Torsion-free $G_{2(2)}^{\ast }$-structures with full holonomy on nilmanifolds

Abstract

We study the existence of invariant metrics with holonomy $G_{2(2)}^{\ast }\subset SO(4,3)$ 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 $G_{2(2)}^{\ast }$-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 $G_{2(2)}^{\ast }$.

16 pages

References in corpus (1)

• Calibrated and parallel structures on almost Abelian Lie algebras

Cited by in corpus (2)

• Calibrated and parallel structures on almost Abelian Lie algebras
• Local Type I Metrics with Holonomy in ${\mathrm{G}}_2^{\ast }$