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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 983069, 12 pages
http://dx.doi.org/10.1155/2011/983069
Research Article

GCR-Lightlike Product of Indefinite Sasakian Manifolds

1University College of Engineering, Punjabi University, Patiala 147002, India
2Department of Applied Sciences, Bhai Gurdas Institute of Engineering & Technology, Sangrur, Punjab 148001, India
3Department of Mathematics, Punjabi University, Patiala 147002, India

Received 21 March 2011; Accepted 12 April 2011

Academic Editor: Klaus Kirsten

Copyright © 2011 Rakesh Kumar 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

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

1. Introduction

The geometry of submanifolds is one of the most important topics of differential geometry. It is well known that the geometry of semi-Riemannian submanifolds have many similarities with their Riemannian case but the geometry of lightlike submanifolds is different since their normal vector bundle intersect with the tangent bundle making it more difficult and interesting to study. The lightlike submanifolds of semi-Riemannian manifolds were introduced and studied by Duggal and Bejancu [1]. Since contact geometry has vital role in the theory of differential equations, optics, and phase spaces of a dynamical system, therefore contact geometry with definite and indefinite metric becomes the topic of main discussion. In the process of establishment of theory of lightlike submanifolds, Duggal and Sahin [2] introduced the theory of contact 𝐶𝑅-lightlike submanifold of indefinite Sasakian manifold. Later on, Duggal and Sahin [3] introduced generalized Cauchy-Riemann (𝐺𝐶𝑅)-lightlike submanifold of indefinite Sasakian manifolds to find an umbrella of invariant, screen, real, contact 𝐶𝑅-lightlike subcases, and real hypersurfaces. In the present paper we study 𝐺𝐶𝑅-lightlike submanifolds extensively and study mixed geodesic 𝐺𝐶𝑅-lightlike submanifolds of indefinite Sasakian manifolds. We also 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 [1] by Duggal and Bejancu.

Let (𝑀,𝑔) be a real (𝑚+𝑛)-dimensional semi-Riemannian manifold of constant index 𝑞 such that 𝑚,𝑛1, 1𝑞𝑚+𝑛1 and (𝑀,𝑔) be an 𝑚-dimensional submanifold of 𝑀 and 𝑔 be 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=𝑢𝑇𝑥𝑀𝑔(𝑢,𝑣)=0,𝑣𝑇𝑥𝑀,𝑥𝑀,(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=𝑇𝑥𝑀𝑇𝑥𝑀 which is known as radical (null) subspace. If the mappingRad𝑇𝑀𝑥𝑀Rad𝑇𝑥𝑀,(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.

Let 𝑆(𝑇𝑀) be a screen distribution which is a semi-Riemannian complementary distribution of Rad(𝑇𝑀) in 𝑇𝑀, that is𝑇𝑀=Rad𝑇𝑀𝑆(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,(2.4)𝑇𝑀|𝑀=𝑇𝑀tr(𝑇𝑀)=(Rad𝑇𝑀ltr(𝑇𝑀))𝑆(𝑇𝑀)𝑆𝑇𝑀.(2.5)

Let 𝑢 be a local coordinate neighborhood of 𝑀 and consider the local quasiorthonormal fields of frames of 𝑀 along 𝑀, on 𝑢 as {𝜉1,,𝜉𝑟,𝑊𝑟+1,,𝑊𝑛,𝑁1,,𝑁𝑟,𝑋𝑟+1,,𝑋𝑚}, where {𝜉1,,𝜉𝑟},{𝑁1,,𝑁𝑟} are local lightlike bases of Γ(Rad𝑇𝑀|𝑢), Γ(ltr(𝑇𝑀)|𝑢) and {W𝑟+1,,𝑊𝑛},{𝑋𝑟+1,,𝑋𝑚} are local orthonormal bases of Γ(𝑆(𝑇𝑀)|𝑢) and Γ(𝑆(𝑇𝑀)|𝑢), respectively. For this quasi-orthonormal fields of frames, we have the following.

Theorem 2.1 (see [1]). Let (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(𝑇𝑀)) be an 𝑟-lightlike submanifold of a semi-Riemannian manifold (𝑀,𝑔). Then there exists a complementary vector bundle ltr(𝑇𝑀) of Rad𝑇𝑀 in 𝑆(𝑇𝑀) and a basis of Γ(ltr(𝑇𝑀)|u) consisting of smooth section {𝑁𝑖} of 𝑆(𝑇𝑀)|u, where u is a coordinate neighborhood of 𝑀 such that 𝑔𝑁𝑖,𝜉𝑗=𝛿𝑖𝑗,𝑔𝑁𝑖,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+(𝑋,𝑌),𝑋,𝑌Γ(𝑇𝑀),(2.7)𝑋𝑈=𝐴𝑈𝑋+𝑋𝑈,𝑋Γ(𝑇𝑀),𝑈Γ(tr(𝑇𝑀)),(2.8)

where {𝑋𝑌,𝐴𝑈𝑋} and {(𝑋,𝑌),𝑋𝑈} belong to Γ(𝑇𝑀) and Γ(tr(𝑇𝑀)), respectively. Here is a torsion-free linear connection on 𝑀, is a symmetric bilinear form on Γ(𝑇𝑀) which is called the second fundamental form, and 𝐴𝑈 is linear a operator on 𝑀 and called a shape operator.

According to (2.4), considering the projection morphisms 𝐿 and 𝑆 of 𝑡𝑟(𝑇𝑀) on ltr(𝑇𝑀) and 𝑆(𝑇𝑀), respectively, (2.11) and (2.12) give 𝑋𝑌=𝑋𝑌+𝑙(𝑋,𝑌)+𝑠(𝑋,𝑌),(2.9)𝑋𝑈=𝐴𝑈𝑋+𝐷𝑙𝑋𝑈+𝐷𝑠𝑋𝑈,(2.10) where one puts 𝑙(𝑋,𝑌)=𝐿((𝑋,𝑌)),𝑠(𝑋,𝑌)=𝑆((𝑋,𝑌)),𝐷𝑙𝑋𝑈=𝐿(𝑋𝑈), 𝐷𝑠𝑋𝑈=𝑆(𝑋𝑈).

As 𝑙 and 𝑠 are Γ(ltr(𝑇𝑀))-valued and Γ(𝑆(𝑇𝑀))-valued, respectively, therefore, these are called as 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 𝑋Γ(𝑇𝑀),𝑁Γ(ltr(𝑇𝑀)) and 𝑊Γ(𝑆(𝑇𝑀)). Using (2.4)-(2.5) and (2.8)–(2.12), one obtains 𝑔𝑠+(𝑋,𝑌),𝑊𝑔𝑌,𝐷𝑙𝐴(𝑋,𝑊)=𝑔𝑊,𝑋,𝑌(2.13)𝑔𝑙+(𝑋,𝑌),𝜉𝑔𝑌,𝑙(𝑋,𝜉)+𝑔𝑌,𝑋𝜉=0,(2.14)𝑔𝐴𝑁𝑋,𝑁+𝑔𝑁,𝐴𝑁𝑋=0,(2.15) for any 𝜉Γ(Rad𝑇𝑀), 𝑊Γ(𝑆(𝑇𝑀)) and 𝑁,𝑁Γ(ltr(𝑇𝑀)).

Let 𝑃 be the projection morphism of 𝑇𝑀 on 𝑆(𝑇𝑀). Then using (2.3), one can induce some new geometric objects on the screen distribution 𝑆(𝑇𝑀) on 𝑀 as 𝑋𝑃𝑌=𝑋𝑃𝑌+(𝑋,𝑌),(2.16)𝑋𝜉=𝐴𝜉𝑋+𝑋𝑡𝜉,(2.17) for any 𝑋,𝑌Γ(𝑇𝑀) and 𝜉Γ(Rad𝑇𝑀), where {X𝑃𝑌,𝐴𝜉𝑋} and {(𝑋,𝑌),𝑋𝑡𝜉} belong to Γ(𝑆(𝑇𝑀)) and Γ(Rad𝑇𝑀), respectively. and 𝑡 are linear connections on complementary distributions 𝑆(𝑇𝑀) and Rad𝑇𝑀, respectively. and 𝐴 are Γ(Rad𝑇𝑀)-valued and Γ(𝑆(𝑇𝑀))-valued bilinear forms and called as the second fundamental forms of distributions 𝑆(𝑇𝑀) and Rad𝑇𝑀, respectively, and one has the following equations: 𝑔𝑙𝐴(𝑋,𝑃𝑌),𝜉=𝑔𝜉,𝑋,𝑃𝑌𝑔𝐴(𝑋,𝑃𝑌),𝑁=𝑔𝑁𝑔𝐴𝑋,𝑃𝑌(2.18)𝜉𝑃𝑋,𝑃𝑌=𝑔𝑃𝑋,𝐴𝜉𝑃𝑌,𝐴𝜉𝜉=0.(2.19)

Next, one recalls some basic definition and results of indefinite Sasakian manifolds [4]. An odd dimensional semi-Riemannian manifolds (𝑀,𝑔) is called an contact metric manifold, if there is a (1,1) tensor field 𝜙, a vector field 𝑉 called characteristic vector field, and a 1-form 𝜂 such that 𝑔(𝜙𝑋,𝜙𝑌)=𝑔(𝑋,𝑌)𝜖𝜂(𝑋)𝜂(𝑌),𝜙𝑔(𝑉,𝑉)=𝜖,(2.20)2(𝑋)=𝑋+𝜂(𝑋)𝑉,𝑔(𝑋,𝑉)=𝜖𝜂(𝑋),(2.21)𝑑𝜂(𝑋,𝑌)=𝑔(𝑋,𝜙𝑌),𝑋,𝑌Γ(𝑇𝑀),(2.22) where 𝜖=±1. Therefore it follows that 𝜙𝑉=0,𝜂𝑜𝜙=0,𝜂(𝑉)=1.(2.23) Then (𝜙,𝑉,𝜂,𝑔) is called contact metric structure of 𝑀. one says that 𝑀 has a normal contact structure if 𝑁𝜙+𝑑𝜂𝑉=0, where 𝑁𝜙 is the Nijenhuis tensor field then 𝑀 is called an indefinite Sasakian manifold and for which one has 𝑋𝑉=𝜙𝑋.(2.24)𝑋𝜙𝑌=𝑔(𝑋,𝑌)𝑉+𝜖𝜂(𝑌)𝑋.(2.25)

3. Generalized Cauchy-Riemann- (GCR-) Lightlike Submanifold

Calin [5] proved that if the characteristic vector field 𝑉 is tangent to (𝑀,𝑔,𝑆(𝑇𝑀)) then it belongs to 𝑆(𝑇𝑀). We assume characteristic vector 𝑉 is tangent to 𝑀, throughout this paper.

Definition 3.1. Let (𝑀,𝑔,𝑆(𝑇𝑀)) be a real lightlike submanifold of an indefinite Sasakian manifold (𝑀,𝑔) then 𝑀 is called generalized Cauchy-Riemann (𝐺𝐶𝑅)-lightlike submanifold if the following conditions are satisfied. (A)There exist two subbundles 𝐷1 and 𝐷2 of Rad(𝑇𝑀) such thatRad(𝑇𝑀)=𝐷1𝐷2𝐷,𝜙1=𝐷1𝐷,𝜙2𝑆(𝑇𝑀).(3.1)(B)There exist two subbundles 𝐷0 and 𝐷 of 𝑆(𝑇𝑀) such that𝑆(𝑇𝑀)=𝜙𝐷2𝐷𝐷0𝑉,𝜙𝐷=𝐿𝑆,(3.2) where 𝐷0 is invariant nondegenerate distribution on 𝑀, {𝑉} is one dimensional distribution spanned by 𝑉, and 𝐿, 𝑆 are vector subbundles of ltr(𝑇𝑀) and 𝑆(𝑇𝑀), respectively.

Then tangent bundle 𝑇𝑀 of 𝑀 is decomposed as𝑇𝑀=𝐷𝐷{𝑉},𝐷=Rad(𝑇𝑀)𝐷0𝐷𝜙2.(3.3) Let 𝑄, 𝑃1, 𝑃2 be the projection morphism on 𝐷, 𝜙𝑆, 𝜙𝐿, respectively, therefore𝑋=𝑄𝑋+𝑉+𝑃1𝑋+𝑃2𝑋,(3.4) for 𝑋Γ(𝑇𝑀). Applying 𝜙 to (3.4), we obtain𝜙𝑋=𝑓𝑋+𝜔𝑃1𝑋+𝜔𝑃2𝑋,(3.5) where 𝑓𝑋Γ(𝐷), 𝜔𝑃1𝑋Γ(𝑆), and 𝜔𝑃2𝑋Γ(𝐿) and we can write (3.5) as𝜙𝑋=𝑓𝑋+𝜔𝑋,(3.6) where 𝑓𝑋 and 𝜔𝑋 are the tangential and transversal components of 𝜙𝑋, respectively. Similarly𝜙𝑈=𝐵𝑈+𝐶𝑈,𝑈Γ(tr(𝑇𝑀)),(3.7) where 𝐵𝑈 and 𝐶𝑈 are the sections of 𝑇𝑀 and 𝑡𝑟(𝑇𝑀), respectively.

Differentiating (3.5) and using (2.11)–(2.13), (2.14), and (3.7), we have 𝐷𝑙𝑋,𝜔𝑃1𝑌=𝑙𝑋𝜔𝑃2𝑌+𝜔𝑃2𝑋𝑌𝑙(𝑋,𝑓𝑌)+𝐶𝑙𝐷(𝑋,𝑌),𝑠𝑋,𝜔𝑃2𝑌=𝑠𝑋𝜔𝑃1𝑌+𝜔𝑃1𝑋𝑌𝑠(𝑋,𝑓𝑌)+𝐶𝑠(𝑋,𝑌),(3.8) for all 𝑋,𝑌Γ(𝑇𝑀). Using Sasakian property of with (2.13) and (2.14), we have the following lemmas.

Lemma 3.2. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then one has 𝑋𝑓𝑌=𝐴𝜔𝑌𝑋+𝐵(𝑋,𝑌)𝑔(𝑋,𝑌)𝑉+𝜖𝜂(𝑌)𝑋,(3.9)𝑡𝑋𝜔𝑌=𝐶(𝑋,𝑌)(𝑋,𝑓𝑌),(3.10) where 𝑋,𝑌Γ(𝑇𝑀) and 𝑋𝑓𝑌=𝑋𝑓𝑌𝑓𝑋𝑌,(3.11)𝑡𝑋𝜔𝑌=𝑡𝑋𝜔𝑌𝜔𝑋𝑌.(3.12)

Lemma 3.3. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then one has 𝑋𝐵𝑈=𝐴𝐶𝑈𝑋𝑓𝐴𝑈𝑋,𝑡𝑋𝐶𝑈=𝜔𝐴𝑈𝑋(𝑋,𝐵𝑈),(3.13) where 𝑋Γ(𝑇𝑀), 𝑈Γ(tr(𝑇𝑀)) and 𝑋𝐵𝑈=𝑋𝐵𝑈𝐵𝑡𝑋𝑈,𝑡𝑋𝐶𝑈=𝑡𝑋𝐶𝑈𝐶𝑡𝑋𝑈.(3.14)

4. Mixed Geodesic GCR-Lightlike Submanifolds

Definition 4.1. A 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold is called mixed geodesic 𝐺𝐶𝑅-lightlike submanifold if its second fundamental form satisfies (𝑋,𝑌)=0 for any 𝑋Γ(𝐷) and 𝑌Γ(𝐷).

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

Theorem 4.3. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then 𝑀 is mixed geodesic if and only if 𝐴𝜉𝑋 and 𝐴𝑊𝑋Γ(𝑀2𝜙𝐷2), for any 𝑋Γ(𝐷𝑉),𝑊Γ(𝑆(𝑇𝑀)) and 𝜉Γ(Rad(𝑇𝑀)).

Proof. Using definition of 𝐺𝐶𝑅-lightlike submanifolds, 𝑀 is mixed geodesic if and only if, 𝑔((𝑋,𝑌),𝑊)=𝑔((𝑋,𝑌),𝜉)=0 for 𝑋Γ(𝐷𝑉),𝑌Γ(𝐷),𝑊Γ(𝑆(𝑇𝑀)), and 𝜉Γ(Rad(𝑇𝑀)). Using (2.12) and (2.17) we get 𝑔((𝑋,𝑌),𝑊)=𝑔𝑋𝑌,𝑊=𝑔𝑌,𝑋𝑊=𝑔𝑌,𝐴𝑊𝑋,𝑔((𝑋,𝑌),𝜉)=𝑔𝑋𝑌,𝜉=𝑔𝑌,𝑋𝜉=𝑔𝑌,𝐴𝜉𝑋.(4.1) Therefore from (4.1), the proof is complete.

Theorem 4.4. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then 𝑀 is 𝐷 geodesic if and only if 𝐴𝜉𝑋 and 𝐴𝑊𝑋Γ(𝑀2𝜙𝐷2), for any 𝑋Γ(𝐷),𝜉ΓRad(𝑇𝑀) and 𝑊Γ(𝑆(𝑇𝑀)).

Proof. Proof is similar to the proof of Theorem 4.3.

Lemma 4.5. Let 𝑀 be a mixed geodesic 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then 𝐴𝜉𝑋Γ(𝜙𝐷2), for any 𝑋Γ(𝐷), 𝜉Γ(𝐷2).

Proof. For 𝑋Γ(𝐷) and 𝜉Γ(𝐷2), using (2.9) we have (𝜙𝜉,𝑋)=𝑋𝜙𝜉𝑋=𝜙𝜉𝑋𝜙𝜉+𝜙𝑋𝜉𝑋𝜙𝜉=𝑔(𝑋,𝜉)𝑉+𝜖𝜂(𝜉)𝑋+𝜙𝑋𝜉𝑋𝜙𝜉=𝜙𝑋𝜉+𝜙(𝑋,𝜉)𝑋𝜙𝜉.(4.2) Since 𝑀 is mixed geodesic therefore 𝜙𝑋𝜉=𝑋𝜙𝜉. Using (2.16) and (2.17) we get 𝜙(𝐴𝜉𝑋+𝑋𝑡𝜉)=𝑋𝜙𝜉+(𝑋,𝜙𝜉) then using (3.6) we obtain 𝑓𝐴𝜉𝑋𝜔𝐴𝜉𝑋+𝜙(𝑋𝑡𝜉)=𝑋𝜙𝜉+(𝑋,𝜙𝜉). Comparing the transversal components we get 𝜔𝐴𝜉𝑋=0,(4.3) or 𝐴𝜉𝐷𝑋Γ0𝐷{𝑉}𝜙2.(4.4) If 𝐴𝜉𝑋𝐷0 then the nondegeneracy of 𝐷0 implies that there must exists a 𝑍0𝐷0 such that 𝑔𝐴𝜉𝑋,𝑍00.(4.5) But from (2.9) and (2.17) we get 𝑔𝐴𝜉𝑋,𝑍0=𝑔𝑋𝜉,𝑍0=𝑔𝑋𝜉,𝑍0=𝑔𝜉,𝑋𝑍0=𝑔𝜉,𝑋𝑍0+𝑋,𝑍0=0.(4.6) Therefore 𝐴𝜉𝐷𝑋Γ0.(4.7) Also using (2.20), (2.21), and (2.24), we get 𝑔𝐴𝜉𝑋,𝑉=𝑔𝑋=𝜉,𝑉𝑔𝜉,𝑋𝑉=𝑔(𝜉,𝜙𝑉)=𝑔(𝑉,𝜙𝜉)=0.(4.8) Therefore 𝐴𝜉𝑋{𝑉}.(4.9) Hence from (4.4), (4.7), and (4.9) the result follows.

Corollary 4.6. Let 𝑀 be a mixed geodesic 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then 𝑔(𝑙(𝑋,𝑌),𝜉)=0, for any 𝑋Γ(𝐷),𝑌Γ(𝑀2) and 𝜉Γ(𝐷2).

Proof. From (2.18) and above lemma, the result follows.

Lemma 4.7. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then 𝑔(𝐴𝑊𝜙𝑋,𝑌)=𝑔(𝐴𝑊𝑌,𝜙𝑋)𝑔(𝜙𝑋,𝐷𝑙(𝑌,𝑊)), for any 𝑋Γ(𝐷{𝑉}),𝑌Γ(𝐷) and 𝑊Γ(𝑆(𝑇𝑀)).

Proof. Using (2.12), we have 𝑔𝐴𝑊𝜙𝑋,𝑌=𝑔𝜙𝑋=𝑊,𝑌=𝑔(𝑊,(𝜙𝑋,𝑌))𝑔𝑌𝜙𝑋,𝑊=𝑔𝜙𝑋,𝑌𝑊=𝑔𝜙𝑋,𝐴𝑊𝑌𝑔𝜙𝑋,𝐷𝑙,(𝑌,𝑊)(4.10) for 𝑋Γ(𝐷{𝑉}),𝑌Γ(𝐷) and 𝑊Γ(𝑆(𝑇𝑀)).

Theorem 4.8. Let 𝑀 be a mixed geodesic 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then 𝐴𝑈𝑋Γ(𝐷{𝑉}) and 𝑡𝑋𝑈Γ(𝐿𝑆), for any 𝑋Γ(𝐷{𝑉}) and 𝑈Γ(𝐿𝑆).

Proof. Since 𝑀 is mixed geodesic, therefore (𝑋,𝑌)=0 for any 𝑋Γ(𝐷{𝑉}),𝑌Γ(𝐷), therefore (2.7) gives 0=𝑋𝑌𝑋𝑌.(4.11) Since 𝐷 is anti-invariant there exist 𝑈Γ(𝐿𝑆) such that 𝜙𝑈=𝑌. Thus from (2.12), (3.6), and (3.7) we get 0=𝑋𝜙𝑈𝑋𝑌=𝑋𝜙𝑈+𝜙𝑋𝑈𝑋𝑌=𝑔(𝑋,𝑈)𝑉+𝜖𝜂(𝑈)𝑋+𝜙𝐴𝑈𝑋+𝑡𝑋𝑈𝑋𝑌=𝑔(𝑋,𝑈)𝑉𝑓𝐴𝑈𝑋𝜔𝐴𝑈𝑋+𝐵𝑡𝑋𝑈+𝐶𝑡𝑋𝑈𝑋𝑌.(4.12) Comparing transversal components we get 𝜔𝐴𝑈𝑋=𝐶𝑡𝑋𝑈,(4.13) since 𝜔𝐴𝑈𝑋Γ(𝐿𝑆) and 𝐶𝑡𝑋𝑈Γ(𝐿𝑆), hence 𝐴𝑈𝑋Γ(𝐷{𝑉}) and 𝑡𝑋𝑈Γ(𝐿𝑆).

5. GCR-Lightlike Product

Definition 5.1. A 𝐺𝐶𝑅-lightlike submanifold 𝑀 of an indefinite Sasakian manifold 𝑀 is called 𝐺𝐶𝑅-lightlike product if both the distributions 𝐷{𝑉} and 𝐷 defines totally geodesic foliation in 𝑀.

Theorem 5.2. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then the distribution 𝐷{𝑉} defines a totally geodesic foliation in 𝑀 if and only if 𝐵(𝑋,𝜙𝑌)=0, for any 𝑋,𝑌𝐷{𝑉}.

Proof. Since 𝐷=𝜙(𝐿𝑆), therefore 𝐷{𝑉} defines a totally geodesic foliation in 𝑀 if and only if 𝑔𝑋𝑌,𝜙𝜉=𝑔𝑋𝑌,𝜙𝑊=0,(5.1) for any 𝑋,𝑌Γ(𝐷{𝑉}), 𝜉Γ(𝐷2) and 𝑊Γ(𝑆). Using (2.9) and (2.25), we have 𝑔𝑋𝑌,𝜙𝜉=𝑔𝑋𝜙𝑌,𝜉=𝑔𝑙,𝑔(𝑋,𝑓𝑌),𝜉𝑋𝑌,𝜙𝑊=𝑔𝑋𝜙𝑌,𝑊=𝑔𝑠.(𝑋,𝑓𝑌),𝑊(5.2) Hence, from (5.2) the assertion follows.

Theorem 5.3. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then the distribution 𝐷 defines a totally geodesic foliation in 𝑀 if and only if 𝐴𝑁𝑋 has no component in 𝜙𝑆𝜙𝐷2 and 𝐴𝜔𝑌𝑋 has no component in 𝐷2𝐷0, for any 𝑋,𝑌Γ(𝐷) and 𝑁Γ(ltr(𝑇𝑀)).

Proof. We know that 𝐷 defines a totally geodesic foliation in 𝑀 if and only if 𝑔𝑋𝑌,𝑁=𝑔𝑋𝑌,𝜙𝑁1=𝑔𝑋𝑌,𝑉=𝑔𝑋𝑌,𝜙𝑍=0,(5.3) for 𝑋,𝑌Γ(𝐷), 𝑁Γ(ltr(𝑇𝑀)), 𝑍Γ(𝐷0), and 𝑁1Γ(𝐿). Using (2.9) and (2.11) we have 𝑔𝑋=𝑌,𝑁𝑔𝑋𝑌,𝑁=𝑔𝑌,𝑋𝑁=𝑔𝑌,𝐴𝑁𝑋.(5.4) Using (2.9), (2.10), and (2.25) we obtain 𝑔𝑋𝑌,𝜙𝑁1𝜙=𝑔𝑋𝑌,𝑁1=𝑔𝑋𝜔𝑌,𝑁1𝐴=𝑔𝜔𝑌𝑋,𝑁1,𝑔𝑋𝜙𝑌,𝜙𝑍=𝑔𝑋𝑌,𝑍=𝑔𝑋𝐴𝜔𝑌,𝑍=𝑔𝜔𝑌,𝑋,𝑍(5.5) also 𝑔𝑋𝑌,𝑉=𝑔𝑋𝑌,𝑉=𝑔𝑌,𝑋𝑉=𝑔(𝑌,𝜙𝑋)=0.(5.6) Thus from (5.4)–(5.6), the result follows.

Theorem 5.4. Let 𝑀 be a 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. If 𝑓=0 then 𝑀 is a 𝐺𝐶𝑅 lightlike product.

Proof. Let 𝑋,𝑌Γ(𝐷) therefore 𝑓𝑌=0, then using (3.11) with the hypothesis, we get 𝑓𝑋𝑌=0 therefore the distribution 𝐷 defines a totally geodesic foliation.Let 𝑋,𝑌𝐷{𝑉} therefore 𝜔𝑌=0 then using (3.9), we get 𝐵(𝑋,𝑌)+𝑔(𝑋,𝑌)𝑉+𝜖𝜂(𝑌)𝑋=0.(5.7) Equating components along the distribution 𝐷 of above equation, we get 𝐵(𝑋,𝑌)=0, therefore 𝐷{𝑉} define a totally geodesic foliation in 𝑀. Hence 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 𝑋𝜉Γ(𝑇𝑀) for any 𝑋Γ(𝑇𝑀) and 𝜉ΓRad(𝑇𝑀). Thus 𝑀 is an irrotational lightlike submanifold if and only if 𝑙(𝑋,𝜉)=0,𝑠(𝑋,𝜉)=0.

Theorem 5.6. Let 𝑀 be an irrotational 𝐺𝐶𝑅-lightlike submanifold of indefinite Sasakian manifold 𝑀. Then 𝑀 is a 𝐺𝐶𝑅 lightlike product if the following conditions are satisfied. (A)𝑋𝑈Γ(𝑆(𝑇𝑀),𝑋Γ(𝑇𝑀),and𝑈Γ(tr(𝑇𝑀)). (B)𝐴𝜉𝑌Γ(𝜙(𝑆)),𝑌Γ(𝐷).

Proof. Let (𝐴) holds, then using (2.11) and (2.12), we get 𝐴𝑁𝑋=0,𝐴𝑊𝑋=0,𝐷𝑙(𝑋,𝑊)=0, and 𝑙𝑋𝑁=0 for 𝑋Γ(𝑇𝑀). These equations imply that the distribution 𝐷 defines a totally geodesic foliation in 𝑀 and with (2.13), we get 𝑔(𝑠(𝑋,𝑌),𝑊)=0. Hence non degeneracy of 𝑆(𝑇𝑀) implies that 𝑠(𝑋,𝑌)=0. Therefore 𝑠(𝑋,𝑌) has no component in 𝑆. Finally, from (2.14) and 𝑀 being irrotational, we have 𝑔(𝑙(𝑋,𝑌),𝜉)=𝑔(𝑌,𝐴𝜉𝑋), for 𝑋Γ(𝑇𝑀) and 𝑌Γ(𝐷). Assume (𝐵) holds, then 𝑙(𝑋,𝑌)=0. Therefore 𝑙(𝑋,𝑌) has no component in 𝐿. Thus the distribution 𝐷{𝑉} defines a totally geodesic foliation in 𝑀. Hence 𝑀 is a 𝐺𝐶𝑅 lightlike product.

Definition 5.7 (see [6]). If the second fundamental form of a submanifold tangent to characteristic vector field 𝑉, of an indefinite Sasakian manifold 𝑀 is of the form (𝑋,𝑌)={𝑔(𝑋,𝑌)𝜂(𝑋)𝜂(𝑌)}𝛼+𝜂(𝑋)(𝑌,𝑉)+𝜂(𝑌)(𝑋,𝑉),(5.8) for any 𝑋,𝑌Γ(𝑇𝑀), where 𝛼 is a vector field transversal to 𝑀, then 𝑀 is called a totally contact umbilical submanifold of an indefinite Sasakian manifold.

Theorem 5.8. Let 𝑀 be a totally contact umbilical 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Then 𝑀 is a 𝐺𝐶𝑅-lightlike product if 𝐵(𝑋,𝑌)=0, for any 𝑋Γ(𝑇𝑀) and 𝑌Γ(𝐷{𝑉}).

Proof. Let 𝑋,𝑌Γ(𝐷{𝑉}), then the hypothesis 𝐵(𝑋,𝑌)=0, implies that the distribution 𝐷{𝑉} defines totally geodesic foliation in 𝑀.Let 𝑋,𝑌Γ(𝐷) then using (3.9), we have 𝑓𝑋𝑌=𝐴𝜔𝑌𝑋+𝐵(𝑋,𝑌)𝑔(𝑋,𝑌)𝑉,(5.9) let 𝑍Γ(𝐷0) and using (2.8) and (2.25), then above equation becomes 𝑔𝑓𝑋𝐴𝑌,𝑍=𝑔𝜔𝑌𝑋+𝐵(𝑋,𝑌)𝑔(𝑋,𝑌)𝑉,𝑍=𝑔𝑋𝜙𝑌,𝑍=𝑔𝑋𝑌,𝜙𝑍=𝑔𝑌,𝑋𝑍,(5.10) where 𝜙𝑍=𝑍Γ(𝐷0). For 𝑋Γ(𝐷) from (3.10), we get 𝜔𝑃𝑋𝑍=(𝑋,𝑓𝑍)𝐶(𝑋,𝑍).(5.11) Using the hypothesis with (5.8), we get 𝜔𝑃𝑋𝑍=0, this implies 𝑋𝑍Γ(𝐷) therefore, (5.10) becomes 𝑔(𝑓𝑋𝑌,𝑍)=0. Then non degeneracy of the distribution 𝐷0 implies that the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Hence the assertion follows.

Theorem 5.9. Let 𝑀 be a totally geodesic 𝐺𝐶𝑅-lightlike submanifold of an indefinite Sasakian manifold 𝑀. Suppose there exists a transversal vector bundle of 𝑀 which is parallel along 𝐷 with respect to Levi-Civita connection on 𝑀, that is, 𝑋𝑈Γ(tr(𝑇𝑀)), for any 𝑈Γ(tr(𝑇𝑀)), 𝑋Γ(𝐷). Then 𝑀 is a 𝐺𝐶𝑅-lightlike product.

Proof. Since 𝑀 is a totally geodesic 𝐺𝐶𝑅-lightlike, therefore 𝐵(𝑋,𝑌)=0, for 𝑋,𝑌Γ(𝐷{𝑉}), this implies 𝐷{𝑉} defines a totally geodesic foliation in 𝑀. Next, 𝑋𝑈Γ(tr(𝑇𝑀)) implies 𝐴𝑈𝑋=0. Hence by Theorem 5.3, the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Thus 𝑀 is a contact 𝐺𝐶𝑅-lightlike product.

References

  1. K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol. 364 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996.
  2. K. L. Duggal and B. Sahin, “Lightlike submanifolds of indefinite Sasakian manifolds,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 57585, 21 pages, 2007. View at Zentralblatt MATH
  3. K. L. Duggal and B. Sahin, “Generalized Cauchy-Riemann lightlike submanifolds of indefinite Sasakian manifolds,” Acta Mathematica Hungarica, vol. 122, no. 1-2, pp. 45–58, 2009. View at Publisher · View at Google Scholar
  4. R. Kumar, R. Rani, and R. K. Nagaich, “On sectional curvatures of ε-Sasakian manifolds,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 93562, 8 pages, 2007.
  5. C. Calin, “On the existence of degenerate hypersurfaces in Sasakian manifolds,” Arab Journal of Mathematical Sciences, vol. 5, no. 1, pp. 21–27, 1999. View at Zentralblatt MATH
  6. K. Yano and M. Kon, Structures on Manifolds, vol. 3 of Series in Pure Mathematics, World Scientific, Singapore, 1984.