One component of the curvature tensor of a Lorentzian manifold

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 $¶(\h)$ 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 $(M,g)$. These results can be applied e.g. to the holonomy classification of the Einstein Lorentzian manifolds.

arXiv commentAn extended version of a part from arXiv:0906.1327

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Cited by in corpus (10)

• Two-symmetric Lorentzian manifolds
• On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines
• Irreducible holonomy algebras of Riemannian supermanifolds
• Some applications of the Lorentzian holonomy algebras
• Classification of third-order symmetric Lorentzian manifolds
• How to find the holonomy algebra of a Lorentzian manifold
• Covariant derivative of the curvature tensor of pseudo-Kählerian manifolds
• Conformally flat Lorentzian manifolds with special holonomy groups
• About the classification of the holonomy algebras of Lorentzian manifolds
• Holonomy algebras of pseudo-hyper-Kählerian manifolds of index 4