Characterization of Holomorphic Bisectional Curvature of GCR-Lightlike Submanifolds

Abstract

We obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite Kaehler manifold. We discuss the boundedness of holomorphic sectional curvature of GCR-lightlike submanifolds of an indefinite complex space form. We establish a condition for a GCR-lightlike submanifold of an indefinite complex space form to be null holomorphically flat. We also obtain some characterization theorems for holomorphic sectional and holomorphic bisectional curvature.

1. Introduction

The study of 𝐶𝑅 submanifolds of Kaehler manifolds was initiated by Bejancu [1], as a generalization of totally real and complex submanifolds and further developed by [2–7]. The 𝐶𝑅 structures on real hypresurfaces of complex manifolds have interesting applications to relativity. Penrose [8] discovered a correspondence, called Penrose correspondence, between points of a Minkowski space and projective lines of a certain real hypersurfaces in a complex projective space, which is an interesting means of passing from the geometry of a Minkowski space to the geometry of a 𝐶𝑅 manifold. Duggal [9, 10] studied the geometry of 𝐶𝑅 submanifolds with Lorentzian metric and obtained their interaction with relativity. The theory of lightlike submanifolds has interaction with some results on Killing horizon, electromagnetic, and radition fields and asymptotically flat spacetimes (for detail see chapters 7, 8, and 9 of [11]). Thus due to the significant applications of 𝐶𝑅 structures in relativity and growing importance of lightlike submanifolds in mathematical physics and relativity, Duggal and Bejancu [11] introduced the notion of 𝐶𝑅-lightlike submanifolds of indefinite Kaehler manifolds which have direct relation with physically important asymptotically flat space time which further lead to Twistor theory of Penrose and Heaven theory of Newman. Moreover, they concluded that, contrary to the 𝐶𝑅-non degenerate submanifolds, 𝐶𝑅-lightlike submanifolds do not include invariant (complex) and totally real lightlike submanifolds. Therefore, Duggal and Sahin [12] introduced 𝑆𝐶𝑅-lightlike submanifolds of indefinite Kaehler manifold which contain complex and totally real subcases but there was no inclusion relation between 𝐶𝑅 and 𝑆𝐶𝑅 cases. Later on, Duggal and Sahin [13] introduced 𝐺𝐶𝑅-lightlike submanifolds of indefinite Kaehler manifolds, which behaves as an umbrella of invariant (complex), screen real and 𝐶𝑅-lightlike submanifolds and also studied the existence (or nonexistence) of this new class in an indefinite space form. R. Kumar et al. [14] studied geodesic 𝐺𝐶𝑅-lightlike submanifolds of indefinite Kaehler manifolds and obtained some characterization theorems for a 𝐺𝐶𝑅-lightlike submanifold to be a 𝐺𝐶𝑅-lightlike product.

Since sectional curvature offers a lot of information concerning the intrinsic geometry of Riemannian manifolds, therefore in this paper, we obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold. In [15], Kulkarni showed that the boundedness of the sectional curvature on a semi-Riemannian manifold implies the constancy of the sectional curvature. In [16], Bonome et al. showed that the boundedness of the holomorphic sectional curvature on indefinite almost Hermitian manifolds leads to the space of pointwise constant holomorphic sectional curvature. Therefore in Section 4, we discuss the boundedness of holomorphic sectional curvature of 𝐺𝐶𝑅-lightlike submanifolds of an indefinite complex space form. In Section 5, we established a condition for a 𝐺𝐶𝑅-lightlike submanifold of an indefinite complex space form to be null holomorphically flat. We also obtain some characterization theorems on holomorphic sectional and holomorphic bisectional curvature.

2. Lightlike Submanifolds

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 (see [11]). For a degenerate metric 𝑔 on 𝑀, 𝑇𝑀^⟂ 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 mapping Rad𝑇𝑀∶𝑥∈𝑀→Rad𝑇_𝑥𝑀 defines a smooth distribution on 𝑀 of rank 𝑟>0, then the submanifold 𝑀 of 𝑀 is called an 𝑟-lightlike submanifold and Rad𝑇𝑀 is called the radical distribution on 𝑀.

Screen distribution 𝑆(𝑇𝑀) is a semi-Riemannian complementary distribution of Rad(𝑇𝑀) in 𝑇𝑀, that is 𝑇𝑀=Rad𝑇𝑀⟂𝑆(𝑇𝑀), (2.1) and 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,𝑀|_𝑀=𝑇𝑀⊕tr(𝑇𝑀)=(Rad𝑇𝑀⊕ltr(𝑇𝑀))⟂𝑆(𝑇𝑀)⟂𝑆𝑇𝑀^⟂. (2.2)(2.3) For a quasi-orthonormal fields of frames on 𝑇𝑀, we have the following.

Theorem 2.1 (see [11]). 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,forany𝑖,𝑗∈{1,2,…,𝑟}, (2.4) 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 (2.3), the Gauss and Weingarten formulas are given by the following: ∇_𝑋𝑌=∇_𝑋𝑌+ℎ(𝑋,𝑌),∇_𝑋𝑈=−𝐴_𝑈𝑋+∇^⟂_𝑋𝑈, (2.5) for any 𝑋,𝑌∈Γ(𝑇𝑀) and 𝑈∈Γ(tr(𝑇𝑀)), 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 a operator on 𝑀 and known as shape operator.

According to (2.2) considering the projection morphisms 𝐿 and 𝑆 of tr(𝑇𝑀) on ltr(𝑇𝑀) and 𝑆(𝑇𝑀^⟂), respectively, then Gauss and Weingarten formulas become ∇_𝑋𝑌=∇_𝑋𝑌+ℎ^𝑙(𝑋,𝑌)+ℎ^𝑠(𝑋,𝑌),∇_𝑋𝑈=−𝐴_𝑈𝑋+𝐷^𝑙_𝑋𝑈+𝐷^𝑠_𝑋𝑈, (2.6) where we put ℎ^𝑙(𝑋,𝑌)=𝐿(ℎ(𝑋,𝑌)),ℎ^𝑠(𝑋,𝑌)=𝑆(ℎ(𝑋,𝑌)),𝐷^𝑙_𝑋𝑈=𝐿(∇^⟂_𝑋𝑈), 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), (2.7) where 𝑋∈Γ(𝑇𝑀),𝑁∈Γ(ltr(𝑇𝑀)), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). Using (2.6) and (2.7), we obtain 𝑔ℎ^𝑠(𝑋,𝑌),𝑊+𝑔𝑌,𝐷^𝑙(𝑋,𝑊)=𝑔𝐴_𝑊𝑋,𝑌,𝑔𝐷^𝑠(𝑋,𝑁),𝑊=𝑔𝐴_𝑊𝑋,𝑁, (2.8)(2.9) for any 𝑋∈Γ(𝑇𝑀), 𝑊∈Γ(𝑆(𝑇𝑀^⟂)), and 𝑁,𝑁^′∈Γ(ltr(𝑇𝑀)).

Let 𝑃 be the projection morphism of 𝑇𝑀 on 𝑆(𝑇𝑀), then, using (2.1), we can induce some new geometric objects on the screen distribution 𝑆(𝑇𝑀) on 𝑀 as follows: ∇_𝑋𝑃𝑌=∇^∗_𝑋𝑃𝑌+ℎ^∗(𝑋,𝑃𝑌),∇_𝑋𝜉=−𝐴^∗_𝜉𝑋+∇_𝑋^∗𝑡𝜉, (2.10) 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. Using (2.6) and (2.10), we obtain 𝑔ℎ^𝑙(𝑋,𝑃𝑌),𝜉=𝑔𝐴^∗_𝜉𝑋,𝑃𝑌,𝑔ℎ^∗(𝑋,𝑃𝑌),𝑁=𝑔𝐴_𝑁𝑋,𝑃𝑌, (2.11) for any 𝑋,𝑌∈Γ(𝑇𝑀),𝜉∈Γ(Rad(𝑇𝑀)) and 𝑁∈Γ(ltr(𝑇𝑀)).

Denote by 𝑅 and 𝑅 the curvature tensors of ∇ and ∇, respectively, then by straightforward calculations ([11]), we have 𝑅(𝑋,𝑌)𝑍=𝑅(𝑋,𝑌)𝑍+𝐴_{ℎ𝑙(𝑋,𝑍)}𝑌−𝐴_{ℎ𝑙(𝑌,𝑍)}𝑋+𝐴_{ℎ𝑠(𝑋,𝑍)}𝑌−𝐴_{ℎ𝑠(𝑌,𝑍)}𝑋+∇_𝑋ℎ^𝑙(𝑌,𝑍)−∇_𝑌ℎ^𝑙(𝑋,𝑍)+𝐷^𝑙𝑋,ℎ^𝑠(𝑌,𝑍)−𝐷^𝑙𝑌,ℎ^𝑠(𝑋,𝑍)+∇_𝑋ℎ^𝑠()−∇𝑌,𝑍_𝑌ℎ^𝑠(𝑋,𝑍)+𝐷^𝑠𝑋,ℎ^𝑙(𝑌,𝑍)−𝐷^𝑠𝑌,ℎ^𝑙(𝑋,𝑍), (2.12) where ∇_𝑋ℎ^𝑠(𝑌,𝑍)=∇^𝑠_𝑋ℎ^𝑠(𝑌,𝑍)−ℎ^𝑠∇_𝑋𝑌,𝑍−ℎ^𝑠𝑌,∇_𝑋𝑍∇_𝑋ℎ^𝑙(𝑌,𝑍)=∇^𝑙_𝑋ℎ^𝑙(𝑌,𝑍)−ℎ^𝑙∇_𝑋𝑌,𝑍−ℎ^𝑙𝑌,∇_𝑋𝑍. (2.13)(2.14) Then Codazzi equation is given, respectively, by the following: 𝑅(𝑋,𝑌)𝑍^⟂=∇_𝑋ℎ^𝑙(𝑌,𝑍)−∇_𝑌ℎ^𝑙(𝑋,𝑍)+𝐷^𝑙𝑋,ℎ^𝑠(𝑌,𝑍)−𝐷^𝑙𝑌,ℎ^𝑠(𝑋,𝑍)+∇_𝑋ℎ^𝑠()−∇𝑌,𝑍_𝑌ℎ^𝑠(𝑋,𝑍)+𝐷^𝑠𝑋,ℎ^𝑙(𝑌,𝑍)−𝐷^𝑠𝑌,ℎ^𝑙(𝑋,𝑍). (2.15) Barros and Romero [17] defined indefinite Kaehler manifolds as follows.

Definition 2.2. Let (𝑀,𝐽,𝑔) be an indefinite almost Hermitian manifold and ∇ the Levi-Civita connection on 𝑀, with respect to an indefinite metric 𝑔. Then 𝑀 is called an indefinite Kaehler manifold if 𝐽 is parallel, with respect to ∇, that is 𝐽²=−𝐼,∇_𝑋𝐽𝑌=0,𝑔(𝐽𝑋,𝐽𝑌)=𝑔(𝑋,𝑌)∀𝑋,𝑌∈Γ𝑇𝑀. (2.16)

3. Generalized Cauchy-Riemann Lightlike Submanifolds

Definition 3.1. Let (𝑀,𝑔,𝑆(𝑇𝑀)) be a real lightlike submanifold of an indefinite Kaehler manifold (𝑀,𝑔,𝐽), then 𝑀 is called a generalized Cauchy-Riemann (𝐺𝐶𝑅)-lightlike submanifold if the following conditions are satisfied.(A) There exist two subbundles 𝐷₁ and 𝐷₂ of Rad(𝑇𝑀) such that Rad(𝑇𝑀)=𝐷₁⊕𝐷₂,𝐽𝐷₁=𝐷₁,𝐽𝐷₂⊂𝑆(𝑇𝑀). (3.1)(B) There exist two subbundles 𝐷₀ and 𝐷^′ of 𝑆(𝑇𝑀) such that 𝑆(𝑇𝑀)=𝐽𝐷₂⊕𝐷^′⟂𝐷₀,𝐽𝐷₀=𝐷₀,𝐽𝐷^′=𝐿₁⟂𝐿₂, (3.2)where 𝐷₀ is a nondegenerate distribution on 𝑀, 𝐿₁ and 𝐿₂ are vector bundle of ltr(𝑇𝑀) and 𝑆(𝑇𝑀)^⟂, respectively.

Then the tangent bundle 𝑇𝑀 of 𝑀 is decomposed as 𝑇𝑀=𝐷⟂𝐷^′, where 𝐷=Rad(𝑇𝑀)⊕𝐷₀⊕𝐽𝐷₂. M is called a proper 𝐺𝐶𝑅-lightlike submanifold if 𝐷₁≠{0},𝐷₂≠{0},𝐷₀≠{0}, and 𝐿₂≠{0}. Let 𝑄,𝑃₁ and 𝑃₂ be the projections on 𝐷, 𝐽(𝐿₁)=𝑀₁, and 𝐽(𝐿₂)=𝑀₂, respectively. Then, for any 𝑋∈Γ(𝑇𝑀), we have 𝑋=𝑄𝑋+𝑃₁𝑋+𝑃₂𝑋, (3.3) applying 𝐽 to (3.3), we obtain 𝐽𝑋=𝑇𝑋+𝑤𝑃₁𝑋+𝑤𝑃₂𝑋, (3.4) and we can write (3.4) as follwos: 𝐽𝑋=𝑇𝑋+𝑤𝑋, (3.5) where 𝑇𝑋 and 𝑤𝑋 are the tangential and transversal components of 𝐽𝑋, respectively. Similarly, 𝐽𝑉=𝐵𝑉+𝐶𝑉, (3.6) for any 𝑉∈Γ(tr(𝑇𝑀)), where 𝐵𝑉 and 𝐶𝑉 are the sections of 𝑇𝑀 and tr(𝑇𝑀), respectively. Applying 𝐽 to (3.5) and (3.6), we get 𝑇²=−𝐼−𝐵𝜔,𝐶²=−𝐼−𝜔𝐵. (3.7)

Differentiating (3.4) and using (2.6), (2.7), and (3.6), we have 𝐷^𝑠𝑋,𝑤𝑃₁𝑌=−∇^𝑠_𝑋𝑤𝑃₂𝑌+𝑤𝑃₂∇_𝑋𝑌−ℎ^𝑠(𝑋,𝑇𝑌)+𝐶ℎ^𝑠(),𝐷𝑋,𝑌_𝑙𝑋,𝑤𝑃₂𝑌=−∇^𝑙_𝑋𝑤𝑃₁𝑌+𝑤𝑃₁∇_𝑋𝑌−ℎ^𝑙(𝑋,𝑇𝑌)+𝐶ℎ^𝑙(𝑋,𝑌). (3.8)(3.9) Using Kaehlerian property of ∇ with (2.7), we have the following lemmas.

Lemma 3.2. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehlerian manifold 𝑀. Then one has ∇_𝑋𝑇𝑌=𝐴_𝑤𝑌𝑋+𝐵ℎ(𝑋,𝑌),∇^𝑡_𝑋𝑤𝑌=𝐶ℎ(𝑋,𝑌)−ℎ(𝑋,𝑇𝑌), (3.10) where 𝑋,𝑌∈Γ(𝑇𝑀) and ∇_𝑋𝑇𝑌=∇_𝑋𝑇𝑌−𝑇∇_𝑋𝑌,∇^𝑡_𝑋𝑤𝑌=∇^𝑡_𝑋𝑤𝑌−𝑤∇_𝑋𝑌. (3.11)

Lemma 3.3. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehlerian manifold 𝑀. Then one has ∇_𝑋𝐵𝑉=𝐴_𝐶𝑉𝑋−𝑇𝐴_𝑉𝑋,∇^𝑡_𝑋𝐶𝑉=−𝑤𝐴_𝑉𝑋−ℎ(𝑋,𝐵𝑉), (3.12) where 𝑋∈Γ(𝑇𝑀), 𝑉∈Γ(tr(𝑇𝑀)), and ∇_𝑋𝐵𝑉=∇_𝑋𝐵𝑉−𝐵∇^𝑡_𝑋𝑉,∇^𝑡_𝑋𝐶𝑉=∇^𝑡_𝑋𝐶𝑉−𝐶∇^𝑡_𝑋𝑉. (3.13)

4. Holomorphic Sectional Curvature of a 𝐺𝐶𝑅-Lightlike Submanifold

Let 𝑀 be a complex space form of constant holomorphic curvature 𝑐. Then the curvature tensor 𝑅 of 𝑀(𝑐) is given by the following: 𝑅()𝑐𝑋,𝑌𝑍=4𝑔(𝑌,𝑍)𝑋−𝑔(𝑋,𝑍)𝑌+𝑔()−𝐽𝑌,𝑍𝐽𝑋𝑔(𝐽𝑋,𝑍)𝐽𝑌+2𝑔(𝑋,𝐽𝑌)𝐽𝑍, (4.1) for 𝑋,𝑌,𝑍 vector fields on 𝑀. Using (4.1) and (2.12), we obtain 𝑔(𝑅())=𝑐𝑋,𝑌𝑍,𝑊4𝑔()𝑔()()𝑔()()𝑔()()𝑔()()𝑔(𝐴𝑌,𝑍𝑋,𝑊−𝑔𝑋,𝑍𝑌,𝑊+𝑔𝐽𝑌,𝑍𝐽𝑋,𝑊−𝑔𝐽𝑋,𝑍𝐽𝑌,𝑊+2𝑔𝑋,𝐽𝑌𝐽𝑍,𝑊)−𝑔_{ℎ𝑙(𝑋,𝑍)}𝐴𝑌,𝑊+𝑔_{ℎ𝑙(𝑌,𝑍)}𝑋,𝑊−𝑔𝐴_{ℎ𝑠(𝑋,𝑍)}𝑌,𝑊+𝑔𝐴_{ℎ𝑠(𝑌,𝑍)}−𝑋,𝑊𝑔∇_𝑋ℎ^𝑙(𝑌,𝑍),𝑊+𝑔∇_𝑌ℎ^𝑙(𝑋,𝑍),𝑊−𝑔𝐷^𝑙𝑋,ℎ^𝑠()+𝑌,𝑍,𝑊𝑔𝐷^𝑙𝑌,ℎ^𝑠(𝑋,𝑍),𝑊. (4.2) Using (2.8) in (4.2), we obtain 𝑔(𝑅())=𝑐𝑋,𝑌𝑍,𝑊4𝑔()𝑔()()𝑔()()𝑔()()𝑔()()𝑔(𝐴𝑌,𝑍𝑋,𝑊−𝑔𝑋,𝑍𝑌,𝑊+𝑔𝐽𝑌,𝑍𝐽𝑋,𝑊−𝑔𝐽𝑋,𝑍𝐽𝑌,𝑊+2𝑔𝑋,𝐽𝑌𝐽𝑍,𝑊)−𝑔_{ℎ𝑙(𝑋,𝑍)}𝐴𝑌,𝑊+𝑔_{ℎ𝑙(𝑌,𝑍)}𝑋,𝑊−𝑔ℎ^𝑠(𝑌,𝑊),ℎ^𝑠(𝑋,𝑍)+𝑔ℎ^𝑠(𝑋,𝑊),ℎ^𝑠()−𝑌,𝑍𝑔∇_𝑋ℎ^𝑙(𝑌,𝑍),𝑊+𝑔∇_𝑌ℎ^𝑙(𝑋,𝑍),𝑊. (4.3) Then the sectional curvature 𝐾_𝑀(𝑋,𝑌)=𝑔(𝑅(𝑋,𝑌)𝑌,𝑋) of 𝑀 determined by orthonormal vectors 𝑋 and 𝑌 of Γ(𝐷₀⊕𝑀₂) and given by the following: 𝐾_𝑀()=𝑐𝑋,𝑌41+3𝑔(𝑋,𝐽𝑌)²−𝑔𝐴_{ℎ𝑙(𝑋,𝑌)}𝑌,𝑋+𝑔𝐴_{ℎ𝑙(𝑌,𝑌)}−𝑋,𝑋𝑔ℎ^𝑠(𝑌,𝑋),ℎ^𝑠(𝑋,𝑌)+𝑔ℎ^𝑠(𝑋,𝑋),ℎ^𝑠(𝑌,𝑌). (4.4)

Corollary 4.1. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Complex space form 𝑀(𝑐). Then sectional curvature of 𝑀 is given by 𝐾_𝑀(𝑋,𝑌)=(𝑐/4){1+3𝑔(𝑋,𝐽𝑌)²}, if (i)𝑀₂ defines a totally geodesic foliation in 𝑀, (ii)𝐷₀ defines a totally geodesic foliation in 𝑀, (iii)𝑀 is totally geodesic in 𝑀.

Definition 4.2. The holomorphic sectional curvature 𝐻(𝑋)=𝑔(𝑅(𝑋,𝐽𝑋)𝐽𝑋,𝑋) of 𝑀 determined by a unit vector 𝑋∈Γ(𝐷₀) is the sectional curvature of a plane section {𝑋,𝐽𝑋}.

Then using (2.11) and (4.4), for a unit vector field 𝑋∈Γ(𝐷₀), we get 𝐻(𝑋)=𝑐−𝑔ℎ^𝑙(𝑋,𝐽𝑋),ℎ^∗(𝐽𝑋,𝑋)+𝑔ℎ^𝑙(𝐽𝑋,𝐽𝑋),ℎ^∗()−𝑋,𝑋𝑔ℎ^𝑠(𝐽𝑋,𝑋),ℎ^𝑠(𝑋,𝐽𝑋)+𝑔ℎ^𝑠(𝑋,𝑋),ℎ^𝑠(𝐽𝑋,𝐽𝑋). (4.5) From (3.8), for any 𝑋,𝑌∈Γ(𝐷₀), we have ℎ^𝑠(𝑋,𝐽𝑌)=𝑤𝑃₂∇_𝑋𝑌+𝐶ℎ^𝑠(𝑋,𝑌), (4.6) and further using (3.7) and (4.6), we have ℎ^𝑠(𝐽𝑋,𝐽𝑌)=𝑤𝑃₂∇_𝐽𝑋𝑌−𝑤𝐵ℎ^𝑠(𝑋,𝑌)−ℎ^𝑠(𝑋,𝑌). (4.7) Hence using (4.6) and (4.7) in (4.5), we obtain the expression for holomorphic sectional curvature as follows: 𝐻(𝑋)=𝑐−𝑔ℎ^𝑙(𝑋,𝐽𝑋),ℎ^∗(𝐽𝑋,𝑋)+𝑔ℎ^𝑙(𝐽𝑋,𝐽𝑋),ℎ^∗(𝑋,𝑋)−𝑤𝑃₂∇_𝑋𝑋²−𝐶ℎ^𝑠(𝑋,𝑋)²+𝑔ℎ^𝑠(𝑋,𝑋),𝑤𝑃₂∇_𝐽𝑋𝑋+𝐵ℎ^𝑠(𝑋,𝑋)²−ℎ^𝑠(𝑋,𝑋)². (4.8)

Theorem 4.3. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite complex space form 𝑀(𝑐). If 𝑀 is totally geodesic in 𝑀(𝑐), then 𝐻(𝑋)≤𝑐, for any unit vector field 𝑋∈Γ(𝐷₀).

Proof. Using the hypothesis in (4.8), we get 𝐻(𝑋)=𝑐−‖𝑤𝑃₂∇_𝑋𝑋‖². Hence the result follows.

Theorem 4.4 4.4 (see [13]). Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀, then the distribution 𝐷 is integrable if and only if ℎ(𝑋,𝐽𝑌)=ℎ(𝑌,𝐽𝑋), for any 𝑋,𝑌∈Γ(𝐷).

Theorem 4.5. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite complex space form 𝑀(𝑐), and 𝐷₀ is integrable, then 𝐻(𝑋)≤𝑐 for any unit vector field 𝑋∈Γ(𝐷₀).

Proof. Since 𝐷₀ is integrable therefore using Theorem 4.4, we have ℎ(𝐽𝑋,𝐽𝑌)=−ℎ(𝑋,𝑌), for any unit vector field 𝑋∈Γ(𝐷₀). Therefore, from (4.5), we obtain 𝐻(𝑋)=𝑐−𝑔ℎ^𝑙(𝑋,𝐽𝑋),ℎ^∗(𝐽𝑋,𝑋)−𝑔ℎ^𝑙(𝑋,𝑋),ℎ^∗()−ℎ𝑋,𝑋_𝑠(𝑋,𝐽𝑋)²−ℎ^𝑠(𝑋,𝑋)²≤𝑐. (4.9)

Theorem 4.6. A 𝐺𝐶𝑅-lightlike submanifold of an indefinite complex space form 𝑀(𝑐) is 𝐷₀-totally geodesic if and only if (i)𝐷₀ is integrable, (ii)𝐻(𝑋)=𝑐, for any unit vector field 𝑋∈Γ(𝐷₀).

Proof. Proof follows from (4.9).

Theorem 4.7 4.7 (see [13]). Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then the distribution 𝐷 defines a totally geodesic foliation in 𝑀 if and only if 𝐵ℎ(𝑋,𝑌)=0, for any 𝑋,𝑌∈Γ(𝐷).

Theorem 4.8. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite complex space form 𝑀(𝑐), and 𝐷₀ defines a totally geodesic foliation in 𝑀, then 𝐻(𝑋)≤𝑐, for any unit vector field 𝑋∈Γ(𝐷₀).

Proof. Since 𝐷₀ defines a totally geodesic foliation in 𝑀, therefore by definition ∇_𝑋𝑋∈Γ(𝐷₀), this implies that ℎ^∗(𝑋,𝑋)=0. Also by using Theorem 4.7, we have 𝐵ℎ(𝑋,𝑋)=0, for any 𝑋∈Γ(𝐷₀); hence, (4.8) becomes 𝐻(𝑋)=𝑐−2‖𝐶ℎ^𝑠(𝑋,𝑋)‖² and the result follows.

Definition 4.9. The horizontal distribution 𝐷 is called parallel with respect to the induced connection ∇ on 𝑀 if ∇_𝑋𝑌∈𝐷 for any 𝑋∈Γ(𝑇𝑀) and 𝑌∈Γ(𝐷).

Theorem 4.10. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite complex space form 𝑀(𝑐) and 𝐷₀ is parallel, with respect to ∇, then 𝐻(𝑋)≤𝑐, for any unit vector field 𝑋∈Γ(𝐷₀).

Proof. Since 𝐷₀ is parallel, with respect to the induced connection ∇ on 𝑀, therefore ∇^∗_𝑋𝑌∈Γ(𝐷₀) and ℎ^∗(𝑋,𝑌)=0, for any 𝑋∈Γ(𝑇𝑀) and 𝑌∈Γ(𝐷₀). Hence, from (4.8), we obtain 𝐻(𝑋)=𝑐+‖𝐵ℎ^𝑠(𝑋,𝑋)‖²−‖ℎ^𝑠(𝑋,𝑋)‖²−‖𝐶ℎ^𝑠(𝑋,𝑋)‖², and then by using (3.6), we get 𝐻(𝑋)=𝑐−2‖𝐶ℎ^𝑠(𝑋,𝑋)‖². Hence the result is complete.

Lemma 4.11. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. If the distribution 𝐷₀ defines a totally geodesic foliation in 𝑀, then 𝑀 is 𝐷₀-geodesic.

Proof. By the definition of 𝐺𝐶𝑅-lightlike submanifold, 𝑀 is 𝐷₀-geodesic if 𝑔(ℎ^𝑙(𝑋,𝑌),𝜉)=𝑔(ℎ^𝑠(𝑋,𝑌),𝑊)=0, for any 𝑋,𝑌∈Γ(𝐷₀), 𝜉∈Γ(Rad(𝑇𝑀)), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). Since 𝐷₀ defines a totally geodesic foliation in 𝑀, therefore 𝑔(ℎ^𝑙(𝑋,𝑌),𝜉)=𝑔(∇_𝑋𝑌,𝜉)=0 and 𝑔(ℎ^𝑠(𝑋,𝑌),𝑊)=𝑔(∇_𝑋𝑌,𝑊)=0. Hence, the assertion follows.

Theorem 4.12. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite complex space form 𝑀(𝑐). If 𝐷₀ defines a totally geodesic foliation in 𝑀, then 𝐻(𝑋)=𝑐, for any unit vector field 𝑋∈Γ(𝐷₀).

Proof. The result follows directly using Lemma 4.11 and (4.8).

5. Null Holomorphically Flat 𝐺𝐶𝑅-Lightlike Submanifold

Let 𝑥∈𝑀 and 𝑈 be a null vector of 𝑇_𝑥𝑀. A plane 𝜋 of 𝑇_𝑥𝑀 is called a null plane directed by 𝑈 if it contains 𝑈, 𝑔_𝑥(𝑈,𝑉)=0, for any 𝑉∈𝜋, and there exists 𝑉₀∈𝜋 such that 𝑔_𝑥(𝑉₀,𝑉₀)≠0. Following Beem-Ehrlich [18], the null sectional curvature of 𝜋, with respect to 𝑈 and ∇, as a real number, is defined as follows: 𝐾_𝑈(𝜋)=𝑔_𝑥𝑅(𝑉,𝑈)𝑈,𝑉𝑔_𝑥(),𝑉,𝑉 (5.1) where 𝑉 is an arbitrary non null vector in 𝜋. Clearly 𝐾_𝑈(𝜋) is independent of 𝑉 but depends in a quadratic fashion on 𝑈.

Consider 𝑢∈𝑀 and a null plane 𝜋 of 𝑇_𝑢𝑀 directed by 𝜉_𝑢∈Rad(𝑇𝑀), then the null sectional curvature of 𝜋, with respect to 𝜉_𝑢 and ∇, as a real number is defined as 𝐾_𝜉𝑢(𝜋)=𝑔_𝑢𝑅𝑉_𝑢,𝜉_𝑢𝜉_𝑢,𝑉_𝑢𝑔_𝑢𝑉_𝑢,𝑉_𝑢, (5.2) where 𝑉_𝑢 is an arbitrary non-null vector in 𝜋.

Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite complex space form 𝑀(𝑐) then using (4.3), the null sectional curvature of 𝜋, with respect to 𝜉, is given by the following: 𝐾_𝜉(𝜋)=1𝑔()𝑔𝐴𝑉,𝑉^{ℎ𝑙(𝜉,𝜉)}𝑉,𝑉−𝑔𝐴_{ℎ𝑙(𝑉,𝜉)}𝜉,𝑉+𝑔ℎ^𝑠(𝑉,𝑉),ℎ^𝑠(−𝜉,𝜉)𝑔ℎ^𝑠(𝜉,𝑉),ℎ^𝑠(𝑉,𝜉)−𝑔∇_𝑉ℎ^𝑙(𝜉,𝜉),𝑉+𝑔∇_𝜉ℎ^𝑙(𝑉,𝜉),𝑉. (5.3) Then, using (2.11), we obtain 𝐾_𝜉(𝜋)=1𝑔()𝑔ℎ𝑉,𝑉^∗(𝑉,𝑉),ℎ^𝑙(𝜉,𝜉)−𝑔ℎ^∗(𝜉,𝑉),ℎ^𝑙(𝑉,𝜉)+𝑔ℎ^𝑠(𝑉,𝑉),ℎ^𝑠(−𝜉,𝜉)𝑔ℎ^𝑠(𝜉,𝑉),ℎ^𝑠(𝑉,𝜉)−𝑔∇_𝑉ℎ^𝑙(𝜉,𝜉),𝑉+𝑔∇_𝜉ℎ^𝑙(𝑉,𝜉),𝑉. (5.4)

We know that a plane 𝜋 is called holomorphic if it remains invariant under the action of the almost complex structure 𝐽, that is, if 𝜋={𝑍,𝐽𝑍}. The sectional curvature associated with the holomorphic plane is called the holomorphic sectional curvature, denoted by 𝐻(𝜋) and given by 𝐻(𝜋)=𝑅(𝑍,𝐽𝑍,𝑍,𝐽𝑍)/𝑔(𝑍,𝑍)². The holomorphic plane 𝜋={𝑍,𝐽𝑍} is called null or degenerate if and only if 𝑍 is a null vector. A manifold (𝑀,𝑔,𝐽) is called null holomorphically flat if the curvature tensor 𝑅 satisfies (see [19]). 𝑅𝑍,𝐽𝑍,𝑍,𝐽𝑍=0, (5.5) for all null vectors 𝑍. Put 𝑔(𝑅(𝑋,𝑌)𝑍,𝑊)=𝑅(𝑋,𝑌,𝑍,𝑊), then, from (5.4), we obtain 𝑅𝜉,𝐽𝜉,𝜉,𝐽𝜉=𝑔ℎ^∗(𝜉,𝜉),ℎ^𝑙𝐽𝜉,𝐽𝜉−𝑔ℎ^∗𝐽𝜉,𝜉,ℎ^𝑙𝜉,+𝐽𝜉𝑔ℎ^𝑠(𝜉,𝜉),ℎ^𝑠𝐽𝜉,𝐽𝜉−𝑔ℎ^𝑠𝐽𝜉,𝜉,ℎ^𝑠𝜉,−𝐽𝜉𝑔∇_𝜉ℎ^𝑙𝐽𝜉,𝐽𝜉,𝜉+𝑔∇_𝐽𝜉ℎ^𝑙𝜉,𝐽𝜉,𝜉. (5.6)

Definition 5.1 (see [20]). A lightlike submanifold (𝑀,𝑔) of a semi-Riemannian manifold (𝑀,𝑔) is said to be a 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, for 𝑋,𝑌∈Γ(𝑇𝑀), ℎ(𝑋,𝑌)=𝐻𝑔(𝑋,𝑌). (5.7) Using (2.6), it is clear that 𝑀 is a totally umbilical if and only if on each coordinate neighborhood 𝑢 there exist smooth vector fields 𝐻^𝑙∈Γ(ltr(𝑇𝑀)) and 𝐻^𝑠∈Γ(𝑆(𝑇𝑀^⟂)) such that ℎ^𝑙(𝑋,𝑌)=𝐻^𝑙𝑔(𝑋,𝑌),ℎ^𝑠(𝑋,𝑌)=𝐻^𝑠𝑔(𝑋,𝑌),𝐷^𝑙(𝑋,𝑊)=0, (5.8) for 𝑋,𝑌∈Γ(𝑇𝑀) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). A lightlike submanifold is said to be totally geodesic if ℎ(𝑋,𝑌)=0, for any 𝑋,𝑌∈Γ(𝑇𝑀). Therefore, in other words, a lightlike submanifold is totally geodesic if 𝐻^𝑙=0 and 𝐻^𝑠=0.
Let 𝑀 be a totally umbilical lightlike submanifold, then, using above definition, we have ℎ(𝐽𝜉,𝐽𝜉)=𝐻𝑔(𝐽𝜉,𝐽𝜉)=𝐻𝑔(𝜉,𝜉)=0 and ℎ(𝜉,𝐽𝜉)=𝐻𝑔(𝜉,𝐽𝜉)=0, for any 𝜉∈Γ(Rad(𝑇𝑀)). Thus, from (5.6), we have the following theorem.

Theorem 5.2. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite complex space form 𝑀(𝑐). If 𝑀 is totally umbilical lightlike submanifold in 𝑀(𝑐), then 𝑀 is null holomorphically flat.

Moreover, from (5.6), it is clear that the expression of 𝑅(𝜉,𝐽𝜉,𝜉,𝐽𝜉) is expressed in terms of screen second fundamental forms of 𝑀, thus 𝐺𝐶𝑅-lightlike submanifold 𝑀 of an indefinite complex space form 𝑀(𝑐) is null holomorphically flat if 𝑀 is totally geodesic.

6. Holomorphic Bisectional Curvature of a 𝐺𝐶𝑅-Lightlike Submanifold

Definition 6.1. The holomorphic bisectional for the pair of unit vector fields {𝑋,𝑌} on 𝑀 is given by 𝐻(𝑋,𝑌)=𝑔(𝑅(𝑋,𝐽𝑋)𝐽𝑌,𝑌).

Theorem 6.2. Let 𝑀 be a mixed totally geodesic 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀 with 𝐷₀ parallel distribution. Then 𝐻(𝑋,𝑍)=0, for any unit vector fields 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂).

Proof. Let 𝑋,𝑌∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂) then, by using that hypothesis that the distribution 𝐷₀ is a parallel, with respect to ∇ on 𝑀, we have 𝑔(𝑇∇_𝑋𝑍,𝑌)=−𝑔(∇_𝑋𝑍,𝑇𝑌)=𝑔(𝑍,∇_𝑋𝑇𝑌)=0. Hence, the non degeneracy of the distribution 𝐷₀ implies that ∇_𝑋𝑍∈Γ𝐷^′, (6.1)
for any 𝑍∈Γ(𝑀₂). Now replacing 𝑌 by 𝐽𝑋, respectively, in (2.15) and then taking inner product with 𝐽𝑍, for any 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂), we get 𝐻(𝑋,𝑍)=𝑔ℎ^𝑠∇_𝑋𝐽𝑋,𝑍,𝐽𝑍−𝑔∇_𝑋ℎ^𝑠(𝐽𝑋,𝑍),𝐽𝑍+𝑔ℎ^𝑠𝐽𝑋,∇_𝑋𝑍+,𝐽𝑍𝑔∇_𝐽𝑋ℎ^𝑠(𝑋,𝑍),𝐽𝑍−𝑔ℎ^𝑠∇_𝐽𝑋𝑋,𝑍,𝐽𝑍−𝑔ℎ^𝑠𝑋,∇_𝐽𝑋𝑍−,𝐽𝑍𝑔𝐷^𝑠𝑋,ℎ^𝑙(𝐽𝑋,𝑍),𝐽𝑍+𝑔𝐷^𝑠𝐽𝑋,ℎ^𝑙(𝑋,𝑍),𝐽𝑍. (6.2) Hence by using that 𝑀 is mixed totally geodesic with (6.1), the assertion follows.

Theorem 6.3. In order that an indefinite complex space form 𝑀(𝑐) may admit a mixed totally geodesic 𝐺𝐶𝑅-lightlike submanifold 𝑀 with parallel horizontal distribution 𝐷₀, it is necessary that 𝑐=0.

Proof. Let 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂) be unit vector fields, then (4.1) implies that 𝐻(𝑋,𝑍)=−(𝑐/2)𝑔(𝑋,𝑋)𝑔(𝑍,𝑍), then the non degeneracy of the distributions 𝐷₀ and 𝑀₂ with the Theorem 6.2, we obtain 𝑐=0. Hence, the result follows.

Lemma 6.4. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then we have the following:(i)if 𝐷^′ defines a totally geodesic foliation in 𝑀 then 𝑔(ℎ^𝑠(𝐷^′,𝐷₀),𝐽𝐷^′)=0,(ii)if 𝐷₀ is a parallel distribution, with respect to ∇, then ℎ(𝑍,𝐽𝑋)=𝐶ℎ(𝑍,𝑋) for any 𝑋∈Γ(𝐷₀),𝑍∈Γ(𝑀₂).

Proof. (i) Let 𝐷^′ define a totally geodesic foliation in 𝑀 this implies that ∇_𝑋𝑌=∇_𝑋𝑌∈Γ(𝐷^′) and ℎ(𝑋,𝑌)=0, for any 𝑋,𝑌∈Γ(𝐷^′). Therefore by using (3.10), we obtain 𝐴_𝑤𝑌𝑋=−𝐵ℎ(𝑋,𝑌)=0. Let 𝑍∈Γ(𝐷₀), then by using (2.8), we get 0=𝑔(𝐴_𝑤𝑌𝑋,𝑍)=𝑔(ℎ^𝑠(𝑋,𝑍),𝑤𝑌). Thus we have 𝑔(ℎ^𝑠(𝐷^′,𝐷₀),𝐽𝐷^′)=0.
(ii) Let 𝐷₀ is a parallel distribution with respect to the induced connection ∇, therefore ∇_𝑋𝑌∈Γ(𝐷₀), for any 𝑌∈Γ(𝐷₀),𝑋∈Γ(𝑇𝑀). Since 𝑀 is Kaehler manifold, therefore for 𝑍∈Γ(𝑀₂) and 𝑋∈Γ(𝐷₀), we have ∇_𝑍𝐽𝑋=𝐽∇_𝑍𝑋. This implies that ∇_𝑍𝐽𝑋+ℎ(𝑍,𝐽𝑋)=𝐽∇_𝑍𝑋+𝐵ℎ(𝑍,𝑋)+𝐶ℎ(𝑍,𝑋), then by equating transversal components on both sides, we get the result.

Theorem 6.5. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. If 𝐷₀ is parallel with respect to the induced connection ∇, and 𝑀₂ defines a totally geodesic foliation in 𝑀, then 𝐻(𝑋,𝑍)=𝑔ℎ(𝐽𝑋,𝑍),𝐽∇_𝑋𝑍−𝑔ℎ(𝑋,𝑍),𝐽∇_𝐽𝑋𝑍+𝑔𝐴_𝐽𝑍𝐽𝑋,∇_𝑋𝑍−𝑔𝐴_𝐽𝑍𝑋,∇_𝐽𝑋𝑍+2𝐶ℎ^𝑠(𝑋,𝑍)², (6.3) for any unit vector fields 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂).

Proof. Let 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂), then the equation of Codazzi (2.15) becomes 𝑔𝑅(𝑋,𝐽𝑋)𝑍,𝐽𝑍=𝑔∇_𝑋ℎ^𝑠(𝐽𝑋,𝑍),𝐽𝑍−𝑔∇_𝐽𝑋ℎ^𝑠()+𝑋,𝑍,𝐽𝑍𝑔𝐷^𝑠𝑋,ℎ^𝑙(𝐽𝑋,𝑍),𝐽𝑍−𝑔𝐷^𝑠𝐽𝑋,ℎ^𝑙(𝑋,𝑍),𝐽𝑍. (6.4) By using (2.13) and (6.1) with the Lemma 6.4 (i), we obtain 𝐻(𝑋,𝑍)=𝑔∇^𝑠_𝐽𝑋ℎ^𝑠(𝑋,𝑍),𝐽𝑍−𝑔∇^𝑠_𝑋ℎ^𝑠()𝐴𝐽𝑋,𝑍,𝐽𝑍+𝑔_𝐽𝑍𝐽𝑋,∇_𝑋𝑍−𝑔𝐴_𝐽𝑍𝑋,∇_𝐽𝑋𝑍−𝑔𝐷^𝑠𝑋,ℎ^𝑙(𝐽𝑋,𝑍),𝐽𝑍+𝑔𝐷^𝑠𝐽𝑋,ℎ^𝑙(𝑋,𝑍),𝐽𝑍. (6.5) Now by using (2.7) with the Lemma 6.4 (ii), we have 𝑔∇^𝑠_𝑋ℎ^𝑠(𝐽𝑋,𝑍),𝐽𝑍=𝑔∇_𝑋ℎ^𝑠(𝐽𝑋,𝑍),𝐽𝑍=−𝑔ℎ^𝑠(𝐽𝑋,𝑍),𝐽∇_𝑋𝑍−‖𝐶ℎ^𝑠(𝑋,𝑍)‖², (6.6) and similarly 𝑔∇^𝑠_𝐽𝑋ℎ^𝑠(𝑋,𝑍),𝐽𝑍=−𝑔ℎ^𝑠(𝑋,𝑍),𝐽∇_𝐽𝑋𝑍+𝐶ℎ^𝑠(𝑋,𝑍)². (6.7) By using (2.7), we have 𝑔𝐷^𝑠𝑋,ℎ^𝑙(𝐽𝑋,𝑍),𝐽𝑍=−𝑔ℎ^𝑙(𝐽𝑋,𝑍),𝐽∇_𝑋𝑍, (6.8)𝑔𝐷^𝑠𝐽𝑋,ℎ^𝑙(𝑋,𝑍),𝐽𝑍=−𝑔ℎ^𝑙(𝑋,𝑍),𝐽∇_𝐽𝑋𝑍. (6.9) Hence by using (6.6)–(6.9) in (6.5), the result follows.

Definition 6.6 6.6 (see [13]). A 𝐺𝐶𝑅-lightlike submanifold 𝑀 of an indefinite Kaehler manifold 𝑀 is called a 𝐺𝐶𝑅-lightlike product if both the distributions 𝐷 and 𝐷^′ define totally geodesic foliations in 𝑀.

Theorem 6.7 (see [14]). Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then, 𝑀 is a 𝐺𝐶𝑅-lightlike product if and only if (∇_𝑋𝑇)𝑌=0, for any 𝑋,𝑌∈Γ(𝐷) or 𝑋,𝑌∈Γ(𝐷^′).

Theorem 6.8. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. If (∇_𝑋𝑇)𝑌=0, for any 𝑋,𝑌∈Γ(𝑇𝑀), then 𝐻(𝑋,𝑍)=2𝑔ℎ^𝑠(𝐽𝑋,𝑍),𝐽ℎ^𝑠(𝑋,𝑍), (6.10) for any unit vector fields 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂).

Proof. Let (∇_𝑋𝑇)𝑌=0, then (3.10) implies that 𝐴_𝑤𝑌𝑋+𝐵ℎ(𝑋,𝑌)=0, (6.11) for any 𝑋,𝑌∈Γ(𝑇𝑀). Let 𝑌∈Γ(𝐷),𝑋∈Γ(𝑇𝑀), then (6.11) gives 𝐵ℎ(𝑋,𝑌)=0. (6.12) Let 𝑍∈Γ(𝐷^′) and 𝑋∈Γ(𝐷), then using (6.11) and (6.12), we obtain 𝐴_𝑤𝑍𝑋=0. (6.13) Particularly choosing 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂) in (2.15), we get 𝑔𝑅(𝑋,𝐽𝑋)𝑍,𝐽𝑍=𝑔∇_𝑋ℎ^𝑠(𝐽𝑋,𝑍),𝐽𝑍−𝑔∇_𝐽𝑋ℎ^𝑠()+𝑋,𝑍,𝐽𝑍𝑔𝐷^𝑠𝑋,ℎ^𝑙(𝐽𝑋,𝑍),𝐽𝑍−𝑔𝐷^𝑠𝐽𝑋,ℎ^𝑙(𝑋,𝑍),𝐽𝑍. (6.14) Since (∇_𝑋𝑇)𝑌=0 therefore, by using Theorem 6.7, the distributions 𝐷 and 𝐷^′ define totally geodesic foliations in 𝑀. Then 𝐷^′ defines totally geodesic foliation in 𝑀 implies that for any 𝑍₁,𝑍₂∈Γ(𝐷^′), we have ∇_𝑍1𝑍₂∈Γ(𝐷^′). Therefore, using (3.10) and (3.11), we get 𝐴_𝑤𝑍2𝑍₁+𝐵ℎ(𝑍₁,𝑍₂)=0. By taking inner product with 𝑋∈Γ(𝐷₀) and using (2.8), we get 𝑔ℎ^𝑠𝑍₁,𝑋,𝑤𝑍₂=0. (6.15) Also, (3.11) implies that 𝑇∇_𝑋𝑍=0, that is, ∇_𝑋𝑍∈Γ(𝐷^′). Therefore using (2.8), (2.13), (6.13), and (6.15) in (6.14), we obtain 𝑔𝑅(𝑋,𝐽𝑋)𝑍,𝐽𝑍=𝑔∇^𝑠_𝑋ℎ^𝑠(𝐽𝑋,𝑍),𝐽𝑍−𝑔∇^𝑠_𝐽𝑋ℎ^𝑠()+𝑋,𝑍,𝐽𝑍𝑔𝐷^𝑠𝑋,ℎ^𝑙(𝐽𝑋,𝑍),𝐽𝑍−𝑔𝐷^𝑠𝐽𝑋,ℎ^𝑙(𝑋,𝑍),𝐽𝑍. (6.16) Now using (2.6), (2.7), (2.8), and (6.15), we have 𝑔∇^𝑠_𝑋ℎ^𝑠(𝐽𝑋,𝑍),𝐽𝑍=𝑔𝐽ℎ^𝑠(𝐽𝑋,𝑍),ℎ^𝑠(𝑋,𝑍). (6.17) Similarly, 𝑔∇^𝑠_𝐽𝑋ℎ^𝑠(𝑋,𝑍),𝐽𝑍=−𝑔𝐽ℎ^𝑠(𝐽𝑋,𝑍),ℎ^𝑠(𝑋,𝑍). (6.18) Also using (2.7), we have 𝑔𝐷^𝑠𝑋,ℎ^𝑙(𝐽𝑋,𝑍),𝐽𝑍=0,𝑔𝐷^𝑠𝐽𝑋,ℎ^𝑙(𝑋,𝑍),𝐽𝑍=0. (6.19) Hence, using (6.17)–(6.19) in (6.16), the result follows.

Lemma 6.9. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold such that (∇_𝑋𝐵)𝑌=0. Then ℎ^𝑠(𝐽𝑋,𝑍)∈Γ(𝐿^⟂₂) for any 𝑋∈Γ(𝐷₀), 𝑍∈Γ(𝑀₂).

Proof. Since (∇_𝑋𝐵)𝑌=0, therefore, from (3.12) we have 𝑇𝐴_𝑊𝑋=0, for any 𝑋∈Γ(𝑇𝑀) and 𝑊∈Γ(𝐿₂), this implies that 𝐴_𝑊𝑋∈Γ𝐷^′, (6.20) for any 𝑊∈Γ(𝐿₂) and 𝑋∈Γ(𝑇𝑀). Since ℎ^𝑠(𝐽𝑋,𝑍)∈Γ(𝑆(𝑇𝑀^⟂)), therefore, to prove that ℎ^𝑠(𝐽𝑋,𝑍)∈Γ(𝐿^⟂₂), it is sufficient to prove that 𝑔(ℎ^𝑠(𝐽𝑋,𝑍),𝑊)=0, for any 𝑊∈Γ(𝐿₂). Let 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂) such that 𝑊=𝑤𝑍, we have 𝑔(∇_𝑈𝑍,𝑋)=𝑔(∇_𝑈𝑍,𝑋)=𝑔(∇_𝑈𝐽𝑍,𝐽𝑋)=−𝑔(𝐴_𝐽𝑍𝑈,𝐽𝑋)=𝑔(𝐽𝐴_𝐽𝑍𝑈,𝑋), then using (2.8) we obtain 𝑔ℎ^𝑠(𝐽𝑋,𝑍),𝑊=−𝑔𝐽𝐴_𝑊𝑍,𝑋=−𝑔∇_𝑍𝑍,𝑋=−𝑔𝑇∇_𝑍𝑍,𝑇𝑋. (6.21) Since, from (3.10), we have 𝑇∇_𝑍𝑍=−𝐴_𝑤𝑍𝑍−𝐵ℎ(𝑍,𝑍), then using (6.20) in (6.21), the result follows.

Theorem 6.10. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold such that (∇_𝑋𝐵)𝑌=0, then 𝐻(𝑋,𝑍)=2𝑔𝐽ℎ^𝑠(𝑋,𝑍),ℎ^𝑠(𝐽𝑋,𝑍)−𝑔𝐴_𝐽𝑍∇_𝐽𝑋𝑋,𝑍+𝑔𝐴_𝐽𝑍∇_𝑋𝐽𝑋,𝑍, (6.22) for any unit vector fields 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂).

Proof. Let 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂), then, from (2.15) and (2.13), we obtain 𝑔𝑅(𝑋,𝐽𝑋)𝑍,𝐽𝑍=𝑔∇^𝑠_𝑋ℎ^𝑠(𝐽𝑋,𝑍),𝐽𝑍−𝑔ℎ^𝑠∇_𝑋−𝐽𝑋,𝑍,𝐽𝑍𝑔ℎ^𝑠𝐽𝑋,∇_𝑋𝑍,𝐽𝑍−𝑔∇^𝑠_𝐽𝑋ℎ^𝑠()+𝑋,𝑍,𝐽𝑍𝑔ℎ^𝑠∇_𝐽𝑋𝑋,𝑍,𝐽𝑍+𝑔ℎ^𝑠𝑋,∇_𝐽𝑋𝑍+,𝐽𝑍𝑔𝐷^𝑠𝑋,ℎ^𝑙(𝐽𝑋,𝑍),𝐽𝑍−𝑔𝐷^𝑠𝐽𝑋,ℎ^𝑙(𝑋,𝑍),𝐽𝑍, (6.23) using (6.20) in (2.9), we obtain 𝑔𝐷^𝑠𝑋,ℎ^𝑙(𝐽𝑋,𝑍),𝐽𝑍=0,𝑔𝐷^𝑠𝐽𝑋,ℎ^𝑙(𝑋,𝑍),𝐽𝑍=0. (6.24) Now consider 𝑔∇^𝑠_𝑋ℎ^𝑠(𝐽𝑋,𝑍),𝐽𝑍=−𝑔ℎ^𝑠(𝐽𝑋,𝑍),𝐽∇_𝑋𝑍=𝑔ℎ^𝑠(𝑋,𝑍),𝐽ℎ^𝑠(𝐽𝑋,𝑍), (6.25) and similarly 𝑔∇^𝑠_𝐽𝑋ℎ^𝑠(𝑋,𝑍),𝐽𝑍=−𝑔ℎ^𝑠(𝑋,𝑍),𝐽ℎ^𝑠(𝐽𝑋,𝑍). (6.26) Also using (2.8), we have 𝑔ℎ^𝑠∇_𝑋𝐽𝑋,𝑍,𝐽𝑍=𝑔𝐴_𝐽𝑍∇_𝑋𝐽𝑋,𝑍,𝑔ℎ^𝑠∇_𝐽𝑋𝑋,𝑍,𝐽𝑍=𝑔𝐴_𝐽𝑍∇_𝐽𝑋𝑋,𝑍. (6.27) Using (6.20) in (2.8), we have 𝑔ℎ^𝑠𝐽𝑋,∇_𝑋𝑍,𝐽𝑍=0,𝑔ℎ^𝑠𝑋,∇_𝐽𝑋𝑍,𝐽𝑍=0. (6.28) Thus using (6.24)–(6.28) in (6.23), the result follows.

Theorem 6.11. Let 𝑀 be a mixed foliate 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀, and 𝑆(𝑇𝑀) is parallel distribution, with respect to the induced connection ∇, then 𝐻(𝑋,𝑍)=2𝑔𝐴_𝐽𝑍𝐽𝑋,𝐽𝐴_𝐽𝑍𝑋, (6.29) for any unit vector fields 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂).

Proof. Since 𝑀 is mixed foliate therefore for any 𝑋∈Γ(𝐷₀) and 𝑍∈Γ(𝑀₂), Codazzi equation (2.15) and (2.13), imply that 𝐻(𝑋,𝑍)=𝑔ℎ^𝑠∇_𝑋𝐽𝑋,𝑍,𝐽𝑍+𝑔ℎ^𝑠𝐽𝑋,∇_𝑋𝑍−,𝐽𝑍𝑔ℎ^𝑠∇_𝐽𝑋𝑋,𝑍,𝐽𝑍−𝑔ℎ^𝑠𝑋,∇_𝐽𝑋𝑍,𝐽𝑍. (6.30) Since using (2.8) and the hypothesis, we have 𝑔(𝐴_𝐽𝑍𝑋,𝐽𝑊)=0 and 𝑔(𝐴_𝐽𝑍𝑋,𝐽𝜉)=𝑔(ℎ^𝑠(𝑋,𝐽𝜉),𝐽𝑍)=𝑔(∇_𝑋𝐽𝜉,𝐽𝑍)=−𝑔(𝜉,∇_𝑋𝑍)=0. Therefore, by the definition of a 𝐺𝐶𝑅-lightlike submanifold, we have 𝐴_𝐽𝑍𝑋∈Γ(𝐷). (6.31) Thus, using (2.7), (2.8), and (6.31) with the hypothesis, we obtain 𝑔ℎ^𝑠𝐽𝑋,∇_𝑋𝑍,𝐽𝑍=𝑔𝐴_𝐽𝑍𝐽𝑋,∇_𝑋𝑍=𝑔𝐽𝐴_𝐽𝑍𝐽𝑋,𝐽∇_𝑋𝑍=𝑔𝐴_𝐽𝑍𝐽𝑋,𝐽𝐴_𝐽𝑍𝑋−𝑔𝐴_𝐽𝑍𝐽𝑋,𝐽𝐷^𝑙()=𝑋,𝐽𝑍𝑔𝐴_𝐽𝑍𝐽𝑋,𝐽𝐴_𝐽𝑍𝑋. (6.32) Similarly, we obtain 𝑔ℎ^𝑠𝑋,∇_𝐽𝑋𝑍,𝐽𝑍=𝑔𝐴_𝐽𝑍𝑋,𝐽𝐴_𝐽𝑍𝐽𝑋,𝑔ℎ^𝑠∇_𝐽𝑋𝑋,𝑍,𝐽𝑍=𝑔𝐴_𝐽𝑍𝑍,∇_𝐽𝑋𝑋−𝑔∇_𝐽𝑋𝑋,𝐷^𝑙(𝐽𝑍,𝑍),𝑔ℎ^𝑠∇_𝑋𝐽𝑋,𝑍,𝐽𝑍=𝑔𝐴_𝐽𝑍𝑍,∇_𝑋𝐽𝑋−𝑔∇_𝑋𝐽𝑋,𝐷^𝑙(𝐽𝑍,𝑍). (6.33) Thus, (6.30) becomes 𝐻(𝑋,𝑍)=2𝑔𝐴_𝐽𝑍𝐽𝑋,𝐽𝐴_𝐽𝑍𝑋−𝑔𝐴_𝐽𝑍𝑍,∇_𝐽𝑋𝑋+𝑔∇_𝐽𝑋𝑋,𝐷^𝑙()𝐴𝐽𝑍,𝑍+𝑔_𝐽𝑍𝑍,∇_𝑋𝐽𝑋−𝑔∇_𝑋𝐽𝑋,𝐷^𝑙(𝐽𝑍,𝑍). (6.34) Since the distribution 𝐷 is integrable, therefore ∇_𝐽𝑋𝑋−∇_𝑋𝐽𝑋=[𝑋,𝐽𝑋]=𝑋^′∈Γ(𝐷), then (6.34) becomes 𝐻(𝑋,𝑍)=2𝑔(𝐴_𝐽𝑍𝐽𝑋,𝐽𝐴_𝐽𝑍𝑋)−𝑔(𝐴_𝐽𝑍𝑍,𝑋^′)+𝑔(𝑋^′,𝐷^𝑙(𝐽𝑍,𝑍)). Hence, using the hypothesis and (2.8), the result follows.

Acknowledgment

The authors would like to thank the anonymous referee for his/her comments that helped us to improve this paper.