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

TL;DR

This paper proves the existence of a unique indecomposable nilpotent Lie algebra with a torsion-free $G_{2(2)}^*$-structure and constructs a 3-parameter family of invariant metrics with full holonomy on the associated compact nilmanifold.

Contribution

It identifies the unique indecomposable nilpotent Lie algebra admitting a torsion-free $G_{2(2)}^*$-structure with a definite center and constructs a family of metrics with full holonomy on the related nilmanifold.

Findings

Unique indecomposable nilpotent Lie algebra with torsion-free $G_{2(2)}^*$-structure

Existence of a 3-parameter family of invariant metrics with full holonomy

Compact nilmanifold admits these special metrics

Abstract

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