Manifolds admitting stable forms

TL;DR

This paper introduces a direct classification method for stable forms on real vector spaces, explores their automorphism groups, and discusses conditions and geometric properties of manifolds admitting such forms.

Contribution

It provides a new direct approach to classify stable forms and analyze their automorphism groups, along with conditions for manifolds to admit these forms.

Findings

Stable forms on R^n are classified explicitly.

In dimensions 6, 7, 8, stable forms are equivalent to non-degenerate forms.

Conditions for manifolds to admit stable forms are established.

Abstract

In this note we give a direct method to classify all stable forms on $\R^n$ as well as to determine their automorphism groups. We show that in dimension 6,7,8 stable forms coincide with non-degnerate forms. We present necessary conditions and sufficient conditions for a manifold to admit a stable form. We also discuss rich properties of the geometry of such manifolds.