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ISRN Geometry
VolumeΒ 2011Β (2011), Article IDΒ 161523, 9 pages
http://dx.doi.org/10.5402/2011/161523
Research Article

Some Results on Lorentzian Para-Sasakian Manifolds

Department of Mathematics, Kuvempu University, Shankaraghatta, Karnataka, Shimoga 577 451, India

Received 3 June 2011; Accepted 3 July 2011

Academic Editor: M.Β Dunajski

Copyright Β© 2011 Venkatesha 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

The object of the present paper is to study Lorentzian para-Sasakian (briefly LP-Sasakian) manifolds satisfying certain conditions on the π‘Š2-curvature tensor.

1. Introduction

In 1989, Matsumoto [1] introduced the notion of Lorentzian para-Sasakian manifold. Then, Mihai and Roşca [2] introduced the same notion independently and they obtained several results on this manifold. LP-Sasakian manifolds have also been studied by Matsumoto and Mihai [3], Mihai et al. [4], and Venkatesha and Bagewadi [5].

On the other hand, Pokhariyal and Mishra [6] have introduced new curvature tensor called π‘Š2-curvature tensor in a Riemannian manifold and studied their properties. Further, Pokhariyal [7] has studied some properties of this curvature tensor in a Sasakian manifold. Matsumoto et al. [8], and YΓ¬ldΓ¬z and De [9] have studied π‘Š2-curvature tensor in P-Sasakian and Kenmotsu manifolds, respectively.

In the present paper, we study some curvature conditions on LP-Sasakian manifolds. Firstly, we study LP-Sasakian manifolds satisfying π‘Š2=0 and π‘Š2-semisymmetric LP-Sasakian manifolds. Further, we study LP-Sasakian manifolds which satisfy π‘ƒβ‹…π‘Š2=0, ξ‚‹π‘€β‹…π‘Š2=0 and πΆβ‹…π‘Š2=0, where 𝑃 is the projective curvature tensor, 𝑀 is the 𝑀-projective curvature tensor, and 𝐢 is the conformal curvature tensor.

2. Preliminaries

An 𝑛-dimensional differentiable manifold 𝑀 is called an LP-Sasakian manifold [1, 2] if it admits a (1,1) tensor field πœ™, a contravariant vector field πœ‰, a 1-form πœ‚, and a Lorentzian metric 𝑔 which satisfyπœ™πœ‚(πœ‰)=βˆ’1,2βˆ‡π‘‹=𝑋+πœ‚(𝑋)πœ‰,𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)+πœ‚(𝑋)πœ‚(π‘Œ),𝑔(𝑋,πœ‰)=πœ‚(𝑋),π‘‹ξ€·βˆ‡πœ‰=πœ™π‘‹,π‘‹πœ™ξ€Έπ‘Œ=𝑔(𝑋,π‘Œ)πœ‰+πœ‚(π‘Œ)𝑋+2πœ‚(𝑋)πœ‚(π‘Œ)πœ‰,(2.1) where βˆ‡ denotes the operator of covariant differentiation with respect to the Lorentzian metric 𝑔.

It can be easily seen that, in an LP-Sasakian manifold, the following relations hold:πœ™πœ‰=0,πœ‚(πœ™π‘‹)=0,rankπœ™=π‘›βˆ’1.(2.2) If we putΞ¦(𝑋,π‘Œ)=𝑔(𝑋,πœ™π‘Œ),(2.3) for any vector fields 𝑋 and π‘Œ, then the tensor field Ξ¦(𝑋,π‘Œ) is a symmetric (0, 2) tensor field [1]. Also, since the 1-form πœ‚ is closed in an LP-Sasakian manifold, we have [1, 4]ξ€·βˆ‡π‘‹πœ‚ξ€Έ(π‘Œ)=Ξ¦(𝑋,π‘Œ),Ξ¦(𝑋,πœ‰)=0,(2.4) for any vector fields 𝑋 and π‘Œ.

Also in an LP-Sasakian manifold, the following relations hold [3, 4]:𝑔(𝑅(𝑋,π‘Œ)𝑍,πœ‰)=πœ‚(𝑅(𝑋,π‘Œ)𝑍)=𝑔(π‘Œ,𝑍)πœ‚(𝑋)βˆ’π‘”(𝑋,𝑍)πœ‚(π‘Œ),(2.5)𝑅(πœ‰,𝑋)π‘Œ=𝑔(𝑋,π‘Œ)πœ‰βˆ’πœ‚(π‘Œ)𝑋,(2.6)𝑅(𝑋,π‘Œ)πœ‰=πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ,(2.7)𝑆(𝑋,πœ‰)=(π‘›βˆ’1)πœ‚(𝑋),(2.8)𝑆(πœ™π‘‹,πœ™π‘Œ)=𝑆(𝑋,π‘Œ)+(π‘›βˆ’1)πœ‚(𝑋)πœ‚(π‘Œ),(2.9) for any vector fields 𝑋,π‘Œ, and 𝑍, where 𝑅 is the Riemannian curvature tensor and 𝑆 is the Ricci tensor of 𝑀.

An LP-Sasakian manifold 𝑀 is said to be Einstein if its Ricci tensor 𝑆 is of the form𝑆(𝑋,π‘Œ)=π‘Žπ‘”(𝑋,π‘Œ),(2.10) for any vector fields 𝑋 and π‘Œ, where π‘Ž is a function on 𝑀.

In [6], Pokhariyal and Mishra have defined the curvature tensor π‘Š2, given byπ‘Š21(𝑋,π‘Œ,π‘ˆ,𝑉)=𝑅(𝑋,π‘Œ,π‘ˆ,𝑉)+[],π‘›βˆ’1𝑔(𝑋,π‘ˆ)𝑆(π‘Œ,𝑉)βˆ’π‘”(π‘Œ,π‘ˆ)𝑆(𝑋,𝑉)(2.11) where 𝑆 is a Ricci tensor of type (0, 2).

Now, consider an LP-Sasakian manifold satisfying π‘Š2=0; then, (2.11) becomes1𝑅(𝑋,π‘Œ,π‘ˆ,𝑉)=[].π‘›βˆ’1𝑔(π‘Œ,π‘ˆ)𝑆(𝑋,𝑉)βˆ’π‘”(𝑋,π‘ˆ)𝑆(π‘Œ,𝑉)(2.12) Taking 𝑋=π‘ˆ=πœ‰ in (2.12) and using (2.7) and (2.8), we have𝑆(π‘Œ,𝑉)=(π‘›βˆ’1)𝑔(π‘Œ,𝑉).(2.13) Therefore, 𝑀 is an Einstein manifold.

Again using (2.13) in (2.12), we get[].𝑅(𝑋,π‘Œ,π‘ˆ,𝑉)=𝑔(π‘Œ,π‘ˆ)𝑔(𝑋,𝑉)βˆ’π‘”(𝑋,π‘ˆ)𝑔(π‘Œ,𝑉)(2.14)

Corollary 2.1. An LP-Sasakian manifold satisfying π‘Š2=0 is a space of constant curvature βˆ’1, that is, it is locally isometric to the hyperbolic space.

Definition 2.2. An LP-Sasakian manifold is called π‘Š2-semisymmetric if it satisfies 𝑅(𝑋,π‘Œ)β‹…π‘Š2=0,(2.15) where 𝑅(𝑋,π‘Œ) is to be considered as a derivation of the tensor algebra at each point of the manifold for tangent vectors 𝑋 and π‘Œ.
It can be easily shown that in an LP-Sasakian manifold the π‘Š2-curvature tensor satisfies the condition πœ‚ξ€·π‘Š2ξ€Έ(𝑋,π‘Œ)𝑍=0.(2.16)

Theorem 2.3. A π‘Š2-semisymmetric LP-Sasakian manifold 𝑀 is an Einstein manifold.

Proof. Since 𝑅(𝑋,π‘Œ)β‹…π‘Š2=0, we have 𝑅(𝑋,π‘Œ)π‘Š2(π‘ˆ,𝑉)π‘βˆ’π‘Š2(𝑅(𝑋,π‘Œ)π‘ˆ,𝑉)π‘βˆ’π‘Š2(π‘ˆ,𝑅(𝑋,π‘Œ)𝑉)π‘βˆ’π‘Š2(π‘ˆ,𝑉)𝑅(𝑋,π‘Œ)𝑍=0.(2.17) Putting 𝑋=πœ‰ in (2.17) and then taking the inner product with πœ‰, we obtain 𝑔𝑅(πœ‰,π‘Œ)π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)𝑍,πœ‰βˆ’π‘”2ξ€Έξ€·π‘Š(𝑅(πœ‰,π‘Œ)π‘ˆ,𝑉)𝑍,πœ‰βˆ’π‘”2(ξ€Έξ€·π‘Šπ‘ˆ,𝑅(πœ‰,π‘Œ)𝑉)𝑍,πœ‰βˆ’π‘”2(ξ€Έπ‘ˆ,𝑉)𝑅(πœ‰,π‘Œ)𝑍,πœ‰=0.(2.18) Using (2.6) in (2.18), we obtain ξ€·βˆ’π‘”π‘Œ,π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)π‘βˆ’πœ‚2ξ€Έπœ‚ξ€·π‘Š(π‘ˆ,𝑉)𝑍(π‘Œ)βˆ’π‘”(π‘Œ,π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(πœ‰,𝑉)π‘βˆ’π‘”(π‘Œ,𝑉)πœ‚2(ξ€Έξ€·π‘Šπ‘ˆ,πœ‰)π‘βˆ’π‘”(π‘Œ,𝑍)πœ‚2(ξ€Έξ€·π‘Šπ‘ˆ,𝑉)πœ‰+πœ‚(π‘ˆ)πœ‚2(ξ€Έξ€·π‘Šπ‘Œ,𝑉)𝑍+πœ‚(𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,π‘Œ)𝑍+πœ‚(𝑍)πœ‚2ξ€Έ(π‘ˆ,𝑉)π‘Œ=0.(2.19) By using (2.16) in (2.19), we get π‘Š2(π‘ˆ,𝑉,𝑍,π‘Œ)=0.(2.20) In view of (2.11) and (2.20), it follows that 1𝑅(π‘ˆ,𝑉,𝑍,π‘Œ)=[].π‘›βˆ’1𝑔(𝑉,𝑍)𝑆(π‘ˆ,π‘Œ)βˆ’π‘”(π‘ˆ,𝑍)𝑆(𝑉,π‘Œ)(2.21) Contracting (2.21), we have 𝑆(𝑉,𝑍)=(π‘›βˆ’1)𝑔(𝑉,𝑍).(2.22) Therefore, 𝑀 is an Einstein manifold.

Again using (2.22) in (2.12), we get[].𝑅(π‘ˆ,𝑉,𝑍,π‘Œ)=𝑔(𝑉,𝑍)𝑔(π‘ˆ,π‘Œ)βˆ’π‘”(π‘ˆ,𝑍)𝑔(𝑉,π‘Œ)(2.23)

Corollary 2.4. A π‘Š2-semisymmetric LP-Sasakian manifold is a space of constant curvature βˆ’1, that is, it is locally isometric to the hyperbolic space.

3. LP-Sasakian Manifolds Satisfying 𝑃(𝑋,π‘Œ)β‹…π‘Š2=0

The projective curvature tensor 𝑃 is defined as [10]1𝑃(𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)π‘βˆ’[].π‘›βˆ’1𝑆(π‘Œ,𝑍)π‘‹βˆ’π‘†(𝑋,𝑍)π‘Œ(3.1) Using (2.6) and (2.8), (3.1) reduces to1𝑃(πœ‰,π‘Œ)𝑍=𝑔(π‘Œ,𝑍)πœ‰βˆ’π‘›βˆ’1𝑆(π‘Œ,𝑍)πœ‰.(3.2) Let us suppose that in an LP-Sasakian manifold 𝑃(𝑋,π‘Œ)β‹…π‘Š2=0.(3.3) This condition implies that𝑃(𝑋,π‘Œ)π‘Š2(π‘ˆ,𝑉)π‘βˆ’π‘Š2(𝑃(𝑋,π‘Œ)π‘ˆ,𝑉)π‘βˆ’π‘Š2(π‘ˆ,𝑃(𝑋,π‘Œ)𝑉)π‘βˆ’π‘Š2(π‘ˆ,𝑉)𝑃(𝑋,π‘Œ)𝑍=0.(3.4) Putting 𝑋=πœ‰ in (3.4) and then taking the inner product with πœ‰, we obtain𝑔𝑃(πœ‰,π‘Œ)π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)𝑍,πœ‰βˆ’π‘”2ξ€Έξ€·π‘Š(𝑃(πœ‰,π‘Œ)π‘ˆ,𝑉)𝑍,πœ‰βˆ’π‘”2(ξ€Έξ€·π‘Šπ‘ˆ,𝑃(πœ‰,π‘Œ)𝑉)𝑍,πœ‰βˆ’π‘”2(ξ€Έπ‘ˆ,𝑉)𝑃(πœ‰,π‘Œ)𝑍,πœ‰=0.(3.5) Using (3.2) in (3.5), we obtainξ€·βˆ’π‘”π‘Œ,π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)π‘βˆ’π‘”(π‘Œ,π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(πœ‰,𝑉)π‘βˆ’π‘”(π‘Œ,𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,πœ‰)π‘βˆ’π‘”(π‘Œ,𝑍)πœ‚2ξ€Έ+1(π‘ˆ,𝑉)πœ‰ξ€Ίπ‘†ξ€·π‘›βˆ’1π‘Œ,π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)𝑍+𝑆(π‘Œ,π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(πœ‰,𝑉)𝑍+𝑆(π‘Œ,𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,πœ‰)𝑍+𝑆(π‘Œ,𝑍)πœ‚2(π‘ˆ,𝑉)πœ‰ξ€Έξ€»=0.(3.6) By using (2.16) in (3.6), we getπ‘”ξ€·π‘Œ,π‘Š2ξ€Έβˆ’1(π‘ˆ,𝑉)π‘π‘†ξ€·π‘›βˆ’1π‘Œ,π‘Š2ξ€Έ(π‘ˆ,𝑉)𝑍=0.(3.7) Taking π‘ˆ=𝑍=πœ‰ in (3.7) and using (2.11) and (2.6), we have𝑆(π‘„π‘Œ,𝑉)=2(π‘›βˆ’1)𝑆(π‘Œ,𝑉)βˆ’(π‘›βˆ’1)2𝑔(π‘Œ,𝑉).(3.8) This implies thatπ‘„π‘Œ=(π‘›βˆ’1)π‘Œ.(3.9) From this, we get𝑆(π‘Œ,𝑉)=(π‘›βˆ’1)𝑔(π‘Œ,𝑉).(3.10) Thus, we can state the following.

Theorem 3.1. An LP-Sasakian manifold 𝑀 satisfying the condition 𝑃(𝑋,π‘Œ)β‹…π‘Š2=0 is an Einstein manifold.

4. LP-Sasakian Manifold Satisfying 𝑀(𝑋,π‘Œ)β‹…π‘Š2=0

The 𝑀-projective curvature tensor 𝑀 is defined as [11]ξ‚‹1𝑀(𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)π‘βˆ’2[].(π‘›βˆ’1)𝑆(π‘Œ,𝑍)π‘‹βˆ’π‘†(𝑋,𝑍)π‘Œ+𝑔(π‘Œ,𝑍)π‘„π‘‹βˆ’π‘”(𝑋,𝑍)π‘„π‘Œ(4.1) Using (2.6) and (2.8), (4.1) reduces toξ‚‹1𝑀(πœ‰,π‘Œ)𝑍=2[]βˆ’1𝑔(π‘Œ,𝑍)πœ‰βˆ’πœ‚(𝑍)π‘Œ2[].(π‘›βˆ’1)𝑆(π‘Œ,𝑍)πœ‰βˆ’πœ‚(𝑍)π‘„π‘Œ(4.2) Suppose that in an LP-Sasakian manifold 𝑀(𝑋,π‘Œ)β‹…π‘Š2=0.(4.3) This condition implies that𝑀(𝑋,π‘Œ)π‘Š2(π‘ˆ,𝑉)π‘βˆ’π‘Š2𝑍𝑀(𝑋,π‘Œ)π‘ˆ,π‘‰βˆ’π‘Š2ξ‚€ξ‚‹ξ‚π‘ˆ,𝑀(𝑋,π‘Œ)π‘‰π‘βˆ’π‘Š2ξ‚‹(π‘ˆ,𝑉)𝑀(𝑋,π‘Œ)𝑍=0.(4.4) Putting 𝑋=πœ‰ in (4.4) and then taking the inner product with πœ‰, we obtain𝑔𝑀(πœ‰,π‘Œ)π‘Š2ξ‚ξ‚€π‘Š(π‘ˆ,𝑉)𝑍,πœ‰βˆ’π‘”2ξ‚€ξ‚‹ξ‚ξ‚ξ‚€π‘Šπ‘€(πœ‰,π‘Œ)π‘ˆ,𝑉𝑍,πœ‰βˆ’π‘”2ξ‚€ξ‚‹ξ‚ξ‚ξ‚€π‘Šπ‘ˆ,𝑀(πœ‰,π‘Œ)𝑉𝑍,πœ‰βˆ’π‘”2(π‘ˆ,𝑉)𝑀(πœ‰,π‘Œ)𝑍,πœ‰=0.(4.5) Using (4.2) in (4.5), we obtain12ξ€Ίξ€·βˆ’π‘”π‘Œ,π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)π‘βˆ’π‘”(π‘Œ,π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(πœ‰,𝑉)π‘βˆ’π‘”(π‘Œ,𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,πœ‰)π‘βˆ’π‘”(π‘Œ,𝑍)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)πœ‰+πœ‚(π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(π‘Œ,𝑉)𝑍+πœ‚(𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,π‘Œ)𝑍+πœ‚(𝑍)πœ‚2+1(π‘ˆ,𝑉)π‘Œξ€Έξ€»ξ€Ίπ‘†ξ€·2(π‘›βˆ’1)π‘Œ,π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)𝑍+𝑆(π‘Œ,π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(πœ‰,𝑉)𝑍+𝑆(π‘Œ,𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,πœ‰)𝑍+𝑆(π‘Œ,𝑍)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)πœ‰βˆ’πœ‚(π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(π‘„π‘Œ,𝑉)π‘βˆ’πœ‚(𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,π‘„π‘Œ)π‘βˆ’πœ‚(𝑍)πœ‚2(π‘ˆ,𝑉)π‘„π‘Œξ€Έξ€»=0.(4.6) By using (2.16) in (4.6), we get12π‘”ξ€·π‘Œ,π‘Š2ξ€Έβˆ’1(π‘ˆ,𝑉)𝑍2𝑆(π‘›βˆ’1)π‘Œ,π‘Š2ξ€Έ(π‘ˆ,𝑉)𝑍=0.(4.7) Taking π‘ˆ=𝑍=πœ‰ in (4.7) and using (2.11) and (2.6), we have𝑆(π‘„π‘Œ,𝑉)=2(π‘›βˆ’1)𝑆(π‘Œ,𝑉)βˆ’(π‘›βˆ’1)2𝑔(π‘Œ,𝑉).(4.8) This implies thatπ‘„π‘Œ=(π‘›βˆ’1)π‘Œ,(4.9) which gives𝑆(π‘Œ,𝑉)=(π‘›βˆ’1)𝑔(π‘Œ,𝑉).(4.10) Thus, we can state the following.

Theorem 4.1. An LP-Sasakian manifold 𝑀 satisfying the condition 𝑀(𝑋,π‘Œ)β‹…π‘Š2=0 is an Einstein manifold.

5. LP-Sasakian Manifolds Satisfying 𝐢(𝑋,π‘Œ)β‹…π‘Š2=0

The conformal curvature tensor 𝐢 is defined as [12]1𝐢(𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)π‘βˆ’[]+π‘Ÿπ‘›βˆ’2𝑆(π‘Œ,𝑍)π‘‹βˆ’π‘†(𝑋,𝑍)π‘Œ+𝑔(π‘Œ,𝑍)π‘„π‘‹βˆ’π‘”(𝑋,𝑍)π‘„π‘Œ([].π‘›βˆ’1)(π‘›βˆ’2)𝑔(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œ(5.1) Using (2.6) and (2.8), (5.1) reduces to𝐢(πœ‰,π‘Œ)𝑍=1βˆ’π‘›+π‘Ÿ[]βˆ’1(π‘›βˆ’1)(π‘›βˆ’2)𝑔(π‘Œ,𝑍)πœ‰βˆ’πœ‚(𝑍)π‘Œ[].π‘›βˆ’2𝑆(π‘Œ,𝑍)πœ‰βˆ’πœ‚(𝑍)π‘„π‘Œ(5.2) Now consider an LP-Sasakian manifold satisfying 𝐢(𝑋,π‘Œ)β‹…π‘Š2=0.(5.3) This condition implies that𝐢(𝑋,π‘Œ)π‘Š2(π‘ˆ,𝑉)π‘βˆ’π‘Š2(𝐢(𝑋,π‘Œ)π‘ˆ,𝑉)π‘βˆ’π‘Š2(π‘ˆ,𝐢(𝑋,π‘Œ)𝑉)π‘βˆ’π‘Š2(π‘ˆ,𝑉)𝐢(𝑋,π‘Œ)𝑍=0.(5.4) Putting 𝑋=πœ‰ in (5.4) and then taking the inner product with πœ‰, we obtain𝑔𝐢(πœ‰,π‘Œ)π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)𝑍,πœ‰βˆ’π‘”2ξ€Έξ€·π‘Š(𝐢(πœ‰,π‘Œ)π‘ˆ,𝑉)𝑍,πœ‰βˆ’π‘”2(ξ€Έξ€·π‘Šπ‘ˆ,𝐢(πœ‰,π‘Œ)𝑉)𝑍,πœ‰βˆ’π‘”2(ξ€Έπ‘ˆ,𝑉)𝐢(πœ‰,π‘Œ)𝑍,πœ‰=0.(5.5) Using (5.2) in (5.5), we obtain1βˆ’π‘›+π‘Ÿξ€Ίξ€·(π‘›βˆ’1)(π‘›βˆ’2)βˆ’π‘”π‘Œ,π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)π‘βˆ’π‘”(π‘Œ,π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(πœ‰,𝑉)π‘βˆ’π‘”(π‘Œ,𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,πœ‰)π‘βˆ’π‘”(π‘Œ,𝑍)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)πœ‰βˆ’πœ‚(π‘Œ)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)𝑍+πœ‚(π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(π‘Œ,𝑉)𝑍+πœ‚(𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,π‘Œ)𝑍+πœ‚(𝑍)πœ‚2+1(π‘ˆ,𝑉)π‘Œξ€Έξ€»ξ€Ίπ‘†ξ€·π‘›βˆ’2π‘Œ,π‘Š2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)𝑍+𝑆(π‘Œ,π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(πœ‰,𝑉)𝑍+𝑆(π‘Œ,𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,πœ‰)𝑍+𝑆(π‘Œ,𝑍)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)πœ‰+(π‘›βˆ’1)πœ‚(π‘Œ)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,𝑉)π‘βˆ’πœ‚(π‘ˆ)πœ‚2ξ€Έξ€·π‘Š(π‘„π‘Œ,𝑉)π‘βˆ’πœ‚(𝑉)πœ‚2ξ€Έξ€·π‘Š(π‘ˆ,π‘„π‘Œ)π‘βˆ’πœ‚(𝑍)πœ‚2(π‘ˆ,𝑉)π‘„π‘Œξ€Έξ€»=0.(5.6) By using (2.16) in (5.6), we getξ‚»1βˆ’π‘›+π‘Ÿξ‚Όπ‘”ξ€·(π‘›βˆ’1)(π‘›βˆ’2)π‘Œ,π‘Š2ξ€Έβˆ’1(π‘ˆ,𝑉)π‘π‘†ξ€·π‘›βˆ’2π‘Œ,π‘Š2ξ€Έ(π‘ˆ,𝑉)𝑍=0.(5.7) Taking π‘ˆ=𝑍=πœ‰ in (5.7) and then using (2.11) and (2.6), we have𝑆(π‘„π‘Œ,𝑉)=2+𝑛(π‘›βˆ’3)+π‘Ÿξ‚Όπ‘›βˆ’1𝑆(π‘Œ,𝑉)βˆ’(π‘›βˆ’1+π‘Ÿ)𝑔(π‘Œ,𝑉).(5.8) This implies thatπ‘„π‘Œ=(π‘›βˆ’1)π‘Œ,(5.9)and it follows that𝑆(π‘Œ,𝑉)=(π‘›βˆ’1)𝑔(π‘Œ,𝑉).(5.10) Thus, we can state the following.

Theorem 5.1. An LP-Sasakian manifold 𝑀 satisfying the condition 𝐢(𝑋,π‘Œ)β‹…π‘Š2=0 is an Einstein manifold.

Acknowledgment

The authors express their thanks to DST (Department of Science and Technology), Government of India for providing financial assistance under major research project.

References

  1. K. Matsumoto, β€œOn Lorentzian paracontact manifolds,” Bulletin of the Yamagata University. Natural Science, vol. 12, no. 2, pp. 151–156, 1989. View at Zentralblatt MATH
  2. I. Mihai and R. Roşca, β€œOn Lorentzian P-Sasakian manifolds,” in Classical Analysis (Kazimierz Dolny, 1991), pp. 155–169, World Scientific, River Edge, NJ, USA, 1992.
  3. K. Matsumoto and I. Mihai, β€œOn a certain transformation in a Lorentzian para-Sasakian manifold,” Tensor, vol. 47, no. 2, pp. 189–197, 1988. View at Zentralblatt MATH
  4. I. Mihai, A. A. Shaikh, and U. C. De, β€œOn Lorentzian para-Sasakian manifolds,” Rendiconti del Seminario Matematico di Messina, no. 3, pp. 149–158, 1999.
  5. Venkatesha and C. S. Bagewadi, β€œOn concircular Ο•-recurrent LP-Sasakian manifolds,” Differential Geometryβ€”Dynamical Systems, vol. 10, pp. 312–319, 2008.
  6. G. P. Pokhariyal and R. S. Mishra, β€œCurvature tensors' and their relativistics significance,” Yokohama Mathematical Journal, vol. 18, pp. 105–108, 1970. View at Zentralblatt MATH
  7. G. P. Pokhariyal, β€œStudy of a new curvature tensor in a Sasakian manifold,” Tensor, vol. 36, no. 2, pp. 222–226, 1982. View at Zentralblatt MATH
  8. K. Matsumoto, S. Ianuş, and I. Mihai, β€œOn P-Sasakian manifolds which admit certain tensor fields,” Publicationes Mathematicae Debrecen, vol. 33, no. 3-4, pp. 199–204, 1986.
  9. A. YΓ­ldΓ­z and U. C. De, β€œOn a type of Kenmotsu manifolds,” Differential Geometryβ€”Dynamical Systems, vol. 12, pp. 289–298, 2010. View at Zentralblatt MATH
  10. K. Yano and M. Kon, Structures on Manifolds, vol. 3 of Series in Pure Mathematics, World Scientific, Singapore, 1984.
  11. G. P. Pokhariyal and R. S. Mishra, β€œCurvature tensors and their relativistic significance. II,” Yokohama Mathematical Journal, vol. 19, no. 2, pp. 97–103, 1971. View at Zentralblatt MATH
  12. L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, NJ, USA, 1949.