  Neutrinos propagating in dense astrophysical environments sustain nonlinear
refractive effects due to neutrino-neutrino forward scattering. We study
geometric phases in neutrino oscillations that arise out of cyclic evolution of
the potential generated by these forward-scattering processes. We perform
several calculations, exact and perturbative, that illustrate the robustness of
such phases, and of geometric effects more broadly, in the flavor evolution of
neutrinos. The scenarios we consider are highly idealized in order to make them
analytically tractable, but they suggest the possible presence of complicated
geometric effects in realistic astrophysical settings. We also point out that
in the limit of extremely high neutrino densities, the nonlinear potential in
three flavors naturally gives rise to non-Abelian geometric phases. This paper
is intended to be accessible to neutrino experts and non-specialists alike.
