One kind of multisymplectic structures on 6-manifolds (Q2733960)

The following is shown: A 6-manifold \(X\) has a multisymplectic structure \(\Omega\) whose local form is \(\omega=\alpha_1 \wedge\alpha_4 \wedge \alpha_5 +\alpha_2 \wedge\alpha_4 \wedge \alpha_6+ \alpha_3 \wedge \alpha_5 \wedge \alpha_6\) if and only if there exists an orientable 3-dimensional vector bundle \(\eta\) of \(X\) such that \(\xi\cong \eta\oplus\eta\), where \(\xi\) is the tangent bundle of \(X\).NEWLINENEWLINENEWLINEA multisymplectic form of order \(k\) on a real vector space \(V\) is a \(k\)-form \(\varphi\) such that the homomorphism NEWLINE\[NEWLINEV\to \bigwedge^{k -1}V^*,\;v\mapsto\iota_v \varphi= \varphi(v,\dots, \cdot)NEWLINE\]NEWLINE is a monomorphism. On \(\mathbb{R}^6\), there exist up to isomorphism, three multisymplectic 3-forms [\textit{B. Capdevielle}, Enseign. Math. 18 (1972), 225-243 (1973; Zbl 0258.15015)]. The above mentioned \(\omega\) is one of them, and in this paper, other two forms are not treated.NEWLINENEWLINENEWLINEIn Sect. 2, algebraic properties related to \(\omega\), epecially, the structure of \(O(\omega)\), the group of \(\omega\) preserving automorphisms is studied. Then, applying the results of section 2 the proposition above is proved. It is also shown that \(\Omega\) exists if and only if the associated frame bundle \(Fr(\xi)\) of \(\xi\) can be reduced to an \(O(\omega)\)-bundle. \(\Omega\) is not closed in general. It is shown that \(\Omega\) is closed if and only if the associated \(O(\omega)\)-structure is integrable. Topological conditions for the existence of multisymplectic structures are not discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00031].