Note on the holonomy groups of pseudo-Riemannian manifolds

For an arbitrary subalgebra $\mathfrak{h}\subset\mathfrak{so}(r,s)$, a polynomial pseudo-Riemannian metric of signature $(r+2,s+2)$ is constructed, the holonomy algebra of this metric contains $\mathfrak{h}$ as a subalgebra. This result shows the essential distinction of the holonomy algebras of pseudo-Riemannian manifolds of index bigger or equal to 2 from the holonomy algebras of Riemannian and Lorentzian manifolds.Comment: 6 pages, final versio

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

Examples of Einstein spacetimes with recurrent null vector fields

The Einstein Equation on 4-dimensional Lorentzian manifolds admitting recurrent null vector fields is discussed. Several examples of a special form are constructed. The holonomy algebras, Petrov types and the Lie algebras of Killing vector fields of the obtained metrics are found.Comment: 7 pages, the final versio

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

Holonomy algebras of Einstein pseudo-Riemannian manifolds

The holonomy algebras of Einstein not Ricci-flat pseudo-Riemannian manifolds of arbitrary signature are classified. As illustrating examples, the cases of Lorentzian manifolds, pseudo-Riemannian manifolds of signature $(2,n)$ and the para-quaternionic-K\"ahlerian manifolds with non-zero scalar curvature are considered. Einstein not Ricci-flat metrics of signature $(2,n)$ with all possible holonomy algebras are given.Comment: final version accepted to Journal of the London Mathematical Societ