Exact G$$_{\mathbf{2}}$$-structures on compact quotients of Lie groups

Abstract

AbstractWe show that the compact quotient $$\Gamma \backslash {\mathrm G}$$ Γ \ G of a seven-dimensional simply connected Lie group $${\mathrm G}$$ G by a co-compact discrete subgroup $$\Gamma \subset {\mathrm G}$$ Γ ⊂ G does not admit any exact $${\mathrm G}_2$$ G 2 -structure which is induced by a left-invariant one on $${\mathrm G}$$ G .