menedore.karimumuryango@imsp-uac Given a null hypersurface of a Lorentzian manifold, we isometrically immerse a null .org; menedore.karimumuryango@ub.edu.bi hypersurface equipped with the Riemannian metric (induced on it by the rigging) Institut de Mathématiques et de into a Riemannian manifold suitably constructed on the Lorentzian manifold. We Sciences Physiques (IMSP), study the intrinsic and extrinsic geometry of such an isometric immersion and we link Porto-Novo, Bénin 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. Keywords: Null hypersurface; Rigging; Closed normalization; Associated Riemannian metric; Ricci inequalities; Minimal hypersurface 1 Introduction Soon after Riemann (1854) introduced the notion of a manifold, Schläﬂi (1873) conjec- tured 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 m n can be isometrically embedded in an Euclidean m-space E with m = (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 intrin- sic 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 δ = τ(p)– inf K(p), (1) where τ(p)is scalar curvature of M and inf(K)(p)= inf{K(π): K(π) is a plane section of T M}. In [5], Chen established the following general optimal inequality involving the new in- trinsic invariant δ and the squared mean curvature H for n-dimensional submanifold © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 2 of 27 M of a real space form of constant sectional curvature c: n (n –2) 1 δ ≤ H + (n +1)(n –2)c.(2) 2(n –1) 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: 2 2 Ric(X) ≤ n H +(n –1)c.(3) In [7], B.Y. Chen estabished the inequality between the shape operator and the mean cur- vature 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]Hongand Tripathi stud- ied this inequality and they presented a general theory for submanifolds of Riemannian manifolds and proved a basic inequality using (3) as follows: 2 2 Ric(X) ≤ n H + Ric (X), (4) (T M) where M is a n-dimensional submanifold of M, Ric (X)isthe n-Ricci curvature of T M (T M) p 1 1 at X ∈ T M with respect to the ambient manifold M and T M is the set of unit vectors in p p T M.The equality case of (4)issatisﬁedby X ∈ T M if and only if α(X, Y)=0, ∀Y ∈ T M, g(X, Y)=0, (5) 2α(X, X)= nH. In [18], this inequality was named a Chen–Ricci inequality by Tripathi where α is the sec- ond fundamental form of M.The equality case of (4) holds for all unit vectors X ∈ T 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 in- volving k-Ricci curvature, k-scalar curvature, the screen scalar curvature on a screen ho- mothetic 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 Rieman- nian metric of a null hypersurface in a Lorentzian manifold as in [3]but arisingfromanull 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 immer- sion (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 n+2 n+1 hypersurface M in a Lorentzian ambient manifold M (Theorem 5.1,Theorem 5.2, Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 3 of 27 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,wegivethe 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 n+1 M equipped with the associated Riemannian metric (induced on it by the rigging) into n+2 a Riemannian manifold suitably constructed on the Lorentzian ambient manifold M and the relations between the Riemannian geometry constructed on the Lorentzian man- ifold 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. 2Preliminaries n+1 Consider a null hypersurface (M , g)of a (n + 2)-dimensional Lorentzian manifold (M, g) of constant index 0 < ν < n+2. The normal bundle of the null hypersurface is the subbundle TM = {V ∈ (TM): g(V, W)=0 ∀W ∈ (TM)} of the tangent bundle TM.Since M is a null hypersurface, dim(T M )=1. In the classical theory of non-degenerate hypersurfaces, we have the following decom- position: ⊥ ⊥ TM = TM ⊕ TM , TM ∩ TM = {0},(6) orth where ⊕ denotes orthogonal direct sum. Any vector ﬁeld of TM splits uniquely into orth a component tangent to M and a component perpendicular to M.However,inthe null hypersurface case, (6)doesnot 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 TM = S(N) ⊕ TM.(7) orth 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 Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 4 of 27 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 satisfying g ¯(N, ξ)=1, g ¯(N, N)= g ¯(N, W)=0, ∀W ∈ (SN).(8) Then TM is decomposed as follows: TM = S(N) ⊕ TM ⊕ tr(TM) = TM ⊕ tr(TM), (9) orth 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) is called a null rigging of M.Itis noteworthy that the choice of a null transversal vector ﬁeld N along M determines the null transversal vector bundle, the screen distribution and a unique radical vector ﬁeld ξ, say the rigged vector ﬁeld, satisfying (8). Deﬁnition 2.1 ([12]) Let M be a null hypersurface of a Lorentzian manifold. A rigging for M is a vector ﬁeld L deﬁned on some open set containing M such that L ∈/ T M for each p p p ∈ M. 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 is the ﬁrst fundamental form and N is a null rigging |M n+2 for M.Let N be a null rigging of a null hypersurface of a Lorentzian manifold (M , g) and θ the 1-form metrically equivalent to N deﬁned on some open set containing M and given by θ = g(N, ·). (10) Suppose that η = i θ (11) 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.Itiseasytocheck 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, ∇ Y = ∇ Y + B (X, Y )N, (12) X X ∇ N =–A X + τ (X)N, (13) X N ∗ N ∇ PY = ∇ PY + C (X, PY )ξ, (14) ∗ N ∇ ξ =–A X – τ (X)ξ (15) ξ Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 5 of 27 for any X, Y ∈ (TM), where ∇ denotes the Levi-Civita connection on (M, g), ∇ denotes the rigged connection on (M, g)inducedfrom ∇ through the projection along N;itsatisﬁes N N (∇ g)(Y , Z)= B (X, Y )η(Z)+ B (X, Z)η(Y ), (16) and ∇ denotes the induced connection on the screen distribution. Here η is an 1-form on TM deﬁned by η(X)= g(N, X), ∀X ∈ (TM), (17) N N B is the null second fundamental form of M and C is the second fundamental on (S(N)), respectively. The second fundamental forms are related by their shape operators, N ∗ ∗ B (X, Y)= g A X, Y , g A X, N = 0, (18) ξ ξ C (X, PY)= g(A X, PY ), g(A Y , N)=0, ∀X, Y ∈ (TM). (19) N N A is the null shape operator with respect to the section ξ and B (X, ξ)=0, A ξ = 0. (20) N N ∗ τ is the 1-form on TM deﬁned by τ (X)= g(∇ N, ξ). From (15)and A ξ =0, we ﬁnd that ∇ ξ =–τ (ξ)ξ, (21) 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 τ (ξ) = 0. (22) n+1 n+2 1 n Example 2.1 Let F be the immersion f : M → R deﬁned by F (x ,..., x ) → 0 1 1 n 1 2 n 2 n+1 [x ,..., x ,(x ) + ··· +(x ) ], and the null hypersurface M = {x =(x ,..., x ), 0 n+1 n+1 n+1 –x + x =0}.Let N be the null rigging of M deﬁned by N =–x ∂ + x ∂ ; 0 0 a a 0 a=1 a a=1 1 n+1 and the null vector ﬁeld ξ = (x ∂ + x ∂ ), x = 0. Hence the null vector ﬁeld ξ is 2 0 0 a a 0 a=1 2x normal to M,thus M is a null hypersurface. n+1 Let X = X ∂ + X ∂ ,the 1-form η is given by 0 0 b b b=1 η(X)= N, X n+1 n+1 = –x ∂ + x ∂ , X ∂ + X ∂ 0 0 a a 0 0 b b a=1 b=1 n+1 = x X + x X 0 0 a a a=1 =2x dx (X). 0 0 Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 6 of 27 dx N 2 0 Hence η =2x dx and dη = 0. This shows that η is closed and τ (X)=–[2x (– )+ 0 0 0 3 2x dx ](X) for all X ∈ (TM). 2 0 0 2x Deﬁnition 2.2 Anullhypersurface 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 N N X, Y ∈ T M, B (x)(X, Y)= ρ(x)g(X, Y)(respectively, B vanishes identically on M). This is equivalent to write A = ρP and A = 0. Also, the screen distribution S(N) is totally ξ ξ umbilical (respectively, totally geodesic) if C (X, PY)= ρ(x)g(X, Y ) for all X, Y ∈ (TM) (respectively, C = 0), which is equivalent to writing A = ρP (respectively, A =0). N N Denote by R and R the Riemann curvature tensors of ∇ and ∇,respectively. Re- call the following Gauss–Codazzi equations ([9]) for all X, Y , Z ∈ (TM), N ∈ tr(TM), ξ ∈ (TM ): N N N N g R(X, Y )Z, ξ = ∇ B (Y , Z)– ∇ B (X, Z)+ B (Y , Z)τ (X) X Y N N – B (X, Z)τ (Y ); (23) g ¯ R(X, Y )ξ, N = g R(X, Y )ξ, N N ∗ N ∗ N = C Y , A X – C X, A Y –2 dτ (X, Y ); (24) ξ ξ N N g R(X, Y )PZ, N = ∇ C (Y , PZ)– ∇ C (X, PZ) X Y N N N N + τ (Y )C (X, PZ)– τ (X)C (Y , PZ). (25) The shape operator A is self-adjoint as the second fundamental form B is symmetric. However, this is not the case for the operator A as show in the following lemma. Lemma 2.1 For all X, Y ∈ (TM) N N A X, Y – A Y , X = τ (X)η(Y)– τ (Y )η(X)–2 dη(X, Y ), (26) N N 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 Wein- garten formula, we have, for all X, Y ∈ (TM), 2 dη(X, Y)=2 dθ(X, Y ) = ∇ N, Y – ∇ N, X X Y =–A X, Y + τ (X)η(Y)+ A Y , YX – τ(Y )η(X). N N Hence N N A X, Y – A Y , X = τ (X)η(Y)– τ (Y )η(X) (27) N N as announced. We shall use the following notations: Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 7 of 27 0 2 0 Notations 2.1 For x ∈ M,weset S (1) = {X ∈ T M, X, X =1 – η(X) }, S (1) = S (1), and for all X ∈ (TM), we set O (X)= {Y ∈ TM, X, Y =–η(X)η(Y )},where x∈M , stands for g or g.Observe that Y ∈ O (X)if and only if X ∈ O (Y). η η Due to the degeneracy of the ﬁrst fundamental form g on the null hypersurface M,itis 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 by : (TM) → TM , X → X = g(X, ·)+ η(X)η(·), ∀X ∈ (TM), (28) X (Y)= g(X, Y)+ η(X)η(Y ). 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 con- sequence the 1-form η vanishes identically and the projection morphism P becomes the identity map on (TM). Let denote the inverse of the isomorphism given by (28). For η η η η X ∈ (TM)(respectively, w ∈ T M), X (respectively, w ) 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)thatif w is a 1-form on M,wehave, for X ∈ (TM), w(X)= g w , X + w(ξ)η(X). (29) Deﬁne a (0, 2)-tensor g by g (X, Y)= X (Y), ∀X, Y ∈ (TM). Clearly, g deﬁnes a non- η η η degenerate 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 is called the associated metric to g on (M, g, N). Also, observe that g coincides with g if the latter is non-degenerate. The (0, 2)- –1 tensor g ,the inverseof g , is called the pseudo-inverse of g with respect to the rigging N. With the quasi-orthonormal local frame ﬁeld {∂ := ξ, ∂ ,..., ∂ , N } adapted to the de- 0 1 n composition (6)and (7)wehave g (X, Y)= g(X, Y ), ∀X, Y ∈ S(N) . g (ξ, X)= η(X), ∀X ∈ (TM). g (ξ, ξ) = 1. (30) Deﬁnition 2.3 ([2]) A normalized null hypersurface (M, g, N) of a pseudo-Riemannian manifold (M, g) is said to have a conformal screen if there exists a non-vanishing smooth function ϕ on M such that A = ϕA holds. N N This is equivalent to the fact that C (X, PY)= ϕB (X, Y ) for all tangent vector ﬁelds X and Y . The function ϕ is called the conformal factor. Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 8 of 27 Remark 2.1 For all x ∈ M 0 2 S (1) = X ∈ T M, X, X =1 – η(X) = X ∈ T M, g (X, X)=1 , (31) x x η that is, S (1) coincides with the unit bundle of M with respect to the associated Rieman- nian metric g from the normalization. Also, for all X ∈ T M, O (X)= X . η x η 3 Relations between null and the associated Riemannian geometry Let (M, g, N) be a normalized null hypersurface of a pseudo-Riemannian manifold n+2 (M , 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 ∇ and ∇. For this purpose, we recall the following. Lemma 3.1 ([3]) For all X, Y , Z ∈ (TM), we have N N (∇ g )(Y , Z)= η(Y ) B (X, PZ)– C (X, PZ) X η N N + η(Z) B (X, PY)– C (X, PY ) +2τ (X)η(Y )η(Z). (32) In this respect and using Lemma 3.1 we give the following result. Proposition 3.1 ([14]) Let (M, g, N) be a normalized null hypersurface with rigged vector ﬁeld ξ. Then, for all X, Y ∈ (TM), we have ∇ Y = ∇ Y + 2g A X, Y – g(A X, Y)– g(A Y , X) X N N X ξ N N η η + η(X)τ (Y)+ η(Y )τ (X) ξ + η(X)(i dη) + η(Y )(i dη) . (33) Y X In particular for a closed normalization ∇ Y = ∇ Y + 2g A X, Y – g(A X, Y)– g(A Y , X) X N N X ξ N N + η(X)τ (Y)+ η(Y )τ (X) ξ (34) with dη(X, Y)= [X · η(Y)– Y · η(X)– η([X, Y ])]. Using Proposition 3.1 we prove the following. Proposition 3.2 Let (M, g, N) be a closed normalized null hypersurface with rigged vector ﬁeld ξ. Then, for all X, Y , W ∈ (TM) and U ∈ (TM ), we have g R (X, Y )Z, PW = g R(X, Y )Z, PW N N + φ B (Y , W)– φ B (X, W) , (35) (X,Z) (Y ,Z) 2 Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 9 of 27 g R (X, Y )Z, U =–g R(X, Y )U, PZ N N – g(A U, Y )B (X, Z)– g(A U, X)B (Y , Z) N N N N N N – τ (X)B (Y , Z)– τ (Y )B (X, Z) η(U), (36) where φ(X, Z) is given by φ =2B (X, Z)– g(A X, Z)– g(A Z, X) (X,Z) N N N N + τ (X)η(Z)+ τ (Z)η(X). (37) Proof The Riemann curvature tensor ﬁeld R of type (1, 3) is deﬁned by η η η η R (X, Y )Z = ∇ , ∇ Z – ∇ Z. (38) X Y [X,Y ] Then (35)and (36) consist of repeated applications of (34)inProposition 3.1. Let Ric and Ric denote the Ricci curvature of (M, g )and (M, g, N), respectively. We shall give the following result, involving the extrinsic Ricci curvature Ric to the associated Ricci curvature Ric . Theorem 3.1 Let (M, g, N) be a closed normalized null hypersurface with rigged vector ﬁeld ξ and τ (ξ)=0 in a (n +2)-pseudo-Riemannian manifold. Then η ∗ N ∗ Ric (X, Y)= Ric(X, Y)– A X, Y – A X, Y + τ (X)η(Y ) tr A ξ ξ + ∇ A (X), Y – (∇ A )(X), Y ξ ξ N N N + ∇ τ (X)η(Y)– ∇ τ (Y ). (39) ξ X Proof Let p ∈ M and (E := ξ, E ,..., E ) be a quasi-orthonormal basis for (T M, g )with 0 1 n p p Span(E ,..., E )= S(N) . When dealing with indices, we adopt the following conventions: 1 n p i, j, k,... ∈{1,..., n}, α, β, γ ∈{0,..., n},and a, b,..., ∈{0,..., n +1}.Thenwehave η αα η Ric (X, Y)= g R (E , X)Y , E . (40) α α α=0 Thus, from (35)and (36), we get η η η R ic(X, Y)= g R (ξ, X)Y , ξ + g R (E , X)Y , E η η i i ∗ ∗ ∗ ∗ ∗ = Ric(X, Y)+ g A X, A Y – g A X, Y – g A X, Y tr A ξ ξ ξ ξ ξ ∗ N – g A X, A Y + g(A X, Y)+ g(A Y , X)– τ (X)η(Y ) N N N N ∗ – τ (Y )η(X) tr A – g R(ξ, X)ξ, Y – g R(ξ, X)Y , N , ξ Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 10 of 27 η ∗ ∗ R ic(X, Y)= Ric(X, Y)– A X, Y – A X, Y tr A ξ ξ N ∗ ∗ N – τ (X)η(Y ) tr A + X, ∇ A (Y ) – ∇ τ (Y ) ξ X ξ ξ N N ∗ – (∇ A )(X), Y +2dτ (ξ, X)η(Y)+ τ A X η(Y ), ξ N where Ric(X, Y)= ε g R(E , X)Y , E + g R(ξ, X)Y , N (41) i i i i=1 is the induced Ricci tensor curvature on a null hypersurface. But X,(∇ A )(Y ) = ∗ N N N N N ∗ (∇ A )(X), Y and 2 dτ (ξ, X)=(∇ τ )(X)–(∇ τ )(ξ)and (∇ τ )(ξ)= τ (A X). Also ξ ξ X X ξ ξ ∗ ∗ ∗ g(R(ξ, X)ξ, Y)= g(A X, A Y)– g(X,(∇ A )(Y )) and ξ ξ ξ g R(ξ, X)Y , N = ∇ τ (Y)+ (∇ A )(X), Y X ξ N N ∗ –2 dτ (ξ, X)η(Y)– A A X, Y . By substituting previous terms in the above expression of Ric (X, Y) weget thedesired formula. For all X, Y ∈ (TM) we deﬁne the (0, 2)-symmetrized Ricci tensor Ric on the null hypersurface by Ric (X, Y)= Ric(X, Y)+ Ric(Y , X) . (42) Theorem 3.2 Let (M, g, N) be a closed normalized null hypersurface with rigged vector ﬁeld ξ and τ (ξ)=0 in a pseudo-Riemannian manifold. Then η 0 ∗ ∗ r = r – tr A – tr A tr A ξ ξ ∗ g N + tr ∇ A – tr(∇ A ) – div τ , (43) ξ ξ N η 0 where r denotes the scalar curvature of the non-degenerate metric g on M, and r is the 0 αβ extrinsic scalar curvature on the rigged null hypersurface (M, g, N) given by r = g Ric , η αβ with respect to a local quasi-orthonormal frame ﬁeld (e := ξ, e ,..., e ) for (M, g ). 0 1 n η η αα η Proof We have r = g Ric in a local quasi-orthonormal frame ﬁeld (e := ξ, e ,..., e ) 0 1 n η αα for (M, g )with span(e ,..., e )= S(N). But η 1 n η 0 N Ric = Ric – A e , e – A e , e + τ (e )η(e ) tr A α α N α α α α αα αα ξ ξ + ∇ A (e ), e – (∇ A )(e ), e ξ α α ξ N α α N N + ∇ τ (e )η(e )– ∇ τ (e ) . ξ α α e α αα Hence, by contracting each side with g and taking into account Proposition 3.1 along with the following facts: (∇ τ)(e )= η(τ )g(A e , e )+ g(∇ τ , e )= g(∇ τ , e ), e i i i e i e i i ξ i i αα N αα N g (∇ τ )(e )η(e )=0, g (∇ A )(e ), e = tr(∇ A )+ g (∇ (τ ), ξ)and ξ α α ξ N α α ξ N η ξ η η αα N αα g τ (e )η(e )=0, g (∇ A )(e ), e = tr(∇ A ), we get Eq. (43). α α ξ α α ξ η η ξ Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 11 of 27 Lemma 3.2 Let (M, g, N) be a closed normalized null hypersurface with rigged ξ and τ (ξ)=0 in a Lorentzian manifold. Then, for all x ∈ M and a non-degenerate plane π ∈ S N, and ξ ∈ (TM) , we have N 2 N N K (π)= K(π)+ B (X, Y ) – B (X, X)B (Y , Y ) N N N N + B (X, X)C (Y , Y)– B (X, Y )C (X, Y ), (44) where X and Y are orthogonal unit vectors in S (N) and π = Span{X, Y }. Proof Suppose π is a non-degenerate plane (for g )in T M.The real number η p g (R (U, V)V, U) K (π)= (45) g (U, U)g (V, V)–(g (U, V)) η η η is the sectional curvature of π (with respect to g ). Observe that a plane π ⊂ S(N)is both non-degenerate with respect to g and g (simultaneously) or not. Now, (44) is a direct consequence of (35)inEq. (45), taking into account the fact, without loss of generality, that we have assumed X and Y to be g -unit and orthogonal in S(N) (and hence also for g). 4 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 n+2 Given a null hypersurface (M, g)in a Lorentzian manifold (M , 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 ν(X)= g(ξ, X), X ∈ (TM), (46) with ξ deﬁned along the null hypersurface, and also ν(N)=1, ν(ξ)=0, θ(N)=0, n+2 θ(ξ) = 1. From now on, we suppose that Eq. (8) holds globally on M ,thatis, g (N , N )= x x n+2 0, g (N , ξ )=1, g (ξ , ξ )=0 ∀x ∈ M . Now, we deﬁne the Riemannian metric g from x x x x x x η the Lorentzian metric g, for all X, Y ∈ (TM), by g (X, Y)= g(X, Y)+ (θ – ν) ⊗ (θ – ν) (X, Y ). (47) For the proof, it is well known that TM = D ⊗ S(N), where D = Span{N, ξ },whichmeans orth that, for all X ∈ (TM), one has S(N) X = X + αξ + βN, (48) S(N) g(X, N)= g(X , N)+ αg(ξ, N)+ βg(N, N), (49) S(N) g(ξ, X)= g(X , ξ)+ αg(ξ, ξ)+ βg(N, ξ), α = θ(X), β = ν(X). (50) Putting (50)in(48), we have S(N) X = X + θ(X)ξ + ν(X)N (51) Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 12 of 27 and S(N) S(N) 2 2 g (X, X)= g X , X + θ(X) + ν(X) . (52) As D = Span{N , ξ } is a hyperbolic, it follows that the screen distribution S(N)on M ⊂ x x n+2 M is Riemannian and the induced metric on S(N) is positive deﬁnite, S(N) S(N) 0 ≤ g X , X = g(X, X)–2θ(X)ν(X). (53) From (52)and (53), we have ∀X ∈ (TM) g (X, X) ≥ 0, and ⎧ ⎧ ⎪ θ(X) =0, ⎪ α =0, ⎨ ⎨ g (X, X)=0 ⇒ ⇒ ν(X) =0, β =0, ⎪ ⎪ ⎩ ⎩ S(N) S(N) S(N) S(N) S(N) g(X , X )=0 g(X , X )=0 ⇒ X =0, which shows that g (X, X)=0 ⇒ X = 0. (54) Clearly g is a Riemannian metric on M and i g = g , which shows that the null hypersur- η η n+1 face M equipped with a Riemannian metric g is isometrically immersed in a Rieman- n+2 nian manifold (M , g ). 4.2 Link between the associated Riemannian geometry of g and Riemannian geometry of g 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 )into(M, g ). Thus, g (N, X)=0, ∀X ∈ (TM), (55) g (N, N) = 1. (56) η η Let ∇ and ∇ be the Levi-Civita connections with respect to g and g . Therefore, the Gauss–Weingarten formulae become ∇ Y = ∇ Y + α(X, Y )N, (57) X X ∇ N =–A X. (58) X N The α from Eq. (57) is rather a (0, 2)-tensor on a line valued in (TM ). Then the following holds: η ⊥ α(X, Y)= g A X, Y , ∀X, Y ∈ (TM), N ∈ TM , (59) where A is the shape operator of M with respect to N. For any X ∈ (TM), we note that the following relations are satisﬁed: g (N, X)= ν(X), g (ξ, X)= θ(X). η η Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 13 of 27 Proposition 4.1 Let (M, g ) be a null hypersurface equipped with associated Riemannian metric g in (n +2)-dimensional Riemannian manifold (M, g ), R and R denote the Rie- mann curvature tensors of ∇ and ∇ , respectively. Then, using the Gauss–Weingarten for- mulae (57) and (58), for all X, Y , Z, W ∈ (TM) and N ∈ (TM ), we prove the following: g R (X, Y )Z, W = g R (X, Y )Z, W + α(X, Z)α(Y , W) – α(Y , Z)α(X, W), (60) η η g R (X, Y )Z, N = ∇ α (Y , Z)– ∇ α (X, Z). (61) η X Y Let {e = ξ, e ,..., e = N } be any orthonormal basis for T M.The sectionalcurvature 0 1 n+1 x of a plane π spanned by orthogonal unit vectors e and e is given by a b K (π)= g R (e , e )e , e . (62) η a b b a For a ﬁxed a ∈{0,..., n +1},the Riccicurvature of e ,isdeﬁnedby n+1 Ric (e )= K (π). (63) a η a=b=0 Theorem 4.1 Let (M, g ) be a null hypersurface equipped with associated Riemannian metric g in (n +2)-dimensional Riemannian manifold (M, g ). Then, for all X, Y , Z, W ∈ (TM) and N ∈ (TM ), we have η η η η Ric (X, Y)= Ric (X, Y)+ g A X, A Y – g A X, Y tr A , (64) η η N N N N η η η η K (π)= K (π)+ g A X, Y g A Y , X – g A Y , Y g A X, X (65) η η η η N N N N for all orthonormal unit vectors X, Y ∈ (TM). We have 2 2 η η η η r (x)= r (x)+ tr A – tr A ; (66) N N η η here Ric and Ric denote the Ricci curvature of ∇ and ∇ , respectively, and K (π) the sec- tional curvature (with respect to g ), and K (π) the sectional curvature of π = Span{X, Y } (with respect to g ). 4.3 Relationship between the geometry of (M, g )and (M, g) First, we relate the Levi-Civita connection of g and g in the following. n+2 Proposition 4.2 Let (M, g) be a null hypersurface of a Lorentzian manifold (M , g), (M, g ) be a Riemannian manifold constructed from the ambient Lorentzian, Nbe a null rigging for M ﬁxed on Msatisfying (8) and ξ be a rigged ﬁeld of M. Then, for all X, Y ∈ (TM), we have ∇ Y = ∇ Y – (L g)(X, Y ) (N – ξ) X N–ξ + θ(Y)– ν(Y ) i (dθ – dν) + θ(X)– ν(X) i (dθ – dν) . (67) Y Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 14 of 27 In particular, for all X, Y ∈ (TM) ⊂ i (TM), with θ closed, we have η 1 ∇ Y = ∇ Y – (L g)(X, Y ) (N – ξ). (68) X N–ξ Proof As the connections ∇ and ∇ are symmetric, we can write ∇ Y = ∇ Y + D (X, Y ), ∀X, Y ∈ (TM). Using the fact that D is a symmetric tensor and ∇ is a g -metric connection, we have η η g D (X, Y ), Z + g Y , D (X, Z) =(∇ g )(Y , Z) η η η = (∇ θ)(Y)–(∇ ν)(Y ) θ(Z) X X + (∇ θ)(Z)–(∇ ν)(Z) θ(Y ) X X – (∇ θ)(Z)–(∇ ν)(Z) ν(Y ) X X – (∇ θ)(Y)–(∇ ν)(Y ) ν(Z). X X η η By a circular permutation, we get similar expressions for g (D (Y , Z), X)+ g (Z, D (Y , X)) η η η η and g (D (Z, X), Y)+ g (X, D (Z, Y )). Summing the ﬁrst two expressions minus the last η η one leads to 2g D (X, Y ), Z = (L g)(X, Y)–(L g)(X, Y ) g(N, Z) N ξ – (L g)(X, Y)–(L g)(X, Y ) g(ξ, Z) N ξ +2θ(Y ) dθ(X, Z)– dν(X, Z) –2ν(Y ) dθ(X, Z)– dν(X, Z) +2θ(X) dθ(Y , Z)– dν(Y , Z) –2ν(X) dθ(Y , Z)– dν(Y , Z) , 2g D (X, Y ), Z = (L g)(X, Y)–(L g)(X, Y ) g (ξ, Z) N ξ η η – (L g)(X, Y)–(L g)(X, Y ) g (N, Z) N ξ +2θ(Y ) g i (dθ – dν) , Z –2ν(Y ) g i (dθ – dν) , Z +2θ(X) g i (dθ – dν) , Z –2ν(X) g i (dθ – dν) , Z . With (L g)(X, Y)=(∇ θ)(Y)+(∇ θ)(X), N X Y (L g)(X, Y)=(∇ ν)(Y)+(∇ ν)(X), ξ X Y 2(dθ)(X, Y)=(∇ θ)(Y)–(∇ θ)(X). X Y Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 15 of 27 It follows that η g D (X, Y)=– (L g)(X, Y ) (N – ξ)+ θ(Y)– ν(Y ) i (dθ – dν) N–ξ X + θ(X)– ν(X) i (dθ – dν) . Remark 4.1 ∀X, Y ∈ (TM), ν(X)=0, and if θ is closed, we have ∇ Y = ∇ Y – (L g)(X, Y ) (N – ξ). (69) X N–ξ For a proper conformal Killing ﬁeld N – ξ, with conformal factor 2λ i.e. L g =2λg and N–ξ 1 g λ = div (N – ξ), we have n+2 D (X, Y)=–λg(X, Y )(N – ξ) (70) and ∇ Y = ∇ Y – λg(X, Y )(N – ξ). (71) From now on, throughout the paper, assume N – ξ, to be proper conformal Killing with conformal factor 2λ with respect to the metric g. n+1 Lemma 4.1 Let (M , g, N) be a closed normalized null hypersurface in Lorentzian man- ifold (M, g), (N – ξ) be a proper conformal Killing with conformal factor 2λ and λ = 1 g N div (N – ξ) and τ (ξ)=0. Then n+2 A = A – τ (·)N + λη(·)(N – ξ). (72) Proof Putting Y = N in (71), we have ∇ N = ∇ N – λg(X, N)(N – ξ). (73) By applying (13)and (58)in(73), we have the desired result. η η Theorem 4.2 Let R and R denote the Riemann curvatures tensors of ∇ and ∇, respec- tively. Then, for all X, Y , Z ∈ (TM) ⊂ i (TM), ξ ∈ (TM ), N be a null rigging for M ﬁxed on Mand N – ξ a proper conformal Killing ﬁeld with θ closed, the following holds: 0 N g R (X, Y )Z, W = g R(X, Y )Z, W + η(W) R (X, Y )Z · λ – ∇ B (Y , Z) N N N N N + ∇ B (X, Z)– τ (X)B (Y , Z)+ τ (Y )B (X, Z) N N N + ∇ C (Y , PZ)– ∇ C (X, PZ)–2 dτ (X, Y )η(Z) X Y N N N N + τ (Y )C (X, PZ)– τ (X)C (Y , PZ) N N + λg(Y , Z) C (X, PW)– B (X, W) – τ (X)η(W)– λη(X)η(W) N N – λg(X, Z) C (Y , PW)– B (Y , W) – τ (Y )η(W)– λη(Y )η(W) (74) Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 16 of 27 for all X, Y , Z, W ∈ (TM). Here R (X, Y )Z · λ = X(λ)g(Y , Z)– Y (λ)g(X, Z). Proof The curvature tensor ﬁeld of ∇ is deﬁned by η η η η R (X, Y )Z = ∇ , ∇ Z – ∇ Z. (75) X Y [X,Y ] By application of (71)in(75), we get (74). η η Theorem 4.3 Let Ric and Ric be the Ricci curvatures of ∇ and ∇, respectively, then let, for all X, Y , Z ∈ (TM) ⊂ i (TM), ξ ∈ TM and N be a null rigging for M ﬁxed on M, and N – ξ a proper conformal Killing ﬁeld with θ closed, then the following holds: 0 N Ric (X, Y)= Ric(X, Y)– R (N, X)Y · λ + ∇ τ (Y)+ (∇ A )(X), Y X ξ N N N – ∇ τ (X)η(Y)– A X, A Y + ∇ τ (X)+ (∇ A )(Y ), X ξ N Y ξ N – ∇ τ (Y )η(X)– A Y , A X – λ A X, Y – A X, Y ξ N N ξ ξ + ξ(λ)– λ + λ tr A – tr A X, Y + λη(Y ) τ(X)– λη(X) . (76) Proof Let x ∈ M and (e = ξ, e ,..., e , e +1 = N) be a quasi-orthonormal basis for 0 1 n n (T M, g )with span(e ,..., e )= S(N) . Suppose that the indices i, j, k run over the range x 1 n | x p 1,..., n, α, β, γ ,... ∈{0,..., n} and a, b,..., ∈{0,..., n +1};then η η Ric (X, Y)= trace Z → R (Z, X)Y n+1 aa = g g R (e , X)Y , e a a η η a=o η η η ii = g R (ξ, X)Y , ξ + g g R (e , X)Y , e + g R (N, X)Y , N . (77) i i η η η η i=1 From (74), with g (N – ξ, ξ)=–1, g (N – ξ, N)=1, and g (A X, e )= g(A X, e ), we get N i N i η η η g R (ξ, X)Y , ξ = g R(ξ, X)Y , ξ – ξ(λ)g(X, Y)– X(λ)g(ξ, Y ) g (N – ξ, ξ) η η η – λg(X, Y ) –g (A ξ, ξ)+ τ (ξ)g (N, ξ)+ g A ξ, ξ η η η ξ + τ (ξ)g (ξ, ξ)– λη(ξ)g (N – ξ, ξ) + λg(ξ, Y ) –g (A X, ξ) η η η N N + τ (X)g (N, ξ)+ g A X, ξ + τ (X)g (ξ, ξ) η η ξ η – λη(X)g (N – ξ, ξ) N N = ∇ τ (Y)+ g (∇ A )(X), Y – ∇ τ (X)η(Y ) X ξ N ξ – g A X, A Y + ξ(λ)– λ g(X, Y ), (78) g R (e , X)Y , e = g R(e , X)Y , e – e (λ)g(X, Y)– X(λ)g(e , Y ) g (N – ξ, e ) i i i i i i i η η η – λg(X, Y ) –g (A e , e )+ τ (e )g (N, e )+ g A e , e N i i i i i i η η η ξ + τ (e )g (ξ, e )– λη(e )g (N – ξ, e ) i i i i η η + λg(e , Y ) –g (A X, e )+ τ (X)g (N, e ) i N i i η η Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 17 of 27 + g A X, e + τ (X)g (ξ, e )– λη(X)g (N – ξ, e ) i i i η ξ η η = g R(e , X)Y , e )+ λg(X, Y ) g(A e , e )– g A e , e i i N i i i i – λg(e , Y ) g(A X, e )– g A X, e , (79) i N i i and N N g R (N, X)Y , N = ∇ τ (X)+ g (∇ A )(Y ), X – ∇ τ (Y )η(X) Y ξ N ξ – g A Y , A X – N(λ)g(X, Y)– X(λ)g(N, Y ) + λg(N, Y ) τ (X)– λη(X) . (80) By substituting (78), (79)and (80)in(77), we have (76). We have a formula relating the scalar curvature r and r of the g and g in x ∈ M as follows. Theorem 4.4 r (x)= r(x)+ div τ + tr(∇ A )– tr A A ξ N N – λ + ξ(λ)– N(λ). (81) Let π be a non-degenerate plane (for g ), we deﬁne the sectional curvature of π (with respect to g )in T M by g (R (X, Y )Y , X) K (π)= , ∀X, Y ∈ (TM) ⊂ i (TM). (82) g (X, X)g (Y , Y)–(g (X, Y )) η η η We give now the relation between K (π)and K(π). Theorem 4.5 For any orthonormal vectors X, Y ∈ (TM) ⊂ i (TM) with respect to g , π = Span{X, Y }, ξ ∈ (TM ), N be a null rigging for M ﬁxed on M, N – ξ aproperconfor- mal Killing ﬁeld with θ closed, 2 2 0 N K (π)= 1– η(X) – η(Y ) K(π)+ η(X) R (X, Y )Y · λ – ∇ B (Y , Y ) η X N N N N N + ∇ B (X, Y)+ B (X, Y )τ (Y)– B (Y , Y )τ (X) N N N + ∇ C (Y , PY)– ∇ C (X, PY)–2 dτ (X, Y )η(Y ) X Y N N N N 2 + τ (Y )C (X, PY)– τ (X)C (Y , PY ) – λη (X) 2 N N N + λ 1– η (Y ) C (X, PX)– B (X, X)– τ (X)η(X) N N N + λη(X)η(Y ) C (Y , PX)– B (Y , X)– τ (Y )η(X) . (83) Proof From g -orthonormal vectors X, Y ∈ (TM) ⊂ i (TM)and π = Span{X, Y },we have g (X, Y)=1 and g (X, Y ) = 0. It follows with (82)that η η K (π)= g R (X, Y )Y , X . (84) η Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 18 of 27 Using Theorem 4.2,weget K (π)= g R(X, Y )Y , X – X(λ)g(Y , Y)– Y (λ)g(X, Y ) g (N – ξ, X) η η – λg(Y , Y ) –g (A X, X)+ τ (X)g (N, X)+ g A X, X η η η ξ + τ (X)g (ξ, X)– λη(X)g (N – ξ, X) + λg(X, Y ) –g (A Y , X) η η η N N + τ (Y )g (N, X)+ g A Y , X + τ (Y )g (ξ, X)– λη(Y )g (N – ξ, X) η η ξ η η = g R(X, Y )Y , X – θ(X) g ξ, R(X, Y )Y – g N, R(X, Y )Y – R (X, Y )Y · λ g (N – ξ, X) – λg(Y , Y ) –g (A X, X)+ τ (X)g (N, X)+ g A X, X η η η ξ + τ (X)g (ξ, X)– λη(X)g (N – ξ, X) + λg(X, Y ) –g (A Y , X) η η η N N + τ (Y )g (N, X)+ g A Y , X + τ (Y )g (ξ, X)– λη(Y )g (N – ξ, X) . (85) η η ξ η η We take into account the fact that we obtain g R(X, Y )N, Y =–g(∇ A )(Y ), Y)+ g(∇ A )(X), Y)– B (X, A Y )η(Y ) X N Y N N N N +2 dτ (X, Y )η(Y)– τ (Y )g(A X, Y ) N N + B (Y , A X)η(Y)+ τ (X)g(A Y , Y ) (86) N N and g (N – ξ, X)=–θ(X), g (ξ, X)= θ(X), g (A X, X)= g(A X, X). (87) N N η η η Putting (23), (86), (87)in(85), we have 2 2 N N K (π)= 1– η (X)– η (Y ) K(π)– η(X) ∇ B (Y , Y)– ∇ B (X, Y ) η X Y N N N N N + τ (Y )B (X, Y)– τ (X)B (Y , Y)– g ∇ C (Y ), Y N N N + B (X, A Y )η(Y)+ ∇ C (X), Y)– B (Y , A X)η(Y ) N Y N N N N – B (X, A Y )η(Y)+2 dτ (X, Y )η(Y)– τ (Y )g(A X, Y ) N N N N 0 + B (Y , A X)η(Y)+ τ (X)g(A Y , Y)– R (X, Y )Y · λ N N – λg(Y , Y ) –g(A X, X)+ g A X, X + τ (X)θ(X)+ λη(X)η(X) + λg(X, Y ) –g(A Y , X)+ g A Y , X + τ (Y )η(X)+ λη(Y )η(X) . N N 2 With g((∇ A )(Y ), Y)=(∇ C )(Y , PY)– B (X, A Y )η(Y), g(X, X)=1 – η (X)and X N X N g(X, Y)= η(X)η(Y ) Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 19 of 27 Corollary 4.1 If the ambient manifold is of a constant sectional curvature k, then we have 2 2 0 K (π)= 1– η(X) – η(Y ) k + η(X) R (X, Y )Y .λ N N N + ∇ C (Y , PY)– ∇ C (X, PY)–2 dτ (X, Y )η(Y ) X Y N N N N 2 + τ (Y )C (X, PY)– τ (X)C (Y , PY ) – λη (X) 2 N N N + λ 1– η (Y ) C (X, PX)– B (X, X)– τ (X)η(X) N N N + λη(X)η(Y ) C (Y , PX)– B (Y , X)– τ (Y )η(X) . (88) Proof Usingthe Gauss–Codazziequation(23) if the ambient manifold has constant sec- N N N N tional curvature k,thenwecan write (∇ B )(Y , Y)–(∇ B )(X, Y)= B (Y , Y )τ (X)– X Y N N B (X, Y )τ (Y ), and substituting this in (83), we have the desired result. Corollary 4.2 For all X, Y ∈ S(N), N N K (π)= K(π)+ λ C (X, PX)– B (X, X) . (89) 5 Chen–Ricci inequality In this section, we establish some basic inequalities between intrinsic invariants namely the Ricci curvature Ric,scalarcurvature,shape operator A , Chen invariant and the ex- trinsicinvariant calledthe squaredmeancurvature fornullhypersurface (M, g)in(n+2)- dimensional Lorentzian manifold (M, g). Theorem 5.1 Let (M, g, N) be aclosednormalizednullhypersurfaceina (n +2)- dimensional Lorentzian manifold (M, g), N – ξ be a proper conformally Killing ﬁeld with 1 0 conformal factor 2λ and λ = div (N – ξ). Then, for all X ∈ S (1), the following holds: n+2 2 2 0 Ric(X) ≤ Ric (X)+ (n +1) H – R (N, X)X · λ –2 A X, A X T M N x g ξ – λ A X, X – A X, X + ξ(λ)– λ + λ tr A – tr A X, X N N ξ ξ N N + λη(X) τ (X)– λη(X) + A X, X – A X, X + τ (X)η(X) tr A ξ ξ N N – ∇ A (X), X +3 ∇ τ (X)+ (∇ A )(X), X – ∇ τ (X)η(X) , (90) ξ X ξ N ξ 1 g where Ric (X) is the Ricci curvature of X at x with respect to (M, g), λ = div (N – ξ) (T M) n+2 and the mean curvature vector H(x) is given by H(x)= (tr A )N . n+1 N The equality case of (90) is satisﬁed by X ∈ S (1) if and only if α(X, Y)=0, ∀Y ∈ TM, g (X, Y)=0, (91) 2α(X, X)=(n +1)H, ∀Y ∈ O (X). The equality case of (90) holds for all unit vectors X ∈ S (1) and for all x ∈ Mif and only if either M is totally geodesic or n =1. Proof The null hypersurface equipped with Riemannian metric g is a Riemannian hyper- surface, and as (M, g ) is isometrically immersed in a Riemannian manifold (M, g ), using η Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 20 of 27 standard techniques as [19], we have this inequality: η 2 2 Ric (X) ≤ Ric (X)+ (n +1) H , (92) (T M) g where X ∈ S (1). If we put the equality (39)ofTheorem 3.1 in (92), we get 2 2 ∗ Ric(X) ≤ Ric (X)+ (n +1) H + A X, X – A X, X T M g ξ x η N ∗ ∗ + τ (X)η(X) tr A – ∇ A (X), X + (∇ A )(X), X ξ ξ N ξ ξ N N – ∇ τ (X)η(X)+ ∇ τ (X); (93) ξ X with (76) plugged into (93), we have the inequality (90) Corollary 5.1 Let (M, g, N) be aclosednormalizednullhypersurfaceina (n +2)- dimensional Lorentzian manifold (M, g) with constant sectional curvature k,(N – ξ) be 1 g a proper conformal Killing ﬁeld with conformal factor 2λ and λ = div (N – ξ). Then for n+2 all X ∈ S (1) the following holds: 2 2 0 Ric(X) ≤ nk + (n +1) H – R (N, X)X · λ –2 A X, A X g ξ – λ A X, X – A X, X + ξ(λ)– λ + λ tr A – tr A X, X N N ξ ξ N N + λη(X) τ (X)– λη(X) + A X, X – A X, X + τ (X)η(X) tr A ξ ξ N N – ∇ A (X), X +3 ∇ τ (X)+ (∇ A )(X), X – ∇ τ (X)η(X) . (94) ξ X ξ N ξ Corollary 5.2 Let (M, g, N) be a closed and conformal screen (with constant conformal fac- tor ϕ) normalized null hypersurface in (n+2)-dimensional Lorentzian manifold (M, g), and 1 g (N – ξ) be a proper conformal Killing ﬁeld with conformal factor 2λ and λ = div (N – ξ). n+2 Then, for all X ∈ S (1), the following holds: 2 2 0 Ric(X) ≤ Ric (X)+ (n +1) H – R (N, X)X · λ –2ϕ A X T M x g ξ 2 2 2 2 + ξ(λ)– λ + λ(ϕ –1) tr A X – λ η (X) ξ g +(3ϕ –1) ∇ A (X), X + 3(ξ · ϕ)– λ(ϕ –1)+(1 – ϕ) tr A A X, X . (95) ξ ξ ξ The equality case holds for all X ∈ S (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 A = ϕA . (96) ξ Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 21 of 27 By Lemma 2.2 in [14], the 1-form τ vanishes identically. Putting (96)in(90) and taking acount of τ =0, we have 2 2 0 Ric(X) ≤ Ric (X)+ (n +1) H – R (N, X)X · λ –2ϕ A X, A X T M x g ξ ξ – λ(ϕ –1) A X, X + ξ(λ)– λ + λ (ϕ –1) tr A X, X ξ ξ 2 2 – λ η (X)+(1 – ϕ) A X, X tr A ξ ξ – ∇ A (X), X +3 ∇ ϕA (X), X . ξ ξ ξ ξ Remark 5.1 In the conformal sreen case with conformal factor ϕ =1 we have τ =0, g g div N = div ξ, λ =0, A = A = A . Hence the following holds: N ξ Corollary 5.3 Let (M, g, N) be a closed and conformal screen (with conformal factor ϕ =1) normalized null hypersurface in (n +2)-dimensional Lorentzian manifold (M, g), (N – ξ) 1 g be a proper conformal Killing ﬁeld with conformal factor 2λ and λ = div (N – ξ). Then, n+2 for all X ∈ S (1), the following holds: 2 2 Ric(X) ≤ Ric (X)+ (n +1) H –2 A X – ∇ A (X), X . (97) T M ξ g ξ ξ The equality case holds for all X ∈ S (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. Theorem 5.2 Let (M, g, N) beaclosednormalizednullhypersurfacein (n+2)-dimensional Lorentzian manifold (M, g), N – ξ be a proper conformal Killing ﬁeld with conformal factor 1 g 0 2λ and λ = div (N – ξ). Then, for all X ∈ S (1), we have the following. n+2 2 2 N (n +1) H ≥ g(A X, X)– τ (X)η(X) tr A – g(A X, A X) N N N N N 2 2 –2λτ (X)η(X)+ λ η(X) . (98) Theequalitycaseistruefor allunitvectors in X ∈ S (1), x ∈ M, if andonlyifeither M is totally geodesic or n =1. Proof The null hypersurface equipped of Riemannian metric g is Riemannian. Using (64) in (92), we have, for all X ∈ S (1), η η η η η η 2 2 Ric (X)– g A X, A X + g A X, X tr A ≤ Ric (X)+ (n +1) H , η η (T M) N N N N (T M) g p p which gives the following inequality: η η η η 2 2 (n +1) H ≥ g A X, X tr A – g A X, A X . (99) η η g N N N N Putting (72) of Lemma 4.1 into (99), we have (98). Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 22 of 27 Corollary 5.4 Let (M, g, N) be a closed and conformal screen (with conformal factor ϕ =1) normalized null hypersurface in (n/+2)-dimensional Lorentzian manifold (M, g), (N – ξ) 1 g be a proper conformal Killing ﬁeld with conformal factor 2λ and λ = div (N – ξ). Then, n+2 for all X ∈ S (1), the following holds: 2 2 (n +1) H ≥ g A X, X tr(A )– g A X, A X . (100) g ξ ξ ξ 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 ,..., a are real numbers then 1 n n n a ≤ a , (101) i=1 i=1 with equality if and only if a = ··· = a . 1 n n+1 Theorem 5.3 Let (M , g, N) be a closed normalized null hypersurface in a Lorentzian (n +2)-manifold (M, g), N – ξ be a proper conformal Killing ﬁeld. Then at every point x ∈ M, the following holds: 0 g N 2 r (x) ≤ r(x)+ div τ + tr(∇ A )– tr A A – λ + ξ(λ)– N(λ) ξ N N 2 g N η η + (tr A ) + tr A – tr A tr A – tr ∇ A + div τ , (102) N N ξ ξ ξ ξ (n +1) where r (x) is the extrinsic scalar curvature of M and r(x) is the scalar curvature of (M, g) at x. Theequalitycaseisobtainedifand only if x ∈ Mis a umbilical point. Proof By Lemma 5.1,wecan write 2 2 η η tr A ≥ tr A . (103) N N n +1 The equality (66)shows that η 2 η 2 η η tr A = r (x)– r (x)+ tr A . (104) N N Putting (104)in(103), we have 2 2 η η η η r (x)– r (x)+ tr A ≥ tr A . (105) N N (n +1) For this purpose, we use η η η –r (x) ≥ – tr A – r (x). (106) (n +1) Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 23 of 27 By substitution of the equality (43)in(106), we have r (x) ≤ r (x)+ tr A + tr A – tr A tr A η N N ξ ξ (n +1) g N – tr ∇ A + tr(∇ A )+ div τ . (107) ξ ξ N ξ η Putting (81)in(107), one has the desired result. n+1 Corollary 5.5 Let (M , g, N) be a closed normalized and conformal screen null hyper- surface in a Lorentzian (n +2)-manifold (M, g) with conformal factor ϕ =1 and N – ξ be a proper conformal Killing ﬁeld. Then, at every point x ∈ M, we have 2 2 r (x) ≤ r(x)+ tr A – tr A , (108) ξ ξ (n +1) where r (x) is the extrinsic scalar curvature of M and r(x) is the scalar curvature of (M, g) at x. Theequalitycaseisobtainedifand 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. n+1 Theorem 5.4 Let (M , g, N) be a closed null hypersurface of (n +2)-dimensional n >2 Lorentzian manifold (M, g) with τ (ξ)=0, N – ξ be a proper conformal Killing ﬁeld with 1 g conformal factor 2λ, λ = div (N – ξ). Then, for each point x ∈ M and for all orthonormal n+2 vectors ﬁelds X, Yand π = Span{X, Y }, the following holds: δ (x) ≤ δ (x)+ n(n +1)H N 2 N N N N + inf B (X, Y ) – B (X, X)B (Y , Y)+ B (X, X)C (Y , Y ) N N N N – B (X, Y )C (X, Y)– λ C (X, PX)– B (X, X) N N N N + C (X, PY )C (Y , PX)– C (Y , PY )C (X, PX)+ tr A – tr A tr A ξ ξ N 2 – tr ∇ A + tr(∇ A )+2 div τ – tr A A – λ + ξ(λ)– N(λ), (109) ξ ξ N N ξ ξ with δ (x)= r (x)– inf K(π) and δ (x)= r(x)– inf K(π). The equality in (109) holds at x ∈ M if and only if there exists an orthonormal basis {e ,..., e } of T Msuch that π = 0 n x Span{e , e } and the shape operator A becomes of the form 1 2 ⎛ ⎞ a 00 η ⎜ ⎟ A = . (110) ⎝ 0 b 0 ⎠ 00 (a + b)I n–1 Proof By the equality (65)and (66)wehave 2 2 η η η η r (x)– K (π)= r (x)– tr A + tr A – K (π) η η N N η η η η + g A X, Y g A Y , X – g A Y , Y g A X, X . (111) η η η η N N N N Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 24 of 27 By Lemma 5.1,wecan write η 2 η 2 tr A ≥ tr A . (112) N N (n +1) In (66), we have η η η η tr A = r (x)– K (π)+ g A X, Y g A Y , X η η η N N N η η η – g A Y , Y g A X, X + tr A – r (x)+ K (π). (113) η η η N N N Equation (112)with(113)gives η η η tr A ≤ r (x)– K (π)+ g A X, Y g A Y , X η η η N N N (n +1) η η η – g A Y , Y g A X, X + tr A – r (p)+ K (π), (114) η η η N N N η η η η r (x)– K (π) ≤ r (x)– K (π)+ g A X, Y g A Y , X η η η η N N η η η 2 η 2 – g A Y , Y g A X, X + tr A – tr A η η N N N N (n +1) η η ≤ r – K (π)+ g A X, Y g A Y , X η η η N N η η η – g A Y , Y g A X, X + tr A , (115) η η N N N (n +1) which gives η η 2 r (x)– K (π) ≤ r (x)– K (π)+ n(n +1)H η η η η η η + g A X, Y g A Y , X – g A Y , Y g A X, X . (116) η η η η N N N N With η 2 2 tr A =(n +1) H . N g Putting (43), (44), (81)and (89)in(116), we have 0 2 r (x)– inf K(π) ≤ r(x)– inf K(π)+ n(n +1)H N 2 N N N N + inf B (X, Y ) – B (X, X)B (Y , Y)+ B (X, X)C (Y , Y ) N N N N – B (X, Y )C (X, Y)– λ C (X, PX)– B (X, X) N N N N + C (X, PY )C (Y , PX)– C (Y , PY )C (X, PX) + tr A – tr A tr A ξ ξ – tr ∇ A + tr(∇ A )+2 div τ – tr A A ξ ξ N N ξ ξ – λ + ξ(λ)– N(λ). (117) By (18), we have the following. Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 25 of 27 n+1 Corollary 5.6 Let (M , g, N) be a closed normalized and conformal screen null hypersur- face of (n +2)-dimensional n >2 Lorentzian manifold (M, g) with conformal factor ϕ =1, τ (ξ)=0 and N – ξ be a proper conformal Killing ﬁeld. Then, for each point x ∈ M, for all orthonormal vectors X, Yand π = Span{X, Y }, the following holds: δ (x) ≤ δ (x)+ n(n +1)H + A X, Y M g ξ – A X, X A Y , Y – tr A (118) ξ ξ ξ with δ (x)= r (x)– inf K(π) and δ (x)= r(x)– inf K(π). The equality in (118) holds at x ∈ M if and only if there exists an orthonormal basis {e ,..., e } of T Msuch that π = 0 n x Span{e , e } and the shape operator A becomes of the form 1 2 ⎛ ⎞ a 00 ⎜ ⎟ A = . (119) ⎝ 0 b 0 ⎠ 00 (a + b)I n–1 Remark 5.2 Since the sectional curvature of null hypersurface equipped with associated Riemannian metric is symmetric, we can denote the scalar curvature by η η ϒ (x)= K (e , e )= r . (120) η i j 1≤i<j≤n+1 By (120), the equalities (43)and (66)become η 0 ∗ ∗ ∗ g N ϒ = r – tr A – tr A tr A – tr ∇ A + tr(∇ A )+ div τ , N ξ ξ N ξ ξ ξ 2 2 η η η r (x)= ϒ (x)+ tr A – tr A ; N N and the inequalities (108)and (118)become n 1 2 2 r (x) ≤ r(x)+ tr A – tr A , (121) ξ ξ 2(n +1) 2 δ (x) ≤ δ (x)+ H + A X, Y M g ξ 2(n +1) – A X, X A Y , Y – tr A . (122) ξ ξ ξ 5.1 Application example One important application of the Chen–Ricci inequalities for null hypersurfaces is to pro- n+1 vide solutions to the basic problem of minimal immersions: given a null hypersurface M n+2 in Lorentzian manifolds R , 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 hypersur- face in Lorentzian manifolds. n+1 Theorem 5.5 Let (M , g, N) be a closed normalized and conformal screen null hypersur- face of (n +2)-dimensional n >2 Lorentzian manifold (M, g) with conformal factor ϕ =1, Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 26 of 27 N – ξ be a proper conformal Killing ﬁeld. If for all x ∈ Mand X ∈ S(N) (1) ∗ N ∗ Ric(X, Y ) ≤ A X, Y – A X, Y + τ (X)η(Y ) tr A ξ ξ – ∇ A (X), Y + (∇ A )(X), Y ξ ξ N N N – ∇ τ (X)η(Y)+ ∇ τ (Y ), (123) ξ X then the null hypersurface admits a minimal isometric immersion in Lorentzian manifold R and the null hypersurface is non-compact. Proof The null hypersurface equipped with associated Riemannian metric g is a Rieman- nian hypersurface. By Nash theorem, we can immerse this in a space Euclidian. By the equality (39), (123)isequivalenttosay that Ric (X, Y ) ≤ 0. Given that the null hypersur- face is a closed normalized and conformal screen with conformal factor ϕ =1, the associ- ated Riemannian metric g agrees with the degenerate metric g,the 1-form τ vanishes identically and by Deﬁnition 2.3 A = A . This shows that Ric(X, X) ≤ 0, which is a neces- sary condition for M to admit a minimal immersion in a R .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 (124) where ϕ : M → E 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. 6Conclusion 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. Acknowledgements The author has greatly beneﬁted from the referee’s report. She is thankful to the referees for remarks and suggestions, which have considerably contributed to the improvement of this work. The author would like to thank Professor Cyriaque Atindogbé for his long-time encouragement and support. Competing interests The author declares that she has no competing interests regarding the publication of this paper. Authors’ contributions The project of this paper lead to a number of long calculations. The author contributed in basic calculations and in the orientation of the results. The author read and approved the ﬁnal manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Received: 24 June 2017 Accepted: 11 May 2018 Ménédore Journal of Inequalities and Applications (2018) 2018:126 Page 27 of 27 References 1. Atindogbe, C., Berard-Bergery, L.: Distinguished normalization on non-minimal null hypersurfaces. Math. Sci. Appl. E-Notes 1(1), 18–35 (2013) 2. Atindogbe, C., Duggal, K.L.: Conformal screen on lightlike hypersurfaces. Int. J. Pure Appl. Math. 11, 421–442 (2004) 3. Atindogbe, C., Ezin, J.-P., Tossa, J.: Pseudo-inversion of degenerate metrics. Int. J. Math. Math. Sci. 55, 3479–3501 (2003) 4. Beltrami, E.: Ricerche di analisi applicata alla geometria. G. Mat. II, 150–162 (1864) 5. Chen, B.Y.: Some pinching and classiﬁcation theorems for minimal submanifolds. Arch. Math. (Basel) 60(6), 568–578 (1993) 6. Chen, B.Y.: A Riemannian invariant for submanifolds in space forms and its applications. In: Geometry and Topology of Submanifolds VI (Leuven, 1993/Brussels, 1993), pp. 58–81. World Scientiﬁc, River Edge (1994) 7. Chen, B.Y.: Mean curvature and shape operator of isometric immersions in real space forms. Glasg. Math. J. 38(1), 87–97 (1996) 8. Chen, B.Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension. Glasg. Math. J. 41, 33–41 (1999) 9. Duggal, K.L., Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Mathematics and Its Applications, vol. 364. Kluwer Academic, Dordrecht (1996) 10. Gülbahar, M., Kiliç, E., Keles, ¸ S.: Chen-like inequalities on lightlike hypersurfaces of a Lorentzian manifold. J. Inequal. Appl. 2013, Article ID 266 (2013) 11. Gülbahar, M., Kılıç, E., Keles, ¸ S.: Some inequalities on screen homothetic lightlike hypersurfaces of a Lorentzian manifold. Taiwan. J. Math. 17(6), 2083–2100 (2013) 12. Gutiérrez, M., Olea, B.: Induced Riemannian structures on null hypersurfaces. Math. Nachr. 289, 1219–1236 (2015). https://doi.org/10.1002/mana.201400355 13. Hong, S.P., Tripathi, M.M.: On Ricci curvature of submanifolds. Int. J. Pure Appl. Math. Sci. 2(2), 227–245 (2005) 14. Karimumuryango, M., Cyriaque, A.: Relationships between curvatures and topology of null hypersurface. Preprint 15. Kılıç, E., Gülbahar, M.: On the sectional curvature of lightlike submanifolds. J. Inequal. Appl. 2016(1), Article ID 57 (2016) 16. Suceava, B.: Some remarks on B.Y. Chen’s inequality involving classical invariants. An. Stiin¸ ¸ t. Univ. ‘Al.I. Cuza’ Iasi, ¸ Mat. XLV(2), 405–412 (1999) 17. Tripathi, M.M.: Certain basic inequalities for submanifolds in (K, μ) space. In: Recent Advances in Riemannian and Lorentzian Geometries, pp. 187–202 (2003) 18. Tripathi, M.M.: Improved Chen–Ricci inequality for curvature like tensors and its applications. Diﬀer. Geom. Appl. 29(5), 685–692 (2011) 19. Tripathi, M.M.: Certain basic inequalities for submanifolds
Journal of Inequalities and Applications – Springer Journals
Published: May 29, 2018
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