Lagrangian Distributions and Connections in Symplectic Geometry

TL;DR

This paper explores the relationship between lagrangian distributions and connections in symplectic and multisymplectic geometry, extending classical theorems to more general fiber bundle contexts relevant for classical field theory.

Contribution

It generalizes Weinstein's tubular neighborhood theorem to poly- and multisymplectic structures, highlighting the role of the Bott connection in these settings.

Findings

Extended Weinstein's theorem to multisymplectic structures

Identified the role of Bott connection in generalized contexts

Applied to covariant Hamiltonian formulations of field theory

Abstract

We discuss the interplay between lagrangian distributions and connections in symplectic geometry, beginning with the traditional case of symplectic manifolds and then passing to the more general context of poly- and multisymplectic structures on fiber bundles, which is relevant for the covariant hamiltonian formulation of classical field theory. In particular, we generalize Weinstein's tubular neighborhood theorem for symplectic manifolds carrying a (simple) lagrangian foliation to this situation. In all cases, the Bott connection, or an appropriately extended version thereof, plays a central role.