino mass
hierarchies. Exploiting this difference, the neutrino signals from a future
galactic supernova can potentially be used to determine the actual neutrino
mass hierarchy.
