  The existence of Hopf fibrations S^{2N+1}/S^1 = CP^N and S^{4K+3}/S^3 = HP^K
allows us to treat the Hilbert space of generic finite-dimensional quantum
systems as the total bundle space with respectively $U(1)$ and $SU(2)$ fibers
and complex and quaternionic projective spaces as base manifolds. This
alternative method of studying quantum states and their evolution reveals the
intimate connection between generic quantum mechanical systems and geometrical
objects. The exact Abelian and non-Abelian geometric phases, and more generally
the geometrical factors for open paths, and their precise correspondence with
geometric Kahler and hyper-Kahler connections will be discussed. Explicit
physical examples are used to verify and exemplify the formalism.
