About this Journal Submit a Manuscript Table of Contents
ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 957176, 14 pages
http://dx.doi.org/10.5402/2012/957176
Research Article

Characterization of CR-Lightlike-Warped Product of Indefinite Kaehler Manifolds

1University College, Moonak 148033, India
2University College of Engineering, Punjabi University, Patiala 147002, India
3Department of Mathematics, Punjabi University, Patiala 147002, India

Received 26 September 2011; Accepted 19 October 2011

Academic Editors: B.Β Gato Rivera, D.Β Gepner, and F.Β Sugino

Copyright Β© 2012 Rachna Rani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we prove that there does not exist a warped product CR-lightlike submanifold in the form 𝑀=π‘βŸ‚Γ—πœ†π‘π‘‡ 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=π‘βŸ‚Γ—πœ†π‘π‘‡ in a Kaehler manifold where π‘βŸ‚ is a totally real submanifold and 𝑁𝑇 is a holomorphic submanifold of 𝑀. He proved if 𝑀=π‘βŸ‚Γ—πœ†π‘π‘‡ 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=π‘βŸ‚Γ—πœ†π‘π‘‡ other than CR-product. Therefore, he called a warped product CR-submanifold in the form 𝑀=π‘π‘‡Γ—πœ†π‘βŸ‚ 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(πœ‡)𝑍 for each π‘‹βˆˆΞ“(𝐷), π‘βˆˆΞ“(π·ξ…ž), πœ‡ a 𝐢∞-function on 𝑀 such that π‘πœ‡=0 for all π‘βˆˆΞ“(π·ξ…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=π‘βŸ‚Γ—πœ†π‘π‘‡ 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 (π‘š+𝑛)-dimensional semi-Riemannian manifold of constant index π‘ž such that π‘š,𝑛β‰₯1, 1β‰€π‘žβ‰€π‘š+π‘›βˆ’1 and let (𝑀,𝑔) be an π‘š-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,π‘‡π‘€βŸ‚ξ‚†=βˆͺπ‘’βˆˆπ‘‡π‘₯π‘€βˆΆπ‘”(𝑒,𝑣)=0,βˆ€π‘£βˆˆπ‘‡π‘₯𝑀,π‘₯βˆˆπ‘€,(2.1) is a degenerate 𝑛-dimensional subspace of 𝑇π‘₯𝑀. Thus, both 𝑇π‘₯𝑀 and 𝑇π‘₯π‘€βŸ‚ are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace Rad𝑇π‘₯𝑀=𝑇π‘₯π‘€βˆ©π‘‡π‘₯π‘€βŸ‚ which is known as radical (null) subspace. If the mappingRadπ‘‡π‘€βˆΆπ‘₯βˆˆπ‘€βŸΆRad𝑇π‘₯𝑀(2.2) defines a smooth distribution on 𝑀 of rank π‘Ÿ>0, then the submanifold 𝑀 of 𝑀 is called π‘Ÿ-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=Radπ‘‡π‘€βŸ‚π‘†(𝑇𝑀),(2.3)𝑆(π‘‡π‘€βŸ‚) is a complementary vector subbundle to Rad𝑇𝑀 in π‘‡π‘€βŸ‚. Let tr(𝑇𝑀) and ltr(𝑇𝑀) be complementary (but not orthogonal) vector bundles to 𝑇𝑀 in π‘‡π‘€βˆ£π‘€ and to Rad𝑇𝑀 in 𝑆(π‘‡π‘€βŸ‚)βŸ‚, respectively. Then, we haveξ€·tr(𝑇𝑀)=ltr(𝑇𝑀)βŸ‚π‘†π‘‡π‘€βŸ‚ξ€Έπ‘‡,(2.4)π‘€βˆ£π‘€ξ€·=π‘‡π‘€βŠ•tr(𝑇𝑀)=(Radπ‘‡π‘€βŠ•ltr(𝑇𝑀))βŸ‚π‘†(𝑇𝑀)βŸ‚π‘†π‘‡π‘€βŸ‚ξ€Έ.(2.5) Let 𝑒 be a local coordinate neighborhood of 𝑀 and consider the local quasiorthonormal fields of frames of 𝑀 along 𝑀, on 𝑒 as {πœ‰1,…,πœ‰π‘Ÿ,π‘Šπ‘Ÿ+1,…,π‘Šπ‘›,𝑁1,…,π‘π‘Ÿ,π‘‹π‘Ÿ+1,…,π‘‹π‘š}, where {πœ‰1,…,πœ‰π‘Ÿ}, {𝑁1,…,π‘π‘Ÿ} are local lightlike bases of Ξ“(Radπ‘‡π‘€βˆ£π‘’), Ξ“(ltr(𝑇𝑀)βˆ£π‘’) and {π‘Šπ‘Ÿ+1,…,π‘Šπ‘›},{π‘‹π‘Ÿ+1,…,π‘‹π‘š} are local orthonormal bases of Ξ“(𝑆(π‘‡π‘€βŸ‚)βˆ£π‘’) and Ξ“(𝑆(𝑇𝑀)βˆ£π‘’), respectively. For this quasiorthonormal fields of frames, we have the following theorem.

Theorem 2.1 (see [5]). Let (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) be an π‘Ÿ-lightlike submanifold of a semi-Riemannian manifold (𝑀,𝑔). Then, there exists a complementary vector bundle ltr(𝑇𝑀) of Rad𝑇𝑀 in 𝑆(π‘‡π‘€βŸ‚)βŸ‚ and a basis of Ξ“(ltr(𝑇𝑀)βˆ£π‘’) consisting of smooth section {𝑁𝑖} of 𝑆(π‘‡π‘€βŸ‚)βŸ‚βˆ£π‘’, where u is a coordinate neighborhood of 𝑀, such that 𝑔𝑁𝑖,πœ‰π‘—ξ€Έ=𝛿𝑖𝑗,𝑔𝑁𝑖,𝑁𝑗=0,(2.6) where {πœ‰1,…,πœ‰r} is a lightlike basis of Ξ“(Rad(𝑇𝑀)).

Let βˆ‡ be the Levi-Civita connection on 𝑀. Then, according to the decomposition (2.5), the Gauss and Weingarten formulas are given byβˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Ž(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀),(2.7)βˆ‡π‘‹π‘ˆ=βˆ’π΄π‘ˆπ‘‹+βˆ‡βŸ‚π‘‹π‘ˆ,βˆ€π‘‹βˆˆΞ“(𝑇𝑀),π‘ˆβˆˆΞ“(tr(𝑇𝑀)),(2.8) where {βˆ‡π‘‹π‘Œ,π΄π‘ˆπ‘‹} and {β„Ž(𝑋,π‘Œ),βˆ‡βŸ‚π‘‹π‘ˆ} belong to Ξ“(𝑇𝑀) and Ξ“(tr(𝑇𝑀)), respectively. Here, βˆ‡ is a torsion-free linear connection on 𝑀, β„Ž is a symmetric bilinear form on Ξ“(𝑇𝑀) which is called second fundamental form, and π΄π‘ˆ 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(π‘‡π‘€βŸ‚), respectively, (2.7) and (2.8) giveβˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Žπ‘™(𝑋,π‘Œ)+β„Žπ‘ (𝑋,π‘Œ),(2.9)βˆ‡π‘‹π‘ˆ=βˆ’π΄π‘ˆπ‘‹+π·π‘™π‘‹π‘ˆ+π·π‘ π‘‹π‘ˆ,(2.10) where we put β„Žπ‘™(𝑋,π‘Œ)=𝐿(β„Ž(𝑋,π‘Œ)), β„Žπ‘ (𝑋,π‘Œ)=𝑆(β„Ž(𝑋,π‘Œ)), π·π‘™π‘‹π‘ˆ=𝐿(βˆ‡βŸ‚π‘‹π‘ˆ), π·π‘ π‘‹π‘ˆ=𝑆(βˆ‡βŸ‚π‘‹π‘ˆ).

As β„Žπ‘™ and β„Žπ‘  are Ξ“(ltr(𝑇𝑀))-valued and Ξ“(𝑆(π‘‡π‘€βŸ‚))-valued, respectively, therefore, they are called the lightlike second fundamental form and the screen second fundamental form on 𝑀. In particular, βˆ‡π‘‹π‘=βˆ’π΄π‘π‘‹+βˆ‡π‘™π‘‹π‘+𝐷𝑠(𝑋,𝑁),(2.11)βˆ‡π‘‹π‘Š=βˆ’π΄π‘Šπ‘‹+βˆ‡π‘ π‘‹π‘Š+𝐷𝑙(𝑋,π‘Š),(2.12) where π‘‹βˆˆΞ“(𝑇𝑀), π‘βˆˆΞ“(ltr(𝑇𝑀)), and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)).

Using (2.4)-(2.5) and (2.9)–(2.12), we obtainπ‘”ξ€·β„Žπ‘ ξ€Έ+(𝑋,π‘Œ),π‘Šπ‘”ξ€·π‘Œ,𝐷𝑙𝐴(𝑋,π‘Š)=π‘”π‘Šξ€Έ,𝑋,π‘Œ(2.13)π‘”ξ€·β„Žπ‘™ξ€Έ+(𝑋,π‘Œ),πœ‰π‘”ξ€·π‘Œ,β„Žπ‘™ξ€Έξ€·(𝑋,πœ‰)+π‘”π‘Œ,βˆ‡π‘‹πœ‰ξ€Έ=0,(2.14)𝑔𝐴𝑁𝑋,π‘ξ…žξ€Έ+𝑔𝑁,𝐴𝑁′𝑋=0,(2.15) for any πœ‰βˆˆΞ“(Rad𝑇𝑀), π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)), and 𝑁,π‘ξ…žβˆˆΞ“(ltr(𝑇𝑀)).

Let 𝑃 be a projection of 𝑇𝑀 on 𝑆(𝑇𝑀). Now, we consider the decomposition (2.3), we can writeβˆ‡π‘‹π‘ƒπ‘Œ=βˆ‡βˆ—π‘‹π‘ƒπ‘Œ+β„Žβˆ—ξ‚€π‘‹,,βˆ‡π‘ƒπ‘Œπ‘‹πœ‰=βˆ’π΄βˆ—πœ‰π‘‹+βˆ‡π‘‹βˆ—π‘‘πœ‰,(2.16) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀), and πœ‰βˆˆΞ“(Rad𝑇𝑀), where {βˆ‡βˆ—π‘‹π‘ƒπ‘Œ,π΄βˆ—πœ‰π‘‹} and {β„Žβˆ—(𝑋,π‘ƒπ‘Œ),βˆ‡π‘‹βˆ—π‘‘πœ‰} belong to Ξ“(𝑆(𝑇𝑀)) and Ξ“(Rad𝑇𝑀), respectively. Here βˆ‡βˆ— and βˆ‡π‘‹βˆ—π‘‘ are linear connections on 𝑆(𝑇𝑀) and Rad𝑇𝑀, respectively. By using (2.9)-(2.10) and (2.16), we obtainπ‘”ξ‚€β„Žπ‘™ξ‚€π‘‹,ξ‚ξ‚ξ‚€π΄π‘ƒπ‘Œ,πœ‰=π‘”βˆ—πœ‰π‘‹,,π‘ƒπ‘Œπ‘”ξ‚€β„Žβˆ—ξ‚€π‘‹,=π‘ƒπ‘Œ,𝑁𝑔𝐴𝑁𝑋,.π‘ƒπ‘Œ(2.17)

Definition 2.2. Let (𝑀,𝐽,𝑔) be a real 2π‘š-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 Radπ‘‡π‘€βˆ©π½(Rad𝑇𝑀)=0;(2.18)(b)there exist vector bundles 𝑆(T𝑀), 𝑆(π‘‡π‘€βŸ‚), ltr(𝑇𝑀), 𝐷0 and π·ξ…ž over 𝑀, such that 𝑆𝐽(𝑇𝑀)=(Rad𝑇𝑀)βŠ•π·ξ…žξ€ΎβŸ‚π·0𝐷;𝐽0ξ€Έ=𝐷0𝐷;π½ξ…žξ€Έ=𝐿1βŸ‚πΏ2,(2.19)where Ξ“(𝐷0) is a nondegenerate distribution on 𝑀, Ξ“(𝐿1) and Ξ“(𝐿2) are vector subbundles of Ξ“(ltr(𝑇𝑀)) and Ξ“(𝑆(π‘‡π‘€βŸ‚)), respectively, and assume that 𝑀1=𝐽(𝐿1) and 𝑀2=𝐽(𝐿2).

Clearly, the tangent bundle of a CR-lightlike submanifold is decomposed as𝑇𝑀=π·βŠ•π·ξ…ž,(2.20) where𝐷=Radπ‘‡π‘€βŸ‚π½(Rad𝑇𝑀)βŸ‚π·0.(2.21)

Now, let 𝑆 and 𝑄 be the projections on 𝐷 and π·ξ…ž, respectively. Then, for any π‘‹βˆˆΞ“(𝑇𝑀), we can write𝑋=𝑆𝑋+𝑄𝑋,(2.22) where π‘†π‘‹βˆˆπ· and π‘„π‘‹βˆˆπ·ξ…ž. Applying 𝐽 to above equation, we get𝐽𝑋=𝑓𝑋+𝑀𝑋,(2.23) where 𝑓𝑋=𝐽𝑆𝑋 and 𝑀𝑋=𝐽𝑄𝑋. Clearly 𝑓 is a tensor field of type (1,1) and 𝑀 is Ξ“(𝐿1βŸ‚πΏ2)-valued 1-form on 𝑀. Clearly, π‘‹βˆˆΞ“(𝐷) if and only if 𝑀𝑋=0. On the other hand, we set𝐽𝑉=𝐡𝑉+𝐢𝑉,(2.24) for any π‘‰βˆˆΞ“(tr(𝑇𝑀)), where 𝐡𝑉 and 𝐢𝑉 are sections of 𝑇𝑀 and tr(𝑇𝑀), respectively.

By using Kaehlerian property of βˆ‡ 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 ξ€·βˆ‡π‘‹π‘“ξ€Έπ‘Œ=π΄π‘€π‘Œξ€·βˆ‡π‘‹+π΅β„Ž(𝑋,π‘Œ),(2.25)π‘‘π‘‹π‘€ξ€Έπ‘Œ=πΆβ„Ž(𝑋,π‘Œ)βˆ’β„Ž(𝑋,π‘“π‘Œ),(2.26) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀), where ξ€·βˆ‡π‘‹π‘“ξ€Έπ‘Œ=βˆ‡π‘‹ξ€·βˆ‡π‘“π‘Œβˆ’π‘“π‘‹π‘Œξ€Έξ€·βˆ‡,(2.27)π‘‘π‘‹π‘€ξ€Έπ‘Œ=βˆ‡π‘‘π‘‹ξ€·βˆ‡π‘€π‘Œβˆ’π‘€π‘‹π‘Œξ€Έ.(2.28)

Lemma 2.4. Let 𝑀 be a CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀 then, one has ξ€·βˆ‡π‘‹π΅ξ€Έπ‘‰=βˆ’π‘“π΄π‘‰π‘‹+π΄πΆπ‘‰ξ€·βˆ‡π‘‹,𝑋𝐢𝑉=βˆ’π‘€π΄π‘‰π‘‹βˆ’β„Ž(𝑋,𝐡𝑉),(2.29) for any π‘‹βˆˆΞ“(𝑇𝑀) and π‘‰βˆˆΞ“(tr(𝑇𝑀)), where ξ€·βˆ‡π‘‹π΅ξ€Έπ‘‰=βˆ‡π‘‹π΅π‘‰βˆ’π΅βˆ‡π‘‘π‘‹ξ€·βˆ‡π‘‰,𝑋𝐢𝑉=βˆ‡π‘‘π‘‹πΆπ‘‰βˆ’πΆβˆ‡π‘‘π‘‹π‘‰.(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 β„Ž(𝑋,π½π‘Œ)=β„Ž(𝐽𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝐷).(2.31)(ii)The totally real distribution π·ξ…ž is integrable if and only if the shape operator of 𝑀 satisfies π΄π½π‘π‘ˆ=π΄π½π‘ˆξ€·π·π‘,βˆ€π‘,π‘ˆβˆˆΞ“ξ…žξ€Έ.(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,π‘ŒβˆˆΞ“(𝐷), β„Ž(𝑋,π‘Œ) has no component in Ξ“(𝐿1βŸ‚πΏ2).

3. CR-Lightlike Warped Product

Warped Product
Let 𝐡 and 𝐹 be two Riemannian manifolds with Riemannian metrics 𝑔𝐡 and 𝑔𝐹, respectively, and πœ†>0 a differentiable function on 𝐡. Assume the product manifold 𝐡×𝐹 with its projection πœ‹βˆΆπ΅Γ—πΉβ†’π΅ and πœ‚βˆΆπ΅Γ—πΉβ†’πΉ. The warped product 𝑀=π΅Γ—πœ†πΉ is the manifold 𝐡×𝐹 equipped with the Riemannian metric 𝑔, where 𝑔=𝑔𝐡+πœ†2𝑔𝐹.(3.1) If 𝑋 is tangent to 𝑀=π΅Γ—πœ†πΉ at (𝑝,π‘ž), then using (3.1), we have ‖𝑋‖2=β€–β€–πœ‹βˆ—π‘‹β€–β€–2+πœ†2β€–β€–πœ‚(πœ‹(𝑋))βˆ—π‘‹β€–β€–2.(3.2) The function πœ† is called the warping function of the warped product. For differentiable function πœ† on M, the gradient βˆ‡πœ† is defined by 𝑔(βˆ‡πœ†,𝑋)=π‘‹πœ†, for all π‘‹βˆˆπ‘‡(𝑀).

Lemma 3.1 (see [7]). Let 𝑀=π΅Γ—πœ†πΉ be a warped product manifold. If 𝑋,π‘Œβˆˆπ‘‡(𝐡) and π‘ˆ,π‘‰βˆˆπ‘‡(𝐹), then βˆ‡π‘‹βˆ‡π‘Œβˆˆπ‘‡(𝐡),(3.3)𝑋𝑉=βˆ‡π‘‰π‘‹=π‘‹πœ†πœ†βˆ‡π‘‰,(3.4)π‘ˆπ‘”π‘‰=βˆ’(π‘ˆ,𝑉)πœ†βˆ‡πœ†.(3.5)

Corollary 3.2. On a warped product manifold 𝑀=π΅Γ—πœ†πΉ one has(i)𝐡 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 π»βˆˆΞ“(tr(𝑇𝑀)) on 𝑀, called the transversal curvature vector field of 𝑀, such that β„Ž(𝑋,π‘Œ)=𝐻𝑔(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀),(3.6) it is easy to see that 𝑀 is a totally umbilical if and only if on each coordinate neighborhood 𝑒, there exist smooth vector fields π»π‘™βˆˆΞ“(ltr(𝑇𝑀)) and π»π‘ βˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)), such that β„Žπ‘™(𝑋,π‘Œ)=𝐻𝑙𝑔(𝑋,π‘Œ),β„Žπ‘ (𝑋,π‘Œ)=𝐻𝑠𝑔(𝑋,π‘Œ),𝐷𝑙(𝑋,π‘Š)=0,(3.7) for any π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)).

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

Proof. Let 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž), then (2.25) and (2.27) imply that π‘“βˆ‡π‘‹π‘Œ=βˆ’π΄π‘€π‘Œπ‘‹βˆ’π΅β„Ž(𝑋,π‘Œ). Let π‘βˆˆΞ“(𝐷0), then π‘”ξ€·π‘“βˆ‡π‘‹ξ€Έξ€·π΄π‘Œ,𝑍=βˆ’π‘”π‘€π‘Œξ€Έ=𝑋,π‘π‘”ξ‚€βˆ‡π‘‹ξ‚π½π‘Œ,𝑍=βˆ’π‘”ξ‚€βˆ‡π‘‹ξ‚π‘Œ,𝐽𝑍=βˆ’π‘”ξ‚€βˆ‡π‘‹π‘Œ,π‘ξ…žξ‚ξ€·=π‘”π‘Œ,βˆ‡π‘‹π‘ξ…žξ€Έ,(3.8) where, π‘ξ…ž=π½π‘βˆˆΞ“(𝐷0). Since π‘‹βˆˆΞ“(π·ξ…ž) and π‘βˆˆΞ“(𝐷0) then (2.26) and (2.28) imply that π‘€βˆ‡π‘‹π‘=β„Ž(𝑋,𝑓𝑍)βˆ’πΆβ„Ž(𝑋,𝑍)=𝐻𝑔(𝑋,𝑓𝑍)βˆ’πΆπ»π‘”(𝑋,𝑍)=0, this implies that βˆ‡π‘‹π‘βˆˆΞ“(𝐷), then (3.8) implies that 𝑔(π‘“βˆ‡π‘‹π‘Œ,𝑍)=0, then the nondegeneracy of the distribution 𝐷0 implies that π‘“βˆ‡π‘‹π‘Œ=0 gives βˆ‡π‘‹π‘ŒβˆˆΞ“(π·ξ…ž) for any 𝑋,π‘ŒβˆˆΞ“(π·ξ…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 π·ξ…ž is integrable.

Proof. Using (2.25) and (2.27) with the above lemma, for any 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž), we get π΄π‘€π‘Œπ‘‹=βˆ’π΅β„Ž(𝑋,π‘Œ),(3.9) this implies π΄π‘€π‘Œπ‘‹βˆˆΞ“(π·ξ…ž) and also π΄π‘€π‘‹π‘Œ=βˆ’π΅β„Ž(π‘Œ,𝑋),(3.10) therefore, using (3.9) and (3.10), we get π΄π‘€π‘Œπ‘‹=π΄π‘€π‘‹π‘Œ, for any 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž). This implies that the distribution π·ξ…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 π·ξ…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=π‘βŸ‚Γ—πœ†π‘π‘‡ 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 π·ξ…ž defines a totally geodesic foliation in 𝑀.
Let β„Žπ‘‡ and 𝐴𝑇 be the second fundamental form and the shape operator of 𝑁𝑇 in 𝑀, then for 𝑋,π‘ŒβˆˆΞ“(𝐷) and π‘βˆˆΞ“(π·ξ…ž), we have 𝑔(β„Žπ‘‡(𝑋,π‘Œ),𝑍)=𝑔(βˆ‡π‘‹π‘Œ,𝑍)=βˆ’π‘”(π‘Œ,βˆ‡π‘‹π‘)=βˆ’π‘”(π‘Œ,βˆ‡π‘‹π‘). Using (3.4), we get π‘”ξ€·β„Žπ‘‡ξ€Έ(𝑋,π‘Œ),𝑍=βˆ’(𝑍lnπœ†)𝑔(𝑋,π‘Œ).(3.11) Now, let ξβ„Ž be the second fundamental form of 𝑁𝑇 in 𝑀, then ξβ„Ž(𝑋,π‘Œ)=β„Žπ‘‡(𝑋,π‘Œ)+β„Žπ‘ (𝑋,π‘Œ)+β„Žπ‘™(𝑋,π‘Œ),(3.12) for any 𝑋,π‘Œ tangent to 𝑁𝑇, then using (3.11), we get π‘”ξ‚€ξξ‚ξ€·β„Žβ„Ž(𝑋,π‘Œ),𝑍=𝑔𝑇(𝑋,π‘Œ),𝑍=βˆ’(𝑍lnπœ†)𝑔(𝑋,π‘Œ).(3.13) Since 𝑁𝑇 is a holomorphic submanifold of 𝑀, then we have ξξξβ„Ž(𝑋,π½π‘Œ)=β„Ž(𝐽𝑋,π‘Œ)=π½β„Ž(𝑋,π‘Œ), therefore, we have π‘”ξ‚€ξξ‚ξ‚€ξξ‚β„Ž(𝑋,π‘Œ),𝑍=βˆ’π‘”β„Ž(𝐽𝑋,π½π‘Œ),𝑍=(𝑍lnπœ†)𝑔(𝑋,π‘Œ).(3.14) Adding (3.13) and (3.14), we get π‘”ξ‚€ξξ‚β„Ž(𝑋,π‘Œ),𝑍=0.(3.15) Using (3.12), we have 𝑔(β„Ž(𝑋,π‘Œ),𝐽𝑍)=𝑔(β„Ž(𝑋,π‘Œ),𝐽𝑍)βˆ’π‘”(β„Žπ‘‡ξξξ(𝑋,π‘Œ),𝐽𝑍)=𝑔(β„Ž(𝑋,π‘Œ),𝐽𝑍)=βˆ’π‘”(π½β„Ž(𝑋,π‘Œ),𝑍)=βˆ’π‘”(β„Ž(𝑋,π½π‘Œ),𝑍)=0. Thus, 𝑔(β„Ž(𝑋,π‘Œ),𝐽𝑍)=0 implies that β„Ž(𝑋,π‘Œ) has no components in 𝐿1βŸ‚πΏ2 for any 𝑋,π‘ŒβˆˆΞ“(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=π‘βŸ‚Γ—πœ†π‘π‘‡ 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=π‘βŸ‚Γ—πœ†π‘π‘‡ other than CR-lightlike product. Thus, for simplicity, we call a warped product CR-lightlike submanifold in the form 𝑀=π‘π‘‡Γ—πœ†π‘βŸ‚ 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=π‘π‘‡Γ—πœ†π‘βŸ‚ be a proper CR-lightlike warped product of an indefinite Kaehler manifold, then 𝑁𝑇 is totally geodesic in 𝑀.

Proof . Let βˆ‡ be a linear connection on 𝑀 induced from βˆ‡. Let 𝑋,π‘Œβˆˆπ‘π‘‡ and π‘βˆˆπ‘βŸ‚, then we have 𝑔(βˆ‡π‘‹π‘Œ,𝑍)=𝑔(βˆ‡π‘‹π‘Œ,𝑍)=βˆ’π‘”(π‘Œ,βˆ‡π‘‹π‘)βˆ’π‘”(π‘Œ,β„Žπ‘™(𝑋,𝑍)), using (3.4), we get 𝑔(βˆ‡π‘‹π‘Œ,𝑍)=βˆ’π‘”(π‘Œ,β„Žπ‘™(𝑋,𝑍)). Since 𝑀 is totally umbilical CR-lightlike submanifold, therefore, β„Žπ‘™(𝑋,𝑍)=β„Žπ‘ (𝑋,𝑍)=0. Hence, 𝑔(βˆ‡π‘‹π‘Œ,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 𝐴𝑀𝑍𝑍=βˆ’π‘“βˆ‡π‘π‘βˆ’π΅β„Žπ‘™(𝑍,𝑍)βˆ’π΅β„Žπ‘ (𝑍,𝑍), for π‘βˆˆΞ“(𝑀2). We obtain 𝑔(𝐴𝐽𝑍𝑍,π½πœ‰)+𝑔(β„Žπ‘™(𝑍,𝑍),πœ‰)=0. Using (2.13) and the hypothesis we obtain 𝑔(𝑍,𝑍)𝑔(𝐻𝑙,πœ‰)=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=π‘π‘‡Γ—πœ†π‘βŸ‚ in an indefinite Kaehler manifold 𝑀, one has (i)𝑔(β„Žπ‘ (𝐷,𝐷),𝐽𝑀2)=0, (ii)𝑔(β„Ž(𝐽𝑋,𝑍),𝐽𝑍1)=(𝑋lnπœ†)𝑔(𝑍,𝑍1), for any π‘‹βˆˆΞ“(𝐷) and 𝑍,𝑍1βˆˆΞ“(𝑀2)βŠ‚Ξ“(π·ξ…ž).

Proof. Since 𝑀 is Kaehlerian, therefore, for π‘‹βˆˆΞ“(𝐷) and π‘βˆˆΞ“(𝑀2), we have π½βˆ‡π‘‹π‘=βˆ‡π‘‹π½π‘, since 𝑀 is totally umbilical, therefore, we have 𝐽(βˆ‡π‘‹π‘)=βˆ’π΄π‘€π‘π‘‹+βˆ‡π‘ π‘‹π‘€π‘, then taking inner product with π½π‘Œ, where π‘ŒβˆˆΞ“(𝐷), we get 𝑔(βˆ‡π‘‹π‘,π‘Œ)=βˆ’π‘”(𝐴𝑀𝑍𝑋,π½π‘Œ). Using (3.4), we obtain 𝑔(𝐴𝑀𝑍𝑋,π½π‘Œ)=0, then using (2.13), we get 𝑔(β„Žπ‘ (𝐷,𝐷),𝐽𝑀2)=0.
Next for any π‘‹βˆˆΞ“(𝐷) and 𝑍,𝑍1βˆˆΞ“(𝑀2)βŠ‚Ξ“(π·ξ…ž), we have 𝑔(β„Ž(𝐽𝑋,𝑍),𝐽𝑍1)=𝑔(βˆ‡π‘π½π‘‹,𝐽𝑍1)=𝑔(βˆ‡π‘π‘‹,𝑍1)=(𝑋lnπœ†)𝑔(𝑍,𝑍1). Hence, the proof is complete.

Corollary 4.2. Let π‘βˆˆΞ“(𝑀1)βŠ‚Ξ“(π·ξ…ž), then clearly 𝑔(β„Žπ‘ (𝐷,𝐷),𝐽𝑍)=0 and also 𝑔(β„Žπ‘™(𝐷,𝐷),𝐽𝑍)=0 for any π‘βˆˆΞ“(π·ξ…ž). Thus, 𝑔(β„Ž(𝐷,𝐷),π½π·ξ…ž)=0, this implies that β„Ž(𝐷,𝐷) has no component in 𝐿1βŸ‚πΏ2, 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)πœ‡)𝑍,(4.1) for π‘‹βˆˆπ·, π‘βˆˆπ·ξ…ž and for some function πœ‡ on 𝑀 satisfying π‘ˆπœ‡=0,π‘ˆβˆˆΞ“(π·ξ…ž).

Proof. Assume that 𝑀 be a proper CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀 satisfying (4.1). Let π‘ŒβˆˆΞ“(𝐷) and π‘βˆˆΞ“(𝑀2)βŠ‚Ξ“(π·ξ…ž), we have 𝑔(𝐴𝐽𝑍𝑋,π½π‘Œ)=𝑔(((𝐽𝑋)πœ‡)𝑍,π½π‘Œ)=0, then using (2.13), we get 𝑔(β„Žπ‘ (𝐷,𝐷),𝐽𝑀2)=0. If π‘βˆˆΞ“(𝑀1)βŠ‚Ξ“(π·ξ…ž), then clearly 𝑔(β„Žπ‘ (𝐷,𝐷),𝐽𝑍)=0 and also 𝑔(β„Žπ‘™(𝐷,𝐷),𝐽𝑍)=0 for any π‘βˆˆΞ“(π·ξ…ž). Thus, π‘”ξ€·β„Ž(𝐷,𝐷),π½π·ξ…žξ€Έ=0,(4.2) that is, β„Ž(𝐷,𝐷) has no component in 𝐿1βŸ‚πΏ2, 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 π‘ˆβˆˆΞ“(π·ξ…ž) and using that 𝑀 is totally umbilical, we get 𝑔(((𝐽𝑋)πœ‡)𝑍,π‘ˆ)=𝑔(𝐴𝐽𝑍𝑋,π‘ˆ)=𝑔(𝐽𝑍,βˆ‡π‘‹π‘ˆ)=𝑔(𝐽𝑍,βˆ‡π‘ˆπ‘‹)=βˆ’π‘”(βˆ‡π‘ˆπ½π‘,𝑋)=𝑔(βˆ‡π‘ˆπ‘,𝐽𝑋), using the definition of gradient 𝑔(βˆ‡πœ™,𝑋)=π‘‹πœ™, we getπ‘”ξ€·βˆ‡π‘ˆξ€Έπ‘,𝐽𝑋=𝑔(βˆ‡πœ‡,𝐽𝑋)𝑔(𝑍,π‘ˆ).(4.3) Let β„Žξ…ž be the second fundamental form of π·ξ…ž in 𝑀 and let βˆ‡ξ…ž be the metric connection of π·ξ…ž in 𝑀 then, particularly for π‘‹βˆˆΞ“(𝐷0), we have π‘”ξ€·β„Žξ…žξ€Έξ€·βˆ‡(π‘ˆ,𝑍),𝐽𝑋=π‘”π‘ˆπ‘βˆ’βˆ‡ξ…žπ‘ˆξ€Έξ€·βˆ‡π‘,𝐽𝑋=π‘”π‘ˆξ€Έπ‘,𝐽𝑋.(4.4) Therefore, from (4.3) and (4.4), we get 𝑔(β„Žξ…ž(π‘ˆ,𝑍),𝐽𝑋)=𝑔(βˆ‡πœ‡,𝐽𝑋)𝑔(𝑍,π‘ˆ), this further implies that β„Žξ…ž(π‘ˆ,𝑍)=βˆ‡πœ‡π‘”(𝑍,π‘ˆ),(4.5) this implies that the distribution π·ξ…ž is totally umbilical in 𝑀. Using Theorem 3.5, the totally real distribution π·ξ…ž is integrable and (4.5) and the condition π‘ˆπœ‡=0 for π‘ˆβˆˆπ·ξ…ž imply that each leaf of π·ξ…ž is an extrinsic sphere in 𝑀. Hence, by a result of [12] which say that β€œif the tangent bundle of a Riemannian manifold 𝑀 splits into an orthogonal sum 𝑇𝑀=𝐸0βŠ•πΈ1 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 𝑀0Γ—πœ†π‘€1,” therefore, we can conclude that 𝑀 is locally a CR-lightlike warped product π‘π‘‡Γ—πœ†π‘βŸ‚ of 𝑀, where πœ†=π‘’πœ‡.
Conversely, let π‘‹βˆˆΞ“(𝑁𝑇) and π‘βˆˆΞ“(π‘βŸ‚), since 𝑀 is a Kaehler manifold so, we have βˆ‡π‘‹π½π‘=π½βˆ‡π‘‹π‘, which further becomes βˆ’π΄π½π‘π‘‹+βˆ‡π‘‘π‘‹π½π‘=((𝐽𝑋)lnπœ†)𝑍, comparing tangential components, we get 𝐴𝐽𝑍𝑋=βˆ’((𝐽𝑋)lnπœ†)𝑍 for each π‘‹βˆˆΞ“(𝐷) and π‘βˆˆ(π·ξ…ž). Since πœ† is a function on 𝑁𝑇, so we also have π‘ˆ(lnπœ†)=0 for all π‘ˆβˆˆΞ“(π·ξ…ž). Hence, the proof is complete.

Lemma 4.4. Let 𝑀=π‘π‘‡Γ—πœ†π‘βŸ‚ be a CR-lightlike warped product of an indefinite Kaehler manifold 𝑀, then ξ€·βˆ‡π‘π‘“ξ€Έξ€·βˆ‡π‘‹=𝑓𝑋(lnπœ†)𝑍.π‘ˆπ‘“ξ€Έπ‘=𝑔(𝑍,π‘ˆ)𝑓(βˆ‡lnπœ†),(4.6) for any π‘ˆβˆˆΞ“(𝑇𝑀),π‘‹βˆˆΞ“(𝑁𝑇), and π‘βˆˆΞ“(π‘βŸ‚).

Proof. For π‘‹βˆˆΞ“(𝑁𝑇) and π‘βˆˆΞ“(π‘βŸ‚), using (2.27) and (3.4), we have (βˆ‡π‘π‘“)𝑋=βˆ‡π‘π‘“π‘‹=𝑓𝑋(lnπœ†)𝑍. Next, again using (2.27), we get (βˆ‡π‘ˆπ‘“)𝑍=βˆ’π‘“βˆ‡π‘ˆπ‘, this implies that (βˆ‡π‘ˆπ‘“)π‘βˆˆΞ“(𝑁𝑇), therefore, for any π‘‹βˆˆΞ“(𝐷0), we have π‘”βˆ‡ξ€·ξ€·π‘ˆπ‘“ξ€Έξ€Έξ€·π‘,𝑋=βˆ’π‘”π‘“βˆ‡π‘ˆξ€Έξ€·βˆ‡π‘,𝑋=π‘”π‘ˆξ€Έ=𝑍,π‘“π‘‹π‘”ξ‚€βˆ‡π‘ˆξ‚ξ€·π‘,𝑓𝑋=βˆ’π‘”π‘,βˆ‡π‘ˆξ€Έπ‘“π‘‹=βˆ’π‘“π‘‹(lnπœ†)𝑔(𝑍,π‘ˆ).(4.7) Hence, using the definition of gradient of πœ† 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 ξ€·βˆ‡π‘ˆπ‘“ξ€Έπ‘‰=(𝑓𝑉(πœ‡))π‘„π‘ˆ+𝑔(π‘„π‘ˆ,𝑄𝑉)𝐽(βˆ‡πœ‡),(4.8) for any π‘ˆ,π‘‰βˆˆΞ“(𝑇𝑀) and for some function πœ‡ on 𝑀 satisfying π‘πœ‡=0,π‘βˆˆΞ“(π·ξ…ž).

Proof. Let 𝑀 be a CR-lightlike submanifold of an indefinite Kaehler manifold 𝑀 satisfying (4.8). Let π‘ˆ,π‘‰βˆˆΞ“(𝐷), then (4.8) implies that (βˆ‡π‘ˆπ‘“)𝑉=0, then (2.25) gives π΅β„Ž(π‘ˆ,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 π‘ˆ,π‘‰βˆˆΞ“(π·ξ…ž), then (4.8) givesξ€·βˆ‡π‘ˆπ‘“ξ€Έπ‘‰=𝑔(π‘„π‘ˆ,𝑄𝑉)𝐽(βˆ‡πœ‡).(4.9) Let π‘‹βˆˆΞ“(𝐷0), then (4.9) implies that π‘”βˆ‡ξ€·ξ€·π‘ˆπ‘“ξ€Έξ€Έπ‘‰,𝑋=𝑔(π‘„π‘ˆ,𝑄𝑉)𝑔(𝐽(βˆ‡πœ‡),𝑋).(4.10) Also π‘”βˆ‡ξ€·ξ€·π‘ˆπ‘“ξ€Έξ€Έξ€·π΄π‘‰,𝑋=𝑔𝑀𝑉=π‘ˆ,π‘‹π‘”ξ‚€βˆ‡π‘ˆξ‚ξ€·βˆ‡π‘‰,𝐽𝑋=π‘”π‘ˆξ€Έπ‘‰,𝐽𝑋,(4.11) therefore, from (4.10) and (4.11), we get π‘”ξ€·βˆ‡π‘ˆξ€Έπ‘‰,𝐽𝑋=βˆ’π‘”(βˆ‡πœ‡,𝐽𝑋)𝑔(π‘ˆ,𝑉).(4.12) Let β„Žξ…ž be the second fundamental form of π·ξ…ž in 𝑀 and let βˆ‡ξ…ž be the metric connection of π·ξ…ž in 𝑀, then π‘”ξ€·β„Žξ…žξ€Έξ€·βˆ‡(π‘ˆ,𝑉),𝐽𝑋=π‘”π‘ˆξ€Έπ‘‰,𝐽𝑋,(4.13) therefore, from (4.12) and (4.13), we get 𝑔(β„Žξ…ž(π‘ˆ,𝑉),𝐽𝑋)=βˆ’π‘”(βˆ‡πœ‡,𝐽𝑋)𝑔(π‘ˆ,𝑉), then the nondegeneracy of the distribution 𝐷0 implies that β„Žξ…ž(π‘ˆ,𝑉)=βˆ’βˆ‡πœ‡π‘”(π‘ˆ,𝑉),(4.14) this gives that the distribution π·ξ…ž is totally umbilical in 𝑀 and using Theorem 3.5, the distribution π·ξ…ž is integrable. Also, π‘πœ‡=0 for π‘βˆˆΞ“(π·ξ…ž), hence as in Theorem 4.3, each leaf of π·ξ…ž is an extrinsic sphere in 𝑀. Thus, 𝑀 is locally a CR-lightlike warped product π‘π‘‡Γ—πœ†π‘βŸ‚ of 𝑀, where πœ†=π‘’πœ‡.
Conversely, let 𝑀 be a CR-lightlike warped product π‘π‘‡Γ—πœ†π‘βŸ‚ of an indefinite Kaehler manifold 𝑀. Using (2.22), we can writeξ€·βˆ‡π‘ˆπ‘“ξ€Έξ€·βˆ‡π‘‰=π‘†π‘ˆπ‘“ξ€Έξ€·βˆ‡π‘†π‘‰+π‘„π‘ˆπ‘“ξ€Έξ€·βˆ‡π‘†π‘‰+π‘ˆπ‘“ξ€Έπ‘„π‘‰.(4.15) Since 𝐷 defines totally geodesic foliation in 𝑀, therefore, using (2.25), we get ξ€·βˆ‡π‘†π‘ˆπ‘“ξ€Έπ‘†π‘‰=0.(4.16) Using (4.6), we have ξ€·βˆ‡π‘„π‘ˆπ‘“ξ€Έξ€·βˆ‡π‘†π‘‰=𝑓𝑉(lnπœ†)π‘„π‘ˆ,(4.17)π‘ˆπ‘“ξ€Έπ‘„π‘‰=𝑔(π‘„π‘ˆ,𝑄𝑉)𝑓(βˆ‡lnπœ†).(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 π‘”βˆ‡ξ€·ξ€·π‘‘π‘ˆπ‘€ξ€Έξ€Έπ‘‰,𝐽𝑍=βˆ’π‘†π‘‰(πœ‡)𝑔(π‘ˆ,𝑍),(4.19) for any π‘ˆ,π‘‰βˆˆΞ“(𝑇𝑀) and for some function πœ‡ on 𝑀 satisfying π‘πœ‡=0,π‘βˆˆΞ“(π·ξ…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 π‘ˆ,π‘‰βˆˆΞ“(𝐷), we get π‘”βˆ‡ξ€·ξ€·π‘‘π‘ˆπ‘€ξ€Έξ€Έξ‚€π‘‰,𝐽𝑍=βˆ’π‘”(β„Ž(π‘ˆ,𝑓𝑉),𝐽𝑍)=βˆ’π‘”βˆ‡π‘ˆξ‚ξ€·βˆ‡π‘‰,𝑍+π‘”π‘ˆξ€Έξ€·βˆ‡π‘“π‘‰,𝐽𝑍=βˆ’π‘”π‘ˆξ€Έξ€·π‘‰,𝑍+π‘”π‘“βˆ‡π‘ˆξ€Έπ‘‰,𝐽𝑍=0.(4.20) For π‘ˆβˆˆΞ“(𝐷),π‘‰βˆˆΞ“(π·ξ…ž) or π‘ˆ,π‘‰βˆˆΞ“(π·ξ…ž), using (2.25), we have π‘”βˆ‡ξ€·ξ€·π‘‘π‘ˆπ‘€ξ€Έξ€Έπ‘‰,𝐽𝑍=0.(4.21) Now let π‘ˆβˆˆΞ“(π·ξ…ž) and π‘‰βˆˆΞ“(𝐷), then using (3.4), we have π‘”βˆ‡ξ€·ξ€·π‘‘π‘ˆπ‘€ξ€Έξ€Έξ€·βˆ‡π‘‰,𝐽𝑍=βˆ’π‘”(β„Ž(π‘ˆ,𝑓𝑉),𝐽𝑍)=βˆ’π‘”π‘ˆξ€Έξ€·π‘‰,𝑍+π‘”π‘“βˆ‡π‘ˆξ€Έπ‘‰,𝐽𝑍=βˆ’π‘‰(lnπœ†)𝑔(π‘ˆ,𝑍).(4.22) Therefore, (4.19) follows from (4.21)–(4.22). Hence, the result is complete.

References

  1. A. Bejancu, β€œCR submanifolds of a Kaehler manifold. II,” Transactions of the American Mathematical Society, vol. 250, pp. 333–345, 1979. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. A. Bejancu, M. Kon, and K. Yano, β€œCR-submanifolds of a complex space form,” Journal of Differential Geometry, vol. 16, no. 1, pp. 137–145, 1981.
  3. B. Y. Chen, β€œCR-submanifolds of a Kaehler manifold. I,” Journal of Differential Geometry, vol. 16, no. 2, pp. 305–322, 1981.
  4. D. E. Blair and B. Y. Chen, β€œOn CR-submanifolds of Hermitian manifolds,” Israel Journal of Mathematics, vol. 34, no. 4, pp. 353–363, 1979. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
  6. R. Kumar, J. Kaur, and R. K. Nagaich, β€œOn CR-lightlike product of an indefinite Kaehler manifold,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 857953, 10 pages, 2011. View at Publisher Β· View at Google Scholar
  7. R. L. Bishop and B. O'Neill, β€œManifolds of negative curvature,” Transactions of the American Mathematical Society, vol. 145, pp. 1–49, 1969.
  8. I. Hasegawa and I. Mihai, β€œContact CR-warped product submanifolds in Sasakian manifolds,” Geometriae Dedicata, vol. 102, pp. 143–150, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  9. B. Sahin, β€œNonexistence of warped product semi-slant submanifolds of Kaehler manifolds,” Geometriae Dedicata, vol. 117, pp. 195–202, 2006. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  10. B. Y. Chen, β€œGeometry of warped product CR-submanifolds in Kaehler manifolds,” Monatshefte für Mathematik, vol. 133, no. 3, pp. 177–195, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  11. K. L. Duggal and D. H. Jin, β€œTotally umbilical lightlike submanifolds,” Kodai Mathematical Journal, vol. 26, no. 1, pp. 49–68, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  12. S. Hiepko, β€œEine innere Kennzeichnung der verzerrten produkte,” Mathematische Annalen, vol. 241, no. 3, pp. 209–215, 1979. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH