Geometric structures associated with a simple Cartan 3-form

Details Geometric structures associated with a simple Cartan 3-form Geometric structures associated with a simple Cartan 3-form

2011-03-07

arxiv.org/abs/1103.1201v2

Mathematics

Differential Geometry

32 pages, Proposition 6.7 is modified, now it is Proposition 6.8. Example 6.7 and a reference are added

Scientific paper

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.

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