Lightlike Submanifolds Research Articles

Half lightlike submanifolds of an indefinite generalized Sasakian space form with a quarter-symmetric metric connection

We study the geometry of half lightlike submanifolds of an indenite generalized Sasakian space form M(f1;f2;f3) with a quarter-symmetric metric connection. Our main result is several characterization theorems for such a half lightlike submanifold of M(f1;f2;f3) endowed with an indenite trans-Sasakian structure of type (

A semi-Riemannian manifold of quasi-constant curvature with half lightlike submanifolds

A semi-Riemannian manifold of quasi-constant curvature with half lightlike submanifolds

Non-screenable generic lightlike submanifolds of an indefinite Kaehler manifold of a quasi-constant curvature

Non-screenable generic lightlike submanifolds of an indefinite Kaehler manifold of a quasi-constant curvature

Non-existence of lightlike submanifolds of an indefinite trans-Sasakian manifold with a non-metric (\phi,\theta)-connection

Non-existence of lightlike submanifolds of an indefinite trans-Sasakian manifold with a non-metric (\phi,\theta)-connection

A semi-Riemannian manifold of quasi-constant curvature admits lightlike submanifolds

A semi-Riemannian manifold of quasi-constant curvature admits lightlike submanifolds

A characterization of half lightlike submanifolds of indefinite generalized Sasakian space forms

A characterization of half lightlike submanifolds of indefinite generalized Sasakian space forms

Non-screenable half lightlike submanifolds of an indefinite Kaehler manifold of a quasi-constant curvature

Non-screenable half lightlike submanifolds of an indefinite Kaehler manifold of a quasi-constant curvature

GEOMETRY OF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KENMOTSU MANIFOLD

Abstract. In this paper, we study the forms of the curvatures oflightlike submanifolds M of an inde nite Kenmotsu manifold Msubject to the conditions: M is locally symmetric or M is semi-symmetric. 1. IntroductionIn the classical theory of Sasakian manifolds, the following result iswell-known: If a Sasakian manifold is locally symmetric, then it is ofconstant positive curvature 1. In 2012, K.L. Duggal and D.H. Jin [4]studied the forms of curvatures of locally symmetric generic lightlikesubmanifolds of an inde nite Sasakian manifold. They obtained thefollowing result: If a locally symmetric generic lightlike submanifold withat lightlike transversal connection of an inde nite Sasakian manifold istotally umbilical, then it is of constant positive curvature 1.Further in 1971, K. Kenmotsu proved the following result: If a Ken-motsu manifold is locally symmetric, then it is of constant negativecurvature 1 [9]. D.H. Jin proved the lightlike version of this result forlightlike hypersurfaces [5, 7] and half lightlike submanifolds [6, 8].The objective of this paper is to study the forms of the curvaturesof general r-lightlike submanifolds of an inde nite Kenmotsu manifoldsubject to the conditions: (1) Mis locally symmetric, i:e:, the curvaturetensor Rof M satis es rR= 0, or (2) M is semi-symmetric, i:e:, Rsatis es RR= 0. We prove the following theorems:

Radical Transversal Lightlike Submanifolds of Indefinite Para-Sasakian Manifolds

Abstract In this paper, we study radical transversal lightlike submanifolds and screen slant radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds giving some non-trivial examples of these submanifolds. Integrability conditions of distributions D and RadTM on radical transversal lightlike submanifolds and screen slant radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds, have been obtained. We also study totally contact umbilical radical transversal lightlike submanifolds of indefinite para-Sasakian manifolds.

LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE SASAKIAN MANIFOLD WITH A NON-METRIC θ-CONNECTION

In this paper, we study two types of 1-lightlike submanifolds, named by lightlike hypersurface and half lightlike submanifold, of an indefinite Sasakian manifold admitting non-metric <TEX>${\theta}$</TEX>-connections. We prove that there exist no such two types of 1-lightlike submanifolds of an indefinite Sasakian manifold.

INDEFINITE GENERALIZED SASAKIAN SPACE FORM ADMITTING A GENERIC LIGHTLIKE SUBMANIFOLD

In this paper, we study the geometry of indefinite generalized Sasakian space form <TEX>$\bar{M}(f_1,f_2,f_3)$</TEX> admitting a generic lightlike submanifold M subject such that the structure vector field of <TEX>$\bar{M}(f_1,f_2,f_3)$</TEX> is tangent to M. The purpose of this paper is to prove a classification theorem of such an indefinite generalized Sasakian space form.

NON-EXISTENCE OF LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KAEHLER MANIFOLDS ADMITTING NON-METRIC π-CONNECTIONS

Abstract. In this paper, we study two types 1-lightlike submanifoldsM, so called lightlike hypersurface and half lightlike submanifold, of anindefinite Kaehler manifold M¯ admitting non-metric π-connection. Weprove that there exist no such two types 1-lightlike submanifolds of anindefinite Kaehler manifold M¯ admitting non-metric π-connections. 1. IntroductionA linear connection ∇¯ on a semi-Riemannian manifold (M,¯ ¯g) is called anon-metric π-connection if, for any vector fields X, Y and Z on M¯, it satisfies(1.1) (∇¯ X g¯)(Y,Z) = −π(Y )¯g(X,Z)−π(Z)¯g(X,Y ),where π is a 1-form, associated with a non-vanishing smooth vector field ζ onM¯ by π(X) = ¯g(X,ζ). We say that ζ is the characteristic vector field of M¯.Two special cases are important for both the mathematical study and theapplications to physics: (1) A non-metric π-connection ∇¯ on M¯ is called asemi-symmetric non-metric connection if its torsion tensor T¯ satisfiesT¯(X,Y ) = π(Y )X −π(X)Y.The notion of semi-symmetric non-metric connections on a Riemannian man-ifold was introduced by Ageshe and Chafle [1] and later studied by many au-thors. The lightlike version of Riemannian manifolds with semi-symmetricnon-metric connections have been studied by some authors [15, 16, 17, 18, 23].(2) A non-metric π-connection ∇¯ on M¯ is called a quarter-symmetric non-connection if its torsion tensor T¯ satisfiesT¯(X,Y) = π(Y )φX −π(X)φY,where φ is a (1,1)-type tensor field. In particular, if φX = X, then the quarter-symmetric connection reduces to the semi-symmetric connection [8]. Thus the

Einstein half lightlike submanifolds of a Lorentzian space form with a semi-symmetric metric connection

In this paper, we prove a classification theorem for Einstein half lightlike sbmanifolds M of a Lorentzian space form (M (c), g) of constant curvature c equipped with a semi-symmetric metric connection subject such that the second fundamental forms of M and its screen distribution S(TM) are conformally related by some non-zero constant and the canonical normal vector field of M is conformal Killing one.

INDEFINITE TRANS-SASAKIAN MANIFOLD ADMITTING AN ASCREEN HALF LIGHTLIKE SUBMANIFOLD

We study the geometry of indefinite trans-Sasakian manifold ¯ M, of type (�,�), admitting a half lightlike submanifold M such that the structure vector field of ¯ M does not belong to the screen and co- screen distributions of M. The purpose of this paper is to prove several classification theorems of such an indefinite trans-Sasakian manifold.

HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE TRANS-SASAKIAN MANIFOLD

We study half lightlike submanifold M of an indefinite trans- Sasakian manifold such that its structure vector field is tangent to M. First we study the general theory for such half lightlike submanifolds. Next we prove some characterization theorems for half lightlike subman- ifolds of an indefinite generalized Sasakian space form.

A Review on Unique Existence Theorems in Lightlike Geometry

This is a review paper of up-to-date research done on the existence of unique null curves, screen distributions, Levi-Civita connection, symmetric Ricci tensor, and scalar curvature for a large variety of lightlike submanifolds of semi-Riemannian (in particular, Lorentzian) manifolds, supported by examples and an extensive bibliography. We also propose some open problems.

GEOMETRY OF HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KAEHLER MANIFOLD WITH A QUARTER-SYMMETRIC METRIC CONNECTION

Abstract. In this paper, we study the geometry of half lightlikesubmanifolds of an inde nite Kaehler manifold equipped with aquarter-symmetric metric connection. The main result is to proveseveral classi cation theorems for such half lightlike submanifolds. 1. IntroductionA linear connection r on a semi-Riemannian manifold (M; g) is calleda quarter-symmetric connection if its torsion tensor T satis esT(X;Y) = ˇ(Y)˚X ˇ(X)˚Yfor any vector elds X and Y on M, where ˚is a (1;1)-type tensor eld, and ˇis a 1-form associated with a non-vanishing smooth vector eld , which is called the torsion vector eld of M, by ˇ(X) = eg(X;).Moreover, if r is satis ed rg = 0, then it is called a quarter-symmetricmetric connection.The notion of the quarter-symmetric metric connection was intro-duced by K. Yano and T. Imai [17], and since then it have been studiedby S.C. Rastogi [15, 16], D. Kamilya and U.C. De [11], R.S. Mishra andS.N. Pandey [12], S. Golab [6] and many others.The study of lightlike submanifolds was initiated by Duggal and Be-jancu [2] and later studied by many authors (see up-to date results in[4, 5]). Its theory is an important topic of research in di erential ge-ometry due to its application in mathematical physics, especially in thegeneral relativity. The class of codimension 2 lightlike submanifoldsof semi-Riemannian manifolds is composed of two classes by virtue of

SINGULAR THEOREMS FOR LIGHTLIKE SUBMANIFOLDS IN A SEMI-RIEMANNIAN SPACE FORM

Abstract. We study the geometry of lightlike submanifolds of a semi-Riemannian manifold. The purpose of this paper is to prove two singulartheorems for irrotational lightlike submanifolds M of a semi-Riemannianspace form M(c) admitting a semi-symmetric non-metric connection suchthat the structure vector eld of M(c) is tangent to M. 1. IntroductionThe theory of lightlike submanifolds is an important topic of research indi erential geometry due to its application in mathematical physics, especiallyin the general relativity. The study of such notion was initiated by Duggal andBejancu [3] and later studied by many authors (see up-to date results in twobooks [4, 7]). Recently many authors have studied lightlike submanifolds Mof inde nite almost contact metric manifolds M (see [5, 6, 7, 8, 14, 16]). Theauthors in above papers principally assumed that the structure vector eld of M is tangent to M. Calin proved the following result in his thesis:Calin’s result [2]: If the structure vector eld of M is tangent to M, thenit belongs to the screen distribution S(TM) of M.After Calin’s work, many earlier works [5, 6, 7, 14, 16], which have been writ-ten on lightlike submanifolds of inde nite almost contact metric manifolds,obtained their results by using the Calin’s result described in above.The notion of a semi-symmetric non-metric connection on a Riemannianmanifold was introduced by Ageshe and Chae [1]. Although now we havelightlike version of a large variety of Riemannian submanifolds, the geometry oflightlike submanifolds of semi-Riemannian manifolds admitting semi-symmetricnon-metric connections has been few known. Several works ([9]˘[13]), whichhave been written on lightlike submanifolds of semi-Riemannian manifolds ad-mitting semi-symmetric non-metric connections, also obtained their results by

ASCREEN LIGHTLIKE HYPERSURFACES OF A SEMI-RIEMANNIAN SPACE FORM WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

Abstract. We study lightlike hypersurfaces of a semi-Riemannian spaceform fM(c) admitting a semi-symmetric non-metric connection. First, weconstruct a type of lightlike hypersurfaces according to the form of thestructure vector field of Mf(c), which is called a ascreen lightlike hyper-surface. Next, we prove a characterization theorem for such an ascreenlightlike hypersurface endow with a totally geodesic screen distribution. 1. IntroductionThe theory of lightlike submanifolds is an important topic of research in dif-ferential geometry due to its application in mathematical physics, especially inthe electromagneticfield theory. The study ofsuch notion wasinitiated by Dug-gal and Bejancu [3] and later studied by many authors (see up-to date resultsin two books [4, 5]). The notion of a semi-symmetric non-metric connectionon a Riemannian manifold was introduced by Ageshe and Chafle [1]. Recentlyseveral authors ([9]-[13]) studied lightlike hypersurfaces in a semi-Riemannianmanifold admitting a semi-symmetric non-metric connection. Most of authorsthat wrote on either lightlike hypersurfaces M of semi-Riemannian manifoldsMfadmitting semi-symmetric non-metric connections or lightlike hypersurfacesM of indefinite almost contact manifolds Mffail to treat with the case the struc-ture vector field ζ of Mfis not tangent to M, but studied only to the case ζis tangent to M (such M is called tangential lightlike submanifold ([9]-[13]) ofMf). There are few papers on non-tangential lightlike submanifolds of indefinitealmost contact manifolds studied by Jin ([6]-[8]).In this paper, we study non-tangential lightlike hypersurfaces of a semi-Riemannian space form admitting a semi-symmetric non-metric connection.There are several different types of non-tangential lightlike hypersurfaces ac-cording to the form of the structure vector field of the ambient manifold. We

ON LIGHTLIKE SUBMANIFOLDS OF A GRW SPACE-TIME

This paper provides a study of lightlike submanifolds of a generalized Robertson-Walker (GRW) space-time. In particular, we investigate lightlike submanifolds with curvature invariance, parallel second fundamental forms, totally umbilical second fundamental forms, null sectional curvatures and null Ricci curvatures, respectively.