Properties of Multisymplectic Manifolds

TL;DR

This paper reviews key properties of multisymplectic manifolds, including their structures, canonical models, Hamiltonian dynamics, and invariants, providing a comprehensive overview of multisymplectic geometry.

Contribution

It introduces characteristic submanifolds, canonical models, and discusses conditions for Darboux coordinates, advancing understanding of multisymplectic structures and their Hamiltonian properties.

Findings

Characterization of multisymplectic manifolds by automorphisms

Identification of conditions for Darboux-type coordinates

Analysis of Hamiltonian structures in multisymplectic geometry

Abstract

This lecture is devoted to review some of the main properties of multisymplectic geometry. In particular, after reminding the standard definition of multisymplectic manifold, we introduce its characteristic submanifolds, the canonical models, and other relevant kinds of multisymplectic manifolds, such as those where the existence of Darboux-type coordinates is assured. The Hamiltonian structures that can be defined in these manifolds are also studied, as well as other important properties, such as their invariant forms and the characterization by automorphisms.