International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 605816 | https://doi.org/10.1155/2012/605816

Varun Jain, Rakesh Kumar, R. K. Nagaich, "On GCR-Lightlike Product of Indefinite Cosymplectic Manifolds", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 605816, 12 pages, 2012. https://doi.org/10.1155/2012/605816

On GCR-Lightlike Product of Indefinite Cosymplectic Manifolds

Academic Editor: Hernando Quevedo
Received26 Mar 2012
Revised15 May 2012
Accepted10 Jul 2012
Published15 Aug 2012

Abstract

We define GCR-lightlike submanifolds of indefinite cosymplectic manifolds and give an example. Then, we study mixed geodesic GCR-lightlike submanifolds of indefinite cosymplectic manifolds and obtain some characterization theorems for a GCR-lightlike submanifold to be a GCR-lightlike product.

1. Introduction

To fill the gaps in the general theory of submanifolds, Duggal and Bejancu [1] introduced lightlike (degenerate) geometry of submanifolds. Since the geometry of 𝐢𝑅-submanifolds has potential for applications in mathematical physics, particularly in general relativity, and the geometry of lightlike submanifolds has extensive uses in mathematical physics and relativity, Duggal and Bejancu [1] clubbed these two topics and introduced the theory of 𝐢𝑅-lightlike submanifolds of indefinite Kaehler manifolds and then Duggal and Sahin [2], introduced the theory of 𝐢𝑅-lightlike submanifolds of indefinite Sasakian manifolds, which were further studied by Kumar et al. [3]. But 𝐢𝑅-lightlike submanifolds do not include the complex and real subcases contrary to the classical theory of 𝐢𝑅-submanifolds [4]. Thus, later on, Duggal and Sahin [5] introduced a new class of submanifolds, generalized-Cauchy-Riemann- (GCR-) lightlike submanifolds of indefinite Kaehler manifolds and then of indefinite Sasakian manifolds in [6]. This class of submanifolds acts as an umbrella of invariant, screen real, contact 𝐢𝑅-lightlike subcases and real hypersurfaces. Therefore, the study of 𝐺𝐢𝑅-lightlike submanifolds is the topic of main discussion in the present scenario. In [7], the present authors studied totally contact umbilical 𝐺𝐢𝑅-lightlike submanifolds of indefinite Sasakian manifolds.

In present paper, after defining 𝐺𝐢𝑅-lightlike submanifolds of indefinite cosymplectic manifolds, we study mixed geodesic 𝐺𝐢𝑅-lightlike submanifolds of indefinite cosymplectic manifolds. In [8, 9], Kumar et al. obtained some necessary and sufficient conditions for a 𝐺𝐢𝑅-lightlike submanifold of indefinite Kaehler and Sasakian manifolds to be a 𝐺𝐢𝑅-lightlike product, respectively. Thus, in this paper, we obtain some characterization theorems for a 𝐺𝐢𝑅-lightlike submanifold of indefinite cosymplectic manifold to be a 𝐺𝐢𝑅-lightlike product.

2. Lightlike Submanifolds

Let 𝑉 be a real π‘š-dimensional vector space with a symmetric bilinear mapping π‘”βˆΆπ‘‰Γ—π‘‰β†’β„œ. The mapping 𝑔 is called degenerate on 𝑉 if there exists a vector πœ‰β‰ 0 of 𝑉 such that 𝑔(πœ‰,𝑣)=0,βˆ€π‘£βˆˆπ‘‰,(2.1) otherwise 𝑔 is called nondegenerate. It is important to note that a non-degenerate symmetric bilinear form on 𝑉 may induce either a non-degenerate or a degenerate symmetric bilinear form on a subspace of 𝑉. Let π‘Š be a subspace of 𝑉 and π‘”βˆ£π‘€β€‰β€‰ degenerate; then π‘Š is called a degenerate (lightlike) subspace of 𝑉.

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. Thus, if 𝑔 is degenerate on the tangent bundle 𝑇𝑀 of 𝑀, then 𝑀 is called a lightlike (degenerate) submanifold of 𝑀 (for detail see [1]). For a degenerate metric 𝑔 on 𝑀, π‘‡π‘€βŸ‚ is also 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 mapping Radπ‘‡π‘€βˆΆπ‘₯βˆˆπ‘€β†’Rad𝑇π‘₯𝑀 defines a smooth distribution on 𝑀 of rank π‘Ÿ>0, then the submanifold 𝑀 of 𝑀 is called an π‘Ÿ-lightlike submanifold and Rad𝑇𝑀 is called the radical distribution on 𝑀. Then, there exists a non-degenerate screen distribution 𝑆(𝑇𝑀) which is a complementary vector subbundle to Rad𝑇𝑀 in 𝑇𝑀. Therefore, 𝑇𝑀=Radπ‘‡π‘€βŸ‚π‘†(𝑇𝑀),(2.2) where βŸ‚ denotes orthogonal direct sum. Let 𝑆(π‘‡π‘€βŸ‚), called screen transversal vector bundle, be a non-degenerate 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 𝑇𝑀|𝑀 and to Rad𝑇𝑀 in 𝑆(π‘‡π‘€βŸ‚)βŸ‚, called transversal vector bundle and lightlike transversal vector bundle of 𝑀, respectively. Then, we have ξ€·tr(𝑇𝑀)=ltr(𝑇𝑀)βŸ‚π‘†π‘‡π‘€βŸ‚ξ€Έπ‘‡,(2.3)𝑀|𝑀=π‘‡π‘€βŠ•tr(𝑇𝑀)=(Radπ‘‡π‘€βŠ•ltr(𝑇𝑀))βŸ‚π‘†(𝑇𝑀)βŸ‚π‘†π‘‡π‘€βŸ‚ξ€Έ.(2.4)

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}and{𝑁1,…,π‘π‘Ÿ} are local lightlike bases of Ξ“(Rad𝑇𝑀|𝑒) and Ξ“(ltr(𝑇𝑀)|𝑒) and {π‘Šπ‘Ÿ+1,…,π‘Šπ‘›}and{π‘‹π‘Ÿ+1,…,π‘‹π‘š} are local orthonormal bases of Ξ“(𝑆(π‘‡π‘€βŸ‚)|𝑒) and Ξ“(𝑆(𝑇𝑀)|𝑒), respectively. For these quasiorthonormal fields of frames, we have the following theorem.

Theorem 2.1 (see [1]). Let (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) be an π‘Ÿ-lightlike submanifold of a semi-Riemannian manifold (𝑀,𝑔). Then there, exist a complementary vector bundle ltr(TM) of Rad TM in 𝑆(π‘‡π‘€βŸ‚)βŸ‚ and a basis of Ξ“(ltr(𝑇𝑀)|𝑒) consisting of smooth section {𝑁𝑖} of 𝑆(π‘‡π‘€βŸ‚)βŸ‚|𝑒, where 𝑒 is a coordinate neighborhood of 𝑀, such that 𝑔𝑁𝑖,πœ‰π‘—ξ€Έ=𝛿𝑖𝑗,𝑔𝑁𝑖,𝑁𝑗=0,forany𝑖,π‘—βˆˆ{1,2,…,π‘Ÿ},(2.5) where {πœ‰1,…,πœ‰π‘Ÿ} is a lightlike basis of Ξ“(Rad(𝑇𝑀)).

Let βˆ‡ be the Levi-Civita connection on 𝑀. Then, according to decomposition (2.4), the Gauss and Weingarten formulas are given by βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Ž(𝑋,π‘Œ),βˆ‡π‘‹π‘ˆ=βˆ’π΄π‘ˆπ‘‹+βˆ‡βŸ‚π‘‹π‘ˆ,(2.6) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀) and π‘ˆβˆˆΞ“(tr(𝑇𝑀)), 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) that is called second fundamental form, and π΄π‘ˆ is a linear operator on 𝑀, known as shape operator.

According to (2.3), considering the projection morphisms 𝐿 and 𝑆 of tr(𝑇𝑀) on ltr(𝑇𝑀) and 𝑆(π‘‡π‘€βŸ‚), respectively, then (2.6) gives βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Žπ‘™(𝑋,π‘Œ)+β„Žπ‘ (𝑋,π‘Œ),βˆ‡π‘‹π‘ˆ=βˆ’π΄π‘ˆπ‘‹+π·π‘™π‘‹π‘ˆ+π·π‘ π‘‹π‘ˆ,(2.7) 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, 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),βˆ‡π‘‹π‘Š=βˆ’π΄π‘Šπ‘‹+βˆ‡π‘ π‘‹π‘Š+𝐷𝑙(𝑋,π‘Š),(2.8) where π‘‹βˆˆΞ“(𝑇𝑀),π‘βˆˆΞ“(ltr(𝑇𝑀)), and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)). By using (2.3)-(2.4) and (2.7)-(2.8), we obtain π‘”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(2.9)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,(2.10) for any πœ‰βˆˆΞ“(Rad𝑇𝑀), π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)), and 𝑁,π‘β€²βˆˆΞ“(ltr(𝑇𝑀)).

Let 𝑃 be the projection morphism of 𝑇𝑀 on 𝑆(𝑇𝑀). Then, using (2.2), we can induce some new geometric objects on the screen distribution 𝑆(𝑇𝑀) on 𝑀 as βˆ‡π‘‹π‘ƒπ‘Œ=βˆ‡βˆ—π‘‹π‘ƒπ‘Œ+β„Žβˆ—(𝑋,π‘Œ),βˆ‡π‘‹πœ‰=βˆ’π΄βˆ—πœ‰π‘‹+βˆ‡π‘‹βˆ—π‘‘πœ‰,(2.11) 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‘Œ),βˆ‡π‘‹βˆ—π‘‘πœ‰} belong to Ξ“(𝑆(𝑇𝑀)) and Ξ“(Rad𝑇𝑀), respectively. βˆ‡βˆ— and βˆ‡βˆ—π‘‘ are linear connections on complementary distributions 𝑆(𝑇𝑀) and Rad𝑇𝑀, respectively. Then, using (2.7), (2.8), and (2.11), we have π‘”ξ€·β„Žπ‘™ξ€Έξ‚€π΄(𝑋,π‘ƒπ‘Œ),πœ‰=π‘”βˆ—πœ‰ξ‚,𝑋,π‘ƒπ‘Œπ‘”ξ€·β„Žβˆ—ξ€Έξ€·π΄(𝑋,π‘ƒπ‘Œ),𝑁=𝑔𝑁𝑋,π‘ƒπ‘Œ.(2.12)

Next, an odd-dimensional semi-Riemannian manifold 𝑀 is said to be an indefinite almost contact metric manifold if there exist structure tensors (πœ™,𝑉,πœ‚,𝑔), where πœ™ is a (1,1) tensor field, 𝑉 is a vector field called structure vector field, πœ‚ is a 1-form, and 𝑔 is the semi-Riemannian metric on 𝑀 satisfying (see [10]) 𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœ‚(𝑋)πœ‚(π‘Œ),πœ™π‘”(𝑋,𝑉)=πœ‚(𝑋),2𝑋=βˆ’π‘‹+πœ‚(𝑋)𝑉,πœ‚βˆ˜πœ™=0,πœ™π‘‰=0,πœ‚(𝑉)=1,(2.13) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀).

An indefinite almost contact metric manifold 𝑀 is called an indefinite cosymplectic manifold if (see [11]) βˆ‡π‘‹πœ™=0,(2.14)βˆ‡π‘‹π‘‰=0.(2.15)

3. Generalized Cauchy-Riemann Lightlike Submanifolds

Calin [12] proved that if the characteristic vector field 𝑉 is tangent to (𝑀,𝑔,𝑆(𝑇𝑀)), then it belongs to 𝑆(𝑇𝑀). We assume that the characteristic vector 𝑉 is tangent to 𝑀 throughout this paper. Thus, we define the generalized Cauchy-Riemann lightlike submanifolds of an indefinite cosymplectic manifold as follows.

Definition 3.1. Let (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) be a real lightlike submanifold of an indefinite cosymplectic manifold (𝑀,𝑔) such that the structure vector field 𝑉 is tangent to 𝑀; 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)=𝐷1βŠ•π·2𝐷,πœ™1ξ€Έ=𝐷1𝐷,πœ™2ξ€ΈβŠ‚π‘†(𝑇𝑀),(3.1)(B)there exist two subbundles 𝐷0 and 𝐷 of 𝑆(𝑇𝑀) such that 𝑆(𝑇𝑀)=πœ™π·2βŠ•π·ξ‚‡βŸ‚π·0ξ‚€βŸ‚π‘‰,πœ™π·ξ‚=πΏβŸ‚π‘†,(3.2) where 𝐷0 is invariant nondegenerate distribution on 𝑀, {𝑉} is one-dimensional distribution spanned by 𝑉, and 𝐿 and 𝑆 are vector subbundles of ltr(𝑇𝑀) and 𝑆(𝑇𝑀)βŸ‚, respectively.

Therefore, the tangent bundle 𝑇𝑀 of 𝑀 is decomposed as 𝑇𝑀=π·βŠ•ξ‚‡π·βŠ•{𝑉},𝐷=Rad(𝑇𝑀)βŠ•π·0ξ€·π·βŠ•πœ™2ξ€Έ.(3.3) A contact 𝐺𝐢𝑅-lightlike submanifold is said to be proper if 𝐷0β‰ {0},𝐷1β‰ {0},𝐷2β‰ {0}, and 𝐿≠{0}. Hence, from the definition of 𝐺𝐢𝑅-lightlike submanifolds, we have that (a)condition (A) implies that dim(Rad𝑇𝑀)β‰₯3,(b)condition (B) implies that dim(𝐷)β‰₯2𝑠β‰₯6 and dim(𝐷2)=dim(𝑆), and thus dim(𝑀)β‰₯9 and dim(𝑀)β‰₯13. (c)any proper 9-dimensional contact 𝐺𝐢𝑅-lightlike submanifold is 3-lightlike, (d)(a) and contact distribution (πœ‚=0) imply that index (𝑀)β‰₯4.The following proposition shows that the class of 𝐺𝐢𝑅-lightlike submanifolds is an umbrella of invariant, contact 𝐢𝑅 and contact 𝑆𝐢𝑅-lightlike submanifolds.

Proposition 3.2. A 𝐺𝐢𝑅-lightlike submanifold 𝑀 of an indefinite cosymplectic manifold 𝑀 is contact 𝐢𝑅-submanifold (resp., contact 𝑆𝐢𝑅-lightlike submanifold) if and only if 𝐷1={0} (resp., 𝐷2={0}).

Proof. Let 𝑀 be a contact 𝐢𝑅-lightlike submanifold; then πœ™Rad𝑇𝑀 is a distribution on 𝑀 such that β‹‚Radπ‘‡π‘€πœ™Rad𝑇𝑀={0}. Therefore, 𝐷2=Rad𝑇𝑀 and 𝐷1={0}. Since β‹‚ltr(𝑇𝑀)πœ™(ltr(𝑇𝑀))={0}, this implies that πœ™(ltr(𝑇𝑀))βŠ‚π‘†(𝑇𝑀). Conversely, suppose that 𝑀 is a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Cosymplectic manifold such that 𝐷1={0}. Then, from (3.1), we have 𝐷2=Rad(𝑇𝑀), and therefore β‹‚Radπ‘‡π‘€πœ™Rad𝑇𝑀={0}. Hence, πœ™Rad𝑇𝑀 is a vector subbundle of 𝑆(𝑇𝑀). This implies that 𝑀 is a contact 𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold. Similarly the other assertion follows.
The following construction helps in understanding the example of 𝐺𝐢𝑅-lightlike submanifold. Let (π‘…π‘ž2π‘š+1,πœ™0,𝑉,πœ‚,𝑔) be with its usual Cosymplectic structure and given by πœ‚=𝑑𝑧,𝑉=πœ•π‘§,𝑔=πœ‚βŠ—πœ‚βˆ’π‘ž/2𝑖=1𝑑π‘₯π‘–βŠ—π‘‘π‘₯𝑖+π‘‘π‘¦π‘–βŠ—π‘‘π‘¦π‘–ξ€Έ+π‘šξ“π‘–=π‘ž+1𝑑π‘₯π‘–βŠ—π‘‘π‘₯𝑖+π‘‘π‘¦π‘–βŠ—π‘‘π‘¦π‘–ξ€Έ,πœ™0𝑋1,𝑋2,…,π‘‹π‘šβˆ’1,π‘‹π‘š,π‘Œ1,π‘Œ2,…,π‘Œπ‘šβˆ’1,π‘Œπ‘šξ€Έ=ξ€·,π‘βˆ’π‘‹2,𝑋1,…,βˆ’π‘‹π‘š,π‘‹π‘šβˆ’1,βˆ’π‘Œ2,π‘Œ1,…,βˆ’π‘Œπ‘š,π‘Œπ‘šβˆ’1ξ€Έ,,0(3.4) where (π‘₯𝑖;𝑦𝑖;𝑧) are the Cartesian coordinates.

Example 3.3. Let 𝑀=(𝑅413,𝑔) be a semi-Euclidean space and 𝑀 a 9-dimensional submanifold of 𝑀 that is given by π‘₯4=π‘₯1cosπœƒβˆ’π‘¦1sinπœƒ,𝑦4=π‘₯1sinπœƒ+𝑦1π‘₯cosπœƒ,2=𝑦3,π‘₯5=𝑦1+5ξ€Έ2,(3.5) where 𝑔 is of signature (βˆ’,βˆ’,+,+,+,+,βˆ’,βˆ’,+,+,+,+,+) with respect to the canonical basis {πœ•π‘₯1,πœ•π‘₯2,πœ•π‘₯3,πœ•π‘₯4,πœ•π‘₯5,πœ•π‘₯6,πœ•π‘¦1,πœ•π‘¦2,πœ•π‘¦3,πœ•π‘¦4,πœ•π‘¦5,πœ•π‘¦6,πœ•π‘§}. Then, the local frame of 𝑇𝑀 is given by πœ‰1=πœ•π‘₯1+cosπœƒπœ•π‘₯4+sinπœƒπœ•π‘¦4,πœ‰2=βˆ’sinπœƒπœ•π‘₯4+πœ•π‘¦1+cosπœƒπœ•π‘¦4,πœ‰3=πœ•π‘₯2+πœ•π‘¦3,𝑋1=πœ•π‘₯3βˆ’πœ•π‘¦2,𝑋2=πœ•π‘₯6,𝑋3=πœ•π‘¦6,𝑋4=𝑦5πœ•π‘₯5+π‘₯5πœ•π‘¦5,𝑋5=πœ•π‘₯3+πœ•π‘¦2,𝑋6=𝑉=πœ•π‘§.(3.6) Hence, 𝑀 is a 3-lightlike as Rad𝑇𝑀=span{πœ‰1,πœ‰2,πœ‰3}. Also, πœ™0πœ‰1=βˆ’πœ‰2 and πœ™0πœ‰3=𝑋1; these imply that 𝐷1=span{πœ‰1,πœ‰2} and 𝐷2=span{πœ‰3}, respectively. Since πœ™0𝑋2=βˆ’π‘‹3,  𝐷0=span{𝑋2,𝑋3}. By straightforward calculations, we obtain π‘†ξ€·π‘‡π‘€βŸ‚ξ€Έξ€½=spanπ‘Š=π‘₯5πœ•π‘₯5βˆ’π‘¦5πœ•π‘¦5ξ€Ύ,(3.7) where πœ™0(π‘Š)=𝑋4; this implies that 𝑆=𝑆(π‘‡π‘€βŸ‚). Moreover, the lightlike transversal bundle ltr(𝑇𝑀) is spanned by 𝑁1=12ξ€·βˆ’πœ•π‘₯1+cosπœƒπœ•π‘₯4+sinπœƒπœ•π‘¦4ξ€Έ,𝑁2=12ξ€·βˆ’sinπœƒπœ•π‘₯4βˆ’πœ•π‘¦1+cosπœƒπœ•π‘¦4ξ€Έ,𝑁3=12ξ€·βˆ’πœ•π‘₯2+πœ•π‘¦3ξ€Έ,(3.8) where πœ™0(𝑁1)=βˆ’π‘2 and πœ™0(𝑁3)=𝑋5. Hence, 𝐿=span{𝑁3}. Therefore, 𝐷=span{πœ™0(𝑁3),πœ™0(π‘Š)}. Thus, 𝑀 is a 𝐺𝐢𝑅-lightlike submanifold of 𝑅413.
Let 𝑄, 𝑃1, 𝑃2 be the projection morphism on 𝐷, πœ™π‘†=𝑀2, πœ™πΏ=𝑀1, respectively; therefore 𝑋=𝑄𝑋+𝑉+𝑃1𝑋+𝑃2𝑋,(3.9) for π‘‹βˆˆΞ“(𝑇𝑀). Applying πœ™ to (3.9), we obtain πœ™π‘‹=𝑓𝑋+πœ”π‘ƒ1𝑋+πœ”π‘ƒ2𝑋,(3.10) where π‘“π‘‹βˆˆΞ“(𝐷), πœ”π‘ƒ1π‘‹βˆˆΞ“(𝐿), and πœ”π‘ƒ2π‘‹βˆˆΞ“(𝑆), or, we can write (3.10) as πœ™π‘‹=𝑓𝑋+πœ”π‘‹,(3.11) where 𝑓𝑋 and πœ”π‘‹ are the tangential and transversal components of πœ™π‘‹, respectively.
Similarly, πœ™π‘ˆ=π΅π‘ˆ+Cπ‘ˆ,π‘ˆβˆˆΞ“(tr(𝑇𝑀)),(3.12) where π΅π‘ˆ and πΆπ‘ˆ are the sections of 𝑇𝑀 and tr(𝑇𝑀), respectively. Differentiating (3.10) and using (2.8)–(2.10) and (3.12), we have 𝐷𝑠𝑋,πœ”π‘ƒ2π‘Œξ€Έ=βˆ’βˆ‡π‘ π‘‹πœ”π‘ƒ1π‘Œ+πœ”π‘ƒ1βˆ‡π‘‹π‘Œβˆ’β„Žπ‘ (𝑋,π‘“π‘Œ)+πΆβ„Žπ‘ π·(𝑋,π‘Œ),𝑙𝑋,πœ”π‘ƒ1π‘Œξ€Έ=βˆ’βˆ‡π‘™π‘‹πœ”π‘ƒ2π‘Œ+πœ”π‘ƒ2βˆ‡π‘‹π‘Œβˆ’β„Žπ‘™(𝑋,π‘“π‘Œ)+πΆβ„Žπ‘™(𝑋,π‘Œ),(3.13) for all 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀). By using, cosymplectic property of βˆ‡ with (2.7), we have the following lemmas.

Lemma 3.4. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀; then one has ξ€·βˆ‡π‘‹π‘“ξ€Έπ‘Œ=π΄πœ”π‘Œξ€·βˆ‡π‘‹+π΅β„Ž(𝑋,π‘Œ),π‘‘π‘‹πœ”ξ€Έπ‘Œ=πΆβ„Ž(𝑋,π‘Œ)βˆ’β„Ž(𝑋,π‘“π‘Œ),(3.14) where 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀) and ξ€·βˆ‡π‘‹π‘“ξ€Έπ‘Œ=βˆ‡π‘‹π‘“π‘Œβˆ’π‘“βˆ‡π‘‹ξ€·βˆ‡π‘Œ,π‘‘π‘‹πœ”ξ€Έπ‘Œ=βˆ‡π‘‘π‘‹πœ”π‘Œβˆ’πœ”βˆ‡π‘‹π‘Œ.(3.15)

Lemma 3.5. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀; then one has ξ€·βˆ‡π‘‹π΅ξ€Έπ‘ˆ=π΄πΆπ‘ˆπ‘‹βˆ’π‘“π΄π‘ˆξ€·βˆ‡π‘‹,𝑑𝑋𝐢U=βˆ’πœ”π΄π‘ˆπ‘‹βˆ’β„Ž(𝑋,π΅π‘ˆ),(3.16) where π‘‹βˆˆΞ“(𝑇𝑀) and π‘ˆβˆˆΞ“(tr(𝑇𝑀)) and ξ€·βˆ‡π‘‹π΅ξ€Έπ‘ˆ=βˆ‡π‘‹π΅π‘ˆβˆ’π΅βˆ‡π‘‘π‘‹ξ€·βˆ‡π‘ˆ,π‘‘π‘‹πΆξ€Έπ‘ˆ=βˆ‡π‘‘π‘‹πΆπ‘ˆβˆ’πΆβˆ‡π‘‘π‘‹π‘ˆ.(3.17)

4. Mixed Geodesic GCR-Lightlike Submanifolds

Definition 4.1. A 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold is called mixed geodesic 𝐺𝐢𝑅-lightlike submanifold if its second fundamental form β„Ž satisfies β„Ž(𝑋,π‘Œ)=0, for any π‘‹βˆˆΞ“(π·βŠ•π‘‰) and π‘ŒβˆˆΞ“(𝐷).

Definition 4.2. A 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold is called 𝐷 geodesic 𝐺𝐢𝑅-lightlike submanifold if its second fundamental form β„Ž satisfies β„Ž(𝑋,π‘Œ)=0, for any 𝑋,π‘ŒβˆˆΞ“(𝐷).

Theorem 4.3. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then, 𝑀 is mixed geodesic if and only if π΄βˆ—πœ‰π‘‹ and π΄π‘Šπ‘‹βˆ‰Ξ“(𝑀2βŸ‚πœ™π·2), for any π‘‹βˆˆΞ“(π·βŠ•π‘‰),π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)) and πœ‰βˆˆΞ“(Rad(𝑇𝑀)).

Proof. Using, definition of 𝐺𝐢𝑅-lightlike submanifolds, 𝑀 is mixed geodesic if and only if 𝑔(β„Ž(𝑋,π‘Œ),π‘Š)=𝑔(β„Ž(𝑋,π‘Œ),πœ‰)=0, for π‘‹βˆˆΞ“(π·βŠ•π‘‰),π‘ŒβˆˆΞ“(𝐷),π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)), and πœ‰βˆˆΞ“(Rad(𝑇𝑀)). Using (2.8) and (2.11), we get 𝑔(β„Ž(𝑋,π‘Œ),π‘Š)=π‘”ξ‚€βˆ‡π‘‹ξ‚ξ‚€π‘Œ,π‘Š=βˆ’π‘”π‘Œ,βˆ‡π‘‹π‘Šξ‚ξ€·=π‘”π‘Œ,π΄π‘Šπ‘‹ξ€Έ,𝑔(β„Ž(𝑋,π‘Œ),πœ‰)=π‘”ξ‚€βˆ‡π‘‹ξ‚ξ€·π‘Œ,πœ‰=βˆ’π‘”π‘Œ,βˆ‡π‘‹πœ‰ξ€Έξ‚€=π‘”π‘Œ,π΄βˆ—πœ‰π‘‹ξ‚.(4.1) Therefore, from (4.1), the proof is complete.

Theorem 4.4. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then, 𝑀 is 𝐷 geodesic if and only if π΄βˆ—πœ‰π‘‹ and π΄π‘Šπ‘‹βˆ‰Ξ“(𝑀2βŸ‚πœ™π·2), for any π‘‹βˆˆΞ“(𝐷),πœ‰βˆˆΞ“Rad(𝑇𝑀), and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)).

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

Lemma 4.5. Let 𝑀 be a mixed geodesic 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then π΄βˆ—πœ‰π‘‹βˆˆΞ“(πœ™π·2), for any π‘‹βˆˆΞ“(𝐷), πœ‰βˆˆΞ“(𝐷2).

Proof. For π‘‹βˆˆΞ“(𝐷) and πœ‰βˆˆΞ“(𝐷2), using (2.7) we have β„Ž(πœ™πœ‰,𝑋)=βˆ‡π‘‹πœ™πœ‰βˆ’βˆ‡π‘‹πœ™πœ‰=πœ™βˆ‡π‘‹πœ‰+πœ™β„Ž(𝑋,πœ‰)βˆ’βˆ‡π‘‹πœ™πœ‰.(4.2) Since 𝑀 is mixed geodesic, we obtain πœ™βˆ‡π‘‹πœ‰=βˆ‡π‘‹πœ™πœ‰. Here, using (2.11), we get πœ™(βˆ’π΄βˆ—πœ‰π‘‹+βˆ‡π‘‹βˆ—π‘‘πœ‰)=βˆ‡βˆ—π‘‹πœ™πœ‰+β„Žβˆ—(𝑋,πœ™πœ‰), and then, by virtue of (3.11), we obtain βˆ’π‘“π΄βˆ—πœ‰π‘‹βˆ’πœ”π΄βˆ—πœ‰π‘‹+πœ™(βˆ‡π‘‹βˆ—π‘‘πœ‰)=βˆ‡βˆ—π‘‹πœ™πœ‰+β„Žβˆ—(𝑋,πœ™πœ‰). Comparing the transversal components, we get πœ”π΄βˆ—πœ‰π‘‹=0; this implies that π΄βˆ—πœ‰ξ€·π·π‘‹βˆˆΞ“0βŠ•ξ€·π·{𝑉}βŸ‚πœ™2.ξ€Έξ€Έ(4.3) If π΄βˆ—πœ‰π‘‹βˆˆπ·0, then the nondegeneracy of 𝐷0 implies that there must exist a 𝑍0∈𝐷0 such that 𝑔(π΄βˆ—πœ‰π‘‹,𝑍0)β‰ 0. But using the hypothesis that 𝑀 is a mixed geodesic with (2.7) and (2.11), we get π‘”ξ‚€π΄βˆ—πœ‰π‘‹,𝑍0=βˆ’π‘”ξ€·βˆ‡π‘‹πœ‰,𝑍0ξ€Έ=π‘”ξ‚€πœ‰,βˆ‡π‘‹π‘0=π‘”ξ€·πœ‰,βˆ‡π‘‹π‘0ξ€·+β„Žπ‘‹,𝑍0ξ€Έξ€Έ=0.(4.4) Therefore, π΄βˆ—πœ‰ξ€·π·π‘‹βˆ‰Ξ“0ξ€Έ.(4.5) Also using (2.13), and (2.15), we get π‘”ξ‚€π΄βˆ—πœ‰ξ‚π‘‹,𝑉=βˆ’π‘”ξ€·βˆ‡π‘‹ξ€Έ=πœ‰,π‘‰π‘”ξ‚€πœ‰,βˆ‡π‘‹π‘‰ξ‚=0.(4.6) Therefore, π΄βˆ—πœ‰π‘‹βˆ‰{𝑉}.(4.7) Hence, from (4.3), (4.5), and (4.7), the result follows.

Corollary 4.6. Let 𝑀 be a mixed geodesic 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then, 𝑔(β„Žπ‘™(𝑋,π‘Œ),πœ‰)=0, for any π‘‹βˆˆΞ“(𝐷),π‘ŒβˆˆΞ“(𝑀2) and πœ‰βˆˆΞ“(𝐷2).

Proof. The result follows from (2.12) and Lemma 4.5.

Theorem 4.7. Let 𝑀 be a mixed geodesic 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then, π΄π‘ˆπ‘‹βˆˆΞ“(π·βŠ•{𝑉}) and βˆ‡π‘‘π‘‹π‘ˆβˆˆΞ“(πΏβŸ‚π‘†), for any π‘‹βˆˆΞ“(π·βŠ•{𝑉}) and π‘ˆβˆˆΞ“(πΏβŸ‚π‘†).

Proof. Since 𝑀 is mixed geodesic 𝐺𝐢𝑅-lightlike submanifold β„Ž(𝑋,π‘Œ)=0 for any π‘‹βˆˆΞ“(π·βŠ•{𝑉}),π‘ŒβˆˆΞ“(𝐷), and thus (2.6) implies that 0=βˆ‡π‘‹π‘Œβˆ’βˆ‡π‘‹π‘Œ.(4.8) Since 𝐷 is an anti-invariant distribution there exists a vector field π‘ˆβˆˆΞ“(πΏβŸ‚π‘†) such that πœ™π‘ˆ=π‘Œ. Thus, from (2.8), (2.14), (3.11), and (3.12), we get 0=βˆ‡π‘‹πœ™π‘ˆβˆ’βˆ‡π‘‹ξ€·π‘Œ=πœ™βˆ’π΄π‘ˆπ‘‹+βˆ‡π‘‘π‘‹π‘ˆξ€Έβˆ’βˆ‡π‘‹π‘Œ=βˆ’π‘“π΄π‘ˆπ‘‹βˆ’πœ”π΄π‘ˆπ‘‹+π΅βˆ‡π‘‘π‘‹π‘ˆ+πΆβˆ‡π‘‘π‘‹π‘ˆβˆ’βˆ‡π‘‹π‘Œ.(4.9) Comparing the transversal components, we get πœ”π΄π‘ˆπ‘‹=πΆβˆ‡π‘‘π‘‹π‘ˆ. Since πœ”π΄π‘ˆπ‘‹βˆˆΞ“(πΏβŸ‚π‘†) and πΆβˆ‡π‘‘π‘‹π‘ˆβˆˆΞ“(πΏβŸ‚π‘†)βŸ‚, this implies that πœ”π΄π‘ˆπ‘‹=0 and πΆβˆ‡π‘‘π‘‹π‘ˆ=0. Hence, π΄π‘ˆπ‘‹βˆˆΞ“(π·βŠ•{𝑉}) and βˆ‡π‘‘π‘‹π‘ˆβˆˆΞ“(πΏβŸ‚π‘†).

5. GCR-Lightlike Product

Definition 5.1. 𝐺𝐢𝑅-lightlike submanifold 𝑀 of an indefinite cosymplectic manifold 𝑀 is called 𝐺𝐢𝑅-lightlike product if both the distributions π·βŠ•{𝑉} and 𝐷 define totally geodesic foliation in 𝑀.

Theorem 5.2. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then, the distribution π·βŠ•{𝑉} define a totally geodesic foliation in 𝑀 if and only if π΅β„Ž(𝑋,πœ™π‘Œ)=0, for any 𝑋,π‘Œβˆˆπ·βŠ•{𝑉}.

Proof. Since 𝐷=πœ™(πΏβŸ‚π‘†), π·βŠ•{𝑉} defines a totally geodesic foliation in 𝑀 if and only if 𝑔(βˆ‡π‘‹π‘Œ,πœ™πœ‰)=𝑔(βˆ‡π‘‹π‘Œ,πœ™π‘Š)=0, for any 𝑋,π‘ŒβˆˆΞ“(π·βŠ•{𝑉}), πœ‰βˆˆΞ“(𝐷2), and π‘ŠβˆˆΞ“(𝑆). Using (2.7) and (2.14), we have π‘”ξ€·βˆ‡π‘‹ξ€Έπ‘Œ,πœ™πœ‰=βˆ’π‘”ξ‚€βˆ‡π‘‹ξ‚πœ™π‘Œ,πœ‰=βˆ’π‘”ξ€·β„Žπ‘™ξ€Έπ‘”ξ€·βˆ‡(𝑋,π‘“π‘Œ),πœ‰,(5.1)π‘‹ξ€Έπ‘Œ,πœ™π‘Š=βˆ’π‘”ξ‚€βˆ‡π‘‹ξ‚πœ™π‘Œ,π‘Š=βˆ’π‘”ξ€·β„Žπ‘ ξ€Έ(𝑋,π‘“π‘Œ),π‘Š.(5.2) Hence, from (5.1) and (5.2), the assertion follows.

Theorem 5.3. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then, the distribution 𝐷 defines a totally geodesic foliation in 𝑀 if and only if 𝐴𝑁𝑋 has no component in πœ™π‘†βŸ‚πœ™π·2 and π΄πœ”π‘Œπ‘‹ has no component in 𝐷2βŸ‚π·0, for any 𝑋,π‘ŒβˆˆΞ“(𝐷) and π‘βˆˆΞ“(ltr(𝑇𝑀)).

Proof. From the definition of a 𝐺𝐢𝑅-lightlike submanifold, we know that 𝐷 defines a totally geodesic foliation in 𝑀 if and only if π‘”ξ€·βˆ‡π‘‹ξ€Έξ€·βˆ‡π‘Œ,𝑁=π‘”π‘‹π‘Œ,πœ™π‘1ξ€Έξ€·βˆ‡=π‘”π‘‹ξ€Έξ€·βˆ‡π‘Œ,𝑉=π‘”π‘‹ξ€Έπ‘Œ,πœ™π‘=0,(5.3) for 𝑋,π‘ŒβˆˆΞ“(𝐷),π‘βˆˆΞ“(ltr(𝑇𝑀)),π‘βˆˆΞ“(𝐷0) and 𝑁1βˆˆΞ“(𝐿). Using (2.7) and (2.8), we have π‘”ξ€·βˆ‡π‘‹ξ€Έ=π‘Œ,π‘π‘”ξ‚€βˆ‡π‘‹ξ‚π‘Œ,𝑁=βˆ’π‘”ξ‚€π‘Œ,βˆ‡π‘‹π‘ξ‚ξ€·=π‘”π‘Œ,𝐴𝑁𝑋.(5.4) Using (2.7), (2.15), and (2.14), we obtain π‘”ξ€·βˆ‡π‘‹π‘Œ,πœ™π‘1ξ€Έξ‚€πœ™=βˆ’π‘”βˆ‡π‘‹π‘Œ,𝑁1=βˆ’π‘”βˆ‡π‘‹πœ”π‘Œ,𝑁1𝐴=π‘”πœ”π‘Œπ‘‹,𝑁1ξ€Έπ‘”ξ€·βˆ‡,(5.5)π‘‹ξ€Έξ‚€πœ™π‘Œ,πœ™π‘=βˆ’π‘”βˆ‡π‘‹ξ‚ξ‚€π‘Œ,𝑍=βˆ’π‘”βˆ‡π‘‹ξ‚ξ€·π΄πœ”π‘Œ,𝑍=π‘”πœ”π‘Œξ€Έπ‘”ξ€·βˆ‡π‘‹,𝑍,(5.6)π‘‹ξ€Έξ‚€π‘Œ,𝑉=π‘”βˆ‡π‘‹ξ‚ξ‚€π‘Œ,𝑉=βˆ’π‘”π‘Œ,βˆ‡π‘‹π‘‰ξ‚=0.(5.7) Thus, from (5.4)–(5.7), the result follows.

Theorem 5.4. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. If (βˆ‡π‘‹π‘“)π‘Œ=0, then 𝑀 is a 𝐺𝐢𝑅 lightlike product.

Proof. Let 𝑋,π‘ŒβˆˆΞ“(𝐷); therefore π‘“π‘Œ=0. Then using (3.15) with the hypothesis, we get π‘“βˆ‡π‘‹π‘Œ=0. Therefore the distribution 𝐷 defines a totally geodesic foliation. Next, let 𝑋,π‘Œβˆˆπ·βŠ•{𝑉}; therefore πœ”π‘Œ=0. Then using (3.14), we get π΅β„Ž(𝑋,π‘Œ)=0. Therefore, π·βŠ•{𝑉} defines a totally geodesic foliation in 𝑀. Hence, 𝑀 is a 𝐺𝐢𝑅 lightlike product.

Definition 5.5. A lightlike submanifold 𝑀 of a semi-Riemannian manifold is said to be an irrotational submanifold if βˆ‡π‘‹πœ‰βˆˆΞ“(𝑇𝑀), for any π‘‹βˆˆΞ“(𝑇𝑀) and πœ‰βˆˆΞ“Rad(𝑇𝑀). Thus, 𝑀 is an irrotational lightlike submanifold if and only if β„Žπ‘™(𝑋,πœ‰)=0 and β„Žπ‘ (𝑋,πœ‰)=0.

Theorem 5.6. Let 𝑀 be an irrotational 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then, 𝑀 is a 𝐺𝐢𝑅 lightlike product if the following conditions are satisfied:(A)βˆ‡π‘‹π‘ˆβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)),forallπ‘‹βˆˆΞ“(𝑇𝑀),andπ‘ˆβˆˆΞ“(tr(𝑇𝑀)), (B)π΄βˆ—πœ‰π‘ŒβˆˆΞ“(πœ™(𝑆)),forallπ‘ŒβˆˆΞ“(𝐷).

Proof. Let (𝐴) hold; then, using (2.8), we get 𝐴𝑁𝑋=0,π΄π‘Šπ‘‹=0, 𝐷𝑙(𝑋,π‘Š)=0, and βˆ‡π‘™π‘‹π‘=0 for π‘‹βˆˆΞ“(𝑇𝑀). These equations imply that the distribution 𝐷 defines a totally geodesic foliation in 𝑀, and, with (2.9), we get 𝑔(β„Žπ‘ (𝑋,π‘Œ),π‘Š)=0. Hence, the non degeneracy of 𝑆(π‘‡π‘€βŸ‚) implies that β„Žπ‘ (𝑋,π‘Œ)=0. Therefore, β„Žπ‘ (𝑋,π‘Œ) has no component in 𝑆. Finally, from (2.10) and the hypothesis that 𝑀 is irrotational, we have 𝑔(β„Žπ‘™(𝑋,π‘Œ),πœ‰)=𝑔(π‘Œ,π΄βˆ—πœ‰π‘‹), for π‘‹βˆˆΞ“(𝑇𝑀) and π‘ŒβˆˆΞ“(𝐷). Assume that (𝐡) holds; then β„Žπ‘™(𝑋,π‘Œ)=0. Therefore, β„Žπ‘™(𝑋,π‘Œ) has no component in 𝐿. Thus, the distribution π·βŠ•{𝑉} defines a totally geodesic foliation in 𝑀. Hence, 𝑀 is a 𝐺𝐢𝑅 lightlike product.

Definition 5.7 (see [13]). If the second fundamental form β„Ž of a submanifold, tangent to characteristic vector field 𝑉, of a Sasakian manifold 𝑀 is of the form β„Ž(𝑋,π‘Œ)={𝑔(𝑋,π‘Œ)βˆ’πœ‚(𝑋)πœ‚(π‘Œ)}𝛼+πœ‚(𝑋)β„Ž(π‘Œ,𝑉)+πœ‚(π‘Œ)β„Ž(𝑋,𝑉),(5.8) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀), where 𝛼 is a vector field transversal to 𝑀, then 𝑀 is called a totally contact umbilical submanifold of a Sasakian manifold.

Theorem 5.8. Let 𝑀 be a totally contact umbilical 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then, 𝑀 is a 𝐺𝐢𝑅-lightlike product if π΅β„Ž(𝑋,π‘Œ)=0, for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀).

Proof. Let 𝑋,π‘ŒβˆˆΞ“(π·βŠ•{𝑉}); then the hypothesis that π΅β„Ž(𝑋,π‘Œ)=0 implies that the distribution π·βŠ•{𝑉} defines a totally geodesic foliation in 𝑀.
If we assume that 𝑋,π‘ŒβˆˆΞ“(𝐷), then, using (3.14), we have βˆ’π‘“βˆ‡π‘‹π‘Œ=π΄πœ”π‘Œπ‘‹+π΅β„Ž(𝑋,π‘Œ), and taking inner product with π‘βˆˆΞ“(𝐷0) and using (2.6) and (2.14), we obtain ξ€·βˆ’π‘”π‘“βˆ‡π‘‹ξ€Έξ€·π΄π‘Œ,𝑍=π‘”πœ”π‘Œξ€Έξ‚€π‘‹+π΅β„Ž(𝑋,π‘Œ),𝑍=π‘”βˆ‡π‘‹ξ‚ξ€·π‘Œ,πœ™π‘=βˆ’π‘”π‘Œ,βˆ‡π‘‹π‘ξ…žξ€Έ,(5.9) where πœ™π‘=π‘β€²βˆˆΞ“(𝐷0). For any π‘‹βˆˆΞ“(𝐷) from (3.14), we have πœ”π‘ƒβˆ‡π‘‹π‘=β„Ž(𝑋,𝑓𝑍)βˆ’πΆβ„Ž(𝑋,𝑍). Therefore, using the hypothesis with (5.8), we get πœ”π‘ƒβˆ‡π‘‹π‘=0; this implies that βˆ‡π‘‹π‘βˆˆΞ“(𝐷), and thus (5.9) becomes 𝑔(π‘“βˆ‡π‘‹π‘Œ,𝑍)=0. Then, the nondegeneracy 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 𝐺𝐢𝑅-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Suppose that there exists a transversal vector bundle of 𝑀 which is parallel along 𝐷 with respect to Levi-Civita connection on 𝑀, that is, βˆ‡π‘‹π‘ˆβˆˆΞ“(tr(𝑇𝑀)), for any π‘ˆβˆˆΞ“(tr(𝑇𝑀)), π‘‹βˆˆΞ“(𝐷). Then, 𝑀 is a 𝐺𝐢𝑅-lightlike product.

Proof. Since 𝑀 is a totally geodesic 𝐺𝐢𝑅-lightlike π΅β„Ž(𝑋,π‘Œ)=0, for 𝑋,π‘ŒβˆˆΞ“(π·βŠ•{𝑉}); this implies π·βŠ•{𝑉} defines a totally geodesic foliation in 𝑀.
Next βˆ‡π‘‹π‘ˆβˆˆΞ“(tr(𝑇𝑀)) implies π΄π‘ˆπ‘‹=0, and hence, by Theorem 5.3, the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Hence, the result follows.

Acknowledgment

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

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Copyright © 2012 Varun Jain 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.


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