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

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-Kähler theory shows that some three-manifold can be presented in this way. We also classify all the homogeneous ones in ${\mathbb{RP}}^7$.