One component of the curvature tensor of a Lorentzian manifold

Details One component of the curvature tensor of a Lorentzian manifold One component of the curvature tensor of a Lorentzian manifold

2010-01-25

arxiv.org/abs/1001.4441v1

J.Geom.Phys.60:962-971,2010

Mathematics

Differential Geometry

An extended version of a part from arXiv:0906.1327

Scientific paper

10.1016/j.geomphys.2010.03.002

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.

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