Polysymplectic spaces, s-Kahler manifolds and lagrangian fibrations

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.