The analysis of problems related to nonlocalized population transfer between quantum levels
requires nontraditional mathematical approaches. Here we study the processes of irremovable nonadiabatic
transitions in a (N − 1)-dimensional system of completely degenerate so-called dark states, taking as a base
model multilevel N-pod systems. It is shown that quantum dynamics of the dark states due to the operator
of nonadiabatic coupling can be described in terms of the Riemannian parallel transport of their geometric
counterparts in the space of control parameters. The results of mathematical modeling of nonadiabatic
effects, presented in the paper for the case of four-level systems interacting with three laser fields (tripod
systems), demonstrate full agreement between the quantum and geometric approaches. These results are
of interest for experimental and laboratory plasma research and atomic and laser physics.