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A multisymplectic manifold not covered by Darboux charts
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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".
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Cited by 1 Pith paper
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Observables of Relative Structures and Lie 2-algebras associated with Quasi-Hamiltonian $G$-spaces
Relative n-plectic structures are claimed to produce L-infinity algebras of observables, yielding Lie 2-algebras and homotopy moment maps for quasi-Hamiltonian G-spaces, though several sign and generality issues remain.
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