Abstract. We study the geometry of lightlike hypersurfaces M of an indefinite cosymplectic manifoldM such that either (1) the characteristic vector field ζ ofM belongs to the screen distribution
IntroductionThe theory of lightlike submanifolds is one of the interesting topics of differential geometry. This theory is relatively new and in a developing stage. Many authors studied the geometry of lightlike submanifolds of indefinite Sasakian manifolds. Recently several authors have studied the geometry of lightlike submanifolds of an indefinite cosymplectic manifold [10].The purpose of this paper is to study the geometry of lightlike hypersurfaces M of an indefinite cosymplectic manifoldM subject to the conditions :
(1) The characteristic vector field ζ ofM belongs to the screen distribution S(T M ) of M , or (2) ζ belongs to the orthogonal complement S(T M )⊥ of S(T M ) in TM . We provide several new results on lightlike hypersurfaces M of this two types by using the structure tensors of M induced by the contact metric structure tensor J ofM .
Lightlike hypersurfacesAn odd dimensional smooth manifold (M ,ḡ) is called a contact metric manifold [1,8] if there exist a (1, 1)-type tensor field J, a vector field ζ, called the characteristic vector field, and its 1-form θ satisfying
Abstract. We study half lightlike submanifolds of an indefinite Sasakian manifold. The aim of this paper is to prove the following result: If a locally symmetric half lightlike submanifold of an indefinite Sasakian manifold is totally umbilical, then it is of constant positive curvature 1. In addition to this result, we prove three characterization theorems for such a half lightlike submanifold.
IntroductionThe class of lightlike submanifolds of codimension 2 is composed entirely of two classes by virtue of the rank of its radical distribution, named by half lightlike or coisotropic submanifolds [3]. Half lightlike submanifold is a special case of r-lightlike submanifold [2] such that r = 1 and its geometry is more general form than that of coisotrophic submanifolds. Much of the works on half lightlike submanifolds will be immediately generalized in a formal way to arbitrary r-lightlike submanifolds. Recently several authors have studied the geometry of lightlike submanifolds of an indefinite Sasakian manifold. Many works of such lightlike submanifolds assumed that M is totally umbilical (or totally geodesic), or M is screen conformal, or its screen distribution S(T M ) is totally umbilical in M .In the theory of Sasakian manifolds, the following result is well-known: If a Sasakian manifold is locally symmetric, then it is of constant positive curvature 1. We proved lightlike hypersurface version of the above classical result: If a locally symmetric lightlike hypersurface of an indefinite Sasakian manifold is totally geodesic, then it is of constant positive curvature 1 [5].
We study lightlike submanifolds M of a semi-Riemannian manifoldM with a semi-symmetric non-metric connection subject to the conditions; (a) the characteristic vector field ofM is tangent to M , (b) the screen distribution on M is totally umbilical in M and (c) the co-screen distribution on M is conformal Killing.
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