Lightlike Hypersurfaces Research Articles

LIGHTLIKE SUBMANIFOLDS OF A SEMI-RIEMANNIAN MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

Abstract. We study the geometry of r-lightlike submanifolds M of asemi-Riemannian manifold M with a semi-symmetric non-metric connec-tion subject to the conditions; (a) the screen distribution of M is to-tally geodesic in M, and (b) at least one among the r-th lightlike secondfundamental forms is parallel with respect to the induced connection ofM. The main result is a classi cation theorem for irrotational r-lightlikesubmanifold of a semi-Riemannian manifold of index r admitting a semi-symmetric non-metric connection. 1. IntroductionThe geometry of lightlike submanifolds is used in mathematical physics, inparticular, in general relativity since lightlike submanifolds produce models ofdi erent types of horizons (event horizons, Cauchy’s horizons, Kruskal’s hori-zons). The universe can be represented as a four dimensional Lorentz subman-ifold (spacetime) embedded in an (n+ 4)-dimensional semi-Riemannian mani-fold. Lightlike hypersurfaces are also studied in the theory of electromagnetism[1]. Thus, large number of applications but limited information available, moti-vated us to do research on this subject matter. Duggal-Bejancu [1] and Kupeli[2] developed the general theory of degenerate (lightlike) submanifolds. Theyconstructed a transversal vector bundle of lightlike submanifold and investi-gated various properties of these manifolds. Duggal-Jin [3] studied totallyumbilical lightlike submanifold of a semi-Riemannian manifold. Ageshe andChae [4] introduced the notion of a semi-symmetric non-metric connection ona Riemannian manifold. Ya˘sar, C˘oken and Yucesan [5] and Jin [6] studied light-like hypersurfaces in semi-Riemannian manifolds admitting a semi-symmetricnon-metric connections. The geometry of half lightlike submanifolds of a semi-Riemannian manifold with semi-symmetric non-metric connection was studied

NON-EXISTENCE OF LIGHTLIKE SUBMANIFOLDS OF INDEFINITE TRANS-SASAKIAN MANIFOLDS WITH NON-METRIC 𝜃-CONNECTIONS

We study two types of 1-lightlike submanifolds, so-called lightlike hypersurface and half lightlike submanifold, of an indefinite trans-Sasakian manifold <TEX>$\bar{M}$</TEX> admitting non-metric <TEX>${\theta}$</TEX>-connection. We prove that there exist no such two types of 1-lightlike submanifolds of an indefinite trans-Sasakian manifold <TEX>$\bar{M}$</TEX> admitting non-metric <TEX>${\theta}$</TEX>-connections.

Lightlike hypersurfaces of a trans-Sasakian manifold with a quarter-symmetric metric connection

In this paper, we study the geometry of lightlike hypersurfaces of an indenite trans-Sasakian manifold with a quarter-symmetric metric connection. The main result is a characterization theorem for such a lightlike hypersurface of an indenite generalized Sasakian space form. Mathematics Subject Classication: 53C25, 53C40, 53C50

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

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

Lightlike Hypersurfaces of Indefinite Generalized Sasakian Space Forms

We study lightlike hypersurfacesMof an indefinite generalized Sasakian space formM-(f1,f2,f3), with indefinite trans-Sasakian structure of type(α,β), subject to the condition that the structure vector field ofM-is tangent toM. First we study the general theory for lightlike hypersurfaces of indefinite trans-Sasakian manifold of type(α,β). Next we prove several characterization theorems for lightlike hypersurfaces of an indefinite generalized Sasakian space form.

Non-existence of screen homothetic lightlike hypersurfaces of an indefinite Kenmotsu manifold

We study screen homothetic lightlike hypersurfaces M of an indefinite Kenmotsu manifold M with flat transversal connection. We prove that there do not exist screen homothetic lightlike hypersurfaces of indefinite Kenmotsu manifolds with flat transversal connection subject to the conditions: (1) M is locally symmetric, or (2) M is semi-symmetric. Mathematics Subject Classification: 53C25, 53C40, 53C50

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.

2-symmetric lightlike hypersurfaces in semi-Riemannian space forms

In this paper, we will investigate 2-symmetry type curvature conditions of a lightlike hypersurface of a semi-Riemannian space form. Then we find some results related with locally symmetric and 2-symmetric lightlike hypersurfaces. Finally the relation between semisymmetric and 2-symmetric lightlike hypersurfaces will be given.

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

LIGHTLIKE HYPERSURFACES OF INDEFINITE KAEHLER MANIFOLDS OF QUASI-CONSTANT CURVATURES

Abstract. We study lightlike hypersurfaces M of an inde nite Kaehlermanifold M of quasi-constant curvature subject to the condition that thecurvature vector eld of M belongs to the screen distribution S(TM).We provide several new results on such lightlike hypersurfaces M. 1. IntroductionB.Y. Chen and K. Yano [1] introduced the notion of a semi-Riemannianmanifold of quasi-constant curvature as a semi-Riemannian manifold (M; g)endowed with the curvature tensor R satisfying the following equation:R(X;Y)Z= ‘fg(Y;Z)X g(X;Z)Yg (1.1)+ ~fg(Y;Z)(X) g(X;Z)(Y)+ (Y)(Z)X (X)(Z)Yg;for any vector elds X;Y and Z, where ‘and ~ are smooth functions, is anon-vanishing smooth unit spacelike vector eld, which is called the curvaturevector eld of M, and is a 1-form associated with by(X) = g(X;):It is well known that if the curvature tensor R is of the form (1.1), then M isconformally at. If ~ = 0, then M is a space of constant curvature ‘.The theory of lightlike hypersurfaces 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 [2] and later studied by many authors (see two books [3, 4]).The purpose of this paper is to study lightlike hypersurfaces Mof an indef-inite Kaehler manifold M of quasi-constant curvature such that the curvaturevector eld of M belongs to the screen distribution S(TM). We provide severalnew results on such lightlike hypersurfaces Msubject such that the structure

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

Lightlike hypersurfaces of a para-Sasakian space form

In this paper, lightlike hypersurfaces of a para-Sasakian space form are studied. We obtain some non-existence results for screen semi-invariant lightlike hypersurfaces of a para-Sasakian space form. It is shown that a lightlike hypersurface is totally geodesic if and only if it is locally symmetric.

Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes

We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle.

A CHARACTERIZATION THEOREM FOR LIGHTLIKE HYPERSURFACES OF SEMI-RIEMANNIAN MANIFOLDS OF QUASI-CONSTANT CURVATURES

In this paper, we study lightlike hypersurfaces M of semi-Riemannian manifolds <TEX>$\bar{M}$</TEX> of quasi-constant curvatures. Our main result is a characterization theorem for screen homothetic Einstein lightlike hypersurfaces of a Lorentzian manifold of quasi-constant curvature subject such that its curvature vector field <TEX>${\zeta}$</TEX> is tangent to M.

Pseudosymmetric lightlike hypersurfaces

We study lightlike hypersurfaces of a semi-Riemannian manifold satisfying pseudosymmetry conditions. We give sufficient conditions for a lightlike hypersurface to be pseudosymmetric and show that there is a close relationship of the pseudosymmetry condition of a lightlike hypersurface and its integrable screen distribution. We obtain that a pseudosymmetric lightlike hypersurface is a semisymmetric lightlike hypersurface or totally geodesic under certain conditions. Moreover, we give an example of pseudosymmetric lightlike hypersurfaces and investigate pseudoparallel lightlike hypersurfaces. Furthermore, we introduce Ricci-pseudosymmetric lightlike hypersurfaces, obtain characterizations, and give an example for such hypersurfaces.

Osserman lightlike hypersurfaces of indefinite $\mathcal S$-manifolds

We mainly deal with the problem of admissibility for screen distributions on a lightlike hypersurface of both a semi-Riemannian manifold and an indefinite S-manifold. In the latter case, we first show that a characteristic screen distribution is never admissible, and then we provide a characterization for admissible screen distributions on proper totally umbilical lightlike hypersurfaces. Finally, in studying Osserman conditions, we characterize Osserman totally umbilical hypersurfaces of a semi-Riemannian manifold, obtaining explicit results on the eigenvalues of the pseudo-Jacobi operators in the case of lightlike hypersurfaces with Lorentzian screen leaves.

On Lightlike Geometry of Para-Sasakian Manifolds

We study lightlike hypersurfaces of para-Sasakian manifolds tangent to the characteristic vector field. In particular, we define invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, respectively, and give examples. Integrability conditions for the distributions on a screen semi-invariant lightlike hypersurface of para-Sasakian manifolds are investigated. We obtain a para-Sasakian structure on the leaves of an integrable distribution of a screen semi-invariant lightlike hypersurface.

Lightlike hypersurfaces along spacelike submanifolds in de Sitter space

Lightlike hypersurfaces along spacelike submanifolds in de Sitter space

Lightlike Hypersurfaces and Canal Hypersurfaces of Lorentzian Surfaces

The lightlike hypersurfaces in semi-Euclidean space are of special interest in Relativity Theory. In particular, the singularities of these lightlike hypersurfaces provide good models for the study of different horizon types. And we obtain some geometrical propositions of the canal hypersurfaces of Lorentzian surfaces. We introduce the notions of flatness for these hypersurfaces and study their singularities.

INDEFINITE TRANS-SASAKIAN MANIFOLD ADMITTING AN ASCREEN LIGHTLIKE HYPERSURFACE

We study indefinite trans-Sasakian manifold admitting an ascreen lightlike hypersurface M. Our main results are several classification theorems of such an indefinite trans-Sasakian manifold.