Geometric structures associated with a simple Cartan 3-form

We introduce the notion of a manifold admitting a simple compact Cartan 3-form ω3. We study algebraic types of such manifolds, in particular those having skew-symmetric torsion, or those associated with a closed or coclosed 3-form ω3. We prove the existence of an algebra of multi-symplectic forms ϕl on these manifolds. Cohomology groups associated with complexes of differential forms on Mn in the presence of such a closed multi-symplectic form ϕl and their relations with the de Rham cohomologies of M are investigated. We show rigidity of a class of strongly associative (resp. strongly coassociative) submanifolds. We include an appendix describing all connected simply connected complete Riemannian manifolds admitting a parallel 3-form.