Some properties of lightlike submanifolds of semi-Riemannian manifolds

We initially obtain various relations and then establish necessary and sufficient condition for the integrability of screen distribution of a lightlike submanifold. We also establish necessary and sufficient condition for a lightlike submanifold to be totally geodesic. In the generalization from Riemannian to semi-Riemannian manifolds, the induced metric may be degenerate (lightlike) therefore there is a natural existence of lightlike submanifolds and for which the local and global geom- etry is completely different than non-degenerate case. In lightlike case the standard text book definitions do not make sense and one fails to use the theory of non-degenerate geometry in the usual way. The primary difference between the lightlike submanifolds and the non-degenerate submanifolds is that in the first case, the normal vector bundle intersects with the tangent bundle. Thus, the study of lightlike submanifolds becomes more difficult and different from the study of non-degenerate submanifolds. Moreover, the geometry of lightlike submanifolds is used in mathematical physics, in par- ticular, in general relativity since lightlike submanifolds produce models of different types of horizons (event horizons, Cauchy's horizons, Kruskal's hori- zons). The universe can be represented as a four dimensional submanifold embedded in a (4 + n)-dimensional spacetime manifold. Lightlike hyper- surfaces are also studied in the theory of electromagnetism (1). Thus, large number of applications but limited information available, motivated us to do research on this subject matter. Kupeli (6) and Bejancu-Duggal (1) developed the general theory of degenerate (lightlike) submanifolds. They constructed a transversal vector bundle of lightlike submanifold and investigated vari-