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 Chebyshevskii Sb., 2020, Volume 21, Issue 2, Pages 362–382 (Mi cheb914)

Classification of $k$-forms on $\mathbb{R}^n$ and the existence of associated geometry on manifolds

Hông Vân Lê, J. Vanžura

Institute of Mathematics of the Czech Academy of Sciences, (Praha, Czech Republic)

Abstract: In this paper we survey methods and results of classification of $k$-forms (resp. $k$-vectors on $\mathbb{R}^n$), understood as description of the orbit space of the standard $\mathrm{GL}(n, \mathbb{R})$-action on $\Lambda^k \mathbb{R}^{n*}$ (resp. on $\Lambda ^k \mathbb{R}^n$). We discuss the existence of related geometry defined by differential forms on smooth manifolds. This paper also contains an Appendix by Mikhail Borovoi on Galois cohomology methods for finding real forms of complex orbits.

Keywords: $\mathrm {GL} (n, {\mathbb R})$-orbits in $\Lambda^k\mathbb{R}^{n*}$, $\theta$-group, geometry defined by differential forms, Galois cohomology.

 Funding Agency Grant Number Czech Science Foundation 18-00496SRVO:67985840 The research of HVL was supported by the GAČR-project 18-00496S and RVO:67985840.

DOI: https://doi.org/10.22405/2226-8383-2018-21-2-362-382

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UDC: 512.64+514.745
Accepted:11.03.2020
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Citation: Hông Vân Lê, J. Vanžura, “Classification of $k$-forms on $\mathbb{R}^n$ and the existence of associated geometry on manifolds”, Chebyshevskii Sb., 21:2 (2020), 362–382

Citation in format AMSBIB
\Bibitem{LeVan20} \by H{\^o}ng~V{\^a}n~L\^e, J.~Van{\v z}ura \paper Classification of $k$-forms on $\mathbb{R}^n$ and the existence of associated geometry on manifolds \jour Chebyshevskii Sb. \yr 2020 \vol 21 \issue 2 \pages 362--382 \mathnet{http://mi.mathnet.ru/cheb914} \crossref{https://doi.org/10.22405/2226-8383-2018-21-2-362-382}