Exact G$_2$-structures on compact quotients of Lie groups

TL;DR

This paper proves that certain compact quotients of seven-dimensional Lie groups cannot have exact G2-structures induced by left-invariant forms, clarifying limitations in geometric structures on these spaces.

Contribution

It establishes a non-existence result for exact G2-structures on compact quotients of Lie groups, specifically those induced by left-invariant forms.

Findings

No exact G2-structures induced by left-invariant forms exist on these quotients.

The result constrains possible geometric structures on compact quotients of Lie groups.

Provides insight into the topology and geometry of G2-structures on Lie group quotients.

Abstract

We show that the compact quotient $\Gamma\backslash\mathrm{G}$ of a seven-dimensional simply connected Lie group $\mathrm{G}$ by a co-compact discrete subgroup $\Gamma\subset\mathrm{G}$ does not admit any exact $\mathrm{G}_2$-structure which is induced by a left-invariant one on $\mathrm{G}$.