In this paper, we investigate lightlike hypersurfaces which are semi-symmetric, Ricci semi-symmetric, parallel or semi-parallel in a semi-Euclidean space. We obtain that every screen conformal lightlike hypersurface of the Minkowski spacetime is semi-symmetric. | Turk J Math 31 (2007) , 139 – 162. ¨ ITAK ˙ c TUB Lightlike Hypersurfaces of Semi-Euclidean Spaces Satisfying Curvature Conditions of Semisymmetry Type Bayram S ¸ ahin Abstract In this paper, we investigate lightlike hypersurfaces which are semi-symmetric, Ricci semi-symmetric, parallel or semi-parallel in a semi-Euclidean space. We obtain that every screen conformal lightlike hypersurface of the Minkowski spacetime is semi-symmetric. For higher dimensions, we show that the semi-symmetry condition of a screen conformal lightlike hypersurface reduces to the semi-symmetry condition of a leaf of its screen distribution. We also obtain that semi-symmetric and Ricci semi-symmetric lightlike hypersurfaces are totally geodesic under certain conditions. Moreover, we show that there exist no non-totally geodesic parallel hypersurfaces in a Lorentzian space. Key Words: Degenerate metric, Screen conformal lightlike hypersurface, Parallel lightlike hypersurface, Semi-symmetric lightlike hypersurface. 1. Introduction The class of semi-Riemannian manifolds, satisfying the condition ∇R = 0, () is a natural generalization of the class of manifolds of constant curvature, where ∇ is the Levi-Civita connection on semi-Riemannian manifold and R is the corresponding 2000 AMS Mathematics Subject Classification: 53C15, 53C40, 53C50. 139 ˙ S ¸ AHIN curvature tensor. For precise definitions of the symbols used, we refer to Section . A semi-Riemannian manifold is called semi-symmetric if R·R = 0, () where R is the curvature operator corresponding to R and the · operation is defined in Section . Semi-symmetric hypersurfaces of Euclidean spaces were classified by Nomizu [15] and a general study of semi-symmetric Riemannian manifolds was made by Szabo [17]. A semi-Riemannian manifold is said to be Ricci semi-symmetric [7], if the following condition is satisfied: R · Ric = 0. () It is clear that every semi-symmetric manifold is Ricci semi-symmetric; the converse .
