Lightlike Hypersurfaces Research Articles

Chen inequalities on lightlike hypersurfaces of a GRW spacetime

In this paper, we introduce [Formula: see text]-Ricci curvature and [Formula: see text]-scalar curvature on lightlike hypersurfaces of a GRW spacetime. Using these curvatures, we establish some inequalities for lightlike hypersurfaces of a GRW spacetime. Using these inequalities, we obtain some characterizations on lightlike hypersurfaces. We also get Chen–Ricci inequality and Chen inequality on a screen homothetic lightlike hypersurfaces of a GRW spacetime.

Extension of Killing vector fields beyond compact Cauchy horizons

We prove that any compact Cauchy horizon with constant non-zero surface gravity in a smooth vacuum spacetime is a smooth Killing horizon. The novelty here is that the Killing vector field is shown to exist on both sides of the horizon. This generalises classical results by Moncrief and Isenberg, by dropping the assumption that the metric is analytic. In previous work by Rácz and the author, the Killing vector field was constructed on the globally hyperbolic side of the horizon. In this paper, we prove a new unique continuation theorem for wave equations through smooth compact lightlike (characteristic) hypersurfaces which allows us to extend the Killing vector field beyond the horizon. The main ingredient in the proof of this theorem is a novel Carleman type estimate. Using a well-known construction, our result applies in particular to smooth stationary asymptotically flat vacuum black hole spacetimes with event horizons with constant non-zero surface gravity. As a special case, we therefore recover Hawking's local rigidity theorem for such black holes, which was recently proven by Alexakis-Ionescu-Klainerman using a different Carleman type estimate.

Lightlike hypersurfaces in spaces with concircular fields

Lightlike hypersurfaces in semi-Riemannian manifolds admitting concircular vector fields are investigated. We prove that such hypersurfaces are generally products of lightlike curves and warped product manifolds. In special cases, we show that these hypersurfaces are totally geodesic or totally screen geodesic provided such concircular fields belong to their normal or transversal bundles. A number of examples are furnished, where possible, to illustrate the main concepts.

Lightlike manifolds and Cartan geometries

Lightlike Cartan geometries are introduced as Cartan geometries modelled on the future lightlike cone in Lorentz-Minkowski spacetime. Then, we provide an approach to the study of lightlike manifolds from this point of view. It is stated that every lightlike Cartan geometry on a manifold N provides a lightlike metric h with radical distribution globally spanned by a vector field Z. For lightlike hypersurfaces of a Lorentz manifold, we give the condition that characterizes when the pull-back of the Levi-Civita connection form of the ambient manifold is a lightlike Cartan connection on such hypersurface. In the particular case that a lightlike hypersurface is properly totally umbilical, this construction essentially returns the original lightlike metric. From the intrinsic point of view, starting from a given lightlike manifold (N, h), we show a method to construct a family of ambient Lorentzian manifolds that realize (N, h) as a hypersurface. This method is inspired on the Feffermann-Graham ambient metric construction in conformal geometry and provides a lightlike Cartan geometry on the original manifold when (N, h) is generic.

On lightlike geometry of indefinite Sasakian statistical manifolds

<abstract><p>In the present study, the concept of Sasakian statistical manifold has been generalized to indefinite Sasakian statistical manifolds. We also introduce lightlike hypersurfaces of an indefinite Sasakian statistical manifold and establish relations between induced geometrical objects with respect to dual connections. Finally, invariant lightlike submanifold of indefinite Sasakian statistical manifold is proved to be an indefinite Sasakian statistical manifold.</p></abstract>

Concurrent Vector Fields on Lightlike Hypersurfaces

Concurrent vector fields lying on lightlike hypersurfaces of a Lorentzian manifold are investigated. Obtained results dealing with concurrent vector fields are discussed for totally umbilical lightlike hypersurfaces and totally geodesic lightlike hypersurfaces. Furthermore, Ricci soliton lightlike hypersurfaces admitting concurrent vector fields are studied and some characterizations for this frame of hypersurfaces are obtained.

Lightlike Hypersurfaces of an Indefinite Nearly Trans-Sasakian Manifold with an (l,m)-type Connection

Lightlike Hypersurfaces of an Indefinite Nearly Trans-Sasakian Manifold with an (l,m)-type Connection

Lightlike Hypersurfaces of an Indefinite Kaehler Manifold with an (<img src=image/13415329_01.gif>)-type Connection

Jin [1] defined an ( )-type connection on semi-Riemannian manifolds. Semi-symmetric nonmetric connection and non-metric ∅-symmetric connection are two important examples of this connection such that ( ) = (1; 0) and ( ) = (0; 1), respectively. In semi-Riemannian geometry, there are few literatures for the lightlike geometry, so we expose new theories for non-degenerate submanifolds in semi-Riemannian geometry. The goal of this paper is to study a characterization of a (Lie) recurrent lightlike hypersurface M of an indefinite Kaehler manifold with an ( )-type connection when the charateristic vector field is tangnet to M. In the special case that an indefinite Kaehler manifold of constant holomorphic sectional curvature is an indefinite complex space form, we investigate a lightlike hypersurface of an indefinite complex space form with an ( )-type connection when the charateristic vector field is tangnet to M. Moreover, we show that the total space, the complex space form, is characterized by the screen conformal lightlike hypersurface with an ( )-type connection. With a semi-symmetric non-metric connection, we show that an indefinite complex space form is flat.

Lightlike hypersurfaces and lightlike focal sets with respect to Bishop frame in 4-dimensional Minkowski Space E1-4.

In this article, light-like hypersurfaces that is derived by null Cartan curves will be examined and discussed. The singularities of lightlike hypersurfaces and light-like focal sets are investigated by using the Bishop frame on the Null Cartan curves. We obtain that the types of these singularities and the order of contact between the null Cartan curves are closely related to the Bishop curvatures of the null Cartan curves. Moreover, two examples of light-like hypersurfaces and light-like focal sets are given to illustrate our theoretical results.

Lightlike submanifolds of metallic semi-Riemannian manifolds

Our aim in this paper is to investigate some special types of lightlike submanifolds in metallic semi-Riemannian manifolds. We study invariant lightlike submanifolds and screen semi-invariant lightlike hypersurfaces of metallic semi-Riemannian manifolds and give examples. We obtain some conditions for the induced connection to be a metric connection and present integrability conditions for the distributions involved in the definitions of such types.

Lightlike Hypersurfaces of an Indefinite Para Sasakian Manifold

In this paper, we initiate the study of lightlike hypersurfaces of an indefinite almost paracontact metric manifold which are tangent to the structure vector field. In particular, we give definitions of invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, and give some examples. Integrability conditions for the distributions involved in the screen semi-invariant lightlike hypersurface of an indefinite para Sasakian manifold are investigated.

Notes on the Lightlike Hypersurfaces Along Spacelike Submanifolds

In connection with the method of construction of lightlike hypersurfaces along spacelike submanifolds, we present a relation between the second fundamental form of a spacelike submanifold and the screen second fundamental form of the corresponding lightlike hypersurface. In addition, we study the conditions for a lightlike hypersurface of this kind to be screen conformal.

Lightlike Hypersurfaces with Planar Normal Sections in $\mathbb{R}^4_1$

In the present paper our aim is to investigate lightlike hypersurfaces of $\mathbb{R}^4_1$ having degenerate or non-degenerate planar normal sections.

Rigged null hypersurfaces in almost paracontact metric manifolds

Given an almost paracontact metric manifold $(\overline {M}, \overline{ \phi},\overline {\eta}, \xi, \overline {g})$, we study lightlike hypersurfaces $M$ of the semi-Riemannian manifold $(\overline {M},\overline{g})$ transversal to the structure vector field $\xi$. The latter is then a rigging for $M$ and defines a null section $E$ of the radical distribution of $M$ and a screen distribution which turns out to be always semi-invariant. We show that leaves of an integrable screen distribution in such hypersurfaces $M$ are almost paracontact metric manifolds too. When the ambient space $(\overline {M}, \overline{ \phi},\overline {\eta}, \xi, \overline {g})$ is para-Sasakian, we show that the screen distribution cannot be conformal and that $E$ is a geodesic vector field in $(M, \nabla),$ where $\nabla$ is the connection induced on $M$ by Levi-Civita connection of $\overline {g}$ and the local null rigging $N=\xi-\frac{1}{2}E$. We also find necessary and sufficient conditions for the leaves of the screen distribution to be para-Sasakian too and finally we investigate integrability conditions for some additional distributions induced on $M$ by the structure $(\overline {M}, \overline{ \phi},\overline {\eta}, \xi, \overline {g})$.

Lightlike hypersurfaces of an indefinite Kaehler manifold of a quasi-constant curvature

Abstract We study lightlike hypersurfaces M of an indefinite Kaehler manifold M̅ of quasi-constant curvature subject to the condition that the characteristic vector field ζ of M̅ is tangent to M. First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface M of M̅ such that (1) the screen distribution S(TM) is totally umbilical or (2) M is screen conformal.

Alpha-Associated metrics on rigged null hypersurfaces

Let $x:(M,g)\to(\Bm,\bg)$ be a null hypersurface isometrically immersed into a proper semi-Riemannian manifold $(\overline{M},\bar g)$. A rigging for $M$ is a vector field $\zeta$ defined on some open subset of $\overline{M}$ containing $M$ such that $\zeta_p\notin T_pM$ for every $p\in M$. Such a vector field induces an everywhere transversal null vector field $N$ defined over $M$ and which induces on $M$ the same geometrical objects as $\zeta$. Let $\bar \eta$ be the 1-form that is $\bar g$-metrically equivalent to $N$ and let $\eta=x^\star\bar\eta$ be its pullback on $M$. For a given nowhere vanishing smooth function $\alpha$ on $M$, we have introduced and studied the so-called $\alpha$-associated (semi-)Riemannian metric $g_{\alpha}=g+\alpha\eta\otimes \eta$. It turns out that this perturbation of the induced metric along a transversal null vector field is always nondegenerate, so we have established some relationships between geometrical objects of the (semi-)Riemannian manifold $(M,{g}_{\alpha})$ and those of the lightlike hypersurface $(M,g)$. For instance, in the case where $N$ is closed, we give a constructive method to find a $\alpha$-associated metric whose Levi-Civita connection coincides with the connection $\nabla$ induced on $M$ through the projection of the Levi-Civita connection $\overline{\nabla}$ of $\overline{M}$ along $N$. As an application, we show that given a null Monge hypersurface $M$ in $\R_q^{n+1},$ there always exists a rigging and a $\alpha$-associated metric whose Levi-Civita connection is the induced connection on $M$.

The Lanczos Equation on Light-Like Hypersurfaces in a Cosmologically Viable Class of Kinetic Gravity Braiding Theories

We discuss junction conditions across null hypersurfaces in a class of scalar–tensor gravity theories (i) with second-order dynamics, (ii) obeying the recent constraints imposed by gravitational wave propagation, and (iii) allowing for a cosmologically viable evolution. These requirements select kinetic gravity braiding models with linear kinetic term dependence and scalar field-dependent coupling to curvature. We explore a pseudo-orthonormal tetrad and its allowed gauge fixing with one null vector standing as the normal and the other being transversal to the hypersurface. We derive a generalization of the Lanczos equation in a 2 + 1 decomposed form, relating the energy density, current, and isotropic pressure of a distributional source to the jumps in the transverse curvature and transverse derivative of the scalar. Additionally, we discuss a scalar junction condition and its implications for the distributional source.

Lightlike hypersurfaces of metallic semi-Riemannian manifolds

In our paper, we introduce and study lightlike hypersurfaces of a metallic semi-Riemannian manifold. We examine some geometric properties of invariant lightlike hypersurfaces. We show that the induced structure on an invariant lightlike hypersurface is also metallic. We also define screen semi-invariant lightlike hypersurfaces, investigate integrability conditions for the distributions and give some examples.

Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection

We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces.

Lightlike hypersurfaces of an (ε)-para Sasakian manifold with a semi-symmetric non-metric connection

In the present paper, we study a lightlike hypersurface, when the ambient manifold is an (?)-para Sasakian manifold endowed with a semi-symmetric non-metric connection. We obtain a condition for such a lightlike hypersurface to be totally geodesic. We define invariant and screen semi-invariant lightlike hypersurfaces of (?)-para Sasakian manifolds with a semi-symmetric non-metric connection. Also, we obtain integrability conditions for the distributions D ? ??? and D' ? ??? of a screen semi-invariant lightlike hypersurface of an (?)-para Sasakian manifolds with a semi-symmetric non-metric connection.