Abstract

In this paper, we prove that there does not exist a warped product CR-lightlike submanifold in the form 𝑀=𝑁⟂×𝜆𝑁𝑇 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 𝑀=𝑁⟂×𝜆𝑁𝑇 in a Kaehler manifold where 𝑁⟂ is a totally real submanifold and 𝑁𝑇 is a holomorphic submanifold of 𝑀. He proved if 𝑀=𝑁⟂×𝜆𝑁𝑇 is a warped product CR-submanifold of a Kaehler manifold 𝑀, then 𝑀 is a CR-product, that is, there do not exist warped product CR-submanifolds of the form 𝑀=𝑁⟂×𝜆𝑁𝑇 other than CR-product. Therefore, he called a warped product CR-submanifold in the form 𝑀=𝑁𝑇×𝜆𝑁⟂ 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 𝑀 of a Kaehler manifold 𝑀 is a CR-warped product if and only if 𝐴𝐽𝑍𝑋=𝐽𝑋(𝜇)𝑍 for each 𝑋∈Γ(𝐷), 𝑍∈Γ(ğ·î…ž), 𝜇 a ğ¶âˆž-function on 𝑀 such that 𝑍𝜇=0 for all 𝑍∈Γ(ğ·î…ž).

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 𝑀=𝑁⟂×𝜆𝑁𝑇 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 (𝑀,𝑔) be a real (𝑚+𝑛)-dimensional semi-Riemannian manifold of constant index ğ‘ž such that 𝑚,𝑛≥1, 1â‰¤ğ‘žâ‰¤ğ‘š+𝑛−1 and let (𝑀,𝑔) be an 𝑚-dimensional submanifold of 𝑀 and 𝑔 the induced metric of 𝑔 on 𝑀. If 𝑔 is degenerate on the tangent bundle 𝑇𝑀 of 𝑀, then 𝑀 is called a lightlike submanifold of 𝑀. For a degenerate metric 𝑔 on 𝑀,𝑇𝑀⟂=∪𝑢∈𝑇𝑥𝑀∶𝑔(𝑢,𝑣)=0,∀𝑣∈𝑇𝑥𝑀,𝑥∈𝑀,(2.1) is a degenerate 𝑛-dimensional subspace of 𝑇𝑥𝑀. Thus, both 𝑇𝑥𝑀 and 𝑇𝑥𝑀⟂ are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace Rad𝑇𝑥𝑀=𝑇𝑥𝑀∩𝑇𝑥𝑀⟂ which is known as radical (null) subspace. If the mappingRad𝑇𝑀∶𝑥∈𝑀⟶Rad𝑇𝑥𝑀(2.2) defines a smooth distribution on 𝑀 of rank 𝑟>0, then the submanifold 𝑀 of 𝑀 is called 𝑟-lightlike submanifold and Rad𝑇𝑀 is called the radical distribution on 𝑀.

Let 𝑆(𝑇𝑀) be a screen distribution which is a semi-Riemannian complementary distribution of Rad(𝑇𝑀) in 𝑇𝑀, that is,𝑇𝑀=Rad𝑇𝑀⟂𝑆(𝑇𝑀),(2.3)𝑆(𝑇𝑀⟂) is a complementary vector subbundle to Rad𝑇𝑀 in 𝑇𝑀⟂. Let tr(𝑇𝑀) and ltr(𝑇𝑀) be complementary (but not orthogonal) vector bundles to 𝑇𝑀 in 𝑇𝑀∣𝑀 and to Rad𝑇𝑀 in 𝑆(𝑇𝑀⟂)⟂, respectively. Then, we havetr(𝑇𝑀)=ltr(𝑇𝑀)⟂𝑆𝑇𝑀⟂𝑇,(2.4)𝑀∣𝑀=𝑇𝑀⊕tr(𝑇𝑀)=(Rad𝑇𝑀⊕ltr(𝑇𝑀))⟂𝑆(𝑇𝑀)⟂𝑆𝑇𝑀⟂.(2.5) Let 𝑢 be a local coordinate neighborhood of 𝑀 and consider the local quasiorthonormal fields of frames of 𝑀 along 𝑀, on 𝑢 as {𝜉1,…,𝜉𝑟,𝑊𝑟+1,…,𝑊𝑛,𝑁1,…,𝑁𝑟,𝑋𝑟+1,…,𝑋𝑚}, where {𝜉1,…,𝜉𝑟}, {𝑁1,…,𝑁𝑟} are local lightlike bases of Γ(Rad𝑇𝑀∣𝑢), Γ(ltr(𝑇𝑀)∣𝑢) and {𝑊𝑟+1,…,𝑊𝑛},{𝑋𝑟+1,…,𝑋𝑚} are local orthonormal bases of Γ(𝑆(𝑇𝑀⟂)∣𝑢) and Γ(𝑆(𝑇𝑀)∣𝑢), respectively. For this quasiorthonormal fields of frames, we have the following theorem.

Theorem 2.1 (see [5]). Let (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(𝑇𝑀⟂)) be an 𝑟-lightlike submanifold of a semi-Riemannian manifold (𝑀,𝑔). Then, there exists a complementary vector bundle ltr(𝑇𝑀) of Rad𝑇𝑀 in 𝑆(𝑇𝑀⟂)⟂ and a basis of Γ(ltr(𝑇𝑀)∣𝑢) consisting of smooth section {𝑁𝑖} of 𝑆(𝑇𝑀⟂)⟂∣𝑢, where u is a coordinate neighborhood of 𝑀, such that 𝑔𝑁𝑖,𝜉𝑗=𝛿𝑖𝑗,𝑔𝑁𝑖,𝑁𝑗=0,(2.6) where {𝜉1,…,𝜉r} is a lightlike basis of Γ(Rad(𝑇𝑀)).

Let ∇ be the Levi-Civita connection on 𝑀. Then, according to the decomposition (2.5), the Gauss and Weingarten formulas are given by∇𝑋𝑌=∇𝑋𝑌+ℎ(𝑋,𝑌),∀𝑋,𝑌∈Γ(𝑇𝑀),(2.7)∇𝑋𝑈=−𝐴𝑈𝑋+∇⟂𝑋𝑈,∀𝑋∈Γ(𝑇𝑀),𝑈∈Γ(tr(𝑇𝑀)),(2.8) where {∇𝑋𝑌,𝐴𝑈𝑋} and {ℎ(𝑋,𝑌),∇⟂𝑋𝑈} belong to Γ(𝑇𝑀) and Γ(tr(𝑇𝑀)), respectively. Here, ∇ is a torsion-free linear connection on 𝑀, ℎ is a symmetric bilinear form on Γ(𝑇𝑀) which is called second fundamental form, and 𝐴𝑈 is a linear operator on 𝑀 and known as shape operator.

According to (2.4), considering the projection morphisms 𝐿 and 𝑆 of tr(𝑇𝑀) on ltr(𝑇𝑀) and 𝑆(𝑇𝑀⟂), respectively, (2.7) and (2.8) give∇𝑋𝑌=∇𝑋𝑌+â„Žğ‘™(𝑋,𝑌)+â„Žğ‘ (𝑋,𝑌),(2.9)∇𝑋𝑈=−𝐴𝑈𝑋+𝐷𝑙𝑋𝑈+𝐷𝑠𝑋𝑈,(2.10) where we put â„Žğ‘™(𝑋,𝑌)=𝐿(ℎ(𝑋,𝑌)), â„Žğ‘ (𝑋,𝑌)=𝑆(ℎ(𝑋,𝑌)), 𝐷𝑙𝑋𝑈=𝐿(∇⟂𝑋𝑈), 𝐷𝑠𝑋𝑈=𝑆(∇⟂𝑋𝑈).

As â„Žğ‘™ and â„Žğ‘  are Γ(ltr(𝑇𝑀))-valued and Γ(𝑆(𝑇𝑀⟂))-valued, respectively, therefore, they are called the lightlike second fundamental form and the screen second fundamental form on 𝑀. In particular, ∇𝑋𝑁=−𝐴𝑁𝑋+∇𝑙𝑋𝑁+𝐷𝑠(𝑋,𝑁),(2.11)∇𝑋𝑊=−𝐴𝑊𝑋+∇𝑠𝑋𝑊+𝐷𝑙(𝑋,𝑊),(2.12) where 𝑋∈Γ(𝑇𝑀), 𝑁∈Γ(ltr(𝑇𝑀)), and 𝑊∈Γ(𝑆(𝑇𝑀⟂)).

Using (2.4)-(2.5) and (2.9)–(2.12), we obtainğ‘”î€·â„Žğ‘ î€¸+(𝑋,𝑌),𝑊𝑔𝑌,𝐷𝑙𝐴(𝑋,𝑊)=𝑔𝑊,𝑋,𝑌(2.13)ğ‘”î€·â„Žğ‘™î€¸+(𝑋,𝑌),𝜉𝑔𝑌,â„Žğ‘™î€¸î€·(𝑋,𝜉)+𝑔𝑌,∇𝑋𝜉=0,(2.14)𝑔𝐴𝑁𝑋,ğ‘î…žî€¸+𝑔𝑁,𝐴𝑁′𝑋=0,(2.15) for any 𝜉∈Γ(Rad𝑇𝑀), 𝑊∈Γ(𝑆(𝑇𝑀⟂)), and 𝑁,ğ‘î…žâˆˆÎ“(ltr(𝑇𝑀)).

Let 𝑃 be a projection of 𝑇𝑀 on 𝑆(𝑇𝑀). Now, we consider the decomposition (2.3), we can write∇𝑋𝑃𝑌=∇∗𝑋𝑃𝑌+â„Žâˆ—î‚€ğ‘‹,,∇𝑃𝑌𝑋𝜉=−𝐴∗𝜉𝑋+∇𝑋∗𝑡𝜉,(2.16) for any 𝑋,𝑌∈Γ(𝑇𝑀), and 𝜉∈Γ(Rad𝑇𝑀), where {∇∗𝑋𝑃𝑌,𝐴∗𝜉𝑋} and {ℎ∗(𝑋,𝑃𝑌),∇𝑋∗𝑡𝜉} belong to Γ(𝑆(𝑇𝑀)) and Γ(Rad𝑇𝑀), respectively. Here ∇∗ and ∇𝑋∗𝑡 are linear connections on 𝑆(𝑇𝑀) and Rad𝑇𝑀, respectively. By using (2.9)-(2.10) and (2.16), we obtainğ‘”î‚€â„Žğ‘™î‚€ğ‘‹,𝐴𝑃𝑌,𝜉=𝑔∗𝜉𝑋,,ğ‘ƒğ‘Œğ‘”î‚€â„Žâˆ—î‚€ğ‘‹,=𝑃𝑌,𝑁𝑔𝐴𝑁𝑋,.𝑃𝑌(2.17)

Definition 2.2. Let (𝑀,𝐽,𝑔) be a real 2𝑚-dimensional indefinite Kaehler manifold and let 𝑀 be an 𝑛-dimensional submanifold of 𝑀. Then 𝑀 is said to be a CR-lightlike submanifold if the following two conditions are fulfilled: (a)𝐽(Rad𝑇𝑀) is distribution on 𝑀 such that Rad𝑇𝑀∩𝐽(Rad𝑇𝑀)=0;(2.18)(b)there exist vector bundles 𝑆(T𝑀), 𝑆(𝑇𝑀⟂), ltr(𝑇𝑀), 𝐷0 and ğ·î…ž over 𝑀, such that 𝑆𝐽(𝑇𝑀)=(Rad𝑇𝑀)âŠ•ğ·î…žî€¾âŸ‚ğ·0𝐷;𝐽0=𝐷0𝐷;ğ½î…žî€¸=𝐿1⟂𝐿2,(2.19)where Γ(𝐷0) is a nondegenerate distribution on 𝑀, Γ(𝐿1) and Γ(𝐿2) are vector subbundles of Γ(ltr(𝑇𝑀)) and Γ(𝑆(𝑇𝑀⟂)), respectively, and assume that 𝑀1=𝐽(𝐿1) and 𝑀2=𝐽(𝐿2).

Clearly, the tangent bundle of a CR-lightlike submanifold is decomposed as𝑇𝑀=ğ·âŠ•ğ·î…ž,(2.20) where𝐷=Rad𝑇𝑀⟂𝐽(Rad𝑇𝑀)⟂𝐷0.(2.21)

Now, let 𝑆 and 𝑄 be the projections on 𝐷 and ğ·î…ž, respectively. Then, for any 𝑋∈Γ(𝑇𝑀), we can write𝑋=𝑆𝑋+𝑄𝑋,(2.22) where 𝑆𝑋∈𝐷 and ğ‘„ğ‘‹âˆˆğ·î…ž. Applying 𝐽 to above equation, we get𝐽𝑋=𝑓𝑋+𝑤𝑋,(2.23) where 𝑓𝑋=𝐽𝑆𝑋 and 𝑤𝑋=𝐽𝑄𝑋. Clearly 𝑓 is a tensor field of type (1,1) and 𝑤 is Γ(𝐿1⟂𝐿2)-valued 1-form on 𝑀. Clearly, 𝑋∈Γ(𝐷) if and only if 𝑤𝑋=0. On the other hand, we set𝐽𝑉=𝐵𝑉+𝐶𝑉,(2.24) for any 𝑉∈Γ(tr(𝑇𝑀)), where 𝐵𝑉 and 𝐶𝑉 are sections of 𝑇𝑀 and tr(𝑇𝑀), respectively.

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

Lemma 2.3. Let 𝑀 be a CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀 then, one has ∇𝑋𝑓𝑌=𝐴𝑤𝑌∇𝑋+ğµâ„Ž(𝑋,𝑌),(2.25)𝑡𝑋𝑤𝑌=ğ¶â„Ž(𝑋,𝑌)−ℎ(𝑋,𝑓𝑌),(2.26) for any 𝑋,𝑌∈Γ(𝑇𝑀), where ∇𝑋𝑓𝑌=∇𝑋∇𝑓𝑌−𝑓𝑋𝑌∇,(2.27)𝑡𝑋𝑤𝑌=∇𝑡𝑋∇𝑤𝑌−𝑤𝑋𝑌.(2.28)

Lemma 2.4. Let 𝑀 be a CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀 then, one has ∇𝑋𝐵𝑉=−𝑓𝐴𝑉𝑋+𝐴𝐶𝑉∇𝑋,𝑋𝐶𝑉=âˆ’ğ‘¤ğ´ğ‘‰ğ‘‹âˆ’â„Ž(𝑋,𝐵𝑉),(2.29) for any 𝑋∈Γ(𝑇𝑀) and 𝑉∈Γ(tr(𝑇𝑀)), where ∇𝑋𝐵𝑉=∇𝑋𝐵𝑉−𝐵∇𝑡𝑋∇𝑉,𝑋𝐶𝑉=∇𝑡𝑋𝐶𝑉−𝐶∇𝑡𝑋𝑉.(2.30)

Theorem 2.5 (see [5]). Let 𝑀 be a CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then, one has the following assertions. (i)The almost complex distribution 𝐷 is integrable if and only if the second fundamental form of 𝑀 satisfies ℎ(𝑋,𝐽𝑌)=ℎ(𝐽𝑋,𝑌),∀𝑋,𝑌∈Γ(𝐷).(2.31)(ii)The totally real distribution ğ·î…ž is integrable if and only if the shape operator of 𝑀 satisfies 𝐴𝐽𝑍𝑈=𝐴𝐽𝑈𝐷𝑍,∀𝑍,ğ‘ˆâˆˆÎ“î…žî€¸.(2.32)

Theorem 2.6 (see [5]). Let 𝑀 be a CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then, 𝐷 defines a totally geodesic foliation on 𝑀 if and only if, for any 𝑋,𝑌∈Γ(𝐷), ℎ(𝑋,𝑌) has no component in Γ(𝐿1⟂𝐿2).

3. CR-Lightlike Warped Product

Warped Product
Let 𝐵 and 𝐹 be two Riemannian manifolds with Riemannian metrics 𝑔𝐵 and 𝑔𝐹, respectively, and 𝜆>0 a differentiable function on 𝐵. Assume the product manifold 𝐵×𝐹 with its projection 𝜋∶𝐵×𝐹→𝐵 and 𝜂∶𝐵×𝐹→𝐹. The warped product 𝑀=𝐵×𝜆𝐹 is the manifold 𝐵×𝐹 equipped with the Riemannian metric 𝑔, where 𝑔=𝑔𝐵+𝜆2𝑔𝐹.(3.1) If 𝑋 is tangent to 𝑀=𝐵×𝜆𝐹 at (𝑝,ğ‘ž), then using (3.1), we have ‖𝑋‖2=‖‖𝜋∗𝑋‖‖2+𝜆2‖‖𝜂(𝜋(𝑋))∗𝑋‖‖2.(3.2) The function 𝜆 is called the warping function of the warped product. For differentiable function 𝜆 on M, the gradient ∇𝜆 is defined by 𝑔(∇𝜆,𝑋)=𝑋𝜆, for all 𝑋∈𝑇(𝑀).

Lemma 3.1 (see [7]). Let 𝑀=𝐵×𝜆𝐹 be a warped product manifold. If 𝑋,𝑌∈𝑇(𝐵) and 𝑈,𝑉∈𝑇(𝐹), then ∇𝑋∇𝑌∈𝑇(𝐵),(3.3)𝑋𝑉=∇𝑉𝑋=𝑋𝜆𝜆∇𝑉,(3.4)𝑈𝑔𝑉=−(𝑈,𝑉)𝜆∇𝜆.(3.5)

Corollary 3.2. On a warped product manifold 𝑀=𝐵×𝜆𝐹 one has(i)𝐵 is totally geodesic in 𝑀, (ii)𝐹 is totally umbilical in 𝑀.

Definition 3.3 (see [11]). A lightlike submanifold (𝑀,𝑔) of a semi-Riemannian manifold (𝑀,𝑔) is said to be totally umbilical in 𝑀 if there is a smooth transversal vector field 𝐻∈Γ(tr(𝑇𝑀)) on 𝑀, called the transversal curvature vector field of 𝑀, such that ℎ(𝑋,𝑌)=𝐻𝑔(𝑋,𝑌),∀𝑋,𝑌∈Γ(𝑇𝑀),(3.6) it is easy to see that 𝑀 is a totally umbilical if and only if on each coordinate neighborhood 𝑢, there exist smooth vector fields 𝐻𝑙∈Γ(ltr(𝑇𝑀)) and 𝐻𝑠∈Γ(𝑆(𝑇𝑀⟂)), such that â„Žğ‘™(𝑋,𝑌)=𝐻𝑙𝑔(𝑋,𝑌),â„Žğ‘ (𝑋,𝑌)=𝐻𝑠𝑔(𝑋,𝑌),𝐷𝑙(𝑋,𝑊)=0,(3.7) for any 𝑊∈Γ(𝑆(𝑇𝑀⟂)).

Lemma 3.4. Let 𝑀 be a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀 then, the distribution ğ·î…ž defines a totally geodesic foliation in 𝑀.

Proof. Let 𝑋,𝑌∈Γ(ğ·î…ž), then (2.25) and (2.27) imply that 𝑓∇𝑋𝑌=âˆ’ğ´ğ‘¤ğ‘Œğ‘‹âˆ’ğµâ„Ž(𝑋,𝑌). Let 𝑍∈Γ(𝐷0), then 𝑔𝑓∇𝑋𝐴𝑌,𝑍=−𝑔𝑤𝑌=𝑋,𝑍𝑔∇𝑋𝐽𝑌,𝑍=−𝑔∇𝑋𝑌,𝐽𝑍=−𝑔∇𝑋𝑌,ğ‘î…žî‚î€·=𝑔𝑌,âˆ‡ğ‘‹ğ‘î…žî€¸,(3.8) where, ğ‘î…ž=𝐽𝑍∈Γ(𝐷0). Since 𝑋∈Γ(ğ·î…ž) and 𝑍∈Γ(𝐷0) then (2.26) and (2.28) imply that 𝑤∇𝑋𝑍=ℎ(𝑋,𝑓𝑍)âˆ’ğ¶â„Ž(𝑋,𝑍)=𝐻𝑔(𝑋,𝑓𝑍)−𝐶𝐻𝑔(𝑋,𝑍)=0, this implies that ∇𝑋𝑍∈Γ(𝐷), then (3.8) implies that 𝑔(𝑓∇𝑋𝑌,𝑍)=0, then the nondegeneracy of the distribution 𝐷0 implies that 𝑓∇𝑋𝑌=0 gives ∇𝑋𝑌∈Γ(ğ·î…ž) for any 𝑋,𝑌∈Γ(ğ·î…ž). Hence, the proof is complete.

Theorem 3.5. Let 𝑀 be a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold, then the totally real distribution ğ·î…ž is integrable.

Proof. Using (2.25) and (2.27) with the above lemma, for any 𝑋,𝑌∈Γ(ğ·î…ž), we get 𝐴𝑤𝑌𝑋=âˆ’ğµâ„Ž(𝑋,𝑌),(3.9) this implies 𝐴𝑤𝑌𝑋∈Γ(ğ·î…ž) and also 𝐴𝑤𝑋𝑌=âˆ’ğµâ„Ž(𝑌,𝑋),(3.10) therefore, using (3.9) and (3.10), we get 𝐴𝑤𝑌𝑋=𝐴𝑤𝑋𝑌, for any 𝑋,𝑌∈Γ(ğ·î…ž). This implies that the distribution ğ·î…ž is integrable.

Definition 3.6 (see [5]). A CR-lightlike submanifold 𝑀 of an indefinite Kaehler manifold 𝑀 is called a CR-lightlike product if both the distribution 𝐷 and ğ·î…ž define totally geodesic foliations in 𝑀.

Theorem 3.7. Let 𝑀 be a totally umbilical CR-lightlike submanifold 𝑀 of an indefinite Kaehler manifold 𝑀. If 𝑀=𝑁⟂×𝜆𝑁𝑇 be a warped product CR-lightlike submanifold, then it is a CR-lightlike product.

Proof. Since 𝑀 is a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold, then using Lemma 3.4, the distribution ğ·î…ž defines a totally geodesic foliation in 𝑀.
Let â„Žğ‘‡ and 𝐴𝑇 be the second fundamental form and the shape operator of 𝑁𝑇 in 𝑀, then for 𝑋,𝑌∈Γ(𝐷) and 𝑍∈Γ(ğ·î…ž), we have 𝑔(â„Žğ‘‡(𝑋,𝑌),𝑍)=𝑔(∇𝑋𝑌,𝑍)=−𝑔(𝑌,∇𝑋𝑍)=−𝑔(𝑌,∇𝑋𝑍). Using (3.4), we get ğ‘”î€·â„Žğ‘‡î€¸(𝑋,𝑌),𝑍=−(𝑍ln𝜆)𝑔(𝑋,𝑌).(3.11) Now, let ℎ be the second fundamental form of 𝑁𝑇 in 𝑀, then ℎ(𝑋,𝑌)=â„Žğ‘‡(𝑋,𝑌)+â„Žğ‘ (𝑋,𝑌)+â„Žğ‘™(𝑋,𝑌),(3.12) for any 𝑋,𝑌 tangent to 𝑁𝑇, then using (3.11), we get ğ‘”î‚€îî‚î€·â„Žâ„Ž(𝑋,𝑌),𝑍=𝑔𝑇(𝑋,𝑌),𝑍=−(𝑍ln𝜆)𝑔(𝑋,𝑌).(3.13) Since 𝑁𝑇 is a holomorphic submanifold of 𝑀, then we have ℎ(𝑋,𝐽𝑌)=ℎ(𝐽𝑋,𝑌)=ğ½â„Ž(𝑋,𝑌), therefore, we have ğ‘”î‚€îî‚î‚€îî‚â„Ž(𝑋,𝑌),𝑍=âˆ’ğ‘”â„Ž(𝐽𝑋,𝐽𝑌),𝑍=(𝑍ln𝜆)𝑔(𝑋,𝑌).(3.14) Adding (3.13) and (3.14), we get ğ‘”î‚€îî‚â„Ž(𝑋,𝑌),𝑍=0.(3.15) Using (3.12), we have 𝑔(ℎ(𝑋,𝑌),𝐽𝑍)=𝑔(ℎ(𝑋,𝑌),𝐽𝑍)−𝑔(â„Žğ‘‡îîî(𝑋,𝑌),𝐽𝑍)=𝑔(ℎ(𝑋,𝑌),𝐽𝑍)=−𝑔(ğ½â„Ž(𝑋,𝑌),𝑍)=−𝑔(ℎ(𝑋,𝐽𝑌),𝑍)=0. Thus, 𝑔(ℎ(𝑋,𝑌),𝐽𝑍)=0 implies that ℎ(𝑋,𝑌) has no components in 𝐿1⟂𝐿2 for any 𝑋,𝑌∈Γ(𝐷). This implies that the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Hence, 𝑀 is a CR-lightlike product.

Theorem 3.7 shows that if 𝑀=𝑁⟂×𝜆𝑁𝑇 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 𝑀=𝑁⟂×𝜆𝑁𝑇 other than CR-lightlike product. Thus, for simplicity, we call a warped product CR-lightlike submanifold in the form 𝑀=𝑁𝑇×𝜆𝑁⟂ a CR-lightlike warped product.

Lemma 3.8. Let 𝑀 be a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Let 𝑀=𝑁𝑇×𝜆𝑁⟂ be a proper CR-lightlike warped product of an indefinite Kaehler manifold, then 𝑁𝑇 is totally geodesic in 𝑀.

Proof . Let ∇ be a linear connection on 𝑀 induced from ∇. Let 𝑋,𝑌∈𝑁𝑇 and 𝑍∈𝑁⟂, then we have 𝑔(∇𝑋𝑌,𝑍)=𝑔(∇𝑋𝑌,𝑍)=−𝑔(𝑌,∇𝑋𝑍)−𝑔(𝑌,â„Žğ‘™(𝑋,𝑍)), using (3.4), we get 𝑔(∇𝑋𝑌,𝑍)=−𝑔(𝑌,â„Žğ‘™(𝑋,𝑍)). Since 𝑀 is totally umbilical CR-lightlike submanifold, therefore, â„Žğ‘™(𝑋,𝑍)=â„Žğ‘ (𝑋,𝑍)=0. Hence, 𝑔(∇𝑋𝑌,𝑍)=0 implies that 𝑁𝑇 is totally geodesic in 𝑀.

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

Theorem 3.10. Let 𝑀 be a totally umbilical proper CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀, then 𝐻𝑙=0.

Proof. Let 𝑀 be a totally umbilical proper CR-lightlike submanifold then using (2.25) and (2.27), we have 𝐴𝑤𝑍𝑍=âˆ’ğ‘“âˆ‡ğ‘ğ‘âˆ’ğµâ„Žğ‘™(𝑍,𝑍)âˆ’ğµâ„Žğ‘ (𝑍,𝑍), for 𝑍∈Γ(𝑀2). We obtain 𝑔(𝐴𝐽𝑍𝑍,𝐽𝜉)+𝑔(â„Žğ‘™(𝑍,𝑍),𝜉)=0. Using (2.13) and the hypothesis we obtain 𝑔(𝑍,𝑍)𝑔(𝐻𝑙,𝜉)=0, then using the non degeneracy of 𝑀2, 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 𝑀 be a totally umbilical CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀, then for a CR-lightlike warped product 𝑀=𝑁𝑇×𝜆𝑁⟂ in an indefinite Kaehler manifold 𝑀, one has (i)𝑔(â„Žğ‘ (𝐷,𝐷),𝐽𝑀2)=0, (ii)𝑔(ℎ(𝐽𝑋,𝑍),𝐽𝑍1)=(𝑋ln𝜆)𝑔(𝑍,𝑍1), for any 𝑋∈Γ(𝐷) and 𝑍,𝑍1∈Γ(𝑀2)⊂Γ(ğ·î…ž).

Proof. Since 𝑀 is Kaehlerian, therefore, for 𝑋∈Γ(𝐷) and 𝑍∈Γ(𝑀2), we have 𝐽∇𝑋𝑍=∇𝑋𝐽𝑍, since 𝑀 is totally umbilical, therefore, we have 𝐽(∇𝑋𝑍)=−𝐴𝑤𝑍𝑋+∇𝑠𝑋𝑤𝑍, then taking inner product with 𝐽𝑌, where 𝑌∈Γ(𝐷), we get 𝑔(∇𝑋𝑍,𝑌)=−𝑔(𝐴𝑤𝑍𝑋,𝐽𝑌). Using (3.4), we obtain 𝑔(𝐴𝑤𝑍𝑋,𝐽𝑌)=0, then using (2.13), we get 𝑔(â„Žğ‘ (𝐷,𝐷),𝐽𝑀2)=0.
Next for any 𝑋∈Γ(𝐷) and 𝑍,𝑍1∈Γ(𝑀2)⊂Γ(ğ·î…ž), we have 𝑔(ℎ(𝐽𝑋,𝑍),𝐽𝑍1)=𝑔(∇𝑍𝐽𝑋,𝐽𝑍1)=𝑔(∇𝑍𝑋,𝑍1)=(𝑋ln𝜆)𝑔(𝑍,𝑍1). Hence, the proof is complete.

Corollary 4.2. Let 𝑍∈Γ(𝑀1)⊂Γ(ğ·î…ž), then clearly 𝑔(â„Žğ‘ (𝐷,𝐷),𝐽𝑍)=0 and also 𝑔(â„Žğ‘™(𝐷,𝐷),𝐽𝑍)=0 for any 𝑍∈Γ(ğ·î…ž). Thus, 𝑔(ℎ(𝐷,𝐷),ğ½ğ·î…ž)=0, this implies that ℎ(𝐷,𝐷) has no component in 𝐿1⟂𝐿2, therefore, using Theorem 2.5, the distribution 𝐷 defines a totally geodesic foliation in 𝑀.

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

Theorem 4.3. A proper totally umbilical CR-lightlike submanifold 𝑀 of an indefinite Kaehler manifold 𝑀 is locally a CR-lightlike warped product if and only if 𝐴𝐽𝑍𝑋=((𝐽𝑋)𝜇)𝑍,(4.1) for 𝑋∈𝐷, ğ‘âˆˆğ·î…ž and for some function 𝜇 on 𝑀 satisfying 𝑈𝜇=0,𝑈∈Γ(ğ·î…ž).

Proof. Assume that 𝑀 be a proper CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀 satisfying (4.1). Let 𝑌∈Γ(𝐷) and 𝑍∈Γ(𝑀2)⊂Γ(ğ·î…ž), we have 𝑔(𝐴𝐽𝑍𝑋,𝐽𝑌)=𝑔(((𝐽𝑋)𝜇)𝑍,𝐽𝑌)=0, then using (2.13), we get 𝑔(â„Žğ‘ (𝐷,𝐷),𝐽𝑀2)=0. If 𝑍∈Γ(𝑀1)⊂Γ(ğ·î…ž), then clearly 𝑔(â„Žğ‘ (𝐷,𝐷),𝐽𝑍)=0 and also 𝑔(â„Žğ‘™(𝐷,𝐷),𝐽𝑍)=0 for any 𝑍∈Γ(ğ·î…ž). Thus, ğ‘”î€·â„Ž(𝐷,𝐷),ğ½ğ·î…žî€¸=0,(4.2) that is, ℎ(𝐷,𝐷) has no component in 𝐿1⟂𝐿2, this implies that the distribution 𝐷 defines totally geodesic foliation in 𝑀 and consequently it is totally geodesic in 𝑀 and using Theorem 3.9, the distribution 𝐷 is integrable.
Taking inner product of (4.1) with 𝑈∈Γ(ğ·î…ž) and using that 𝑀 is totally umbilical, we get 𝑔(((𝐽𝑋)𝜇)𝑍,𝑈)=𝑔(𝐴𝐽𝑍𝑋,𝑈)=𝑔(𝐽𝑍,∇𝑋𝑈)=𝑔(𝐽𝑍,∇𝑈𝑋)=−𝑔(∇𝑈𝐽𝑍,𝑋)=𝑔(∇𝑈𝑍,𝐽𝑋), using the definition of gradient 𝑔(∇𝜙,𝑋)=𝑋𝜙, we get𝑔∇𝑈𝑍,𝐽𝑋=𝑔(∇𝜇,𝐽𝑋)𝑔(𝑍,𝑈).(4.3) Let ℎ be the second fundamental form of ğ·î…ž in 𝑀 and let ∇ be the metric connection of ğ·î…ž in 𝑀 then, particularly for 𝑋∈Γ(𝐷0), we have ğ‘”î€·â„Žî…žî€¸î€·âˆ‡(𝑈,𝑍),𝐽𝑋=ğ‘”ğ‘ˆğ‘âˆ’âˆ‡î…žğ‘ˆî€¸î€·âˆ‡ğ‘,𝐽𝑋=𝑔𝑈𝑍,𝐽𝑋.(4.4) Therefore, from (4.3) and (4.4), we get 𝑔(ℎ(𝑈,𝑍),𝐽𝑋)=𝑔(∇𝜇,𝐽𝑋)𝑔(𝑍,𝑈), this further implies that ℎ(𝑈,𝑍)=∇𝜇𝑔(𝑍,𝑈),(4.5) this implies that the distribution ğ·î…ž is totally umbilical in 𝑀. Using Theorem 3.5, the totally real distribution ğ·î…ž is integrable and (4.5) and the condition 𝑈𝜇=0 for ğ‘ˆâˆˆğ·î…ž imply that each leaf of ğ·î…ž is an extrinsic sphere in 𝑀. Hence, by a result of [12] which say that “if the tangent bundle of a Riemannian manifold 𝑀 splits into an orthogonal sum 𝑇𝑀=𝐸0⊕𝐸1 of nontrivial vector subbundles such that 𝐸1 is spherical and its orthogonal complement 𝐸0 is autoparallel, then the manifold 𝑀 is locally isometric to a warped product 𝑀0×𝜆𝑀1,” therefore, we can conclude that 𝑀 is locally a CR-lightlike warped product 𝑁𝑇×𝜆𝑁⟂ of 𝑀, where 𝜆=𝑒𝜇.
Conversely, let 𝑋∈Γ(𝑁𝑇) and 𝑍∈Γ(𝑁⟂), since 𝑀 is a Kaehler manifold so, we have ∇𝑋𝐽𝑍=𝐽∇𝑋𝑍, which further becomes −𝐴𝐽𝑍𝑋+∇𝑡𝑋𝐽𝑍=((𝐽𝑋)ln𝜆)𝑍, comparing tangential components, we get 𝐴𝐽𝑍𝑋=−((𝐽𝑋)ln𝜆)𝑍 for each 𝑋∈Γ(𝐷) and 𝑍∈(ğ·î…ž). Since 𝜆 is a function on 𝑁𝑇, so we also have 𝑈(ln𝜆)=0 for all 𝑈∈Γ(ğ·î…ž). Hence, the proof is complete.

Lemma 4.4. Let 𝑀=𝑁𝑇×𝜆𝑁⟂ be a CR-lightlike warped product of an indefinite Kaehler manifold 𝑀, then ∇𝑍𝑓∇𝑋=𝑓𝑋(ln𝜆)𝑍.𝑈𝑓𝑍=𝑔(𝑍,𝑈)𝑓(∇ln𝜆),(4.6) for any 𝑈∈Γ(𝑇𝑀),𝑋∈Γ(𝑁𝑇), and 𝑍∈Γ(𝑁⟂).

Proof. For 𝑋∈Γ(𝑁𝑇) and 𝑍∈Γ(𝑁⟂), using (2.27) and (3.4), we have (∇𝑍𝑓)𝑋=∇𝑍𝑓𝑋=𝑓𝑋(ln𝜆)𝑍. Next, again using (2.27), we get (∇𝑈𝑓)𝑍=−𝑓∇𝑈𝑍, this implies that (∇𝑈𝑓)𝑍∈Γ(𝑁𝑇), therefore, for any 𝑋∈Γ(𝐷0), we have 𝑔∇𝑈𝑓𝑍,𝑋=−𝑔𝑓∇𝑈∇𝑍,𝑋=𝑔𝑈=𝑍,𝑓𝑋𝑔∇𝑈𝑍,𝑓𝑋=−𝑔𝑍,∇𝑈𝑓𝑋=−𝑓𝑋(ln𝜆)𝑔(𝑍,𝑈).(4.7) Hence, using the definition of gradient of 𝜆 and the nondegeneracy of the distribution 𝐷0, the result follows.

Theorem 4.5. A proper totally umbilical CR-lightlike submanifold 𝑀 of an indefinite Kaehler manifold 𝑀 is locally a CR-lightlike warped product if and only if ∇𝑈𝑓𝑉=(𝑓𝑉(𝜇))𝑄𝑈+𝑔(𝑄𝑈,𝑄𝑉)𝐽(∇𝜇),(4.8) for any 𝑈,𝑉∈Γ(𝑇𝑀) and for some function 𝜇 on 𝑀 satisfying 𝑍𝜇=0,𝑍∈Γ(ğ·î…ž).

Proof. Let 𝑀 be a CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀 satisfying (4.8). Let 𝑈,𝑉∈Γ(𝐷), then (4.8) implies that (∇𝑈𝑓)𝑉=0, then (2.25) gives ğµâ„Ž(𝑈,𝑉)=0. Thus 𝐷 defines a totally geodesic foliation in 𝑀 and consequently it is totally geodesic in 𝑀 and integrable using Theorem 3.9.
Let 𝑈,𝑉∈Γ(ğ·î…ž), then (4.8) gives∇𝑈𝑓𝑉=𝑔(𝑄𝑈,𝑄𝑉)𝐽(∇𝜇).(4.9) Let 𝑋∈Γ(𝐷0), then (4.9) implies that 𝑔∇𝑈𝑓𝑉,𝑋=𝑔(𝑄𝑈,𝑄𝑉)𝑔(𝐽(∇𝜇),𝑋).(4.10) Also 𝑔∇𝑈𝑓𝐴𝑉,𝑋=𝑔𝑤𝑉=𝑈,𝑋𝑔∇𝑈∇𝑉,𝐽𝑋=𝑔𝑈𝑉,𝐽𝑋,(4.11) therefore, from (4.10) and (4.11), we get 𝑔∇𝑈𝑉,𝐽𝑋=−𝑔(∇𝜇,𝐽𝑋)𝑔(𝑈,𝑉).(4.12) Let ℎ be the second fundamental form of ğ·î…ž in 𝑀 and let ∇ be the metric connection of ğ·î…ž in 𝑀, then ğ‘”î€·â„Žî…žî€¸î€·âˆ‡(𝑈,𝑉),𝐽𝑋=𝑔𝑈𝑉,𝐽𝑋,(4.13) therefore, from (4.12) and (4.13), we get 𝑔(ℎ(𝑈,𝑉),𝐽𝑋)=−𝑔(∇𝜇,𝐽𝑋)𝑔(𝑈,𝑉), then the nondegeneracy of the distribution 𝐷0 implies that ℎ(𝑈,𝑉)=−∇𝜇𝑔(𝑈,𝑉),(4.14) this gives that the distribution ğ·î…ž is totally umbilical in 𝑀 and using Theorem 3.5, the distribution ğ·î…ž is integrable. Also, 𝑍𝜇=0 for 𝑍∈Γ(ğ·î…ž), hence as in Theorem 4.3, each leaf of ğ·î…ž is an extrinsic sphere in 𝑀. Thus, 𝑀 is locally a CR-lightlike warped product 𝑁𝑇×𝜆𝑁⟂ of 𝑀, where 𝜆=𝑒𝜇.
Conversely, let 𝑀 be a CR-lightlike warped product 𝑁𝑇×𝜆𝑁⟂ of an indefinite Kaehler manifold 𝑀. Using (2.22), we can write∇𝑈𝑓∇𝑉=𝑆𝑈𝑓∇𝑆𝑉+𝑄𝑈𝑓∇𝑆𝑉+𝑈𝑓𝑄𝑉.(4.15) Since 𝐷 defines totally geodesic foliation in 𝑀, therefore, using (2.25), we get ∇𝑆𝑈𝑓𝑆𝑉=0.(4.16) Using (4.6), we have ∇𝑄𝑈𝑓∇𝑆𝑉=𝑓𝑉(ln𝜆)𝑄𝑈,(4.17)𝑈𝑓𝑄𝑉=𝑔(𝑄𝑈,𝑄𝑉)𝑓(∇ln𝜆).(4.18) Hence, from (4.15)–(4.18), the result follows.

Theorem 4.6. Let 𝑀 be a locally CR-lightlike warped product of an indefinite Kaehler manifold 𝑀, then 𝑔∇𝑡𝑈𝑤𝑉,𝐽𝑍=−𝑆𝑉(𝜇)𝑔(𝑈,𝑍),(4.19) for any 𝑈,𝑉∈Γ(𝑇𝑀) and for some function 𝜇 on 𝑀 satisfying 𝑍𝜇=0,𝑍∈Γ(ğ·î…ž).

Proof. Let 𝑀 be a CR-lightlike warped product of an indefinite Kaehler manifold 𝑀. Therefore, the distribution 𝐷 defines totally geodesic foliation in 𝑀, then using (2.25) for 𝑈,𝑉∈Γ(𝐷), we get 𝑔∇𝑡𝑈𝑤𝑉,𝐽𝑍=−𝑔(ℎ(𝑈,𝑓𝑉),𝐽𝑍)=−𝑔∇𝑈∇𝑉,𝑍+𝑔𝑈∇𝑓𝑉,𝐽𝑍=−𝑔𝑈𝑉,𝑍+𝑔𝑓∇𝑈𝑉,𝐽𝑍=0.(4.20) For 𝑈∈Γ(𝐷),𝑉∈Γ(ğ·î…ž) or 𝑈,𝑉∈Γ(ğ·î…ž), using (2.25), we have 𝑔∇𝑡𝑈𝑤𝑉,𝐽𝑍=0.(4.21) Now let 𝑈∈Γ(ğ·î…ž) and 𝑉∈Γ(𝐷), then using (3.4), we have 𝑔∇𝑡𝑈𝑤∇𝑉,𝐽𝑍=−𝑔(ℎ(𝑈,𝑓𝑉),𝐽𝑍)=−𝑔𝑈𝑉,𝑍+𝑔𝑓∇𝑈𝑉,𝐽𝑍=−𝑉(ln𝜆)𝑔(𝑈,𝑍).(4.22) Therefore, (4.19) follows from (4.21)–(4.22). Hence, the result is complete.