%0 Journal Article
%T A Review on Unique Existence Theorems in Lightlike Geometry
%A K. L. Duggal
%J Geometry
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/835394
%X This is a review paper of up-to-date research done on the existence of unique null curves, screen distributions, Levi-Civita connection, symmetric Ricci tensor, and scalar curvature for a large variety of lightlike submanifolds of semi-Riemannian (in particular, Lorentzian) manifolds, supported by examples and an extensive bibliography. We also propose some open problems. 1. Introduction The theory of Riemannian and semi-Riemannian manifolds and their submanifold is one of the most interesting areas of research in differential geometry. Most of the work on the Riemannian, semi-Riemannian, and Lorentzian manifolds has been described in the standard books by Chen [1], Beem and Ehrlich [2], and O¡¯Neill [3]. Berger¡¯s book [4] includes the major developments of Riemannian geometry since 1950, covering the works of differential geometers of that time and many cited therein. In general, an inner product on a vector space V is of type , where , with for all nonzero , and with for all nonzero . Kupeli [5] called a manifold of this type a singular semi-Riemannian manifold if admits a Koszul derivative; that is, is Lie parallel along the degenerate vector fields on . Based on this, he studied the intrinsic geometry of such degenerate manifolds. On the other hand, a degenerate submanifold of a semi-Riemannian manifold may not be studied intrinsically since due to the degenerate tensor field on one cannot use, in general, the geometry of . To overcome this difficulty, Kupeli used the quotient space and the canonical projection for the study of intrinsic geometry of , where is its radical distribution. For a general study of extrinsic geometry of degenerate submanifolds (popularly known as lightlike submanifolds) of a semi-Riemannian manifold, we refer to three books [6¨C8] published in 1996, 2007, and 2010, respectively. A submanifold of a semi-Riemannian manifold is called lightlike submanifold if it is a lightlike manifold with respect to the degenerate metric induced from and is a nondegenerate screen distribution which is complementary of the radical distribution in ; that is, where is a symbol for orthogonal direct sum. The technique of using a nondegenerate was first introduced by Bejancu [9] for null curves and then by Bejancu and Duggal [10] for hypersurfaces to study the induced geometry of lightlike submanifolds. Unfortunately, (i) the induced objects on depend on which, in general, is not unique. This raises the question of the existence of unique or canonical null curves and screen distributions in lightlike geometry. (ii) The induced connection on is not
%U http://www.hindawi.com/journals/geometry/2014/835394/