Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 567
Berger algebras, weak-Berger algebras and Lorentzian holonomy. Thomas Leistner
The holonomy algebra of an indecomposable but non--irreducible Lorentzian manifold of dimension $n+2$ is contained in $( mathbb{R} oplus mathfrak{so}(n) ) ltimes mathbb{R}^n$. In contrary to the whole holonomy the $mathfrak{so}(n)$--projection decomposes in irreducible acting Lie algebras. We derive an algebraic criterion to this projection and its irreducible parts, which is weaker then the usual Berger criterion, but also based on the first Bianchi identity. Using this weak--Berger criterion we show, that every such projection which is contained in $mathfrak{u} (n/2)$ is the holonomy algebra of a Riemannian manifold.
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