The geometry of lightlike hypersurfaces

This thesis consists of four chapter.The first chapter has been designed as an introduction.The second chapter is devoted to the basic materials such as vector bundles, distributions, semi-Riemannian manifolds, lightlike submanifolds of semi-Riemannian manifold and lightlike manifolds, which will be useful for other chapters. In the third chapter, lightlike hypersurfaces have been introduced and theorems about lightlike hypersurfaces have been given. After investigating unique existence of screen distributions, it was found basic results on unique existence of screen distributions. Besides, the induced scalar curvature which is a new concept is examined in a lightlike hypersurface of a semi-Riemanian manifold. In the fourth chapter, we introduced lightlike Einstein hypersurfaces and the geometry of distributions is investigated on the lightlike Einstein hypersurfaces. Besides, Gauss equation for lightlike hypersurface of a semi-Euclidean space has been given. Then, we obtained that every screen conformal lightlike hypersurface of the Minkowski spacetime is semi-symmetric. We showed that the semi-symmetry condition of a screen conformal lightlike hypersurface reduced to the semi-symmetry condition of a leaf of its screen distribütion. We also obtained that semi-symmetric and Ricci semi-symmetric lightlike hypersurfaces are totally geodesic under certain conditions. Morever, we showed that there exist no non-totally geodesic parallel hypersurfaces in a Lorentzian space. KEY WORDS: Semi-Riemanian manifold, Lightlike manifold, Lightlike hypersurface, Screen conformal hypersurface, Lightlike Einstain hypersurface, Induced scalar curvature, Totally geodesic, Totally umbilical, Semi-symmetry metric connection, Ricci curvature tensor