On Parallelism of Half-Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

We mainly investigate the parallelism of half-lightlike submanifolds of indefinite Kenmotsu manifolds. It is proved that a tangential half-lightlike submanifold of an indefinite Kenmotsu space form with semiparallel second fundamental form either satisfies or is -mixed geodesic. 1. Introduction As the intersection of normal bundle and tangent bundle of a submanifold of a semi-Riemannian manifold may not be trivial, 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 [1], Duggal and Bejancu [2], Duggal and Jin [3], and Duggal and Sahin [4], respectively. Let be a lightlike submanifold immersed in a semi-Riemannian manifold, it is obvious to see that there are two cases of codimension 2 lightlike submanifolds, since for this type the dimension of their radical distributions is either 1 or 2. A codimension 2 lightlike submanifold of a semi-Riemannian manifold is called a half-lightlike submanifold [5] if , where denotes the degenerate radical distribution of . For more results about half-lightlike submanifolds, we refer the reader to [4, 6, 7]. In the theory of submanifolds of Riemannian manifolds, the parallel and semiparallel immersions were studied by Ferus [8] and Deprez [9], respectively. Recently, Massamba [10–13] and Upadhyay and Gupta [14] studied the parallel and semiparallel lightlike hypersurfaces of an indefinite Sasakian, Kenmotsu, and cosymplectic manifolds, respectively. However, the parallel and semiparallel half-lightlike submanifolds of an indefinite Kenmotsu manifolds have not yet been considered. The aim of this paper is to investigate the parallelism of half-lightlike submanifolds of indefinite Kenmotsu manifolds. This paper is organized in the following way. In Section 2, we provide some well-known basic formulas and properties of indefinite Kenmotsu manifolds and half-lightlike submanifolds. Section 3 is devoted to presenting some main results on semiparallel half-lightlike submanifolds of indefinite Kenmotsu space form. Finally, in Section 4, some properties of parallel half-lightlike submanifolds of indefinite Kenmotsu manifolds are investigated. 2. Preliminaries In this section, we follow Duggal and Sahin [4] for the notation and fundamental equations for half-lightlike submanifolds of indefinite Kenmotsu manifolds. A -dimensional semi-Riemannian is said to be an indefinite Kenmotsu manifold if it admits a normal almost contact metric structure , where is a tensor field of type