Chen’s Ricci inequalities and topological obstructions on null hypersurfaces of a Lorentzian manifold

Given a null hypersurface of a Lorentzian manifold, we isometrically immerse a null hypersurface equipped with the Riemannian metric (induced on it by the rigging) into a Riemannian manifold suitably constructed on the Lorentzian manifold. We study the intrinsic and extrinsic geometry of such an isometric immersion and we link them to the null geometry of the null hypersurface in the Lorentzian manifold. In the course of this immersion, we find the basic relationships between the main extrinsic invariants and the main intrinsic invariants, named Chen-Ricci inequalities of the null hypersurface in the Lorentzian manifold. The findings prove a topological implication of these relationships.


Introduction
Soon after Riemann (1854) introduced the notion of a manifold, Schläfli (1873) conjectured that every Riemannian manifold could be locally considered as a submanifold of an Euclidean space with sufficiently high codimension. This was later proved in different steps by Janet (1926), Cartan (1927), Burstin (1931) and extended to semi-Riemannian manifolds by Friedmann (1965). In 1956, Nash proved that every n-Riemannian manifold can be isometrically embedded in an Euclidean m-space E m with m = n 2 (n + 1)(3n + 11). From the aforementioned, it is difficult to apply Nash' theorem, because it requires a very large codimension and there is no general optimal relationships between the known intrinsic invariants and the main extrinsic invariants for Riemannian submanifold of Euclidean spaces. To overcome the difficulties Chen introduced in 1993 a new type of Riemannian invariants for a Riemannian manifold M [6]. We have where τ (p) is scalar curvature of M and inf(K)(p) = inf{K(π) : K(π) is a plane section of T p M}.
In [5], Chen established the following general optimal inequality involving the new intrinsic invariant δ M and the squared mean curvature H 2 for n-dimensional submanifold M of a real space form of constant sectional curvature c: δ M ≤ n 2 (n -2) 2(n -1) H 2 + 1 2 (n + 1)(n -2)c. ( 2 ) In [8], Chen proved a basic inequality involving the Ricci curvature and the squared mean curvature of a submanifold of a real space form: Ric(X) ≤ 1 4 n 2 H 2 + (n -1)c.
In [7], B.Y. Chen estabished the inequality between the shape operator and the mean curvature of isometric immersions in real space forms. In [16], the autors give some remarks on B.Y. Chen's inequality involving classical invariants. In [13] Hong and Tripathi studied this inequality and they presented a general theory for submanifolds of Riemannian manifolds and proved a basic inequality using (3) as follows: where M is a n-dimensional submanifold of M, Ric (T p M) (X) is the n-Ricci curvature of T p M at X ∈ T 1 p M with respect to the ambient manifold M and T 1 p M is the set of unit vectors in T p M. The equality case of (4) is satisfied by X ∈ T 1 p M if and only if α(X, Y ) = 0, ∀Y ∈ T p M, g(X, Y ) = 0, 2α(X, X) = nH. (5) In [18], this inequality was named a Chen-Ricci inequality by Tripathi where α is the second fundamental form of M. The equality case of (4) holds for all unit vectors X ∈ T 1 p M and for all p ∈ M if and only if either M is totally geodesic, or n = 2 and p is a totally umbilical point. In the degenerate submanifolds, Gülbahar, Kiliç and Keleş introduced k-Ricci curvature, k-scalar curvature, k-degenerate Ricci curvature, k-degenerate scalar curvature and they established some inequalities that characterize lightlike hypersurfaces of a Lorentzian manifold ( [10]). Afterward, in [11] they established some inequalities involving k-Ricci curvature, k-scalar curvature, the screen scalar curvature on a screen homothetic lightlike hypersurface of a Lorentzian manifold and they computed the Chen-Ricci inequality and the Chen inequality on a screen homothetic lightlike hypersurface of a Lorentzian manifold. In [15], they established some inequalities on the sectional curvature of lightlike submanifolds. In this present paper, we first consider the associated Riemannian metric of a null hypersurface in a Lorentzian manifold as in [3] but arising from a null rigging defined on a neighborhood of the null hypersurface, and we isometrically immerse the null hypersurface equipped with the associated Riemannian metric (induced on it by the rigging) into a Riemannian manifold suitably constructed on the Lorentzian manifold. We establish the link between intrinsic and extrinsic geometry of such isometric immersion (Proposition 4.1, Theorem 4.1). We connect the geometry of the Riemannian metric constructed on the Lorentzian ambient manifold with the Lorentzian geometry (Theorem 4.2, Theorem 4.3, Theorem 4.4, Theorem 4.5). We establish the Chen inequalities of a null hypersurface M n+1 in a Lorentzian ambient manifold M n+2 (Theorem 5.1, Theorem 5.2, Figure 1 Links between the null and Lorentzian geometry Theorem 5.3). It is here important first to give the organizational structure of this paper. Section 2 gives the necessary preliminaries about null hypersurfaces. In Sect. 3, we give the links between a null geometry and its associated Riemannian geometry. The relationships between the intrinsic and extrinsic geometries of the immersion of the null hypersurface M n+1 equipped with the associated Riemannian metric (induced on it by the rigging) into a Riemannian manifold suitably constructed on the Lorentzian ambient manifold M n+2 and the relations between the Riemannian geometry constructed on the Lorentzian manifold and the Lorentzian geometry are discussed in Sect. 4. We connect the intrinsic and extrinsic geometries of this immersion in order to link those two geometries to the null geometry and the Lorentzian one (see Fig. 1).
In the last section, we establish Chen's inequalities for a null hypersurface in Lorentzian manifold and we discover a topological obstruction to the minimal isometric immersion of the null hypersurface in an ambient Lorentzian manifold.
In the classical theory of non-degenerate hypersurfaces, we have the following decomposition: where ⊕ orth denotes orthogonal direct sum. Any vector field of TM splits uniquely into a component tangent to M and a component perpendicular to M. However, in the null hypersurface case, (6) does not hold because TM and TM ⊥ have a non-trivial intersection. Therefore, the introduction of the main induced geometric objects on M as the Levi-Civita connection, the second fundamental form, the operator form, have different properties from the non-degenerate case. In [9], the authors introduced a complementary bundle of TM ⊥ in TM which is a rank n non-degenerate distribution over M, called a screen distribution of M, which we denote by S(N), such as The existence of S(N) is secured provided that M is paracompact. A null hypersurface with a specific screen distribution is given by (M, g, S(N)). It is well known from [9] that, for such a triplet (M, g, S(N)), there exists a unique rank 1 vector subbundle tr(TM) of TM over M, such that, for any non-zero section ξ of TM ⊥ on a coordinate neighborhood U ⊂ M, there exists a unique section N of tr(TM) on U satisfyinḡ Then TM is decomposed as follows: tr(TM) is called a (null) transversal vector bundle along M. In fact, from (8) and (9) one shows that, conversely, a choice of a transversal bundle tr(TM) determines uniquely the distribution S(N). A vector field N as defined in (8)  An outstanding property of a rigging is that it allows a definition of geometric objects globally on M. We say that we have a null rigging when the restriction of L to the null hypersurface is a null vector field. Throughout the paper, we fix a null rigging N for M on M. In particular this rigging fixes a unique null vector field ξ ∈ (TM ⊥ ) called the rigged vector field. From now on, we denote the normalized (or rigged) null hypersurface by a triplet (M, g, N) where g = g |M is the first fundamental form and N is a null rigging for M. Let N be a null rigging of a null hypersurface of a Lorentzian manifold (M n+2 , g) and θ the 1-form metrically equivalent to N defined on some open set containing M and given by Suppose that is a restriction to M, i : M → M being the inclusion map. The normalization N will be said to be closed if the 1-form θ is closed on M. It is easy to check that S(N) = ker(η) and the screen distribution S(N) is integrable whenever η is closed. On a normalized null hypersurface (M, g, N), the Gauss and Weingarten type formulae are given by, respectively, for any X, Y ∈ (TM), where ∇ denotes the Levi-Civita connection on (M, g), ∇ denotes the rigged connection on (M, g) induced from ∇ through the projection along N ; it satisfies and ∇ * denotes the induced connection on the screen distribution.
Here η is an 1-form on TM defined by B N is the null second fundamental form of M and C N is the second fundamental on (S(N)), respectively. The second fundamental forms are related by their shape operators, A * ξ is the null shape operator with respect to the section ξ and τ N is the 1-form on TM defined by τ N (X) = g(∇ X N, ξ ). From (15) and A * ξ ξ = 0, we find that which means that integral curves of ξ are pregeodesic. Throughout the paper, we consider the integral curves of ξ to be geodesics in M and M, which means that . , x n , (x 1 ) 2 + · · · + (x n ) 2 ], and the null hypersurface M n+1 Let N be the null rigging of M defined by N = -x 0 ∂ 0 + n+1 a=1 x a ∂ a ; and the null vector field ξ = 1 Hence the null vector field ξ is normal to M, thus M is a null hypersurface.
Hence η = 2x 0 dx 0 and dη = 0. This shows that η is closed and τ N (X) = -[2x 2 0 (-dx 0 Definition 2.2 A null hypersurface M is said to be totally umbilical (respectively, totally geodesic) if there exists a smooth function ρ on M such that at each x ∈ M and for all X, Y ∈ T x M, B N (x)(X, Y ) = ρ(x)g(X, Y ) (respectively, B N vanishes identically on M). This is equivalent to write A ξ = ρP and A ξ = 0. Also, the screen distribution S(N) is totally umbilical (respectively, totally geodesic) if C N (X, PY ) = ρ(x)g(X, Y ) for all X, Y ∈ (TM) (respectively, C N = 0), which is equivalent to writing A N = ρP (respectively, A N = 0).
Denote by R and R the Riemann curvature tensors of ∇ and ∇, respectively. Recall the following Gauss-Codazzi equations ( [9]) for all X, Y , Z ∈ (TM), N ∈ tr(TM), ξ ∈ (TM ⊥ ): The shape operator A ξ is self-adjoint as the second fundamental form B N is symmetric. However, this is not the case for the operator A N as show in the following lemma.

Lemma 2.1 For all X, Y ∈ (TM)
where (throughout) , = g stands for the Lorentzian metric.
Proof Recall that η = i θ where θ = N, · . Taking the differential of θ and using the Weingarten formula, we have, for all X, Y ∈ (TM), Hence as announced.
We shall use the following notations: Due to the degeneracy of the first fundamental form g on the null hypersurface M, it is not possible to define the natural dual isomorphism between the tangent vector bundle TM and the cotangent vector bundle T M following the usual Riemannian way. However, this construction is possible by setting a rigging N (see [3], [1] for further details). Consider a normalized null hypersurface (M, g, N) and 1-form defined by (17). For all X ∈ (TM), X = PX + η(X)ξ and η(X) = 0 if and ony if X ∈ (S(N)). Now, we define η by η is an isomorphism of (TM) on to (T * M) and can be used to generalize the usual non-degenerate theory. In the latter case, (S(N)) coincides with (TM), and as a consequence the 1-form η vanishes identically and the projection morphism P becomes the identity map on (TM). Let η denote the inverse of the isomorphism η given by (28). For X ∈ (TM) (respectively, w ∈ T * M), X η (respectively, w η ) is called the dual 1-form of X (respectively, the dual vector field of w) with respect to the degenerate metric g. It follows from (28) that if w is a 1-form on M, we have, for X ∈ (TM), Define a (0, 2)-tensor g η by g η (X, Y ) = X η (Y ), ∀X, Y ∈ (TM). Clearly, g η defines a nondegenerate metric on M which plays an important role in defining the usual differential operators (gradient, divergence, Laplacian) with respect to degenerate metric g on null hypersurface (for details see [3]). In this case, g η is called the associated metric to g on (M, g, N). Also, observe that g η coincides with g if the latter is non-degenerate. The (0, 2)tensor g -1 η , the inverse of g η , is called the pseudo-inverse of g with respect to the rigging N . With the quasi-orthonormal local frame field {∂ 0 := ξ , ∂ 1 , . . . , ∂ n , N} adapted to the decomposition (6) and (7) we have This is equivalent to the fact that C N (X, PY ) = ϕB N (X, Y ) for all tangent vector fields X and Y . The function ϕ is called the conformal factor.
that is, S 0 (1) coincides with the unit bundle of M with respect to the associated Riemannian metric g η from the normalization. Also, for all X ∈ T x M, O η (X) = X ⊥ gη .

Relations between null and the associated Riemannian geometry
Let (M, g, N) be a normalized null hypersurface of a pseudo-Riemannian manifold (M n+2 , g), ∇ the induced connection on M. In order to relate the main geometric objects of both null and associated non-degenerate geometry on the null hypersurface, we first need to relate the covariant derivatives ∇ η and ∇. For this purpose, we recall the following.
In this respect and using Lemma 3.1 we give the following result.
In particular for a closed normalization Using Proposition 3.1 we prove the following.
where φ(X, Z) is given by Proof The Riemann curvature tensor field R η of type (1, 3) is defined by Then (35) and (36) consist of repeated applications of (34) in Proposition 3.1.
Let Ric η and Ric denote the Ricci curvature of (M, g η ) and (M, g, N), respectively. We shall give the following result, involving the extrinsic Ricci curvature Ric to the associated Ricci curvature Ric η .
Proof Let p ∈ M and (E 0 := ξ , E 1 , . . . , E n ) be a quasi-orthonormal basis for (T p M, g p ) with Span(E 1 , . . . , E n ) = S(N) p . When dealing with indices, we adopt the following conventions: , and a, b, . . . , ∈ {0, . . . , n + 1}. Then we have Thus, from (35) and (36), we get where is the induced Ricci tensor curvature on a null hypersurface. But X, By substituting previous terms in the above expression of Ric η (X, Y ) we get the desired formula.
For all X, Y ∈ (TM) we define the (0, 2)-symmetrized Ricci tensor Ric 0 on the null hypersurface by Theorem 3.2 Let (M, g, N) be a closed normalized null hypersurface with rigged vector field ξ and τ N (ξ ) = 0 in a pseudo-Riemannian manifold. Then where r η denotes the scalar curvature of the non-degenerate metric g η on M, and r 0 is the extrinsic scalar curvature on the rigged null hypersurface (M, g, N) given by r 0 = g αβ η Ric 0 αβ , with respect to a local quasi-orthonormal frame field (e 0 := ξ , e 1 , . . . , e n ) for (M, g η ).
Proof We have r η = g αα η Ric η αα in a local quasi-orthonormal frame field (e 0 := ξ , e 1 , . . . , e n ) for (M, g η ) with span (e 1 , . . . , e n ) = S(N). But Hence, by contracting each side with g αα η and taking into account Proposition 3.1 along with the following facts: (M, g, N) be a closed normalized null hypersurface with rigged ξ and τ N (ξ ) = 0 in a Lorentzian manifold. Then, for all x ∈ M and a non-degenerate plane π ∈ S x N , and ξ ∈ (TM) ⊥ , we have

Lemma 3.2 Let
where X and Y are orthogonal unit vectors in S x (N) and π = Span{X, Y }.
Proof Suppose π is a non-degenerate plane (for g η ) in T p M. The real number is the sectional curvature of π (with respect to g η ). Observe that a plane π ⊂ S(N) is both non-degenerate with respect to g η and g (simultaneously) or not. Now, (44) is a direct consequence of (35) in Eq. (45), taking into account the fact, without loss of generality, that we have assumed X and Y to be g η -unit and orthogonal in S(N) (and hence also for g).

Relations between the associated Riemannian geometry and Riemannian geometry constructed on Lorentzian manifold 4.1 Isometric immersion of a null hypersurface equipped with Riemannian associated metric
Given a null hypersurface (M, g) in a Lorentzian manifold (M n+2 , g), first, we fix a null rigging N fixed on M, θ is an 1-form given by (10). Let ν be an 1-form given by with ξ defined along the null hypersurface, and also ν(N) = 1, ν(ξ ) = 0, θ (N) = 0, θ (ξ ) = 1. From now on, we suppose that Eq. (8) holds globally on M n+2 , that is, . Now, we define the Riemannian metric g η from the Lorentzian metric g, for all X, Y ∈ (TM), by For the proof, it is well known that TM = D⊗ orth S(N), where D = Span{N, ξ }, which means that, for all X ∈ (TM), one has Putting (50) in (48), we have and g η (X, X) = g X S(N) , X S(N) + θ (X) 2 + ν(X) 2 .
As D = Span{N x , ξ x } is a hyperbolic, it follows that the screen distribution S(N) on M ⊂ M n+2 is Riemannian and the induced metric on S(N) is positive definite, From (52) and (53), we have ∀X ∈ (TM) g η (X, X) ≥ 0, and Clearly g η is a Riemannian metric on M and i g η = g η , which shows that the null hypersurface M n+1 equipped with a Riemannian metric g η is isometrically immersed in a Riemannian manifold (M n+2 , g η ).

Link between the associated Riemannian geometry of g η and Riemannian geometry of g η
A striking fact is that the null rigging of the null hypersurface M is the normal unit vector field to the immersion of (M, g η ) into (M, g η ). Thus, Let ∇ η and ∇ η be the Levi-Civita connections with respect to g η and g η . Therefore, the Gauss-Weingarten formulae become The α from Eq. (57) is rather a (0, 2)-tensor on a line valued in (TM ⊥ ). Then the following holds: where A η N is the shape operator of M with respect to N . For any X ∈ (TM), we note that the following relations are satisfied: g η (N, X) = ν(X), g η (ξ , X) = θ (X). Proposition 4.1 Let (M, g η ) be a null hypersurface equipped with associated Riemannian metric g η in (n + 2)-dimensional Riemannian manifold (M, g η ), R η and R η denote the Riemann curvature tensors of ∇ η and ∇ η , respectively. Then, using the Gauss-Weingarten formulae (57) and (58), for all X, Y , Z, W ∈ (TM) and N ∈ (TM ⊥ ), we prove the following: Let {e 0 = ξ , e 1 , . . . , e n+1 = N} be any orthonormal basis for T x M. The sectional curvature of a plane π spanned by orthogonal unit vectors e a and e b is given by For a fixed a ∈ {0, . . . , n + 1}, the Ricci curvature of e a , is defined by Theorem 4.1 Let (M, g η ) be a null hypersurface equipped with associated Riemannian metric g η in (n + 2)-dimensional Riemannian manifold (M, g η ). Then, for all X, Y , Z, W ∈ (TM) and N ∈ (TM ⊥ ), we have for all orthonormal unit vectors X, Y ∈ (TM). We have here Ric η and Ric η denote the Ricci curvature of ∇ η and ∇ η , respectively, and K η (π) the sectional curvature (with respect to g η ), and K η (π) the sectional curvature of π = Span{X, Y } (with respect to g η ).

Relationship between the geometry of (M, g η ) and (M, g)
First, we relate the Levi-Civita connection of g η and g in the following.
In particular, for all X, Y ∈ (TM) ⊂ i (TM), with θ closed, we have Proof As the connections ∇ η and ∇ are symmetric, we can write Using the fact that D η is a symmetric tensor and ∇ η is a g η -metric connection, we have By a circular permutation, we get similar expressions for g η (D η (Y , Z), X) + g η (Z, D η (Y , X)) and g η (D η (Z, X), Y ) + g η (X, D η (Z, Y )). Summing the first two expressions minus the last one leads to It follows that For a proper conformal Killing field Nξ , with conformal factor 2λ i.e. L N-ξ g = 2λg and λ = 1 n+2 div g (Nξ ), we have and From now on, throughout the paper, assume Nξ , to be proper conformal Killing with conformal factor 2λ with respect to the metric g.
Proof The curvature tensor field of ∇ η is defined by By application of (71) in (75), we get (74).
We have a formula relating the scalar curvature r η and r of the g η and g in x ∈ M as follows.
Let π be a non-degenerate plane (for g η ), we define the sectional curvature of π (with respect to g η ) in T p M by We give now the relation between K η (π) and K(π).

Corollary 4.1 If the ambient manifold is of a constant sectional curvature k, then we have
Proof Using the Gauss-Codazzi equation (23) if the ambient manifold has constant sectional curvature k, then we can write ( , and substituting this in (83), we have the desired result.

Corollary 4.2
For all X, Y ∈ S(N),

Chen-Ricci inequality
In this section, we establish some basic inequalities between intrinsic invariants namely the Ricci curvature Ric, scalar curvature, shape operator A N , Chen invariant and the extrinsic invariant called the squared mean curvature for null hypersurface (M, g) in (n + 2)dimensional Lorentzian manifold (M, g). (M, g, N) be a closed normalized null hypersurface in a (n + 2)dimensional Lorentzian manifold (M, g), Nξ be a proper conformally Killing field with conformal factor 2λ and λ = 1 n+2 div g (Nξ ). Then, for all X ∈ S 0 x (1), the following holds:

Theorem 5.1 Let
where Ric (T x M) (X) is the Ricci curvature of X at x with respect to (M, g), λ = 1 n+2 div g (Nξ ) and the mean curvature vector H(x) is given by The equality case of (90) holds for all unit vectors X ∈ S 0 x (1) and for all x ∈ M if and only if either M is totally geodesic or n = 1.
Proof The null hypersurface equipped with Riemannian metric g η is a Riemannian hypersurface, and as (M, g η ) is isometrically immersed in a Riemannian manifold (M, g η ), using standard techniques as [19], we have this inequality: where X ∈ S 0 x (1). If we put the equality (39) of Theorem 3.1 in (92), we get with (76)  Then for all X ∈ S 0 x (1) the following holds: (Nξ ) be a proper conformal Killing field with conformal factor 2λ and λ = 1 n+2 div g (Nξ ). Then, for all X ∈ S 0 x (1), the following holds: The equality case holds for all X ∈ S 0 x (1) if and only if either x is a totally geodesic point or n = 1.
Proof In the closed normalized and conformal screen case, it well-known in Definition 2.3 that By Lemma 2.2 in [14], the 1-form τ N vanishes identically. Putting (96) in (90) and taking acount of τ N = 0, we have Remark 5.1 In the conformal sreen case with conformal factor ϕ = 1 we have τ N = 0, Hence the following holds: g, N) be a closed and conformal screen (with conformal factor ϕ = 1) normalized null hypersurface in (n + 2)-dimensional Lorentzian manifold (M, g), (Nξ ) be a proper conformal Killing field with conformal factor 2λ and λ = 1 n+2 div g (Nξ ). Then, for all X ∈ S 0 x (1), the following holds: The equality case holds for all X ∈ S 0 x (1) if and only if either x is totally geodesic point or n = 1.
We give now a basic inequality for shape operator and the squared mean curvature of a null hypersurface in the following. (M, g, N) be a closed normalized null hypersurface in (n+2)-dimensional Lorentzian manifold (M, g), Nξ be a proper conformal Killing field with conformal factor 2λ and λ = 1 n+2 div g (Nξ ). Then, for all X ∈ S 0 x (1), we have the following.

Theorem 5.2 Let
The equality case is true for all unit vectors in X ∈ S 0 x (1), x ∈ M, if and only if either M is totally geodesic or n = 1.

Corollary 5.4 Let
x (1), the following holds: Now, to establish inequality between the extrinsic scalar curvature of M and the scalar curvature of (M, g), we shall need the following lemma.
Lemma 5.1 [17] If a 1 , . . . , a n are real numbers then with equality if and only if a 1 = · · · = a n . (M n+1 , g, N) be a closed normalized null hypersurface in a Lorentzian (n + 2)-manifold (M, g), Nξ be a proper conformal Killing field. Then at every point x ∈ M, the following holds:

Theorem 5.3 Let
where r 0 (x) is the extrinsic scalar curvature of M and r(x) is the scalar curvature of (M, g) at x. The equality case is obtained if and only if x ∈ M is a umbilical point.
Proof By Lemma 5.1, we can write The equality (66) shows that Putting (104) in (103), we have For this purpose, we use By substitution of the equality (43) in (106), we have Putting (81) in (107), one has the desired result. (M n+1 , g, N) be a closed normalized and conformal screen null hypersurface in a Lorentzian (n + 2)-manifold (M, g) with conformal factor ϕ = 1 and Nξ be a proper conformal Killing field. Then, at every point x ∈ M, we have

Corollary 5.5 Let
where r 0 (x) is the extrinsic scalar curvature of M and r(x) is the scalar curvature of (M, g) at x. The equality case is obtained if and only if x ∈ M is a totally umbilical point.
Now, we establish the inequality which gives a general optimal relationship involving the squared mean curvature of the null hypersurface and the Chen invariants. (M n+1 , g, N) be a closed null hypersurface of (n + 2)-dimensional n > 2 Lorentzian manifold (M, g) with τ N (ξ ) = 0, Nξ be a proper conformal Killing field with conformal factor 2λ, λ = 1 n+2 div g (N -ξ ). Then, for each point x ∈ M and for all orthonormal vectors fields X, Y and π = Span{X, Y }, the following holds:

Theorem 5.4 Let
Proof By the equality (65) and (66) we have Equation (112) with (113) gives which gives By (18), we have the following. (M n+1 , g, N) be a closed normalized and conformal screen null hypersurface of (n + 2)-dimensional n > 2 Lorentzian manifold (M, g) with conformal factor ϕ = 1, τ N (ξ ) = 0 and Nξ be a proper conformal Killing field. Then, for each point x ∈ M, for all orthonormal vectors X, Y and π = Span{X, Y }, the following holds:

Application example
One important application of the Chen-Ricci inequalities for null hypersurfaces is to provide solutions to the basic problem of minimal immersions: given a null hypersurface M n+1 in Lorentzian manifolds R n+2 1 , what are the necessary conditions for M to admit a minimal isometric immersion in a semi-Euclidean space? As an application of our results, we have the following topological obstruction to minimal isometric immersions of null hypersurface in Lorentzian manifolds. (M n+1 , g, N) be a closed normalized and conformal screen null hypersurface of (n + 2)-dimensional n > 2 Lorentzian manifold (M, g) with conformal factor ϕ = 1, Nξ be a proper conformal Killing field. If for all x ∈ M and X ∈ S(N) 0

Theorem 5.5 Let
x (1) then the null hypersurface admits a minimal isometric immersion in Lorentzian manifold R n 1 and the null hypersurface is non-compact.
Proof The null hypersurface equipped with associated Riemannian metric g η is a Riemannian hypersurface. By Nash theorem, we can immerse this in a space Euclidian. By the equality (39), (123) is equivalent to say that Ric η (X, Y ) ≤ 0. Given that the null hypersurface is a closed normalized and conformal screen with conformal factor ϕ = 1, the associated Riemannian metric g η agrees with the degenerate metric g, the 1-form τ N vanishes identically and by Definition 2.3 A N = A ξ . This shows that Ric(X, X) ≤ 0, which is a necessary condition for M to admit a minimal immersion in a R n 1 .The equality hold if and only if the submanifold is totally geodesic. A well-known result of Beltrami [4] proves that the Laplacian of M satisfies ϕ = -(n + 1)H, n + 1, n + 1 = dim M where ϕ : M → E m denotes the immersion and H is the mean curvature vector of M. Since the only harmonic functions on compact Riemannian manifolds are constant functions, it follows immediately from Eq. (124) that each minimal hypersurface is non-compact.

Conclusion
In this paper, using rigging techniques we isometrically immerse the null hypersurface equipped with the associated Riemannian metric into a Riemannian manifold suitably constructed on the Lorentzian manifold. Through this immersion, we establish the Chen-Ricci inequalities of the null hypersurface in Lorentzian manifold. As an application, we give a topological obstruction to the minimal isometric immersion of the null hypersurface in an ambient Lorentzian.