Generalized Transversal Lightlike Submanifolds of Indefinite Sasakian Manifolds

We introduce and study generalized transversal lightlike submanifold of indefinite Sasakian manifolds which includes radical and transversal lightlike submanifolds of indefinite Sasakian manifolds as its trivial subcases. A characteristic theorem and a classification theorem of generalized transversal lightlike submanifolds are obtained. 1. Introduction The theory of submanifolds in Riemannian geometry is one of the most important topics in differential geometry for years. We see from [1] that semi-Riemannian submanifolds have many similarities with the Riemannian counterparts. However, it is well known that the intersection of the normal bundle and the tangent bundle of a submanifold of a semi-Riemannian manifold may be not trivial, so it is more difficult and interesting to study the geometry of lightlike submanifolds than nondegenerate submanifolds. The two standard methods to deal with the above difficulties were developed by Kupeli [2] and Duggal-Bejancu [3, 4], respectively. The study of CR-lightlike submanifolds of an indefinite Kaehler manifold was initiated by Duggal-Bejancu [3]. Since the book was published, many geometers investigated the lightlike submanifolds of indefinite Kaehler manifolds by generalizing the CR-lightlike submanifold [3], SCR-lightlike submanifolds [5] to GCR-lightlike submanifolds [6], and discussing the integrability and umbilication of these lightlike submanifolds. We also refer the reader to [7] for invariant lightlike submanifolds and to [8] for totally real lightlike submanifolds of indefinite Kaehler manifolds. On the other hand, after Duggal-Sahin introduced screen real lightlike submanifolds and contact screen CR-lightlike submanifolds [9] of indefinite Sasakian manifolds by studying the integrability of distributions and the geometry of leaves of distributions as well as other properties of this submanifolds, the generalized CR-lightlike submanifold which contains contact CR and SCR-lightlike submanifolds were introduced in [4]. However, all these submanifolds of indefinite Sasakian manifolds mentioned above have the same geometric condition , where is the almost contact structure on indefinite Sasakian manifolds, is the radical distribution, and is the tangent bundle. Until recently Y？ld？r？m and ？ahin [10] introduced radical transversal and transversal lightlike submanifold of indefinite Sasakian manifolds for which the action of the almost contact structure on radical distribution of such submanifolds does not belong to the tangent bundle, more precisely, , where is the lightlike transversal bundle of lightlike