  We present a formal geometric framework for the study of adiabatic quantum
mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This
framework generalizes earlier holonomy interpretations of the geometric phase
to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an
investigation of the space of non-degenerate operators on a finite-dimensional
state space. We then show how the energy bands of a Hamiltonian family form a
covering space. Likewise, we show that the eigenrays form a bundle, a
generalization of a principal bundle, which admits a natural connection
yielding the (generalized) geometric phase. This bundle provides in addition a
natural generalization of the quantum geometric tensor and derived tensors, and
we show how it can incorporate the non-geometric dynamical phase as well. We
finish by demonstrating how the bundle can be recast as a principal bundle, so
that both the geometric phases and the permutations of eigenstates can be
expressed simultaneously by means of standard holonomy theory.
