#### Abstract

In this paper, we prove that there does not exist a warped product CR-lightlike submanifold in the form other than CR-lightlike product in an indefinite Kaehler manifold. We also obtain some characterizations for a CR-lightlike submanifold to be locally a CR-lightlike warped product.

#### 1. Introduction

The general theory of Cauchy-Riemann (CR-) submanifolds of Kaehler manifolds, being generalization of holomorphic and totally real submanifolds of Kaehler manifolds, was initiated in Bejancu [1] and has been further developed in [2–4] and others. Later on, Duggal and Bejancu [5] introduced a new class called CR-lightlike submanifolds of indefinite Kaehler manifolds. A special class of CR-lightlike submanifolds is the class of CR-lightlike product submanifolds. Duggal and Bejancu [5] and Kumar et al. [6] characterized a CR-lightlike submanifold to be a CR-lightlike product. In [7], the notion of warped product manifolds was introduced by Bishop and O’ Neill in 1969 and it was studied by many mathematicians and physicists. These manifolds are generalization of Riemannian product manifolds. This generalized product metric appears in differential geometric studies in a natural way. For instance, a surface of revolution is a warped product manifold. Moreover, many important submanifolds in real and complex space forms are expressed as warped product submanifolds. In view of its physical applications, many research articles have recently appeared exploring existence (or nonexistence) of warped product submanifolds in known spaces (cf. [8, 9], etc.). Chen [10] introduced warped product CR-submanifolds and showed that there does not exist a warped product CR-submanifold in the form in a Kaehler manifold where is a totally real submanifold and is a holomorphic submanifold of . He proved if is a warped product CR-submanifold of a Kaehler manifold , then is a CR-product, that is, there do not exist warped product CR-submanifolds of the form other than CR-product. Therefore, he called a warped product CR-submanifold in the form a CR-warped product. Chen also obtained a characterization for CR-submanifold of a Kaehler manifold to be locally a warped product submanifold. He showed that a CR-submanifold of a Kaehler manifold is a CR-warped product if and only if for each , , a -function on such that for all .

The growing importance of lightlike submanifolds and hypersurfaces in mathematical physics, especially in relativity, motivated us to club the concept of CR-warped product with lightlike geometry. In this paper, we showed that there does not exist a warped product CR-lightlike submanifold in the form other than CR-lightlike product in an indefinite Kaehler manifold. We also obtained some characterizations for a CR-lightlike submanifold to be locally a CR-lightlike warped product.

#### 2. Lightlike Submanifolds

We recall notations and fundamental equations for lightlike submanifolds, which are due to [5] by Duggal and Bejancu.

Let be a real -dimensional semi-Riemannian manifold of constant index such that , and let be an -dimensional submanifold of and the induced metric of on . If is degenerate on the tangent bundle of , then is called a lightlike submanifold of . For a degenerate metric on , is a degenerate -dimensional subspace of . Thus, both and are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace which is known as radical (null) subspace. If the mapping defines a smooth distribution on of rank , then the submanifold of is called -lightlike submanifold and is called the radical distribution on .

Let be a screen distribution which is a semi-Riemannian complementary distribution of in , that is, is a complementary vector subbundle to in . Let and be complementary (but not orthogonal) vector bundles to in and to in , respectively. Then, we have Let be a local coordinate neighborhood of and consider the local quasiorthonormal fields of frames of along , on as , where , are local lightlike bases of , and are local orthonormal bases of and , respectively. For this quasiorthonormal fields of frames, we have the following theorem.

Theorem 2.1 (see [5]). * Let be an -lightlike submanifold of a semi-Riemannian manifold . Then, there exists a complementary vector bundle of in and a basis of consisting of smooth section of , where is a coordinate neighborhood of , such that
**
where is a lightlike basis of .*

Let be the Levi-Civita connection on . Then, according to the decomposition (2.5), the Gauss and Weingarten formulas are given by where and belong to and , respectively. Here, is a torsion-free linear connection on , is a symmetric bilinear form on which is called second fundamental form, and is a linear operator on and known as shape operator.

According to (2.4), considering the projection morphisms and of on and , respectively, (2.7) and (2.8) give where we put , , , .

As and are -valued and -valued, respectively, therefore, they are called the lightlike second fundamental form and the screen second fundamental form on . In particular, where , , and .

Using (2.4)-(2.5) and (2.9)–(2.12), we obtain for any , , and .

Let be a projection of on . Now, we consider the decomposition (2.3), we can write for any , and , where and belong to and , respectively. Here and are linear connections on and , respectively. By using (2.9)-(2.10) and (2.16), we obtain

*Definition 2.2. *Let be a real -dimensional indefinite Kaehler manifold and let be an -dimensional submanifold of . Then is said to be a CR-lightlike submanifold if the following two conditions are fulfilled: (a) is distribution on such that
(b)there exist vector bundles , , , and over , such that
where is a nondegenerate distribution on , and are vector subbundles of and , respectively, and assume that and .

Clearly, the tangent bundle of a CR-lightlike submanifold is decomposed as where

Now, let and be the projections on and , respectively. Then, for any , we can write where and . Applying to above equation, we get where and . Clearly is a tensor field of type and is -valued 1-form on . Clearly, if and only if . On the other hand, we set for any , where and are sections of and , respectively.

By using Kaehlerian property of with (2.7) and (2.8), we have the following lemmas.

Lemma 2.3. *Let be a CR-lightlike submanifold of an indefinite Kaehler manifold then, one has
**
for any , where
*

Lemma 2.4. *Let be a CR-lightlike submanifold of an indefinite Kaehler manifold then, one has
**
for any and , where
*

Theorem 2.5 (see [5]). *Let be a CR-lightlike submanifold of an indefinite Kaehler manifold . Then, one has the following assertions. *(i)*The almost complex distribution is integrable if and only if the second fundamental form of satisfies
*(ii)*The totally real distribution is integrable if and only if the shape operator of satisfies
*

Theorem 2.6 (see [5]). *Let be a CR-lightlike submanifold of an indefinite Kaehler manifold . Then, defines a totally geodesic foliation on if and only if, for any , has no component in .*

#### 3. CR-Lightlike Warped Product

*Warped Product*

Let and be two Riemannian manifolds with Riemannian metrics and , respectively, and a differentiable function on . Assume the product manifold with its projection and . The warped product is the manifold equipped with the Riemannian metric , where
If is tangent to at , then using (3.1), we have
The function is called the warping function of the warped product. For differentiable function on *M*, the gradient is defined by , for all .

Lemma 3.1 (see [7]). *Let be a warped product manifold. If and , then
*

Corollary 3.2. *On a warped product manifold one has*(i)* is totally geodesic in , *(ii)* is totally umbilical in . *

*Definition 3.3 (see [11]). *A lightlike submanifold of a semi-Riemannian manifold is said to be totally umbilical in if there is a smooth transversal vector field on , called the transversal curvature vector field of , such that
it is easy to see that is a totally umbilical if and only if on each coordinate neighborhood , there exist smooth vector fields and , such that
for any .

Lemma 3.4. *Let be a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold then, the distribution defines a totally geodesic foliation in .*

*Proof. *Let , then (2.25) and (2.27) imply that . Let , then
where, . Since and then (2.26) and (2.28) imply that , this implies that , then (3.8) implies that , then the nondegeneracy of the distribution implies that gives for any . Hence, the proof is complete.

Theorem 3.5. *Let be a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold, then the totally real distribution is integrable. *

*Proof. *Using (2.25) and (2.27) with the above lemma, for any , we get
this implies and also
therefore, using (3.9) and (3.10), we get , for any . This implies that the distribution is integrable.

*Definition 3.6 (see [5]). *A CR-lightlike submanifold of an indefinite Kaehler manifold is called a CR-lightlike product if both the distribution and define totally geodesic foliations in .

Theorem 3.7. *Let be a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold . If be a warped product CR-lightlike submanifold, then it is a CR-lightlike product.*

*Proof. *Since is a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold, then using Lemma 3.4, the distribution defines a totally geodesic foliation in .

Let and be the second fundamental form and the shape operator of in , then for and , we have . Using (3.4), we get
Now, let be the second fundamental form of in , then
for any tangent to , then using (3.11), we get
Since is a holomorphic submanifold of , then we have , therefore, we have
Adding (3.13) and (3.14), we get
Using (3.12), we have . Thus, implies that has no components in for any . This implies that the distribution defines a totally geodesic foliation in . Hence, is a CR-lightlike product.

Theorem 3.7 shows that if is a warped product CR-lightlike submanifold of an indefinite Kaehler manifold, then it is CR-lightlike product, that is, there does not exist warped product CR-lightlike submanifolds of the form other than CR-lightlike product. Thus, for simplicity, we call a warped product CR-lightlike submanifold in the form a CR-lightlike warped product.

Lemma 3.8. *Let be a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold . Let be a proper CR-lightlike warped product of an indefinite Kaehler manifold, then is totally geodesic in .*

*Proof . *Let be a linear connection on induced from . Let and , then we have , using (3.4), we get . Since is totally umbilical CR-lightlike submanifold, therefore, . Hence, implies that is totally geodesic in .

Theorem 3.9 (see[6]). *Let be a CR-lightlike submanifold of an indefinite Kaehler manifold . Then distribution defines totally geodesic foliation if and only if is integrable.*

Theorem 3.10. *Let be a totally umbilical proper CR-lightlike submanifold of an indefinite Kaehler manifold , then .*

*Proof. *Let be a totally umbilical proper CR-lightlike submanifold then using (2.25) and (2.27), we have , for . We obtain . Using (2.13) and the hypothesis we obtain , then using the non degeneracy of , the result follows.

#### 4. A Characterization of CR-Lightlike Warped Products

For a CR-lightlike warped products in indefinite Kaehler manifolds, we have

Lemma 4.1. *Let be a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold , then for a CR-lightlike warped product in an indefinite Kaehler manifold , one has *(i)*, *(ii)*, **for any and .*

*Proof. *Since is Kaehlerian, therefore, for and , we have , since is totally umbilical, therefore, we have , then taking inner product with , where , we get . Using (3.4), we obtain , then using (2.13), we get .

Next for any and , we have . Hence, the proof is complete.

Corollary 4.2. *Let , then clearly and also for any . Thus, , this implies that has no component in , therefore, using Theorem 2.5, the distribution defines a totally geodesic foliation in .*

We have the following some characterizations of CR-lightlike warped product.

Theorem 4.3. *A proper totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold is locally a CR-lightlike warped product if and only if
**
for , and for some function on satisfying .*

*Proof. *Assume that be a proper CR-lightlike submanifold of an indefinite Kaehler manifold satisfying (4.1). Let and , we have , then using (2.13), we get . If , then clearly and also for any . Thus,
that is, has no component in , this implies that the distribution defines totally geodesic foliation in and consequently it is totally geodesic in and using Theorem 3.9, the distribution is integrable.

Taking inner product of (4.1) with and using that is totally umbilical, we get , using the definition of gradient , we get
Let be the second fundamental form of in and let be the metric connection of in then, particularly for , we have
Therefore, from (4.3) and (4.4), we get , this further implies that
this implies that the distribution is totally umbilical in . Using Theorem 3.5, the totally real distribution is integrable and (4.5) and the condition for imply that each leaf of is an extrinsic sphere in . Hence, by a result of [12] which say that “if the tangent bundle of a Riemannian manifold splits into an orthogonal sum of nontrivial vector subbundles such that is spherical and its orthogonal complement is autoparallel, then the manifold is locally isometric to a warped product ,” therefore, we can conclude that is locally a CR-lightlike warped product of , where .

Conversely, let and , since is a Kaehler manifold so, we have , which further becomes , comparing tangential components, we get for each and . Since is a function on , so we also have for all . Hence, the proof is complete.

Lemma 4.4. *Let be a CR-lightlike warped product of an indefinite Kaehler manifold , then
**
for any , and .*

*Proof. *For and , using (2.27) and (3.4), we have . Next, again using (2.27), we get , this implies that , therefore, for any , we have
Hence, using the definition of gradient of and the nondegeneracy of the distribution , the result follows.

Theorem 4.5. *A proper totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold is locally a CR-lightlike warped product if and only if
**
for any and for some function on satisfying .*

*Proof. *Let be a CR-lightlike submanifold of an indefinite Kaehler manifold satisfying (4.8). Let , then (4.8) implies that , then (2.25) gives . Thus defines a totally geodesic foliation in and consequently it is totally geodesic in and integrable using Theorem 3.9.

Let , then (4.8) gives
Let , then (4.9) implies that
Also
therefore, from (4.10) and (4.11), we get
Let be the second fundamental form of in and let be the metric connection of in , then
therefore, from (4.12) and (4.13), we get , then the nondegeneracy of the distribution implies that
this gives that the distribution is totally umbilical in and using Theorem 3.5, the distribution is integrable. Also, for , hence as in Theorem 4.3, each leaf of is an extrinsic sphere in . Thus, is locally a CR-lightlike warped product of , where .

Conversely, let be a CR-lightlike warped product of an indefinite Kaehler manifold . Using (2.22), we can write
Since defines totally geodesic foliation in , therefore, using (2.25), we get
Using (4.6), we have
Hence, from (4.15)–(4.18), the result follows.

Theorem 4.6. *Let be a locally CR-lightlike warped product of an indefinite Kaehler manifold , then
**
for any and for some function on satisfying .*

*Proof. *Let be a CR-lightlike warped product of an indefinite Kaehler manifold . Therefore, the distribution defines totally geodesic foliation in , then using (2.25) for , we get
For or , using (2.25), we have
Now let and , then using (3.4), we have
Therefore, (4.19) follows from (4.21)–(4.22). Hence, the result is complete.