Lightlike Hypersurfaces Research Articles

Some Applications of Cauchy-Schwarz Inequality on Lightlike Warped Product Submanifolds

The well-known Cauchy-Schwarz inequality is a fundamental tool of analysis in Hilbert spaces. It is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is the most important and widely used inequalities in mathematics. We explore that inequality in case of lightlike warped product hypersurface of lorentzian space form to reach to the existence of minimal screen conformal lightlike hypersurface.

On some basic curvature invariants of screen homothetic lightlike hypersurfaces in a GRW spacetime

This study is focused on the investigation of lightlike hypersurfaces of a generalized Robertson-Walker (GRW) spacetime. Recently, Poyraz (2022) [51,52] established some basic inequalities involving various curvature invariants of screen homothetic lightlike hypersurfaces of GRW spacetimes, like k-scalar curvature and k-Ricci curvature. In this work, we consider other basic curvature invariants, namely the scalar curvature and δ-Casorati curvatures, and derive new inequalities for such hypersurfaces of a GRW spacetime. We also find the conditions for which the equality cases in these inequalities hold and give some applications in Lorentzian geometry.

Screen Semi-Invariant Lightlike Hypersurfaces on Hermite-Like Manifolds

Hermite-like manifolds, which admit two different, almost complex structures, can be considered a general concept of Hermitian manifolds. Factoring in the effects of these two complex structures on the radical, screen, and transversal spaces, a new classification of lightlike hypersurfaces of Hermite-like manifolds is proposed in the present paper. Moreover, an example of screen semi-invariant lightlike hypersurfaces of Hermite-like manifolds is provided. Besides, some results on these hypersurfaces admitting a statistical structure are obtained. Further, screen semi-invariant lightlike hypersurfaces are investigated on Kaehler-like statistical manifolds. In addition, several characteristics of totally geodesic, mixed geodesic, and totally umbilical screen lightlike hypersurfaces are obtained. Finally, the need for further research is discussed.

Lightlike Hypersurfaces of Meta-Golden Semi-Riemannian Manifolds

In this research, we embark on the examination of lightlike hypersurfaces within an almost meta-Golden semi-Riemannian manifold. We investigate the properties of the induced structure on a lightlike hypersurface by meta-Golden semi-Riemannian structure. Then, we introduce invariant lightlike hypersurfaces, anti-invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces of almost meta-Golden semi-Riemannian manifolds and give examples.

Lightlike Hypersurfaces of an Indefinite Kaehler Manifold with an $(\ell,\,m)$-type Metric Connection

Jin introduced a non-symmetric metric connection, called an {\it $(\ell,m)$-type metric connection} \cite{Jin1, Jin2}. There are two examples of $(\ell, m)$-type: a semi-symmetric metric connection when ${\ell}=1$ and $m=0$ and a quater-symmetric connection for ${\ell}=0$ and $m=1$ . Our purpose is to investigate lightlike hypersurfaces of an indefinite (complex) Kaehler manifolds with an $(\ell,m)$-type metric connection under the tangent characteristic vector field on such hypersurfaces.

Screen invariant lightlike hypersurfaces of almost product-like statistical manifolds and locally product-like statistical manifolds

Screen invariant lightlike hypersurfaces of almost product-like statistical manifolds and locally product-like statistical manifolds are introduced. The main formulas and relations are presented for these hypersurfaces. Concurrent and recurrent vector fields are investigated and some characterizations are obtained for screen invariant lightlike hypersurfaces.

Concircular Vector Fields on Radical Anti-Invariant Lightlike Hypersurfaces of Almost Product-like Statistical Manifolds

The motivation of the present study is to describe the main relations of the radical anti-invariant lightlike hypersurfaces of almost product-like statistical manifolds. We provide concircular vector fields on radical anti-invariant lightlike hypersurfaces and obtain some results involving these vector fields.

Geometry of Lightlike Hypersurfaces of a Statistical Manifold

In this paper, we introduced the lightlike hypersurfaces of a statistical manifold. It is shown that a lightlike hypersurface of a statistical manifold is not a statistical manifold with respect to the induced connections, but the screen distribution has a canonical statistical structure. Some relations between induced geometric objects with respect to dual connections in a lightlike hypersurface of a statistical manifold are obtained. An example is presented. Induced Ricci tensors for lightlike hypersurface of a statistical manifold are computed.

Lightlike Hypersurfaces in Semi-Riemmanian Manifolds Admitting Affine Conformal Vector Fields

Lightlike hypersurfaces with integrable screen distributions are very important as far as lightlike geometry is concerned. They include, among others, screen conformal and screen totally umbilic ones. In this paper, we show that any lightlike hypersurface of a semi-Riemannian manifold admitting a certain closed affine conformal vector field has an integrable screen distribution. Several examples are furnished in support of the main results.

ON RECURRENT LIGHTLIKE HYPERSURFACE OF KENMOTSU MANIFOLD

The object of present paper is to study the properties of recurrent lightlike hypersurfaces of Kenmotsu manifold with (l, m)−type connection.

Lightlike Hypersurfaces of Almost Productlike Semi-Riemannian Manifolds

The main purpose of this paper is to investigate lightlike hypersurfaces of almost productlike semi-Riemannian manifolds. For this purpose, screen-semi-invariant, screen-invariant, radical-anti-invariant, and radical-invariant lightlike hypersurfaces of almost productlike semi-Riemannian manifolds are introduced and some examples of these classifications are presented. Furthermore, various characterizations dealing screen semi-invariant lightlike hypersurfaces are obtained.

Rigidity results for complete spacelike submanifolds in plane fronted waves

New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.

Strongly minimal complex lightlike hypersurfaces

In this paper, complex lightlike hypersurfaces of an indefinite Kähler manifold are studied. An optimal inequality characterized to strongly minimality for coisotropic lightlike submanifolds is proved. Strongly minimal Monge-type hypersurfaces in $\mathcal{C}_{1}^{4}$ are examined and some examples of these hypersurfaces are given.

Topological structures of event horizons along framed null Cartan curves in Minkowski space

In this work, we model two event horizons associated with a null Cartan curve as two lightlike hypersurfaces, respectively. We define a lightlike surface and a spacelike surface whose images coincide with the sets of critical values of two event horizons, and meanwhile we present two curves whose images coincide with the sets of critical values of these two surfaces, respectively. Using the singularity theory, we characterize the local topological structures of two event horizons, two surfaces and two curves at their singularities by means of two new invariants. Moreover, we also present a spacelike braneworld model along the particle as a spacelike surface in hyperbolic 3-space. An important fact shows that from the viewpoint of Legendrian dualities, this surface is [Formula: see text]-dual to the tangent trajectory [Formula: see text] of the null Cartan curve in Lorentz–Minkowski space-time. Meanwhile, we also consider a curve whose image is the set of critical values of this surface in hyperbolic 3-space. The third invariant of the null Cartan curve characterizes the singularities of the surface [Formula: see text] and the curve [Formula: see text] in hyperbolic 3-space. A result indicates that surface [Formula: see text] is locally diffeomorphic to the swallowtail [Formula: see text] or cuspidal edge [Formula: see text] and [Formula: see text] is locally diffeomorphic to the [Formula: see text]-cusp at certain a singular point. It is also shown that there exist deep relationships between the singularities of the surface [Formula: see text] and the curve [Formula: see text] and the order of contact between [Formula: see text] and elliptic quadric [Formula: see text] or the order of contact between [Formula: see text] and spacelike hyperplane [Formula: see text]. Finally, we present several examples to describe the main results.

Main Curvatures Identities on Lightlike Hypersurfaces of Statistical Manifolds and Their Characterizations

In this study, some identities involving the Riemannian curvature invariants are presented on lightlike hypersurfaces of a statistical manifold in the Lorentzian settings. Several inequalities characterizing lightlike hypersurfaces are obtained. These inequalities are also investigated on lightlike hypersurfaces of Lorentzian statistical space forms.

Quasi-Yamabe solitons and almost quasi-Yamabe solitons on lightlike hypersurfaces

In the present paper, we study the quasi-Yamabe solitons and almost quasi-Yamabe solitons on the lightlike hypersurfaces of the semi-Riemannian manifolds endowed with a torse-forming vector field. We show some conditions for the lightlike hypersurfaces to be quasi-Yamabe solitons and almost quasi- Yamabe solitons with the tangential component of the torse-forming vector field on the semi-Riemannian manifolds as the soliton field. In particular, we also specify the conditions for lightlike hypersurfaces of (n + 2)-dimension semi-Riemannian manifolds of constant curvature to be quasi-Yamabe solitons and almost quasi-Yamabe solitons. Besides, we provide some geometric properties of the lightlike hypersurfaces satisfying quasi-Yamabe solitons, quasi-Yamabe gradient solitons, almost quasi-Yamabe solitons and almost quasi-Yamabe gradient solitons. Furthermore, we investigate properties of screen homothetic lightlike hypersurfaces admitting quasi-Yamabe solitons and almost quasi-Yamabe solitons.

On certain principal directions in lightlike hypersurfaces of an indefinite Sasakian manifold

On certain principal directions in lightlike hypersurfaces of an indefinite Sasakian manifold

Minimal homothetical and translation lightlike graphs in $ \mathbb{R} _{q}^{n+2} $

<abstract><p>In this paper, homothetical and translation lightlike graphs, which are generalizations of homothetical and translation lightlike hypersurfaces are investigated in the semi-Euclidean space $ \mathbb{R}_{q}^{n+2} $, respectively. We prove that all homothetical and all translation lightlike graphs are locally the hyperplanes. According to this fact, both of these graphs are minimal.</p></abstract>

On recurrent Light like Hypersurfaces of Indefinite Sasakian Manifold

The object of present paper is to study the properties of recurrent lightlike hypersurfaces of indefinite Sasakian manifold with (l, m) –type connection.

Chen-like inequalities on lightlike hypersurfaces of semi-Riemannian manifolds

Chen-like inequalities on lightlike hypersurfaces of semi-Riemannian manifolds