Local Type I Metrics with Holonomy in ${\rm G}_{2}^*$

Abstract

By [arXiv:1604.00528], a list of possible holonomy algebras for pseudo-Riemannian manifolds with an indecomposable torsion free ${\rm G}_{2}^*$-structure is known. Here indecomposability means that the standard representation of the algebra on ${\mathbb R}^{4,3}$ does not leave invariant any proper non-degenerate subspace. The dimension of the socle of this representation is called the type of the Lie algebra. It is equal to one, two or three. In the present paper, we use Cartan's theory of exterior differential systems to show that all Lie algebras of Type I from the list in [arXiv:1604.00528] can indeed be realised as the holonomy of a local metric. All these Lie algebras are contained in the maximal parabolic subalgebra $\mathfrak p_1$ that stabilises one isotropic line of ${\mathbb R}^{4,3}$. In particular, we realise $\mathfrak p_1$ by a local metric.