One component of the curvature tensor of a Lorentzian manifold
arXiv:1001.4441 · doi:10.1016/j.geomphys.2010.03.002
Abstract
The holonomy algebra $\g$ of an -dimensional Lorentzian manifold 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 $¶(\h)$ consisting of linear maps from $\Real^n$ to $\h$ satisfying an identity similar to the Bianchi one. In the present paper the spaces $¶(\h)$ are computed for each possible $\h$. This gives the complete description of the values of the curvature tensor of the manifold . These results can be applied e.g. to the holonomy classification of the Einstein Lorentzian manifolds.
An extended version of a part from arXiv:0906.1327