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

Authors: D. H. Jin; J. W. Lee
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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.


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## Body

## 1. 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+n)-dimensional 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 (M̃,g̃) with the curvature tensor R̃ satisfying the condition(1.1)g̃(R̃(X,Y)Z,W)=α{g̃(Y,Z)g̃(X,W)-g̃(X,Z)g̃(Y,W)}+β{g̃(X,W)θ(Y)θ(Z)-g̃(X,Z)θ(Y)θ(W)+g̃(Y,Z)θ(X)θ(W)-g̃(Y,W)θ(X)θ(Z)},for any vector fields X,Y,Z, and W on M̃, where  α,β are scalar functions and θ is a 1-form defined by(1.2)θ(X)=g̃(X,ζ),
where ζ is a unit vector field on M̃ which called the curvature vector field. It is well known that if the curvature tensor R̃ 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 dimensionn(>2) is defined to be a quasi-Einstein manifold [4] if its Ricci tensor satisfies the condition(1.3)Ric̃(X,Y)=ag̃(X,Y)+bϕ(X)ϕ(Y),wherea,b are scalar functions such that b≠0, and ϕ is a nonvanishing 1-form such that g̃(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(M̃,g̃) of quasiconstant curvature. We prove two characterization theorems for such a lightlike submanifold (M,g,S(TM),S(TM⊥)) as follows.Theorem 1.1.
LetM be an  r-lightlike submanifold of a semi-Riemannian manifold  (M̃,g̃) of quasiconstant curvature. If the curvature vector field  ζ of  M̃ is tangent to  M and  S(TM) is totally geodesic in  M, then one has  the following results: (1)
ifS(TM⊥) is a Killing distribution, then the functions  α and  β, defined by (1.1), vanish identically. Furthermore,  M̃,  M, and the leaf  M* of  S(TM) are flat manifolds;(2)
ifS(TM⊥) is a conformal Killing distribution, then the function  β vanishes identically. Furthermore,  M̃ 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.
LetM be an irrotational  r-lightlike submanifold of a semi-Riemannian manifold  (M̃,g̃) 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,  M̃ 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.


## 2. Lightlike Submanifolds

Let(M,g) be an m-dimensional lightlike submanifold of an (m+n)-dimensional semi-Riemannian manifold (M̃,g̃). We follow 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:(2.1)TM=Rad(TM)⊕orthS(TM);TM⊥=Rad(TM)⊕orthS(TM⊥),
where the symbol ⊕orth denotes the orthogonal direct sum. We denote such a lightlike submanifold by M=(M,g,S(TM),S(TM⊥)). Let tr⁡(TM) and ltr(TM) be complementary (but not orthogonal) vector bundles to TM in TM̃|M and TM⊥ in S(TM)⊥, respectively, and let {Ni} 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(2.2)g̃(Ni,ξj)=δij,g̃(Ni,Nj)=0,where{ξ1,…,ξr} is a lightlike basis of Γ(Rad(TM)). Then,(2.3)TM̃=TM⊕tr⁡(TM)={Rad(TM)⊕tr⁡(TM)}⊕orthS(TM)={Rad(TM)⊕ltr(TM)}⊕orthS(TM)⊕orthS(TM⊥).
We say that a lightlike submanifold (M,g,S(TM),S(TM⊥)) of M̃ is(1)r-lightlike submanifold if 1≤r<min{m,n},(2)coisotropic submanifold if 1≤r=n<m,(3)isotropic submanifold if 1≤r=m<n,(4)totally lightlike submanifold if 1≤r=m=n.The above three classes (2)~(4) are particular cases of the class (1) as follows: S(TM⊥)={0},S(TM)={0}, and S(TM)=S(TM⊥)={0}, respectively.Example 2.1.
Consider inℝ24 the 1-lightlike submanifold M given by equations
(2.4)x3=12(x1+x2),x4=12log⁡(1+(x1-x2)2),
then we have TM=span⁡{U1,U2} and TM⊥={H1,H2}, where we set
(2.5)U1=2(1+(x1-x2)2)∂x1+(1+(x1-x2)2)∂x3+2(x1-x2)∂x4,U2=2(1+(x1-x2)2)∂x2+(1+(x1-x2)2)∂x3+2(x1-x2)∂x4,H1=∂x1+∂x2+2∂x3,H2=2(1+(x2-x1)2)∂x2+2(x1-x2)∂x3+(1+(x1-x2)2)∂x4.
It follows that Rad(TM) is a distribution on M of rank 1 spanned by ξ=H1. Choose S(TM) and S(TM⊥) spanned by U2 and H2 where are timelike and spacelike, respectively. Finally, the lightlike transversal vector bundle
(2.6)ltr(TM)=Span⁡{N=12∂x1+12∂x2+12∂x3}
and the transversal vector bundle
(2.7)tr⁡(TM)=Span⁡{N,H2}
are obtained.
Let∇̃ be the Levi-Civita connection of M̃ 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(2.8)∇̃XY=∇XY+∑i=1rhil(X,Y)Ni+∑α=r+1nhαs(X,Y)Wα,(2.9)∇̃XNi=-ANiX+∑j=1rτij(X)Nj+∑α=r+1nρiα(X)Wα,(2.10)∇̃XWα=-AWαX+∑i=1rϕαi(X)Ni+∑β=r+1nθαβ(X)Wβ,(2.11)∇XPY=∇X*PY+∑i=1rhi*(X,PY)ξi,(2.12)∇Xξi=-Aξi*X-∑j=1rτji(X)ξj,
for any X,Y∈Γ(TM), where ∇ and ∇* are induced linear connections on TM and S(TM), respectively, the bilinear forms hiℓ and hαs on M are called the local lightlike second fundamental form and local screen second fundamental form on TM, respectively, and hi* is called the local radical second fundamental form on S(TM). ANi,Aξi*, and AWα are linear operators on Γ(TM), and τij,ρiα,ϕαi, and θαβ are 1-forms on TM.
Since∇̃ is torsion-free, ∇ is also torsion-free and both hiℓ and hαs are symmetric. From the fact that hiℓ(X,Y)=g̃(∇̃XY,ξi), we know that hiℓ are independent of the choice of a screen distribution. Note that hiℓ,τij, and ρiα depend on the section ξ∈Γ(Rad(TM)|𝒰). Indeed, take ξ¯i=∑j=1raijξj, then we have d(tr⁡(τij))=d(tr⁡(τ̃ij)) [5].
The induced connection∇ on TM is not metric and satisfies
(2.13)(∇Xg)(Y,Z)=∑i=1r{hil(X,Y)ηi(Z)+hil(X,Z)ηi(Y)},
where ηi is the 1-form such that
(2.14)ηi(X)=g̃(X,Ni),∀X∈Γ(TM),i∈{1,…,r}.
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
(2.15)hil(X,Y)=g(Aξi*X,Y)-∑k=1rhkl(X,ξi)ηk(Y),(2.16)hil(X,PY)=g(Aξi*X,PY),g̃(Aξi*X,Nj)=0,(2.17)ϵαhαs(X,Y)=g(AWαX,Y)-∑i=1rϕαi(X)ηi(Y),(2.18)ϵαhαs(X,PY)=g(AWαX,PY),g̃(AWαX,Ni)=ϵαρiα(X),(2.19)hi*(X,PY)=g(ANiX,PY),ηj(ANiX)+ηi(ANjX)=0,
and ϵβθαβ=-ϵαθβα, where X,Y∈Γ(TM). From (2.19), we know that the operators ANi are shape operators related to hi* for each i, called the radical shape operators on S(TM). From (2.16), we know that the operators Aξi* are Γ(S(TM)) valued. Replace Y by ξj in (2.15), then we have hiℓ(X,ξj)+hjℓ(X,ξi)=0 for all X∈Γ(TM). It follows that
(2.20)hil(X,ξi)=0,hil(ξj,ξk)=0.
Also, replace X by ξj in (2.15) and use (2.20), then we have
(2.21)hil(X,ξj)=g(X,Aξi*ξj),Aξi*ξj+Aξj*ξi=0,Aξi*ξi=0.
Thus ξi is an eigenvector field of Aξi* corresponding to the eigenvalue 0. For an r-lightlike submanifold, replace Y by ξi in (2.17), then we have
(2.22)hαs(X,ξi)=-ϵαϕαi(X).
From (2.15)~(2.18), we show that the operators Aξi* and AWα 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  (M̃,g̃), then the following assertions are equivalent: (i)
Aξi* are self-adjoint on  Γ(TM) with respect to  g,  for all  i,(ii)
hiℓ satisfy  hiℓ(X,ξj)=0 for all  X∈Γ(TM),i and  j,(iii)
Aξi*ξj=0 for all  i and  j, that is, the image of  Rad(TM) with respect to  Aξi* for each  i is a trivial vector bundle,(iv)
hiℓ(X,Y)=g(Aξi*X,Y) for all  X,Y∈Γ(TM) and  i, that is,  Aξi* is a shape operator  on M, related by the second fundamental form hiℓ.Proof.
From (2.15) and the fact that hiℓ are symmetric, we have
(2.23)g(Aξi*X,Y)-g(X,Aξi*Y)=∑j=1r{hkl(X,ξi)ηk(Y)-hkl(Y,ξi)ηk(X)}.
(i)⇔(ii). If hiℓ(X,ξj)=0 for all X∈Γ(TM),i and j, then we have g(Aξi*X,Y)=g(Aξi*Y,X) for all X,Y∈Γ(TM), that is, Aξi* are self-adjoint on Γ(TM) with respect to g. Conversely, if Aξi* are self-adjoint on Γ(TM) with respect to g, then we have
(2.24)hkl(X,ξi)ηk(Y)=hkl(Y,ξi)ηk(X),
for all X,Y∈Γ(TM). Replace Y by ξj in this equation and use the second equation of (2.20), then we have hjℓ(X,ξi)=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 hiℓ(X,ξj)=0⇔Aξi*ξj=0, for all i and j.
(ii)⇔(iv). From (2.16), we have hiℓ(X,Y)=g(Aξi*X,Y)⇔hjℓ(X,ξi)=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  (M̃,g̃), then the operators  Aξi* are self-adjoint on  Γ(TM) with respect to  g.Definition 2.4.
Anr-lightlike submanifold (M,g,S(TM),S(TM⊥)) of a semi-Riemannian manifold (M̃,g̃) is said to be irrotational if ∇̃Xξi∈Γ(TM) for any X∈Γ(TM) and i.For anr-lightlike submanifold M of M̃, the above definition is equivalent to hjℓ(X,ξi)=0 and hαs(X,ξi)=0 for any X∈Γ(TM). In this case, Aξi* 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 R̃,R, and R* the curvature tensors of the Levi-Civita connection ∇̃ of M̃, the induced connection ∇ of M, and the induced connection ∇* on S(TM), respectively:(2.25)g̃(R̃(X,Y)Z,PW)=g(R(X,Y)Z,PW)+∑i=1r{hil(X,Z)hi*(Y,PW)-hil(Y,Z)hi*(X,PW)}+∑α=r+1nϵα{hαs(X,Z)hαs(Y,PW)-hαs(Y,Z)hαs(X,PW)},(2.26)ϵαg̃(R̃(X,Y)Z,Wα)=(∇Xhαs)(Y,Z)-(∇Yhαs)(X,Z)+∑i=1r{hil(Y,Z)ρiα(X)-hil(X,Z)ρiα(Y)}+∑β=r+1n{hβs(Y,Z)θβα(X)-hβs(X,Z)θβα(Y)},(2.27)g̃(R̃(X,Y)Z,Ni)=g̃(R(X,Y)Z,Ni)+∑j=1r{hjl(X,Z)ηi(ANjY)-hjl(Y,Z)ηi(ANjX)}+∑α=r+1nϵα{hαs(X,Z)ρiα(Y)-hαs(Y,Z)ρiα(X)},(2.28)g̃(R̃(X,Y)ξi,Nj)=g̃(R(X,Y)ξi,Nj)+∑k=1r{hkl(X,ξi)ηj(ANkY)-hkl(Y,ξi)ηj(ANkX)}+∑α=r+1n{ρjα(X)ϕαi(Y)-ρjα(Y)ϕαi(X)}=g(Aξi*X,ANjY)-g(Aξi*Y,ANjX)-2dτji(X,Y)+∑k=1r{hkl(X,ξi)ηj(ANkY)-hkl(Y,ξi)ηj(ANkX)}+∑k=1r{τjk(X)τki(Y)-τjk(Y)τki(X)}+∑α=r+1n{ρjα(X)ϕαi(Y)-ρjα(Y)ϕαi(X)},(2.29)g̃(R(X,Y)PZ,PW)=g(R*(X,Y)PZ,PW)+∑i=1r{hi*(X,PZ)hil(Y,PW)-hi*(Y,PZ)hil(X,PW)},(2.30)g(R(X,Y)PZ,Ni)=(∇Xhi*)(Y,PZ)-(∇Yhi*)(X,PZ)+∑j=1r{hj*(X,PZ)τij(Y)-hj*(Y,PZ)τij(X)}.The Ricci tensor ofM̃ is given by(2.31)Ric̃(X,Y)=trace{Z⟶R̃(Z,X)Y},∀X,Y∈Γ(TM̃),
for any X,Y∈Γ(TM̃). Let dim⁡M̃ = m+n. Locally, Ric̃ is given by(2.32)Ric̃(X,Y)=∑i=1m+nϵig̃(R̃(Ei,X)Y,Ei),
where  {E1,…,Em+n} is an orthonormal frame field of TM̃. If dim⁡(M̃)>2 and(2.33)Ric̃=κ̃g̃,κ̃isaconstant,
then M̃ is an Einstein manifold. If dim⁡(M̃)=2, any M̃ is Einstein, but κ̃ in (2.33) is not necessarily constant. The scalar curvaturer̃ is defined by(2.34)r̃=∑i=1m+nϵiRic̃(Ei,Ei).
Putting (2.33) in (2.34) implies that M̃ is Einstein if and only if(2.35)Ric̃=r̃m+ng̃.


## 3. The Tangential Curvature Vector Field

LetR(0,2) denote the induced Ricci tensor of type (0,2) on M, given by(3.1)R(0,2)(X,Y)=trace{Z⟶R(Z,X)Y},∀X,Y∈Γ(TM̃).
Consider an induced quasiorthonormal frame field(3.2){ξ1,…,ξr;N1,…,Nr;Xr+1,…,Xm;Wr+1,…,Wn},where{Ni,Wα} is a basis of Γ(tr⁡(TM)|𝒰) on a coordinate neighborhood 𝒰 of M such that Ni∈Γ(ltr(TM)|𝒰) and Wα∈Γ(S(TM⊥)|𝒰). By using (2.29) and (3.1), we obtain the following local expression for the Ricci tensor:(3.3)Ric̃(X,Y)=∑a=r+1nϵag̃(R̃(Wa,X)Y,Wa)+∑i=1rg̃(R̃(ξi,X)Y,Ni)+∑b=r+1mϵbg̃(R̃(Xb,X)Y,Xb)+∑i=1rg̃(R̃(Ni,X)Y,ξi),(3.4)R(0,2)(X,Y)=∑a=r+1mϵag(R(Xa,X)Y,Xa)+∑i=1rg̃(R(ξi,X)Y,Ni).
Substituting (2.25) and (2.27) in (3.3) and using (2.15)~(2.18) and (3.4), we obtain(3.5)R(0,2)(X,Y)=Ric̃(X,Y)+∑i=1rhil(X,Y)tr⁡ANi+∑α=r+1nhαs(X,Y)tr⁡AWα-∑i=1rg(ANiX,Aξi*Y)-∑α=r+1nϵαg(AWαX,AWαY)-∑i,j=1rhjl(ξi,Y)ηi(ANjX)+∑i=1r∑α=r+1nρiα(X)ϕαi(Y)-∑α=r+1nϵαg̃(R̃(Wα,X)Y,Wα)-∑i=1rg̃(R̃(ξi,Y)X,Ni),for anyX, 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)R(0,2)(X,Y)-R(0,2)(Y,X)=∑i=1r{g(Aξi*X,ANiY)-g(Aξi*Y,ANiX)}s+∑i,j=1r{hjl(X,ξi)ηi(ANjY)-hjl(Y,ξi)ηi(ANjY)}+∑i=1r∑α=r+1n{ρiα(X)ϕαi(Y)-ρiα(Y)ϕαi(X)}-∑i=1rg̃(R̃(X,Y)ξi,Ni).From this equation and (2.28), we have(3.7)R(0,2)(X,Y)-R(0,2)(Y,X)=2d(tr⁡(τij))(X,Y).Theorem 3.1 (see[5]).
LetM be a lightlike submanifold of a semi-Riemannian manifold  (M̃,g̃), then the tensor field  R(0,2) is a symmetric Ricci tensor  Ric if and only if  each 1-form   tr(τij) is closed, that is,  d(tr(τij))=0, on any  𝒰⊂M.Note 1.
Suppose that the tensorR(0,2) is symmetric Ricci tensor Ric, then  the 1-form tr⁡(τij) is closed by Theorem 3.1. Thus, there exist a smooth function f on 𝒰 such that tr⁡(τij)=df. Consequently, we get tr⁡(τij)(X)=X(f). If we take ξ̃i=∑j=1rαijξj, it follows that tr⁡(τij)(X)=tr⁡(τ̃ij)(X)+X(ln⁡Δ). Setting Δ=exp⁡(f) in this equation, we get tr⁡(τ̃ij)(X)=0 for any X∈Γ(TM|𝒰). We call the pair {ξi,Ni}i on 𝒰 such that the corresponding 1-form tr⁡(τij) vanishes the canonical null pair of M.For the rest of this paper, letM be a lightlike submanifold of a semi-Riemannian manifold M̃ of quasiconstant curvature. We may assume that the curvature vector field ζ of M̃ is a unit spacelike tangent vector field of M and dim⁡M̃>4,(3.8)Ric̃(X,Y)={(n+m-1)α+β}g(X,Y)+(n+m-2)βθ(X)θ(Y),(3.9)g̃(R̃(ξi,Y)X,Ni)=αg(X,Y)+βθ(X)θ(Y),(3.10)ϵαg̃(R̃(Wα,Y)X,Wα)=αg(X,Y)+βθ(X)θ(Y),
for all X,Y∈Γ(TM). Substituting (3.8)~(3.10) into (3.5), we have(3.11)R(0,2)(X,Y)={(m-1)α+β}g(X,Y)+(m-2)βθ(X)θ(Y)+∑i=1rhil(X,Y)tr⁡ANi+∑α=r+1nhαs(X,Y)tr⁡AWα-∑i=1rg(ANiX,Aξi*Y)-∑α=r+1nϵαg(AWαX,AWαY)-∑i,j=1rhjl(ξi,Y)ηi(ANjX)+∑i=1r∑α=r+1nρiα(X)ϕαi(Y).Definition 3.2.
We say that the screen distributionS(TM) of M is totally umbilical [1] in M if, on any coordinate neighborhood 𝒰⊂M, there is a smooth function γi such that ANiX=γiPX for any X∈Γ(TM), or equivalently,(3.12)hi*(X,PY)=γig(X,Y),∀X,Y∈Γ(TM).
In case γi=0 on 𝒰, we say that S(TM) is totally geodesic in M.A vector fieldX on M̃ is said to be a conformal Killing vector field [6] if ℒ̃Xg̃=-2δg̃ for any smooth function δ, where ℒ̃X denotes the Lie derivative with respect to X, that is,(3.13)(L̃Xg̃)(Y,Z)=X(g̃(Y,Z))-g̃([X,Y],Z)-g̃(Y,[X,Z]),∀X,Y,Z∈Γ(TM̃).In particular, ifδ=0, then X is called a Killing vector field [7]. A distribution 𝒢 on M̃ is called a conformal Killing (resp., Killing)distribution on M̃ if each vector field belonging to 𝒢 is a conformal Killing (resp., Killing) vector field on M̃. If the coscreen distribution S(TM⊥) is a Killing distribution, using (2.10) and (2.17), we have(3.14)g̃(∇̃XWα,Y)=-g(AWαX,Y)+∑i=1rϕαi(X)ηi(Y)=-ϵαhαs(X,Y).Therefore, sincehαs are symmetric, we obtain(3.15)(L̃Wαg̃)(Y,Z)=-2ϵαhαs(X,Y).Theorem 3.3.
LetM be an  r-lightlike submanifold of a semi-Riemannian manifold  (M̃,g̃), then the coscreen distribution  S(TM⊥) is a conformal Killing  (resp., Killing) distribution if and only if there exists a smooth function  δα such that
(3.16)hαs(X,Y)=ϵαδαg(X,Y),{resp.hαs(X,Y)=0,}∀X,Y∈Γ(TM).Theorem 3.4.
LetM be an irrotational  r-lightlike submanifold of a semi-Riemannian manifold  (M̃,g̃) of quasiconstant curvature. If the curvature vector field  ζ of  M̃ 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.
From (2.17)~(2.20), (2.22), (3.16), and (3.11), we have
(3.17)hαs(X,Y)=ϵαδαg(X,Y),ϕαi(X)=0,AWαX=δαPX+∑i=1rϵαρiα(X)ξi,(3.18)R(0,2)(X,Y)={(m-1)α+β+(m-r-1)∑α=r+1nϵαδα2+∑α=r+1n∑i=1rδαρiα(ξi)}g(X,Y)+(m-2)βθ(X)θ(Y)+(m-r-1)∑i=1rγig(Aξi*X,Y),∀X,Y∈Γ(TM).
Using (3.17), we show that R(0,2) is symmetric.


## 4. Proof of Theorem1.1

Ashi*=0, we get g̃(R(X,Y)PZ,Ni)=0 by (2.30). From (2.27) and (3.16), we have(4.1)g̃(R̃(X,Y)PZ,Ni)=∑α=r+1nδα{g(X,PZ)ρiα(Y)-g(Y,PZ)ρiα(X)}.By Theorems3.1 and 3.4, we get dτ=0 on TM. Thus, we have g̃(R̃(X,Y)ξi,Ni)=0 due to (2.28). From the above results, we deduce the following equation:(4.2)g̃(R̃(X,Y)Z,Ni)=∑α=r+1nδα{g(X,PZ)ρiα(Y)-g(Y,PZ)ρiα(X)}.
Taking X=ξi and Z=X to (4.2) and then  comparing with (3.9), we have(4.3)βθ(X)θ(Y)=-{α+∑α=r+1nδαρiα(ξi)}g(X,Y),∀X,Y∈Γ(TM).Case 1.
IfS(TM⊥) is a Killing distribution, that is, δα=0, then we have
(4.4)βθ(X)θ(Y)=-αg(X,Y),∀X,Y∈Γ(TM).
Substituting (4.3) into (1.1) and using (2.25) and the facts g̃(R̃(X,Y)Z,ξi)=0 and  g̃(R̃(X,Y)Z,Ni)=0 due to (1.1), we have
(4.5)R(X,Y)Z=-α{g(Y,Z)X-g(X,Z)Y},∀X,Y,Z∈Γ(TM).
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 δα=γi=0, we have
(4.6)Ric(X,Y)=0,∀X,Y∈Γ(TM).
On the other hand, substituting (4.5) and g(R(ξi,Y)X,Ni)=0 into (3.4), we have
(4.7)Ric(X,Y)=-(m-1)αg(X,Y),∀X,Y∈Γ(TM).
From the last two equations, we get α=0 as m>1. Thus, β=0, and M̃ and M are flat manifolds by (1.1) and (4.5). From this result and (2.29), we show that M* is also flat.Case 2.
IfS(TM⊥) is a conformal Killing distribution, assume that β≠0. Taking X=Y=ζ to (4.3), we have β=-{α+∑α=r+1nδαρiα(ξi)}. From this and (4.3), we show that
(4.8)g(X,Y)=θ(X)θ(Y),∀X,Y∈Γ(TM).
Substituting (4.8) into (1.1) and using (2.25) with hi*=0 and (3.16), we have
(4.9)g(R(X,Y)Z,W)=(α+2β+∑α=r+1nϵαδα2){g(Y,Z)g(X,W)-g(X,Z)g(Y,W)},
for all X,Y,Z,W∈Γ(TM). Substituting (4.8) into (3.18) with γi=0, we have
(4.10)Ric(X,Y)=(m-r-1){α+β+∑α=r+1nϵαδα2}g(X,Y),∀X,Y∈Γ(TM),
by the fact that  ∑α=r+1nδαρiα(ξi)=-(α+β). On the other hand, from (2.27), (3.9), and (4.3), we have g(R(ξi,Y)X,Ni)=0. Substituting this result and (4.9) into (3.4), we have
(4.11)Ric(X,Y)=(m-r-1){α+2β+∑α=r+1nϵαδα2}g(X,Y),∀X,Y∈Γ(TM).
The last two equations imply β=0 as m-r>1. It is a contradiction. Thus, β=0 and M̃ is a space of constant curvature α. From (2.29) and (4.9), we show that M* is a space of constant curvature (α+∑α=r+1nϵαδα2). But M is not a space of constant curvature by (3.17)3. Let κ=(m-r-1)(α+∑α=r+1nϵαδα2), then the last two equations reduce to
(4.12)R(0,2)(X,Y)=Ric(X,Y)=κg(X,Y),∀X,Y∈Γ(TM).
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)r=∑i=1rR(0,2)(ξi,ξi)+∑a=r+1mϵaR(0,2)(Xa,Xa).
Since M is an Einstein manifold satisfying (4.12), we obtain
(4.14)r=κ∑i=1rg(ξi,ξi)+κ∑a=r+1mϵag(Xa,Xa)=κ(m-r).
Thus, we have
(4.15)Ric(X,Y)=rm-rg(X,Y),
which provides a geometric interpretation of half lightlike Einstein submanifold (the same as in Riemannian case) as we have shown that the constant κ=r/(m-r).


## 5. Proof of Theorem1.2

Assume thatζ 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(5.1)(∇Xhαs)(Y,Z)-(∇Yhαs)(X,Z)=∑i=1r{hil(X,Z)ρiα(Y)-hil(Y,Z)ρiα(X)}+∑β=r+1n{hβs(X,Z)θβα(Y)-hβs(Y,Z)θβα(X)},for allX,Y,Z∈Γ(TM). Replacing W by N to (1.1), we have(5.2)g̃(R̃(X,Y)Z,Ni)={αηi(X)+eiβθ(X)}g(Y,Z)-{αηi(Y)+eiβθ(Y)}g(X,Z)+β{θ(Y)ηi(X)-θ(X)ηi(Y)}θ(Z),for allX,Y,Z∈Γ(TM) and where ei=θ(Ni). Applying ∇X to (3.12) and using (2.13), we have(5.3)(∇Xhi*)(Y,PZ)=(X[γi])g(Y,PZ)+γi∑j=1rhjl(X,PZ)ηj(Y),for allX,Y,Z∈Γ(TM). Substituting this equation into (2.30), we obtain(5.4)g̃(R(X,Y)PZ,Ni)={X[γi]-∑j=1rγjτij(X)}g(Y,PZ)-{Y[γi]-∑j=1rγjτij(Y)}g(X,PZ)+γi∑j=1rhjl(X,PZ)ηj(Y)-γi∑j=1rhjl(Y,PZ)ηj(X),∀X,Y,Z∈Γ(TM).Substituting this equation and (5.2) into (2.27) and using θ(ξi)=0, we obtain(5.5){X[γi]-∑j=1rγjτij(X)-αηi(X)-eiβθ(X)-∑α=r+1nδαρiα(X)}g(Y,Z)-{Y[γi]-∑j=1rγjτij(Y)-αηi(Y)-eiβθ(Y)-∑α=r+1nδαρiα(Y)}g(X,Z)=γi{∑j=1rhjl(Y,PZ)ηj(X)-∑j=1rhjl(X,PZ)ηj(Y)}+β{θ(Y)ηi(X)-θ(X)ηi(Y)}θ(Z),∀X,Y,Z∈Γ(TM).ReplacingY by ξi to this and using (2.20)1 and the fact θ(ξi)=0, we have(5.6)γihil(X,Y)={ξi[γi]-∑j=1rγjτij(ξi)-α-∑α=r+1nδαρiα(ξi)}g(X,Y)-βθ(X)θ(Y),for all X,Y∈Γ(TM). Differentiating (3.16) and using (5.1), we have(5.7)∑i=1r{δαηi(X)-ϵαρiα(X)}hil(Y,Z)-∑i=1r{δαηi(Y)-ϵαρiα(Y)}hil(X,Z)={X[δα]+ϵα∑β=r+1nϵβδβθβα(X)}g(Y,Z)-{Y[δα]+ϵα∑β=r+1nϵβδβθβα(Y)}g(X,Z).ReplacingY by ξi in the last equation and using (2.20)1, we obtain(5.8){δα-ϵαρiα(ξi)}hil(X,Z)={ξi[δα]+ϵα∑β=r+1nϵβδβθβα(ξi)}g(X,Z).As the conformal factorδα is nonconstant, we show that δα-ϵαρiα(ξi)≠0. Thus, we have(5.9)hil(X,Y)=σig(X,Y),∀X,Y∈Γ(TM),
where σi={ξi[δα]+ϵα∑β=r+1nϵβδβθβα(ξi)}(δα-ϵαρiα(ξi))-1. From (3.17)1 and (5.9), we show that the second fundamental form tensor h, given by h(X,Y)=∑i=1rhiℓ(X,Y)Ni+∑α=r+1nhαs(X,Y)Wα, satisfies(5.10)h(X,Y)=Hg(X,Y),∀X,Y∈Γ(TM).Thus,M is totally umbilical [5]. Substituting (5.9) into (5.6), we have(5.11){ξi[γi]-∑j=1rγjτij(ξi)-γiσi-α-∑α=r+1nδαρiα(ξi)}g(X,Y)=βθ(X)θ(Y),for allX,Y∈Γ(TM). Taking X=Y=ζ to this equation, we have(5.12)β=ξi[γi]-∑j=1rγjτij(ξi)-γiσi-α-∑α=r+1nδαρiα(ξi).Assume thatβ≠0, then we have(5.13)g(X,Y)=θ(X)θ(Y),∀X,Y∈Γ(TM).
Substituting (5.13) into (1.1) and using (2.25), (3.12), (3.17)1, and (5.9), we have(5.14)g(R(X,Y)Z,W)=(α+2β+∑i=1rσiγi+∑α=r+1nϵαδα2){g(Y,Z)g(X,W)-g(X,Z)g(Y,W)},
for all X,Y,Z,W∈Γ(TM). Substituting (5.9) and (5.13) into (3.18), we have(5.15)Ric(X,Y)={(m-1)(α+β)+(m-r-1)(∑i=1rσiγi+∑α=r+1nϵαδα2)+∑α=r+1n∑i=1rδαρiα(ξi)}g(X,Y).
On the other hand, substituting (5.14) and the fact that(5.16)g̃(R(ξi,Y)X,Ni)={α+β+∑α=r+1nϵαδαρiα(ξi)}g(X,Y)into (3.4), we have(5.17)Ric(X,Y)={(m-1)α+2(m-1)β+(m-r-1)(∑i=1rσiγi+∑α=r+1nϵαδα2)+∑α=r+1n∑i=1rδαρiα(ξi)}g(X,Y).
Comparing (5.15) 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 M̃ and M* are spaces of constant curvatures α and (α+2∑i=1rσiγi+∑α=r+1nϵαδα2), respectively. Let(5.18)κ={(m-1)α+(m-r-1)(∑i=1rσiγi+∑α=r+1nϵαδα2)+∑α=r+1n∑i=1rδαρiα(ξi)},then (5.15) and (5.17) reduce to(5.19)R(0,2)(X,Y)=Ric(X,Y)=κg(X,Y),∀X,Y∈Γ(TM).Thus,M is an Einstein manifold. The scalar quantity c of M is given by(5.20)c=∑i=1rR(0,2)(ξi,ξi)+∑a=r+1mϵaR(0,2)(Xa,Xa)=∑i=1rκg(ξi,ξi)+κ∑a=r+1mϵag(Xa,Xa)=κ(m-r).Thus, we have(5.21)Ric(X,Y)=cm-rg(X,Y).Example 5.1.
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
(5.22)J2X=-X+ϑ(X)ζ,Jζ=0,ϑ∘J=0,ϑ(ζ)=1,ϑ(X)=g¯(ζ,X),g¯(JX,JY)=g¯(X,Y)-ϑ(X)ϑ(Y),∇¯Xζ=-X+ϑ(X)ζ,(∇¯XJ)Y=-g¯(JX,Y)ζ+ϑ(Y)JX,
for any vector 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 [9])
(5.23)g¯(R¯(X,Y)Z,PW)=g(R(X,Y)Z,PW)+B(X,Z)C(Y,PW)-B(Y,Z)C(X,PW),g(R(X,Y)PZ,PW)=g(R*(X,Y)PZ,PW)+C(X,PZ)B(Y,PW)-C(Y,PZ)B(X,PW),
for 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 [10])
(5.24)R¯(X,Y)Z=g(X,Z)Y-g(Y,Z)X.
Assume that M is almost screen conformal, that is,
(5.25)C(X,PY)=φB(X,PY)+η(X)ϑ(Y),
where φ is a nonvanishing function on a neighborhood 𝒰 in M, and ζ is tangent to M, then, by the method in Section 2 of [9], we have
(5.26)B(X,Y)=ρ{g(X,Y)-ϑ(X)ϑ(Y)},
where ρ is a nonvanishing function on a neighborhood 𝒰. Then the leaf M* of S(TM) is a semi-Riemannian manifold of quasiconstant curvature such that α=-1+2φρ2,  β=-2φρ2, and θ=ϑ in (1.1).



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