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 is the study of the curvature 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  P. Gilkey studied geometric properties of natural operators defined by the Riemann curvature tensor and Osserman proposed in  a characterization of Riemannian rank 1-symmetric spaces in terms of the spectrum of the Jacobi operator. Many other central works have been done by Ivanova, Stanilov, Videv and Szabo     .
On lightlike geometry of hypersurfaces, C. Atindogbe and K. L. Duggal have studied Pseudo-Jacobi operators and considered Osserman conditions , and in , the authors introduced the notion of r-lightlike Osserman submanifolds.
Let be a semi-Riemannian manifold and . An element is said to be an algebraic curvature tensor on if R has the following symmetries:
The Riemannian curvature tensor of a Levi-Civita connection is algebraic on for all . If R is an algebraic curvature tensor on , the associated Jacobi operator with respect to is the self-adjoint linear map on characterized by the identity
It is obvious that and the domain of is the unit pseudo-sphere of unit timelike or unit spacelike vectors
Due to the algebraic properties (1) and (2) of the curvature, we have and . Then, the Jacobi operator naturally reduces to the endomorphism .
The Riemannian curvature tensor R of a semi-Riemannian manifold is said to be a spacelike (resp. timelike) Osserman tensor on if the spectrum spec ( ) is constant on (resp. ). If this is the case at each , we say that is pointwise Osserman semi-Riemannian manifold.
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 ..
Let be a -dimensional semi-Riemannian manifold of constant index q such that and be an m-dimensional submanifold of . We assume that both m and k are . At each point ,
is the normal space at p. In case is non-degenerate on , both and are non-degenerate and we have . If the mapping
is a smooth distribution with constant rank , M is said to be lightlike (or null) submanifold of , with nullity degree r. This mapping is called the radical distribution on M. Any complementary (and hence orthogonal) distribution of in TM is called a screen distribution. For a fixed screen distribution on M, the tangent bundle splits as
where is the orthogonal direct sum.
A screen transversal vector bundle on M is any (semi-Riemannian) complementary vector bundle of in . It is obvious that both and is non-degenerate with respect to and
A null submanifold M with nullity degree r equipped with a screen distribution and a screen transversal vector bundle is denoted . It is said to be
• r-lightlike if ;
• coisotropic if (hence );
• isotropic if , (hence );
• totally null if , (hence ).
For any local frame of , there exists a local frame of sections with values in the orthogonal complement of in such that
and it follows that there exists a lightlike transversal vector bundle locally spanned by .
If we denote by a (not orthogonal) complementary vector bundle to in , the following relations hold
The Gauss and Weingarten formulas are
. The components and belong to , and to . and are linear connections on TM and the vector bundle respectively. According to the decomposition (8), let L and S denote the projection morphisms of onto and respectively, , where is the composition law, , . The transformations and do not define linear connections but Otsuki connections on with respect to the vector bundle morphisms L and S. Then,
Since is a metric connection, using (12)-(14) we have
Let P the projection morphism of TM onto . Using the decomposition (6) we get
and is a metric connection on .
It follows from (17) and (18) that
Let and R denote the Riemannian curvature tensors on and M respectively. The Gauss equation is given by
Definition 2.1 ( ). A lightlike submanifold of a semi-Riemannian manifold is totally umbilical in if there is a smooth transversal vector field on M called the transversal curvature vector field of M such that, for all
Using (10) and (12) it is easy to see that M is totally umbilical if and only if on each coordinate neighbourhood there exist smooth vector fields and such that
Definition 2.2 ( ). Let be a r-lightlike (i.e. ) or a coisotropic m-dimensional submanifold of a -dimensional semi-Riemannian manifold . We say that the screen distribution is totally umbilical if for any section N of on a coordinate neighbourhood , there exists a smooth function on such that
Definition 2.3. A coisotropic submanifold of a semi-Riemannian manifold is screen locally conformal if the local second fundamental forms of the screen distribution are related with the local second fundamental form of M as follows:
where is a conformal smooth function on a coordinate neighbourhood in M. In particular, we say that M is screen homothetic if is a non-zero constant.
Let be a null submanifold with nullity degree r of a semi-Riemannian manifold , and local frames of and respectively satisfying . Consider the 1-forms metrically equivalent to the i.e. . Then, each tangent vector field X has the splitting,
From now on, we assume that the frames and are globally defined on M. Consider the values mapping defined on by
where denotes the interior product with respect to X. The mapping is an isomorphisme of onto and we let denote its reverse mapping. (resp), is called the dual 1-form of X and the dual field of with respect to the pair of frames and. Define a -tensor on M by
It is straightforward to check that defines a non-degenerate metric on M and that for it coincides with g. The -tensor is called the pseudo-inverse of g. Let be a quasi-orthonormal field of frames on M with respect to the decomposition (6). Using (30) we have
Definition 2.4. Let and be semi-Riemannian and be positive smooth functions. The multiply warped product is the product manifold furnished with the metric tensor
where, are the projection morphisms. The functions are called the warping functions and the base manifold of the multiply warped product. Each is called a fiber manifold.
• If then we obtain a singly warped product
• If for then we have a multiple product manifold.
• If all are Riemanniann then is also a Riemannian multiply warped product manifold. is Lorentzian multiply warped product if are Riemannian and either is Lorentzian or a one-dimensional manifold with a negative definite metric.
• is lightlike with nullity degree r if is degenerate with of rank r. still has rank r and all screen structure on M has dimension where is the dimension of any screen structure on.
Proposition 1.  On, if;, then
1) is the lift of,
3) is the lift of,
Corollary 1. The leaves of the warped product are totally geodesic; the fibers are totally umbilical.
3. 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 is not an algebraic curvature map in general as it can be seen from (23). In case this requirement is satisfied, the pair of screens is said to be admissible.
In semi-Riemannian case, the relation (3) characterizes the Jacobi operator associated to an algebraic curvature tensor. For (or), , we have
For degenerate warped product setting, we consider the associated non-degenerate metric defined by (30) of a lightlike warped product metric. We denote by and the natural isomorphisms with respect to. The equivalent relation of (3) is given by
Definition 3.1. Let be a lightlike warped product submanifold of a semi-Riemannian manifold, , (or) and an algebraic curvature tensor on. A pseudo-Jacobi operator associated to R with respect to X is the self-adjoint linear map on defined by
Definition 3.2. A lightlike warped product submanifold of a semi-Riemannian manifold is called spacelike (resp. timelike) Osserman at if for each admissible pair of screens and associate induced Riemann curvature R, the characteristic polynomial of is independent of (resp.). If this is the case for each, then is called pointwise Osserman. If in addition there is no dependence with respect to then is said to be globally Osserman.
Theorem 2. Let and be a totally lightlike manifold and a conformally Osserman semi-Riemannian manifold respectively. Let f be an isometric immersion of in a semi-Riemannian space form where and. Then is a conformally Osserman Lightlike warped product submanifold of.
Proof. Let and R be the Riemannian curvature tensors of and N respectively. being totally degenerate, the Riemannian curvature tensor and its Weyl tensor vanish identically. Moreover is conformally Osserman. By Theorem 5 in , 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 which is conformally Osserman lightlike
product submanifold. Since the Weyl tensor is invariant in the conformal class of a metric, we conclude that is a conformally Osserman lightlike warped product.
From definition 2.2, it is obvious that if a screen distribution is totally umbilical, the bilinear form is symmetric on. By theorem 2.5 in  p.161, is integrable and by theorem 5.3 in , if the ambiant space is of constant sectional curvature, the induced Ricci tensor is symmetric.
Due to Proposition 2 in , we establish the following two results for coisotropic warped product with totally degenerate isometrically immersed in a semi-Riemannian space form.
Theorem 3. Let be a coisotropic isometric immersion of a warped product of a totally lightlike manifold and a semi-Riemannian manifold in a semi-Riemannian space form that is screen conformal. Then the associated Ricci tensor is symetric and N is locally Einstein. Also, N is pointwise Osserman.
Proof. From (22), the induced Riemannian curvature tensor is
Using (20) and (27), for all, we get
Let be a quasi-orthonormal field of frames
on where is a orthonormal field of frames on. Then, for all we compute the induced Ricci curvator tensor as follow
Thus the induced Ricci curvature tensor is symmetric and N is locally Einstein. Consider and. By (37) we have
The pseudo-Jacobi operator is given by
and its characteristic polynomial is
Therefore N is pointwise Osserman. ■
From Proposition 2, theorem 5 in  and Theorem 4.3 in , 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.
Theorem 4. Let, be a coisotropic warped product submanifold of a semi-Riemannian space forme where is totally degenerate. Then any screen distribution is admissible and totally umbilical on N. In addition N is locally Einstein and pointwise Osserman.
Let be a semi-Riemannian manifold, where is semi-Euclidean space of signature with respect to the canonical basis. Let M be submanifold of given by
Then is spanned by
. Thus M is a 2-lightlike submanifold of with. Choose, we construct two null vectors
Since rank equals codimension of M, we conclude that M is a coisotropic submanifold of. By direct calculations, and are integrable in M. The induced metric tensor on M is given by
and we get
It is obvious that M is a coisotropic warped product submanifold of with warping function. Using (12) and (17) we obtain
From (38), we have
and we conclude that is a totally umbilical null warped product submanifold of and it is obvious that it is mixed totally geodesic.
From (39) we have
and we conclude that the screen distribution of M is totally umbilical and M is screen homothetic.
M being coisotropic, taking into account has constant sectional curvature, consider (40) and (41), from (42) we get
Let’s consider, (or),. Then by using (43), we have
The pseudo-Jacobi operator is given by and its characteristic polynomial is given by
that is independent of. Therefore, M is spacelike (timelike) pointwise Osserman null warped product.
Remark. From (44), it is obvious that for a lightlike warped product manifold, to be spacelike Osserman or timelike Osserman are equivalent.
5. Conclusion and Suggestions
Osserman conditions on lightlike warped product manifolds have been considered in this paper. The case of lightlike warped product with the first factor totally degenerate has been explored. Especially in coisotropic case, we have proved that this class consists of Einstein and locally Osserman lightlike warped product. In perspective, we are going to extend this study to other classes of lightlike warped product in order to get later a certain characterization of lightlike warped product Osserman manifolds.
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