LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

Abstract

Abstract. We study the geometry of r-lightlike submanifolds M of asemi-Riemannian manifold M with a semi-symmetric non-metric connec-tion subject to the conditions; (a) the screen distribution of M is to-tally geodesic in M, and (b) at least one among the r-th lightlike secondfundamental forms is parallel with respect to the induced connection ofM. The main result is a classi cation theorem for irrotational r-lightlikesubmanifold of a semi-Riemannian manifold of index r admitting a semi-symmetric non-metric connection. 1. IntroductionThe geometry of lightlike submanifolds is used in mathematical physics, inparticular, in general relativity since lightlike submanifolds produce models ofdi erent types of horizons (event horizons, Cauchy’s horizons, Kruskal’s hori-zons). The universe can be represented as a four dimensional Lorentz subman-ifold (spacetime) embedded in an (n+ 4)-dimensional semi-Riemannian mani-fold. Lightlike hypersurfaces are also studied in the theory of electromagnetism[1]. Thus, large number of applications but limited information available, moti-vated us to do research on this subject matter. Duggal-Bejancu [1] and Kupeli[2] developed the general theory of degenerate (lightlike) submanifolds. Theyconstructed a transversal vector bundle of lightlike submanifold and investi-gated various properties of these manifolds. Duggal-Jin [3] studied totallyumbilical lightlike submanifold of a semi-Riemannian manifold. Ageshe andChae [4] introduced the notion of a semi-symmetric non-metric connection ona Riemannian manifold. Ya˘sar, C˘oken and Yucesan [5] and Jin [6] studied light-like hypersurfaces in semi-Riemannian manifolds admitting a semi-symmetricnon-metric connections. The geometry of half lightlike submanifolds of a semi-Riemannian manifold with semi-symmetric non-metric connection was studied