A multisymplectic manifold not covered by Darboux charts

TL;DR

This paper constructs a specific example of a multisymplectic manifold that cannot be covered by Darboux charts, highlighting fundamental differences from symplectic geometry.

Contribution

It provides the first explicit counterexample of a multisymplectic manifold lacking Darboux charts, demonstrating limitations of Darboux-type theorems beyond symplectic geometry.

Findings

Existence of multisymplectic forms on R^6 without Darboux charts

Counterexample with non-constant linear type forms

Shows fundamental difference from symplectic case

Abstract

The Darboux theorem in symplectic geometry implies that any two points in a connected symplectic manifold have neighbourhoods symplectomorphic to each other. The impossibility of such a theorem in the more general multisymplectic framework appears to be, at least, folkloristic, but no explicit counterexample seems to exist in the literature. In this note we provide such an example by constructing multisymplectic three-forms on the connected manifold $\mathbb R^6$, which do not even have constant linear type and therefore can not allow for an atlas consisting of "Darboux charts".