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.