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Annals of Global Analysis and Geometry

, Volume 56, Issue 1, pp 113–136 | Cite as

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

  • Christian VolkhausenEmail author
Article
  • 39 Downloads

Abstract

Fino and Kath determined all possible holonomy groups of seven-dimensional pseudo-Riemannian manifolds contained in the exceptional, non-compact, simple Lie group \(\mathrm {G}_2^*\) 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.

Keywords

Holonomy Pseudo-Riemannian manifold Exterior differential systems Exceptional Lie group 

Mathematics Subject Classification

53C29 53C50 53C10 

Notes

Acknowledgements

I am very grateful to Ines Kath for introducing me into the field of holonomy theory as well as her support and useful advices on this article.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institut für Mathematik und InformatikUniversität GreifswaldGreifswaldGermany
  2. 2.Max-Planck-Institut für PlasmaphysikGreifswaldGermany

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