Geometric structures associated with a simple Cartan 3-form

TL;DR

This paper explores the geometric and algebraic properties of manifolds with a simple compact Cartan 3-form, including their cohomology, submanifold rigidity, and classification of certain Riemannian manifolds.

Contribution

It introduces the concept of manifolds with a simple compact Cartan 3-form and studies their algebraic structures, cohomology, and submanifold rigidity, extending understanding of multi-symplectic forms.

Findings

Existence of multi-symplectic form algebras on these manifolds

Relations between cohomology groups and de Rham cohomology

Rigidity results for certain submanifolds

Abstract

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.