Geodesic Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

The aim of the present paper is to study geodesic contact screen Cauchy Riemannian (SCR-) lightlike submanifolds, geodesic screen transversal lightlike, and geodesic transversal lightlike submanifolds of indefinite Kenmotsu manifolds. 1. Introduction The study of the geometry of submanifolds of a Riemannian or semi-Riemannian manifold is one of the interesting topics of differential geometry. Despite of some similarities between semi-Riemannian manifolds and Riemannian manifolds, the lightlike submanifolds are different since their normal vector bundle intersect with the tangent bundle making it more interesting and difficult to study. These submanifolds were introduced and studied by Duggal and Bejancu [1]. On the other hand, geodesic CR-lightlike submanifolds in K？hler manifolds were studied by Sahin and Gunes [2], and geodesic lightlike submanifolds of indefinite Sasakian manifolds were investigated by Dong and Liu [3]. In 2006, Sahin [4] initiated the study of transversal lightlike submanifolds of an indefinite K？hler manifold which are different from CR-lightlike [1] and screen CR-lightlike submanifolds [5]. Recently, Sahin [6] introduced the notion of screen transversal lightlike submanifolds of indefinite K？hler manifolds and obtained some useful results. In this paper, we study geometric conditions under which some lightlike submanifolds of an indefinite Kenmotsu manifold are totally geodesic. 2. Preliminaries We follow [1] for the notation and fundamental equations for lightlike submanifolds used in this paper. A submanifold immersed in a semi-Riemannian manifold is called a lightlike submanifold if it admits a degenerate metric induced from whose radical distribution is of rank , where and Let be a screen distribution which is a semi-Riemannian complementary distribution of in , that is, Consider a screen transversal vector bundle , which is a semi-Riemannian complementary vector bundle of in . Since for any local basis of , there exists a local null frame of sections with values in the orthogonal complement of in such that and , it follows that there exists a lightlike transversal vector bundle locally spanned by [1, page 144]. Let be the complementary (but not orthogonal) vector bundle to in . Then, Let be the Levi-Civita connection on . Then, in view of the decomposition (2.3), the Gauss and Weingarten formulas are given by where and belong to and , respectively, and are linear connection on and on the vector bundle , respectively. Moreover, we have for all, , and . If we denote the projection of on by , then by using (2.6)–(2.8) and the