Algebraicity of Induced Riemannian Curvature Tensor on Lightlike Warped Product Manifolds

Lightlike warped product manifolds are considered in this paper. The geometry of lightlike submanifolds is difficult to study since the normal vector bundle intersects with the tangent bundle. Due to the degenerate metric, the induced connection is not metric and it follows that the Riemannian curvature tensor is not algebraic. In this situation, some basic techniques of calulus are not useable. In this paper, we consider lightlike warped product as submanifold of semi-Riemannian manifold and establish some remarkable geometric properties from which we establish some conditions on the algebraicity of the induced Riemannian curvature tensor.


Introduction
Semi-Riemannian geometry is the study of smooth manifolds with non-degenerate metric signature [1]. Semi-Riemannian geometry includes the Riemannian geometry with a positive definite metric and Lorentzian geometry which is the mathematical theory used in General Relativity.
In 1969, Bishop and O'Neill [2] introduced a new concept of warped product manifolds to construct a rich variety of manifolds with useful applications in General Relativity on the study of cosmological models and black holes. For example, it has been pointed out in [3] that some well-known exact solutions to Einstein field equations are semi-Riemannian warped products.
It is well-known that for any semi-Riemannian (warped product) manifold, case of submanifolds namely lightlike (degenerate) [4]. The geometry of lightlike submanifolds is different from the non-lightlike one and rather difficult since its normal vector bundle intersects with the tangent bundle. Due to the degenerate metric induced on a lightlike manifold, the induced connection is not metric and it follows that the Riemannian curvature tensor is not algebraic. Thus, one can not use, in the usual way, the habitual submanifold theory to define any induced object on a degenerate submanifold.
A Riemannian curvature tensor of a semi-Riemannian manifold ( ) , M g is algebraic if it has the following symetry properties The notion of curvature is one of the central concepts of differential geometry, one could argue that is the one central on, distinguishing the geometrical core of the subject from those aspects that are analytic, algebraic, or topological [5]. Curvature also plays a key role in physics. The motion of a body in a gravitational field is determined, according to Einstein, by the curvature of space-time.
Since the whole curvature tensor is difficult to handle, the investigation usually focuses on different objects whose properties allow us to recover curvature tensor. One can associate to R an endomorphism on tangent bundle of a manifold [6]. In lightlike geometry, to make such study, we have to ensure that the Riemannian tensor has the algebraic proprties.
Although the lightlike geometry is difficult to study, there are important applications in Physic. In [7] the author used the warped product technique to study a problem concerning of finding a warping function such that the degenerate metric of a globally lightlike warped product manifold admits constant scalar curvature and discovered that this approach has an interplay with the static vaccum solutions of Einstein equation of general relativity.
In this paper, we examine some conditions on lightlike warped product (sub-)manifolds to admit an algebraic curvature tensor. We particularly consider single lightlike warped product (sub-)manifolds and present some technical and characterization results (Proposition 2, Proposition 3, Proposition 4). We establish algebraicity condition for the (induced) Riemannian curvature tensor on lightlike warped product submanifold (Theorm 5, Theorem 6).

Basic Notions on Lightlike Geometry
For more details see [4]. Let ( ) is the normal space at p. In

S TM S TM
A lightlike submanifold M with lightlikeity degree r equipped with a screen distribution ( )

S TM and a screen transversal vector bundle ( )
For any local frame { } i ξ of ( ) Rad TM , there exists a local frame { } i N of sections with values in the orthogonal complement of ( ) The Gauss and Weingarten formulas are  (7), let L and S denote the projection morphisms of ( ) Since ∇ is a metric connection, using (11)-(13) we have Let P the projection morphism of TM onto ( ) S TM . Using the decomposi- It follows from (16) and (17) Using (9) and (11) where i φ is a conformal smooth function in a coordinate neighbourhood  in M. In particular, we say that M is sreen homothetic if i φ is a non-zero constant. For a singly warped product, we have the following: From the previous proposition, one can see that

Our Main Results
In the following, we consider a lightlike warped product ( ) Proof. In case of coisotropic submanifold we have  , , , From (19) and (30) We give the following result on the algebraic properties of the induced Riemannian tensor on lightlike warped product with the first factor totally degenerate. with the first factor 1 N totally degenerate. Then the induced Riemannian curvature is an algebraic tensor.
Proof. The result hold from Theorem 3.2 in [9] and proposition 4. In case of coisotropic warped product of a semi-Riemannian manifold with constant sectional curvature which is conformal screen, we establish the following. : be a coisotropic isometric immersion of a lightlike warped product into a semi-Riemannian manifold which is a space form such that the lightlike warped product M is conformal screen. Then the induced Riemannian curvature R is an algebraic curvature tensor. Proof

Conclusion and Suggestions
The algebraicity conditions of the induced Riemannian curvature tensor have been explored in this paper. Some remarkable geometric properties of lightlike warped product submanifolds have been given. From the above results, one can see that the induced Riemannian curvature tensor on lightlike warped product submanifolds with totally null first factor is an algebraic curvator tensor. In the future, we will be studying Osserman conditions on lightlike warped product manifolds.