Non orientable three-submanifolds of $\mathrm{G}_2-$manifolds

TL;DR

This paper introduces a new class of non-orientable three-dimensional submanifolds in almost G2-manifolds, classifies homogeneous examples in real projective 7-space, and applies Cartan-Kähler theory to demonstrate their existence.

Contribution

It defines non-orientable G2-submanifolds modeled on special planes and classifies all homogeneous cases in RP^7, expanding understanding of G2-geometry.

Findings

Existence of non-orientable G2-submanifolds demonstrated.

Classification of homogeneous examples in RP^7.

Application of Cartan-Kähler theory to construct examples.

Abstract

By analogy with associative and co-associative cases we introduce a class of three-dimensional non-orientable submanifolds, of almost $\mathrm{G}_2-$manifolds, modelled on planes lying in a special $\mathrm{G}_2-$orbit. An application of the Cartan-K\"ahler theory shows that some three-manifold can be presented in this way. We also classify all the homogeneous ones in $\mathbb{RP}^7$.