  For neutrinos streaming from a supernova (SN) core, dense matter suppresses
self-induced flavor transformations if the electron density n_e significantly
exceeds the neutrino density n_nu in the conversion region. If n_e is
comparable to n_nu one finds multi-angle decoherence, whereas the standard
self-induced transformation behavior requires that in the transformation region
n_nu is safely above n_e. This condition need not be satisfied in the early
phase after supernova core bounce. Our new multi-angle effect is a subtle
consequence of neutrinos traveling on different trajectories when streaming
from a source that is not point-like.
