# Generalized Transversal Lightlike Submanifolds of Indefinite Sasakian Manifolds

Authors: Yaning Wang; Ximin Liu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We introduce and study generalized transversal lightlike submanifold of indefinite Sasakian manifolds which includes radical and transversal lightlike submanifolds of indefinite Sasakian manifolds as
its trivial subcases. A characteristic theorem and a classification theorem of generalized transversal lightlike submanifolds are obtained.


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## Body

## 1. Introduction

The theory of submanifolds in Riemannian geometry is one of the most important topics in differential geometry for years. We see from [1] that semi-Riemannian submanifolds have many similarities with the Riemannian counterparts. However, it is well known that the intersection of the normal bundle and the tangent bundle of a submanifold of a semi-Riemannian manifold may be not trivial, so it is more difficult and interesting to study the geometry of lightlike submanifolds than nondegenerate submanifolds. The two standard methods to deal with the above difficulties were developed by Kupeli [2] and Duggal-Bejancu [3, 4], respectively.The study of CR-lightlike submanifolds of an indefinite Kaehler manifold was initiated by Duggal-Bejancu [3]. Since the book was published, many geometers investigated the lightlike submanifolds of indefinite Kaehler manifolds by generalizing the CR-lightlike submanifold [3], SCR-lightlike submanifolds [5] to GCR-lightlike submanifolds [6], and discussing the integrability and umbilication of these lightlike submanifolds. We also refer the reader to [7] for invariant lightlike submanifolds and to [8] for totally real lightlike submanifolds of indefinite Kaehler manifolds.On the other hand, after Duggal-Sahin introduced screen real lightlike submanifolds and contact screen CR-lightlike submanifolds [9] of indefinite Sasakian manifolds by studying the integrability of distributions and the geometry of leaves of distributions as well as other properties of this submanifolds, the generalized CR-lightlike submanifold which contains contact CR and SCR-lightlike submanifolds were introduced in [4]. However, all these submanifolds of indefinite Sasakian manifolds mentioned above have the same geometric condition ϕ(Rad(TM))⊂TM, where ϕ is the almost contact structure on indefinite Sasakian manifolds, Rad(TM) is the radical distribution, and TM is the tangent bundle. Until recently Yıldırım and Şahin [10] introduced radical transversal and transversal lightlike submanifold of indefinite Sasakian manifolds for which the action of the almost contact structure on radical distribution of such submanifolds does not belong to the tangent bundle, more precisely, ϕ(Rad(TM))=ltr(TM), where ltr(TM) is the lightlike transversal bundle of lightlike submanifolds.The purpose of this paper is to generalize the radical and transversal lightlike submanifolds of indefinite Sasakian manifolds by introducing generalized transversal lightlike submanifolds. The paper is arranged as follows. In Section2, we give the preliminaries of lightlike geometry of Sasakian manifolds needed for this paper. In Section 3, we introduce the generalized transversal lightlike submanifolds and obtain a characterization theorem for such lightlike submanifolds. Section 4 is devoted to discuss the integrability and geodesic foliation of distributions of generalized transversal lightlike submanifolds. In Section 5, we investigate the geometry of totally contact umbilical generalized transversal lightlike submanifolds and obtain a classification theorem for such lightlike submanifolds.


## 2. Preliminaries

In this section, we follow [4, 10] developed by Duggal-Sahin and Yıldırım-Şahin, respectively, for the notations and fundamental equations for lightlike submanifolds of indefinite Sasakian manifolds.A submanifold(M,g) of dimension m immersed in a semi-Riemannian manifold (M¯,g¯) of dimension (m+n) is called a lightlike submanifold if the metric g induced from ambient space is degenerate and its radical distribution Rad(TM) is of rank r, where m≥2 and 1≤r≤m. It is well known that the radical distribution is given by Rad(TM)=TM∩TM⊥, where TM⊥ is called normal bundle of M in M¯. Thus there exist the nondegenerate complementary distribution S(TM) and S(TM⊥) of Rad(TM) in TM and TM⊥, respectively, which are called the screen and screen transversal distribution on M, respectively. Thus we have
(2.1)TM=Rad(TM)⊕orthS(TM),(2.2)TM⊥=Rad(TM)⊕orthS(TM⊥),
where ⊕orth denotes the orthogonal direct sum.Considering the orthogonal complementary distributionS(TM)⊥ of S(TM) in TM¯, it is easy to see that TM⊥ is a subbundle of S(TM)⊥. As S(TM⊥) is a nondegenerate subbundle of S(TM)⊥, the orthogonal complementary distribution S(TM⊥)⊥ of S(TM⊥) in S(TM)⊥ is also a nondegenerate distribution. Clearly, Rad(TM) is a subbundle of S(TM⊥)⊥. Since for any local basis {ξi} of Rad(TM), there exists a local null frame {Ni} of sections with values in the orthogonal complement of S(TM⊥) in S(TM)⊥ such that g¯(ξi,Nj)=δij and g¯(Ni,Nj)=0, it follows that there exists a lightlike transversal vector bundle ltr(TM) locally spanned by {Ni} (see [3]). Then we have that S(TM⊥)⊥=Rad(TM)⊕ltr(TM). Let tr(TM)=S(TM⊥)⊕orthltr(TM). We call {Ni}, ltr(TM) and tr(TM) the lightlike transversal vector fields, lightlike transversal vector bundle and transversal vector bundle of M with respect to the chosen screen distribution S(TM) and S(TM⊥), respectively. Then TM¯ is decomposed as follows:
(2.3)TM¯=TM⊕tr(TM)=Rad(TM)⊕tr(TM)⊕orthS(TM)=Rad(TM)⊕ltr(TM)⊕orthS(TM)⊕orthS(TM⊥).
A lightlike submanifold (M,g,S(TM),S(TM⊥)) of g¯ is said to be as follows.Case  1:r-lightlike if r<minm,n.Case  2: coisotropic ifr=n<m, S(TM⊥)=0.Case  3: isotropic ifr=m<n, S(TM)=0.Case  4: totally lightlike ifr=m=n, S(TM⊥)=S(TM)=0.Let∇¯, ∇, and ∇t denote the linear connections on M¯, M, and tr(TM), respectively. Then the Gauss and Weingarten formulas are given by
(2.4)∇¯XY=∇XY+h(X,Y),∀X,Y∈Γ(TM),(2.5)∇¯XU=-AUX+∇XtU,∀X∈Γ(TM),∀U∈Γ(tr(TM)),
where {∇XY,AUX} and {h(X,Y),∇XtU} belong to Γ(TM) and Γ(tr(TM)), respectively, AU is the shape operator of M with respective to U. Moreover, according to the decomposition (2.3), denoting by hl and hs the Γ(ltr(TM))-valued and Γ(S(TM⊥))-valued lightlike second fundamental form and screen second fundamental form of M, respectively, we have
(2.6)∇¯XY=∇XY+hl(X,Y)+hs(X,Y),∀X,Y∈Γ(TM),(2.7)∇¯XN=-ANX+∇XlN+Ds(X,N),∀N∈Γ(ltr(TM)),(2.8)∇¯XW=-AWX+∇XsW+Dl(X,W),∀W∈Γ(S(TM⊥)).Then by using (2.4), (2.6)–(2.8), and the fact that ∇¯ is a metric connection, we have
(2.9)g¯(hs(X,Y),W)+g¯(Dl(X,W),Y)=g(AWX,Y),g¯(Ds(X,N),W)=g(AWX,N).LetP be the projection morphism of TM on S(TM) with respect to the decomposition (2.1), then we have
(2.10)∇XPY=∇X*PY+h*(X,PY),∇Xξ=-Aξ*X+∇X*ξ,
for any X,Y∈Γ(TM) and ξ∈Γ(Rad(TM)), where h* and A* are the second fundamental form and shape operator of distribution S(TM) and Rad(TM), respectively. Then we have the following:
(2.11)g¯(hl(X,PY),ξ)=g(AX*,PY),g¯(h*(X,PY),N)=g(ANX,PY).g¯(hl(X,ξ),ξ)=0,Aξ*ξ=0.It follows from (2.6) that
(2.12)(∇Xg)(Y,Z)=g¯(hl(X,Y),Z)+g¯(hl(X,Z),Y),
for any X,Y,Z∈Γ(TM). Thus the induced connection ∇ on M is torsion free but is not metric, the induced connection ∇* on S(TM) is metric.Finally, we recall some basic definitions and results of indefinite Sasakian manifolds following from [4, 11]. An odd dimensional semi-Riemannian manifold (M¯,g¯) of dimension 2n+1 is said to be with an almost contact structure (ϕ,η,V) if there exist a (1,1)-type tensor ϕ, a vector field V called the characteristic vector field and a 1-form η such that
(2.13)ϕ2(X)=-X+η(X)V,η(V)=ϵ=±1.
It follows that ϕ(V)=0, η∘ϕ=0 and rank(ϕ)=2n. A Riemannian metric g¯ on (M¯,g¯) is called an associated or compatible metric of an almost constant structure (ϕ,η,V) of M if
(2.14)g¯(ϕX,ϕY)=g¯(X,Y)-ϵη(X)η(Y).
A semi-Riemannian manifold endowed with an almost contact structure is said to be an almost contact metric manifold if the semi-Riemannian metric is associated or compatible to almost contact structure and is denoted by (M¯,ϕ,η,V,g¯). It is known that g¯ is a normal contact structure if Nϕ+dη⊗V=0, where Nϕ is the Nijenhuis tensor field of ϕ. A normal contact metric manifolds is called a Sasakian manifold. We know from [12] that an almost contact metric manifold is Sasakian if and only if
(2.15)(∇¯Xϕ)Y=g¯(X,Y)V-ϵη(Y)X.
It follows from (2.13) and (2.15) that
(2.16)∇¯XV=-ϕX.It is well known thatRq2m+1 can be regarded as a Sasakian manifold. Denoting by (Rq2m+1,ϕ,V,η,g¯) the manifold Rq2m+1 with its usual Sasakian structure given by
(2.17)η=12(dz-∑i=1m‍yidxi),V=2∂z,g¯=η⊗η+14(-∑i=1q/2‍dxi⊗dxi+dyi⊗dyi+∑i=q+1m‍dxi⊗dxi+dyi⊗dyi),ϕ(∑i=1m‍(Xi∂xi+Yi∂yi)+Z∂z)=∑i=1m‍(Yi∂xi-Xi∂yi)+∑i=1m‍Yiyi∂z,
where (xi,yi,z) are the Cartesian coordinates in Rq2m+1.


## 3. Generalized Transversal Lightlike Submanifolds

In this section, we define a class of lightlike submanifolds of indefinite Sasakian manifolds and study the geometry of such lightlike submanifolds. Firstly, we recall the following lemma.Lemma 3.1 (see [4]).
LetM be a lightlike submanifold of an indefinite almost contact metric manifold M¯. If V is tangent to M, then the structure vector field V does not belong to Rad(TM).Definition 3.2 (see [10]).
Let(M,g,S(TM),S(TM⊥)) be a lightlike submanifold, tangent to the structure vector field V, immersed in an indefinite Sasakian manifold (M¯,g¯). We say that M is a radical transversal lightlike submanifold of M¯ if the following conditions are satisfied:
(3.1)ϕ(Rad(TM))=ltr(TM),ϕ(S(TM))=S(TM).
It follows from Lemma3.1 and (2.1) that the structure vector field V∈Γ(S(TM)). Suppose that ϕ(S(TM))=S(TM), then we have X∈Γ(S(TM)) such that V=ϕ(X). since (2.13) implies that ϕ(V)=0, using (2.13), we have X=η(X)V. Substituting X=η(X)V into V=ϕ(X), we have V=0. So it is impossible for ϕ(S(TM))=S(TM) in Definition 3.2. Thus, we modify the above definition as the following one.Definition 3.3.
Let(M,g,S(TM),S(TM⊥)) be a lightlike submanifold, tangent to the structure vector field V, immersed in an indefinite Sasakian manifold (M¯,g¯). M is said to be a radical transversal lightlike submanifold of M¯ if there exists an invariant nondegenerate vector subbundle D such
(3.2)ϕ(Rad(TM))=ltr(TM),ϕ(D)=D,
where D is a subbundle of S(TM) and S(TM)=D⊕{V}.Definition 3.4 (see [10]).
Let(M,g,S(TM),S(TM⊥)) be a lightlike submanifold, tangent to the structure vector field V, immersed in an indefinite Sasakian manifold (M¯,g¯). We say that M is a transversal lightlike submanifold of M¯ if the following conditions are satisfied:
(3.3)ϕ(Rad(TM))=ltr(TM),ϕ(S(TM))⊆S(TM⊥).
From Definitions 3.3 and 3.4, we introduce generalized lightlike submanifold as follows.Definition 3.5.
Let(M,g,S(TM),S(TM⊥)) be a lightlike submanifold, tangent to the structure vector field V, immersed in an indefinite Sasakian manifold (M¯,g¯). M is said to be a generalized transversal lightlike submanifold of M¯ if there exist vector subbundle D and D⊥ such
(3.4)ϕ(Rad(TM))=ltr(TM),ϕ(D)=D,ϕ(D⊥)⊆S(TM⊥),
where D and D⊥ are nondegenerate subbundles of S(TM) and S(TM)=D⊕D⊥⊕{V}.Remark 3.6.
It is easy to see that a generalized transversal lightlike submanifold is a radical transversal lightlike submanifold or a transversal lightlike submanifold [10] if and only if D⊥={0} or D={0}, respectively.
We say thatM is a proper generalized transversal lightlike submanifold of M¯ if D≠{0} and D⊥≠{0}. We denote by μ the orthogonal complement of ϕ(D⊥) in S(TM⊥). The following properties of a generalized transversal lightlike submanifolds are easy to obtain.(1)
There do not exist 1-lightlike generalized transversal lightlike submanifolds of an indefinite Sasakian manifolds. The proof of the above assertion is similar to Proposition 3.1 of [10], so we omit it here. Then, dim(Rad(TM))≥2. From (3.4), we know that there exists no a generalized transversal lightlike hypersurface of M¯ as dim(ltr(TM))≥2.(2)
Since (3.4) implies that dim(D)≥2, we have dim(M)≥6 if M is a proper generalized transversal lightlike submanifold of M¯. It follows that any 5-dimensional generalized transversal lightlike submanifold of M¯ must be a 2-lightlike submanifold.(3)
dim(M¯)≥9 if M is a proper generalized transversal lightlike submanifold of an indefinite Sasakian manifold M¯.
In this paper, we assume that the characteristic vector field is a spacelike vector filed, that is,ϵ=1. If V is a timelike vector field, then one can obtain similar results.Proposition 3.7.
There exist no isotropic, coisotropic, and totally lightlike proper generalized transversal lightlike submanifolds of indefinite Sasakian manifolds.Proof.
IfM is isotropic or totally lightlike submanifolds of M¯, we have D={0} as S(TM)={0}, which is a contradiction with definition of proper generalized transversal lightlike submanifolds. Similarly, if M is coisotropic or totally lightlike submanifolds of M¯, we have D⊥={0} as S(TM⊥)={0}, there is a contradiction, which proves the assertion.Lemma 3.8.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds M¯, then we have
(3.5)g(∇XY,V)=g¯(Y,ϕX),
where X,Y∈Γ(S(TM)-{V}).Proof.
Noticing the fact that∇¯ on M¯ is a metric connection and Definition 3.4, we have
(3.6)g(∇XY,V)=g¯(∇¯XY,V)=-g¯(Y,∇¯XV)+X(g(Y,V))=-g¯(Y,∇¯XV).
Thus, the proof follows from (2.14), (2.16), and (3.9).Theorem 3.9.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds M¯, then μ is an invariant distribution with respect to ϕ.Proof.
For anyX∈Γ(μ), ξ∈Γ(Rad(TM)) and N∈Γ(ltr(TM)), it follows from (2.14) that g¯(ϕX,N)=-g¯(X,ϕN)=0, which implies that ϕX has no components in ltr(TM). Similarly, from (3.4) and (2.14), we have g¯(ϕX,ξ)=-g¯(X,ϕξ)=0, which implies that ϕX has no components in Rad(TM). For any W∈Γ(D⊥), we have g¯(ϕX,W)=-g¯(X,ϕW)=0 as ϕW∈Γ(ϕ(D⊥)). Thus, ϕX has no components in D⊥. Finally, suppose that ϕX=αV, where α is a smooth function on M¯, then we get X=0 by replacing X in (2.13) by ϕX. Thus, we have ϕ(μ)∈Γ(μ), which completes the proof.Next, we give a characterization theorem for generalized transversal lightlike submanifolds.Theorem 3.10.
LetM be a lightlike submanifold of indefinite Sasakian space form (M¯(c),g¯), c≠1. Then M a generalized transversal lightlike submanifolds of M¯ if and only if(1)
the maximal invariant subspaces ofTpM(p∈M) define a nondegenerate distribution D with respect to ϕ;(2)
g¯(R¯(X,Y)Z,W)=0, for all X,Y∈Γ(D⊕Rad(TM)) and for all Z,W∈Γ(D⊥).Proof.
Suppose thatM is a lightlike submanifold of indefinite Sasakian space form (M¯(c),g¯). It follows from (5.1) that
(3.7)g¯(R¯(X,Y)Z,W)=1-c2g(X,Y)g¯(ϕZ,W),
for any X,Y∈Γ(D⊕Rad(TM)) and Z,W∈Γ(D⊥). Noticing that ϕZ∈Γ(S(TM⊥)) implies g¯(ϕZ,W)=0, then (2) is satisfied. Also, (1) holds naturally by using the definition of generalized transversal lightlike submanifolds.
Conversely, sinceD is a nondegenerate distribution, we may choose X,Y∈Γ(D) such that g(X,Y)≠0. Thus, from (3.7), we have g(ϕZ,W)=0 for any Z,W∈Γ(D⊥), which implies that ϕZ have components in D⊥. For any X∈Γ(D), g¯(ϕZ,X)=-g¯(Z,ϕX)=0 implies that ϕZ have no components in D. Noticing condition (1), we also have g¯(ϕZ,ξ)=-g¯(Z,ϕξ)=0 and g¯(ϕZ,N)=-g¯(Z,ϕN)=0, respectively. Thus, we get ϕ(D⊥)⊆S(TM⊥), which completes the proof.For anyX∈Γ(TM), we denote by P0, P1 and P2 and P3 are the projection morphisms of TM on Rad(TM), D and D⊥ and {V}, respectively. Let ϕX=TX+QX+FX, where TX, QX, and FX are the components of ϕX on TM, ltr(TM) and S(TM⊥), respectively. Moreover, we have TX=ϕP1∈Γ(D), QX=ϕP0∈Γ(ltr(TM)) and FX=ϕP2∈Γ(ϕ(D⊥)).Similarly, for anyU∈Γ(tr(TM)), we denote by U1, U2 and U3 are the components of U on ltr(TM), ϕ(D⊥) and μ, respectively. Using Theorem 3.9, then the components of ϕU on TM and tr(TM) are denoted by SU=ϕU1+ϕU2∈Γ(Rad(TM)⊕D⊥) and LU=ϕU3∈Γ(μ).LetM be a lightlike submanifold of indefinite Sasakian manifolds (M¯,g¯). It follows from (2.15) that ∇¯XϕY-ϕ∇¯XY=g(X,Y)V-η(Y)X. Substituting X=TX+QX+FX∈Γ(TM) and Y=TY+QY+FY∈Γ(TM) into the above equation and taking the tangential, screen transversal, and lightlike transversal parts, respectively, yield that
(3.8)(∇XT)Y=AQYX+AFYX+ϕhl(X,Y)+Shs(X,Y)+g(X,Y)V-η(Y)X,(3.9)Q∇XY=hl(X,TY)+∇XlQY+Dl(X,FY),(3.10)F∇XY=hs(X,TY)+Ds(X,QY)+∇XsFY-Lhs(X,Y),
where (∇XT)Y=∇XTY-T∇XY.Proposition 3.11.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds M¯. Then we have
(3.11)∇XV=-ϕX,hl(X,V)=0,hs(X,V)=0,∀X∈Γ(D),(3.12)∇XV=0,hl(X,V)=0,hs(X,V)=-ϕX-ϕ(Lhs(X,V)),∀X∈Γ(D⊥),(3.13)∇XV=0,hl(X,V)=-ϕX,hs(X,V)=0,∀X∈Γ(Rad(TM)),
where V is the characteristic vector filed.Proof.
ReplacingY by V in (3.8), it follows from ϕV=0 and (3.8) that
(3.14)T∇XV=X-ϕhl(X,V)-Shs(X,V)-η(X)V.
Similarly, replacing Y by V in (3.9) and (3.10), respectively, we get Q∇XV=0 and F∇XV=-Lhs(X,V). Noticing that FX∈Γ(ϕD⊥) for any X∈Γ(TM) and LW∈Γ(μ) for any W∈Γ(tr(TM)), we have F∇XV=Lhs(X,V)=0.
LetX∈Γ(D), since η(X)=0 and Shs(X,V)∈Γ(D⊥), we have T∇XV=X and ϕhl(X,V)=Shs(X,V)=0, which proves (3.11).
LetX∈Γ(D⊥), from (3.14), we have T∇XV=ϕhl(X,V)=0 and Shs(X,V)=X, which proves (3.12).
LetX∈Γ(Rad(TM)), it follows from (3.14) that T∇XV=Shs(X,V)=0 and ϕhl(X,V)=X, which proves (3.13).We know from (2.12) that the induced connection ∇ on M is not a metric connection, the following theorem gives a necessary and sufficient condition for the ∇ to be a metric connection.Theorem 3.12.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then the following assertions are equivalent.(1)
The induced connection∇ on M is a metric connection.(2)
AϕXY has no components in Rad(TM) for any X∈Γ(S(TM)) and Y∈Γ(TM).(3)
AϕξX has no components in D for any X∈Γ(TM) and Ds(X,N)∈Γ(μ) for any X∈Γ(TM).Proof.
From (2.13) and (2.15), we have
(3.15)g(∇Xξ,V)=g¯(∇¯Xξ,V)=g¯(ϕ∇¯Xξ,ϕV)=0,
which implies that ∇Xξ has not component in {V} for any X∈Γ(TM).
(
1
)
⇔
(
2
). For X1∈Γ(D) and X∈Γ(TM), it follows from (2.6), (2.14), (2.15), and the fact that ∇¯ is metric connection, we have
(3.16)g(∇Xξ,X1)=g¯(∇¯Xξ,X1)=g¯(ϕ∇¯Xξ,ϕX1)+η(∇¯Xξ)η(X1)=g¯(∇¯Xϕξ+η(ξ)X-g(X,ξ)V,X1)=-g¯(ϕξ,∇¯XϕX1)=g¯(ϕξ,AϕX1X).
For X2∈Γ(D⊥), Together with (2.8), (2.14), and (2.15), we have
(3.17)g(∇Xξ,X2)=g(∇¯Xξ,X2)=g¯(ϕ∇¯Xξ,ϕX2)+η(∇¯Xξ)η(X2)=g¯(∇¯Xϕξ+η(ξ)X-g(X,ξ)V,X2)=-g¯(ϕξ,∇¯XϕX2)=g¯(ϕξ,AϕX2X).
Thus, we prove the equivalence between (1) and (2).
(1)⇔(3). Using the similar method shown in the above, from (3.16) and (3.17), we have
(3.18)g(∇Xξ,X1)=g¯(∇¯Xξ,X1)=-g(AϕξX,ϕX1),g(∇Xξ,X2)=g(∇¯Xξ,X2)=-g¯(Ds(X,ϕξ),ϕX2).
Thus, the proof follows from the above equations.


## 4. Integrability and Geodesic of Distributions

Theorem 4.1.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then D⊕{V} is integrable if and only if h(X,ϕY)=h(Y,ϕX) for any X,Y∈Γ(D).Proof.
LetX,Y∈Γ(D), we have ϕY=TY and QY=FY=0. Thus, it follows from (3.9) that Q∇XY=hl(X,ϕY). Interchanging the roles of X and Y in the above equation and subtracting, we have
(4.1)Q[X,Y]=hl(X,ϕY)-hl(Y,ϕX),∀X,Y∈Γ(D).
Similarly, it follows from (3.10) that F∇XY=hs(X,ϕY)-Lhs(X,Y). Noticing that h is symmetric and interchanging the roles of X and Y in the above equation and subtracting, we have
(4.2)F[X,Y]=hs(X,ϕY)-hs(Y,ϕX),∀X,Y∈Γ(D).
Thus, our assertion follows from (4.1) and (4.2).Corollary 4.2.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then D is not integrable.Proof.
It follows from Lemma3.8 that g([X,Y],V)=2g¯(Y,ϕX) for any X,Y∈Γ(D). Suppose that D is integrable, then we have g¯(Y,ϕX)=0, which is a contradiction to the fact that D is a nondegenerate distribution of M.Theorem 4.3.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then D⊥ is integrable if and only if Dl(X,FY)=Dl(Y,FX) and AFXY=AFYX for any X,Y∈Γ(D⊥).Proof.
ForX,Y∈Γ(D⊥), we have ϕY=FY and TY=QY=0. Thus, it follows from (3.8) that T∇XY=-AFYX-Qhl(X,Y)-Shs(X,Y)-g(X,Y)V. As h and g are symmetric, by interchanging the roles of X and Y in the above equation and subtracting the resulting equations, we have
(4.3)T[X,Y]=AFXY-AFYX,∀X,Y∈Γ(D⊥).
Similarly, it follows from (3.9) that Q∇XY=Dl(X,FY). Interchanging the roles of X and Y in the above equation and subtracting the resulting equations, we have
(4.4)Q[X,Y]=Dl(X,FY)-Dl(Y,FX),∀X,Y∈Γ(D⊥).
Noticing Lemma 3.8, we have g([X,Y],V)=2g¯(Y,ϕX)=0 for any X,Y∈Γ(D⊥). Thus, our assertion follows from (4.3) and (4.4).Remark 4.4.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). If D={0}, then M is a transversal lightlike submanifolds [9]. It follows from Theorem 4.3 that S(TM) is integrable if and only if Dl(X,FY)=Dl(Y,FX) for all X,Y∈Γ(S(TM)), which is just one of the conclusions shown in [10].Theorem 4.5.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then Rad(TM)⊕{V} is integrable if and only if AϕXY=AϕYX and Ds(X,ϕY)-Ds(Y,ϕX) has no components in ϕ(D⊥) for any X,Y∈Γ(Rad(TM)).Proof.
Together with (2.14), (2.15), and the fact that ∇¯ is a metric connection, we have
(4.5)g([X,Y],Z)=g¯(∇¯XϕY-∇¯YϕX,ϕZ)=g(AϕXY-AϕYX,ϕZ),
where Z∈Γ(D). Similarly, for W∈Γ(D⊥), we have
(4.6)g([X,Y],W)=g¯(∇¯XϕY-∇¯YϕX,ϕW)=g(Ds(X,ϕY)-Ds(Y,ϕX),ϕW),
Thus, our assertion follows from (4.5) and (4.6).Corollary 4.6.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then Rad(TM) is not integrable.Proof.
It follows from Lemma3.8 that g([X,Y],V)=2g¯(Y,ϕX) holds for any X,Y∈Γ(Rad(TM)). Suppose that Rad(TM) is integrable, then we have g¯(Y,ϕX)=0. On the other hand, for any X∈Γ(Rad(TM)), there must exists Y∈Γ(Rad(TM)) such that g(X,ϕY)≠0 as ϕ(Rad(TM))=ltr(TM), a contradiction. Then we complete the proof.Using the similar method in the proof of Theorems4.1–4.5 and Corollaries 4.2 and 4.6, we have the following results.Theorem 4.7.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then,(1)
Rad(TM)⊕D⊕{V} is integrable if and only if hs(X,ϕY)-hs(Y,ϕX) has no components in ϕ(D⊥) for any X,Y∈Γ(Rad(TM)⊕D).(2)
Rad(TM)⊕D⊥⊕{V} is integrable if and only if AϕYX-AϕXY has no components in D for any X,Y∈Γ(Rad(TM)⊕D⊥).(3)
Rad(TM)⊕D is not integrable.(4)
Rad(TM)⊕D⊥ is not integrable.We know from [3] that a distribution on M is said to define a totally geodesic foliation if any leaf of D is geodesic. we focus on this property of generalized transversal lightlike submanifolds in the following of this section.Theorem 4.8.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then the screen distribution defines a totally geodesic foliation if and only if g(TY,AϕN*X)=g¯(FY,hs(X,ϕN)) for any X,Y∈Γ(S(TM)).Proof.
It is known thatS(TM) defines a totally geodesic foliation if and only if ∇XY∈Γ(S(TM)) for any X,Y∈Γ(S(TM)). For X∈Γ(S(TM)), we have QX=0. Also, we have
(4.7)g(∇XY,N)=g¯(∇¯XϕY,ϕN)=-g¯(ϕY,∇¯XϕN)=g¯(ϕY,AϕN*)X-hs(X,ϕN)=g(TY,AϕN*X)-g¯(FY,hs(X,ϕN)).
Thus we complete the proof.Remark 4.9.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). If D⊥={0}, then M is radical transversal lightlike submanifolds. For X∈Γ(S(TM)), we have FX=0. Thus, it follows from Theorem 4.8 that S(TM) defines a totally geodesic foliation if and only if AϕN*X has no components in S(TM), which is just the Theorem 3.6 proved in [10].Theorem 4.10.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then, The following assertions are equivalent.(1)
D⊥ defines a totally geodesic foliation.(2)
Dl(X,ϕY)=0 and AϕYX has no components in D.(3)
hs(X,ϕZ) and hs(X,ξ) has no components ϕ(D⊥), where X,Y,Z∈Γ(D⊥) and ξ∈Γ(Rad(TM)).Proof.
It is easy to see thatD⊥ defines a totally geodesic foliation if and only if ∇XY∈Γ(D⊥) for any X,Y∈Γ(D⊥).
(
1
)
⇔
(
2
). From (2.14) and (2.15), we have that
(4.8)g¯(∇XY,N)=g¯(ϕ∇¯XY,ϕN)=g¯(∇¯XϕY,ϕN)=g¯(Dl(X,ϕY),ϕN),
where X,Y∈Γ(D⊥) and N∈Γ(ltr(TM)). Also, for Z∈Γ(D), we have
(4.9)g¯(∇XY,Z)=g¯(ϕ∇¯XY,ϕZ)=g¯(∇¯XϕY,ϕZ)=-g¯(AϕYX,ϕZ).
Noticing Lemma 3.8, we have g([X,Y],V)=2g¯(Y,ϕX)=0 for all X,Y∈Γ(D⊥). Then the equivalence between (1) and (2) follows from (4.8) and (4.9). Noticing that ∇¯ is a metric connection, then from (4.8) and (4.9) we have that
(4.10)g¯(∇XY,N)=-g¯(ϕY,hs(X,ϕN)),g¯(∇XY,Z)=-g¯(hs(X,ϕZ),ϕY).
Thus, the equivalence between (1) and (3) follows from (4.10).From (4.9), we know that g¯(∇XY,Z)=-g¯(AϕYX,ϕZ) for any X,Y∈Γ(D⊥⊕Rad(TM)) and Z∈Γ(D), then the following corollary holds.Corollary 4.11.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then D⊥⊕Rad(TM)⊕{V} defines a totally geodesic foliation if and only if AϕYX has no components in D for any X,Y∈Γ(D⊥⊕Rad(TM){V}).Theorem 4.12.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then D⊕{V} defines a totally geodesic foliation if and only if AϕN*X and AϕWX has no components in D for any X∈Γ(D) and W∈Γ(D⊥) and N∈Γ(ltr(TM)).Proof.
Noticing that∇¯ is a metric connection and (2.14) and (2.15), we have
(4.11)g¯(∇XY,N)=g¯(ϕ∇¯XY,ϕN)=g¯(∇¯XϕY,ϕN)=g¯(AϕN*X,ϕY).
Similarly, for W∈Γ(D⊥), we have
(4.12)g¯(∇XY,W)=g¯(ϕ∇¯XY,ϕW)=g¯(∇¯XϕY,ϕ)=g¯(AϕWX,ϕY).
Then the proof follows from (4.11) and (4.12) and Lemma 3.8.It follows from (2.13) that ϕV=0, then (4.11) and (4.12) hold for any X,Y∈Γ(D⊕{V}). Thus it is easy to get the following corollary.Corollary 4.13.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then D cannot define a totally geodesic foliation.Proof.
It follows from Lemma3.8 that g(∇XY,V)=g¯(Y,ϕX) for any X,Y∈Γ(D). Suppose that D defines a totally geodesic foliation, then we have g¯(Y,ϕX)=0. Which is a contradiction to the fact that D is a nondegenerate distribution of M.We say thatM is a contact generalized transversal lightlike product manifold if D⊕{V} and D⊥⊕Rad(TM) define totally geodesic foliations in M.Theorem 4.14.
LetM be a generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then, M is a contact generalized transversal lightlike product manifold if and only if(1)
h(X,ϕY)=Lhs(X,Y) for any X,Y∈Γ(D⊕{V});(2)
AϕXY has no components in D for any X,Y∈Γ(D⊥⊕Rad(TM)).Proof.
ForX,Y∈Γ(D⊕{V}), we have ϕX=TX and QX=FX=0 as ϕV=0. It follows from (3.9) and (3.10), respectively, that
(4.13)Q∇XY=hl(X,TY),F∇XY=hs(X,TY)-Lhs(X,Y).
On the other hand, for X,Y∈Γ(D⊥⊕Rad(TM)), we have TX=0. Then it follows from (3.8) that
(4.14)AϕYX=-T∇XY-ϕhl(X,Y)-Shs(X,Y)-g(X,Y)V.
Noticing that ∇XY belongs to Γ(D⊕{V}) or Γ(D⊥⊕Rad(TM)) if and only if Q∇XY=F∇XY=0 or T∇XY=0, respectively, then our assertion follows from (4.13) and (4.14).


## 5. Totally Contact Umbilical Lightlike Submanifolds

In this section, we study totally contact umbilical generalized transversal lightlike submanifolds of indefinite Sasakian manifolds defined by Duggal and Sahin [9]. We mainly obtain a classification theorem for such lightlike submanifolds.A plane section of a Sasakian manifold(M¯,ϕ,η,V,g¯) is called a ϕ-section if it is spanned by a unit vector X orthogonal to V and ϕX, where X is a non-null vector field on M¯. The sectional curvature K(X,ϕX) of a ϕ-section is called a ϕ-sectional curvature. If M¯ has a ϕ-sectional curvature c which is not depend on the ϕ-section at each point, then c is a constant and M¯ is called a Sasakian space form, denoted by M¯(c). The curvature tensor R¯ of a Sasakian space form M¯(c) is given in [13] as follows:
(5.1)R¯(X,Y)Z=c+34{g¯(Y,Z)X-g¯(X,Z)Y}+c-14{ϵη(X)η(Z)Y-ϵη(Y)η(Z)X+g¯(X,Z)η(Y)V-g¯(Y,Z)η(X)V+g¯(ϕY,Z)ϕX+g¯(ϕZ,X)ϕY-2g¯(ϕX,Y)ϕZ},
where X,Y,Z∈Γ(TM¯).Definition 5.1 (see [9]).
A lightlike submanifold of an indefinite Sasakian manifold is contact totally umbilical if(5.2)hl(X,Y)={g(X,Y)-η(X)η(Y)}αL+η(X)hl(Y,V)+η(Y)hl(X,V),(5.3)hs(X,Y)={g(X,Y)-η(X)η(Y)}αS+η(X)hs(Y,V)+η(Y)hs(X,V),
where X,Y∈Γ(TM), αL∈Γ(ltr(TM)), and αS∈Γ(S(TM⊥)).Theorem 5.2.
LetM be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then αL=0 if and only if D is integrable.Proof.
LetX,Y∈Γ(D), it follows from (2.15) and (5.2) that
(5.4)g¯([X,Y],N)=g¯(ϕ∇¯XY-ϕ∇¯YX,ϕN)=g¯(∇¯XϕY-∇¯YϕX,ϕN)=g¯(hl(X,ϕY)-hl(Y,ϕX),ϕN)=(g(X,ϕY)-g(Y,ϕX))g¯(αL,ϕN).
Noticing that (2.15) implies g(X,ϕY)+g(Y,ϕX)=0 for any X,Y∈Γ(TM), then it follows from (5.4) that g¯([X,Y],N)=2g(X,ϕY)g¯(αL,ϕN). On the other hand, for any Z∈Γ(D⊥) it follows from (2.14), (2.15), and (5.3) that
(5.5)g¯([X,Y],Z)=g¯(ϕ∇¯XY-ϕ∇¯YX,ϕZ)=g¯(∇¯XϕY-∇¯YϕX,ϕZ)=g¯(hs(X,ϕY)-hs(Y,ϕX),ϕZ)=0,
which means that [X,Y] have components in D⊥ for any X,Y∈Γ(D). Noticing Lemma 3.8, we know that [X,Y] has no components in {V} for any X,Y∈Γ(D). Thus, the proof is complete.Theorem 5.3.
LetM be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then αS=0.Proof.
LetX∈Γ(D), from (5.3), we have hs(X,Y)=g(X,Y)αS, which implies hs(X,ϕY)-hs(Y,ϕX)=2g(X,ϕX)αS. Also, it follows from (3.10) that
(5.6)hs(X,ϕY)=F∇XY+Lhs(X,Y).
Interchanging the role of X and Y in the above equation and subtracting, we have hs(X,ϕY)-hs(Y,ϕX)=F[X,Y]. Using the same method as shown in the proof of Theorem 5.2, we have αS=0. Which proves the theorem.Lemma 5.4.
LetM be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then, for any X,Y∈Γ(D), we have g(∇XϕX,V)=g(X,X) and g(∇ϕXX,V)=-g(X,X).Proof.
Since∇¯ is a metric connection, we have
(5.7)X(g¯(ϕX,V))-g¯(∇¯XϕX,V)-g¯(ϕX,∇¯XV)=0.
Noticing (2.16) and the fact that X∈Γ(D) implies g(ϕX,V)=0 and ϕX∈Γ(D), then from (5.7) and (2.14), we have g(∇XϕX,V)=g(X,X). Interchanging the role of X and ϕX in (5.7) we get g(∇ϕXX,V)=-g(X,X).Lemma 5.5.
LetM be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold (M¯,g¯). Then for any X,Y∈Γ(D), one have g(∇Xξ,ϕX)=0 and g(∇ϕXξ,X)=0.Proof.
Since∇¯ is a metric connection, we have
(5.8)ϕX(g¯(X,ξ))-g¯(∇¯ϕXX,ξ)-g¯(X,∇¯ϕXξ)=0.
For X∈Γ(D), it is easy to see that g(ϕX,ξ)=0 and ϕX∈Γ(D), then g(∇ϕXξ,X)=-g¯(hl(ϕX,X),ξ). Using (5.2) we have g(∇Xξ,ϕX)=0. Interchanging the role of X and ϕX in (5.8) we get g(∇ϕXξ,X)=0.At last, we complete this paper by a classification theorem for generalized transversal lightlike submanifolds.Theorem 5.6.
LetM be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian space form (M¯(c),g¯). Then c=-3.Proof.
For anyX,Y,Z∈Γ(TM), it follows from (2.6)–(2.8) that
(5.9)R¯(X,Y)Z=R(X,Y)Z+Ahl(X,Z)Y-Ahl(Y,Z)X+Ahs(X,Z)Y-Ahs(Y,Z)X+(∇Xhl)(Y,Z)-(∇Yhl)(X,Z)+Dl(X,hs(Y,Z))-Dl(Y,hs(X,Z))+(∇Xhs)(Y,Z)-(∇Yhs)(X,Z)+Ds(X,hl(Y,Z))-Dl(Y,hl(X,Z)),
where (∇Xhl)(Y,Z)=∇Xlhl(Y,Z)-hl(∇XY,Z)-hl(∇XZ,Y) and (∇Xhs)(Y,Z)=∇Xshs(Y,Z)-hs(∇XY,Z)-hs(∇XZ,Y).
Noticing Theorem5.3 then we have hs(X,Y)=0 for any X,Y∈Γ(TM-{V}). Let X,Y∈Γ(D) and ξ,ξ'∈Γ(Rad(TM)), replacing Z by ξ in (5.9) and using (5.2), we have
(5.10)g¯(R¯(X,Y)ξ,ξ′)=g¯((∇Xhl)(Y,ξ)-(∇Yhl)(X,ξ),ξ′)=g¯(hl(∇YX,ξ)-hl(∇XY,ξ)+hl(X,∇Xξ)-hl(Y,∇Xξ),ξ′)=g¯(η(∇YX)hl(ξ,V)-η(∇XY)hl(ξ,V)+g(X,∇Yξ)αL-g(Y,∇Xξ)αL,ξ′).
Replacing Y by ϕX in (5.10) and using Proposition 3.11 and Lemmas 5.4 and 5.5, we have
(5.11)g¯(R¯(X,Y)ξ,ξ′)=2g(X,X)g(ϕξ,ξ′).
On the other hand, it follows from (5.1) that
(5.12)g¯(R¯(X,Y)ξ,ξ′)=1-c2g(ϕX,Y)g(ϕξ,ξ′).
ReplacingY by ϕX in (5.12) and using (5.11), we have that (c+3)g(X,X)g(ϕξ,ξ')=0 for all X∈Γ(D). As D is nondegenerate, so we may choose X such that g(X,X)≠0 in the above equation. Thus, we complete the proof.



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*Source: 361794-2012-11-26.xml*