International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 251213 | https://doi.org/10.5402/2012/251213

S. M. Khursheed Haider, Mamta Thakur, Advin, "Hemi-Slant Lightlike Submanifolds of Indefinite Kenmotsu Manifolds", International Scholarly Research Notices, vol. 2012, Article ID 251213, 16 pages, 2012. https://doi.org/10.5402/2012/251213

Hemi-Slant Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

Academic Editor: T. Friedrich
Received25 Apr 2012
Accepted25 Jun 2012
Published13 Sep 2012

Abstract

We introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. We give an example of hemi-slant lightlike submanifold and establish two characterization theorems for the existence of such submanifolds. We prove some theorems which ensure the existence of minimal hemi-slant lightlike submanifolds and obtain a condition under which the induced connection βˆ‡ on M is a metric connection. An example of proper minimal hemi-slant lightlike submanifolds is also given.

1. Introduction

Given a semi-Riemannian manifold, one can consider its lightlike submanifold whose study is important from application point of view and difficult in the sense that the intersection of normal vector bundle and tangent bundle of these submanifolds is nonempty. This unique feature makes the study of lightlike submanifolds different from the study of nondegenerate submanifolds. The general theory of lightlike submanifolds was developed by Kupeli [1]; Duggal and Bejancu [2]. On the other hand, the concepts of transversal and screen transversal lightlike submanifolds of an indefinite Kaehler manifold were given by Sahin [3, 4]. In Sasakian setting, these submanifolds were investigated by Yildirim and Sahin [5–7].

In the present paper, we introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold. The paper is arranged as follows. In Section 2, we recall definitions for indefinite Kenmotsu manifolds and give basic information on the lightlike geometry needed for this paper. In Section 3, after defining hemi-slant lightlike submanifolds immersed in an indefinite Kenmotsu manifold, we give an example which ensures the existence of hemi-slant lightlike submanifolds. We obtain integrability conditions of distributions involved, investigate the geometry of leaves of distributions, and establish two characterization theorems for the existence of hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold. We prove that there does not exist curvature invariant hemi-slant lightlike submanifold in an indefinite Kenmotsu space form with some condition on 𝑐 and obtain a geometric condition under which the induced connection on 𝑀 is a metric connection. In Section 4, we study minimal hemi-slant lightlike submanifolds and give an example of proper minimal hemi-slant lightlike submanifolds immersed in 𝑅92.

2. Preliminaries

An odd dimensional semi-Riemannian manifold (𝑀,𝑔) is said to be an indefinite contact metric manifold [8] if there exists a (1,1) tensor field πœ™, a vector field 𝑉, called the characteristic vector field, and its 1-form πœ‚ satisfying 𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœ–πœ‚(𝑋)πœ‚(π‘Œ),πœ™π‘”(𝑉,𝑉)=πœ–2𝑋=βˆ’π‘‹+πœ‚(𝑋)𝑉,𝑔(𝑋,𝑉)=πœ–πœ‚(𝑋),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀),(2.1) where πœ–=Β±1. It follows that πœ™π‘‰=0,πœ‚π‘œπœ™=0,πœ‚(𝑉)=πœ–.

An indefinite almost contact metric manifold 𝑀 is said to be an indefinite Kenmotsu manifold [9] if βˆ‡π‘‹ξ‚€π‘‰=βˆ’π‘‹+πœ‚(𝑋)𝑉,(2.2)βˆ‡π‘‹πœ™ξ‚π‘Œ=βˆ’π‘”(πœ™π‘‹,π‘Œ)𝑉+πœ–πœ‚(π‘Œ)πœ™π‘‹.(2.3) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀).

We follow [2] for the notation and formulae used in this paper. A submanifold (π‘€π‘š,𝑔) immersed in a semi-Riemannian manifold (π‘€π‘š+𝑛,𝑔) is called a lightlike submanifold if the metric 𝑔 induced from 𝑔 is degenerate and the radical distribution Rad𝑇𝑀 is of rank π‘Ÿ, where 1β‰€π‘Ÿβ‰€π‘š. Let 𝑆(𝑇𝑀) be a screen distribution which is a semi-Riemannian complementary distribution of Rad𝑇𝑀 in 𝑇𝑀, that is, 𝑇𝑀=Radπ‘‡π‘€βŸ‚π‘†(𝑇𝑀).(2.4) Consider a screen transversal vector bundle 𝑆(π‘‡π‘€βŸ‚), which is a semi-Riemannian complementry vector bundle of Rad𝑇𝑀 in π‘‡π‘€βŸ‚. Since for any local basis {πœ‰π‘–} of Rad𝑇𝑀, there exist a local null frame {𝑁𝑖} of sections with values in the orthogonal complement of 𝑆(π‘‡π‘€βŸ‚) in [𝑆(𝑇𝑀)]βŸ‚ such that 𝑔(πœ‰π‘–,𝑁𝑗)=𝛿𝑖𝑗, it follows that there exist a lightlike transversal vector bundle ltr(𝑇𝑀) locally spanned by {𝑁𝑖} [2, page 144]. Let tr(𝑇𝑀) be complementary (but not orthogonal) vector bundle to 𝑇𝑀 in 𝑇𝑀|𝑀. Then ξ€·tr(𝑇𝑀)=ltr(𝑇𝑀)βŸ‚π‘†π‘‡π‘€βŸ‚ξ€Έ.𝑇𝑀||𝑀[]ξ€·=𝑆(𝑇𝑀)βŸ‚(Rad𝑇𝑀)βŠ•ltr(𝑇𝑀)βŸ‚π‘†π‘‡π‘€βŸ‚ξ€Έ.(2.5) The Gauss and Weingarten equations are as follows: βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Ž(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀),βˆ‡π‘‹π‘ˆ=βˆ’π΄π‘ˆπ‘‹+βˆ‡π‘‘π‘‹π‘ˆ,βˆ€π‘‹βˆˆΞ“(𝑇𝑀),π‘ˆβˆˆΞ“(tr(𝑇𝑀)),(2.6) where {βˆ‡π‘‹π‘Œ,Aπ‘ˆπ‘‹} and {β„Ž(𝑋,π‘Œ),βˆ‡π‘‘π‘‹π‘ˆ} belong to Ξ“(𝑇𝑀) and Ξ“(tr(𝑇𝑀)), respectively, βˆ‡ and βˆ‡π‘‘ are linear connections on 𝑀 and on the vector bundle tr(𝑇𝑀), respectively. Moreover, we have βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Žπ‘™(𝑋,π‘Œ)+β„Žπ‘ (𝑋,π‘Œ),(2.7)βˆ‡π‘‹π‘=βˆ’π΄π‘π‘‹+βˆ‡π‘™π‘‹π‘+𝐷𝑠(𝑋,𝑁),(2.8)βˆ‡π‘‹π‘Š=βˆ’π΄π‘Šπ‘‹+βˆ‡π‘ π‘‹π‘Š+𝐷𝑙(𝑋,π‘Š),(2.9) for each 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀), π‘βˆˆΞ“(ltr(𝑇𝑀)) and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)). If we denote the projection of 𝑇𝑀 on 𝑆(𝑇𝑀) by 𝑃, then by using (2.7)–(2.9) and the fact that βˆ‡ is a metric connection, 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(2.10)𝑔(𝐷𝑠(𝑋,𝑁),π‘Š)=𝑔𝑁,π΄π‘Šπ‘‹ξ€Έ.(2.11) From the decomposition of the tangent bundle of a lightlike submanifold, we have βˆ‡π‘‹π‘ƒπ‘Œ=βˆ‡βˆ—π‘‹π‘ƒπ‘Œ+β„Žβˆ—βˆ‡(𝑋,π‘ƒπ‘Œ),(2.12)π‘‹πœ‰=βˆ’π΄βˆ—πœ‰π‘‹+βˆ‡π‘‹βˆ—π‘‘πœ‰,(2.13) for 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀) and πœ‰βˆˆΞ“(Rad𝑇𝑀). By using the above equation, we get π‘”ξ€·β„Žπ‘™ξ€Έξ‚€π΄(𝑋,π‘ƒπ‘Œ),πœ‰=π‘”βˆ—πœ‰ξ‚π‘‹,π‘ƒπ‘Œ.(2.14) It is important to note that the induced connection βˆ‡ on 𝑀 is not a metric connection whereas βˆ‡βˆ— is a metric connection on 𝑆(𝑇𝑀).

We denote curvature tensors of 𝑀 and 𝑀 by 𝑅 and 𝑅, respectively. The Gauss equation for 𝑀 is given by 𝑅(𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)𝑍+π΄β„Žπ‘™(𝑋,𝑍)π‘Œβˆ’π΄β„Žπ‘™(π‘Œ,𝑍)𝑋+π΄β„Žπ‘ (𝑋,𝑍)π‘Œβˆ’π΄β„Žπ‘ (π‘Œ,𝑍)𝑋+ξ€·βˆ‡π‘‹β„Žπ‘™ξ€Έξ€·βˆ‡(π‘Œ,𝑍)βˆ’π‘Œβ„Žπ‘™ξ€Έ(𝑋,𝑍)+𝐷𝑙𝑋,β„Žπ‘ ξ€Έ(π‘Œ,𝑍)βˆ’π·π‘™ξ€·π‘Œ,β„Žπ‘ ξ€Έ+ξ€·βˆ‡(𝑋,𝑍)π‘‹β„Žπ‘ ξ€Έξ€·βˆ‡(π‘Œ,𝑍)βˆ’π‘Œβ„Žπ‘ ξ€Έ(𝑋,𝑍)+𝐷𝑠𝑋,β„Žπ‘™ξ€Έ(π‘Œ,𝑍)βˆ’π·π‘ ξ€·π‘Œ,β„Žπ‘™ξ€Έ,(𝑋,𝑍)(2.15) for all 𝑋,π‘Œ,π‘βˆˆΞ“(𝑇𝑀).

The curvature tensor 𝑅 of an indefinite Kenmotsu space form 𝑀(𝑐) is given by [9] as follows: 𝑅(𝑋,π‘Œ)𝑍=π‘βˆ’34𝑔(π‘Œ,𝑍)π‘‹βˆ’ξ€Ύ+𝑔(𝑋,𝑍)π‘Œπ‘+14ξ€½πœ‚(𝑋)πœ‚(𝑍)π‘Œβˆ’πœ‚(π‘Œ)πœ‚(𝑍)𝑋+𝑔(𝑋,𝑍)πœ‚(π‘Œ)π‘‰βˆ’+𝑔(π‘Œ,𝑍)πœ‚(𝑋)𝑉𝑔(πœ™π‘Œ,𝑍)πœ™π‘‹βˆ’π‘”(πœ™π‘‹,𝑍)πœ™π‘Œβˆ’2ξ€Ύ,𝑔(πœ™π‘‹,π‘Œ)πœ™π‘(2.16) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀).

From now on, we denote (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(π‘‡π‘€βŸ‚)) by 𝑀 in this paper.

3. Hemi-Slant Lightlike Submanifolds

The concept of pseudoslant (antislant) submanifolds with definite metric is given by Carriazo [10] which was renamed by Sahin as hemi-slant submanifolds and studied their warped product in Kaehler manifolds [11]. In this section, we introduce and study hemi-slant lightlike submanifolds of an indefinite Kenmotsu manifold for which we need the following lemma from [3].

Lemma 3.1. Let 𝑀 be a 2π‘ž-lightlike submanifold of an indefinite Kaehler manifold 𝑀 with constant index 2π‘ž such that 2π‘ž<dim(𝑀). Then the screen distribution 𝑆(𝑇𝑀) of lightlike submanifold 𝑀 is Riemannian.

In view of the above lemma, we can define hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold as follows.

Definition 3.2. Let 𝑀 be a 2π‘ž-lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2π‘ž with the structure vector field 𝑉 tangent to 𝑀 such that 2π‘ž<dim𝑀. Then, the submanifold 𝑀 is said to be hemi-slant lightlike submanifold of 𝑀 if the following conditions are satisfied: (i)Rad𝑇𝑀 is a distribution on 𝑀 such that πœ™(Rad𝑇𝑀)=ltr(𝑇𝑀);(3.1)(ii)for all π‘₯βˆˆπ‘ˆβŠ‚π‘€ and for each non zero vector field 𝑋 tangent to 𝑆(𝑇𝑀)=π·πœƒβŸ‚{𝑉} if 𝑋 and 𝑉 are linearly independent, then the angle πœƒ(𝑋) between πœ™π‘‹ and the vector space 𝑆(𝑇𝑀) is constant, where π·πœƒ is the complementary distribution to 𝑉 in the screen distribution 𝑆(𝑇𝑀).

The angle πœƒ(𝑋) is called the slant angle of the distribution π·πœƒ. A hemi-slant lightlike submanifold is said to be proper if π·πœƒβ‰ 0 and πœƒβ‰ 0,πœ‹/2.

From Definition 3.2, we have the following decomposition: 𝑇𝑀=Radπ‘‡π‘€βŸ‚π·πœƒβŸ‚{𝑉},tr(𝑇𝑀)=ltr(𝑇𝑀)βŸ‚πΉπ·πœƒπ‘‡βŸ‚πœ‡,𝑀=(Radπ‘‡π‘€βŠ•ltr(𝑇𝑀))βŸ‚π·πœƒβŸ‚πΉπ·πœƒβŸ‚πœ‡βŸ‚{𝑉}.(3.2)

Denoting (π‘…π‘ž2π‘š+1,πœ™π‘œ,𝑉,πœ‚,𝑔) as a manifold π‘…π‘ž2π‘š+1 with its usual Kenmotsu structure given by πœ‚=𝑑𝑧,𝑉=πœ•π‘,𝑔=πœ‚βŠ—πœ‚+𝑒2π‘§ξ‚†βˆ’ξ‚€Ξ£π‘ž/2𝑖=1𝑑π‘₯π‘–βŠ—π‘‘π‘₯𝑖+π‘‘π‘¦π‘–βŠ—π‘‘π‘¦π‘–ξ‚+Ξ£π‘šπ‘–=π‘ž+1𝑑π‘₯π‘–βŠ—π‘‘π‘₯𝑖+π‘‘π‘¦π‘–βŠ—π‘‘π‘¦π‘–ξ‚‡,πœ™π‘œξ€·Ξ£π‘šπ‘–=1ξ€·π‘‹π‘–πœ•π‘₯𝑖+π‘Œπ‘–πœ•π‘¦π‘–ξ€Έξ€Έ+π‘πœ•π‘§=Ξ£π‘šπ‘–=1ξ€·π‘Œπ‘–πœ•π‘₯π‘–βˆ’π‘‹π‘–πœ•π‘¦π‘–ξ€Έ,(3.3) where (π‘₯𝑖,𝑦𝑖,𝑧) are the cartesian coordinates.

The existence of proper hemi-slant lightlike submanifolds in indefinite Kenmotsu manifolds is ensured by the following example.

Example 3.3. Let 𝑀 be a submanifold of 𝑀=(𝑅92,𝑔) defined by π‘₯1=𝑠,π‘₯2=𝑑,π‘₯3=𝑒sin𝑀,π‘₯4𝑦=sin𝑒,1=𝑑,𝑦2=𝑠,𝑦3=𝑒cos𝑀,𝑦4=cos𝑒,(3.4) where 𝑒,π‘€βˆˆ(0,πœ‹/2) and 𝑅92 is a semi-Euclidean space of signature (βˆ’,βˆ’,+,+,+,+,+,+,+).
Then, a local frame of 𝑇𝑀 is given by πœ‰1=π‘’βˆ’π‘§ξ€·πœ•π‘₯1+πœ•π‘¦2ξ€Έ,πœ‰2=π‘’βˆ’π‘§ξ€·πœ•π‘₯2+πœ•π‘¦1ξ€Έ,𝑋1=π‘’βˆ’π‘§ξ€·sinπ‘€πœ•π‘₯3+cosπ‘’πœ•π‘₯4+cosπ‘€πœ•π‘¦3βˆ’sinπ‘’πœ•π‘¦4ξ€Έ,𝑋2=π‘’βˆ’π‘§ξ€·π‘’cosπ‘€πœ•π‘₯3βˆ’π‘’sinπ‘€πœ•π‘¦3ξ€Έ,𝑉=πœ•π‘.(3.5) We see that 𝑀 is a 2-lightlike submanifold with Rad𝑇𝑀=span{πœ‰1,πœ‰2} and 𝑆(𝑇𝑀)=span{𝑋1,𝑋2}βŸ‚{𝑉} which is Riemannian and can be easily proved that 𝑆(𝑇𝑀) is a slant distribution with slant angle πœƒ=πœ‹/4. Moreover, the screen transversal bundle 𝑆(π‘‡π‘€βŸ‚) is spanned by π‘Š1=π‘’βˆ’π‘§ξ€·sinπ‘’πœ•π‘₯4+cosπ‘’πœ•π‘¦4ξ€Έ,π‘Š2=π‘’βˆ’π‘§ξ€·sinπ‘€πœ•π‘₯3βˆ’cosπ‘’πœ•π‘₯4+cosπ‘€πœ•π‘¦3+sinπ‘’πœ•π‘¦4ξ€Έ,(3.6) and the transversal lightlike bundle ltr(𝑇𝑀) is spanned by 𝑁1𝑒=βˆ’βˆ’π‘§2ξ€·πœ•π‘₯1βˆ’πœ•π‘¦2ξ€Έ,𝑁2=π‘’βˆ’π‘§2ξ€·πœ•π‘₯2βˆ’πœ•π‘¦1ξ€Έ.(3.7) It is not difficult to see that πœ™πœ‰1=𝑁2,πœ™πœ‰2=𝑁1.(3.8) Hence, 𝑀 is a hemi-slant lightlike submanifold of 𝑅92.

For any π‘‹βˆˆΞ“(𝑇𝑀), we write πœ™π‘‹=𝑇𝑋+𝐹𝑋,(3.9) where 𝑇𝑋 and 𝐹𝑋 are the tangential and transversal components of πœ™π‘‹, respectively. Similarly, for π‘ŠβˆˆΞ“(tr(𝑇𝑀)), we write πœ™π‘Š=π΅π‘Š+πΆπ‘Š,(3.10) where π΅π‘Š and πΆπ‘Š are the tangential and transversal components of πœ™π‘Š, respectively. If the projections on the distributions 𝑆(𝑇𝑀)=π·πœƒβŸ‚{𝑉} and Rad𝑇𝑀 are denoted by 𝑄1 and 𝑄2, respectively, then any 𝑋 tangent to 𝑀 can be written as 𝑋=𝑄1𝑋+𝑄2𝑋.(3.11) We observe that 𝑄1𝑋=𝑄1𝑋+πœ‚(𝑋)𝑉. By applying πœ™ to (3.11) and using (3.9), we conclude that πœ™π‘‹=𝑇𝑄1𝑋+𝐹𝑄1𝑋+𝐹𝑄2𝑋,(3.12)πœ™π‘„2𝑋=𝐹𝑄2𝑋,𝑇𝑄2𝑋=0,𝑇𝑄1π‘‹βˆˆΞ“(𝑆(𝑇𝑀)).(3.13)

The following two theorems are the characterizations of hemi-slant lightlike submanifolds which are similar to the characterization of slant submanifolds in indefinite Hermitian manifolds given by Sahin [12].

Theorem 3.4. The necessary and sufficient condition for a 2π‘ž-lightlike submanifold 𝑀 of an indefinite Kenmotsu manifold 𝑀 of index 2π‘ž with structure vector field 𝑉 tangent to 𝑀 to be hemi-slant lightlike submanifold is that (i)πœ™(ltr(𝑇𝑀)) is a distribution on 𝑀; (ii)for any vector field 𝑋 tangent to 𝑀 linearly independent of structure vector field 𝑉, there exists a constant πœ†βˆˆ[βˆ’1,0] such that 𝑄1𝑇2𝑄1𝑋=πœ†π‘„1ξ‚€π‘‹βˆ’πœ‚π‘„1𝑋𝑉,(3.14)where πœ†=βˆ’cos2πœƒ and πœƒ is the slant angle of 𝑀.

Proof. Assume that 𝑀 is a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then πœ™(Rad𝑇𝑀)=ltr(𝑇𝑀), from which we have πœ™(ltr(𝑇𝑀))=Rad(𝑇𝑀). Hence πœ™(ltr(𝑇𝑀)) is a distribution on 𝑀. In order to prove (ii), we observe that, for any π‘‹βˆˆΞ“(𝑇𝑀) linearly independent of structure vector field 𝑉 and 𝑄1π‘‹βˆˆΞ“(π·πœƒ), one has 𝑄cosπœƒ1𝑋=ξ‚€π‘„βˆ’π‘”1𝑄𝑋,1𝑇2𝑄1𝑋||𝑄1𝑋||||𝑇𝑄1𝑋||.(3.15) But cosπœƒ(𝑄1𝑋) can also be computed as 𝑄cosπœƒ1𝑋=||𝑇𝑄1𝑋||||πœ™π‘„1𝑋||.(3.16) From (3.15) and (3.16), we arrive at cos2πœƒξ€·π‘„1𝑋=ξ‚€π‘„βˆ’π‘”1𝑄𝑋,1𝑇2𝑄1𝑋||𝑄1𝑋||2.(3.17) Taking into account that πœƒ(𝑄1𝑋) is constant on π·πœƒ, from (3.17) we infer that 𝑄1𝑇2𝑄1𝑋=πœ†π‘„1𝑋=πœ†π‘„1ξ‚€π‘‹βˆ’πœ‚π‘„1𝑋𝑉,(3.18) where we have used πœ†=βˆ’cos2πœƒ(𝑄1𝑋). Clearly πœ†βˆˆ[βˆ’1,0]. Converse part directly follows from (i) and (ii).

Theorem 3.5. The necessary and sufficient condition for a 2π‘ž-lightlike submanifold 𝑀 of an indefinite Kenmotsu manifold 𝑀 of index 2π‘ž with structure vector field 𝑉 tangent to 𝑀 to be hemi-slant lightlike submanifold is that (i)πœ™(ltr(𝑇𝑀)) is a distribution on 𝑀, (ii)for any vector field X tangent to 𝑀 linearly independent of structure vector field 𝑉, there exists a constant πœ‡βˆˆ[βˆ’1,0] such that 𝐡𝐹𝑄1𝑋=πœ‡π‘„1ξ‚€π‘‹βˆ’πœ‚π‘„1𝑋𝑉,(3.19)
where πœ†=βˆ’cos2πœƒ and πœƒ is the slant angle of 𝑀.

Proof. If 𝑀 is a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2π‘ž, then πœ™(ltr(𝑇𝑀)) is a distribution on 𝑀. For the proof of (ii), applying πœ™ to (3.12) and using (3.9) and (3.10), we arrive at ξ€·π‘„βˆ’π‘‹+πœ‚(𝑋)𝑉=1𝑇2𝑄1𝑋+𝐹𝑄1𝑇𝑄1𝑋+𝐡𝐹𝑄1𝑋+𝐢𝐹𝑄1𝑋+𝐡𝐹𝑄2𝑋.(3.20) Comparing the components of screen distribution on both sides of the above equation, we get βˆ’π‘„1𝑋+πœ‚π‘„1𝑋𝑄𝑉=1𝑇2𝑄1𝑋+𝐡𝐹𝑄1𝑋.(3.21) In view of Theorem 3.4 and the fact that 𝑀 is a hemi-slant lightlike submanifold, we conclude that 𝑄1𝑇2𝑄1𝑋=βˆ’cos2πœƒξ‚€π‘„1ξ‚€π‘‹βˆ’πœ‚π‘„1𝑋𝑉.(3.22) Hence (ii) follows from (3.21) and (3.22) together with 𝐹𝑉=0. Conversely, assume that (i) and (ii) hold. From (ii) and (3.21), we observe that 𝑄1𝑇2𝑄1𝑋=βˆ’(1+πœ‡)𝑄1𝑋.(3.23) If we take βˆ’(1+πœ‡)=πœ†, then πœ†βˆˆ[βˆ’1,0]. Hence the proof of Theorem follows from Theorem 3.4.

Corollary 3.6. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑔𝑇𝑄1𝑋,𝑇𝑄1π‘Œξ‚=cos2πœƒ||𝑆(𝑇𝑀)𝑔𝑄1𝑋,𝑄1π‘Œξ‚ξ‚€βˆ’πœ‚π‘„1π‘‹ξ‚πœ‚ξ‚€π‘„1π‘Œπ‘”ξ‚€πΉξ‚ξ‚„,(3.24)𝑄1𝑋,𝐹𝑄1π‘Œξ‚=sin2πœƒ||𝑆(𝑇𝑀)𝑔𝑄1𝑋,𝑄1π‘Œξ‚ξ‚€βˆ’πœ‚π‘„1π‘‹ξ‚πœ‚ξ‚€π‘„1π‘Œξ‚ξ‚„(3.25) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀).

Differentiating (3.12), taking into account that 𝑀 is an indefinite Kenmotsu manifold and then comparing tangential, lightlike transversal and screen transversal parts, we obtain ξ€·βˆ‡π‘‹π‘‡π‘„1ξ€Έπ‘Œ=𝐴𝐹𝑄1π‘Œπ‘‹+𝐴𝐹𝑄2π‘Œπ‘‹+π΅β„Žπ‘™(𝑋,π‘Œ)+π΅β„Žπ‘ (𝑋,π‘Œ)βˆ’π‘”(πœ™π‘‹,π‘Œ)𝑉+πœ‚(π‘Œ)𝑇𝑄1π‘‹ξ€·βˆ‡(3.26)𝑋𝐹𝑄2π‘Œ=βˆ’β„Žπ‘™ξ€·π‘‹,𝑇𝑄1π‘Œξ€Έβˆ’π·π‘™ξ€·π‘‹,𝐹𝑄1π‘Œξ€Έ+πœ‚(π‘Œ)𝐹𝑄2π‘‹ξ€·βˆ‡(3.27)𝑋𝐹𝑄1π‘Œ=πΆβ„Žπ‘ (𝑋,π‘Œ)βˆ’β„Žπ‘ ξ€·π‘‹,𝑇𝑄1π‘Œξ€Έβˆ’π·π‘ ξ€·π‘‹,𝐹𝑄2π‘Œξ€Έ+πœ‚(π‘Œ)𝐹𝑄1𝑋.(3.28)

The integrability conditions of the distributions involved in the definition of hemi-slant lightlike submanifolds are given by the following theorems.

Theorem 3.7. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2π‘ž. Then the distribution Rad𝑇𝑀 is integrable if and only if(i)𝐴𝐹𝑄2π‘Œπ‘‹βˆ’π΄πΉπ‘„2π‘‹π‘Œ=0;(ii)𝐷𝑠(𝑋,𝐹𝑄2π‘Œ)=𝐷𝑠(π‘Œ,𝐹𝑄2𝑋)for any 𝑋,π‘ŒβˆˆΞ“(Rad𝑇𝑀).

Proof. For any 𝑋,π‘ŒβˆˆΞ“(Rad𝑇𝑀), from (3.26) we have βˆ’π‘‡π‘„1βˆ‡π‘‹π‘Œ=𝐴𝐹𝑄2π‘Œπ‘‹+π΅β„Žπ‘™(𝑋,π‘Œ)+π΅β„Žπ‘ (𝑋,π‘Œ)βˆ’π‘”(πœ™π‘‹,π‘Œ)𝑉.(3.29) Interchanging 𝑋 and π‘Œ in (3.29) and subtracting (3.29) from the resulting equation, we get 𝑇𝑄1[]𝑋,π‘Œ=𝐴𝐹𝑄2π‘‹π‘Œβˆ’π΄πΉπ‘„2π‘Œπ‘‹+2𝑔(πœ™π‘‹,π‘Œ)𝑉.(3.30) Taking inner product of the above equation with πœ™π‘ for π‘βˆˆΞ“(π·πœƒ), after simplification, we obtain []𝑔(𝑋,π‘Œ,𝑍)=sec2πœƒξ€½π‘”ξ€·π΄πΉπ‘„2π‘‹π‘Œβˆ’π΄πΉπ‘„2π‘Œπ‘‹ξ€Έξ€Ύ.,πœ™π‘(3.31) Also, from (3.28) we have βˆ’πΉπ‘„1βˆ‡π‘‹π‘Œ=πΆβ„Žπ‘ (𝑋,π‘Œ)βˆ’π·π‘ ξ€·π‘‹,𝐹𝑄2π‘Œξ€Έ.(3.32) Interchanging the role of 𝑋 and π‘Œ in (3.32) and subtracting (3.32) from the resulting equation, we get 𝐹𝑄1[]𝑋,π‘Œ=𝐷𝑠𝑋,𝐹𝑄2π‘Œξ€Έβˆ’π·π‘ ξ€·π‘Œ,𝐹𝑄2𝑋.(3.33) Taking inner product of above equation with πœ™π‘, we arrive at []𝑔(𝑋,π‘Œ,𝑍)=cosec2ξ€·π·πœƒπ‘”π‘ ξ€·π‘‹,𝐹𝑄2π‘Œξ€Έβˆ’π·π‘ ξ€·π‘Œ,𝐹𝑄2𝑋.,πœ™π‘(3.34) Hence our assertion follows from (3.31) and (3.34).

Theorem 3.8. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2π‘ž. Then the distribution π·πœƒβŸ‚{𝑉} is integrable if and only if β„Žπ‘™ξ€·π‘‹,𝑇𝑄1π‘Œξ€Έβˆ’β„Žπ‘™ξ€·π‘Œ,𝑇𝑄1𝑋+𝐷𝑙𝑋,𝐹𝑄1π‘Œξ€Έβˆ’π·π‘™ξ€·π‘Œ,𝐹𝑄1𝑋=0,(3.35) for any 𝑋,π‘ŒβˆˆΞ“(𝑆(𝑇𝑀)).

Proof. For any 𝑋,π‘ŒβˆˆΞ“(π·πœƒβŸ‚{𝑉}), from (3.27) we have βˆ’πΉπ‘„2βˆ‡π‘‹π‘Œ=βˆ’β„Žπ‘™ξ€·π‘‹,𝑇𝑄1Yξ€Έβˆ’π·π‘™ξ€·π‘‹,𝐹𝑄1π‘Œξ€Έ.(3.36) Interchanging 𝑋 and π‘Œ in (3.36) and subtracting (3.36) from the resulting equation, we obtain 𝐹𝑄2[]𝑋,π‘Œ=β„Žπ‘™ξ€·π‘‹,𝑇𝑄1π‘Œξ€Έβˆ’β„Žπ‘™ξ€·π‘Œ,𝑇𝑄1𝑋+𝐷𝑙𝑋,𝐹𝑄1π‘Œξ€Έβˆ’π·π‘™ξ€·π‘Œ,𝐹𝑄1𝑋,(3.37) which proves our assertion.

In view of the above argument, if we restrict the vector fields 𝑋 and π‘Œ to π·πœƒ, then we see that the distribution π·πœƒ is not integrable.

The following two theorems deals with the totally geodesic foliations of the leaves of the distributions 𝑆(𝑇𝑀)=π·πœƒβŸ‚{𝑉} and Rad𝑇𝑀.

Theorem 3.9. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then the distribution π·πœƒβŸ‚{𝑉} defines totally geodesic foliation in 𝑀 if and only if 𝐴𝐹𝑄1𝑇𝑄1π‘Œπ‘‹βˆ’π΅π·π‘™(𝑋,𝐹𝑄1Y) has no component in Rad𝑇𝑀 for any 𝑋,π‘ŒβˆˆΞ“(𝑆(𝑇𝑀)).

Proof. Using (2.7), (2.9), (3.12), and (3.14), for any 𝑋,π‘ŒβˆˆΞ“(𝑆(𝑇𝑀)) and π‘βˆˆΞ“(ltr(𝑇𝑀)), we obtain π‘”ξ‚€βˆ‡π‘‹ξ‚π‘Œ,𝑁=cos2π‘”ξ‚€βˆ‡π‘‹ξ‚+π‘Œ,𝑁𝑔𝐴𝐹𝑄1𝑇𝑄1π‘Œπ‘‹βˆ’π΅π·π‘™ξ€·π‘‹,𝐹𝑄1π‘Œξ€Έξ€Έ,𝑁,(3.38) which can be written as sin2πœƒπ‘”ξ‚€βˆ‡π‘‹ξ‚=π‘Œ,𝑁𝑔𝐴𝐹𝑄1𝑇𝑄1π‘Œπ‘‹βˆ’π΅π·π‘™ξ€·π‘‹,𝐹𝑄1π‘Œξ€Έξ€Έ,𝑁.(3.39) Thus, our assertion follows from (3.39).

Theorem 3.10. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 of index 2π‘ž. Then the distribution Rad𝑇𝑀 defines totally geodesic foliation in 𝑀 if and only if β„Žβˆ—(πœ‰1,𝑇𝑄1𝑋)βˆ’π΄πΉπ‘„1π‘‹πœ‰1 has no component in Rad𝑇𝑀 for any πœ‰1βˆˆΞ“(Rad𝑇𝑀) and π‘‹βˆˆΞ“(π·πœƒ).

Proof. For any πœ‰1,πœ‰2βˆˆΞ“(Rad𝑇𝑀) and π‘‹βˆˆΞ“(π·πœƒ), from (2.7), (2.9), (2.12), and (3.12), we obtain π‘”ξ€·βˆ‡πœ‰1πœ‰2ξ€Έ,𝑋=βˆ’π‘”ξ€·πœ™πœ‰2,β„Žβˆ—ξ€·πœ‰1,𝑇𝑄1π‘‹ξ€Έβˆ’π΄πΉπ‘„1π‘‹πœ‰1ξ€Έ.(3.40) Hence our assertion follows from (3.40).

The necessary and sufficient conditions under which the induced connection βˆ‡ on a hemi-slant lightlike submanifold immersed in indefinite Kenmotsu manifold to be metric connection is given by the following result.

Theorem 3.11. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifolds 𝑀. Then the induced connection βˆ‡ on 𝑀 is a metric connection if and only if 𝑇𝑄1𝐴𝐹𝑄2π‘Œπ‘‹βˆ’π΅π·π‘ ξ€·π‘‹,𝐹𝑄2π‘Œξ€Έ=0,(3.41) for any π‘‹βˆˆΞ“(𝑇𝑀) and π‘ŒβˆˆΞ“(Rad𝑇𝑀).

Proof. For π‘‹βˆˆΞ“(𝑇𝑀) and π‘ŒβˆˆΞ“(Rad𝑇𝑀), from (3.1), we have βˆ‡π‘‹π‘Œ=βˆ’πœ™βˆ‡π‘‹ξ‚€πœ™π‘Œβˆ’βˆ‡π‘‹πœ™ξ‚πœ™π‘Œ.(3.42) Making use of (2.3), (2.7), and (2.9) in the above equation, we arrive at βˆ‡π‘‹π‘Œ+β„Žπ‘™(𝑋,π‘Œ)+β„Žπ‘ ξ€·(𝑋,π‘Œ)=βˆ’πœ™βˆ’π΄πΉπ‘„2π‘Œπ‘‹+βˆ‡π‘™π‘‹πΉπ‘„2π‘Œ+𝐷𝑠𝑋,𝐹𝑄2π‘Œξ€Έξ€Έ+𝑔(πœ™π‘‹,πœ™π‘Œ)𝑉.(3.43) Using (3.9) and (3.10) in (3.43) and comparing the tangential components of the resulting equation, we obtain βˆ‡π‘‹π‘Œ=𝑇𝑄1𝐴𝐹𝑄2π‘Œπ‘‹βˆ’π΅βˆ‡π‘™π‘‹πΉπ‘„2π‘Œβˆ’π΅π·π‘ ξ€·π‘‹,𝐹𝑄2π‘Œξ€Έ,(3.44) from which our assertion follows.

Theorem 3.12. Let 𝑀 be a hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑇 is never parallel on 𝑀.

Proof. For π‘‹βˆˆΞ“(𝑇𝑀), from (2.2) and (2.7), we have βˆ‡π‘‹π‘‰+β„Žπ‘™(𝑋,𝑉)+β„Žπ‘ (𝑋,𝑉)=βˆ’π‘‹+πœ‚(𝑋)𝑉.(3.45) Comparing the tangential Components of the above equation, we get βˆ‡π‘‹π‘‰=βˆ’π‘‹+πœ‚(𝑋)𝑉.(3.46) Since 𝑇𝑉=0, so differentiating 𝑇𝑉=0 covariantly with respect to 𝑋 we obtain ξ€·βˆ‡π‘‹π‘‡ξ€Έξ€·βˆ‡π‘‰+𝑇𝑋𝑉=0.(3.47) Using (3.46) in the above equation, we get ξ€·βˆ‡π‘‹π‘‡ξ€Έπ‘‰=𝑇𝑋,(3.48) from which our assertion follows.

Similar to the notion of curvature invariant submanifolds given by Atceken and Kilic [13], we define curvature invariant lightlike submanifolds as follows.

Definition 3.13. Let 𝑀 be a 2π‘ž-lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is said to be curvature invariant lightlike submanifold of 𝑀 if the covariant derivatives of the second fundamental forms β„Žπ‘™ and β„Žπ‘  of 𝑀 satisfy ξ‚€βˆ‡π‘‹β„Žπ‘™ξ‚ξ‚€(π‘Œ,𝑍)βˆ’βˆ‡π‘Œβ„Žπ‘™ξ‚ξ‚€(𝑋,Z)=0,βˆ‡π‘‹β„Žπ‘ ξ‚ξ‚€(π‘Œ,𝑍)βˆ’βˆ‡π‘Œβ„Žπ‘ ξ‚(𝑋,𝑍)=0,(3.49) for any 𝑋,π‘ŒβˆˆΞ“(𝑇𝑀).

Theorem 3.14. There does not exist any curvature-invariant proper hemi-slant lightlike submanifold of an indefinite Kenmotsu space form M(𝑐) with π‘β‰ βˆ’1.

Proof. Using (2.15) and (2.16), for any 𝑋,π‘ŒβˆˆΞ“(𝑆(𝑇𝑀)) and π‘βˆˆΞ“(Rad𝑇𝑀), we obtain ξ‚€βˆ‡π‘‹β„Žπ‘ ξ‚ξ‚€(π‘Œ,𝑍)βˆ’βˆ‡π‘Œβ„Žπ‘ ξ‚ξ‚€(𝑋,𝑍)=0,(3.50)βˆ‡π‘‹β„Žπ‘™ξ‚ξ‚€(π‘Œ,𝑍)βˆ’βˆ‡π‘Œβ„Žπ‘™ξ‚(𝑋,𝑍)=βˆ’π‘+12𝑔𝑇𝑄1𝑋,π‘ŒπΉπ‘„2𝑍.(3.51) Thus, our assertion follows from (3.50) and (3.51).

4. Minimal Hemi-Slant Submanifolds

In this section, we study minimal hemi-slant lightlike submanifolds immersed in an indefinite Kenmotsu manifold. First we recall the following definition of minimal lightlike submanifolds of a semi-Riemannian manifold 𝑀 given by Bejan and Duggal [14].

Definition 4.1. A lightlike submanifold (𝑀,𝑔,𝑆(𝑇𝑀)) isometrically immersed in a semi-Riemannian manifold (𝑀,𝑔) is said to be minimal if(i)β„Žπ‘ =0 on Rad𝑇𝑀 and(ii)trace β„Ž=0,
where trace is written with respect to 𝑔 restricted to 𝑆(𝑇𝑀).

In [14], it has been shown that the above definition is independent of 𝑆(T𝑀) and 𝑆(π‘‡π‘€βŸ‚), but it depends on the choice of transversal bundle tr(𝑇𝑀).

For the existence of proper minimal hemi-slant lightlike submanifolds, we give the following example.

Example 4.2. Let 𝑀=(𝑅92,𝑔) be a semi-Euclidean space of signature (βˆ’,βˆ’,+,+,+,+,+,+,+) with respect to canonical basis (πœ•π‘₯1,πœ•π‘¦1,πœ•π‘₯2,πœ•π‘¦2,πœ•π‘₯3,πœ•π‘¦3,πœ•π‘₯4,πœ•π‘¦4,πœ•π‘§).
Consider an almost contact structure πœ™ on 𝑅92 defined by πœ™(π‘₯1, 𝑦1, π‘₯2, 𝑦2, π‘₯3, 𝑦3, π‘₯4, 𝑦4, 𝑧)=(βˆ’π‘¦1,π‘₯1, βˆ’π‘¦2,π‘₯2, βˆ’π‘₯4cosπ›Όβˆ’π‘¦3sin𝛼, βˆ’π‘¦4cos𝛼+π‘₯3sin𝛼, π‘₯3cos𝛼+𝑦4sin𝛼, 𝑦3cosπ›Όβˆ’π‘₯4sin𝛼,0).
For some π›Όβˆˆ(0,πœ‹/2). Let 𝑀 be a submanifold of (𝑅92,πœ™) defined by the following equations: π‘₯1=𝑒1,π‘₯2=𝑒2,π‘₯3=𝑒3,π‘₯4=𝑒4,𝑦1=𝑒2,𝑦2=𝑒1,𝑦3=sin𝑒3sinh𝑒4,𝑦4=cos𝑒3cosh𝑒4.(4.1) Then the tangent bundle 𝑇𝑀 of 𝑀 is spanned by πœ‰1=π‘’βˆ’π‘§ξ€·πœ•π‘₯1+πœ•π‘¦2ξ€Έ,πœ‰2=π‘’βˆ’π‘§ξ€·πœ•π‘₯2+πœ•π‘¦1ξ€Έ,𝑋1=π‘’βˆ’π‘§ξ€·πœ•π‘₯3+cos𝑒3sinh𝑒4πœ•π‘¦3βˆ’sin𝑒3cosh𝑒4πœ•π‘¦4ξ€Έ,𝑋2=π‘’βˆ’π‘§ξ€·πœ•π‘₯4+sin𝑒3cosh𝑒4πœ•π‘¦3+cos𝑒3sinh𝑒4πœ•π‘¦4ξ€Έ,𝑉=πœ•π‘§.(4.2) Suppose that the distribution Rad𝑇𝑀=span{πœ‰1,πœ‰2}. Then 𝑀 is a 2-lightlike submanifold of 𝑀. If we take 𝑆(𝑇𝑀)=span{𝑋1,𝑋2βŸ‚{𝑉}}, then 𝑆(𝑇𝑀) is Riemannian vector subbundle and one can easily see that 𝑆(𝑇𝑀) is a slant distribution with slant angle 𝛼. A direct calculation shows that the screen transversal bundle is spanned by π‘Š1=π‘’βˆ’π‘§ξ€·βˆ’cosh𝑒4sinh𝑒4πœ•π‘₯3+cos𝑒3cosh𝑒4πœ•π‘¦3βˆ’sin𝑒3cos𝑒3πœ•π‘₯4+sin𝑒3sinh𝑒4πœ•π‘¦4ξ€Έ,π‘Š2=π‘’βˆ’π‘§ξ€·cos𝑒3sin𝑒3πœ•π‘₯3+sin𝑒3sinh𝑒4πœ•π‘¦3βˆ’cosh𝑒4sinh𝑒4πœ•π‘₯4+cos𝑒3cosh𝑒4πœ•π‘¦4ξ€Έ,(4.3) and ltr(𝑇𝑀) is spanned by 𝑁1𝑒=βˆ’βˆ’π‘§2ξ€·βˆ’πœ•π‘¦2+πœ•π‘₯1ξ€Έ,𝑁2=π‘’βˆ’π‘§2ξ€·βˆ’πœ•π‘¦1+πœ•π‘₯2ξ€Έ.(4.4) For a vector field 𝑋 tangent to 𝑀, from Gauss formula and after calculations, we get β„Žπ‘™=0,β„Žπ‘ ξ€·π‘‹,πœ‰1ξ€Έ=0,β„Žπ‘ ξ€·π‘‹,πœ‰2ξ€Έβ„Ž=0,𝑠𝑋1,𝑋1ξ€Έ=βˆ’π‘’βˆ’π‘§ξ€·sinh2𝑒4+cos2𝑒3ξ€Έcosh4𝑒4βˆ’sin4𝑒3π‘Š2,β„Žπ‘ ξ€·π‘‹2,𝑋2ξ€Έ=π‘’βˆ’π‘§ξ€·sinh2𝑒4+cos2𝑒3ξ€Έcosh4𝑒4βˆ’sin4𝑒3π‘Š2,β„Žπ‘ ξ€·π‘‹1,𝑋2ξ€Έ=π‘’βˆ’π‘§ξ€·sinh2𝑒4+cos2𝑒3ξ€Έcosh4𝑒4βˆ’sin4𝑒3π‘Š1.(4.5) Hence, 𝑀 is a minimal proper hemi-slant lightlike submanifold of 𝑅92, and the induced connection on 𝑀 is a metric connection. We note that 𝑀 is neither totally geodesic nor totally umbilical.

Now, we prove a lemma which will be helpful for the study of minimal hemi-slant lightlike submanifolds.

Lemma 4.3. Let 𝑀 be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 such that dim(π·πœƒ)=dim(𝑆(π‘‡π‘€βŸ‚)). If {𝑒1,𝑒2,…,π‘’π‘š} is a local orthonormal basis of Ξ“(π·πœƒ), then {cscπœƒπΉπ‘’1,…,cscπœƒπΉπ‘’π‘š} is an orthonormal basis of 𝑆(π‘‡π‘€βŸ‚).

Proof. Suppose that {𝑒1,𝑒2,…,π‘’π‘š} is a local orthonormal basis of π·πœƒ. From (3.25) together with the fact that π·πœƒ is Riemannian, we conclude that 𝑔cscπœƒπΉπ‘’π‘–,cscπœƒπΉπ‘’π‘—ξ€Έ=𝛿𝑖𝑗,(4.6) where 𝑖,𝑗=1,2,3,…,π‘š. Hence our assertion follows from (4.6).

The existence of minimal hemi-slant lightlike submanifolds in an indefinite Kenmotsu manifold is given by the following two theorems.

Theorem 4.4. Let 𝑀 be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is minimal if and only if traceπ΄βˆ—πœ‰π‘—|||𝑆(𝑇𝑀)=0,traceπ΄π‘Šπ›Ό||𝑆(𝑇𝑀)𝑔𝐷=0,𝑙(𝑋,π‘Š),π‘Œ=0,(4.7) for any 𝑋,π‘ŒβˆˆΞ“(Rad𝑇𝑀) and π‘ŠβˆˆΞ“(𝑆(𝑇MβŸ‚)), where {πœ‰π‘—}π‘Ÿπ‘—=1 is a basis of Rad𝑇𝑀 and {π‘Šπ›Ό}π‘šπ›Ό=1 is a basis of 𝑆(π‘‡π‘€βŸ‚).

Proof. Using βˆ‡π‘‰π‘‰=0 and (2.2), we obtain β„Žπ‘™(𝑉,𝑉)=β„Žπ‘ (𝑉,𝑉)=0. It is known that β„Žπ‘™=0 on Rad𝑇𝑀 [14, Proposition 4.1]. If we take {𝑒1,𝑒2,…,π‘’π‘š} as an orthonormal basis of π·πœƒ, then 𝑀 is minimal if and only if Ξ£π‘šπ‘˜=1β„Žξ€·π‘’π‘˜,π‘’π‘˜ξ€Έ=0(4.8) and β„Žπ‘ =0 on Rad𝑇𝑀. From (2.10) and (2.14), we have Ξ£π‘šπ‘˜=1β„Žξ€·π‘’π‘˜,π‘’π‘˜ξ€Έ=Ξ£π‘šπ‘˜=11π‘ŸΞ£π‘šπ‘—=1π‘”ξ‚€π΄βˆ—πœ‰π‘—π‘’π‘˜,π‘’π‘˜ξ‚π‘π‘—+1π‘šΞ£π‘šπ›Ό=1π‘”ξ€·π΄π‘Šπ›Όπ‘’π‘˜,π‘’π‘˜ξ€Έπ‘Šπ›Ό.(4.9) On the other hand, from (2.10), we infer that β„Žπ‘ =0 on Rad𝑇𝑀 if 𝑔𝐷𝑙(𝑋,π‘Š),π‘Œ=0,(4.10) for 𝑋,π‘ŒβˆˆΞ“(Rad𝑇𝑀) and π‘ŠβˆˆΞ“(𝑆(π‘‡π‘€βŸ‚)). Thus, our assertion follows from (4.9) and (4.10).

Theorem 4.5. Let 𝑀 be a proper hemi-slant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 such that dim(π·πœƒ)=dim(𝑆(π‘‡π‘€βŸ‚)). Then 𝑀 is minimal if and only if traceπ΄βˆ—πœ‰π‘—|||𝑆(𝑇𝑀)=0,traceπ΄π‘Šπ‘’π‘–||𝑆(𝑇𝑀)𝑔𝐷=0,𝑙X,𝐹𝑒𝑖,π‘Œ=0.(4.11) for 𝑋,π‘ŒβˆˆΞ“(Rad𝑇𝑀), where {πœ‰π‘˜}π‘Ÿπ‘˜=1 is a basis of Rad𝑇𝑀 and {𝑒𝑗}π‘šπ‘—=1 is a basis of π·πœƒ.

Proof. From βˆ‡π‘‰π‘‰=0 and (2.2), we obtain β„Žπ‘™(𝑉,𝑉)=β„Žπ‘ (𝑉,𝑉)=0. Recall that β„Žπ‘™=0 on Rad𝑇𝑀 [14, Proposition 4.1]. In view of Lemma 4.3, we observe that {cscπœƒπΉπ‘’1,…,cscπœƒπΉπ‘’π‘š} is an orthonormal basis of 𝑆(π‘‡π‘€βŸ‚). Then, for any π‘‹βˆˆΞ“(𝑇𝑀) and for some function 𝐴𝑖, π‘–βˆˆ(1,2,…,π‘š), we can write β„Žπ‘ (𝑋,𝑋)=Ξ£π‘šπ‘–=1𝐴𝑖cscπœƒπΉπ‘’π‘–,(4.12) from which we have β„Žπ‘ (𝑋,𝑋)=Ξ£π‘šπ‘–=1𝐴cscπœƒπ‘”πΉπ‘’π‘–ξ€Έπ‘‹,𝑋𝐹𝑒𝑖,(4.13) for any π‘‹βˆˆΞ“(𝑆(𝑇𝑀)). Thus our assertion follows from the above equation and Theorem 4.4.

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Copyright © 2012 S. M. Khursheed Haider 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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