Hamiltonian systems on complex Grassmann manifolds. Holonomy and Schrodinger equation

Abstract

Differential geometric structures such as principal bundles for canonical vector bundles on complex Grassmann manifolds, canonical connection forms on these bundles, canonical symplectic forms on complex Grassmann manifolds, and the corresponding dynamical systems are investigated. Grassmann manifolds are considered as orbits of the co-adjoint action and symplectic forms are described as the restrictions of the canonical Poisson structure to Lie coalgebras. Holonomies of connections on principal bundles over Grassmannians and their relation with Berry phases is considered and investigated for integral curves of Hamiltonian dynamical systems.