Lightlike Submanifolds of a Semi-Riemannian Manifold of Quasi-Constant Curvature

We study the geometry of lightlike submanifolds (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(𝑇𝑀⟂)) of a semi-Riemannian manifold (𝑀,𝑔) of quasiconstant curvature subject to the following conditions: (1) the curvature vector field ζ of 𝑀 is tangent to 𝑀, (2) the screen distribution 𝑆(𝑇𝑀) of 𝑀 is totally geodesic in 𝑀, and (3) the coscreen distribution 𝑆(𝑇𝑀⟂) of 𝑀 is a conformal Killing distribution.


Introduction
In the generalization from the theory of submanifolds in Riemannian to the theory of submanifolds in semi-Riemannian manifolds, the induced metric on submanifolds may be degenerate lightlike .Therefore, there is a natural existence of lightlike submanifolds and for which the local and global geometry is completely different than nondegenerate case.In lightlike case, the standard text book definitions do not make sense, and one fails to use the theory of nondegenerate geometry in the usual way.The primary difference between the lightlike submanifolds and nondegenerate submanifolds is that in the first case, the normal vector bundle intersects with the tangent bundle.Thus, the study of lightlike submanifolds becomes more difficult and different from the study of nondegenerate submanifolds.Moreover, the geometry of lightlike submanifolds is used in mathematical physics, in particular, in general relativity since lightlike submanifolds produce models of different types of horizons event horizons, Cauchy's horizons, and Kruskal's horizons .The universe can be represented as a four-dimensional submanifold embedded in a 4 ndimensional spacetime manifold.Lightlike hypersurfaces are also studied in the theory of electromagnetism 1 .Thus, large number of applications but limited information available motivated us to do research on this subject matter.Kupeli 2 and Duggal and Bejancu 1 developed the general theory of degenerate lightlike submanifolds.They constructed a transversal vector bundle of lightlike submanifold and investigated various properties of these manifolds.
In the study of Riemannian geometry, Chen and Yano 3 introduced the notion of a Riemannian manifold of a quasiconstant curvature as a Riemannian manifold M, g with the curvature tensor R satisfying the condition 1.1 for any vector fields X, Y, Z, and W on M, where α, β are scalar functions and θ is a 1-form defined by where ζ is a unit vector field on M which called the curvature vector field.It is well known that if the curvature tensor R is of the form 1.1 , then the manifold is conformally flat.If β 0, then the manifold reduces to a space of constant curvature.
A nonflat Riemannian manifold of dimension n > 2 is defined to be a quasi-Einstein manifold 4 if its Ricci tensor satisfies the condition where a, b are scalar functions such that b / 0, and φ is a nonvanishing 1-form such that g X, U φ X for any vector field X, where U is a unit vector field.If b 0, then the manifold reduces to an Einstein manifold.It can be easily seen that every Riemannian manifold of quasiconstant curvature is a quasi-Einstein manifold.
The subject of this paper is to study the geometry of lightlike submanifolds of a semi-Riemannian manifold M, g of quasiconstant curvature.We prove two characterization theorems for such a lightlike submanifold M, g, S TM , S TM ⊥ as follows.
Theorem 1.1.Let M be an r-lightlike submanifold of a semi-Riemannian manifold M, g of quasiconstant curvature.If the curvature vector field ζ of M is tangent to M and S TM is totally geodesic in M, then one has the following results:

Lightlike Submanifolds
Let M, g be an m-dimensional lightlike submanifold of an m n -dimensional semi-Riemannian manifold M, g .We follow

2.5
It Let ∇ be the Levi-Civita connection of M and P the projection morphism of Γ TM on Γ S TM with respect to the decomposition 2.1 .For an r-lightlike submanifold, the local Gauss-Weingartan formulas are given by for any X, Y ∈ Γ TM , where ∇ and ∇ * are induced linear connections on TM and S TM , respectively, the bilinear forms h i and h s α on M are called the local lightlike second fundamental form and local screen second fundamental form on TM, respectively, and h * i is called the local radical second fundamental form on S TM .A N i , A * ξ i , and A W α are linear operators on Γ TM , and τ ij , ρ iα , φ αi , and θ αβ are 1-forms on TM.
Since ∇ is torsion-free, ∇ is also torsion-free and both h i and h s α are symmetric.From the fact that h i X, Y g ∇ X Y, ξ i , we know that h i are independent of the choice of a screen distribution.Note that h i , τ ij , and ρ iα depend on the section ξ ∈ Γ Rad TM | U .Indeed, take ξ i r j 1 a ij ξ j , then we have d tr τ ij d tr τ ij 5 .The induced connection ∇ on TM is not metric and satisfies where η i is the 1-form such that

2.14
But the connection ∇ * on S TM is metric.The above three local second fundamental forms of M and S TM are related to their shape operators by and β θ αβ − α θ βα , where X, Y ∈ Γ TM .From 2.19 , we know that the operators A N i are shape operators related to h * i for each i, called the radical shape operators on S TM .From 2.16 , we know that the operators A * ξ i are Γ S TM valued.Replace Y by ξ j in 2.15 , then we have h i X, ξ j h j X, ξ i 0 for all X ∈ Γ TM .It follows that Also, replace X by ξ j in 2.15 and use 2.20 , then we have Thus ξ i is an eigenvector field of A * ξ i corresponding to the eigenvalue 0. For an r-lightlike submanifold, replace Y by ξ i in 2.17 , then we have

2.22
From 2.15 ∼ 2.18 , we show that the operators A * ξ i and A W α are not self-adjoint on Γ TM but self-adjoint on Γ S TM .Theorem 2.2.Let M, g, S TM , S TM ⊥ be an r-lightlike submanifold of a semi-Riemannian manifold M, g , then the following assertions are equivalent: i A * ξ i are self-adjoint on Γ TM with respect to g, for all i, ii h i satisfy h i X, ξ j 0 for all X ∈ Γ TM , i and j, iii A * ξ i ξ j 0 for all i and j, that is, the image of Rad TM with respect to A * ξ i for each i is a trivial vector bundle, Proof.From 2.15 and the fact that h i are symmetric, we have ξ i are self-adjoint on Γ TM with respect to g, then we have for all X, Y ∈ Γ TM .Replace Y by ξ j in this equation and use the second equation of 2.20 , then we have h j X, ξ i 0 for all X ∈ Γ TM , i and j. ii ⇔ iii .Since S TM is nondegenerate, from the first equation of 2.21 , we have h i X, ξ j 0 ⇔ A * ξ i ξ j 0, for all i and j. ii ⇔ iv .From 2.16 , we have h i X, Y g A * ξ i X, Y ⇔ h j X, ξ i 0 for any X, Y ∈ Γ TM and for all i and j.Corollary 2.3.Let M, g, S TM , S TM ⊥ be a 1-lightlike submanifold of a semi-Riemannian manifold M, g , then the operators A * ξ i are self-adjoint on Γ TM with respect to g. Definition 2.4.An r-lightlike submanifold M, g, S TM , S TM ⊥ of a semi-Riemannian manifold M, g is said to be irrotational if ∇ X ξ i ∈ Γ TM for any X ∈ Γ TM and i.
For an r-lightlike submanifold M of M, the above definition is equivalent to h j X, ξ i 0 and h s α X, ξ i 0 for any X ∈ Γ TM .In this case, A * ξ i are self-adjoint on Γ TM with respect to g, for all i.
We need the following Gauss-Codazzi equations for a full set of these equations see 1, chapter 5 for M and S TM .Denote by R, R, and R * the curvature tensors of the Levi-Civita connection ∇ of M, the induced connection ∇ of M, and the induced connection ∇ * on S TM , respectively:

2.30
The Ricci tensor of M is given by

The Tangential Curvature Vector Field
Let R 0,2 denote the induced Ricci tensor of type 0, 2 on M, given by Substituting 2.25 and 2.27 in 3.3 and using 2.15 ∼ 2.18 and 3.4 , we obtain for any X, Y ∈ Γ TM .This shows that R 0,2 is not symmetric.A tensor field R 0,2 of M, given by 3.1 , is called its induced Ricci tensor if it is symmetric.From now and in the sequel, a symmetric R 0,2 tensor will be denoted by Ric.Using 2.28 , 3.5 , and the first Bianchi identity, we obtain

3.6
From this equation and 2.28 , we have

3.7
Theorem 3.1 see 5 .Let M be a lightlike submanifold of a semi-Riemannian manifold M, g , then the tensor field R 0,2 is a symmetric Ricci tensor Ric if and only if each 1-form tr τ ij is closed, that is, d tr τ ij 0, on any U ⊂ M.
Note 1. Suppose that the tensor R 0,2 is symmetric Ricci tensor Ric, then the 1-form tr τ ij is closed by Theorem 3.1.Thus, there exist a smooth function f on U such that tr τ ij df.Consequently, we get tr τ ij X X f .If we take ξ i r j 1 α ij ξ j , it follows that tr τ ij X tr τ ij X X ln Δ .Setting Δ exp f in this equation, we get tr τ ij X 0 for any X ∈ Γ TM |U .We call the pair {ξ i , N i } i on U such that the corresponding 1-form tr τ ij vanishes the canonical null pair of M.
For the rest of this paper, let M be a lightlike submanifold of a semi-Riemannian manifold M of quasiconstant curvature.We may assume that the curvature vector field ζ of M is a unit spacelike tangent vector field of M and dim M > 4, for all X, Y ∈ Γ TM .Substituting 3.8 ∼ 3.10 into 3.5 , we have

3.11
Definition 3.2.We say that the screen distribution S TM of M is totally umbilical 1 in M if, on any coordinate neighborhood U ⊂ M, there is a smooth function γ i such that A N i X γ i P X for any X ∈ Γ TM , or equivalently,

3.12
In case γ i 0 on U, we say that S TM is totally geodesic in M.
A vector field X on M is said to be a conformal Killing vector field 6 if L X g −2δ g for any smooth function δ, where L X denotes the Lie derivative with respect to X, that is,

3.13
In particular, if δ 0, then X is called a Killing vector field 7 .A distribution G on M is called a conformal Killing resp., Killing distribution on M if each vector field belonging to G is a conformal Killing resp., Killing vector field on M. If the coscreen distribution S TM ⊥ is a Killing distribution, using 2.10 and 2.17 , we have

3.14
Therefore, since h s α are symmetric, we obtain

3.15
Theorem 3.3.Let M be an r-lightlike submanifold of a semi-Riemannian manifold M, g , then the coscreen distribution S TM ⊥ is a conformal Killing (resp., Killing) distribution if and only if there exists a smooth function δ α such that

3.16
Theorem 3.4.Let M be an irrotational r-lightlike submanifold of a semi-Riemannian manifold M, g of quasiconstant curvature.If the curvature vector field ζ of M is tangent to M, S TM is totally umbilical in M, and S TM ⊥ is a conformal Killing distribution, then the tensor field R 0,2 is an induced symmetric Ricci tensor of M.

Proof of Theorem 1.1
As h * i 0, we get g R X, Y PZ, N i 0 by 2.30 .From 2.27 and 3.16 , we have By Theorems 3.1 and 3.4, we get dτ 0 on TM.Thus, we have g R X, Y ξ i , N i 0 due to 2.28 .From the above results, we deduce the following equation: Taking X ξ i and Z X to 4.2 and then comparing with 3.9 , we have Substituting 4.3 into 1.1 and using 2.25 and the facts g R X, Y Z, Thus, M is a space of constant curvature −α.Taking X Y ζ to 4.3 , we have β −α.Substituting 4.3 into 3.18 with δ α γ i 0, we have Ric X, Y 0, ∀X, Y ∈ Γ TM .

4.6
On the other hand, substituting 4.5 and g R ξ i , Y X, N i 0 into 3.4 , we have From the last two equations, we get α 0 as m > 1.Thus, β 0, and M and M are flat manifolds by 1.1 and 4.5 .From this result and 2.29 , we show that M * is also flat.The last two equations imply β 0 as m − r > 1.It is a contradiction.Thus, β 0 and M is a space of constant curvature α.From 2.29 and 4.9 , we show that M * is a space of constant curvature α n α r 1 α δ 2 α .But M is not a space of constant curvature by 3.17 Thus M is an Einstein manifold.The scalar quantity r of M 8 , obtained from R 0,2 by the method of 2.34 , is given by

4.13
Since M is an Einstein manifold satisfying 4.12 , we obtain

Proof of Theorem 1.2
Assume that ζ is tangent to M, S TM is totally umbilical, and S TM ⊥ is a conformal Killing vector field.Using 1.1 , 2.26 reduces to for all X, Y, Z ∈ Γ TM and where e i θ N i .Applying ∇ X to 3.12 and using 2.13 , we have for all X, Y, Z ∈ Γ TM .Substituting this equation into 2.30 , we obtain

5.4
Substituting this equation and 5.2 into 2.27 and using θ ξ i 0, we obtain

5.5
Replacing Y by ξ i to this and using 2.20 1 and the fact θ ξ i 0, we have for all X, Y ∈ Γ TM .Differentiating 3.16 and using 5.1 , we have

5.7
Replacing Y by ξ i in the last equation and using 2.20 1 , we obtain As the conformal factor δ α is nonconstant, we show that δ α − α ρ iα ξ i / 0. Thus, we have where σ i {ξ i δ α α n β r 1 β δ β θ βα ξ i } δ α − α ρ iα ξ i −1 .From 3.17 1 and 5.9 , we show that the second fundamental form tensor h, given by h X, Y r Thus, M is totally umbilical 5 .Substituting 5.9 into 5.6 , we have for all X, Y ∈ Γ TM .Taking X Y ζ to this equation, we have

5.15
On the other hand, substituting 5.14 and the fact that δ α ρ iα ξ i g X, Y .
5.17 Let M, g be a lightlike hypersurface of an indefinite Kenmotsu manifold M equipped with a screen distribution S TM , then there exist an almost contact metric structure J, ζ, ϑ, g on M, where J is a 1, 1 -type tensor field, ζ is a vector field, ϑ is a 1-form, and g is the semi-Riemannian metric on M such that

Comparing 5 .r i 1 σ i γ i n α r 1 α δ 2 α
15 and 5.17 , we obtain m − 1 β 0. As m > 1, we have β 0, which is a contradiction.Thus, we have β 0. Consequently, by 1.1 , 2.29 , and 5.14 , we show that M and M * are spaces of constant curvatures α and α 2 and 5.17 reduce toR 0,2 X, Y Ric X, Y κg X, Y , ∀X, Y ∈ Γ TM .5.19Thus, M is an Einstein manifold.The scalar quantity c of M is given by fields X, Y on M, where ∇ is the Levi-Civita connection of M. Using the local second fundamental forms B and C of M and S TM , respectively, and the projection morphism P of M on S TM , the curvature tensors R, R, and R * of the connections ∇, ∇, and ∇ * on M, M, and S TM , respectively, are given by see 9g R X, Y Z, P W g R X, Y Z, P W B X, Z C Y, P W − B Y, Z C X, P W , g R X, Y PZ, PW g R * X, Y PZ, PW C X, P Z B Y, P W − C Y, P Z B X, P W ,5.23for any X, Y, Z, W ∈ Γ TM .In case the ambient manifold M is a space form M c of constant J-holomorphic sectional curvature c, R is given by see 10R X, Y Z g X, Z Y − g Y, Z X.5.24Assume that M is almost screen conformal, that is, C X, P Y ϕB X, P Y η X ϑ Y , 5.25 then the functions α and β, defined by 1.1 , vanish identically.Furthermore, M, M, and the leaf M * of S TM are flat manifolds; 2 if S TM ⊥ is a conformal Killing distribution, then the function β vanishes identically.Furthermore, M and M * are space of constant curvatures, and M is an Einstein manifold such that Ric r/ m − r g, where r is the induced scalar curvature of M. Theorem 1.2.Let M be an irrotational r-lightlike submanifold of a semi-Riemannian manifold M, g of quasiconstant curvature.If ζ is tangent to M, S TM is totally umbilical in M, and S TM ⊥ is a conformal Killing distribution with a nonconstant conformal factor, then the function β vanishes identically.Moreover, M and M * are space of constant curvatures, and M is a totally umbilical Einstein manifold such that Ric c/ m − r g, where c is the scalar quantity of M.
Duggal and Bejancu 1 for notations and results used in this paper.The radical distribution Rad TM TM ∩ TM ⊥ is a vector subbundle of the tangent bundle TM and the normal bundle TM ⊥ , of rank r 1 ≤ r ≤ min{m, n} .Then, in general, there exist two complementary nondegenerate distributions S TM and S TM ⊥ of Rad TM in TM and TM ⊥ , respectively, called the screen and coscreen distribution on M, and we have the following decompositions: TM ⊥ .Let tr TM and ltr TM be complementary but not orthogonal vector bundles to TM in T M |M and TM ⊥ in S TM ⊥ , respectively, and let {N i } be a lightlike basis of Γ ltr TM | U consisting of smooth sections of S TM ⊥ | U , where U is a coordinate neighborhood of M, such that TM span{U 1 , U 2 } and TM ⊥ {H 1 , H 2 }, where we set follows that Rad TM is a distribution on M of rank 1 spanned by ξ H 1 .Choose S TM and S TM ⊥ spanned by U 2 and H 2 where are timelike and spacelike, respectively.
, W α } is a basis of Γ tr TM | U on a coordinate neighborhood U of M such that N i ∈ Γ ltr TM | U and W α ∈ Γ S TM ⊥ | U .By using 2.29 and 3.1 , we obtain the following local expression for the Ricci tensor: 1 Consider an induced quasiorthonormal frame field {ξ 1 , . . ., ξ r ; N 1 , . . ., N r ; X r 1 , . . ., X m ; W r 1 , . . ., W n }, 3.2 where {N i