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

TL;DR

This paper constructs explicit metrics of signature (4,3) on 7-dimensional manifolds with Type III holonomy groups in the exceptional Lie group G_2*, demonstrating their realizability.

Contribution

It provides explicit constructions of metrics with Type III holonomy in G_2*, filling a gap in the classification of such geometries.

Findings

All Type III holonomy groups in G_2* are realizable by explicit metrics.

Metrics are constructed using Cartan's exterior differential systems.

The work confirms the theoretical classification with concrete examples.

Abstract

Fino and Kath determined all possible holonomy groups of seven-dimensional pseu\-do-Rie\-man\-nian manifolds contained in the exceptional, non-compact, simple Lie group $\mathrm{G}_2^*$ 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 $(4,3)$. 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.