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