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.

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 , where is the almost contact structure on indefinite Sasakian manifolds, is the radical distribution, and 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, , where 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.

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 of dimension immersed in a semi-Riemannian manifold of dimension is called a lightlike submanifold if the metric induced from ambient space is degenerate and its radical distribution is of rank , where and . It is well known that the radical distribution is given by , where is called normal bundle of in . Thus there exist the nondegenerate complementary distribution and of in and , respectively, which are called the screen and screen transversal distribution on , respectively. Thus we have where denotes the orthogonal direct sum.

Considering the orthogonal complementary distribution of in , it is easy to see that is a subbundle of . As is a nondegenerate subbundle of , the orthogonal complementary distribution of in is also a nondegenerate distribution. Clearly, is a subbundle of . Since for any local basis of , there exists a local null frame of sections with values in the orthogonal complement of in such that and , it follows that there exists a lightlike transversal vector bundle locally spanned by (see [3]). Then we have that . Let . We call , and the lightlike transversal vector fields, lightlike transversal vector bundle and transversal vector bundle of with respect to the chosen screen distribution and , respectively. Then is decomposed as follows: A lightlike submanifold of is said to be as follows. Case  1: -lightlike if . Case  2: coisotropic if , . Case  3: isotropic if , . Case  4: totally lightlike if , .

Let , , and denote the linear connections on , , and , respectively. Then the Gauss and Weingarten formulas are given by where and belong to and , respectively, is the shape operator of with respective to . Moreover, according to the decomposition (2.3), denoting by and the -valued and -valued lightlike second fundamental form and screen second fundamental form of , respectively, we have

Then by using (2.4), (2.6)–(2.8), and the fact that is a metric connection, we have

Let be the projection morphism of on with respect to the decomposition (2.1), then we have for any and , where and are the second fundamental form and shape operator of distribution and , respectively. Then we have the following:

It follows from (2.6) that for any . Thus the induced connection on is torsion free but is not metric, the induced connection on is metric.

Finally, we recall some basic definitions and results of indefinite Sasakian manifolds following from [4, 11]. An odd dimensional semi-Riemannian manifold of dimension is said to be with an almost contact structure if there exist a -type tensor , a vector field called the characteristic vector field and a -form such that It follows that , and . A Riemannian metric on is called an associated or compatible metric of an almost constant structure of if 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 . It is known that is a normal contact structure if , where is the Nijenhuis tensor field of . 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 It follows from (2.13) and (2.15) that

It is well known that can be regarded as a Sasakian manifold. Denoting by the manifold with its usual Sasakian structure given by where are the Cartesian coordinates in .

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]). Let be a lightlike submanifold of an indefinite almost contact metric manifold . If is tangent to , then the structure vector field does not belong to .

Definition 3.2 (see [10]). Let be a lightlike submanifold, tangent to the structure vector field , immersed in an indefinite Sasakian manifold . We say that is a radical transversal lightlike submanifold of if the following conditions are satisfied:
It follows from Lemma 3.1 and (2.1) that the structure vector field . Suppose that , then we have such that . since (2.13) implies that , using (2.13), we have . Substituting into , we have . So it is impossible for in Definition 3.2. Thus, we modify the above definition as the following one.

Definition 3.3. Let be a lightlike submanifold, tangent to the structure vector field , immersed in an indefinite Sasakian manifold . is said to be a radical transversal lightlike submanifold of if there exists an invariant nondegenerate vector subbundle such where is a subbundle of and .

Definition 3.4 (see [10]). Let be a lightlike submanifold, tangent to the structure vector field , immersed in an indefinite Sasakian manifold . We say that is a transversal lightlike submanifold of if the following conditions are satisfied: From Definitions 3.3 and 3.4, we introduce generalized lightlike submanifold as follows.

Definition 3.5. Let be a lightlike submanifold, tangent to the structure vector field , immersed in an indefinite Sasakian manifold . is said to be a generalized transversal lightlike submanifold of if there exist vector subbundle and such where and are nondegenerate subbundles of and .

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 or , respectively.
We say that is a proper generalized transversal lightlike submanifold of if and . We denote by the orthogonal complement of in . 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, . From (3.4), we know that there exists no a generalized transversal lightlike hypersurface of as .(2)Since (3.4) implies that , we have if is a proper generalized transversal lightlike submanifold of . It follows that any 5-dimensional generalized transversal lightlike submanifold of must be a 2-lightlike submanifold.(3) if is a proper generalized transversal lightlike submanifold of an indefinite Sasakian manifold .
In this paper, we assume that the characteristic vector field is a spacelike vector filed, that is, . If 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. If is isotropic or totally lightlike submanifolds of , we have as , which is a contradiction with definition of proper generalized transversal lightlike submanifolds. Similarly, if is coisotropic or totally lightlike submanifolds of , we have as , there is a contradiction, which proves the assertion.

Lemma 3.8. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds , then we have where .

Proof. Noticing the fact that on 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 be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds , then is an invariant distribution with respect to .

Proof. For any , and , it follows from (2.14) that , which implies that has no components in . Similarly, from (3.4) and (2.14), we have , which implies that has no components in . For any , we have as . Thus, has no components in . Finally, suppose that , where is a smooth function on , then we get by replacing in (2.13) by . Thus, we have , which completes the proof.

Next, we give a characterization theorem for generalized transversal lightlike submanifolds.

Theorem 3.10. Let be a lightlike submanifold of indefinite Sasakian space form , . Then a generalized transversal lightlike submanifolds of if and only if(1)the maximal invariant subspaces of define a nondegenerate distribution with respect to ;(2), for all and for all .

Proof. Suppose that is a lightlike submanifold of indefinite Sasakian space form . It follows from (5.1) that for any and . Noticing that implies , then (2) is satisfied. Also, (1) holds naturally by using the definition of generalized transversal lightlike submanifolds.
Conversely, since is a nondegenerate distribution, we may choose such that . Thus, from (3.7), we have for any , which implies that have components in . For any , implies that have no components in . Noticing condition (1), we also have and , respectively. Thus, we get , which completes the proof.

For any , we denote by , and and are the projection morphisms of on , and and , respectively. Let , where , , and are the components of on , and , respectively. Moreover, we have , and .

Similarly, for any , we denote by , and are the components of on , and , respectively. Using Theorem 3.9, then the components of on and are denoted by and .

Let be a lightlike submanifold of indefinite Sasakian manifolds . It follows from (2.15) that . Substituting and into the above equation and taking the tangential, screen transversal, and lightlike transversal parts, respectively, yield that where .

Proposition 3.11. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds . Then we have where is the characteristic vector filed.

Proof. Replacing by in (3.8), it follows from and (3.8) that Similarly, replacing by in (3.9) and (3.10), respectively, we get and . Noticing that for any and for any , we have .
Let , since and , we have and , which proves (3.11).
Let , from (3.14), we have and , which proves (3.12).
Let , it follows from (3.14) that and , which proves (3.13).

We know from (2.12) that the induced connection on 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 be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then the following assertions are equivalent.(1)The induced connection on is a metric connection.(2) has no components in for any and .(3) has no components in for any and for any .

Proof. From (2.13) and (2.15), we have which implies that has not component in for any .
. For and , it follows from (2.6), (2.14), (2.15), and the fact that is metric connection, we have For , Together with (2.8), (2.14), and (2.15), we have 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 Thus, the proof follows from the above equations.

4. Integrability and Geodesic of Distributions

Theorem 4.1. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then is integrable if and only if for any .

Proof. Let , we have and . Thus, it follows from (3.9) that . Interchanging the roles of and in the above equation and subtracting, we have Similarly, it follows from (3.10) that . Noticing that is symmetric and interchanging the roles of and in the above equation and subtracting, we have Thus, our assertion follows from (4.1) and (4.2).

Corollary 4.2. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then is not integrable.

Proof. It follows from Lemma 3.8 that for any . Suppose that is integrable, then we have , which is a contradiction to the fact that is a nondegenerate distribution of .

Theorem 4.3. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then is integrable if and only if and for any .

Proof. For , we have and . Thus, it follows from (3.8) that . As and are symmetric, by interchanging the roles of and in the above equation and subtracting the resulting equations, we have Similarly, it follows from (3.9) that . Interchanging the roles of and in the above equation and subtracting the resulting equations, we have Noticing Lemma 3.8, we have for any . Thus, our assertion follows from (4.3) and (4.4).

Remark 4.4. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . If , then is a transversal lightlike submanifolds [9]. It follows from Theorem 4.3 that is integrable if and only if for all , which is just one of the conclusions shown in [10].

Theorem 4.5. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then is integrable if and only if and has no components in for any .

Proof. Together with (2.14), (2.15), and the fact that is a metric connection, we have where . Similarly, for , we have Thus, our assertion follows from (4.5) and (4.6).

Corollary 4.6. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then is not integrable.

Proof. It follows from Lemma 3.8 that holds for any . Suppose that is integrable, then we have . On the other hand, for any , there must exists such that as , 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 be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then,(1) is integrable if and only if has no components in for any .(2) is integrable if and only if has no components in for any .(3) is not integrable.(4) is not integrable.

We know from [3] that a distribution on is said to define a totally geodesic foliation if any leaf of is geodesic. we focus on this property of generalized transversal lightlike submanifolds in the following of this section.

Theorem 4.8. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then the screen distribution defines a totally geodesic foliation if and only if for any .

Proof. It is known that defines a totally geodesic foliation if and only if for any . For , we have . Also, we have Thus we complete the proof.

Remark 4.9. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . If , then is radical transversal lightlike submanifolds. For , we have . Thus, it follows from Theorem 4.8 that defines a totally geodesic foliation if and only if has no components in , which is just the Theorem 3.6 proved in [10].

Theorem 4.10. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then, The following assertions are equivalent.(1) defines a totally geodesic foliation.(2) and has no components in .(3) and has no components , where and .

Proof. It is easy to see that defines a totally geodesic foliation if and only if for any .
. From (2.14) and (2.15), we have that where and . Also, for , we have Noticing Lemma 3.8, we have for all . 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 Thus, the equivalence between and follows from (4.10).

From (4.9), we know that for any and , then the following corollary holds.

Corollary 4.11. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then defines a totally geodesic foliation if and only if has no components in for any .

Theorem 4.12. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then defines a totally geodesic foliation if and only if and has no components in for any and and .

Proof. Noticing that is a metric connection and (2.14) and (2.15), we have Similarly, for , we have Then the proof follows from (4.11) and (4.12) and Lemma 3.8.

It follows from (2.13) that , then (4.11) and (4.12) hold for any . Thus it is easy to get the following corollary.

Corollary 4.13. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then cannot define a totally geodesic foliation.

Proof. It follows from Lemma 3.8 that for any . Suppose that defines a totally geodesic foliation, then we have . Which is a contradiction to the fact that is a nondegenerate distribution of .

We say that is a contact generalized transversal lightlike product manifold if and define totally geodesic foliations in .

Theorem 4.14. Let be a generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then, is a contact generalized transversal lightlike product manifold if and only if(1) for any ;(2) has no components in for any .

Proof. For , we have and as . It follows from (3.9) and (3.10), respectively, that On the other hand, for , we have . Then it follows from (3.8) that Noticing that belongs to or if and only if or , 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 is called a -section if it is spanned by a unit vector orthogonal to and , where is a non-null vector field on . The sectional curvature of a -section is called a -sectional curvature. If has a -sectional curvature which is not depend on the -section at each point, then is a constant and is called a Sasakian space form, denoted by . The curvature tensor of a Sasakian space form is given in [13] as follows: where .

Definition 5.1 (see [9]). A lightlike submanifold of an indefinite Sasakian manifold is contact totally umbilical if where , , and .

Theorem 5.2. Let be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then if and only if is integrable.

Proof. Let , it follows from (2.15) and (5.2) that
Noticing that (2.15) implies for any , then it follows from (5.4) that . On the other hand, for any it follows from (2.14), (2.15), and (5.3) that which means that have components in for any . Noticing Lemma 3.8, we know that has no components in for any . Thus, the proof is complete.

Theorem 5.3. Let be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then .

Proof. Let , from (5.3), we have , which implies . Also, it follows from (3.10) that Interchanging the role of and in the above equation and subtracting, we have . Using the same method as shown in the proof of Theorem 5.2, we have . Which proves the theorem.

Lemma 5.4. Let be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then, for any , we have and .

Proof. Since is a metric connection, we have Noticing (2.16) and the fact that implies and , then from (5.7) and (2.14), we have . Interchanging the role of and in (5.7) we get .

Lemma 5.5. Let be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold . Then for any , one have and .

Proof. Since is a metric connection, we have For , it is easy to see that and , then . Using (5.2) we have . Interchanging the role of and in (5.8) we get .

At last, we complete this paper by a classification theorem for generalized transversal lightlike submanifolds.

Theorem 5.6. Let be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian space form . Then .

Proof. For any , it follows from (2.6)–(2.8) that where and .
Noticing Theorem 5.3 then we have for any . Let and , replacing by in (5.9) and using (5.2), we have Replacing by in (5.10) and using Proposition 3.11 and Lemmas 5.4 and 5.5, we have
On the other hand, it follows from (5.1) that
Replacing by in (5.12) and using (5.11), we have that for all . As is nondegenerate, so we may choose such that in the above equation. Thus, we complete the proof.

Acknowledgments

The authors would like to thank the referee for his or her valuable suggestions and comments on the improvement of this paper. This work is supported by NSFC (no. 10931005) and Natural Science Foundation of Guangdong Province of China (no. S2011010000471).