2016/08/26 by Leonid Ryvkin, Ryvkin, Leonid
Mathematics · #53B99 #53C15 #53D05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1608.07424
openalex publication_date 2016/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 R6, which do not even have constant linear type and therefore can not allow for an atlas consisting of "Darboux charts".