On compact homogeneous $\mathrm{G}_{2(2)}$-manifolds

TL;DR

This paper proves that the only compact homogeneous space with an invariant torsion-free $ ext{G}_{2(2)}$-structure, under certain Lie group conditions, is the seven-dimensional flat torus.

Contribution

It establishes a uniqueness result for compact homogeneous spaces admitting invariant torsion-free $ ext{G}_{2(2)}$-structures, specifically identifying the flat torus as the sole example.

Findings

The seven-dimensional flat torus admits an invariant torsion-free $ ext{G}_{2(2)}$-structure.

No other compact homogeneous space under the specified conditions admits such a structure.

The result constrains the geometry of homogeneous spaces with special $ ext{G}_{2(2)}$-structures.

Abstract

We prove that among all compact homogeneous spaces for an effective transitive action of a Lie group whose Levi subgroup has no compact simple factors, the seven-dimensional flat torus is the only one that admits an invariant torsion-free $\mathrm{G}_{2(2)}$-structure.