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Local Type II metrics with holonomy in \(\mathrm{G}_2^*\). (English) Zbl 1432.53074

This paper is concerned with pseudo-Riemannian manifolds whose metric has signature of type \((4,3)\) and whose holonomy group is the exceptional non-compact Lie group \(G_2\) in Berger’s classification. The question is which Lie subgroups of \(G_2\) can be realised as holonomy groups of manifolds as such, with the restriction that the holonomy representation on the tangent bundle is indecomposable.
Fino and Kath classified all subgroups of \(G_2\) which fulfill a criterion of Berger, to Types I, II and III. They gave an affirmative answer to the above question for Type I. The author gave an affirmative answer for Type III in [Ann. Global Anal. Geom. 56, No. 1, 113–136 (2019; Zbl 1422.53041)].
In this paper an affirmative answer is given also for Type II. Thus, the fact that all Berger subalgebras of \(G_2\) can be realised by a pseudo-Riemannian metric with indecomposable representation, is established.

MSC:

53C29 Issues of holonomy in differential geometry
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C10 \(G\)-structures

Citations:

Zbl 1422.53041
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References:

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