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Mathematical Problems in Engineering
VolumeΒ 2011, Article IDΒ 516238, 9 pages
http://dx.doi.org/10.1155/2011/516238
Research Article

Totally Umbilical Proper Slant and Hemislant Submanifolds of an LP-Cosymplectic Manifold

1Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia
2Department of Mathematics, University of Tabouk, Tabouk, Saudi Arabia
3School of Mathematics and Computer Applications, Thapar University, Patiala 147 004, India

Received 13 December 2010; Accepted 15 April 2011

Academic Editor: AlexΒ Elias-Zuniga

Copyright Β© 2011 Siraj Uddin 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

In the present note, we study slant and hemislant submanifolds of an LP-cosymplectic manifold which are totally umbilical. We prove that every totally umbilical proper slant submanifold 𝑀 of an LP-cosymplectic manifold 𝑀 is either totally geodesic or if 𝑀 is not totally geodesic in 𝑀 then we derive a formula for slant angle of 𝑀. Also, we obtain the integrability conditions of the distributions of a hemi-slant submanifold, and then we give a result on its classification.

1. Introduction

A manifold 𝑀 with Lorentzian paracontact metric structure (πœ™,πœ‰,πœ‚,𝑔) satisfying (βˆ‡π‘‹πœ™)π‘Œ=0 is called an LP-cosymplectic manifold, where βˆ‡ is the Levi-Civita connection corresponding to the Lorentzian metric 𝑔 on 𝑀. The study of slant submanifolds was initiated by Chen [1]. Since then, many research papers have appeared in this field. Slant submanifolds are the natural generalization of both holomorphic and totally real submanifolds. Lotta [2] defined and studied these submanifolds in contact geometry. Later on, Cabrerizo et al. studied slant, semi-slant, and bislant submanifolds in contact geometry [3, 4]. In particular, totally umbilical proper slant submanifold of a Kaehler manifold has also been studied in [5]. Recently, Khan et al. [6] studied these submanifolds in the setting of Lorentzian paracontact manifolds.

The idea of hemi-slant submanifolds was introduced by Carriazo as a particular class of bislant submanifolds, and he called them antislant submanifolds [7]. Recently, these submanifolds are studied by Sahin for their warped products [8]. In this paper, we study slant and hemi-slant submanifolds of an LP-cosymplectic manifold. We prove that a totally umbilical proper slant submanifold 𝑀 is either totally geodesic in 𝑀 or if it is not totally geodesic, then the slant angle πœƒ=tanβˆ’1(βˆšπ‘”(𝑋,π‘Œ)/πœ‚(𝑋)πœ‚(π‘Œ)). Also, we define hemi-slant submanifolds of an LP-contact manifold. After we find integrability conditions of the distributions, we investigate a classification of totally umbilical hemi-slant submanifolds of an LP-cosymplectic manifold.

2. Preliminaries

Let 𝑀 be a 𝑛-dimensional paracontact manifold with the Lorentzian paracontact metric structure (πœ™,πœ‰,πœ‚,𝑔), that is, πœ™ is a (1,1) tensor field, πœ‰ is a contravariant vector field, πœ‚ is a 1-form, and 𝑔 is a Lorentzian metric with signature (βˆ’,+,+,…,+) on 𝑀, satisfying [9], πœ™2=𝑋+πœ‚(𝑋)πœ‰,πœ‚(πœ‰)=βˆ’1,πœ™πœ‰=0,πœ‚βˆ˜πœ™=0,rank(πœ™)=π‘›βˆ’1,(2.1)𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)+πœ‚(𝑋)πœ‚(π‘Œ),πœ‚(𝑋)=𝑔(𝑋,πœ‰),(2.2) for all 𝑋,π‘Œβˆˆπ‘‡π‘€.

A Lorentzian paracontact metric structure on 𝑀 is called a Lorentzian para-cosymplectic structure if βˆ‡πœ™=0, where βˆ‡ denotes the Levi-Civita connection with respect to 𝑔. The manifold 𝑀 in this case is called a Lorentzian para-cosymplectic (in brief, an LP-cosymplectic) manifold [10]. From formula βˆ‡πœ™=0, it follows that βˆ‡π‘‹πœ‰=0.

Let 𝑀 be a submanifold of a Lorentzian almost paracontact manifold 𝑀 with Lorentzian almost paracontact structure (πœ™,πœ‰,πœ‚,𝑔). Let the induced metric on 𝑀 also be denoted by 𝑔, then Gauss and Weingarten formulae are given by βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+β„Ž(𝑋,π‘Œ),(2.3)βˆ‡π‘‹π‘=βˆ’π΄π‘π‘‹+βˆ‡βŸ‚π‘‹π‘,(2.4) for any 𝑋,π‘Œ in 𝑇𝑀 and 𝑁 in π‘‡βŸ‚π‘€, where 𝑇𝑀 is the Lie algebra of vector field in 𝑀 and π‘‡βŸ‚π‘€ is the set of all vector fields normal to 𝑀. βˆ‡βŸ‚ is the connection in the normal bundle, β„Ž is the second fundamental form, and 𝐴𝑁 is the Weingarten endomorphism associated with 𝑁. It is easy to see that 𝑔𝐴𝑁𝑋,π‘Œ=𝑔(β„Ž(𝑋,π‘Œ),𝑁).(2.5)

For any π‘‹βˆˆπ‘‡π‘€, we write πœ™π‘‹=𝑃𝑋+𝐹𝑋,(2.6) where 𝑃𝑋 is the tangential component and 𝐹𝑋 is the normal component of πœ™π‘‹. Similarly for π‘βˆˆπ‘‡βŸ‚π‘€, we write πœ™π‘=𝐡𝑁+𝐢𝑁,(2.7) where 𝐡𝑁 is the tangential component and 𝐢𝑁 is the normal component of πœ™π‘.

The covariant derivatives of the tensor fields πœ™, 𝑃, and 𝐹 are defined as ξ‚€βˆ‡π‘‹πœ™ξ‚π‘Œ=βˆ‡π‘‹πœ™π‘Œβˆ’πœ™βˆ‡π‘‹π‘Œ,βˆ€π‘‹,π‘Œβˆˆπ‘‡π‘€,(2.8)ξ‚€βˆ‡π‘‹π‘ƒξ‚π‘Œ=βˆ‡π‘‹π‘ƒπ‘Œβˆ’π‘ƒβˆ‡π‘‹π‘Œ,βˆ€π‘‹,π‘Œβˆˆπ‘‡π‘€,(2.9)ξ‚€βˆ‡π‘‹πΉξ‚π‘Œ=βˆ‡βŸ‚π‘‹πΉπ‘Œβˆ’πΉβˆ‡π‘‹π‘Œ,βˆ€π‘‹,π‘Œβˆˆπ‘‡π‘€.(2.10) Moreover, for an LP-cosymplectic manifold, one has ξ‚€βˆ‡π‘‹π‘ƒξ‚π‘Œ=π΄πΉπ‘Œπ‘‹+π΅β„Ž(𝑋,π‘Œ),(2.11)ξ‚€βˆ‡π‘‹πΉξ‚π‘Œ=πΆβ„Ž(𝑋,π‘Œ)βˆ’β„Ž(𝑋,π‘ƒπ‘Œ).(2.12)

A submanifold 𝑀 is said to be totally umbilical if β„Ž(𝑋,π‘Œ)=𝑔(𝑋,π‘Œ)𝐻,(2.13) where 𝐻 is the mean curvature vector. Furthermore, if β„Ž(𝑋,π‘Œ)=0 for all 𝑋,π‘Œβˆˆπ‘‡π‘€, then 𝑀 is said to be totally geodesic, and if 𝐻=0, then 𝑀 is minimal in 𝑀.

A submanifold 𝑀 of a paracontact manifold 𝑀 is said to be a slant submanifold if for any π‘₯βˆˆπ‘€ and π‘‹βˆˆπ‘‡π‘₯π‘€βˆ’βŸ¨πœ‰βŸ©, the angle between πœ™π‘‹ and 𝑇π‘₯𝑀 is constant. The constant angle πœƒβˆˆ[0,πœ‹/2] is then called slant angle of 𝑀. The tangent bundle 𝑇𝑀 of 𝑀 is decomposed as 𝑇𝑀=π·βŠ•βŸ¨πœ‰βŸ©,(2.14) where the orthogonal complementary distribution 𝐷 of βŸ¨πœ‰βŸ© is known as the slant distribution on 𝑀. If πœ‡ is πœ™-invariant subspace of the normal bundle π‘‡βŸ‚π‘€, then π‘‡βŸ‚π‘€=πΉπ‘‡π‘€βŠ•πœ‡.(2.15)

Khan et al. [6] proved the following theorem for a slant submanifold 𝑀 of a Lorentzian paracontact manifold 𝑀 with slant angle πœƒ.

Theorem 2.1. Let 𝑀 be a submanifold of an 𝐿𝑃-contact manifold 𝑀 such that πœ‰βˆˆπ‘‡π‘€, then 𝑀 is slant submanifold if and only if there exists a constant πœ†βˆˆ[0,1] such that 𝑃2=πœ†(𝐼+πœ‚βŠ—πœ‰).(2.16) Furthermore, if πœƒ is slant angle of 𝑀, then πœ†=π‘π‘œπ‘ 2πœƒ.
Thus, one has the following consequences of formula (2.16): 𝑔(𝑃𝑋,𝑃𝑋)=cos2πœƒ[],𝑔(𝑋,π‘Œ)+πœ‚(𝑋)πœ‚(π‘Œ)(2.17)𝑔(𝐹𝑋,πΉπ‘Œ)=sin2πœƒ[],𝑔(𝑋,π‘Œ)+πœ‚(𝑋)πœ‚(π‘Œ)(2.18) for any 𝑋,π‘Œβˆˆπ‘‡π‘€.

3. Totally Umbilical Proper Slant Submanifold

In this section, we consider 𝑀 as a totally umbilical proper slant submanifold of an LP-cosymplectic manifold 𝑀. Such submanifolds we always consider tangent to the structure vector field πœ‰.

Theorem 3.1. A nontrivial totally umbilical proper slant submanifold 𝑀 of an LP-cosymplectic manifold 𝑀 is either totally geodesic or if it is not totally geodesic in 𝑀, then the slant angle πœƒ=tanβˆ’1(βˆšπ‘”(𝑋,π‘Œ)/πœ‚(𝑋)πœ‚(π‘Œ)), for any 𝑋,π‘Œβˆˆπ‘‡π‘€.

Proof. For any 𝑋,π‘Œβˆˆπ‘‡π‘€, (2.11) gives ξ‚€βˆ‡π‘‹π‘ƒξ‚π‘Œ=π΄πΉπ‘Œπ‘‹+π΅β„Ž(𝑋,π‘Œ).(3.1) Taking the product with πœ‰ and using (2.9), we obtain π‘”ξ€·βˆ‡π‘‹ξ€Έξ€·π΄π‘ƒπ‘Œ,πœ‰=π‘”πΉπ‘Œξ€Έπ‘‹,πœ‰+𝑔(π΅β„Ž(𝑋,π‘Œ),πœ‰).(3.2) Using (2.5) and the fact that 𝑀 is totally umbilical, the above equation takes the form ξ€·βˆ’π‘”π‘ƒπ‘Œ,βˆ‡π‘‹πœ‰ξ€Έ=𝑔(𝐻,πΉπ‘Œ)πœ‚(𝑋)+𝑔(𝑋,π‘Œ)𝑔(𝐡𝐻,πœ‰).(3.3) Then, from the characteristic equation of LP-cosymplectic manifold, we obtain 0=𝑔(𝐻,πΉπ‘Œ)πœ‚(𝑋).(3.4) Thus, from (3.4), it follows that either π»βˆˆπœ‡ or 𝑀 is trivial.
Now, for an LP-cosymplectic manifold, one has, from (2.8), βˆ‡π‘‹πœ™π‘Œ=πœ™βˆ‡π‘‹π‘Œ,(3.5) for any 𝑋,π‘Œβˆˆπ‘‡π‘€. From (2.3) and (2.6), we obtain βˆ‡π‘‹π‘ƒπ‘Œ+βˆ‡π‘‹ξ€·βˆ‡πΉπ‘Œ=πœ™π‘‹ξ€Έπ‘Œ+β„Ž(𝑋,π‘Œ).(3.6) Again using (2.3), (2.4), and (2.6), we get βˆ‡π‘‹π‘ƒπ‘Œ+β„Ž(𝑋,π‘ƒπ‘Œ)βˆ’π΄πΉπ‘Œπ‘‹+βˆ‡βŸ‚π‘‹πΉπ‘Œ=π‘ƒβˆ‡π‘‹π‘Œ+πΉβˆ‡π‘‹π‘Œ+πœ™β„Ž(𝑋,π‘Œ).(3.7) As 𝑀 is totally umbilical, then βˆ‡π‘‹π‘ƒπ‘Œ+β„Ž(𝑋,π‘ƒπ‘Œ)βˆ’π΄πΉπ‘Œπ‘‹+βˆ‡βŸ‚π‘‹πΉπ‘Œ=π‘ƒβˆ‡π‘‹π‘Œ+πΉβˆ‡π‘‹π‘Œ+𝑔(𝑋,π‘Œ)πœ™π».(3.8) Taking the inner product with πœ™π» and using the fact that π»βˆˆπœ‡, we obtain π‘”ξ€·βˆ‡(β„Ž(𝑋,π‘ƒπ‘Œ),πœ™π»)+π‘”βŸ‚π‘‹ξ€Έξ€·πΉπ‘Œ,πœ™π»=π‘”πΉβˆ‡π‘‹ξ€Έπ‘Œ,πœ™π»+𝑔(𝑋,π‘Œ)𝑔(πœ™π»,πœ™π»).(3.9) Then from (2.2) and (2.13), we get π‘”ξ€·βˆ‡(𝑋,π‘ƒπ‘Œ)𝑔(𝐻,πœ™π»)+π‘”βŸ‚π‘‹ξ€Έξ€·πΉπ‘Œ,πœ™π»=π‘”πΉβˆ‡π‘‹ξ€Έπ‘Œ,πœ™π»+𝑔(𝑋,π‘Œ)‖𝐻‖2.(3.10) Again, using (2.2) and the fact that π»βˆˆπœ‡, then πœ™π» is also lies in πœ‡; thus, we obtain π‘”ξ€·βˆ‡βŸ‚π‘‹ξ€ΈπΉπ‘Œ,πœ™π»=𝑔(𝑋,π‘Œ)‖𝐻‖2.(3.11) Then, from (2.4), we derive π‘”ξ‚€βˆ‡π‘‹ξ‚πΉπ‘Œ,πœ™π»=𝑔(𝑋,π‘Œ)‖𝐻‖2.(3.12) Now, for any π‘‹βˆˆπ‘‡π‘€, one has ξ‚€βˆ‡π‘‹πœ™ξ‚π»=βˆ‡π‘‹πœ™π»βˆ’πœ™βˆ‡π‘‹π».(3.13) Using the fact that as 𝑀 is an LP-cosymplectic manifold, we obtain βˆ‡π‘‹πœ™π»=πœ™βˆ‡π‘‹π».(3.14) Using (2.4), (2.6), and (2.7), we obtain βˆ’π΄πœ™π»π‘‹+βˆ‡βŸ‚π‘‹πœ™π»=βˆ’π‘ƒπ΄π»π‘‹βˆ’πΉπ΄π»π‘‹+π΅βˆ‡βŸ‚π‘‹π»+πΆβˆ‡βŸ‚π‘‹π».(3.15) Taking the product in (3.15) with πΉπ‘Œ for any π‘Œβˆˆπ‘‡π‘€ and using the fact πΆβˆ‡βŸ‚π‘‹π»βˆˆπœ‡, the above equation gives π‘”ξ€·βˆ‡βŸ‚π‘‹ξ€Έξ€·πœ™π»,πΉπ‘Œ=βˆ’π‘”πΉπ΄π»ξ€Έπ‘‹,πΉπ‘Œ.(3.16) Using (2.18), we obtain π‘”ξ‚€βˆ‡π‘‹ξ‚πΉπ‘Œ,πœ™π»=sin2πœƒξ€Ίπ‘”ξ€·π΄π»ξ€Έξ€·π΄π‘‹,π‘Œ+πœ‚π»π‘‹ξ€Έξ€»πœ‚(π‘Œ),(3.17) then, from (2.5) and (2.13), we get π‘”ξ‚€βˆ‡π‘‹ξ‚πΉπ‘Œ,πœ™π»=sin2πœƒ[]‖𝑔(𝑋,π‘Œ)+πœ‚(𝑋)πœ‚(π‘Œ)𝐻‖2.(3.18) Thus, from (3.12) and (3.18), we derive ξ€Ίcos2πœƒπ‘”(𝑋,π‘Œ)βˆ’sin2ξ€»πœƒπœ‚(𝑋)πœ‚(π‘Œ)‖𝐻‖2=0.(3.19) Hence, (3.19) gives either 𝐻=0 or if 𝐻≠0, then the slant angle of 𝑀 is πœƒ=tanβˆ’1(βˆšπ‘”(𝑋,π‘Œ)/πœ‚(𝑋)πœ‚(π‘Œ)). This proves the theorem completely.

4. Hemislant Submanifolds

In the following section, we assume that 𝑀 is a hemi-slant submanifold of an LP-cosymplectic manifold 𝑀 such that the structure vector field πœ‰ tangent to 𝑀. First, we define a hemi-slant submanifold, and then we obtain the integrability conditions of the involved distributions 𝐷1 and 𝐷2 in the definition of a hemi-slant submanifold 𝑀 of an LP-cosymplectic manifold 𝑀.

Definition 4.1. A submanifold M of an LP-contact manifold 𝑀 is said to be a hemi-slant submanifold if there exist two orthogonal complementary distributions 𝐷1 and 𝐷2 satisfying(i)𝑇𝑀=𝐷1βŠ•π·2βŠ•βŸ¨πœ‰βŸ©,(ii)𝐷1 is a slant distribution with slant angle πœƒβ‰ πœ‹/2,(iii)𝐷2 is totally real that is, πœ™π·2βŠ†π‘‡βŸ‚π‘€.

If πœ‡ is πœ™-invariant subspace of the normal bundle π‘‡βŸ‚π‘€, then in case of hemi-slant submanifold, the normal bundle π‘‡βŸ‚π‘€ can be decomposed as π‘‡βŸ‚π‘€=𝐹𝐷1βŠ•πΉπ·2βŠ•πœ‡.(4.1)

In the following, we obtain the integrability conditions of involved distributions in the definition of hemi-slant submanifold.

Proposition 4.2. Let 𝑀 be a hemi-slant submanifold of an LP-cosymplectic manifold 𝑀, then the anti-invariant distribution 𝐷2 is integrable if and only if π΄πΉπ‘π‘Š=π΄πΉπ‘Šπ‘,(4.2) for any 𝑍,π‘Šβˆˆπ·2.

Proof. For any 𝑍,π‘Šβˆˆπ·2, one has πœ™[]𝑍,π‘Š=πœ™βˆ‡π‘π‘Šβˆ’πœ™βˆ‡π‘Šπ‘.(4.3) Using (2.8), we obtain πœ™[]=𝑍,π‘Šβˆ‡π‘πœ™π‘Šβˆ’βˆ‡π‘Šπœ™π‘.(4.4) Then, from (2.4), we derive πœ™[]𝑍,π‘Š=βˆ’π΄πΉπ‘Šπ‘+βˆ‡βŸ‚π‘πΉπ‘Š+π΄πΉπ‘π‘Šβˆ’βˆ‡βŸ‚π‘ŠπΉπ‘.(4.5) As 𝐷2 is an anti-invariant distribution, then the tangential part of (4.5) should be identically zero; hence, we obtain the required result.

Proposition 4.3. Let 𝑀 be a hemi-slant submanifold of an LP-cosymplectic manifold 𝑀, then the invariant distribution 𝐷1βŠ•βŸ¨πœ‰βŸ© is integrable if and only if π‘”ξ€·β„Ž(𝑋,π‘ƒπ‘Œ)βˆ’β„Ž(π‘Œ,𝑃𝑋)+βˆ‡βŸ‚π‘‹πΉπ‘Œβˆ’βˆ‡βŸ‚π‘Œξ€ΈπΉπ‘‹,𝐹𝑍=0,(4.6) for any 𝑋,π‘Œβˆˆπ·1βŠ•βŸ¨πœ‰βŸ© and π‘βˆˆπ·2.

Proof. For any 𝑋,π‘Œβˆˆπ·1βŠ•βŸ¨πœ‰βŸ©, one has πœ™[]𝑋,π‘Œ=πœ™βˆ‡π‘‹π‘Œβˆ’πœ™βˆ‡π‘Œπ‘‹.(4.7) Then, from (2.8) and the fact that 𝑀 is LP-cosymplectic, we obtain πœ™[]=𝑋,π‘Œβˆ‡π‘‹πœ™π‘Œβˆ’βˆ‡π‘Œπœ™π‘‹.(4.8) Using (2.6), we get πœ™[]=𝑋,π‘Œβˆ‡π‘‹π‘ƒπ‘Œ+βˆ‡π‘‹πΉπ‘Œβˆ’βˆ‡π‘Œπ‘ƒπ‘‹βˆ’βˆ‡π‘ŒπΉπ‘‹.(4.9) Thus, from (2.3) and (2.4), we derive πœ™[]𝑋,π‘Œ=βˆ‡π‘‹π‘ƒπ‘Œ+β„Ž(𝑋,π‘ƒπ‘Œ)βˆ’π΄πΉπ‘Œπ‘‹+βˆ‡βŸ‚π‘‹πΉπ‘Œβˆ’βˆ‡π‘Œπ‘ƒπ‘‹βˆ’β„Ž(π‘Œ,𝑃𝑋)+π΄πΉπ‘‹π‘Œβˆ’βˆ‡βŸ‚π‘ŒπΉπ‘‹.(4.10) Taking the product in (4.10) with 𝐹𝑍, for any π‘βˆˆπ·2, we obtain 𝑔[]ξ€·β„Ž(πœ™π‘‹,π‘Œ,𝐹𝑍)=𝑔(𝑋,π‘ƒπ‘Œ)+βˆ‡βŸ‚π‘‹πΉπ‘Œβˆ’β„Ž(π‘Œ,𝑃𝑋)βˆ’βˆ‡βŸ‚π‘Œξ€ΈπΉπ‘‹,𝐹𝑍.(4.11) Thus, the assertion follows from (4.11) after using (2.2) and the fact that πœ‰ is tangential to 𝐷1.

Now, we consider 𝑀 as a totally umbilical hemi-slant submanifold of an LP-cosymplectic manifold 𝑀. For any 𝑋,π‘Œβˆˆπ‘‡π‘€, one has βˆ‡π‘‹πœ™π‘Œ=πœ™βˆ‡π‘‹π‘Œ.(4.12) Using this fact, if we take for any 𝑍,π‘Šβˆˆπ·2, then from (2.3) and (2.4), the above equation takes the form βˆ’π΄πΉπ‘Šπ‘+βˆ‡βŸ‚π‘ξ€·βˆ‡πΉπ‘Š=πœ™π‘ξ€Έπ‘Š+β„Ž(𝑍,π‘Š).(4.13) Thus, on using (2.6) and (2.7), we obtain βˆ’π΄πΉπ‘Šπ‘+βˆ‡βŸ‚π‘πΉπ‘Š=π‘ƒβˆ‡π‘π‘Š+πΉβˆ‡π‘π‘Š+π΅β„Ž(𝑍,π‘Š)+πΆβ„Ž(𝑍,π‘Š).(4.14) Equating the tangential components, we get π‘ƒβˆ‡π‘π‘Š=βˆ’π΄πΉπ‘Šπ‘βˆ’π΅β„Ž(𝑍,π‘Š).(4.15) Taking the product with π‘‰βˆˆπ·2, we obtain π‘”ξ€·π‘ƒβˆ‡π‘ξ€Έξ€·π΄π‘Š,𝑉=βˆ’π‘”πΉπ‘Šξ€Έπ‘,π‘‰βˆ’π‘”(π΅β„Ž(𝑍,π‘Š),𝑉).(4.16) Using (2.2), (2.5), and the fact that π‘ƒπ‘Š=0, for any π‘Šβˆˆπ·2, thus, the above equation takes the form 0=𝑔(β„Ž(𝑍,𝑉),πΉπ‘Š)+𝑔(π΅β„Ž(𝑍,π‘Š),𝑉).(4.17) As 𝑀 is totally umbilical, we derive 0=𝑔(𝑍,𝑉)𝑔(𝐻,πΉπ‘Š)+𝑔(𝑍,π‘Š)𝑔(𝐡𝐻,𝑉).(4.18) Thus, (4.18) has a solution if either 𝑍=π‘Š=𝑉=πœ‰, that is, dim𝐷2=1 or π»βˆˆπœ‡ or 𝐷2={0}. Hence, we state the following theorem.

Theorem 4.4. Let 𝑀 be a totally umbilical hemi-slant submanifold of an LP-cosymplectic manifold 𝑀, then at least one of the following statements is true: (i)the dimension of anti-invariant distribution is one, that is, dim𝐷2=1,(ii)the mean curvature vector π»βˆˆπœ‡,(iii)𝑀 is proper slant submanifold of 𝑀.

Acknowledgments

The authors are thankful to the referee for his valuable suggestion and comments. The first author is supported by the research Grant no. RG117/10AFR (University Malaya).

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