Abstract

The object of the present paper is to study Lorentzian para-Sasakian (briefly LP-Sasakian) manifolds satisfying certain conditions on the š‘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 š‘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 YƬldƬz and De [9] have studied š‘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 š‘Š2=0 and š‘Š2-semisymmetric LP-Sasakian manifolds. Further, we study LP-Sasakian manifolds which satisfy š‘ƒā‹…š‘Š2=0, ī‚‹š‘€ā‹…š‘Š2=0 and š¶ā‹…š‘Š2=0, where š‘ƒ is the projective curvature tensor, ī‚‹š‘€ is the š‘€-projective curvature tensor, and š¶ is the conformal curvature tensor.

2. Preliminaries

An š‘›-dimensional differentiable manifold š‘€ is called an LP-Sasakian manifold [1, 2] if it admits a (1,1) tensor field šœ™, a contravariant vector field šœ‰, a 1-form šœ‚, and a Lorentzian metric š‘” which satisfyšœ™šœ‚(šœ‰)=āˆ’1,2āˆ‡š‘‹=š‘‹+šœ‚(š‘‹)šœ‰,š‘”(šœ™š‘‹,šœ™š‘Œ)=š‘”(š‘‹,š‘Œ)+šœ‚(š‘‹)šœ‚(š‘Œ),š‘”(š‘‹,šœ‰)=šœ‚(š‘‹),š‘‹ī€·āˆ‡šœ‰=šœ™š‘‹,š‘‹šœ™ī€øš‘Œ=š‘”(š‘‹,š‘Œ)šœ‰+šœ‚(š‘Œ)š‘‹+2šœ‚(š‘‹)šœ‚(š‘Œ)šœ‰,(2.1) where āˆ‡ denotes the operator of covariant differentiation with respect to the Lorentzian metric š‘”.

It can be easily seen that, in an LP-Sasakian manifold, the following relations hold:šœ™šœ‰=0,šœ‚(šœ™š‘‹)=0,rankšœ™=š‘›āˆ’1.(2.2) If we putĪ¦(š‘‹,š‘Œ)=š‘”(š‘‹,šœ™š‘Œ),(2.3) for any vector fields š‘‹ and š‘Œ, then the tensor field Ī¦(š‘‹,š‘Œ) is a symmetric (0, 2) tensor field [1]. Also, since the 1-form šœ‚ is closed in an LP-Sasakian manifold, we have [1, 4]ī€·āˆ‡š‘‹šœ‚ī€ø(š‘Œ)=Ī¦(š‘‹,š‘Œ),Ī¦(š‘‹,šœ‰)=0,(2.4) for any vector fields š‘‹ and š‘Œ.

Also in an LP-Sasakian manifold, the following relations hold [3, 4]:š‘”(š‘…(š‘‹,š‘Œ)š‘,šœ‰)=šœ‚(š‘…(š‘‹,š‘Œ)š‘)=š‘”(š‘Œ,š‘)šœ‚(š‘‹)āˆ’š‘”(š‘‹,š‘)šœ‚(š‘Œ),(2.5)š‘…(šœ‰,š‘‹)š‘Œ=š‘”(š‘‹,š‘Œ)šœ‰āˆ’šœ‚(š‘Œ)š‘‹,(2.6)š‘…(š‘‹,š‘Œ)šœ‰=šœ‚(š‘Œ)š‘‹āˆ’šœ‚(š‘‹)š‘Œ,(2.7)š‘†(š‘‹,šœ‰)=(š‘›āˆ’1)šœ‚(š‘‹),(2.8)š‘†(šœ™š‘‹,šœ™š‘Œ)=š‘†(š‘‹,š‘Œ)+(š‘›āˆ’1)šœ‚(š‘‹)šœ‚(š‘Œ),(2.9) for any vector fields š‘‹,š‘Œ, and š‘, where š‘… is the Riemannian curvature tensor and š‘† is the Ricci tensor of š‘€.

An LP-Sasakian manifold š‘€ is said to be Einstein if its Ricci tensor š‘† is of the formš‘†(š‘‹,š‘Œ)=š‘Žš‘”(š‘‹,š‘Œ),(2.10) for any vector fields š‘‹ and š‘Œ, where š‘Ž is a function on š‘€.

In [6], Pokhariyal and Mishra have defined the curvature tensor š‘Š2, given byš‘Š21(š‘‹,š‘Œ,š‘ˆ,š‘‰)=š‘…(š‘‹,š‘Œ,š‘ˆ,š‘‰)+[],š‘›āˆ’1š‘”(š‘‹,š‘ˆ)š‘†(š‘Œ,š‘‰)āˆ’š‘”(š‘Œ,š‘ˆ)š‘†(š‘‹,š‘‰)(2.11) where š‘† is a Ricci tensor of type (0, 2).

Now, consider an LP-Sasakian manifold satisfying š‘Š2=0; then, (2.11) becomes1š‘…(š‘‹,š‘Œ,š‘ˆ,š‘‰)=[].š‘›āˆ’1š‘”(š‘Œ,š‘ˆ)š‘†(š‘‹,š‘‰)āˆ’š‘”(š‘‹,š‘ˆ)š‘†(š‘Œ,š‘‰)(2.12) Taking š‘‹=š‘ˆ=šœ‰ in (2.12) and using (2.7) and (2.8), we haveš‘†(š‘Œ,š‘‰)=(š‘›āˆ’1)š‘”(š‘Œ,š‘‰).(2.13) Therefore, š‘€ is an Einstein manifold.

Again using (2.13) in (2.12), we get[].š‘…(š‘‹,š‘Œ,š‘ˆ,š‘‰)=š‘”(š‘Œ,š‘ˆ)š‘”(š‘‹,š‘‰)āˆ’š‘”(š‘‹,š‘ˆ)š‘”(š‘Œ,š‘‰)(2.14)

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

Definition 2.2. An LP-Sasakian manifold is called š‘Š2-semisymmetric if it satisfies š‘…(š‘‹,š‘Œ)ā‹…š‘Š2=0,(2.15) where š‘…(š‘‹,š‘Œ) is to be considered as a derivation of the tensor algebra at each point of the manifold for tangent vectors š‘‹ and š‘Œ.
It can be easily shown that in an LP-Sasakian manifold the š‘Š2-curvature tensor satisfies the condition šœ‚ī€·š‘Š2ī€ø(š‘‹,š‘Œ)š‘=0.(2.16)

Theorem 2.3. A š‘Š2-semisymmetric LP-Sasakian manifold š‘€ is an Einstein manifold.

Proof. Since š‘…(š‘‹,š‘Œ)ā‹…š‘Š2=0, we have š‘…(š‘‹,š‘Œ)š‘Š2(š‘ˆ,š‘‰)š‘āˆ’š‘Š2(š‘…(š‘‹,š‘Œ)š‘ˆ,š‘‰)š‘āˆ’š‘Š2(š‘ˆ,š‘…(š‘‹,š‘Œ)š‘‰)š‘āˆ’š‘Š2(š‘ˆ,š‘‰)š‘…(š‘‹,š‘Œ)š‘=0.(2.17) Putting š‘‹=šœ‰ in (2.17) and then taking the inner product with šœ‰, we obtain š‘”ī€·š‘…(šœ‰,š‘Œ)š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘,šœ‰āˆ’š‘”2ī€øī€·š‘Š(š‘…(šœ‰,š‘Œ)š‘ˆ,š‘‰)š‘,šœ‰āˆ’š‘”2(ī€øī€·š‘Šš‘ˆ,š‘…(šœ‰,š‘Œ)š‘‰)š‘,šœ‰āˆ’š‘”2(ī€øš‘ˆ,š‘‰)š‘…(šœ‰,š‘Œ)š‘,šœ‰=0.(2.18) Using (2.6) in (2.18), we obtain ī€·āˆ’š‘”š‘Œ,š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘āˆ’šœ‚2ī€øšœ‚ī€·š‘Š(š‘ˆ,š‘‰)š‘(š‘Œ)āˆ’š‘”(š‘Œ,š‘ˆ)šœ‚2ī€øī€·š‘Š(šœ‰,š‘‰)š‘āˆ’š‘”(š‘Œ,š‘‰)šœ‚2(ī€øī€·š‘Šš‘ˆ,šœ‰)š‘āˆ’š‘”(š‘Œ,š‘)šœ‚2(ī€øī€·š‘Šš‘ˆ,š‘‰)šœ‰+šœ‚(š‘ˆ)šœ‚2(ī€øī€·š‘Šš‘Œ,š‘‰)š‘+šœ‚(š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘Œ)š‘+šœ‚(š‘)šœ‚2ī€ø(š‘ˆ,š‘‰)š‘Œ=0.(2.19) By using (2.16) in (2.19), we get š‘Š2(š‘ˆ,š‘‰,š‘,š‘Œ)=0.(2.20) In view of (2.11) and (2.20), it follows that 1š‘…(š‘ˆ,š‘‰,š‘,š‘Œ)=[].š‘›āˆ’1š‘”(š‘‰,š‘)š‘†(š‘ˆ,š‘Œ)āˆ’š‘”(š‘ˆ,š‘)š‘†(š‘‰,š‘Œ)(2.21) Contracting (2.21), we have š‘†(š‘‰,š‘)=(š‘›āˆ’1)š‘”(š‘‰,š‘).(2.22) Therefore, š‘€ is an Einstein manifold.

Again using (2.22) in (2.12), we get[].š‘…(š‘ˆ,š‘‰,š‘,š‘Œ)=š‘”(š‘‰,š‘)š‘”(š‘ˆ,š‘Œ)āˆ’š‘”(š‘ˆ,š‘)š‘”(š‘‰,š‘Œ)(2.23)

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

3. LP-Sasakian Manifolds Satisfying š‘ƒ(š‘‹,š‘Œ)ā‹…š‘Š2=0

The projective curvature tensor š‘ƒ is defined as [10]1š‘ƒ(š‘‹,š‘Œ)š‘=š‘…(š‘‹,š‘Œ)š‘āˆ’[].š‘›āˆ’1š‘†(š‘Œ,š‘)š‘‹āˆ’š‘†(š‘‹,š‘)š‘Œ(3.1) Using (2.6) and (2.8), (3.1) reduces to1š‘ƒ(šœ‰,š‘Œ)š‘=š‘”(š‘Œ,š‘)šœ‰āˆ’š‘›āˆ’1š‘†(š‘Œ,š‘)šœ‰.(3.2) Let us suppose that in an LP-Sasakian manifold š‘ƒ(š‘‹,š‘Œ)ā‹…š‘Š2=0.(3.3) This condition implies thatš‘ƒ(š‘‹,š‘Œ)š‘Š2(š‘ˆ,š‘‰)š‘āˆ’š‘Š2(š‘ƒ(š‘‹,š‘Œ)š‘ˆ,š‘‰)š‘āˆ’š‘Š2(š‘ˆ,š‘ƒ(š‘‹,š‘Œ)š‘‰)š‘āˆ’š‘Š2(š‘ˆ,š‘‰)š‘ƒ(š‘‹,š‘Œ)š‘=0.(3.4) Putting š‘‹=šœ‰ in (3.4) and then taking the inner product with šœ‰, we obtainš‘”ī€·š‘ƒ(šœ‰,š‘Œ)š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘,šœ‰āˆ’š‘”2ī€øī€·š‘Š(š‘ƒ(šœ‰,š‘Œ)š‘ˆ,š‘‰)š‘,šœ‰āˆ’š‘”2(ī€øī€·š‘Šš‘ˆ,š‘ƒ(šœ‰,š‘Œ)š‘‰)š‘,šœ‰āˆ’š‘”2(ī€øš‘ˆ,š‘‰)š‘ƒ(šœ‰,š‘Œ)š‘,šœ‰=0.(3.5) Using (3.2) in (3.5), we obtainī€·āˆ’š‘”š‘Œ,š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘āˆ’š‘”(š‘Œ,š‘ˆ)šœ‚2ī€øī€·š‘Š(šœ‰,š‘‰)š‘āˆ’š‘”(š‘Œ,š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,šœ‰)š‘āˆ’š‘”(š‘Œ,š‘)šœ‚2ī€ø+1(š‘ˆ,š‘‰)šœ‰ī€ŗš‘†ī€·š‘›āˆ’1š‘Œ,š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘+š‘†(š‘Œ,š‘ˆ)šœ‚2ī€øī€·š‘Š(šœ‰,š‘‰)š‘+š‘†(š‘Œ,š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,šœ‰)š‘+š‘†(š‘Œ,š‘)šœ‚2(š‘ˆ,š‘‰)šœ‰ī€øī€»=0.(3.6) By using (2.16) in (3.6), we getš‘”ī€·š‘Œ,š‘Š2ī€øāˆ’1(š‘ˆ,š‘‰)š‘š‘†ī€·š‘›āˆ’1š‘Œ,š‘Š2ī€ø(š‘ˆ,š‘‰)š‘=0.(3.7) Taking š‘ˆ=š‘=šœ‰ in (3.7) and using (2.11) and (2.6), we haveš‘†(š‘„š‘Œ,š‘‰)=2(š‘›āˆ’1)š‘†(š‘Œ,š‘‰)āˆ’(š‘›āˆ’1)2š‘”(š‘Œ,š‘‰).(3.8) This implies thatš‘„š‘Œ=(š‘›āˆ’1)š‘Œ.(3.9) From this, we getš‘†(š‘Œ,š‘‰)=(š‘›āˆ’1)š‘”(š‘Œ,š‘‰).(3.10) Thus, we can state the following.

Theorem 3.1. An LP-Sasakian manifold š‘€ satisfying the condition š‘ƒ(š‘‹,š‘Œ)ā‹…š‘Š2=0 is an Einstein manifold.

4. LP-Sasakian Manifold Satisfying ī‚‹š‘€(š‘‹,š‘Œ)ā‹…š‘Š2=0

The š‘€-projective curvature tensor ī‚‹š‘€ is defined as [11]ī‚‹1š‘€(š‘‹,š‘Œ)š‘=š‘…(š‘‹,š‘Œ)š‘āˆ’2[].(š‘›āˆ’1)š‘†(š‘Œ,š‘)š‘‹āˆ’š‘†(š‘‹,š‘)š‘Œ+š‘”(š‘Œ,š‘)š‘„š‘‹āˆ’š‘”(š‘‹,š‘)š‘„š‘Œ(4.1) Using (2.6) and (2.8), (4.1) reduces toī‚‹1š‘€(šœ‰,š‘Œ)š‘=2[]āˆ’1š‘”(š‘Œ,š‘)šœ‰āˆ’šœ‚(š‘)š‘Œ2[].(š‘›āˆ’1)š‘†(š‘Œ,š‘)šœ‰āˆ’šœ‚(š‘)š‘„š‘Œ(4.2) Suppose that in an LP-Sasakian manifold ī‚‹š‘€(š‘‹,š‘Œ)ā‹…š‘Š2=0.(4.3) This condition implies thatī‚‹š‘€(š‘‹,š‘Œ)š‘Š2(š‘ˆ,š‘‰)š‘āˆ’š‘Š2ī‚€ī‚‹ī‚š‘š‘€(š‘‹,š‘Œ)š‘ˆ,š‘‰āˆ’š‘Š2ī‚€ī‚‹ī‚š‘ˆ,š‘€(š‘‹,š‘Œ)š‘‰š‘āˆ’š‘Š2ī‚‹(š‘ˆ,š‘‰)š‘€(š‘‹,š‘Œ)š‘=0.(4.4) Putting š‘‹=šœ‰ in (4.4) and then taking the inner product with šœ‰, we obtainš‘”ī‚€ī‚‹š‘€(šœ‰,š‘Œ)š‘Š2ī‚ī‚€š‘Š(š‘ˆ,š‘‰)š‘,šœ‰āˆ’š‘”2ī‚€ī‚‹ī‚ī‚ī‚€š‘Šš‘€(šœ‰,š‘Œ)š‘ˆ,š‘‰š‘,šœ‰āˆ’š‘”2ī‚€ī‚‹ī‚ī‚ī‚€š‘Šš‘ˆ,š‘€(šœ‰,š‘Œ)š‘‰š‘,šœ‰āˆ’š‘”2ī‚‹ī‚(š‘ˆ,š‘‰)š‘€(šœ‰,š‘Œ)š‘,šœ‰=0.(4.5) Using (4.2) in (4.5), we obtain12ī€ŗī€·āˆ’š‘”š‘Œ,š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘āˆ’š‘”(š‘Œ,š‘ˆ)šœ‚2ī€øī€·š‘Š(šœ‰,š‘‰)š‘āˆ’š‘”(š‘Œ,š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,šœ‰)š‘āˆ’š‘”(š‘Œ,š‘)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘‰)šœ‰+šœ‚(š‘ˆ)šœ‚2ī€øī€·š‘Š(š‘Œ,š‘‰)š‘+šœ‚(š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘Œ)š‘+šœ‚(š‘)šœ‚2+1(š‘ˆ,š‘‰)š‘Œī€øī€»ī€ŗš‘†ī€·2(š‘›āˆ’1)š‘Œ,š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘+š‘†(š‘Œ,š‘ˆ)šœ‚2ī€øī€·š‘Š(šœ‰,š‘‰)š‘+š‘†(š‘Œ,š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,šœ‰)š‘+š‘†(š‘Œ,š‘)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘‰)šœ‰āˆ’šœ‚(š‘ˆ)šœ‚2ī€øī€·š‘Š(š‘„š‘Œ,š‘‰)š‘āˆ’šœ‚(š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘„š‘Œ)š‘āˆ’šœ‚(š‘)šœ‚2(š‘ˆ,š‘‰)š‘„š‘Œī€øī€»=0.(4.6) By using (2.16) in (4.6), we get12š‘”ī€·š‘Œ,š‘Š2ī€øāˆ’1(š‘ˆ,š‘‰)š‘2š‘†ī€·(š‘›āˆ’1)š‘Œ,š‘Š2ī€ø(š‘ˆ,š‘‰)š‘=0.(4.7) Taking š‘ˆ=š‘=šœ‰ in (4.7) and using (2.11) and (2.6), we haveš‘†(š‘„š‘Œ,š‘‰)=2(š‘›āˆ’1)š‘†(š‘Œ,š‘‰)āˆ’(š‘›āˆ’1)2š‘”(š‘Œ,š‘‰).(4.8) This implies thatš‘„š‘Œ=(š‘›āˆ’1)š‘Œ,(4.9) which givesš‘†(š‘Œ,š‘‰)=(š‘›āˆ’1)š‘”(š‘Œ,š‘‰).(4.10) Thus, we can state the following.

Theorem 4.1. An LP-Sasakian manifold š‘€ satisfying the condition ī‚‹š‘€(š‘‹,š‘Œ)ā‹…š‘Š2=0 is an Einstein manifold.

5. LP-Sasakian Manifolds Satisfying š¶(š‘‹,š‘Œ)ā‹…š‘Š2=0

The conformal curvature tensor š¶ is defined as [12]1š¶(š‘‹,š‘Œ)š‘=š‘…(š‘‹,š‘Œ)š‘āˆ’[]+š‘Ÿš‘›āˆ’2š‘†(š‘Œ,š‘)š‘‹āˆ’š‘†(š‘‹,š‘)š‘Œ+š‘”(š‘Œ,š‘)š‘„š‘‹āˆ’š‘”(š‘‹,š‘)š‘„š‘Œ([].š‘›āˆ’1)(š‘›āˆ’2)š‘”(š‘Œ,š‘)š‘‹āˆ’š‘”(š‘‹,š‘)š‘Œ(5.1) Using (2.6) and (2.8), (5.1) reduces toš¶(šœ‰,š‘Œ)š‘=1āˆ’š‘›+š‘Ÿ[]āˆ’1(š‘›āˆ’1)(š‘›āˆ’2)š‘”(š‘Œ,š‘)šœ‰āˆ’šœ‚(š‘)š‘Œ[].š‘›āˆ’2š‘†(š‘Œ,š‘)šœ‰āˆ’šœ‚(š‘)š‘„š‘Œ(5.2) Now consider an LP-Sasakian manifold satisfying š¶(š‘‹,š‘Œ)ā‹…š‘Š2=0.(5.3) This condition implies thatš¶(š‘‹,š‘Œ)š‘Š2(š‘ˆ,š‘‰)š‘āˆ’š‘Š2(š¶(š‘‹,š‘Œ)š‘ˆ,š‘‰)š‘āˆ’š‘Š2(š‘ˆ,š¶(š‘‹,š‘Œ)š‘‰)š‘āˆ’š‘Š2(š‘ˆ,š‘‰)š¶(š‘‹,š‘Œ)š‘=0.(5.4) Putting š‘‹=šœ‰ in (5.4) and then taking the inner product with šœ‰, we obtainš‘”ī€·š¶(šœ‰,š‘Œ)š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘,šœ‰āˆ’š‘”2ī€øī€·š‘Š(š¶(šœ‰,š‘Œ)š‘ˆ,š‘‰)š‘,šœ‰āˆ’š‘”2(ī€øī€·š‘Šš‘ˆ,š¶(šœ‰,š‘Œ)š‘‰)š‘,šœ‰āˆ’š‘”2(ī€øš‘ˆ,š‘‰)š¶(šœ‰,š‘Œ)š‘,šœ‰=0.(5.5) Using (5.2) in (5.5), we obtain1āˆ’š‘›+š‘Ÿī€ŗī€·(š‘›āˆ’1)(š‘›āˆ’2)āˆ’š‘”š‘Œ,š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘āˆ’š‘”(š‘Œ,š‘ˆ)šœ‚2ī€øī€·š‘Š(šœ‰,š‘‰)š‘āˆ’š‘”(š‘Œ,š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,šœ‰)š‘āˆ’š‘”(š‘Œ,š‘)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘‰)šœ‰āˆ’šœ‚(š‘Œ)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘+šœ‚(š‘ˆ)šœ‚2ī€øī€·š‘Š(š‘Œ,š‘‰)š‘+šœ‚(š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘Œ)š‘+šœ‚(š‘)šœ‚2+1(š‘ˆ,š‘‰)š‘Œī€øī€»ī€ŗš‘†ī€·š‘›āˆ’2š‘Œ,š‘Š2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘+š‘†(š‘Œ,š‘ˆ)šœ‚2ī€øī€·š‘Š(šœ‰,š‘‰)š‘+š‘†(š‘Œ,š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,šœ‰)š‘+š‘†(š‘Œ,š‘)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘‰)šœ‰+(š‘›āˆ’1)šœ‚(š‘Œ)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘‰)š‘āˆ’šœ‚(š‘ˆ)šœ‚2ī€øī€·š‘Š(š‘„š‘Œ,š‘‰)š‘āˆ’šœ‚(š‘‰)šœ‚2ī€øī€·š‘Š(š‘ˆ,š‘„š‘Œ)š‘āˆ’šœ‚(š‘)šœ‚2(š‘ˆ,š‘‰)š‘„š‘Œī€øī€»=0.(5.6) By using (2.16) in (5.6), we getī‚»1āˆ’š‘›+š‘Ÿī‚¼š‘”ī€·(š‘›āˆ’1)(š‘›āˆ’2)š‘Œ,š‘Š2ī€øāˆ’1(š‘ˆ,š‘‰)š‘š‘†ī€·š‘›āˆ’2š‘Œ,š‘Š2ī€ø(š‘ˆ,š‘‰)š‘=0.(5.7) Taking š‘ˆ=š‘=šœ‰ in (5.7) and then using (2.11) and (2.6), we haveī‚»š‘†(š‘„š‘Œ,š‘‰)=2+š‘›(š‘›āˆ’3)+š‘Ÿī‚¼š‘›āˆ’1š‘†(š‘Œ,š‘‰)āˆ’(š‘›āˆ’1+š‘Ÿ)š‘”(š‘Œ,š‘‰).(5.8) This implies thatš‘„š‘Œ=(š‘›āˆ’1)š‘Œ,(5.9)and it follows thatš‘†(š‘Œ,š‘‰)=(š‘›āˆ’1)š‘”(š‘Œ,š‘‰).(5.10) Thus, we can state the following.

Theorem 5.1. An LP-Sasakian manifold š‘€ satisfying the condition š¶(š‘‹,š‘Œ)ā‹…š‘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.