Osserman Lightlike Hypersurfaces on a Foliated Class of Lorentzian Manifolds

This paper deals with a family of Osserman lightlike hypersurfaces (Mu) of a class of Lorentzian manifolds M̄ such that its each null normal vector is defined on some open subset of M̄ around Mu. We prove that a totally umbilical family of lightlike hypersurfaces of a connected Lorentzian pointwise Osserman manifold of constant curvature is locally Einstein and pointwise F−Osserman, where our foliation approach provides the required algebraic symmetries of the induced curvature tensor. Also we prove two new characterization theorems for the family of Osserman lightlike hypersurfaces, supported by a physical example of Osserman lightlike hypersurfaces of the Schwarzschild spacetime.


Introduction
A primary interest in differential geometry is to determine the curvature and the metric of a given smooth manifold, which distinguishes the geometry of this subject from the others that are analytic, algebraic or topological.It is now well-known that the research on the Osserman condition (which involves sectional curvature and Jacobi operator) has provided substantial information on the curvature and metric tensors of Riemannian manifolds.An up-to-date account on Riemannian and semi-Riemannian Osserman geometry is available in a book (Garc ía-R ío et. al., 2002).Let ( M, ḡ) be a semi-Riemannian manifold.We say that R ∈ ⊗ 4 T ⋆ p M is an algebraic curvature map (tensor) on T p M if it satisfies the following symmetries: for all X, Y, Z, W ∈ T p M, p ∈ M. The Riemann curvature tensor is an algebraic curvature tensor on the tangent space T p M, for every p ∈ M. For an algebraic curvature map R the associated Jacobi operator J(X) is the linear map on T p M characterized by the identity ḡ(J(X)Y, Z) = R(Y, X, X, Z), (1.2) J(X) is a self-adjoint map and R is spacelike (respectively timelike) Osserman tensor if Spec{J(X)} is constant on the pseudo-sphere of unit spacelike (respectively unit timelike) vectors in T p M. These are equivalent notions (Gilkey, 2001) and such a tensor is called an Osserman tensor.The basic problem is to what extent general sectional curvatures can provide information on the curvature and metric tensors.Since any semi-Riemannian manifold has lightlike subspaces, one reasonably expects a role of Jacobi and Szabó type operators (Gilkey, 2001) in the study of lightlike hypersurfaces say (M, g) of an (n + 2)-dimensional semi-Riemannian manifold ( M, ḡ).According to (Duggal-Bejancu, 1996) approach, for a lightlike (M, g) there exists a non-vanishing null vector field ξ entirely in T M such that g(ξ, X) = 0, ∀X ∈ T M and T M = Rad(T M) ⊕ orth S (T M), (1.3) where Rad(T M) = {ξ} and S (T M) are a 1-dimensional radical null distribution and n-dimension non-degenerate complementary screen distribution, respectively, and ⊕ orth is a symbol for orthogonal sum.Throughout (Duggal-Bejancu, 1996) approach the null normal vector field ξ is defined entirely on M .For up-to date information on a general study of lightlike hypersurfaces of semi-Riemannian manifolds using the distribution equation ( 1.3) we refer to books (Duggal-Bejancu, chapter 4, 1996) and (Duggal-Jin, chapter 7, 2007).Since Jacobi and Szabó operators need the use of an inverse of a metric, due to degenerate g of M the definition of these operators is not possible in the usual way for the lightlike case.
To deal with this (Atindogbe-Duggal, 2009) used the concept of pseudo-inverse of a degenerate metric (Atindogbe et. el., 2003) and defined pseudo-Jacobi operators.Besides this anomaly, in general, induced Riemann curvature tensors on lightlike hypersurfaces are not algebraic curvature tensors, i.e (1.1) does not hold.For this anomaly, they proved the following result.
Theorem 1. (Atindogbe-Duggal, 2009) If the induced curvature tensor of M is an an algebraic curvature tensor, then, locally at least one of the following holds.
(a) M is totally geodesic.
(b) M is null transversally closed, that is, its transversal bundle is parallel along the radical direction.
Also, since in Duggal-Bejancu approach the lightlike geometry depends on a choice of screen distribution S (T M) which is not unique, a well-defined concept of Osserman condition is not possible for an arbitrary lightlike hypersurface.Therefore, they looked for an admissible S (T M) for which the associated induced curvature tensor of M is an algebraic curvature.
To the best of our knowledge, at the time of writing this paper, there are the (Atindogbe-Duggal's, 2009) paper and (Atindogbe et. el., 2011, Brunette, 2014) on the Osserman lightlike geometry, using an admissible screen distribution.Since lightlike (also called degenerate) geometry has been studied by several ways other than using a screen for specific problems (for example, see (Akivis-Goldberg 2000, Leistner, 2006) one may ask the following question.Is there a better way to deal with the lightlike Osserman geometry to improve on previous works on this topic and also find some new results?In this paper we answer both of these questions by using the following different approach: Consider a class of lightlike hypersurfaces M for which we assume that the null normal ξ is defined on some open subset of M around M. Following ( Carter, 1997), a simple way is to consider a foliation of M (in the vicinity of M) by a family (M u ) so that each ξ u is in the part of M foliated by this family such that at each point in this region, ξ u is a null normal to M u for some value of u.Although the family (M u ) is not unique, for our purpose we can manage (with some reasonable condition(s)) to involve only those quantities which are independent of the choice of the family (M u ) once evaluated at, say, M u=const. .We highlight that since ξ is not entirely in M the distribution equation (1.3) will not hold.However, due to degenerate metric g there is no canonical transversal direction since the normal vector ξ coincides with the tangent vector of M. Thus, a projector mapping II : T p M → T p M (needed to obtain well-defined induced objects on M) can not be defined from M alone.There is a need for some extra structure on M. In this paper we consider an extra structure of a class of Lorentzian manifolds ( M, ḡ) whose metric ḡ is prescribed by (2.5).This Lorentzian structure and its hypersurfaces have an added advantage of its uses in a variety of physical problems (see some references in Section 2 and 5).In Section 2 we obtain the normalized expression for any ξ u , its corresponding transversal vector field N u of T p M (see Theorem 3) and a well-defined projector onto M. Using the transversal vector field and the projector onto M, in Section 3 we write the local extrinsic form of the Gauss and Weingarten equations and the three local Gauss-Codazzi equations (3.6).It is important to notice from these three Gauss-Codazzi equations that the induced Riemannian curvature tensor on M may not be an algebraic curvature tensor, i.e. (1.1) does not hold.In Section 4, we first recall the following Osserman conjecture (see Chi, 1988).
Any Osserman manifold is either a locally flat space or a locally rank-one symmetric space.
In (Garc ía-R ío et.el., 2002) we have historical development on solving the Osserman conjecture for Riemannian, semi-Riemannian and Lorentzian manifolds.In the Lorentzian case they have presented the following positive answer to this conjecture: "If ( M, ḡ) is a connected (dim( M ≥ 3) Lorentzian pointwise Osserman manifold, then it is a real space form."(see Garc ía-R ío et. el., 2002, Theorem 3.1.2, page 42).
In this paper, we prove "A totally umbilical family of lightlike hypersurfaces (M u ) of a connected Lorentzian pointwise manifold of constant curvature is locally Einstein and pointwise F −Osserman".Contrary to a condition in Theorem 1 of a previous work of ( Atindogbe-Duggal, 2009), we show that in Theorem 11 our foliation approach provides the required algebraic symmetries of the induced curvature tensor of any member of (M u ), which is our first new result.Also, we prove two new characterization theorems (see Theorems 12 and 13) on totally geodesic lightlike Osserman hypersurfaces.In Theorem 12, using a symmetry condition we prove that the induced Riemann curvature tensor of the family (M u ) of lightlike hypersurfaces has the required symmetries if and only if the local second fundamental form of its each member is a Codazzi tensor.In Theorem 13 we use stationary coordinate system for a spacetime M to present a physical example of Osserman lightlike hypersurfaces of the Schwarzschild spacetime.

Projector Mapping onto (M, g)
To make this paper self contained, we have taken some material from a paper (Duggal, 2014).Recall that a hypersurface (M, g) of a semi-Riemannian manifold ( M, ḡ) is lightlike if there exists a non-vanishing null vector field ξ in T M which is orthogonal (with respect to g) to all vector fields in T M, that is, where g is the degenerate metric of M. In this paper we assume that the null normal ξ is not entirely in M, but, is defined in some open subset of M around M. This will permit to well-define the covariant derivative ∇ξ, where ∇ denotes the Levi-Civita connection on M. As explained in introduction, we have a foliation of M (in the vicinity of M) by a family (M u ) so that each ξ u is in the part of M foliated by this family such that at each point in this region, ξ u is a null normal to M u for some value of u.For simplicity, in this paper we consider (M, g) as a member of the family ((M u ), (g u )) and its respective metric g for some value of u, with the understanding that the results are same for any other member.The "bending" of M in M is described by the Weingarten map: W ξ associates each X of M a variation of ξ along X, with respect to the connection ∇.The second fundamental form, say B, of M is the symmetric bilinear form and is related to the Weingarten map by B(X, ξ) = 0 for any null normal ξ and for any X ∈ T M and this implies that B has the same ξ degeneracy as that of the induced metric g. (M, g) is called totally umbilical in M if and only if there is a smooth function ρ on M such that The totally umbilical notion does not depend on particular choice of ξ.M is proper totally umbilical if and only if ρ is non-zero on entire M. In particular, M is totally geodesic if and only if B vanishes, i.e., if and only if ρ vanishes on M. From (2.2), B(X, ξ) = 0 for any null normal ξ and (2.3), we conclude that M is totally umbilical in M if and only if, on each neighborhood U the conformal function ρ satisfies W ξ X − ρX ∈ ker(g), for all X ∈ Γ(T M). (2.4) Following result is important in the study of lightlike hypersurfaces: Proposition 2. Let ((M u ), (g u )) be a family of hypersurfaces of a semi-Riemannian manifold.Then its each member (M, g) is totally umbilical if and only if its each null normal ξ is a conformal Killing vector of the degenerate metric g.
Proof.Consider a member (M, g) of ((M u ), (g u )).Using the expression £ ξ g(X, Y) = g( ∇X ξ, Y) + g( ∇Y ξ, X) and B(X, Y) symmetric in above equation we obtain which is well defined up to conformal rescaling (related to the choice of ξ).Suppose M is totally umbilical, i.e., (2.3) holds.Using this in above equation we have £ ξ g = 2ρg on M. Therefore, ξ is conformal Killing vector of the metric g.
which implies that (2.3) holds so M is totally umbilical.
Throughout this paper, for simplicity, we denote by (M, g, ξ) a member of the family of lightlike hypersurfaces ((M u ), (g u ), (ξ u )) for some value of u of an (n + 2)−dimensional Lorentzian manifold ( M, ḡ).Due to degenerate metric g of M and not using any screen distribution, there is no canonical way for the degenerate structure alone of M to define a projector mapping II : T p M → T p M, to get induced extrinsic objects of M. Thus, there is a need for some extra structure on M.
For this purpose, consider (M, g, ξ) a lightlike hypersurface of M evolved from a spacelike hypersurface H t , at a coordinate time t to another spacelike hypersurface H t+dt at coordinate time t + dt, whose metric ḡ is given by where x 0 = t, and ) is the lapse function and U a are the components of a spacelike vector U, called the shift vector.In this way we also have a family of spacelike hypersurfaces (H t ) of M. The above choice of spacetime metric (2.5) is physically important frame work.For example, see (Gourgoulhon-Jaramillo, 1006) on event and isolated horizons (also see Duggal, 2008Duggal, , 2012Duggal, , 2014)), including several related references.The coordinate time vector t = ∂ ∂t is such that ḡ(dt, t) = 1, which we can write where n denotes the timelike unit vector field.We assume that each H t intersects each corresponding lightlike hypersurface M of M u on some n-dimensional submanifold of M, we denote by S t,u = M u ∩ H t an element of this family (S t,u ) of submanifolds of M. Let s ∈ Γ(T H t ) be a unit spacelike vector normal to S t,u defined in some open neighborhood of M u .
(a) The vector field N of T M along M given by is a null transversal normalizing vector field of M.
(b) The corresponding normalized expression for each ξ of M is Proof.Let s ∈ Γ(T H t ) be a unit vector normal to S t defined in some open neighborhood of M. Taking (S t ) a foliation of M, the coordinate t can be used as a non-affine parameter along each null geodesic generating M. The question is how to find some (null) direction transverse to M so that the corresponding normalized null tangent vector field ξ is tangent vector associated with this parameterization of the null generators, i.e., This means that ξ is a vector field "dual" to the 1−form dt.Equivalently, the function t can be regarded as a coordinate compatible with ξ, i.e., g(dt, ξ) = ∇ξ t = 1.
As n and s are timelike and spacelike respectively and they both do not belong to M, but also are normal to the ndimensional spacelike submanifold S t in M of M, it is easy to see that both ξ and N are linear combination of n and s.
Based on above facts, it is easy to see that ξ has the normalized form (2.7) which is tangent to M and it has the property of Lie dragging the family of submanifolds (S t,u ) of M. As any M is defined by u = a constant, the gradient du is its normal, i.e. g(du, X) = 0, ∀X ∈ Γ(T M).
Thus, the 1−form ξ associated to the null normal ξ is collinear to du, i.e. ξ = e ρ du, where ρ ∈ C ∞ (M) is a scalar field.It follows that N as given in (2.6) is a null transversal normalizing vector field of M satisfying the second equality in (2.7).This completes the proof.
We say that two null normals ξ and ξ of M belong to the same equivalence class [ξ] if ξ = cξ for some positive constant c.Then, it follows from (2.3) that with respect to change of ξ to ξ there is another Ñ = (1/c)N satisfying (2.3).Now we define the projector onto M along N by Above mapping is well defined, i.e., its image is in T p M. Indeed, II leaves invariant any vector in T p M and II(N) = 0.Moreover, the definition of the projector II does not depend on the normalization of ξ and N if they satisfy the relation (2.7).In other words, II is determined only by the foliation of the family (S t,u ) of M u and not by any rescaling of ξ.
Example 4. Consider a null cone ) are the Minkowskian coordinates with origin 0. Exclude 0 to keep null cone smooth.Let Λ n+1 0 be a member at the level u = b 0.Then, the scalar u generates a family of null cones ((Λ n+1 0 ) u ) given by The components of ξ are where λ = e ρ .The two unit normals n and s in the expression of ξ are Therefore, it follows from (2.7) that Example 5. Consider a smooth function F : Ω → R, where Ω is an open set of R n+1 .Then a hypersurface M of R n+2 1 is called a Monge hypersurface (Duggal-Bejancu, 1996, page 129) given by the equation ) are the standard Minkowskian coordinates with origin 0. The scalar u generates a family of Monge hypersurfaces (M u ) as the level sets of u and is given by where b is a constant and we take M b = M a member of the family (M u ).
Then, the null normal ξ of M is Thefore, it follows from (2.7) that Induced metric on (S t,u ).Let h be the induced Riemannian metric on an n-dimensional element S t,u of the family (S t,u ) of co-dimension 2 submanifolds of M. Using the spacelike normal s and timelike normal n it is easy to see that the expression of h is given by h (2.8) Proposition 6.The positive definite metric h induced by ḡ on T p (S t ) coincides with the degenerate metric g induced by ḡ on T p (M).
Proof.Let x and y be the projections along ξ on T p (S t ) of their respective pair of vectors (X, Y) in T p (M).Then, we have the unique decompositions X = x + aξ, Y = y + bξ, for two real numbers a and b.Using n.x = n.y= s.x= s.y= 0 in the righthand side of the relation (2.8) it is straightforward to get h(X, Y) = ḡ(X, Y).
This means that h and ḡ coincide on T p (M) which further means that h coincides with the degenerate metric g of T p (M) that completes the proof.
It follows from above that g can replace the notation h given by (2.9) The endomorphism T p ( M) → T p ( M) canonically associated with h by the metric ḡ is the orthogonal projector onto S t given by h = I + n⟨n, •⟩ − s⟨s, •⟩. (2.10)

The Induced Extrinsic Objects
Recall that (Duggal-Bejancu, 1996) used a screen distribution to obtain induced extrinsic objects of a lightlike hypersurface.Although we are not using any screen for M, we do have a vector bundle T S of the family of n-dimensional co-dimension 2 submanifolds of M. For the extrinsic structure equations needed in this paper we replace the role of screen by the vector bundle T S of M which has the added advantage that it is obviously integrable.With this understanding, from Theorem 3 we have the following decomposition.
where tr(T M) = span{N} denotes a null transversal vector bundle of rank 1 and is complementary to T M in T M|M .
Using above decomposition and the second fundamental form B, we obtain the following extrinsic Gauss and Weingarten formulas: where A N is the shape operator of T p M in M and τ is a 1−form on M and ∇ the linear connection on (M, ξ).In general, ∇ is not a Levi-Civita connection and it satisfies where η(X) = ḡ(X, N) ∀X ∈ Γ(T M |U ).Let R and R denote the curvature tensors of the Levi-Civita connection ∇ on M and the linear connection ∇ on M, respectively.Using (3.2) and (3.3) we obtain (3.5) The local Gauss-Codazzi equations are The induced Ricci tensor of (M, g) is given by the following formula: Since ∇ on M is not a Levi-Civita connection, in general, Ricci tensor is not symmetric.Indeed, let F = {ξ, N, e 1 , • • • , e n } be a quasi-orthonormal frame on M.Then, we obtain Using Gauss-Codazzi equations and the first Bianchi identity we get Also, the 1−form τ in (3.3) depends on the the normalizing field N and, requiring that Ric is symmetric is important to both geometric and physical purpose.So, the following fact is noteworthy.
Proposition 7. Let (M, g, N) be a lightlike hypersurface of a semi-Riemannian manifold ( M, ḡ) with g given by (2.5) and null transversal vector field (2.7).Then, the induced Ricci tensor of M is symmetric if and only if the connection 1−form on M given by τ , where ∇ denotes the Levi-Civita connection of g.

Results
To interpret the relation (1.2) we need to get a pseudo-Jacobi operator associated to an algebraic curvature R of (M, g).Consider on M a normalized pair {ξ, N} satisfying Theorem 3 and the 1−form η(X) = ḡ(N, X), ∀X ∈ Γ(T M).Define an isomorphic map ♭ by Clearly, g is a Riemannian metric on M and the following holds: Let S p (M) = {V ∈ T p M||g(V, V)| = 1} be a unit bundle at any point p ∈ M.Then, in terms of the above isomorphisms ♭ g and ♯ g , the following is equivalence to the relation (1.2) for V in the unit bundle, X, Y ∈ T p M and g Riemannian.
Definition 8. Let (M, g, ξ, N) be a member of a family of lightlike hypersurfaces F = ((M u ), (g u )) of a Lorentzian manifold ( M, ḡ) defined by the metric (2.5) such that the induced Riemannian curvature tensor R of each M ∈ F is an algebraic curvature tensor and V ∈ S p (M).By a pseudo-Jacobi operator of R with respect to V we mean the self-adjoint linear map where ♯ g is the dual isomorphism on the triplet (M, g, N).
Remark.Let R be the induced (algebraic) Riemann curvature tensor of (M, g), (p ∈ M) and ξ ∈ T M ⊥ .Then, we have and since g is non-degenerate on T M, we have J R (V)ξ = 0.
Definition 9. A lightlike hypersurface (M, g) of a Lorentzian manifold (M, g) with metric g given by (2.5) is said to be timelike (resp.spacelike) F −Osserman at p ∈ M if the normalization (2.7) induces an algebraic curvature map and the characteristic polynomial of the associate Jacobi operator J R (x) is independent of x ∈ S − p (M) (resp.x ∈ S − p (M)).Moreover, if this holds at each p ∈ M, then (M, g) is called pointwise F −Osserman.
It is easy to check that timelike F −Osserman and spacelike F −Osserman are equivalent notions and we will use indistinctly F −Osserman for both.Recall that the Osserman Conjecture is true in Lorentzian manifolds, which we quote Theorem 10. ( Garc ía-R ío et. al., 2002, Th. 3.1.2, page 42) Osserman manifold, then it is a real space form.
Thus, the Lorentzian Osserman manifolds have constant curvaure.In our foliated approach fixing the ambiguities inherent to null hypersurfaces normalizations, we have the following important result in characterizing F −Osserman lightlike hypersurfaces of Lorentzian manifolds of constant curvature.
Theorem 11.Let F = ((M u ), (g u ), (ξ u ), (N u )) be a family of totally umbilical lightlike hypersurfaces of an n + 2 ≥ 3 connected Lorentzian pointwise manifold ( M, ḡ) of constant curvature and defined by the metric (2.5).Then, each lightlike hypersurface of the family F is locally Einstein and pointwise F −Osserman.
Proof.We first point out that, as per Theorem 10, for dim M = n + 2 ≥ 3, the Lorentzian manifold (M, g) given by (2.5) being pointwise Osserman is necessarily a real space form.Pick M = M u 0 a member of the family F and let g denote the induced degenerate metric on M. By hypothesis M being totally umbilical in M implies that B(X, Y) = ρg(X, Y) for some smooth function ρ.Let c denote the (constant) ambient sectional curvature and R the induced Riemann curvature on M.Then, For any X ∈ Γ(T M) denote by x its corresponding projection along ξ on T S .Then, Since each M u is totally umbilical in M, it follows from the Proposition 6 that the codimension 2 family (S t = M u 0 ∩ H t ) of M gives a totally umbilical nondegenerate foliation.Hence, from total umbilicity of the foliation (S t ), there is a smooth function µ with where we set Thus, it is clear that the induced Riemann tensor has required algebraic symmetries.By using (2.6) and (2.7) we compute the (0, 2)−tensor of Ricci: This shows that M is locally Einstein.Let p ∈ M, x ∈ T p S and Z ∈ x ⊥ .Then, In adapted quasi-orthonormal basis the matrix of J R (x) has the form It follows that the characteristic polynomial f x of J R (x) is given by Hence, M is pointwise F −Osserman, which completes the proof.
Theorem 12. Let F = ((M u ), (g u ), (N u )) be a family of lightlike hypersurfaces of a Lorentzian manifold ( M, ḡ) defined by the metric (2.5).Assume that the lapse function satisfies dλ = ⟨n, ∇ • s⟩ and the symmetry for all X and Y tangent to M. In particular, as η vanishes on Γ(T S t ), On the other hand, we compute the left hand side of (4.7) for the normalization N = 1 2λ (s − n).We have (4.9) So, combining (4.8) and (4.9) leads to It follows from above that for a fixed u = u 0 , the foliation S t = M u 0 ∩ H t is totally geodesic in M. In particular, the Gauss-Codazzi equations reduce to for X, Y, Z and W tangent to M u 0 and we have Therefore, as the induced Riemann curvature tensor always satisfies the first Bianchi identity, the only obstruction for R to satisfy the remaining required algebraic symmetries is that R(X, Y)ξ be proportional to ξ.Note that the connection 1−form is given by Hence the hypothesis that the given lapse function satisfies dλ = ⟨n, ∇ • s⟩ means that the connection 1−form τ vanishes identically.But by Gauss-Codazzi equations, we also have It follows that the induced Riemann curvature tensor satisfies the required symmetries if and only if the second fundamental form B is a Codazzi tensor.Now, let R, R ′ and ⋆ R denote the algebraic curvature tensor induced on M = M u 0 by the normalization (2.7), the restriction of R on S t and the Riemann curvature tensor given by the Levi-Civita connection ⋆ ∇ on the S t , respectively.We show that R ′ = ⋆ R at any p in the domain of the foliation.Let x, y, z be tangent to S t .By straightforward calculation using structure equations of Gauss and Weingarten, we have Thus, we get R ′ (x, y)z = ⋆ R (x, y)z from £ n g − £ s g = 0 using (4.10).But for a given x ∈ T p M with p ∈ S t , x ∈ S p (M) if and only if where ⊥ T p S t denotes the orthogonality symbol restricted to T p S t .Hence, Therefore, as from (4.2) we have J R (x)ξ = 0, if we let f x (t) and h⋆ x (t) denote the characteristic polynomials of J R (x) . Hence, M u 0 is (timelike) F −Osserman at p if and only if the leaf S t which is a Riemannian due to the Lorentzian signature of ( M, ḡ) is (timelike) Osserman at p.

Physical Application
It is well-known that the metric symmetry (an important concept in mathematics and physics) is based on the existence of Killing or conformal Killing vector (CKV) fields (see Duggal-Sharma, 1999) and references therein).Related to this paper, we concentrate on the Killing symmetry extensively used in general relativity, in particular in reference to black hole physics and then prove a characterization theorem for totally geodesic Osserman lightlike hypersurfaces of a stationary spacetime, supported by a physical example.For this purpose we recall the following.A spacetime admitting a timelike Killing vector field is called stationary.For example, see a paper by (Makoto-Kei-ichi, 2007) in which they found a way to construct the black brane solutions from intersecting M-branes via torus compactification.They proved that five dimensional black brane solutions with asymptotically flatness and regularity at rotating axis are stationary spacetimes.Also we know (Gromoll-Grove, 1985) that in any space of constant curvature, line fields with bundle-like metric (i.e.Riemannian flows) are always flat (i.e the transversal distribution to the foliation defines a totally geodesic foliation) or homogeneous (i.e. the orbit foliation of a group of isometries).Moreover, time-independent gravitational fields play an important role in General Relativity (Schwarzschild solutions (exterior and interior) and the Kerr metric of a rotating black hole are common examples of stationary space-times.All these exact solutions share two properties: namely asymptotic flatness and time-independence which implies the existence of a timelike Killing vector field.Furthermore, Killing symmetry of totally geodesic lightlike hypersurfaces has been widely studied in the literature in connection with the isolated and event black hole horizons (see Gourgoulhon-Jaramillo, 2006, Hawking, 1972).For a physical application, we need the following stationary coordinate systems of our landing spacetime.
Stationary coordinate systems.Let (M, g, ξ, N) be a member of the family ((M u ), (g u )) of lightlike hypersurfaces of a spacetime ( M, ḡ) whose metric ḡ is given by (2.5).With respect to each S t , the shift vector U can be expressed as U = αs − V, α = s • U, V ∈ T p (S t ).
(3) ( M, ḡ) is called null Osserman with respect to a unit timelike vector z ∈ T p ( M) if the characteristic polynomial of Rξ is independent of ξ ∈ N(z).Let L be a timelike line subbundle of T M. Then ( M, ḡ) is called globally null Osserman with respect to L if it is pointwise null Osserman with resepect to L and the characteristic polynomial of Rξ is independent of unit z ∈ L. Using the methodology as adapted in this paper and above Theorem 14, we propose the study on Osserman lightlike hypersurfaces of warped product globally null Osserman Lorentzian manifolds.
(B) General study.There is a need to extend the results of this paper for a general study on Osserman lightlike hypersurfaces of an ambient semi-Riemannian manifold ( M, ḡ).It is quite a challenging open problem to define a well-defined projector mapping II : T p M → T p M for an arbitrary M, which is needed to get induced extrinsic objects of M and a theorem similar to Theorem 3 of this paper.Initially, one may try to work by choosing a prescribed semi-Riemannian manifold.

Theorem 14 .
(Garc ía-R ío et.al., 2002, page 56) Let ( M, ḡ) be a Lorentzian manifold of dimension ≥ 4. If ( M, ḡ) is globally null Osserman with resepect to a timelike line bundle L of T M and is not of constant sectional curvature, then it is locally a warped product (I × N, −dt 2 ⊕ f g ′ ), whereI ⊆ R is an open interval and (N, g ′ ) is a Riemannian real space form.
6)holds when restricted to the intersecting data.Then the induced Riemann curvature tensor of each member (M, g, ξ, N) of F has the required symmetries if and only if the local second fundamental form of each member is a Codazzi tensor.Moreover, each member of the family is F −Osserman if and only if the non-degenerate intersecting data S t,u are Osserman.Proof.Recall from Proposition 6 that the positive definite induced metrics on S t by both ḡ and (the degenerate metric) g coincide on each T p S t , (p ∈ S t ).We first show that each member M = M u 0 of the family is geodesicaly foliated by the (S t ).Let C denote the local second fundamental form of the integrable distribution S t .