Calibrated and parallel structures on almost Abelian Lie algebras

Abstract

In this article, we determine the seven-dimensional almost Abelian Lie algebras which admit calibrated or parallel G_2-/G_2^*-structures. Along the way, we show that certain well-established curvature restrictions for calibrated and parallel G_2-structures are not valid in the G_2^* case. In more detail, we provide the first example of a Ricci-flat calibrated G_2^*-structure on a compact manifold whose holonomy is not contained in G_2^*. Moreover, we get examples of non-flat parallel G_2^*-structures on almost Abelian Lie algebras g. We give a full classification of these G_2^*-structures if g is additionally nilpotent.