Abstract

We study geodesic 𝐺𝐶𝑅-lightlike submanifolds of indefinite Kaehler manifolds and obtain some necessary and sufficient conditions for a 𝐺𝐶𝑅-lightlike submanifold to be a 𝐺𝐶𝑅-lightlike product.

1. Introduction

The geometry of 𝐶𝑅-submanifolds of Kaehler manifolds was initiated by Bejancu [1], which includes holomorphic and totally real submanifolds as subcases, and further developed by Bejancu [2], Bejancu et al. [3], Blair and Chen [4], Chen [5], Yano and Kon [6, 7], and many others. They all studied the geometry of 𝐶𝑅-submanifolds with positive definite metric. Therefore this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite. Thus the geometry of 𝐶𝑅-submanifolds with indefinite metric became a topic of chief discussion and Duggal [8, 9] played a very crucial role. Duggal and Bejancu [10] introduced the notion of 𝐶𝑅-lightlike submanifolds which exclude the totally real and complex subcases. Then Duggal and Sahin [11] introduced 𝑆𝐶𝑅-lightlike submanifolds which contain complex and totally real subcases but there was no inclusion relation between 𝐶𝑅 and 𝑆𝐶𝑅-cases. Thus to find a class of submanifolds which would behave as an umbrella for 𝐶𝑅-lightlike and 𝑆𝐶𝑅-lightlike submanifolds of an indefinite Kaehler manifold, Duggal and Sahin [12] introduced 𝐺𝐶𝑅-lightlike submanifolds of indefinite Kaehler manifolds. This paper starts with a very brief introduction about lightlike geometry and 𝐺𝐶𝑅-lightlike submanifolds which will be needed throught the paper and then we study geodesic 𝐺𝐶𝑅-lightlike submanifolds and obtain some necessary and sufficient conditions for a 𝐺𝐶𝑅-lightlike submanifold to be a 𝐺𝐶𝑅-lightlike product.

2. Lightlike Submanifolds

We recall notations and fundamental equations for lightlike submanifolds, which are due to the book [8] 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 (𝑀,𝑔) is an 𝑚-dimensional submanifold of 𝑀 and 𝑔 is 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 𝑀𝑇𝑀⟂=∪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 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),(2.3) 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,(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}and {𝑁1,…,𝑁𝑟} are local lightlike bases of Γ(Rad𝑇𝑀|𝑢) and Γ(ltr(𝑇𝑀)|𝑢), and {𝑊𝑟+1,…,𝑊𝑛} and {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 2.1 (see [8]). Let (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(𝑇𝑀⟂)) be an 𝑟-lightlike submanifold of a semi-Riemannian manifold (𝑀,𝑔). Then there exist 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,forany𝑖,𝑗∈{1,2,…,𝑟},(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 a 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, then (2.7) and (2.8) become∇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), and 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 (2.9)–(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,𝑁′∈Γ(ltr(𝑇𝑀)).

Let 𝑃 be the projection morphism of 𝑇𝑀 on 𝑆(𝑇𝑀); then using (2.3), we can induce some new geometric objects on the screen distribution 𝑆(𝑇𝑀) on 𝑀 as∇𝑋𝑃𝑌=∇∗𝑋𝑃𝑌+ℎ∗∇(𝑋,𝑌),(2.16)𝑋𝜉=−𝐴∗𝜉𝑋+∇𝑋∗𝑡𝜉,(2.17) for any 𝑋,𝑌∈Γ(𝑇𝑀) and 𝜉∈Γ(Rad𝑇𝑀), where {∇∗𝑋𝑃𝑌,𝐴∗𝜉𝑋} and {ℎ∗(𝑋,𝑌),∇𝑋∗𝑡𝜉} belong to Γ(𝑆(𝑇𝑀)) and Γ(Rad𝑇𝑀), respectively. ∇∗ and ∇∗𝑡 are linear connections on complementary distributions 𝑆(𝑇𝑀) and Rad𝑇𝑀, respectively. ℎ∗ and 𝐴∗ are Γ(Rad𝑇𝑀)-valued and Γ(𝑆(𝑇𝑀))-valued bilinear forms and are called as second fundamental forms of distributions 𝑆(𝑇𝑀) and Rad𝑇𝑀, respectively.

From the geometry of Riemannian submanifolds and nondegenerate submanifolds, it is known that the induced connection ∇ on a nondegenerate submanifold is a metric connection. Unfortunately, this is not true for lightlike submanifolds. Indeed considering ∇ a metric connection, we have∇𝑋𝑔(𝑌,𝑍)=ğ‘”î€·â„Žğ‘™î€¸+(𝑋,𝑌),ğ‘ğ‘”î€·â„Žğ‘™î€¸,(𝑋,𝑍),𝑌(2.18) for any 𝑋,𝑌,𝑍∈Γ(𝑇𝑀). From [8, page 171], using the properties of linear connection, we haveî€·âˆ‡ğ‘‹â„Žğ‘™î€¸(𝑌,𝑍)=âˆ‡ğ‘™ğ‘‹î€·â„Žğ‘™î€¸(𝑌,𝑍)âˆ’â„Žğ‘™î€·âˆ‡ğ‘‹î€¸ğ‘Œ,ğ‘âˆ’â„Žğ‘™î€·ğ‘Œ,∇𝑋𝑍,î€·âˆ‡ğ‘‹â„Žğ‘ î€¸(𝑌,𝑍)=âˆ‡ğ‘ ğ‘‹î€·â„Žğ‘™î€¸(𝑌,𝑍)âˆ’â„Žğ‘ î€·âˆ‡ğ‘‹î€¸ğ‘Œ,ğ‘âˆ’â„Žğ‘ î€·ğ‘Œ,∇𝑋𝑍.(2.19) Barros and Romero [13] defined indefinite Kaehler manifolds as follows.

Definition 2.2. Let (𝑀,𝐽,𝑔) be an indefinite almost Hermitian manifold and let ∇ be the Levi-Civita connection on 𝑀 with respect to 𝑔. Then 𝑀 is called an indefinite Kaehler manifold if 𝐽 is parallel with respect to ∇, that is, ∇𝑋𝐽𝑇𝑌=0,∀𝑋,𝑌∈Γ𝑀.(2.20)

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,𝐽𝐷1=𝐷1,𝐽𝐷2⊂𝑆(𝑇𝑀).(3.1)(B)There exist two subbundles 𝐷0 and ğ·î…ž of 𝑆(𝑇𝑀) such that 𝑆(𝑇𝑀)=𝐽𝐷2âŠ•ğ·î…žî‚‡âŸ‚ğ·0,𝐽𝐷0=𝐷0,ğ½î€·ğ·î…žî€¸=𝐿1⟂𝐿2,(3.2) where 𝐷0 is a nondegenerate distribution on 𝑀, and 𝐿1 and 𝐿2 are vector bundle of ltr(𝑇𝑀) and 𝑆(𝑇𝑀)⟂, respectively.

Then the tangent bundle 𝑇𝑀 of 𝑀 is decomposed as𝑇𝑀=ğ·âŸ‚ğ·î…ž,𝐷=Rad(𝑇𝑀)⊕𝐷0⊕𝐽𝐷2.(3.3)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 have𝑋=𝑄𝑋+𝑃1𝑋+𝑃2𝑋,(3.4) applying 𝐽 to (3.4), we obtain𝐽𝑋=𝑇𝑋+𝑤𝑃1𝑋+𝑤𝑃2𝑋,(3.5) and we can write (3.5) as𝐽𝑋=𝑇𝑋+𝑤𝑋,(3.6) where 𝑇𝑋 and 𝑤𝑋 are the tangential and transversal components of 𝐽𝑋, respectively.

Similarly𝐽𝑉=𝐵𝑉+𝐶𝑉,(3.7) for any 𝑉∈Γ(tr(𝑇𝑀)), where 𝐵𝑉 and 𝐶𝑉 are the sections of 𝑇𝑀 and tr(𝑇𝑀), respectively.

Differentiating (3.5) and using (2.9)–(2.12) and (3.7) we have𝐷𝑠𝑋,𝑤𝑃1𝑌=−∇𝑠𝑋𝑤𝑃2𝑌+𝑤𝑃2âˆ‡ğ‘‹ğ‘Œâˆ’â„Žğ‘ (𝑋,𝑇𝑌)+ğ¶â„Žğ‘ ğ·(𝑋,𝑌),𝑙𝑋,𝑤𝑃2𝑌=−∇𝑙𝑋𝑤𝑃1𝑌+𝑤𝑃1âˆ‡ğ‘‹ğ‘Œâˆ’â„Žğ‘™(𝑋,𝑇𝑌)+ğ¶â„Žğ‘™(𝑋,𝑌).(3.8) Using Kaehlerian property of ∇ with (2.11) and (2.12), we have the following lemmas.

Lemma 3.2. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehlerian manifold 𝑀. Then one has ∇𝑋𝑇𝑌=𝐴𝑤𝑌∇𝑋+ğµâ„Ž(𝑋,𝑌),(3.9)𝑡𝑋𝑤𝑌=ğ¶â„Ž(𝑋,𝑌)−ℎ(𝑋,𝑇𝑌),(3.10) where 𝑋,𝑌∈Γ(𝑇𝑀) and ∇𝑋𝑇𝑌=∇𝑋𝑇𝑌−𝑇∇𝑋∇𝑌,(3.11)𝑡𝑋𝑤𝑌=∇𝑡𝑋𝑤𝑌−𝑤∇𝑋𝑌.(3.12)

Lemma 3.3. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehlerian manifold 𝑀. Then one has ∇𝑋𝐵𝑉=𝐴𝐶𝑉𝑋−𝑇𝐴𝑉∇𝑋,𝑡𝑋𝐶𝑉=âˆ’ğ‘¤ğ´ğ‘‰ğ‘‹âˆ’â„Ž(𝑋,𝐵𝑉),(3.13) where 𝑋∈Γ(𝑇𝑀), 𝑉∈Γ(tr(𝑇𝑀)), and ∇𝑋𝐵𝑉=∇𝑋𝐵𝑉−𝐵∇𝑡𝑋∇𝑉,𝑡𝑋𝐶𝑉=∇𝑡𝑋𝐶𝑉−𝐶∇𝑡𝑋𝑉.(3.14)

Duggal and Sahin [12] investigated the conditions to define totally geodesic foliations by the distributions 𝐷 and ğ·î…ž in 𝑀 as follows.

Theorem 3.4 (see [12]). 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 ğµâ„Ž(𝑋,𝑌)=0, for any 𝑋,𝑌∈Γ(𝐷).

Theorem 3.5 (see [12]). Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then the distribution ğ·î…ž defines a totally geodesic foliation in 𝑀 if and only if 𝐴𝑤𝑌𝑋∈Γ(ğ·î…ž), for any 𝑋,𝑌∈Γ(ğ·î…ž).

4. Geodesic 𝐺𝐶𝑅-Lightlike Submanifolds

Definition 4.1. A 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold is called mixed geodesic 𝐺𝐶𝑅-lightlike submanifold if its second fundamental form ℎ satisfies ℎ(𝑋,𝑌)=0 for any 𝑋∈Γ(𝐷) and 𝑌∈Γ(ğ·î…ž).

Definition 4.2. A 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold is called 𝐷 geodesic 𝐺𝐶𝑅-lightlike submanifold if its second fundamental form ℎ satisfies ℎ(𝑋,𝑌)=0 for any 𝑋,𝑌∈Γ(𝐷).

Definition 4.3. A 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold is called ğ·î…ž geodesic 𝐺𝐶𝑅-lightlike submanifold if its second fundamental form ℎ satisfies ℎ(𝑋,𝑌)=0 for any 𝑋,𝑌∈Γ(ğ·î…ž).

Theorem 4.4. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is 𝐷-geodesic if and only if 𝑔𝐴𝑊=𝑋,𝑌𝑔𝐷𝑙,∇(𝑋,𝑊),𝑌∗𝑋𝐷𝐽𝜉∉Γ0⟂𝐽𝐿1,𝐴∗𝜉𝑌∉Γ𝐽𝐿1,â„Žğ‘™î€·ğ¿(𝑋,𝜉)∉Γ1,(4.1) for any 𝑋,𝑌∈Γ(𝐷),𝜉∈Γ(Rad(𝑇𝑀)), and 𝑊∈Γ(𝑆(𝑇𝑀⟂)).

Proof. Using the definition of 𝐺𝐶𝑅-lightlike submanifolds, 𝑀 is 𝐷-geodesic, if and only if ğ‘”î€·â„Žğ‘™î€¸,(𝑋,𝑌),𝜉=0ğ‘”î€·â„Žğ‘ î€¸(𝑋,𝑌),𝑊=0,(4.2) for any 𝑋,𝑌∈Γ(𝐷),𝜉∈Γ(Rad(𝑇𝑀)), and 𝑊∈Γ(𝑆(𝑇𝑀⟂)). Thus for 𝑋,𝑌∈Γ(𝐷), first part of the assertion follows from (2.13).
Now for 𝑋,𝑌∈Γ(𝐷),𝜉∈Γ(Rad(𝑇𝑀)) using (2.16), we have ğ‘”î€·â„Žğ‘™î€¸=(𝑋,𝑌),𝜉𝑔∇𝑋𝑌,𝜉=−𝑔𝐽𝑌,∇𝑋𝐽𝜉=−𝑔𝐽𝑌,∇𝑋−𝐽𝜉𝑔𝐽𝑌,â„Žğ‘™î‚€ğ‘‹,𝐽𝜉=−𝑔𝐽𝑌,∇∗𝑋−𝐽𝜉𝑔𝐽𝑌,â„Žğ‘™î‚€ğ‘‹,.𝐽𝜉(4.3) Since 𝑌∈Γ(𝐷), this implies that 𝑌∈Γ(𝐷0), 𝑌∈Γ(𝐷1), 𝑌∈Γ(𝐷2), or 𝑌∈Γ(𝐽𝐷2). If 𝑌∈Γ(𝐷0) or 𝑌∈Γ(𝐷2), then we have 𝑔𝐽𝑌,â„Žğ‘™î‚€ğ‘‹,𝐽𝜉=0,(4.4) and if 𝑌∈Γ(𝐷1) or 𝑌∈Γ(𝐽𝐷2), then we have 𝑔𝐽𝑌,â„Žğ‘™î‚€ğ‘‹,𝐴𝐽𝜉=𝑔∗𝜉′𝑋,+ğ½ğœ‰ğ‘”î‚€â„Žğ‘™î€·ğ‘‹,ğœ‰î…žî€¸,𝐽𝜉,(4.5) for any 𝜉′=𝐽𝑌∈Γ(Rad(𝑇𝑀)).
Now using (4.4) and (4.5) in (4.3), we obtain ğ‘”î€·â„Žğ‘™î€¸î‚€(𝑋,𝑌),𝜉=−𝑔𝐽𝑌,∇∗𝑋𝐴𝐽𝜉−𝑔∗𝜉′𝑋,î‚âˆ’ğ½ğœ‰ğ‘”î‚€â„Žğ‘™î€·ğ‘‹,ğœ‰î…žî€¸,𝐽𝜉.(4.6) Hence the second part of the assertion follows from (4.6).

Theorem 4.5. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is ğ·î…ž-geodesic if and only if 𝐴𝑊𝑋 and 𝐴∗𝜉𝑋 have no components in 𝑀2⟂𝐽𝐷2 for any 𝑋∈Γ(ğ·î…ž), 𝜉∈Γ(Rad(𝑇𝑀)), and 𝑊∈Γ(𝑆(𝑇𝑀⟂)).

Proof. For 𝑋,𝑌∈Γ(ğ·î…ž) and 𝑊∈Γ(𝑆(𝑇𝑀⟂)) using (2.13), we obtain ğ‘”î€·â„Žğ‘ î€¸î€·ğ´(𝑋,𝑌),𝑊=𝑔𝑊𝑋,𝑌,(4.7) and for 𝜉∈Γ(Rad(𝑇𝑀)) using (2.14) and (2.17) we obtain ğ‘”î€·â„Žğ‘™î€¸î‚€ğ´(𝑋,𝑌),𝜉=𝑔∗𝜉𝑋,𝑌.(4.8) Hence the assertion follows from (4.7) and (4.8).

Theorem 4.6. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is mixed geodesic if and only if 𝐴∗𝜉𝐷𝑋∈Γ0⟂𝐽𝐿1,𝐴𝑊𝐷𝑋∈Γ0⟂Rad(𝑇𝑀)⟂𝐽𝐿1,(4.9) for any 𝑋∈Γ(𝐷),𝜉∈Γ(Rad(𝑇𝑀)), and 𝑊∈Γ(𝑆(𝑇𝑀⟂)).

Proof. For any 𝑋∈Γ(𝐷),𝑌∈Γ(ğ·î…ž), and 𝜉∈Γ(Rad(𝑇𝑀)) using (2.14) and (2.17) we obtain ğ‘”î€·â„Žğ‘™î€¸î‚€ğ´(𝑋,𝑌),𝜉=𝑔∗𝜉𝑋,𝑌,(4.10) and for 𝑊∈Γ(𝑆(𝑇𝑀⟂)) with (2.13), we obtain ğ‘”î€·â„Žğ‘ î€¸î€·ğ´(𝑋,𝑌),𝑊=𝑔𝑊𝑋,𝑌.(4.11) Hence the result follows from (4.10) and (4.11).

Theorem 4.7. Let 𝑀 be a mixed geodesic 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then one has 𝐴∗𝜉𝑋∈Γ𝐽𝐷2,(4.12) for any 𝑋∈Γ(ğ·î…ž) and 𝜉∈Γ(𝐷2).

Proof. For 𝑋∈Γ(ğ·î…ž) and 𝜉∈Γ(𝐷2) we have â„Žî‚€ğ‘‹,=𝐽𝜉∇𝑋𝐽𝜉−∇𝑋=𝐽𝜉𝐽∇𝑋𝜉−∇𝑋=ğ½ğœ‰ğ½âˆ‡ğ‘‹ğœ‰âˆ’ğ½â„Ž(𝑋,𝜉)−∇𝑋𝐽𝜉.(4.13) Since 𝑀 is mixed geodesic, therefore 𝐽∇𝑋𝜉=∇𝑋𝐽𝜉.(4.14) Using (2.16) and (2.17) we obtain −𝑇𝐴∗𝜉𝑋−𝑤𝐴∗𝜉𝑋+𝐽∇𝑋∗𝑡𝜉=∇∗𝑋𝐽𝜉+â„Žâˆ—î‚€ğ‘‹,𝐽𝜉.(4.15) Equating the transversal components we have 𝑤𝐴∗𝜉𝑋=0.(4.16) Thus 𝐴∗𝜉𝑋∈Γ𝐽𝐷2⟂𝐷0.(4.17) Now, for 𝑍∈Γ(𝐷0) and 𝜉∈Γ(𝐷2) we have 𝑔𝐴∗𝜉=𝑋,𝑍𝑔−∇𝑋𝜉+∇𝑋∗𝑡𝜉,𝑍=−𝑔∇𝑋𝜉,𝑍=−𝑔∇𝑋=𝜉,𝑍𝑔𝜉,∇𝑋𝑍=𝑔𝜉,∇𝑋𝑍+ℎ(X,𝑍)=0.(4.18) If 𝐴∗𝜉𝑋∈Γ(𝐷0), then using the nondegeneracy of 𝐷0 for any 𝑍∈Γ(𝐷0), we have 𝑔(𝐴∗𝜉𝑋,𝑍)≠0. Therefore 𝐴∗𝜉𝑋∉Γ(𝐷0). Hence the assertion is proved.

Theorem 4.8. Let 𝑀 be a mixed geodesic 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then the transversal section 𝑉∈Γ(ğ½ğ·î…ž) is 𝐷-parallel if and only if ∇𝑋𝐽𝑉∈Γ(𝐷), for any 𝑋∈Γ(𝐷).

Proof. Let 𝑌∈Γ(ğ·î…ž) such that 𝐽𝑌=𝑤𝑌=𝑉∈Γ(𝐿1⟂𝐿2) and 𝑋∈Γ(𝐷); then using hypothesis in (3.9) we have 𝑇∇𝑋𝑌=−𝐴𝑤𝑌𝑋=−𝐴𝑉𝑋. Now ∇𝑡𝑋𝑉=∇𝑋𝑉+𝐴𝑉𝑋=∇𝑋𝐽𝑌−𝑇∇𝑋𝑌. Since ∇ is a Kaehlerian connection and 𝑀 is mixed geodesic, therefore we have ∇𝑡𝑋𝑉=𝑤∇𝑋𝑌 or consequently ∇𝑡𝑋𝑉=−𝑤∇𝑋𝐽𝑉, which clearly proves the theorem.

Theorem 4.9. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀 such that 𝐷𝑠(𝑋,𝑉)∈Γ(𝐿⟂2). Then 𝐴𝐽𝑉𝑋=𝐽𝐴𝑉𝑋 for any 𝑋∈Γ(𝐷) and 𝑉∈Γ(𝐿⟂1).

Proof. For 𝑋∈Γ(𝐷),𝑌∈Γ(ğ·î…ž), and 𝑉∈Γ(𝐿⟂1), we have 𝑔𝐴𝐽𝑉𝑋−𝐽𝐴𝑉𝐴𝑋,𝑌=𝑔𝐽𝑉𝑋,𝑌−𝑔𝐽𝐴𝑉𝐴𝑋,𝑌=𝑔𝐽𝑉𝐴𝑋,𝑌+𝑔𝑉𝑋,𝐽𝑌=−𝑔∇𝑋𝐽𝑉,𝑌−𝑔∇𝑋𝑉,𝐽𝑌=−𝑔𝐽∇𝑋𝑉,𝑌+𝑔𝐽∇𝑋𝑉,𝑌=0.(4.19) For 𝑋∈Γ(𝐷),𝑍∈Γ(𝐷0), and 𝑉∈Γ(𝐿⟂1), we have 𝑔𝐴𝐽𝑉𝑋−𝐽𝐴𝑉𝐴𝑋,𝑍=𝑔𝐽𝑉𝑋,𝑍−𝑔𝐽𝐴𝑉𝐴𝑋,𝑍=𝑔𝐽𝑉𝐴𝑋,𝑍+𝑔𝑉𝑋,𝐽𝑍=−𝑔∇𝑋𝐽𝑉,𝑍−𝑔∇𝑋𝑉,𝐽𝑍=−𝑔𝐽∇𝑋𝑉,𝑍+𝑔𝐽∇𝑋𝑉,𝑍=0.(4.20) For 𝑋∈Γ(𝐷),𝑁∈Γ(ltr(𝑇𝑀)), and 𝑉∈Γ(𝐿⟂1), we have 𝑔𝐴𝐽𝑉𝑋−𝐽𝐴𝑉𝐴𝑋,𝑁=𝑔𝐽𝑉𝑋,𝑁−𝑔𝐽𝐴𝑉𝐴𝑋,𝑁=𝑔𝐽𝑉𝐴𝑋,𝑁+𝑔𝑉𝑋,𝐽𝑁=−𝑔∇𝑋𝐽𝑉,𝑁−𝑔∇𝑋𝑉,𝐽𝑁=−𝑔𝐽∇𝑋𝑉,𝑁+𝑔𝐽∇𝑋𝑉,𝑁=0.(4.21) For 𝑋∈Γ(𝐷),𝐽𝑁∈Γ(𝐽𝐿1), and 𝑉∈Γ(𝐿⟂1), we have 𝑔𝐴𝐽𝑉𝑋−𝐽𝐴𝑉𝑋,𝐴𝐽𝑁=𝑔𝐽𝑉𝑋,𝐽𝑁−𝑔𝐽𝐴𝑉𝑋,𝐴𝐽𝑁=𝑔𝐽𝑉𝑋,𝐴𝐽𝑁−𝑔𝑉𝑋,𝑁=−𝑔∇𝑋𝐽𝑉,𝐽𝑁+𝑔∇𝑋𝑉,𝑁=−𝑔𝐽∇𝑋𝑉,𝐽𝑁+𝑔∇𝑋𝑉,𝑁=−𝑔∇𝑋𝑉,𝑁+𝑔∇𝑋𝑉,𝑁=0.(4.22) Hence the assertion follows from (4.19)–(4.22).

5. 𝐺𝐶𝑅-Lightlike Product

Definition 5.1. A 𝐺𝐶𝑅-lightlike submanifold 𝑀 of an indefinite Kaehler manifold 𝑀 is called a 𝐺𝐶𝑅-lightlike product if both the distributions 𝐷 and ğ·î…ž define totally geodesic foliations in 𝑀.

Lemma 5.2. Let 𝑀 be a totally umbilical 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀; then the distribution ğ·î…ž defines a totally geodesic foliation in 𝑀.

Proof. For any 𝑋,𝑌∈Γ(ğ·î…ž), (3.9) implies that 𝑇∇𝑋𝑌=âˆ’ğ´ğ‘¤ğ‘Œğ‘‹âˆ’ğµâ„Ž(𝑋,𝑌); then for 𝑍∈Γ(𝐷0) we have 𝑔𝑇∇𝑋𝐴𝑌,𝑍=−𝑔𝑤𝑌𝑋,𝑍−𝑔(ğµâ„Ž(𝑋,𝑌),𝑍)=𝑔∇𝑋w𝑌,𝑍=𝑔∇𝑋𝐽𝑌,𝑍=−𝑔∇𝑋𝑌,𝐽𝑍=−𝑔∇𝑋𝑌,ğ‘î…žî‚î€·=𝑔𝑌,âˆ‡ğ‘‹ğ‘î…žî€¸,(5.1) where 𝑍′=𝐽𝑍∈Γ(𝐷0). Since 𝑋∈Γ(ğ·î…ž) and 𝑍∈Γ(𝐷0), then from (3.8) we have 𝑤𝑃∇𝑋𝑍=ℎ(𝑋,𝑇𝑍)âˆ’ğ¶â„Ž(𝑋,𝑍)=𝐻𝑔(𝑋,𝑇𝑍)−𝐶𝐻𝑔(𝑋,𝑍)=0, therefore 𝑤𝑃∇𝑋Z=0, and this implies that ∇𝑋𝑍∈Γ(𝐷). Therefore (5.1) implies that 𝑔(𝑇∇𝑋𝑌,𝑍)=0; then the nondegeneracy of 𝐷0 implies that 𝑇∇𝑋𝑌=0. Hence ∇𝑋𝑌∈Γ(ğ·î…ž), for any 𝑋,𝑌∈Γ(ğ·î…ž). Thus the result follows.

Theorem 5.3. Let 𝑀 be a totally umbilical 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is a 𝐺𝐶𝑅-lightlike product if and only if ğµâ„Ž(𝑋,𝑌)=0, for any 𝑋∈Γ(𝑇𝑀) and 𝑌∈Γ(𝐷).

Proof. Let 𝑀 be a 𝐺𝐶𝑅-lightlike product; therefore the distributions 𝐷 and ğ·î…ž define a totally geodesic foliation in 𝑀. Therefore using Theorem 3.4, ğµâ„Ž(𝑋,𝑌)=0 for any 𝑋,𝑌∈Γ(𝐷). Now let 𝑋∈Γ(ğ·î…ž) and 𝑌∈Γ(𝐷); then ğµâ„Ž(𝑋,𝑌)=𝑔(𝑋,𝑌)𝐵𝐻=0. Hence ğµâ„Ž(𝑋,𝑌)=0, for any 𝑋∈Γ(𝑇𝑀) and 𝑌∈Γ(𝐷).
Conversely, let 𝑋,𝑌∈Γ(𝐷); then ğµâ„Ž(𝑋,𝑌)=0 implies that 𝐷 defines a totally geodesic foliation in 𝑀. Let 𝑋,𝑌∈Γ(ğ·î…ž); then (3.9) and (3.11) imply that 𝐴𝑤𝑌𝑋=âˆ’ğ‘‡âˆ‡ğ‘‹ğ‘Œâˆ’ğµâ„Ž(𝑋,𝑌). Using Lemma 5.2, we obtain 𝑇𝐴𝑤𝑌𝑋+𝑤𝐴𝑤𝑌𝑋=−ℎ(𝑋,𝑌), we compare the tangential components, we get 𝑇𝐴𝑤𝑌𝑋=0, and this implies that 𝐴𝑤𝑌𝑋∈Γ(ğ·î…ž). Hence using Theorem 3.5, the distribution ğ·î…ž defines a totally geodesic foliation in 𝑀. Consequently, 𝑀 is a 𝐺𝐶𝑅-lightlike product of an indefinite Kaehler manifold.

Theorem 5.4. Let 𝑀 be a totally geodesic 𝐺𝐶𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Suppose that there exists a transversal vector bundle of 𝑀, which is parallel along ğ·î…ž with respect to the Levi-Civita connection on 𝑀; that is, one has ∇𝑋𝑉∈Γ(tr(𝑇𝑀)), for any 𝑉∈Γ(tr(𝑇𝑀)) and 𝑋∈Γ(ğ·î…ž). Then M is a 𝐺𝐶𝑅-lightlike product.

Proof. Since 𝑀 is a totally geodesic 𝐺𝐶𝑅-lightlike submanifold, therefore ğµâ„Ž(𝑋,𝑌)=0, for any 𝑋,𝑌∈Γ(𝐷). Hence the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Next, since ∇𝑋𝑉∈Γ(tr(𝑇𝑀)) for any 𝑉∈Γ(tr(𝑇𝑀)) and 𝑋∈Γ(ğ·î…ž), therefore using (2.8), we have 𝐴𝑉𝑋=0 then using (3.9), we obtain 𝑇∇𝑋𝑌=0 for any 𝑋,𝑌∈Γ(ğ·î…ž) and this implies that ∇𝑋𝑌∈Γ(ğ·î…ž). Hence the distribution ğ·î…ž defines a totally geodesic foliation in 𝑀. Thus 𝑀 is a 𝐺𝐶𝑅-lightlike product.

Definition 5.5. A lightlike submanifold 𝑀 of a semi-Riemannian manifold is said to be an irrotational submanifold if ∇𝑋𝜉∈Γ(𝑇𝑀) for any 𝑋∈Γ(𝑇𝑀) and 𝜉∈ΓRad(𝑇𝑀). Thus 𝑀 is an irrotational lightlike submanifold if and only if â„Žğ‘™(𝑋,𝜉)=0,â„Žğ‘ (𝑋,𝜉)=0.

Theorem 5.6. Let 𝑀 be an irrotational lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is a 𝐺𝐶𝑅-lightlike product if the following conditions are satisfied: (A)∇𝑋𝑈∈Γ(𝑆(𝑇𝑀⟂)forall𝑋∈Γ(𝑇𝑀)and𝑈∈Γtr(𝑇𝑀), (B)𝐴∗𝜉𝑌∈Γ(𝐽𝐿2)forall𝑌∈Γ(𝐷).

Proof. Using (2.11) and (2.12) with (A), we get 𝐴𝑊𝑋=0,𝐷𝑙(𝑋,𝑊)=0, and ∇𝑙𝑋𝑊=0 for any 𝑋∈Γ(𝑇𝑀) and 𝑊∈Γ(𝑆(𝑇𝑀⟂)). Therefore using (2.13) we have 𝑔(â„Žğ‘ (𝑋,𝑌),𝑊)=0; then nondegeneracy of 𝑆(𝑇𝑀⟂) implies that â„Žğ‘ (𝑋,𝑌)=0. Hence ğµâ„Žğ‘ (𝑋,𝑌)=0. Now, let 𝑋,𝑌∈Γ(𝐷) and 𝜉∈Γ(Rad(𝑇𝑀)); then using (B), we have 𝑔(â„Žğ‘™(𝑋,𝑌),𝜉)=−𝑔(∇𝑋𝜉,𝑌)=𝑔(𝐴∗𝜉𝑋,𝑌)=0. Then using (2.6), we get â„Žğ‘™(𝑋,𝑌)=0. Hence ğµâ„Žğ‘™(𝑋,𝑌)=0. Thus the distribution 𝐷 defines a totally geodesic foliation in 𝑀.
Next, let 𝑋,𝑌∈Γ(ğ·î…ž); then 𝐽𝑌=𝑤𝑌∈Γ(𝐿1⟂𝐿2)⊂tr(TM). Using (3.9) we obtain 𝑇∇𝑋𝑌=âˆ’ğµâ„Ž(𝑋,𝑌), comparing the components along 𝐷 we get 𝑇∇𝑋𝑌=0, and this implies that ∇𝑋𝑌∈Γ(ğ·î…ž). Thus the distribution ğ·î…ž defines a totally geodesic foliation in 𝑀. Hence 𝑀 is a 𝐺𝐶𝑅-lightlike product.

Theorem 5.7. 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(ğ·î…ž).

Proof. Let (∇𝑋𝑇)𝑌=0, for any 𝑋,Y∈Γ(𝐷) or 𝑋,𝑌∈Γ(ğ·î…ž). Let 𝑋,𝑌∈Γ(𝐷), then 𝑤𝑌=0 and (3.9) gives that ğµâ„Ž(𝑋,𝑌)=0. Hence using Theorem 3.4, the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Next, let 𝑋,𝑌∈Γ(ğ·î…ž). Since 𝐵𝑉∈Γ(ğ·î…ž) for any 𝑉∈Γ(tr(𝑇𝑀)), then (3.9) implies that 𝐴𝑤𝑌𝑋∈Γ(ğ·î…ž). Hence using Theorem 3.5, the distribution ğ·î…ž defines a totally geodesic foliation in 𝑀. Since both the distributions 𝐷 and ğ·î…ž define totally geodesic foliations in 𝑀, hence 𝑀 is a 𝐺𝐶𝑅-lightlike product.
Conversely, let 𝑀 be a 𝐺𝐶𝑅-lightlike product; therefore the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Using Kaehlerian property of ∇, for any 𝑋,𝑌∈Γ(𝐷) we have ∇𝑋𝐽𝑌=𝐽∇𝑋𝑌; then comparing transversal components, we obtain ℎ(𝑋,𝐽𝑌)=ğ½â„Ž(𝑋,𝑌) and then (∇𝑋𝑇)𝑌=∇𝑋𝑇𝑌−𝑇∇𝑋𝑌=âˆ‡ğ‘‹ğ½ğ‘Œâˆ’â„Ž(𝑋,𝐽𝑌)−𝐽∇𝑋𝑌+ℎ(𝑋,𝐽𝑌)=0, that is, (∇𝑋𝑇)𝑌=0, for any 𝑋,𝑌∈Γ(𝐷). Let ğ·î…ž defines a totally geodesic foliation in 𝑀, and using Kaehlerian property of ∇, we have ∇𝑋𝐽𝑌=𝐽∇𝑋𝑌; then comparing tangential components on both sides, we obtain −𝐴𝑤𝑌𝑋=ğµâ„Ž(𝑋,𝑌); then (3.9) implies that (∇𝑋𝑇)𝑌=0, which completes the proof.