Generalized Transversal Lightlike Submanifolds of Indefinite Sasakian Manifolds

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.


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.

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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 S ¸ahin 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 Section 2, 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.

Preliminaries
In this section, we follow 4, 10 developed by Duggal-Sahin and Yıldırım-S ¸ahin, 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 where ⊕ orth denotes the orthogonal direct sum.Considering the orthogonal complementary distribution S TM ⊥ of S TM in T M, 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 {ξ i } of Rad TM , there exists a local null frame {N i } of sections with values in the orthogonal complement of S TM ⊥ in S TM ⊥ such that g ξ i , N j δ ij and g N i , N j 0, it follows that there exists a lightlike transversal vector bundle ltr TM locally spanned by {N i } see 3 .Then we have that S TM ⊥ ⊥ Rad TM ⊕ ltr TM .Let tr TM S TM ⊥ ⊕ orth ltr TM .We call {N i }, 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 T M is decomposed as follows:

2.3
A lightlike submanifold M, g, S TM , S TM ⊥ of g is said to be as follows.
Case 3: isotropic if r m < n, S TM 0.
Case 4: totally lightlike if r 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 where {∇ X Y, A U X} and {h X, Y , ∇ t X U} belong to Γ TM and Γ tr TM , respectively, A U is the shape operator of M with respective to U.Moreover, according to the decomposition 2.3 , denoting by h l and h s the Γ ltr TM -valued and Γ S TM ⊥ -valued lightlike second fundamental form and screen second fundamental form of M, respectively, we have Then by using 2.4 , 2.6 -2.8 , and the fact that ∇ is a metric connection, we have

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Let P be the projection morphism of TM on S TM with respect to the decomposition 2.1 , then we have 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
It follows from 2.6 that 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
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 g φX, φY g X, Y − η X η Y .

2.14
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 N φ dη ⊗ V 0, where N φ 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
It follows from 2.13 and 2.15 that

2.16
It is well known that R 2m 1 q can be regarded as a Sasakian manifold.Denoting by R 2m 1 q , φ, V, η, g the manifold R 2m 1 q with its usual Sasakian structure given by 2.17 where x i , y i , z are the Cartesian coordinates in R 2m 1 q .

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.It follows from Lemma 3.1 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.In this paper, we assume that the characteristic vector field is a spacelike vector filed, that is, 1.If V is a timelike vector field, then one can obtain similar results.
Proof.Noticing the fact that ∇ on M is a metric connection and Definition 3.4, we have Thus, the proof follows from 2.14 , 2.16 , and 3.9 .
Theorem 3.9.Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds M, then μ is an invariant distribution with respect to φ.
Proof.For any X ∈ 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.Let M 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 of T p M p ∈ M define a nondegenerate distribution D with respect to φ; 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, since D 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 International Journal of Mathematics and Mathematical Sciences g φZ, N −g Z, φN 0, respectively.Thus, we get φ D ⊥ ⊆ S TM ⊥ , which completes the proof.
For any X ∈ Γ TM , we denote by P 0 , P Proposition 3.11.Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds M. Then we have 11 where V is the characteristic vector filed.
Proof.Replacing Y by V in 3.8 , it follows from φV 0 and 3.8 that

3.14
Similarly, replacing Y by V in 3.9 and 3.10 , respectively, we get Q∇ X V 0 and F∇ X V −Lh s 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∇ X V Lh s X, V 0. Let X ∈ Γ D , since η X 0 and Sh s X, V ∈ Γ D ⊥ , we have T ∇ X V X and φh l X, V Sh s X, V 0, which proves 3.11 .Let X ∈ Γ D ⊥ , from 3.14 , we have T ∇ X V φh l X, V 0 and Sh s X, V X, which proves 3.12 .
Let X ∈ Γ Rad TM , it follows from 3.14 that T ∇ X V Sh s X, V 0 and φh l 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.Let M 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.Proof.From 2.13 and 2.15 , we have For X 1 ∈ Γ D and X ∈ Γ TM , it follows from 2.6 , 2.14 , 2.15 , and the fact that ∇ is metric connection, we have

3.17
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
Thus, the proof follows from the above equations.

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Proof.Let X, Y ∈ Γ D , we have φY TY and QY FY 0. Thus, it follows from 3.9 that Q∇ X Y h l X, φY .Interchanging the roles of X and Y in the above equation and subtracting, we have Similarly, it follows from 3.10 that F∇ X Y h s X, φY − Lh s X, Y .Noticing that h is symmetric and interchanging the roles of X and Y in the above equation and subtracting, we have 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 Similarly, it follows from 3.9 that Q∇ X Y D l X, FY .Interchanging the roles of X and Y in the above equation and subtracting the resulting equations, we have 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 Theorems 4.1-4.5 and Corollaries 4.2 and 4.6, we have the following results.Theorem 4.7.Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g .Then, 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.Let M 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, A * φN X g FY, h s X, φN for any X, Y ∈ Γ S TM .

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Proof.It is known that S TM defines a totally geodesic foliation if and only if ∇ X Y ∈ Γ S TM for any X, Y ∈ Γ S TM .For X ∈ Γ S TM , we have QX 0. Also, we have

4.7
Thus we complete the proof.
Remark 4.9.Let M 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 A * φN X has no components in S TM , which is just the Theorem 3.6 proved in 10 .
Theorem 4.10.Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g .Then, The following assertions are equivalent.

D l X, φY
0 and A φY X has no components in D. 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.We say that M is a contact generalized transversal lightlike product manifold if D⊕{V } and D ⊥ ⊕ Rad TM define totally geodesic foliations in M. Theorem 4.14.Let M 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 Proof.For X, 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 On the other hand, for X, Y ∈ Γ D ⊥ ⊕ Rad TM , we have TX 0. Then it follows from 3.8 that

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Noticing that ∇ X Y belongs to Γ D ⊕{V } or Γ D ⊥ ⊕Rad TM if and only if Q∇ X Y F∇ X Y 0 or T ∇ X Y 0, respectively, then our assertion follows from 4.13 and 4.14 .

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: where X, Y, Z ∈ Γ T M .
Definition 5.1 see 9 .A lightlike submanifold of an indefinite Sasakian manifold is contact totally umbilical if Interchanging the role of X and Y in the above equation and subtracting, we have h s X, φY − h s Y, φX F X, Y .Using the same method as shown in the proof of Theorem 5.2, we have α S 0. Which proves the theorem.Lemma 5.4.Let M 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 ∇ X φX, V g X, X and g ∇ φX X, V −g X, X .
Proof.Since ∇ is a metric connection, we have X g φX, V − g ∇ X φX, V − g φX, ∇ X V 0. 5.7 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 ∇ X φX, V g X, X .Interchanging the role of X and φX in 5.7 we get g ∇ φX X, V −g X, X .

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For X ∈ Γ D , it is easy to see that g φX, ξ 0 and φX ∈ Γ D , then g ∇ φX ξ, X −g h l φX, X , ξ .Using 5.2 we have g ∇ X ξ, φX 0. Interchanging the role of X and φX in 5.8 we get g ∇ φX ξ, X 0.
At last, we complete this paper by a classification theorem for generalized transversal lightlike submanifolds.Theorem 5.6.Let M be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian space form M c , g .Then c −3.

5.10
Replacing Y by φX in 5.10 and using Proposition 3.11 and Lemmas 5.4 and 5.5, we have g R X, Y ξ, ξ 2g X, X g φξ, ξ .

5.11
On the other hand, it follows from 5.1 that g R X, Y ξ, ξ 1 − c 2 g φX, Y g φξ, ξ .

5.12
Replacing Y 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.

Lemma 3 .1 see 4 .
Let M 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: φ Rad TM ltr TM , φ S TM S TM .3.1

Definition 3 . 3 .where
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 φ Rad TM ltr TM D is a subbundle of S TM and S TM D ⊕ {V }.
1 and P 2 and P 3 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 φP 1 ∈ Γ D , QX φP 0 ∈ Γ ltr TM and FX φP 2 ∈ Γ φ D ⊥ .Similarly, for any U ∈ Γ tr TM , we denote by U 1 , U 2 and U 3 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 φU 1 φU 2 ∈ Γ Rad TM ⊕ D ⊥ and LU φU 3 ∈ Γ μ .Let M be a lightlike submanifold of indefinite Sasakian manifolds M, g .It follows from 2.15 that ∇ X φY −φ∇ X Y 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

2
A φX Y has no components in Rad TM for any X ∈ Γ S TM and Y ∈ Γ TM .3A φξ X has no components in D for any X ∈ Γ TM and D s X, N ∈ Γ μ for any X ∈ Γ TM .

Corollary 4 . 13 .
Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g .Then D cannot define a totally geodesic foliation.Proof.It follows from Lemma 3.8 that g ∇ X Y, 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.
International Journal of Mathematics and Mathematical SciencesDefinition 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: 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 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.
⊥ ⊆ S TM ⊥ , 3.4 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 that M is a proper generalized transversal lightlike submanifold of M if D / {0} and D ⊥ / {0}. 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.
Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g .Then D ⊥ is integrable if and only if D l X, FY D l Y, FX and A FX Y A FY X for any X, Y ∈ Γ D ⊥ .
Corollary 4.2.Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g .Then D is not integrable.Proof.It follows from Lemma 3.8 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.
Let M 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 D l X, FY D l Y, FX for all X, Y ∈ Γ S TM , which is just one of the conclusions shown in 10 .Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g .Then Rad TM ⊕ {V } is integrable if and only if A φX Y A φY X and D s X, φY − D s Y, φX has no components in φ D ⊥ for any X, Y ∈ Γ Rad TM .
Corollary 4.6.Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g .Then Rad TM is not integrable.Proof.It follows from Lemma 3.8 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 ∈ Γ D ⊥ and N ∈ Γ ltr TM .Also, for Z ∈ Γ D , we have A φY X, φZ for any X, Y ∈ Γ D ⊥ ⊕Rad TM and Z ∈ Γ D , then the following corollary holds.Let M 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 A φY X has no components in D for any X, Y ∈ Γ D ⊥ ⊕ Rad TM {V } .Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g .Then D ⊕ {V } defines a totally geodesic foliation if and only if A * φN X and A φW X has no components in D for any X ∈ Γ D and W ∈ Γ D ⊥ and N ∈ Γ ltr TM .
3 h s X, φZ and h s X, ξ has no components φ D ⊥ , where X, Y, Z ∈ Γ D ⊥ and ξ ∈ Γ Rad TM .Proof.It is easy to see that D ⊥ defines a totally geodesic foliation if and only if ∇ X Y ∈ Γ D ⊥ for any X, Y ∈ Γ D ⊥ . 1 ⇔ 2 .From 2.14 and 2.15 , we have that ltr TM , and α S ∈ Γ S TM ⊥ .∈ Γ TM , then it follows from 5.4 that g X, Y , N 2g X, φY g α L , φN .On the other hand, for any Z ∈ Γ D ⊥ it follows from 2.14 , 2.15 , and 5.3 that 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.