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

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 ${\mathrm{G}}_2^{\ast }$ 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.

24 pages, 1 table, typos removed, 2 references added, 1 reference updated

References in corpus (4)

• Holonomy groups of Lorentzian manifolds
• Holonomy classification of Lorentz-Kähler manifolds
• Geometry of $G$-Structures via the Intrinsic Torsion
• Local Type I Metrics with Holonomy in ${\mathrm{G}}_2^{\ast }$

Cited by in corpus (1)

• Parallel spinors for ${\mathrm{G}}_2^{\ast }$ and isotropic structures