tailieunhanh - Pseudosymmetric lightlike hypersurfaces

We study lightlike hypersurfaces of a semi-Riemannian manifold satisfying pseudosymmetry conditions. We give sufficient conditions for a lightlike hypersurface to be pseudosymmetric and show that there is a close relationship of the pseudosymmetry condition of a lightlike hypersurface and its integrable screen distribution. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 1050 – 1070 ¨ ITAK ˙ c TUB ⃝ doi: Pseudosymmetric lightlike hypersurfaces ˙ ∗ Sema KAZAN, Bayram S ¸ AHIN ˙ Department of Mathematics, Faculty of Science and Arts, In¨ on¨ u University, 44280, Malatya, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: We study lightlike hypersurfaces of a semi-Riemannian manifold satisfying pseudosymmetry conditions. We give sufficient conditions for a lightlike hypersurface to be pseudosymmetric and show that there is a close relationship of the pseudosymmetry condition of a lightlike hypersurface and its integrable screen distribution. We obtain that a pseudosymmetric lightlike hypersurface is a semisymmetric lightlike hypersurface or totally geodesic under certain conditions. Moreover, we give an example of pseudosymmetric lightlike hypersurfaces and investigate pseudoparallel lightlike hypersurfaces. Furthermore, we introduce Ricci-pseudosymmetric lightlike hypersurfaces, obtain characterizations, and give an example for such hypersurfaces. Key words: Semisymmetric lightlike hypersurface, Ricci-semisymmetric lightlike hypersurface, pseudosymmetric lightlike hypersurface, pseudoparallel lightlike hypersurface 1. Introduction Let (M, g) be a Riemannian manifold of dimension n and ∇ be the Levi-Civita connection. A Riemannian manifold is called locally symmetric if ∇R = 0, where R is the Riemannian curvature tensor of M [6]. Locally symmetric Riemannian manifolds are a generalization of manifolds of constant curvature. As a generalization of locally symmetric Riemannian manifolds, semisymmetric Riemannian manifolds were defined by the condition R · R = 0. It is known that locally symmetric manifolds are semisymmetric manifolds but the converse is not true [28]. Such manifolds were investigated by Cartan and they were locally .