Manifolds admitting a $\tilde G_2$-structure

TL;DR

This paper establishes precise conditions under which 7-manifolds can admit a $ ilde G_2$-structure, advancing understanding of special geometric structures in differential geometry.

Contribution

It provides a necessary and sufficient criterion for compact 7-manifolds to admit a $ ilde G_2$-structure and a sufficient condition for open 7-manifolds to admit a closed 3-form of this type.

Findings

Characterization of compact 7-manifolds with $ ilde G_2$-structure

Conditions for open 7-manifolds to admit closed 3-forms of $ ilde G_2$-type

Advancement in understanding $ ilde G_2$-geometries

Abstract

We find a necessary and sufficient condition for a compact 7-manifold to admit a $\tilde G_2$-structure. As a result we find a sufficient condition for an open 7-manifold to admit a closed 3-form of $\tilde G_2$-type.