Table of Contents
ISRN Geometry
VolumeΒ 2011, Article IDΒ 531281, 13 pages
http://dx.doi.org/10.5402/2011/531281
Research Article

GCR-Lightlike Product of Indefinite Kaehler Manifolds

1University College of Engineering, Punjabi University, Patiala, India
2Rayat Institute of Engineering & Information Technology, Railmajra, Ropar, India
3Department of Mathematics, Punjabi University, Patiala, India

Received 7 April 2011; Accepted 27 May 2011

Academic Editor: E.Β Previato

Copyright Β© 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 geodesic 𝐺𝐢𝑅-lightlike submanifolds of indefinite Kaehler manifolds and obtain some necessary and sufficient conditions for a 𝐺𝐢𝑅-lightlike submanifold to be a 𝐺𝐢𝑅-lightlike product.

1. Introduction

The geometry of 𝐢𝑅-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 𝐢𝑅-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 𝐢𝑅-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 𝐢𝑅-lightlike submanifolds which exclude the totally real and complex subcases. Then Duggal and Sahin [11] introduced 𝑆𝐢𝑅-lightlike submanifolds which contain complex and totally real subcases but there was no inclusion relation between 𝐢𝑅 and 𝑆𝐢𝑅-cases. Thus to find a class of submanifolds which would behave as an umbrella for 𝐢𝑅-lightlike and 𝑆𝐢𝑅-lightlike submanifolds of an indefinite Kaehler manifold, Duggal and Sahin [12] introduced 𝐺𝐢𝑅-lightlike submanifolds of indefinite Kaehler manifolds. This paper starts with a very brief introduction about lightlike geometry and 𝐺𝐢𝑅-lightlike submanifolds which will be needed throught the paper and then we study geodesic 𝐺𝐢𝑅-lightlike submanifolds and obtain some necessary and sufficient conditions for a 𝐺𝐢𝑅-lightlike submanifold to be a 𝐺𝐢𝑅-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 (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 (𝑀,𝑔) is an π‘š-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 π‘€π‘‡π‘€βŸ‚ξ‚†=βˆͺπ‘’βˆˆπ‘‡π‘₯π‘€βˆΆπ‘”(𝑒,𝑣)=0,βˆ€π‘£βˆˆπ‘‡π‘₯𝑀,π‘₯βˆˆπ‘€(2.1) is 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 mappingRadπ‘‡π‘€βˆΆπ‘₯βˆˆπ‘€βŸΆRad𝑇π‘₯𝑀(2.2) 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.

Screen distribution 𝑆(𝑇𝑀) is a semi-Riemannian complementary distribution of Rad(𝑇𝑀) in 𝑇𝑀, that is,𝑇𝑀=Radπ‘‡π‘€βŸ‚π‘†(𝑇𝑀),(2.3) and 𝑆(π‘‡π‘€βŸ‚) is a 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‚, respectively. Then we haveξ€·tr(𝑇𝑀)=ltr(𝑇𝑀)βŸ‚π‘†π‘‡π‘€βŸ‚ξ€Έπ‘‡,(2.4)𝑀|𝑀=𝑇MβŠ•tr(𝑇𝑀)=(Radπ‘‡π‘€βŠ•ltr(𝑇𝑀))βŸ‚π‘†(𝑇𝑀)βŸ‚π‘†π‘‡π‘€βŸ‚ξ€Έ.(2.5)

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 this quasiorthonormal fields of frames, we have the following.

Theorem 2.1 (see [8]). 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 Rad𝑇𝑀 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},(2.6) 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 the decomposition (2.5), the Gauss and Weingarten formulas are given byβˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Ž(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀),(2.7)βˆ‡π‘‹π‘ˆ=βˆ’π΄π‘ˆπ‘‹+βˆ‡βŸ‚π‘‹π‘ˆ,βˆ€π‘‹βˆˆΞ“(𝑇𝑀),π‘ˆβˆˆΞ“(tr(𝑇𝑀)),(2.8) 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) which is called second fundamental form, and π΄π‘ˆ 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(π‘‡π‘€βŸ‚), respectively, then (2.7) and (2.8) becomeβˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Žπ‘™(𝑋,π‘Œ)+β„Žπ‘ (𝑋,π‘Œ),(2.9)βˆ‡π‘‹π‘ˆ=βˆ’π΄π‘ˆπ‘‹+π·π‘™π‘‹π‘ˆ+π·π‘ π‘‹π‘ˆ,(2.10) 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), and 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, therefore they are called the lightlike second fundamental form and the screen second fundamental form on 𝑀. In particularβˆ‡π‘‹π‘=βˆ’π΄π‘π‘‹+βˆ‡π‘™π‘‹π‘+𝐷𝑠(𝑋,𝑁),(2.11)βˆ‡π‘‹π‘Š=βˆ’π΄π‘Šπ‘‹+βˆ‡π‘ π‘‹π‘Š+𝐷𝑙(𝑋,π‘Š),(2.12) where π‘‹βˆˆΞ“(𝑇𝑀), π‘βˆˆΞ“(ltr(𝑇𝑀)), and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)). Using (2.9)–(2.12) we obtainπ‘”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(2.13)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,(2.14)𝑔𝐴𝑁𝑋,π‘ξ…žξ€Έ+𝑔𝑁,𝐴𝑁′𝑋=0,(2.15) 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 (2.3), we can induce some new geometric objects on the screen distribution 𝑆(𝑇𝑀) on 𝑀 asβˆ‡π‘‹π‘ƒπ‘Œ=βˆ‡βˆ—π‘‹π‘ƒπ‘Œ+β„Žβˆ—βˆ‡(𝑋,π‘Œ),(2.16)π‘‹πœ‰=βˆ’π΄βˆ—πœ‰π‘‹+βˆ‡π‘‹βˆ—π‘‘πœ‰,(2.17) 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. β„Žβˆ— and π΄βˆ— are Ξ“(Rad𝑇𝑀)-valued and Ξ“(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 βˆ‡ on a nondegenerate submanifold is a metric connection. Unfortunately, this is not true for lightlike submanifolds. Indeed considering βˆ‡ a metric connection, we haveξ€·βˆ‡π‘‹π‘”ξ€Έ(π‘Œ,𝑍)=π‘”ξ€·β„Žπ‘™ξ€Έ+(𝑋,π‘Œ),π‘π‘”ξ€·β„Žπ‘™ξ€Έ,(𝑋,𝑍),π‘Œ(2.18) for any 𝑋,π‘Œ,π‘βˆˆΞ“(𝑇𝑀). From [8, page 171], using the properties of linear connection, we haveξ€·βˆ‡π‘‹β„Žπ‘™ξ€Έ(π‘Œ,𝑍)=βˆ‡π‘™π‘‹ξ€·β„Žπ‘™ξ€Έ(π‘Œ,𝑍)βˆ’β„Žπ‘™ξ€·βˆ‡π‘‹ξ€Έπ‘Œ,π‘βˆ’β„Žπ‘™ξ€·π‘Œ,βˆ‡π‘‹π‘ξ€Έ,ξ€·βˆ‡π‘‹β„Žπ‘ ξ€Έ(π‘Œ,𝑍)=βˆ‡π‘ π‘‹ξ€·β„Žπ‘™ξ€Έ(π‘Œ,𝑍)βˆ’β„Žπ‘ ξ€·βˆ‡π‘‹ξ€Έπ‘Œ,π‘βˆ’β„Žπ‘ ξ€·π‘Œ,βˆ‡π‘‹π‘ξ€Έ.(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 βˆ‡ 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 βˆ‡, that is, ξ‚€βˆ‡π‘‹π½ξ‚ξ‚€π‘‡π‘Œ=0,βˆ€π‘‹,π‘ŒβˆˆΞ“π‘€ξ‚.(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 (𝐺𝐢𝑅)-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,𝐽𝐷1ξ€Έ=𝐷1,𝐽𝐷2ξ€ΈβŠ‚π‘†(𝑇𝑀).(3.1)(B)There exist two subbundles 𝐷0 and π·ξ…ž of 𝑆(𝑇𝑀) such that 𝑆(𝑇𝑀)=𝐽𝐷2βŠ•π·ξ…žξ‚‡βŸ‚π·0,𝐽𝐷0ξ€Έ=𝐷0,π½ξ€·π·ξ…žξ€Έ=𝐿1βŸ‚πΏ2,(3.2) where 𝐷0 is a nondegenerate distribution on 𝑀, and 𝐿1 and 𝐿2 are vector bundle of ltr(𝑇𝑀) and 𝑆(𝑇𝑀)βŸ‚, respectively.

Then the tangent bundle 𝑇𝑀 of 𝑀 is decomposed as𝑇𝑀=π·βŸ‚π·ξ…ž,𝐷=Rad(𝑇𝑀)βŠ•π·0βŠ•π½π·2.(3.3)𝑀 is called a proper 𝐺𝐢𝑅-lightlike submanifold if 𝐷1β‰ {0}, 𝐷2β‰ {0}, 𝐷0β‰ {0}, and 𝐿2β‰ {0}.

Let 𝑄, 𝑃1, and 𝑃2 be the projections on 𝐷, 𝐽(𝐿1)=𝑀1 and 𝐽(𝐿2)=𝑀2, respectively. Then for any π‘‹βˆˆΞ“(𝑇𝑀) we have𝑋=𝑄𝑋+𝑃1𝑋+𝑃2𝑋,(3.4) applying 𝐽 to (3.4), we obtain𝐽𝑋=𝑇𝑋+𝑀𝑃1𝑋+𝑀𝑃2𝑋,(3.5) and we can write (3.5) as𝐽𝑋=𝑇𝑋+𝑀𝑋,(3.6) where 𝑇𝑋 and 𝑀𝑋 are the tangential and transversal components of 𝐽𝑋, respectively.

Similarly𝐽𝑉=𝐡𝑉+𝐢𝑉,(3.7) for any π‘‰βˆˆΞ“(tr(𝑇𝑀)), where 𝐡𝑉 and 𝐢𝑉 are the sections of 𝑇𝑀 and tr(𝑇𝑀), respectively.

Differentiating (3.5) and using (2.9)–(2.12) and (3.7) we have𝐷𝑠𝑋,𝑀𝑃1π‘Œξ€Έ=βˆ’βˆ‡π‘ π‘‹π‘€π‘ƒ2π‘Œ+𝑀𝑃2βˆ‡π‘‹π‘Œβˆ’β„Žπ‘ (𝑋,π‘‡π‘Œ)+πΆβ„Žπ‘ π·(𝑋,π‘Œ),𝑙𝑋,𝑀𝑃2π‘Œξ€Έ=βˆ’βˆ‡π‘™π‘‹π‘€π‘ƒ1π‘Œ+𝑀𝑃1βˆ‡π‘‹π‘Œβˆ’β„Žπ‘™(𝑋,π‘‡π‘Œ)+πΆβ„Žπ‘™(𝑋,π‘Œ).(3.8) Using Kaehlerian property of βˆ‡ with (2.11) and (2.12), we have the following lemmas.

Lemma 3.2. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehlerian manifold 𝑀. Then one has ξ€·βˆ‡π‘‹π‘‡ξ€Έπ‘Œ=π΄π‘€π‘Œξ€·βˆ‡π‘‹+π΅β„Ž(𝑋,π‘Œ),(3.9)π‘‘π‘‹π‘€ξ€Έπ‘Œ=πΆβ„Ž(𝑋,π‘Œ)βˆ’β„Ž(𝑋,π‘‡π‘Œ),(3.10) 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,(3.11)π‘‘π‘‹π‘€ξ€Έπ‘Œ=βˆ‡π‘‘π‘‹π‘€π‘Œβˆ’π‘€βˆ‡π‘‹π‘Œ.(3.12)

Lemma 3.3. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehlerian manifold 𝑀. Then one has ξ€·βˆ‡π‘‹π΅ξ€Έπ‘‰=π΄πΆπ‘‰π‘‹βˆ’π‘‡π΄π‘‰ξ€·βˆ‡π‘‹,𝑑𝑋𝐢𝑉=βˆ’π‘€π΄π‘‰π‘‹βˆ’β„Ž(𝑋,𝐡𝑉),(3.13) where π‘‹βˆˆΞ“(𝑇𝑀), π‘‰βˆˆΞ“(tr(𝑇𝑀)), and ξ€·βˆ‡π‘‹π΅ξ€Έπ‘‰=βˆ‡π‘‹π΅π‘‰βˆ’π΅βˆ‡π‘‘π‘‹ξ€·βˆ‡π‘‰,𝑑𝑋𝐢𝑉=βˆ‡π‘‘π‘‹πΆπ‘‰βˆ’πΆβˆ‡π‘‘π‘‹π‘‰.(3.14)

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

Theorem 3.4 (see [12]). Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then the distribution 𝐷 defines a totally geodesic foliation in 𝑀 if and only if π΅β„Ž(𝑋,π‘Œ)=0, for any 𝑋,π‘ŒβˆˆΞ“(𝐷).

Theorem 3.5 (see [12]). Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then the distribution π·ξ…ž defines a totally geodesic foliation in 𝑀 if and only if π΄π‘€π‘Œπ‘‹βˆˆΞ“(π·ξ…ž), for any 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž).

4. Geodesic 𝐺𝐢𝑅-Lightlike Submanifolds

Definition 4.1. A 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler 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“(𝐷) and π‘ŒβˆˆΞ“(π·ξ…ž).

Definition 4.2. A 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler 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).

Definition 4.3. A 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold is called π·ξ…ž geodesic 𝐺𝐢𝑅-lightlike submanifold if its second fundamental form β„Ž satisfies β„Ž(𝑋,π‘Œ)=0 for any 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž).

Theorem 4.4. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is 𝐷-geodesic if and only if π‘”ξ€·π΄π‘Šξ€Έ=𝑋,π‘Œπ‘”ξ€·π·π‘™ξ€Έ,βˆ‡(𝑋,π‘Š),π‘Œβˆ—π‘‹ξ‚€π·π½πœ‰βˆ‰Ξ“0βŸ‚π½πΏ1,π΄βˆ—πœ‰ξ‚€π‘Œβˆ‰Ξ“π½πΏ1,β„Žπ‘™ξ€·πΏ(𝑋,πœ‰)βˆ‰Ξ“1ξ€Έ,(4.1) for any 𝑋,π‘ŒβˆˆΞ“(𝐷),πœ‰βˆˆΞ“(Rad(𝑇𝑀)), and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)).

Proof. Using the definition of 𝐺𝐢𝑅-lightlike submanifolds, 𝑀 is 𝐷-geodesic, if and only if π‘”ξ€·β„Žπ‘™ξ€Έ,(𝑋,π‘Œ),πœ‰=0π‘”ξ€·β„Žπ‘ ξ€Έ(𝑋,π‘Œ),π‘Š=0,(4.2) for any 𝑋,π‘ŒβˆˆΞ“(𝐷),πœ‰βˆˆΞ“(Rad(𝑇𝑀)), and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)). Thus for 𝑋,π‘ŒβˆˆΞ“(𝐷), first part of the assertion follows from (2.13).
Now for 𝑋,π‘ŒβˆˆΞ“(𝐷),πœ‰βˆˆΞ“(Rad(𝑇𝑀)) using (2.16), we have π‘”ξ€·β„Žπ‘™ξ€Έ=(𝑋,π‘Œ),πœ‰π‘”ξ‚€βˆ‡π‘‹ξ‚π‘Œ,πœ‰=βˆ’π‘”ξ‚€π½π‘Œ,βˆ‡π‘‹ξ‚ξ‚€π½πœ‰=βˆ’π‘”π½π‘Œ,βˆ‡π‘‹ξ‚βˆ’π½πœ‰π‘”ξ‚€π½π‘Œ,β„Žπ‘™ξ‚€π‘‹,ξ‚€π½πœ‰ξ‚ξ‚=βˆ’π‘”π½π‘Œ,βˆ‡βˆ—π‘‹ξ‚βˆ’π½πœ‰π‘”ξ‚€π½π‘Œ,β„Žπ‘™ξ‚€π‘‹,.π½πœ‰ξ‚ξ‚(4.3) Since π‘ŒβˆˆΞ“(𝐷), this implies that π‘ŒβˆˆΞ“(𝐷0), π‘ŒβˆˆΞ“(𝐷1), π‘ŒβˆˆΞ“(𝐷2), or π‘ŒβˆˆΞ“(𝐽𝐷2). If π‘ŒβˆˆΞ“(𝐷0) or π‘ŒβˆˆΞ“(𝐷2), then we have π‘”ξ‚€π½π‘Œ,β„Žπ‘™ξ‚€π‘‹,π½πœ‰ξ‚ξ‚=0,(4.4) and if π‘ŒβˆˆΞ“(𝐷1) or π‘ŒβˆˆΞ“(𝐽𝐷2), then we have π‘”ξ‚€π½π‘Œ,β„Žπ‘™ξ‚€π‘‹,ξ‚€π΄π½πœ‰ξ‚ξ‚=π‘”βˆ—πœ‰β€²π‘‹,+π½πœ‰π‘”ξ‚€β„Žπ‘™ξ€·π‘‹,πœ‰ξ…žξ€Έ,ξ‚π½πœ‰,(4.5) for any πœ‰β€²=π½π‘ŒβˆˆΞ“(Rad(𝑇𝑀)).
Now using (4.4) and (4.5) in (4.3), we obtain π‘”ξ€·β„Žπ‘™ξ€Έξ‚€(𝑋,π‘Œ),πœ‰=βˆ’π‘”π½π‘Œ,βˆ‡βˆ—π‘‹ξ‚ξ‚€π΄π½πœ‰βˆ’π‘”βˆ—πœ‰β€²π‘‹,ξ‚βˆ’π½πœ‰π‘”ξ‚€β„Žπ‘™ξ€·π‘‹,πœ‰ξ…žξ€Έ,ξ‚π½πœ‰.(4.6) Hence the second part of the assertion follows from (4.6).

Theorem 4.5. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is π·ξ…ž-geodesic if and only if π΄π‘Šπ‘‹ and π΄βˆ—πœ‰π‘‹ have no components in 𝑀2βŸ‚π½π·2 for any π‘‹βˆˆΞ“(π·ξ…ž), πœ‰βˆˆΞ“(Rad(𝑇𝑀)), and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)).

Proof. For 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž) and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)) using (2.13), we obtain π‘”ξ€·β„Žπ‘ ξ€Έξ€·π΄(𝑋,π‘Œ),π‘Š=π‘”π‘Šξ€Έπ‘‹,π‘Œ,(4.7) and for πœ‰βˆˆΞ“(Rad(𝑇𝑀)) using (2.14) and (2.17) we obtain π‘”ξ€·β„Žπ‘™ξ€Έξ‚€π΄(𝑋,π‘Œ),πœ‰=π‘”βˆ—πœ‰ξ‚π‘‹,π‘Œ.(4.8) Hence the assertion follows from (4.7) and (4.8).

Theorem 4.6. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is mixed geodesic if and only if π΄βˆ—πœ‰ξ‚€π·π‘‹βˆˆΞ“0βŸ‚π½πΏ1,π΄π‘Šξ‚€π·π‘‹βˆˆΞ“0βŸ‚Rad(𝑇𝑀)βŸ‚π½πΏ1,(4.9) for any π‘‹βˆˆΞ“(𝐷),πœ‰βˆˆΞ“(Rad(𝑇𝑀)), and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)).

Proof. For any π‘‹βˆˆΞ“(𝐷),π‘ŒβˆˆΞ“(π·ξ…ž), and πœ‰βˆˆΞ“(Rad(𝑇𝑀)) using (2.14) and (2.17) we obtain π‘”ξ€·β„Žπ‘™ξ€Έξ‚€π΄(𝑋,π‘Œ),πœ‰=π‘”βˆ—πœ‰ξ‚π‘‹,π‘Œ,(4.10) and for π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)) with (2.13), we obtain π‘”ξ€·β„Žπ‘ ξ€Έξ€·π΄(𝑋,π‘Œ),π‘Š=π‘”π‘Šξ€Έπ‘‹,π‘Œ.(4.11) Hence the result follows from (4.10) and (4.11).

Theorem 4.7. Let 𝑀 be a mixed geodesic 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then one has π΄βˆ—πœ‰ξ‚€π‘‹βˆˆΞ“π½π·2,(4.12) for any π‘‹βˆˆΞ“(π·ξ…ž) and πœ‰βˆˆΞ“(𝐷2).

Proof. For π‘‹βˆˆΞ“(π·ξ…ž) and πœ‰βˆˆΞ“(𝐷2) we have β„Žξ‚€π‘‹,=π½πœ‰βˆ‡π‘‹π½πœ‰βˆ’βˆ‡π‘‹=π½πœ‰π½βˆ‡π‘‹πœ‰βˆ’βˆ‡π‘‹=π½πœ‰π½βˆ‡π‘‹πœ‰βˆ’π½β„Ž(𝑋,πœ‰)βˆ’βˆ‡π‘‹π½πœ‰.(4.13) Since 𝑀 is mixed geodesic, therefore π½βˆ‡π‘‹πœ‰=βˆ‡π‘‹π½πœ‰.(4.14) Using (2.16) and (2.17) we obtain βˆ’π‘‡π΄βˆ—πœ‰π‘‹βˆ’π‘€π΄βˆ—πœ‰π‘‹+π½βˆ‡π‘‹βˆ—π‘‘πœ‰=βˆ‡βˆ—π‘‹π½πœ‰+β„Žβˆ—ξ‚€π‘‹,ξ‚π½πœ‰.(4.15) Equating the transversal components we have π‘€π΄βˆ—πœ‰π‘‹=0.(4.16) Thus π΄βˆ—πœ‰ξ‚€π‘‹βˆˆΞ“π½π·2βŸ‚π·0.(4.17) Now, for π‘βˆˆΞ“(𝐷0) and πœ‰βˆˆΞ“(𝐷2) we have π‘”ξ‚€π΄βˆ—πœ‰ξ‚=𝑋,π‘π‘”ξ€·βˆ’βˆ‡π‘‹πœ‰+βˆ‡π‘‹βˆ—π‘‘ξ€Έπœ‰,𝑍=βˆ’π‘”ξ€·βˆ‡π‘‹ξ€Έπœ‰,𝑍=βˆ’π‘”ξ‚€βˆ‡π‘‹ξ‚=πœ‰,π‘π‘”ξ‚€πœ‰,βˆ‡π‘‹π‘ξ‚=π‘”ξ€·πœ‰,βˆ‡π‘‹ξ€Έπ‘+β„Ž(X,𝑍)=0.(4.18) If π΄βˆ—πœ‰π‘‹βˆˆΞ“(𝐷0), then using the nondegeneracy of 𝐷0 for any π‘βˆˆΞ“(𝐷0), we have 𝑔(π΄βˆ—πœ‰π‘‹,𝑍)β‰ 0. Therefore π΄βˆ—πœ‰π‘‹βˆ‰Ξ“(𝐷0). Hence the assertion is proved.

Theorem 4.8. Let 𝑀 be a mixed geodesic 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then the transversal section π‘‰βˆˆΞ“(π½π·ξ…ž) is 𝐷-parallel if and only if βˆ‡π‘‹π½π‘‰βˆˆΞ“(𝐷), for any π‘‹βˆˆΞ“(𝐷).

Proof. Let π‘ŒβˆˆΞ“(π·ξ…ž) such that π½π‘Œ=π‘€π‘Œ=π‘‰βˆˆΞ“(𝐿1βŸ‚πΏ2) and π‘‹βˆˆΞ“(𝐷); then using hypothesis in (3.9) we have π‘‡βˆ‡π‘‹π‘Œ=βˆ’π΄π‘€π‘Œπ‘‹=βˆ’π΄π‘‰π‘‹. Now βˆ‡π‘‘π‘‹π‘‰=βˆ‡π‘‹π‘‰+𝐴𝑉𝑋=βˆ‡π‘‹π½π‘Œβˆ’π‘‡βˆ‡π‘‹π‘Œ. Since βˆ‡ is a Kaehlerian connection and 𝑀 is mixed geodesic, therefore we have βˆ‡π‘‘π‘‹π‘‰=π‘€βˆ‡π‘‹π‘Œ or consequently βˆ‡π‘‘π‘‹π‘‰=βˆ’π‘€βˆ‡π‘‹π½π‘‰, which clearly proves the theorem.

Theorem 4.9. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀 such that 𝐷𝑠(𝑋,𝑉)βˆˆΞ“(πΏβŸ‚2). Then 𝐴𝐽𝑉𝑋=𝐽𝐴𝑉𝑋 for any π‘‹βˆˆΞ“(𝐷) and π‘‰βˆˆΞ“(πΏβŸ‚1).

Proof. For π‘‹βˆˆΞ“(𝐷),π‘ŒβˆˆΞ“(π·ξ…ž), and π‘‰βˆˆΞ“(πΏβŸ‚1), we have π‘”ξ‚€π΄π½π‘‰π‘‹βˆ’π½π΄π‘‰ξ‚ξ€·π΄π‘‹,π‘Œ=𝑔𝐽𝑉𝑋,π‘Œβˆ’π‘”π½π΄π‘‰ξ‚ξ€·π΄π‘‹,π‘Œ=𝑔𝐽𝑉𝐴𝑋,π‘Œ+𝑔𝑉𝑋,ξ‚ξ‚€π½π‘Œ=βˆ’π‘”βˆ‡π‘‹ξ‚ξ‚€π½π‘‰,π‘Œβˆ’π‘”βˆ‡π‘‹π‘‰,ξ‚ξ‚€π½π‘Œ=βˆ’π‘”π½βˆ‡π‘‹ξ‚ξ‚€π‘‰,π‘Œ+π‘”π½βˆ‡π‘‹ξ‚π‘‰,π‘Œ=0.(4.19) For π‘‹βˆˆΞ“(𝐷),π‘βˆˆΞ“(𝐷0), and π‘‰βˆˆΞ“(πΏβŸ‚1), we have π‘”ξ‚€π΄π½π‘‰π‘‹βˆ’π½π΄π‘‰ξ‚ξ€·π΄π‘‹,𝑍=𝑔𝐽𝑉𝑋,π‘βˆ’π‘”π½π΄π‘‰ξ‚ξ€·π΄π‘‹,𝑍=𝑔𝐽𝑉𝐴𝑋,𝑍+𝑔𝑉𝑋,𝐽𝑍=βˆ’π‘”βˆ‡π‘‹ξ‚ξ‚€π½π‘‰,π‘βˆ’π‘”βˆ‡π‘‹π‘‰,𝐽𝑍=βˆ’π‘”π½βˆ‡π‘‹ξ‚ξ‚€π‘‰,𝑍+π‘”π½βˆ‡π‘‹ξ‚π‘‰,𝑍=0.(4.20) For π‘‹βˆˆΞ“(𝐷),π‘βˆˆΞ“(ltr(𝑇𝑀)), and π‘‰βˆˆΞ“(πΏβŸ‚1), we have π‘”ξ‚€π΄π½π‘‰π‘‹βˆ’π½π΄π‘‰ξ‚ξ€·π΄π‘‹,𝑁=𝑔𝐽𝑉𝑋,π‘βˆ’π‘”π½π΄π‘‰ξ‚ξ€·π΄π‘‹,𝑁=𝑔𝐽𝑉𝐴𝑋,𝑁+𝑔𝑉𝑋,𝐽𝑁=βˆ’π‘”βˆ‡π‘‹ξ‚ξ‚€π½π‘‰,π‘βˆ’π‘”βˆ‡π‘‹π‘‰,𝐽𝑁=βˆ’π‘”π½βˆ‡π‘‹ξ‚ξ‚€π‘‰,𝑁+π‘”π½βˆ‡π‘‹ξ‚π‘‰,𝑁=0.(4.21) For π‘‹βˆˆΞ“(𝐷),π½π‘βˆˆΞ“(𝐽𝐿1), and π‘‰βˆˆΞ“(πΏβŸ‚1), we have π‘”ξ‚€π΄π½π‘‰π‘‹βˆ’π½π΄π‘‰π‘‹,𝐴𝐽𝑁=𝑔𝐽𝑉𝑋,ξ‚ξ‚€π½π‘βˆ’π‘”π½π΄π‘‰π‘‹,𝐴𝐽𝑁=𝑔𝐽𝑉𝑋,ξ‚ξ€·π΄π½π‘βˆ’π‘”π‘‰ξ€Έξ‚€π‘‹,𝑁=βˆ’π‘”βˆ‡π‘‹π½π‘‰,𝐽𝑁+π‘”βˆ‡π‘‹ξ‚ξ‚€π‘‰,𝑁=βˆ’π‘”π½βˆ‡π‘‹π‘‰,𝐽𝑁+π‘”βˆ‡π‘‹ξ‚ξ‚€π‘‰,𝑁=βˆ’π‘”βˆ‡π‘‹ξ‚ξ‚€π‘‰,𝑁+π‘”βˆ‡π‘‹ξ‚π‘‰,𝑁=0.(4.22) Hence the assertion follows from (4.19)–(4.22).

5. 𝐺𝐢𝑅-Lightlike Product

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

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

Proof. For any 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž), (3.9) implies that π‘‡βˆ‡π‘‹π‘Œ=βˆ’π΄π‘€π‘Œπ‘‹βˆ’π΅β„Ž(𝑋,π‘Œ); then for π‘βˆˆΞ“(𝐷0) we have π‘”ξ€·π‘‡βˆ‡π‘‹ξ€Έξ€·π΄π‘Œ,𝑍=βˆ’π‘”π‘€π‘Œξ€Έξ‚€π‘‹,π‘βˆ’π‘”(π΅β„Ž(𝑋,π‘Œ),𝑍)=π‘”βˆ‡π‘‹ξ‚ξ‚€wπ‘Œ,𝑍=π‘”βˆ‡π‘‹ξ‚ξ‚€π½π‘Œ,𝑍=βˆ’π‘”βˆ‡π‘‹π‘Œ,𝐽𝑍=βˆ’π‘”βˆ‡π‘‹π‘Œ,π‘ξ…žξ‚ξ€·=π‘”π‘Œ,βˆ‡π‘‹π‘ξ…žξ€Έ,(5.1) where 𝑍′=π½π‘βˆˆΞ“(𝐷0). Since π‘‹βˆˆΞ“(π·ξ…ž) and π‘βˆˆΞ“(𝐷0), then from (3.8) we have π‘€π‘ƒβˆ‡π‘‹π‘=β„Ž(𝑋,𝑇𝑍)βˆ’πΆβ„Ž(𝑋,𝑍)=𝐻𝑔(𝑋,𝑇𝑍)βˆ’πΆπ»π‘”(𝑋,𝑍)=0, therefore π‘€π‘ƒβˆ‡π‘‹Z=0, and this implies that βˆ‡π‘‹π‘βˆˆΞ“(𝐷). Therefore (5.1) implies that 𝑔(π‘‡βˆ‡π‘‹π‘Œ,𝑍)=0; then the nondegeneracy of 𝐷0 implies that π‘‡βˆ‡π‘‹π‘Œ=0. Hence βˆ‡π‘‹π‘ŒβˆˆΞ“(π·ξ…ž), for any 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž). Thus the result follows.

Theorem 5.3. Let 𝑀 be a totally umbilical 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is a 𝐺𝐢𝑅-lightlike product if and only if π΅β„Ž(𝑋,π‘Œ)=0, for any π‘‹βˆˆΞ“(𝑇𝑀) and π‘ŒβˆˆΞ“(𝐷).

Proof. Let 𝑀 be a 𝐺𝐢𝑅-lightlike product; therefore the distributions 𝐷 and π·ξ…ž define a totally geodesic foliation in 𝑀. Therefore using Theorem 3.4, π΅β„Ž(𝑋,π‘Œ)=0 for any 𝑋,π‘ŒβˆˆΞ“(𝐷). Now let π‘‹βˆˆΞ“(π·ξ…ž) and π‘ŒβˆˆΞ“(𝐷); then π΅β„Ž(𝑋,π‘Œ)=𝑔(𝑋,π‘Œ)𝐡𝐻=0. Hence π΅β„Ž(𝑋,π‘Œ)=0, for any π‘‹βˆˆΞ“(𝑇𝑀) and π‘ŒβˆˆΞ“(𝐷).
Conversely, let 𝑋,π‘ŒβˆˆΞ“(𝐷); then π΅β„Ž(𝑋,π‘Œ)=0 implies that 𝐷 defines a totally geodesic foliation in 𝑀. Let 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž); then (3.9) and (3.11) imply that π΄π‘€π‘Œπ‘‹=βˆ’π‘‡βˆ‡π‘‹π‘Œβˆ’π΅β„Ž(𝑋,π‘Œ). Using Lemma 5.2, we obtain π‘‡π΄π‘€π‘Œπ‘‹+π‘€π΄π‘€π‘Œπ‘‹=βˆ’β„Ž(𝑋,π‘Œ), we compare the tangential components, we get π‘‡π΄π‘€π‘Œπ‘‹=0, and this implies that π΄π‘€π‘Œπ‘‹βˆˆΞ“(π·ξ…ž). Hence using Theorem 3.5, the distribution π·ξ…ž defines a totally geodesic foliation in 𝑀. Consequently, 𝑀 is a 𝐺𝐢𝑅-lightlike product of an indefinite Kaehler manifold.

Theorem 5.4. Let 𝑀 be a totally geodesic 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Suppose that there exists a transversal vector bundle of 𝑀, which is parallel along π·ξ…ž with respect to the Levi-Civita connection on 𝑀; that is, one has βˆ‡π‘‹π‘‰βˆˆΞ“(tr(𝑇𝑀)), for any π‘‰βˆˆΞ“(tr(𝑇𝑀)) and π‘‹βˆˆΞ“(π·ξ…ž). Then M is a 𝐺𝐢𝑅-lightlike product.

Proof. Since 𝑀 is a totally geodesic 𝐺𝐢𝑅-lightlike submanifold, therefore π΅β„Ž(𝑋,π‘Œ)=0, for any 𝑋,π‘ŒβˆˆΞ“(𝐷). Hence the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Next, since βˆ‡π‘‹π‘‰βˆˆΞ“(tr(𝑇𝑀)) for any π‘‰βˆˆΞ“(tr(𝑇𝑀)) and π‘‹βˆˆΞ“(π·ξ…ž), therefore using (2.8), we have 𝐴𝑉𝑋=0 then using (3.9), we obtain π‘‡βˆ‡π‘‹π‘Œ=0 for any 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž) and this implies that βˆ‡π‘‹π‘ŒβˆˆΞ“(π·ξ…ž). Hence the distribution π·ξ…ž defines a totally geodesic foliation in 𝑀. Thus 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,β„Žπ‘ (𝑋,πœ‰)=0.

Theorem 5.6. Let 𝑀 be an irrotational lightlike submanifold of an indefinite Kaehler 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“(𝐽𝐿2)forallπ‘ŒβˆˆΞ“(𝐷).

Proof. Using (2.11) and (2.12) with (A), we get π΄π‘Šπ‘‹=0,𝐷𝑙(𝑋,π‘Š)=0, and βˆ‡π‘™π‘‹π‘Š=0 for any π‘‹βˆˆΞ“(𝑇𝑀) and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)). Therefore using (2.13) we have 𝑔(β„Žπ‘ (𝑋,π‘Œ),π‘Š)=0; then nondegeneracy of 𝑆(π‘‡π‘€βŸ‚) implies that β„Žπ‘ (𝑋,π‘Œ)=0. Hence π΅β„Žπ‘ (𝑋,π‘Œ)=0. Now, let 𝑋,π‘ŒβˆˆΞ“(𝐷) and πœ‰βˆˆΞ“(Rad(𝑇𝑀)); then using (B), we have 𝑔(β„Žπ‘™(𝑋,π‘Œ),πœ‰)=βˆ’π‘”(βˆ‡π‘‹πœ‰,π‘Œ)=𝑔(π΄βˆ—πœ‰π‘‹,π‘Œ)=0. Then using (2.6), we get β„Žπ‘™(𝑋,π‘Œ)=0. Hence π΅β„Žπ‘™(𝑋,π‘Œ)=0. Thus the distribution 𝐷 defines a totally geodesic foliation in 𝑀.
Next, let 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž); then π½π‘Œ=π‘€π‘ŒβˆˆΞ“(𝐿1βŸ‚πΏ2)βŠ‚tr(TM). Using (3.9) we obtain π‘‡βˆ‡π‘‹π‘Œ=βˆ’π΅β„Ž(𝑋,π‘Œ), comparing the components along 𝐷 we get π‘‡βˆ‡π‘‹π‘Œ=0, and this implies that βˆ‡π‘‹π‘ŒβˆˆΞ“(π·ξ…ž). Thus the distribution π·ξ…ž defines a totally geodesic foliation in 𝑀. Hence 𝑀 is a 𝐺𝐢𝑅-lightlike product.

Theorem 5.7. Let 𝑀 be a 𝐺𝐢𝑅-lightlike submanifold of an indefinite Kaehler manifold 𝑀. Then 𝑀 is a 𝐺𝐢𝑅-lightlike product if and only if (βˆ‡π‘‹π‘‡)π‘Œ=0, for any 𝑋,π‘ŒβˆˆΞ“(𝐷) or 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž).

Proof. Let (βˆ‡π‘‹π‘‡)π‘Œ=0, for any 𝑋,YβˆˆΞ“(𝐷) or 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž). Let 𝑋,π‘ŒβˆˆΞ“(𝐷), then π‘€π‘Œ=0 and (3.9) gives that π΅β„Ž(𝑋,π‘Œ)=0. Hence using Theorem 3.4, the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Next, let 𝑋,π‘ŒβˆˆΞ“(π·ξ…ž). Since π΅π‘‰βˆˆΞ“(π·ξ…ž) for any π‘‰βˆˆΞ“(tr(𝑇𝑀)), then (3.9) implies that π΄π‘€π‘Œπ‘‹βˆˆΞ“(π·ξ…ž). Hence using Theorem 3.5, the distribution π·ξ…ž defines a totally geodesic foliation in 𝑀. Since both the distributions 𝐷 and π·ξ…ž define totally geodesic foliations in 𝑀, hence 𝑀 is a 𝐺𝐢𝑅-lightlike product.
Conversely, let 𝑀 be a 𝐺𝐢𝑅-lightlike product; therefore the distribution 𝐷 defines a totally geodesic foliation in 𝑀. Using Kaehlerian property of βˆ‡, for any 𝑋,π‘ŒβˆˆΞ“(𝐷) we have βˆ‡π‘‹π½π‘Œ=π½βˆ‡π‘‹π‘Œ; then comparing transversal components, we obtain β„Ž(𝑋,π½π‘Œ)=π½β„Ž(𝑋,π‘Œ) and then (βˆ‡π‘‹π‘‡)π‘Œ=βˆ‡π‘‹π‘‡π‘Œβˆ’π‘‡βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π½π‘Œβˆ’β„Ž(𝑋,π½π‘Œ)βˆ’π½βˆ‡π‘‹π‘Œ+β„Ž(𝑋,π½π‘Œ)=0, that is, (βˆ‡π‘‹π‘‡)π‘Œ=0, for any 𝑋,π‘ŒβˆˆΞ“(𝐷). Let π·ξ…ž defines a totally geodesic foliation in 𝑀, and using Kaehlerian property of βˆ‡, we have βˆ‡π‘‹π½π‘Œ=π½βˆ‡π‘‹π‘Œ; then comparing tangential components on both sides, we obtain βˆ’π΄π‘€π‘Œπ‘‹=π΅β„Ž(𝑋,π‘Œ); then (3.9) implies that (βˆ‡π‘‹π‘‡)π‘Œ=0, which completes the proof.

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