Local Type III metrics with holonomy in $$\\mathrm {G}_2^*$$G2∗

Abstract

Fino and Kath determined all possible holonomy groups of seven-dimensional pseudo-Riemannian manifolds contained in the exceptional, non-compact, simple Lie group $$\\mathrm {G}_2^*$$G2∗ via the corresponding Lie algebras. They are distinguished by the dimension of their maximal semi-simple subrepresentation on the tangent space, the socle. An algebra is called of Type I, II or III if the socle has dimension 1, 2 or 3, respectively. This article proves that each possible holonomy group of Type III can indeed be realized by a metric of signature (4,\xa03). For this purpose, metrics are explicitly constructed, using Cartan’s methods of exterior differential systems, such that the holonomy of the manifold has the desired properties.