Lagrangian submanifolds of standard multisymplectic manifolds

TL;DR

This paper provides a comprehensive proof of Martin's normal form theorem for Lagrangian submanifolds in multisymplectic manifolds, extending Weinstein's classical results in symplectic geometry.

Contribution

It offers a detailed, self-contained proof of a key theorem in multisymplectic geometry, including foundational results in foliated differential topology.

Findings

Proves Martin's normal form theorem for multisymplectic manifolds.

Establishes foliated tubular neighborhood theorem.

Provides foliated relative Poincaré lemma.

Abstract

We give a detailed, self-contained proof of Geoffrey Martin's normal form theorem for Lagrangian submanifolds of standard multisymplectic manifolds (that generalises Alan Weinstein's famous normal form theorem in symplectic geometry), providing also complete proofs for the necessary results in foliated differential topology, i.e., a foliated tubular neighborhood theorem and a foliated relative Poincar\'e lemma.