Calibrated and parallel structures on almost Abelian Lie algebras

2013/07/09 by Freibert, Marco
#53C10 (primary) #53C25 #53C30 #53C50 (secondary) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1307.2542

Abstract

In this article, we determine the seven-dimensional almost Abelian Lie algebras which admit calibrated or parallel G₂-/G₂^*-structures. Along the way, we show that certain well-established curvature restrictions for calibrated and parallel G₂-structures are not valid in the G₂^* case. In more detail, we provide the first example of a Ricci-flat calibrated G₂^*-structure on a compact manifold whose holonomy is not contained in G₂^*. Moreover, we get examples of non-flat parallel G₂^*-structures on almost Abelian Lie algebras g. We give a full classification of these G₂^*-structures if g is additionally nilpotent.

Related

• Almost Parallel Structures
• The shear construction
• Almost abelian pseudo-Kähler Lie algebras
• Coclosed G₂-structures inducing nilsolitons
• Odd dimensional counterparts of abelian complex and hypercomplex structures