Hamiltonian systems on complex Grassmann manifolds. Holonomy and Schrodinger equation

Abstract

Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over Grassmannian and its relation with Berry phase is considered and investigated for the integral curves of Hamiltonian dynamical systems.