Geometric structures associated with a simple Cartan 3-form

We introduce the notion of a manifold admitting a simple compact Cartan 3-form ${\text{om}}^3$. We study algebraic types of such manifolds specializing on those having skew-symmetric torsion, or those associated with a closed or coclosed 3-form ${\text{om}}^3$. We prove the existence of an algebra of multi-symplectic forms ${\varphi }^l$ on these manifolds. Cohomology groups associated with complexes of differential forms on $M^n$ in presence of such a closed multi-symplectic form ${\varphi }^l$ 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.