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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. ...

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. ...

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}.

В этой статье мы рассмотрим методы и результаты классификации $k$-форм (соотв. $k$-векторов на $ \R ^ n $), понимаемых как описание пространства орбит стандартного $\GL(n, \R)$-действие на $\Lambda^k \R^{n*}$ (соотв. на $\Lambda ^k \R^n$). Мы обсудим существование связанной геометрии, определяемой дифференциальными формами на гладких многообразиях. Эта статья также содержит Приложение, написанное Михаилом Боровым, о методах когомологии Галуа для нахождения вещественных форм комплексных орбит.

A nearly parallel G2-structure on a seven-dimensional Riemannian manifold is equivalent to a spin structure with a Killing spinor. We prove general results about the automorphism group of such structures and we construct new examples. We classify all nearly parallel G2-manifolds with large symmetry group and in particular all homogeneous nearly parallel G2-structures.

In this thesis, we are concerned with the study of cohomology with local coefficients and applications to (non-orientable) real vector bundles. The thesis consists of an introduction and three separate sections. The introduction gives some motivation for considering cohomology with local coefficients and an outline of the results obtained. The first section deals with a general discussion of such cohomology groups and contains a Künneth Theorem for such groups. The second section is devoted to some computations which are needed later and the final section gives a complete description of the integral cohomology of the spaces $BO(n)$ and $BSO(n)$.

We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine the group structures of the fixed points subgroup. They correspond to the classification of all irreducible compact symmetric spaces of exceptional type, and that they also correspond to classification of all non-compact exceptionalsimple Lie groups. Finally, we determined the group structures of the maximal subgroups of maximal rank. At any rate, we would like this book to be used in mathematics and physics.

Hông-Vân Lê: Manifolds admitting a ˜ G 2 -structure, arXiv:0704.0503v1 [MS] Milnor, Stasheff: Characteristic classes

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[Hu] Hussemoller: Fibre Bundles, Graduate Texts in Mathematics, Springer-Verlag
[Le] Hông-Vân Lê: Manifolds admitting a ˜
G 2 -structure, arXiv:0704.0503v1
[MS] Milnor, Stasheff: Characteristic classes, Princeton University Press and University of Tokyo
Press
[T] E. Thomas: Vector Fields on Low Dimensional Manifolds, Math. Zeitschr. 103, 85–93 (1968)
[Th] E. Thomas: Seminar on Fibre Bundles, Lecture Notes in Mathematics, Springer-Verlag

Structures defined by 3-forms on 7-dimensional manifolds

- Martin Doubek

Martin Doubek: Structures defined by 3-forms on 7-dimensional manifolds, Diploma Thesis,
Charles University 2008

Semmelmann: On nearly parallel G 2 -structures

- T Fkms
- I Friedrich
- A Kath
- U Moroianu

[FKMS] T. Friedrich, I. Kath, A. Moroianu, U. Semmelmann: On nearly parallel G 2 -structures,
Journal of Geometry and Physics 23 (1997) 259-286 " DOI : 10.1016/S0393-0440(97)80004-6