Polysymplectic spaces, s-Kähler manifolds and Lagrangian Fibrations

In 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-Kähler and almost s-Kähler manifolds. These are a generalization of the usual notion of Kähler and almost Kähler manifold, and they reduce to them for s = 1. The basic properties of Kähler manifolds, and their Hodge theory, can be generalized to s-Kähler manifolds, with some modifications. The most interesting examples come from semi-flat special lagrangian fibrations of Calabi-Yau manifolds.