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

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.

minor clarification in second (published) version