$G_2$-structures as Octonion Algebras

TL;DR

This paper establishes a categorical equivalence between $G_2$-structures on 7-manifolds and octonion algebras over smooth functions, linking geometric structures to algebraic frameworks.

Contribution

It introduces an isomorphism between $G_2$-structures and octonion algebras over $C^inity(M)$, enabling algebraic methods to study geometric $G_2$-structures.

Findings

Classification of $G_2$-structures aligns with octonion algebra parametrization.

Many properties of real octonion algebras extend to octonion algebras over rings.

The local structure of octonion algebras over $C^inity(M)$ resembles that over $ eal$.

Abstract

We define the category of $G_2$-structures over a Riemannian 7-manifold $M$ and present an isomorphism between this category and a full subcategory of the category of octonion algebras over the ring of smooth real-valued functions $C^\infty(M)$ of the same manifold $M$. A classification of $G_2$-structures in the same metric class is shown to agree with a parametrisation of octonion algebras with isometric norm. A short study of the local structure of octonion algebras over $C^\infty(M)$ shows similarities to the theory of octonion algebras over $\mathbb{R}$. Thus, many of the results on real octonion algebras, and in general octonion algebras over rings, can be applied to $G_2$-structures viewed as octonion algebras, under the aforementioned isomorphism of categories.