  Bargmann invariants and null phase curves are known to be important
ingredients in understanding the essential nature of the geometric phase in
quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are
submanifolds made up entirely of null phase curves, and so are equally
important for geometric phase considerations. It is shown that the complete
characterization of null phase manifolds involves both the Riemannian metric
structure and the symplectic structure of ray space in equal measure, which
thus brings together these two aspects in a natural manner.
