One component of the curvature tensor of a Lorentzian manifold

The holonomy algebra gg of an n+2n+2-dimensional Lorentzian manifold (M,g)(M,g) admitting a parallel distribution of isotropic lines is contained in the subalgebra sim(n)=(R⊕so(n))⋉Rn⊂so(1,n+1)sim(n)=(R⊕so(n))⋉Rn⊂so(1,n+1). An important invariant of gg is its so(n)so(n)-projection h⊂so(n)h⊂so(n), which is a Riemannian holonomy algebra. One component of the curvature tensor of the manifold belongs to the space P(h)P(h) consisting of linear maps from RnRn to hh satisfying an identity similar to the Bianchi one. In the present paper the spaces P(h)P(h) are computed for each possible hh. This gives the complete description of the values of the curvature tensor of the manifold (M,g)(M,g). These results can be applied e.g. to the holonomy classification of the Einstein Lorentzian manifolds.