G2-structures as Octonion Algebras

Abstract

We define the category of G2-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 of the same manifold M. A classification of G2-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 the real numbers shows similarities to the theory of octonion algebras over the ring of real-valued smooth functions on M. Thus, many of the results on real octonion algebras, and in general octonion algebras over rings, can be applied to G2-structures viewed as octonion algebras, under the aforementioned isomorphism of categories.