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Starting from a $2$-form $\omega$ which is nondegenerate and closed on a $2n$-dimensional manifold, it is always possible to find local coordinates $x_1,y_1,\ldots,x_n,y_n$ so that the form $\omega$ can be written locally as $$\sum_{k:1}^n a_Idx_k\wedge dy_k.$$ It is well-known that we can also say more on the functions $a_I$ (Darboux theorem).

Take $\psi$, a $4$-form nondegenerate and closed on a $4n$-dimensional manifold. It is not true, in general, that locally $\psi$ can be written as $$\sum_{k:1}^n a_Idx_{4k-3}\wedge dx_{4k-2}\wedge dx_{4k-1}\wedge dx_{4k}$$ where $a_I$ are in $C^{\infty}(M)$ when n>1. Someone knows if there are some conditions on the manifold that assure that this can be done?

Travis Willse
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rubi
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  • The condition is local, so any such condition besides the value of $n$ itself, must actually be on the $4$-form, rather than on the underlying manifold. – Travis Willse May 12 '15 at 10:51
  • Yes you are right, I am asking whether, for example, the Kraines form on quaternionic-Kaehler manifolds has this property or not? – rubi May 12 '15 at 13:11
  • No, I don't think so. (As I recall) the stabilizer of a $4$-form of the same algebraic type as a Kraines form under the standard action of $GL(4n, \Bbb R)$ on $\Lambda^4 \Bbb R^{4n}$ is $Sp(n) \cdot Sp(1)$, but probably one can show that (at least for $n > 1$) that this is not the stabilizer of any form of the given type. (Of course, for $n = 1$ all $4$-forms are volume forms, so the claim in that case is true.) – Travis Willse May 12 '15 at 14:20

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