One component of the curvature tensor of a Lorentzian manifold

The holonomy algebra $\text{g}$ of an $n+2$-dimensional Lorentzian manifold $(M,g)$ admitting a parallel distribution of isotropic lines is contained in the subalgebra $\text{simil}(n)=(\text{Real}\oplus \text{so}(n))\text{zr}{\text{Real}}^n\subset \text{so}(1,n+1)$. An important invariant of $\text{g}$ is its $\text{so}(n)$-projection $\text{h}\subset \text{so}(n)$, which is a Riemannian holonomy algebra. One component of the curvature tensor of the manifold belongs to the space $\mathrm{\P }(\text{h})$ consisting of linear maps from ${\text{Real}}^n$ to $\text{h}$ satisfying an identity similar to the Bianchi one. In the present paper the spaces $\mathrm{\P }(\text{h})$ are computed for each possible $\text{h}$. 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.