One component of the curvature tensor of a Lorentzian manifold

TL;DR

This paper computes the space of certain curvature tensor components for Lorentzian manifolds with specific holonomy properties, aiding in the classification of Einstein Lorentzian manifolds.

Contribution

It provides a complete calculation of the space (\u03b7) for all possible (), enabling a detailed understanding of the curvature tensor in these manifolds.

Findings

Computed () for all () cases

Provided a full description of the curvature tensor values

Facilitated holonomy classification of Einstein Lorentzian manifolds

Abstract

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.