Screen conformal lightlike submanifolds of semi-Riemannian manifolds

Since the induced objects on a lightlike submanifold depend on its screen distribution which, in general, is not unique and hence we can not use the classical submanifold theory on a lightlike submanifold in the usual way. Therefore, in present paper, we study screen conformal lightlike submanifolds of a semi-Riemannian manifold, which are essential for the existence of unique screen distribution. We obtain a characterization theorem for the existence of screen conformal lightlike submanifolds of a semi-Riemannian manifold. We prove that if the differential operator D^s is a metric Otsuki connection on transversal lightlike bundle for a screen conformal lightlike submanifold then semi-Riemannian manifold is a semi-Euclidean space. We also obtain some characterization theorems for a screen conformal totally umbilical lightlike submanifold of a semi-Riemannian space form. Further, we obtain a necessary and sufficient condition for a screen conformal lightlike submanifold of constant curvature to be a semi-Euclidean space. Finally, we prove that for an irrotational screen conformal lightlike submanifold of a semi-Riemannian space form, the induced Ricci tensor is symmetric and the null sectional curvature vanishes.     

On a Submersion Between Reinhart Lightlike Manifolds and Semi-Riemannian Manifolds

In this paper, we introduce a new submersion, namely, screen lightlike submersion between a lightlike manifold and a semi-Riemannian manifold. We give an example and obtain a characterization for lightlike manifold to be Reinhart under such submersion. Then, we investigate the geometry of a screen lightlike submersion when the total manifold is a Reinhart lightlike manifold. Received: March 6, 2007 Revised: October 4, 2007 and November 2, 2007 Accepted: December 6, 2007     

Regularized Lardy Scheme for Solving Singularly Perturbed Elliptic and Parabolic PDEs

Mediterranean Journal of Mathematics - In the present paper, we introduce screen generic lightlike submanifold of golden semi-Riemannian manifolds and give non-trivial examples of such...     

Submanifolds in Manifolds with Metric Mixed 3-Structures

Mixed 3-structures are odd-dimensional analogues of paraquaternionic structures. They appear naturally on lightlike hypersurfaces of almost paraquaternionic hermitian manifolds. We study invariant and anti-invariant submanifolds in a manifold endowed with a mixed 3-structure and a compatible (semi-Riemannian) metric. Particular attention is given to two cases of ambient space: mixed 3-Sasakian and mixed 3-cosymplectic.     

Radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric

In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. Such class of lightlike hypersurfaces cannot exist for indefinite almost Hermitian manifolds. The considered lightlike hypersurfaces have two important properties. The first one is the uniqueness of their screen distributions, which implies that the induced geometric objects are well-defined. The second property is that the induced Ricci tensor on radical transversal lightlike hypersurface of a Kähler manifold with Norden metric is symmetric. This allows to define an induced scalar curvature of the hypersurface. We obtain new results about lightlike hypersurfaces concerning their relations with non-degenerate hypersurfaces of almost complex manifolds with Norden metric. Examples of the considered hypersurfaces are given.     

A Report on Canonical Null Curves and Screen Distributions for Lightlike Geometry

The general theory of lightlike submanifolds makes use of a non-degenerate screen distribution which is not unique and, therefore, the induced objects (starting from null curves) depend on the choice of a screen, which creates a problem. The purpose of this paper is to report on the existence of a canonical representation of null curves of Lorentzian manifolds and the choice of a canonical or a good screen for large classes of lightlike hypersurfaces of semi-Riemannian manifolds. We also prove a new theorem on the existence of an integrable canonical screen, subject to a geometric condition, and supported by a physical application.      

Lightlike real hypersurfaces of indefinite quaternion Kaehler manifold

In this paper, we introduce real lightlike hypersurfaces of indefinite quaternion Kaehler manifold. Fundamental properties of real lightlike hypersurfaces of an indefinite quaternion Kaehler manifold are investigated. We prove the non existence of real lightlike hypersurfaces in indefinite qaternionic space form under some conditions. Received 31 October 2000; revised 20 June 2001.     

Null screen quasi-conformal hypersurfaces in semi-Riemannian manifolds and applications

We introduce a class of null hypersurfaces of a semi-Riemannian manifold, namely, screen quasi-conformal hypersurfaces, whose geometry may be studied through the geometry of its screen distribution. In particular, this notion allows us to extend some results of previous works to the case in which the sectional curvature of the ambient space is different from zero. As applications, we study umbilical, isoparametric and Einstein null hypersurfaces in Lorentzian space forms and provide several classification results.     

Geometry of lightlike hypersurfaces of an indefinite Sasakian manifold

In this paper, we study the geometry of lightlike hypersurfaces of an indefinite Sasakian manifold. The main result is to prove three characterization theorems for such a lightlike hypersurface. In addition to these main theorems, we study the geometry of totally geodesic lightlike hypersurfaces of an indefinite Sasakian manifold.     

On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds

We study semi-parallel lightlike hypersurfaces of an indefinite Kenmotsu manifold, tangent to the structure vector field. Some Theorems on parallel and semi-parallel vector field, geodesibility of lightlike hypersurfaces are obtained. The geometrical configuration of such lightlike hypersurfaces is established. We prove that, in totally contact umbilical lightlike hypersurfaces of an indefinite Kenmotsu manifold which has constant ${\overline{\phi}}$ -holomorphic sectional curvature c, tangent to the structure vector field and such that its distribution is parallel, the parallelism and semi-parallelism notions are equivalent.     

On lightlike geometry in indefinite Kenmotsu manifolds

We investigate some geometric aspects of lightlike hypersurfaces of indefinite Kenmotsu manifolds, tangent to the structure vector field, by paying attention to the geometry of leaves of integrable distributions. Theorems on parallel vector fields, Killing distribution, geodesibility of their leaves are obtained. The geometric configuration of such lightlike hypersurfaces and leaves of its screen integrable distributions are established. We show that no totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface can be an extrinsic sphere. We also prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle.     

Real Lightlike Hypersurfaces of Paraquaternionic Kähler Manifolds

The main purpose of this paper is to give basic properties of real lightlike hypersurfaces of paraquaternionic manifold and to prove the nonexistence of real lightlike hypersurfaces in paraquaternionic space forms under some conditions. Dedicated to the memory of Professor Aldo Cossu     

Einstein-Weyl structures on lightlike hypersurfaces

We study Weyl structures on lightlike hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambient Lorentzian space ? ₁ ⁿ⁺² and a totally umbilical screen foliation, there is a strong interplay with the induced (Riemannian) Weyl-structure on the leaves.     

Lightlike Hypersurfaces in Indefinite Trans-Sasakian Manifolds

This paper deals with lightlike hypersurfaces of indefinite trans-Sasakian manifolds of type (α, β), tangent to the structure vector field. Characterization Theorems on parallel vector fields, integrable distributions, minimal distributions, Ricci-semi symmetric, geodesibility of lightlike hypersurfaces are obtained. The geometric configuration of lightlike hypersurfaces is established. We prove, under some conditions, that there are no parallel and totally contact umbilical lightlike hypersurfaces of trans-Sasakian space forms, tangent to the structure vector field. We show that there exists a totally umbilical distribution in an Einstein parallel lightlike hypersurface which does not contain the structure vector field. We characterize the normal bundle along any totally contact umbilical leaf of an integrable screen distribution. We finally prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle and its image under ${\overline{\phi}}$ .     

Generalized Cauchy-Riemann lightlike submanifolds of indefinite Sasakian manifolds

We study a class of submanifolds, called Generalized Cauchy-Riemann (GCR) lightlike submanifolds of indefinite Sasakian manifolds as an umbrella of invariant, screen real, contact CR lightlike subcases [8] and real hypersurfaces [9]. We prove existence and non-existence theorems and a characterization theorem on minimal GCR-lightlike submanifolds.     

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

On canonical screen for lightlike submanifolds of codimension two

In this paper we study two classes of lightlike submanifolds of codimension two of semi-Riemannian manifolds, according as their radical subspaces are 1-dimensional or 2-dimensional. For a large variety of both these classes, we prove the existence of integrable canonical screen distributions subject to some reasonable geometric conditions and support the results through examples.      

Semi-Riemannian hypersurfaces in manifolds with metric mixed 3-structures

The mixed 3-structures are the counterpart of paraquaternionic structures in odd dimension. A compatible metric with a mixed 3-structure is necessarily semi-Riemann and mixed 3-Sasakian manifolds are Einstein. We investigate the differential geometry of the semi-Riemannian hypersurfaces of co-index both 0 and 1 in a manifold endowed with a mixed 3-structure and a compatible metric.     

Conformally Flat Semi-Riemannian Manifolds with Nilpotent Ricci Operators and Affine Differential Geometry

We investigate the conformally flat semi-Riemannian manifolds withnilpotent Ricci operators. We construct a lot of complete orhomogeneous, conformally flat semi-Riemannian manifolds with nilpotentRicci operators. In this construction, we show interesting relationsbetween the semi-Riemannian geometry and the affine differentialgeometry of centro-affine hypersurfaces.     

一类浸入在半黎曼卷积空间的完备超曲面

We deal with complete hypersurfaces immersed in a semi-Riemannian warped product of the type eI×f M~n,where M~n is a connected n-dimensional oriented Riemannian manifold.When the fiber M~n is complete with sectional curvature-k≤K_M for some positive constant k,under appropriate restrictions on the norm of the gradient of the height function h,we proceed with our technique in order to guarantee that complete hypersurface immersed in a semi-Riemannian warped product is a slice.Our approach is based on the well known generalized maximum principle and another suitable maximum principle at the infinity due to Yau.