Polysymplectic spaces, s-Kahler manifolds and lagrangian fibrations

Details Polysymplectic spaces, s-Kahler manifolds and lagrangian fibrations Polysymplectic spaces, s-Kahler manifolds and lagrangian fibrations

2000-06-21

arxiv.org/abs/math/0006154v1

Mathematics

Differential Geometry

56 pages, LaTeX2e, no figures

Scientific paper

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.

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