One component of the curvature tensor of a Lorentzian manifold
| Authors | |
|---|---|
| Year of publication | 2010 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Geometry and Physics |
| MU Faculty or unit | |
| Citation | |
| Field | General mathematics |
| Keywords | Lorentzian manifold; Holonomy algebra; Curvature tensor; Einstein manifold |
| Description | 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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