The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds

Details The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds

2003-04-25

arxiv.org/abs/math/0304407v3

Differential Geom. Appl. 22 no. 1 (2005) 1--18

Mathematics

Differential Geometry

21 pages

Scientific paper

The holonomy algebra $\g$ of an indecomposable Lorentzian (n+2)-dimensional manifold $M$ is a weakly-irreducible subalgebra of the Lorentzian algebra $\so_{1,n+1}$. L. Berard Bergery and A. Ikemakhen divided weakly-irreducible not irreducible subalgebras into 4 types and associated with each such subalgebra $\g$ a subalgebra $\h\subset \so_n$ of the orthogonal Lie algebra. We give a description of the spaces $\R(\g)$ of the curvature tensors for algebras of each type in terms of the space $\P(\h)$ of $\h$-valued 1-forms on $\Real^n$ that satisfy the Bianchi identity and reduce the classification of the holonomy algebras of Lorentzian manifolds to the classification of irreducible subalgebras $\h$ of $so(n)$ with $L(\P(\h))=\h$. We prove that for $n\leq 9$ any such subalgebra $\h$ is the holonomy algebra of a Riemannian manifold. This gives a classification of the holonomy algebras for Lorentzian manifolds $M$ of dimension $\leq 11$.

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