Bargmann invariants and geometric phases: A generalized connection

We develop the broadest possible generalization of the well known connection between quantum-mechanical Bargmann invariants and geometric phases. The key concept is that of null phase curves in quantum-mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalization is shown to be essential for properly understanding geometric phase results in the cases of coherent states and of Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.