One component of the curvature tensor of a Lorentzian manifold

The holonomy algebra \g of an $n+2$-dimensional Lorentzian manifold $(M,g)$ admitting a parallel distribution of isotropic lines is contained in the subalgebra \simil(n)=(\Real\oplus\so(n))\zr\Real^n\subset\so(1,n+1). An important invariant of \g is its \so(n)-projection \h\subset\so(n), which is a Riemannian holonomy algebra. One component of the curvature tensor of the manifold belongs to the space \P(\h) consisting of linear maps from \Real^n to \h satisfying an identity similar to the Bianchi one. In the present paper the spaces \P(\h) are computed for each possible \h. This gives the complete description of the values of the curvature tensor of the manifold $(M,g)$. These results can be applied e.g. to the holonomy classification of the Einstein Lorentzian manifolds.Comment: An extended version of a part from arXiv:0906.132

Isometry groups of Lobachevskian spaces, similarity transformation groups of Euclidean spaces and Lorentzian holonomy groups

Weakly-irreducible not irreducible subalgebras of \so(1,n+1) were classified by L. Berard Bergery and A. Ikemakhen. In the present paper a geometrical proof of this result is given. Transitively acting isometry groups of Lobachevskian spaces and transitively acting similarity transformation groups of Euclidean spaces are classified.Comment: 12 page

About the classification of the holonomy algebras of Lorentzian manifolds

The classification of the holonomy algebras of Lorentzian manifolds can be reduced to the classification of irreducible subalgebras $\mathfrak{h}\subset\mathfrak{so}(n)$ that are spanned by the images of linear maps from $\mathbb{R}^n$ to $\mathfrak{h}$ satisfying an identity similar to the Bianchi one. T. Leistner found all such subalgebras and it turned out that the obtained list coincides with the list of irreducible holonomy algebras of Riemannian manifolds. The natural problem is to give a simple direct proof to this fact. We give such proof for the case of semisimple not simple Lie algebras $\mathfrak{h}$.Comment: 9 pages, the final versio