Metrics that realize all Lorentzian holonomy algebras. (English) Zbl 1112.53039

Summary: All candidates to the weakly-irreducible not irreducible holonomy algebras of the Lorentzian manifolds are known. In the present paper metrics that realize all these candidates as holonomy algebras are given. This completes the classification of the Lorentzian holonomy algebras. Also new examples of metrics with the holonomy algebras \(g_{2} \ltimes \mathbb R^{7} \subset \mathfrak{so}(1, 8)\) and \(\mathfrak{spin}(7) \ltimes \mathbb R^8 \subset \mathfrak{so}(1, 9)\) are constructed.


53C29 Issues of holonomy in differential geometry
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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