Multisymplectic 3-forms on 7-dimensional manifolds

Abstract

A 3-form ω∈Λ3R7⁎ is called multisymplectic if it satisfies some natural non-degeneracy requirement. It is well known that there are 8 orbits (or types) of multisymplectic 3-forms on R7 under the canonical action of GL(7,R) and that two types are open. This leads to 8 types of global multisymplectic 3-forms on 7-dimensional manifolds without boundary. The existence of a global multisymplectic 3-form of a fixed type is a classical problem in differential topology which is equivalent to the existence of a certain G-structure. The open types are the most interesting cases as they are equivalent to a G2 and G˜2-structure, respectively. The existence of these two structures is a well known and solved problem. In this article is solved (under some convenient assumptions) the problem of the existence of multisymplectic 3-forms of the remaining types.