  We discuss adiabatic quantum phenomena at a level crossing. Given a path in
the parameter space which passes through a degeneracy point, we find a
criterion which determines whether the adiabaticity condition can be satisfied.
For paths that can be traversed adiabatically we also derive a differential
equation which specifies the time dependence of the system parameters, for
which transitions between distinct energy levels can be neglected. We also
generalize the well-known geometric connections to the case of adiabatic paths
containing arbitrarily many level-crossing points and degenerate levels.
