2010/01/25 by Anton S. Galaev
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Geometric Analysis and Curvature Flows #math.DG #msc:53B30 #msc:53C29 #msc:53C50
paper · pdf · doi:10.1016/j.geomphys.2010.03.002
published as J.Geom.Phys.60:962-971,2010 · An extended version of a part from arXiv:0906.1327
arxiv created 2010/01/25 · openalex publication_date 2010/03/16 · arxiv updated 2010/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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⊕\so(n))\zr\Realn⊂\so(1,n+1). An important invariant of \g is its \so(n)-projection \h⊂\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 \Realn 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.