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# Multisymplectic 3-forms on 7-dimensional manifolds

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## 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.

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... See [10,Section 4.4] and the references cited there for the case of the tangent bundle. ...
... The first completes the characterization of 7-dimensional vector bundles by characteristic classes in [12]. It has been already used in [10] to obtain results on the existence of multisymplectic 3-forms on 7-dimensional manifolds. ...
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This paper gives a uniform, self-contained and direct approach to a variety of obstruction-theoretic problems on manifolds of dimension 7 and 6. We give necessary and sufficient cohomological criteria for the existence of various G-structures on vector bundles over such manifolds especially using low dimensional representations of the group U(2).
... The first completes the characterization of 7-dimensional vector bundles by characteristic classes in [13]. It has been already used in [11] to obtain results on the existence of multisymplectic 3-forms on 7-dimensional manifolds. Proof. ...
Article
Full-text available
This paper gives a uniform, self-contained, and direct approach to a variety of obstruction-theoretic problems on manifolds of dimension 7 and 6. We give necessary and sufficient cohomological criteria for the existence of various G-structures on vector bundles over such manifolds especially using low dimensional representations of U(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{U}(2)$$\end{document}.
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