On CR-Lightlike Product of an Indefinite Kaehler Manifold

Abstract

We have studied mixed foliate CR-lightlike submanifolds and CR-lightlike product of an indefinite Kaehler manifold and also obtained relationship between them. Mixed foliate CR-lightlike submanifold of indefinite complex space form has also been discussed and showed that the indefinite Kaehler manifold becomes the complex semi-Euclidean space.

1. Introduction

The geometry of CR-submanifolds of Kaehler manifolds was initiated by Bejancu [1] and has been developed by [2–5] and others. They studied the geometry of CR-submanifolds with positive definite metric. Thus, this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite. Moreover, because of growing importance of lightlike submanifolds and hypersurfaces in mathematical physics, especially in relativity, make the geometry of lightlike submanifolds and hypersurfaces a topic of chief discussion in the present scenario. In the establishment of the general theory of lightlike submanifolds and hypersurfaces, Kupeli [6] and Duggal and Bejancu [7] played a very crucial role. The objective of this paper is to study CR-lightlike submanifolds of an indefinite Kaehler manifold. This motivated us to study CR-lightlike submanifolds extensively.

2. Lightlike Submanifolds

We recall notations and fundamental equations for lightlike submanifolds, which are due to [7] by Duggal and Bejancu.

Let (𝑀,𝑔) be a real (𝑚+𝑛)-dimensional semi-Riemannian manifold of constant index 𝑞 such that 𝑚,𝑛≥1, 1≤𝑞≤𝑚+𝑛−1, (𝑀,𝑔) 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∈𝑇_𝑥𝑀∶𝑔(𝑢,𝑣)=0,∀𝑣∈𝑇_𝑥𝑀,𝑥∈𝑀 (2.1) is a degenerate 𝑛-dimensional subspace of 𝑇_𝑥𝑀. Thus, both 𝑇_𝑥𝑀 and 𝑇_𝑥𝑀^⟂ are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace Rad𝑇_𝑥𝑀=𝑇_𝑥𝑀∩𝑇_𝑥𝑀^⟂ which is known as radical (null) subspace. If the mappingRad𝑇𝑀∶𝑥∈𝑀⟶Rad𝑇_𝑥𝑀 (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) where 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 havetr(𝑇𝑀)=ltr(𝑇𝑀)⟂𝑆𝑇𝑀^⟂𝑇,𝑀∣_𝑀=𝑇𝑀⊕tr(𝑇𝑀)=(Rad𝑇𝑀⊕ltr(𝑇𝑀))⟂𝑆(𝑇𝑀)⟂𝑆𝑇𝑀^⟂. (2.4)(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,…,𝜉_𝑟,𝑊_𝑟+1,…,𝑊_𝑛,𝑁₁,…,𝑁_𝑟,𝑋_𝑟+1,…,𝑋_𝑚}, where {𝜉₁,…,𝜉_𝑟}, {𝑁₁,…,𝑁_𝑟} are local lightlike bases of Γ(Rad𝑇𝑀∣_𝑢), Γ(ltr(𝑇𝑀)|_𝑢) and {𝑊_𝑟+1,…,𝑊_𝑛},{𝑋_𝑟+1,…,𝑋_𝑚} are local orthonormal bases of Γ(𝑆(𝑇𝑀^⟂)|_𝑢) and Γ(𝑆(𝑇𝑀)|_𝑢), respectively. For this quasi-orthonormal fields of frames, we have the following.

Theorem 2.1 (see [7]). 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 {𝑁_𝑖} of 𝑆(𝑇𝑀^⟂)^⟂|_u, where u is a coordinate neighborhood of 𝑀, such that 𝑔𝑁_𝑖,𝜉_𝑗=𝛿_𝑖𝑗,𝑔𝑁_𝑖,𝑁_𝑗=0, (2.6) where {𝜉₁,…,𝜉_𝑟} 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),∇_𝑋𝑈=−𝐴_𝑈𝑋+∇^⟂_𝑋𝑈,∀𝑋∈Γ(𝑇𝑀),𝑈∈Γ(tr(𝑇𝑀)), (2.7)(2.8) where {∇_𝑋𝑌,𝐴_𝑈𝑋} 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 𝐴_U 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+ℎ^𝑙(𝑋,𝑌)+ℎ^𝑠(𝑋,𝑌),∇_𝑋𝑈=−𝐴_𝑈𝑋+𝐷^𝑙_𝑋𝑈+𝐷^𝑠_𝑋𝑈, (2.9)(2.10) where we put ℎ^𝑙(𝑋,𝑌)=𝐿(ℎ(𝑋,𝑌)), ℎ^𝑠(𝑋,𝑌)=𝑆(ℎ(𝑋,𝑌)), 𝐷^𝑙_𝑋𝑈=𝐿(∇^⟂_𝑋𝑈),and 𝐷^𝑠_𝑋𝑈=𝑆(∇^⟂_𝑋𝑈).

As ℎ^𝑙 and ℎ^𝑠 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=−𝐴_𝑁𝑋+∇^𝑙_𝑋𝑁+𝐷^𝑠(𝑋,𝑁),∇_𝑋𝑊=−𝐴_𝑊𝑋+∇^𝑠_𝑋𝑊+𝐷^𝑙(𝑋,𝑊), (2.11)(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 obtain𝑔ℎ^𝑠(𝑋,𝑌),𝑊+𝑔𝑌,𝐷^𝑙(𝑋,𝑊)=𝑔𝐴_𝑊𝑋,𝑌,𝑔ℎ^𝑙(𝑋,𝑌),𝜉+𝑔𝑌,ℎ^𝑙(𝑋,𝜉)+𝑔𝑌,∇_𝑋𝜉=0,𝑔𝐴_𝑁𝑋,𝑁^′+𝑔𝑁,𝐴_𝑁′𝑋=0, (2.13) for any 𝜉∈Γ(Rad𝑇𝑀), 𝑊∈Γ(𝑆(𝑇𝑀^⟂)), and 𝑁,𝑁′∈Γ(ltr(𝑇𝑀)).

Let 𝑃 is a projection of 𝑇𝑀 on 𝑆(𝑇𝑀). Now, considering the decomposition (2.3), we can write∇_𝑋𝑃𝑌=∇^∗_𝑋𝑃𝑌+ℎ^∗𝑋,,∇𝑃𝑌_𝑋𝜉=−𝐴^∗_𝜉𝑋+∇_𝑋^∗𝑡𝜉, (2.14) 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 (2.14), we obtain 𝑔ℎ^𝑙𝑋,𝑃𝑌,𝜉=𝑔𝐴^∗_𝜉𝑋,𝑃𝑌,𝑔ℎ^∗𝑋,𝑃𝑌,𝑁=𝑔𝐴_𝑁𝑋,𝑃𝑌. (2.15)

Definition 2.2. Let (𝑀,𝐽,𝑔) be a real 2𝑚-dimensional indefinite Kaehler manifold and 𝑀 an 𝑛-dimensional submanifold of 𝑀. Then, 𝑀 is said to be a CR-lightlike submanifold if the following two conditions are fulfilled:A𝐽(Rad𝑇𝑀) is distribution on 𝑀 such that Rad𝑇𝑀∩𝐽(Rad𝑇𝑀)=0, (2.16)Bthere exist vector bundles 𝑆(𝑇𝑀),𝑆(𝑇𝑀^⟂),ltr(𝑇𝑀),𝐷₀ and 𝐷′ over 𝑀 such that𝑆(𝑇𝑀)=𝐽(Rad𝑇𝑀)⊕𝐷^′⟂𝐷₀,𝐽𝐷₀=𝐷₀,𝐽𝐷^′=𝐿₁⟂𝐿₂, (2.17) where Γ(𝐷₀) is a nondegenerate distribution on 𝑀, Γ(𝐿₁) and Γ(𝐿₂) are vector sub-bundles of Γ(ltr(𝑇𝑀)) and Γ(𝑆(𝑇𝑀^⟂)), respectively.
Clearly, the tangent bundle of a CR-lightlike submanifold is decomposed as𝑇𝑀=𝐷⊕𝐷′, (2.18) where 𝐷=Rad𝑇𝑀⟂𝐽(Rad𝑇𝑀)⟂𝐷₀. (2.19)

Theorem 2.3. Let 𝑀 be a 1-lightlike submanifold of codimension 2 of a real 2 m-dimensional indefinite almost Hermitian manifold (𝑀,𝐽,𝑔) such that 𝐽(Rad𝑇𝑀) is a distribution on 𝑀. Then, 𝑀 is a CR-lightlike submanifold.

Proof. Since 𝑔(𝐽𝜉,𝜉)=0, therefore, 𝐽𝜉 is tangent to 𝑀. Moreover, Rad𝑇𝑀 and 𝐽(Rad𝑇𝑀) are distributions of rank 1 on 𝑀, therefore, Rad𝑇𝑀∩𝐽(Rad𝑇𝑀)={0}. This enables one to choose a screen distribution 𝑆(𝑇𝑀) such that it contains 𝐽(Rad𝑇𝑀). For any 𝑊∈Γ(𝑆(𝑇𝑀^⟂)), we have 𝑔(𝐽𝑊,𝜉)=−𝑔(𝑊,𝐽𝜉)=0 and 𝑔(𝐽𝑊,𝑊)=0. As 𝑆(𝑇𝑀^⟂) is of rank 1, therefore, 𝐽(𝑆(𝑇𝑀^⟂)) is also a distribution on 𝑀 such that 𝐽(𝑆(𝑇𝑀^⟂))∩{𝑇𝑀^⟂⟂𝐽(𝑇𝑀^⟂)}={0} and 𝑆(𝑇𝑀)={𝐽(Rad𝑇𝑀)⟂𝐽(𝑆(𝑇𝑀^⟂))}⊕𝐸, where 𝐸 is a distribution of rank 2 m–5. Now, for any 𝑁∈Γ(ltr(𝑇𝑀)), we have 𝑔(𝐽𝑁,𝜉)=−𝑔(𝑁,𝐽𝜉)=0 and 𝑔(𝐽𝑁,𝑊)=0, therefore, 𝐽𝑁 is tangent to 𝑀 and 𝑔(𝐽𝑁,𝑁)=𝑔(𝐽𝑁,𝐽𝑁)=𝑔(𝐽𝑁,𝐽𝑊)=0. This implies that 𝐽𝑁 has no component in Rad𝑇𝑀, 𝐽(Rad𝑇𝑀), and 𝐽(𝑆(𝑇𝑀^⟂)). Thus, 𝐽𝑁 lies in Γ(𝐸) and we have 𝑆(𝑇𝑀)=𝐽(Rad𝑇𝑀)⊕𝐽(ltr(𝑇𝑀))⟂𝐽𝑆𝑇𝑀^⟂⟂𝐷₀, (2.20) where 𝐷₀ is a nondegenerate distribution; otherwise 𝑆(𝑇𝑀) would be degenerate.
Now, denote 𝐷′=𝐽(ltr(𝑇𝑀))⟂𝐽(𝑆(𝑇𝑀^⟂)), then condition (B) is satisfied, where 𝐽𝐷′=𝐿₁⟂𝐿₂, 𝐿₁=ltr𝑇𝑀, and 𝐿₂=𝑆(𝑇𝑀^⟂). Hence the result is present.

3. Mixed Geodesic and CR-Lightlike Product

Definition 3.1. A CR-lightlike submanifold of an indefinite almost Hermitian manifold is called mixed geodesic CR-lightlike submanifolds if the second fundamental form ℎ satisfies ℎ(𝑋,𝑌)=0, for any 𝑋∈Γ(𝐷) and 𝑌∈Γ(𝐷′).

Definition 3.2. A CR-lightlike submanifold of an indefinite almost Hermitian manifold is called 𝐷-geodesic (resp., 𝐷′-geodesic) CR-lightlike submanifolds if its second fundamental form ℎ, satisfies ℎ(𝑋,𝑌)=0, for any 𝑋,𝑌∈Γ(𝐷) (resp., 𝑋,𝑌∈Γ(𝐷′)).

Definition 3.3 (see [8]). A CR-lightlike submanifold 𝑀 in a Kaehler manifold is said to be a mixed foliate if the distribution 𝐷 is integrable and 𝑀 is mixed totally geodesic CR-lightlike submanifold.
Now, suppose {𝑁₁,𝑁₂,…,𝑁_𝑟} be a basis of Γ(ltr(𝑇𝑀)) with respect to the basis {𝜉₁,𝜉₂,…,𝜉_𝑟} of Γ(Rad𝑇𝑀), such that {𝑁₁,𝑁₂,…,𝑁_𝑝} is a basis of Γ(𝐿₁). Also, consider an orthonormal basis {𝑊₁,𝑊₂,…,𝑊_𝑠} of Γ(𝐿₂).

Theorem 3.4. Let 𝑀 be a CR -lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then, the distribution 𝐷 defines a totally geodesic foliation if 𝑀 is 𝐷-geodesic.

Proof. Distribution 𝐷 defines a totally geodesic foliation, if and only if, ∇_𝑋𝑌∈Γ(𝐷),∀𝑋,𝑌∈Γ(𝐷). (3.1)
Since 𝐷′=𝐽(𝐿₁⟂𝐿₂), therefore (3.1) holds, if and only if, 𝑔(∇_𝑋𝑌,𝐽𝜉_𝑖)=0, 𝑔(∇_𝑋𝑌,𝐽𝑊_𝛼)=0, for all 𝑖∈{1,…,𝑝},𝛼∈{1,…,𝑠}.
Now let 𝑋,𝑌∈Γ(𝐷), then 𝑔(∇_𝑋𝑌,𝐽𝜉_𝑖)=𝑔(∇_𝑋𝑌,𝐽𝜉_𝑖)=−𝑔(ℎ(𝑋,𝐽𝑌),𝜉_𝑖), similarly 𝑔(∇_𝑋𝑌,𝐽𝑊_𝛼)=−𝑔(ℎ(𝑋,𝐽𝑌),𝑊_𝛼). Then, the result follows directly by using hypothesis.

Corollary 3.5. Let 𝑀 be a CR -lightlike submanifold of an indefinite Kaehler manifold 𝑀. If the distribution 𝐷 defines a totally geodesic foliation, then ℎ^𝑠(𝑋,𝐽𝑌)=0, for all 𝑋,𝑌∈Γ(𝐷).

Theorem 3.6 (See [7]). Let 𝑀 be a CR -lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then, the almost complex distribution 𝐷 is integrable, if and only if, the second fundamental form of 𝑀 satisfies ℎ𝑋,𝐽𝑌=ℎ𝐽𝑋,𝑌, (3.2) for any 𝑋,𝑌∈Γ(𝐷).
Now, let 𝑆 and 𝑄 be the projections on 𝐷 and 𝐷′, respectively. Then, one has𝐽𝑋=𝑓𝑋+𝑤𝑋,∀𝑋∈Γ(𝑇𝑀), (3.3) 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, one sets 𝐽𝑉=𝐵𝑉+𝐶𝑉,∀𝑉∈Γ(tr(𝑇𝑀)), (3.4) where 𝐵𝑉 and 𝐶𝑉 are sections of 𝑇𝑀 and tr(𝑇𝑀), respectively.
By using Kaehlerian property of ∇ with (2.7) and (2.8), one has the following lemmas.

Lemma 3.7. Let 𝑀 be a CR -lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then, one has ∇_𝑋𝑓𝑌=𝐴_𝑤𝑌()∇𝑋+𝐵ℎ𝑋,𝑌,_𝑡𝑋𝑤𝑌=𝐶ℎ(𝑋,𝑌)−ℎ𝑋,𝑓𝑌, (3.5)(3.6) for any 𝑋,𝑌∈Γ(𝑇𝑀), where ∇_𝑋𝑓𝑌=∇_𝑋𝑓𝑌−𝑓∇_𝑋𝑌∇_𝑡𝑋,𝑤𝑌=∇^𝑡_𝑋𝑤𝑌−𝑤∇_𝑋𝑌. (3.7)(3.8)

Lemma 3.8. Let 𝑀 be a CR -lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then, one has ∇_𝑋𝐵𝑉=−𝑓𝐴_𝑉𝑋+𝐴_𝐶𝑉∇𝑋,_𝑋𝐶𝑉=−𝑤𝐴_𝑉𝑋−ℎ(𝑋,𝐵𝑉), (3.9)(3.10) for any 𝑋∈Γ(𝑇𝑀) and 𝑉∈Γ(tr(𝑇𝑀)), where ∇_𝑋𝐵𝑉=∇_𝑋𝐵𝑉−𝐵∇^𝑡_𝑋∇𝑉,_𝑋𝐶𝑉=∇^𝑡_𝑋𝐶𝑉−𝐶∇^𝑡_𝑋𝑉. (3.11)(3.12)

Theorem 3.9. 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.

Proof. From (3.6), we obtain ℎ𝑋,𝐽𝑌=𝐶ℎ(𝑋,𝑌)+𝑤∇_𝑋𝑌, (3.13) for all 𝑋,𝑌∈Γ(𝐷). Then, taking into account that ℎ is symmetric and ∇ is torsion free, we obtain ℎ𝑋,𝐽𝑌−ℎ𝑌,𝐽𝑋=𝑤∇_𝑋𝑌−𝑤∇_𝑌𝑋, (3.14) which proves the assertion.

Corollary 3.10. If CR -lightlike submanifold 𝑀 in an indefinite Kaehler manifold is mixed foliate, then, using Theorem 3.9 and Corollary 3.5, it is clear that ℎ(𝑋,𝑌)=0, for any 𝑋∈Γ(𝐷),𝑌∈Γ(𝐷′), and ℎ^𝑠(𝑋,𝐽𝑌)=0, for any 𝑋,𝑌∈Γ(𝐷).

Theorem 3.11. Let 𝑀 be a CR -lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then, 𝑀 is mixed geodesic, if and only if, 𝐴_𝑤𝑍𝑋∈Γ(𝐷), ∇^𝑡_𝑋𝑤𝑍∈Γ(𝐿₁⟂𝐿₂), for any 𝑋∈Γ(𝐷), 𝑍∈Γ(𝐷′).

Proof. Since 𝑀 is a Kaehler manifold, therefore ℎ(𝑋,𝑌)=−𝐽∇𝑋𝐽𝑍−∇_𝑋𝑍=−𝐽{−𝐴_𝐽𝑍𝑋+∇^𝑡_𝑋𝐽𝑍}−∇_𝑋𝑍. Since 𝑍∈Γ(𝐷′), this implies that 𝐽𝑍=𝑤𝑍, 𝑓𝑍=0, therefore, ℎ(𝑋,𝑌)=𝐽(𝐴_𝐽𝑍𝑋)−𝐽(∇^𝑡_𝑋𝐽𝑍)−∇_𝑋𝑍. This gives ℎ(𝑋,𝑌)=𝑓𝐴_𝑤𝑍𝑋+𝑤𝐴_𝑤𝑍𝑋−𝐵∇^𝑡_𝑋𝑤𝑍−𝐶∇^𝑡_𝑋𝑤𝑍−∇_𝑋𝑍; comparing transversal parts, we obtain ℎ(𝑋,𝑌)=𝑤𝐴_𝑤𝑍𝑋−𝐶∇^𝑡_𝑋𝑤𝑍. Thus, 𝑀 is mixed geodesic, if and only if, 𝑤𝐴_𝑤𝑍𝑋=0,𝐶∇^𝑡_𝑋𝑤𝑍=0. This implies 𝐴_𝑤𝑍𝑋∈Γ(𝐷),∇^𝑡_𝑋𝑤𝑍∈Γ(𝐿₁⟂𝐿₂).

Lemma 3.12. Let 𝑀 be a CR -lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then, ∇_𝑋𝑍=−𝑓𝐴_𝑊𝑋+𝐵∇^𝑠_𝑋𝑊+𝐵𝐷^𝑙(𝑋,𝑊), (3.15) for any 𝑋∈Γ(𝑇𝑀),𝑍∈Γ(𝐽𝐿₂) and 𝑊∈Γ(𝐿₂).

Proof. Let 𝑊∈Γ(𝐿₂) such that 𝑍=𝐽𝑊. Since 𝑀 is a Kaehler manifold, therefore, ∇_𝑋𝑍=𝐽∇𝑋𝑊, for any 𝑋∈Γ(𝑇𝑀). Then, ∇_𝑋𝑍+ℎ(𝑋,𝑍)=𝐽(−𝐴_𝑊𝑋+∇^𝑠_𝑋𝑊+𝐷^𝑙(𝑋,𝑊)=−𝑓𝐴_𝑊𝑋−𝑤𝐴_𝑊𝑋+𝐵∇^𝑠_𝑋𝑊+𝐶∇^𝑠_𝑋𝑊+𝐵𝐷^𝑙(𝑋,𝑊)+𝐶𝐷^𝑙(𝑋,𝑊). Comparing the tangential parts, we obtain (3.9).
Let 𝑀 be an indefinite Kaehler manifold of constant holomorphic sectional curvature 𝑐, then curvature tensor is given by𝑅()𝑐𝑋,𝑌𝑍=4𝑔(𝑌,𝑍)𝑋−𝑔(𝑋,𝑍)𝑌+𝑔𝐽𝑌,𝑍−𝐽𝑋𝑔𝐽𝑋,𝑍𝐽𝑌+2𝑔𝑋,𝐽𝑌𝐽𝑍. (3.16)

Theorem 3.13. Let 𝑀 be a mixed foliate CR -lightlike submanifold of an indefinite complex space form 𝑀(𝑐). Then, 𝑀 is a complex semi-Euclidean space.

Proof. From (3.16), we have 𝑔𝑅𝑋,𝐽𝑋𝑍,𝑐𝐽𝑍=−2𝑔(𝑋,𝑋)𝑔(𝑍,𝑍), (3.17) for any 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝐽𝐿₂). Since, 𝑀 is a mixed foliate, therefore, for 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝐽𝐿₂), we have 𝑔𝑅𝑋,𝐽𝑋𝑍,𝐽𝑍=𝑔∇_𝑋ℎ^𝑠𝐽𝑋,𝑍−∇_𝐽𝑋ℎ^𝑠(𝑋,𝑍),𝐽𝑍, (3.18) where (∇_𝑋ℎ^𝑠)(𝐽𝑋,𝑍)= ∇^𝑠_𝑋(ℎ^𝑠(𝐽𝑋,𝑍))− ℎ^𝑠(∇_𝑋𝐽𝑋,𝑍)− ℎ^𝑠(𝐽𝑋,∇_𝑋𝑍) and (∇_𝐽𝑋ℎ^𝑠)(𝑋,𝑍)=  ∇^𝑠_𝐽𝑋(ℎ^𝑠(𝑋,𝑍))− ℎ^𝑠(∇_𝐽𝑋𝑋,𝑍)− ℎ^𝑠(𝑋,∇_𝐽𝑋𝑍). By hypothesis, 𝑀 is mixed geodesic, therefore, (∇_𝑋ℎ^𝑠)(𝐽𝑋,𝑍)− (∇_𝐽𝑋ℎ^𝑠)(𝑋,𝑍)=  ℎ^𝑠(∇_𝐽𝑋𝑋,𝑍)+ ℎ^𝑠(𝑋,∇_𝐽𝑋𝑍)− ℎ^𝑠(∇_𝑋𝐽𝑋,𝑍)− ℎ^𝑠(𝐽𝑋,∇_𝑋𝑍). Again, by using hypothesis and Theorem 3.9, the distribution 𝐷 defines totally geodesic foliation. Hence, ∇_𝐽𝑋𝑋,∇_𝑋𝐽𝑋∈Γ(𝐷). Therefore, we have (∇_𝑋ℎ^𝑠)(𝐽𝑋,𝑍)− (∇_𝐽𝑋ℎ^𝑠)(𝑋,𝑍)= ℎ^𝑠(𝑋,∇_𝐽𝑋𝑍)− ℎ^𝑠(𝐽𝑋,∇_𝑋𝑍); using Lemma 3.12 here, we obtain (∇_𝑋ℎ^𝑠)(𝐽𝑋,𝑍)− (∇_𝐽𝑋ℎ^𝑠)(𝑋,𝑍)= −ℎ^𝑠(𝑋,𝑓𝐴_𝑊𝐽𝑋)+ ℎ^𝑠(𝑋,𝐵∇^𝑠_𝐽𝑋𝑊)+ ℎ^𝑠(𝑋,𝐵𝐷^𝑙(𝐽𝑋,𝑊))+ ℎ^𝑠(𝐽𝑋,𝑓𝐴_𝑊𝑋)− ℎ^𝑠(𝐽𝑋,𝐵∇^𝑠_𝑋𝑊)− ℎ^𝑠(𝐽𝑋,𝐵𝐷^𝑙(𝑋,𝑊)).
By using, 𝑀 is mixed geodesic and Corollary 3.10, we obtain∇_𝑋ℎ^𝑠𝐽𝑋,𝑍−∇_𝐽𝑋ℎ^𝑠(𝑋,𝑍)=0. (3.19) Therefore, from (3.17)–(3.19), we get (𝑐/2)𝑔(𝑋,𝑋)𝑔(𝑍,𝑍)=0. Since 𝐷₀ and 𝐽𝐿₂ are non-degenerate, therefore, we have 𝑐=0, which completes the proof.

Definition 3.14 (see [7]). A CR-lightlike submanifold 𝑀 of a Kaehler manifold 𝑀 is called a CR-lightlike product if both the distribution 𝐷 and 𝐷′ define totally geodesic foliations on 𝑀.

Theorem 3.15 (see [7]). 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).

Theorem 3.16 (see [7]). 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′), one has ℎ^∗()𝐴𝑋,𝑌=0,_𝐽𝑌𝑋hasnocomponentin𝐷₀,𝐴_𝐽𝑌𝑋+𝐵ℎ(𝑋,𝑌)hasnocomponentinRad𝑇𝑀. (3.20)(3.21)(3.22)

Theorem 3.17 (see [3]). A CR -submanifold of a Kaehler manifold 𝑀 is a CR -product, if and only if, 𝑃 is parallel, that is, ∇𝑃=0, where 𝐽𝑋=𝑃𝑋+𝐹𝑋.

Now, we will give the characterization of CR-lightlike product in the following form.

Theorem 3.18. A CR -lightlike submanifold of an indefinite Kaehler manifold 𝑀 is a CR -lightlike product, if and only if, 𝑓 is parallel, that is, ∇𝑓=0.

Proof. If 𝑓 is parallel, then (3.5), gives 𝐵ℎ(𝑋,𝑌)=−𝐴_𝑤𝑌𝑋, (3.23) for any 𝑋,𝑌∈Γ(𝑇𝑀). In particular, if 𝑋∈Γ(𝐷), then 𝑤𝑋=0. Hence, (3.23) implies 𝐵ℎ(𝑋,𝑌)=0, thus, Theorem 3.15 follows.
Let 𝑋,𝑌∈Γ(𝐷′), then 𝐽𝑌=𝑤𝑌, then (3.23) implies 𝐴_𝐽𝑌𝑋+𝐵ℎ(𝑋,𝑌)=0, and, hence, (3.22) follows. For any 𝑍∈Γ(𝐷₀), 𝑔(𝐴_𝐽𝑌𝑋,𝑍)=𝑔(𝐴_𝑤𝑌𝑋,𝑍)=−𝑔(𝐵ℎ(𝑋,𝑌),𝑍)=𝑔(ℎ(𝑋,𝑌),𝐽𝑍)=0. Thus, (3.21) holds well. Since (∇_𝑋𝑓)𝑌=0 for any 𝑋,𝑌∈Γ(𝑇𝑀), let 𝑋,𝑌∈Γ(𝐷′), then we have 𝑓∇_𝑋𝑌=0. Hence, ∇_𝑋𝑌∈Γ(𝐷′). Therefore, 𝑔(∇_𝑋𝑌,𝑁)=0, for any 𝑋,𝑌∈Γ(𝐷′) and 𝑁∈Γ(ltr(𝑇𝑀)) and consequently ℎ^∗(𝑋,𝑌)=0. Hence, (3.20) holds well. Thus, 𝑀 is a CR-lightlike product in 𝑀.
Conversely, if 𝑀 is a CR-lightlike product, then by definition, 𝐷 and 𝐷′ both defines totally geodesic foliation, that is, ∇_𝑋𝑌∈Γ(𝐷), for any 𝑋,𝑌∈Γ(𝐷) and ∇_𝑋𝑌∈Γ(𝐷′), for any 𝑋,𝑌∈Γ(𝐷′). If ∇_𝑋𝑌∈Γ(𝐷) for any 𝑋,𝑌∈Γ(𝐷), then by comparing the transversal parts in the simplification of Kaehlerian property of ∇, we get ℎ𝑋,𝐽𝑌=𝐽ℎ(𝑋,𝑌). (3.24) Therefore, from (2.7), (3.7), and (3.24), we may prove that (∇_𝑋𝑓)𝑌=0 or ∇𝑓=0.
Let ∇_𝑋𝑌∈Γ(𝐷′), for any 𝑋,𝑌∈Γ(𝐷′), then by comparing the tangential parts in the simplification of Kaehlerian property of ∇, we get𝐵ℎ(𝑋,𝑌)+𝐴_𝑤𝑌𝑋=0. (3.25) Thus, from (3.5) and (3.25), we obtain (∇_𝑋𝑓)𝑌=0 or ∇𝑓=0. Hence, the proof is complete.

Theorem 3.19. Let 𝑀 be a CR -lightlike submanifold of an indefinite Kaehler manifold 𝑀. If 𝑀 is a CR -lightlike product, then it is a mixed foliate CR-lightlike submanifold.

Proof. Since 𝑀 is a CR-lightlike product then by Theorem 3.9, it is clear that the distribution 𝐷 is integrable. Also, by using Theorem 3.18, for CR-lightlike product, ∇𝑓=0, then (3.5) gives 𝐵ℎ(𝑋,𝑌)=−𝐴_𝑤𝑌𝑋, (3.26) for any 𝑋,𝑌∈Γ(𝑇𝑀). Then, in particular, if 𝑋∈Γ(𝐷), then 𝑤𝑋=0. Hence, (3.26) implies 𝐵ℎ(𝑋,𝑌)=0, that is, 𝐴_𝐽𝑍𝑋=0, (3.27) for any 𝑍∈Γ(𝐷′) and 𝑋∈Γ(𝐷).
Next, by the definition of CR-lightlike submanifold 𝑀 is mixed geodesic, if and only if, 𝑔(ℎ(𝑋,𝑌),𝜉)=0 and 𝑔(ℎ(𝑋,𝑌),𝑊)=0, for any 𝑋∈Γ(𝐷), 𝑌∈Γ(𝐷′), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). By hypothesis, we have (∇_𝑋𝑓)𝑌=0 for any 𝑋,𝑌∈Γ(𝑇𝑀); let 𝑋∈Γ(𝐷) and 𝑌∈Γ(𝐷′), then we have 𝑓∇_𝑋𝑌=0. Hence, ∇_𝑋𝑌∈Γ(𝐷′). Therefore, 𝑔(ℎ(𝑋,𝑌),𝜉)=𝑔(∇_𝑋𝑌,𝜉), since 𝑌∈Γ(𝐷′), therefore, there exists 𝑉∈Γ(𝐿₁⟂𝐿₂) such that 𝑌=𝐽𝑉; this gives 𝑔(ℎ(𝑋,𝑌),𝜉)=𝑔(∇_𝑋𝐽𝑉,𝜉)=−𝑔(∇_𝑋𝑉,𝐽𝜉)=𝑔(𝐴_𝑉𝑋,𝐽𝜉)=−𝑔(𝐴_𝐽𝑌𝑋,𝐽𝜉)=0, by virtue of (3.27).
Similarly, 𝑔(ℎ(𝑋,𝑌),𝑊)=𝑔(𝐽∇_𝑋𝑉,𝑊)=−𝑔(𝐶∇^𝑡_𝑋𝑉,𝑊). Now, from (3.10) and (3.12), we have 𝐶∇^𝑡_𝑋𝑉=∇^𝑡_𝑋𝐶𝑉+𝑤𝐴_𝑉𝑋+ℎ(𝑋,𝐵𝑉); since 𝑉∈Γ(𝐿₁⟂𝐿₂), therefore 𝐶𝑉=0, so we have 𝐶∇^𝑡_𝑋𝑉=𝑤𝐴_𝑉𝑋+ℎ(𝑋,𝐵𝑉). Therefore, by using (3.27), we have 𝑔(ℎ(𝑋,𝑌),𝑊)=−𝑔(ℎ(𝑋,𝐵𝑉),𝑊)=−𝑔(ℎ(𝑋,𝐽𝑌),𝑊); this gives 𝑔(ℎ(𝑋,𝑌),𝑊)=0, for all 𝑋∈Γ(𝐷),𝑌∈Γ(𝐷′). Hence, the proof is complete.