Nonexistence of Totally Contact Umbilical Slant Lightlike Submanifolds of Indefinite Cosymplectic Manifolds

We study totally contact umbilical slant lightlike submanifolds of indefinite cosymplectic manifolds. We prove that there do not exist totally contact umbilical proper slant lightlike submanifolds in indefinite cosymplectic manifolds other than totally contact geodesic proper slant lightlike submanifolds. We also prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic space forms. Finally we give characterization theorems on minimal slant lightlike submanifolds. 1. Introduction The notion of slant submanifolds was initiated by Chen [1, 2], as a generalization of both holomorphic and totally real submanifolds in complex geometry. Since then such submanifolds have been studied by many authors. Lotta [3, 4] defined and studied slant submanifolds in contact geometry. Cabrerizo et al. studied slant, semislant, and bislant submanifolds in contact geometry [5, 6]. They all studied the geometry of slant submanifolds with positive definite metric. Therefore this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite. Thus the geometry of slant submanifolds with indefinite metric became a topic of chief discussion, and ？ahin [7] played a very crucial role in this direction by introducing the notion of slant lightlike submanifolds of indefinite Hermitian manifolds. Recently Gupta et al. [8] introduced the notion of slant lightlike submanifolds of an indefinite cosymplectic manifold and obtained necessary and sufficient conditions for the existence of slant lightlike submanifolds. In [9], Jain et al. studied -lightlike submanifolds of indefinite cosymplectic manifolds and proved that every totally contact umbilical GCR-lightlike submanifold of an indefinite cosymplectic manifold is a totally contact geodesic -lightlike submanifold. In this paper we prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic manifolds other than totally contact geodesic. We also prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic space forms. Finally, we give characterization theorems on minimal slant lightlike submanifolds. 2. Preliminaries An odd-dimensional semi-Riemannian manifold is said to be an indefinite almost contact metric manifold if there exist structure tensors , where is a tensor field, is a vector field, called structure vector field, is a -form, and is the semi-Riemannian metric on satisfying (see [10]) the