  In the first part of this paper we begin the study of polysymplectic
manifolds, and of their relationship with PDE's. This notion provides a
generalization of symplectic manifolds which is very well suited for the
geometric study of PDE's with values in a smooth manifold. Some of the standard
tools of analytical mechanics, such as the Legendre transformation and
Hamilton's equations, are shown to generalize to this new setting. There is a
strong link with lagrangian fibrations, which can be used to build
polysymplectic manifolds. We then provide the definition and some basic
properties of s-Kahler and almost s-Kahler manifolds. These are a
generalization of the usual notion of Kahler and almost Kahler manifold, and
they reduce to them for s=1. The basic properties of Kahler manifolds, and
their Hodge theory, can be generalized to s-Kahler manifolds, with some
modifications. The most interesting examples come from semi-flat special
lagrangian fibrations of Calabi-Yau manifolds.
