Osserman Conditions in Lightlike Warped Product Geometry

In this paper, we consider Osserman conditions on lightlike warped product (sub-)manifolds with respect to the Jacobi Operator. We define the Jacobi operator for lightlike warped product manifold and introduce a study of lightlike warped product Osserman manifolds. For the coisotropic case with totally degenerates first factor, we prove that this class consists of Einstein and locally Osserman lightlike warped product.


Introduction
The Riemann curvature tensor is one of the central concepts in the mathematical field of differential geometry. It assigns a tensor to each point of a (semi-)Riemannian manifold that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space. It expresses the curvature of (semi-)Riemannian. Curvature tensor is a central mathematical tool in the theory of general relativity and gravity.
The geometry of a pseudo-Riemannian manifold ( ) , M g is the study of the curvature 4 *

R T M ∈⊗
which is defined by the Levi-Civita connection ∇ .
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 for example associate to R an endomorphism on tangent bundle of a manifold. In [1] P. Gilkey The Riemannian curvature tensor of a Levi-Civita connection is algebraic on It is obvious that Motivated by the recent works on lightlike geometry, we consider in this paper lightlike warped product (sub-)manifolds and examine Osserman conditions depending on geometric properties of the factors.
In Section 2, we present background materials of lightlike geometry. In Section 3 we define lightlike warped product Osserman (definition 3.2) and present some important results of our research (Theorem 2, Theorem 3, Theorem 4). Section 4 is concerned with an example given in the neutral semi-Riemannian space 6 3 R ..

Preliminaries
Let ( ) A null submanifold M with nullity degree r equipped with a screen distribution The Gauss and Weingarten formulas are , Since ∇ is a metric connection, using (12)-(14) we have Let P the projection morphism of TM onto ( ) It follows from (17) and (18) that Let R and R denote the Riemannian curvature tensors on M and M respectively. The Gauss equation is given by Using (10) and (12) We say that the screen distribution ( ) S TM is totally umbilical if for any section N of ( ) ltr TM on a coordinate neighbourhood From now on, we assume that the frames ( ) i.e.
It is straightforward to check that g  defines a non-degenerate metric on M and that for 0 r = it coincides with g. The ( )

Lightlike Warped Product Geometry and Osserman Conditions
As it is well known, Jacobi operators are associated to algebraic curvature maps (tensors). But contrary to non-lightlike manifolds, the induced Riemann curvature tensor of a lightlike submanifold For degenerate warped product setting, we consider the associated non-degenerate metric g  defined by (30) of a lightlike warped product metric. We denote by g   and # g  the natural isomorphisms with respect to g  . The equivalent relation of (3) is given by , N g being totally degenerate, the Riemannian curvature tensor 1 R and its Weyl tensor vanish identically. Moreover 1 N is conformally Osserman. By Theorem 5 in [12], R is an algebraic curvature tensor. If we restrict our study on the product ( ) , it is obvious that N is a conformally Osserman manifold. The lightlike warped product metric g belongs to the conformal class of Due to Proposition 2 in [12], we establish the following two results for coisotropic warped product ( ) Proof. From (22), the induced Riemannian curvature tensor is Using (20) and (27), for all The pseudo-Jacobi operator ( ) R J X is given by Therefore N is pointwise Osserman.
■ From Proposition 2, theorem 5 in [12] and Theorem 4.3 in [9], we proved the following result that characterizes any screen distribution of a coisotropic warped product of a semi-Riemannian space form with the first factor totally null. This case consists of a class of null warped products that is Einstein and pointwise Osserman.  , , ,

Example
Let M be submanifold of 6 3  given by , ; sin