On some \(n\)-forms in dimension \(2n\) (Q2761876)

Let \(V\) be a vector space of dimension \(2n\). A nondegenerate \(n\)-form \(\omega\) on \(V\) is said to be binary if there exist an endomorphism \(J\) and another \(n\)-form \(\omega_1\), linearly independent of \(\omega\), such that \(\omega(v_1,\dots,v_n)=\omega_1(Jv_1,v_2,\dots,v_n)\). NEWLINENEWLINENEWLINEIf the \(n\)-form \(\omega\) is 0-deformable and \(\omega(p),\) \(p\in M\) is binary it is proved that if \(\dim\;M\geq 8,\) then the form \(\omega\) is flat (namely there are local coordinates in which it is written with constant coordinates), if and only if it is closed. NEWLINENEWLINENEWLINEAnother result is the following theorem: Consider an \(n\)-form \(\omega\) of Liouville type (i.e. \(\omega\) is flat and \(J\) satisfies \(J^2=0\)) defined on the manifold \(M\) of dimension \(2n\). Let \(N\) be an \(n\)-dimensional submanifold immersed in \(M\) and transverse to the foliation \({\mathcal{F}}_{\omega}\). If the restriction of \(\omega\) to \(N\) vanishes, then \(N\) has a neighborhood in \(M\) which is diffeomorphic to a neighborhood of the zero section in the bundle \(\Lambda^{n-1} T^* N\) such that the pull-back of the Liouville \(n\)-form is \(\omega\) and the diffeomorphism is the identity on \(N\).