How to find the holonomy algebra of a Lorentzian manifold

Details How to find the holonomy algebra of a Lorentzian manifold How to find the holonomy algebra of a Lorentzian manifold

2011-10-10

arxiv.org/abs/1110.1998v1

Mathematics

Differential Geometry

18 pages

Scientific paper

If the holonomy algebra $\mathfrak{g}\subset\mathfrak{so}(1,n-1)$ of a locally indecomposable Lorentzian manifold $(M,g)$ of dimension $n$ is different from $\mathfrak{so}(1,n-1)$, then it is contained in the similitude algebra $\mathfrak{sim}(n-2)$. There are 4 types of such holonomy algebras. We give criterion how to find the type of $\mathfrak{g}$. To each $\mathfrak{g}$ there is a canonically associated subalgebra $\mathfrak{h}\subset\mathfrak{so}(n-2)$. We provide an algorithm how to find $\mathfrak{h}$. We also give algorithms for obtaining the de Rham-Wu decomposition for Riemannian and Lorentzian manifolds. These results show how one can find the holonomy algebra of an arbitrary Lorentzian manifold.

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