On Darboux theorems for geometric structures induced by closed forms

TL;DR

This paper reviews and extends Darboux theorems for various geometric structures induced by closed forms, crucial in classical field theories, including new results for singular and multisymplectic cases.

Contribution

It introduces new Darboux theorems for k-symplectic, k-cosymplectic, and pre-structures, expanding the geometric understanding of classical and singular field theories.

Findings

Extended Darboux theorems to k-symplectic and k-cosymplectic manifolds.

Established Darboux theorems for classes of pre-structures.

Provided approaches based on flat connections and new polarization results.

Abstract

This work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe regular and singular mechanical systems), and certain cases of multisymplectic manifolds, and extends it in new ways to k-symplectic and k-cosymplectic manifolds (all these structures appear in the geometric formulation of first-order classical field theories). Moreover, we discuss the existence of Darboux theorems for classes of precosymplectic, k-presymplectic, k-precosymplectic, and premultisymplectic manifolds, which are the geometrical structures underlying some kinds of singular field theories. Approaches to Darboux theorems based on flat connections associated with geometric structures are given, while new results on polarisations for (k-)(pre)(co)symplectic structures arise.