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

TL;DR

This paper surveys classification methods for k-forms on R^n, exploring their orbit spaces under GL(n,R) actions and the related geometric structures on manifolds, including Galois cohomology techniques.

Contribution

It provides a comprehensive overview of classification results for k-forms and discusses the existence of geometric structures derived from these forms on manifolds.

Findings

Classification of k-forms via orbit space analysis

Existence criteria for geometric structures on manifolds

Application of Galois cohomology to real forms

Abstract

In this paper we survey methods and results of classification of $k$-forms (resp. $k$-vectors on ${\bf R}^n$), understood as description of the orbit space of the standard ${\bf GL}(n, {\bf R})$-action on $\Lambda^k {\bf R}^{n*}$ (resp. on $\Lambda ^k {\bf R}^n$). 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.