Conditions for Integrability of a 3-form

We find necessary and sufficient conditions for the integrability of one type of multisymplectic 3-forms on a 6-dimensional manifold. Let V be a 6-dimensional real vector space. The general linear group GL(V ) operates naturally on the space of 3-forms Λ3V ∗ by φα(v, v′, v′′) = α(φ−1v, φ−1v′, φ−1v′′) , α ∈ Λ3V ∗, φ ∈ GL(V ) . This action has six orbits, see e.g. [1]. They can be described by their representatives. Let us choose a basis v1, . . . , v6 of V , and let α1, . . . , α6 be the corresponding dual basis. Let us recall that a 3-form α ∈ Λ3V ∗ is called regular or multisymplectic if the linear mapping ι : V → Λ2V ∗, ι(v) = ιvα is injective. All the other forms are then called singular. Obviously, all forms belonging to an orbit are either regular or singular. We then speak about regular orbits and singular orbits. We denote R+, R− and R0 the regular orbits and by ρ+, ρ−, ρ0 their representatives. Similarly we denote S1, S2 and S3 the singular orbits and by σ1, σ2, σ3 their representatives. ρ+ = α1 ∧ α2 ∧ α3 + α4 ∧ α5 ∧ α6 , (R+) ρ− = α1 ∧ α2 ∧ α3 + α1 ∧ α4 ∧ α5 + α2 ∧ α4 ∧ α6 − α3 ∧ α5 ∧ α6 , (R−) ρ0 = α1 ∧ α4 ∧ α5 + α2 ∧ α5 ∧ α6 + α3 ∧ α6 ∧ α4 , (R0) σ1 = 0 , (S1) σ2 = α1 ∧ α2 ∧ α3 , (S2) σ3 = α1 ∧ (α2 ∧ α3 + α4 ∧ α5) . (S3) We recall that a 2-form β on a vector space is called decomposable if there exist 1-forms γ and γ′ such that β = γ ∧ γ′. It is well known that a 2-form β is decomposable if and only if β ∧ β = 0. With every 3-form α ∈ Λ3V ∗ we can associate a subset ∆(α) ⊂ V defined by ∆(α) = {v ∈ V ; ιvα ∧ ιvα = 0} . 2010 Mathematics Subject Classification: primary 37J30; secondary 53C10.