On the geometry of multisymplectic manifolds

A multisymplectic structure on a manifold is defined by a closed differenti al form with zero characteristic distribution. Starting from the linear cas e, some of the basic properties of multisymplectic structures are described . Various examples of multisymplectic manifolds are considered, and special attention is paid to the canonical multisymplectic structure living on a b undle of exterior k-forms on a manifold. For a class of multisymplectic man ifolds admitting a 'Lagrangian' fibration, a general structure theorem is g iven which, in particular, leads to a classification of these manifolds in terms of a prescribed family of cohomology classes.