  The theory of geometric phase is generalized to a cyclic evolution of the
eigenspace of an invariant operator with $N$-fold degeneracy.
  The corresponding geometric phase is interpreted as a holonomy inherited from
the universal connection of a Stiefel U(N)-bundle over a Grassmann manifold.
Most significantly, for an arbitrary initial state, this geometric phase
captures the inherent geometric feature of the state evolution. Moreover, the
geometric phase in the evolution of the eigenspace of an adiabatic action
operator is also addressed, which is elaborated by a pullback U(N)-bundle.
Several intriguing physical examples are illustrated.
