Torsion-free G*2(2)-structures with full holonomy on nilmanifolds

Abstract

Abstract We study the existence of invariant metrics with holonomy G* 2(2) ⊂ SO(4, 3) on compact nilmanifolds, i.e. on compact quotients of nilpotent Lie groups by discrete subgroups. We prove that, up to isomorphism, there exists only one indecomposable nilpotent Lie algebra admitting a torsion-free G* 2(2)-structure such that the center is definite with respect to the induced inner product. In particular, we show that the associated compact nilmanifold admits a 3-parameter family of invariant metrics with full holonomy G* 2(2).