Local Type I Metrics with Holonomy in G₂*

Abstract

By [arXiv:1604.00528], a list of possible holonomy algebras for pseudo-Riemannian manifolds with an indecomposable torsion-free G₂*-structure is known. Here, indecomposability means that the standard representation of the algebra on R⁴,³ 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 p₁ that stabilises one isotropic line of R⁴,³. In particular, we realise p₁ by a local metric.