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

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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.

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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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Correspondence to Christian Volkhausen.

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Volkhausen, C. Local Type III metrics with holonomy in \(\mathrm {G}_2^*\). Ann Glob Anal Geom 56, 113–136 (2019). https://doi.org/10.1007/s10455-019-09659-8

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