A multisymplectic manifold not covered by Darboux charts

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".