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Kyungpook Mathematical Journal 2022; 62(3): 485-495

Published online September 30, 2022

Copyright © Kyungpook Mathematical Journal.

Generalized Quasi-Einstein Metrics and Contact Geometry

Gour Gopal Biswas, Uday Chand De*, Ahmet Yıldız

Department of Mathematics, University of Kalyani, Kalyani-741235, West Bengal, India
e-mail : ggbiswas6@gmail.com

Department of Pure Mathematics, University of Calcutta, 35 Ballygaunge Circular Road, Kolkata -700019, West Bengal, India
e-mail : uc_de@yahoo.com

Education Faculty, Department of Mathematics, Inonu University, 44280, Malatya, Turkey
e-mail : a.yildiz@inonu.edu.tr

Received: February 15, 2021; Revised: January 30, 2022; Accepted: February 10, 2022

The aim of this paper is to characterize K-contact and Sasakian manifolds whose metrics are generalized quasi-Einstein metric. It is proven that if the metric of a K-contact manifold is generalized quasi-Einstein metric, then the manifold is of constant scalar curvature and in the case of a Sasakian manifold the metric becomes Einstein under certain restriction on the potential function. Several corollaries have been provided. Finally, we consider Sasakian 3-manifold whose metric is generalized quasi-Einstein metric.

Keywords: GQE metrics, Almost contact manifolds, Contact manifolds, K-contact manifolds, Sasakian manifolds

If the Ricci tensor S of a Riemannian manifold (Mn,g), n>2, satisfies the condition Ric=λ g, λ being a constant, then the manifold is named an Einstein manifold. According to Besse [4] this condition is called Einstein metric condition. The study of Einstein manifolds and their generalizations are very interesting in Riemannian and semi-Riemannian geometry. There are several generalizations of Einstein metric such as quasi-Einstein metric [8], m-quasi-Einstein metric [9], (m,ρ)-quasi-Einstein metric [18], generalized quasi-Einstein metric [10] and many others.

The idea of generalized quasi-Einstein metric in a Riemannian manifold of dimension n is introduced by Catino [10]. A metric g of an n(>2) dimensional Riemannian manifold Mn is called a generalized quasi-Einstein metric (shortly, GQE metric) if there exist three smooth functions ψ,α,β such that

S+Hψαdψdψ=βg,

where Hψ is the Hessian of the function ψ defined by

Hψ(E,F)=g(Egradψ,F)

for all vector fields E,F in Mn. Here ∇ is the Riemannian connection and grad denotes the gradient operator. Obviously, when ψ is a constant, the metric becomes an Einstein metric.

For individual values of α and β, we get different type of metrics. They are

  • i) Gradient Ricci soliton [7] for α=0 and β,

  • ii) Gradient almost Ricci soliton [1] for α=0 and βC(Mn),

  • iii) Gradient ρ-Einstein soliton [11] for α=0, β=ρr+λ and λ, r being the scalar curvature,

  • iv) m-quasi-Einstein metric for α=1m,m and β,

  • v) gradient generalized m-quasi metric [2] for α=1m,m and βC(Mn),

  • vi) (m,ρ)-quasi-Einstein metric for α=1m, m>0, β=ρr+λ and λ.

The idea of a gradient ρ-Einstein soliton is introduced by Catino and Mazzieri [11] and studied in the papers ([12], [19]). In the paper [27], Venkatesha and Kumara studied gradient ρ-Einstein solitons on almost Kenmotsu manifolds. In [13], Chen studied m-quasi-Einstein structure in almost cosymplectic manifolds.

In the paper [10], Catino gave a local characterization of GQE metric with harmonic Weyl tensor and C(gradψ,,)=0, where C is the Weyl tensor. He proved that if the metric of a manifold (Mn,g),,n≥ 3 is a GQE metric with harmonic Weyl tensor and C(gradψ,,)=0, then M is locally warped product with (m-1)-dimensional Einstein fibers around any regular point of ψ. Recently, GQE manifolds have been studied by Mirshafeazadeh and Bidabad ([22], [23]). So far our knowledge goes, contact or paracontact manifolds whose metrics are GQE metric have not been investigated. In the present paper we attempt to characterize K-contact and Sasakian manifolds whose metrics are GQE metric.

At first we obtain the expression of Riemannian curvature tensor and Ricci tensor in a Riemannian manifold whose metric is GQE metric. Then we provide our main theorems. In the proof of the theorems we assume that the potential function ψ remains invariant under the characteristic vector field ξ, that is, £ξψ=0, which implies that ξψ=0, £ξ being the Lie-derivative in the direction of ξ. Precisely we prove the following theorems:

Theorem 1.1. The scalar curvature of a K-contact manifold with GQE metric is constant, provided the potential function ψ remains invariant under the characteristic vector field ξ.

Theorem 1.2. A Sasakian manifold with GQE metric is an Einstein manifold, provided the potential function ψ remains invariant under the Reeb vector field ξ.

Let M2n+1 be a smooth manifold. Let η be a 1-form, ξ be a vector field and φ be a (1,1)-tensor field. The triple (η,ξ,φ) is called an almost contact structure (acs) if

I=φ2+ηξ,η(ξ)=1,

where I is the identity map. Obviously φξ=0 and η°φ=0. The acs is called normal if the almost complex structure J on the product manifold M2n+1× defined by

JE,γddt=φEγξ,η(E)ddt

for all Eχ(M2n+1) and γC(M2n+1×), is integrable. Here χ(M2n+1) denotes the tangent space of M2n+1. Blair [5] proved that the acs is normal if and only if [φ,φ]+2ηξ=0, where [φ,φ] denotes the Nijenhuis tensor of φ defined by

[φ,φ](E,F)=φ2[E,F]+[φE,φF]φ[φE,F]φ[E,φF],E,Fχ(M2n+1).

If there exists a Riemannian metric g on M2n+1 such that

g=g(φ,φ)+ηη,

then the manifold M2n+1 together with (η,ξ,φ,g) is said to be an almost contact metric manifold (shortly, acm manifold). On acm manifold we can define the fundamental 2-form 𝚽 defined by Φ=g(,φ). When dη=Φ, the acm manifold is called a contact metric (cm) manifold. On a cm manifolds η(dη)n is a non-vanishing (2n+1)-form. Contact metric manifolds have been studied by several authors such as ([14], [15], [20], [24]-[26], [28]) and many others.

Given a cm manifold M2n+1 we can define a symmetric (1,1)-tensor field h=12£ξφ, where £ξ denotes the Lie derivative along the vector field ξ, which satisfy

hξ=0,hφ+φh=0
Eξ=φEφhE
(Eφ)F+(φEφ)φF=2g(E,F)ξη(F)(E+hE+η(E)ξ)

for all E,Fχ(M2n+1). We denote R for Riemannian curvature tensor and Q for Ricci operator defined by

R(E,F)=[E,F][E,F],
S(E,F)=g(QE,F).

According to Blair [5] h=0 if and only if the Reeb vector field ξ is Killing. If ξ is a Killing vector field, then the cm manifold M2n+1 is called K-contact manifold [5]. On a K-contact manifold M2n+1 the following relations hold:

Eξ=φE
Qξ=2nξ
R(ξ,E)F=(Eφ)F
(Eφ)F+(φEφ)φF=2g(E,F)ξη(F)(E+η(E)ξ)

for all E,Fχ(M2n+1). Taking covariant derivative of (2.8) along Eχ(M2n+1), we obtain

(EQ)ξ=QφE2nφE.

Since ξ is Killing, £ξQ=0. By direct computation

(ξQ)E=QφEφQE.

A normal cm manifold is said to be a Sasakian manifold. A necessary and sufficient condition for an acm manifold M2n+1 to be Sasakian is that

(Eφ)F=g(E,F)ξη(F)E

for all E,Fχ(M2n+1). A cm manifold is Sasakian if and only if

R(E,F)ξ=η(F)Eη(E)F.

Every Sasakian manifold is K-contact manifold, but the converse is not true, in general. However in 3-dimensional manifold K-contact and Sasakian manifolds are equivalent [21]. The relations (2.7)-(2.10) also hold in Sasakian manifolds. The Ricci tensor in a Sasakian 3-manifold is given by [6]

S=r22g+6r2ηη.

From the above we see that if r=6 then the manifold is an Einstein manifold and conversely. Since in a three dimensional manifold, Einstein and space of constant sectional curvature are equivalent, a Sasakian 3-manifold is of constant sectional curvature 1 if and only if r=6.

In this section we deduce the expression of R and S on a Riemannian manifold with GQE metric.

Proposition 3.1. In a Riemannian manifold (M2n+1,g) with GQE metric, the tensors R and S satisfy

R(E,F)gradψ=(FQ)E(EQ)F+(Eβ)F(Fβ)E      +{(Eα)(Fψ)(Fα)(Eψ)}gradψ      α{(Fψ)QE(Eψ)QF}+αβ{(Fψ)E(Eψ)F}

and

(1α)S(F,gradψ)=12(Fr)2n(Fβ)g(gradψ,gradψ)(Fα)        +{g(gradα,gradψ)αr+2nαβ}(Fψ)

for all E,Fχ(M2n+1).

Proof. From (1.1) it follows that

Fgradψ=QF+βF+αg(gradψ,F)gradψ.

where Hψ= Hessian of the function ψ is defined by

Hψ(E,F)=g(Egradψ,F)

for all vector fields E,F in M2n+1.

Taking covariant derivative of (3.3) in the direction Eχ(M2n+1), we obtain

EFgradψ=E(QF)+(Eβ)F+βEF+(Eα)g(gradψ,F)gradψ    +α(Eg(gradψ,F))gradψ+αg(gradψ,F)Egradψ.

Interchanging E and F in the foregoing equation, we have

FEgradψ=F(QE)+(Fβ)E+βFE+(Fα)g(gradψ,E)gradψ    +α(Fg(gradψ,E))gradψ+αg(gradψ,E)Fgradψ.

Using (3.3)-(3.5) in (2.6), we get (3.1). Contracting the equation (3.1) and applying the well known formulas Er=tr{F(EQ)F} and 12Er=divQ, we get the second result.

Proof of the Theorem 1.1. Replacing E by ξ in (3.1) and using (2.11) and (2.12), we have

R(ξ,F)gradψ=φQF2nφF+(ξβ)F(Fβ)ξ    +{(ξα)(Fψ)(Fα)(ξψ)}gradψ    α{2n(Fψ)ξ(ξψ)QF}+αβ{(Fψ)ξ(ξψ)F}.

Taking inner product of the foregoing equation with E and using (2.9), we infer

g((Fφ)E,gradψ)=g(φQF,E)2ng(E,φF)        +(ξβ)g(E,F)(Fβ)η(E)        +(ξα)(Eψ)(Fψ)(ξψ)(Eψ)(Fα)        α{2n(Fψ)η(E)(ξψ)g(QF,E)}        +αβ{(Fψ)η(E)(ξψ)g(E,F)}.

Replacing E by φE and F by φF in (4.2), entail that

g((φFφ)φE,gradψ)=g(QφF,E)2ng(E,φF)        +(ξβ)g(φE,φF)+(ξα)g(φE,gradψ)g(φF,gradψ)        (ξψ)g(φE,gradψ)g(φF,gradα)        +α(ξψ)g(QφF,φE)αβ(ξψ)g(φE,φF).

Adding (4.2) and (4.3) and using (2.10), we get

2g(E,F)(ξψ)+η(E)((Fψ)+η(F)(ξψ))=g(φQF+QφF,E)4ng(E,φF)+(ξβ)(g(E,F)+g(φE,φF))(Fβ)η(E)+(ξα)((Eψ)(Fψ)+g(φE,gradψ)g(φF,gradψ))(ξψ)((Eψ)(Fα)+g(φE,gradψ)g(φF,gradα))α{2n(Fψ)η(E)(ξψ)g(QF,E)}+αβ{(Fψ)η(E)(ξψ)g(E,F)}+α(ξψ)g(QφF,φE)αβ(ξψ)g(φE,φF).

Anti-symmetrizing the above equation, it follows that

(1+2nααβ)((Fψ)η(E)(Eψ)η(F))=2g(φQF+QφF,E)8ng(E,φF)+(Eβ)η(F)(Fβ)η(E)+(ξψ)((Eα)(Fψ)(Fα)(Eψ))+(ξψ)(g(φE,gradα)g(φF,gradψ)g(φF,gradα)g(φE,gradψ)).

Now we assume that the potential function ψ remains invariant under the characteristic vector field ξ, that is, ξ ψ=0. Then the equation (4.5) reduces to

(1+2nααβ)((Fψ)η(E)(Eψ)η(F))=2g(φQF+QφF,E)8ng(E,φF)+(Eβ)η(F)(Fβ)η(E).

Replacing E by φE and F by φF in the equation (4.6), we infer

φQF+QφF=4nφF

for all vector field F on M2n+1. Suppose {e1,e2,,en,φe1,φe2,,φen,ξ} is a φ-basis of (M2n+1,g).

Then g(φQei,φei)=g(Qei,ei) for i=1,2,,n. We compute

r=i=1ng(Qei,ei)+i=1ng(Qφei,φei)+g(Qξ,ξ)=i=1ng(φQei+Qφei,φei)+2n=2n(2n+1).

This finishes the proof.

Suppose dαdψ=0. Then

(Eα)(Fψ)(Fα)(Eψ)=0

for all E,Fχ(M2n+1), which implies (Eα)gradψ(Eψ)gradα=0, that is, gradα and gradψ are collinear. Conversely, if gradα and gradψ are collinear then dαdψ=0. Using (4.8) in (4.5), we get

(1+2nααβ)((Fψ)η(E)(Eψ)η(F))=2g(φQF+QφF,E)8ng(E,φF)+(Eβ)η(F)(Fβ)η(E).

Proceeding in the similar way as in the proof of Theorem 1.1, it follows that the manifold is of constant scalar curvature. Hence, we can state the following:

Corollary 4.1. The scalar curvature of a K-contact manifold with GQE metric is constant, provided gradα and gradψ are collinear.

Proof of the Theorem 1.2. Let (M2n+1,g) be a Sasakian manifold with GQE metric. In a Sasakian manifold the relation φQ=Qφ holds. Therefore ξQ=0. Using (2.13) in (4.2), we get

g(E,F)(ξψ)+(Fψ)η(E)=g(φQF,E)2ng(E,φF)          +(ξβ)g(E,F)(Fβ)η(E)          +(ξα)(Eψ)(Fψ)(ξψ)(Eψ)(Fα)          α{2n(Fψ)η(E)(ξψ)g(QF,E)}          +αβ{(Fψ)η(E)(ξψ)g(E,F)}.

Anti-symmetrizing the equation (4.9), we infer

(1+2nααβ)((Fψ)η(E)(Eψ)η(F))=2g(φQF,E)4ng(E,φF)+(Eβ)η(F)(Fβ)η(E)+(ξψ){(Eα)(Fψ)(Fα)(Eψ)}.

Replacing E by φE and F by φF in (4.10), we have

0=2g(φQF,E)4ng(E,φF)  +(ξψ){g(φE,gradα)g(φF,gradψ)g(φF,gradα)g(φE,gradψ)}.

Again replacing E by φE in (4.11) and applying (2.8), we obtain

S(E,F)=2ng(E,F)12(ξψ){g(φ2E,gradα)g(φF,gradψ)    g(φF,gradα)g(φ2E,gradψ)}

for all vector fields E,F on M2n+1. Contracting the equation, we get

r=2n(2n+1)+(ξψ)g(φ(gradα),gradψ).

Suppose that, the potential function ψ remains invariant under the characteristic vector field ξ, that is, ξ ψ=0. Then from (4.12), we see that S=2ng.

This finishes the proof.

If α=0 and β, from (4.12) we see that S=2ng and the manifold is an Einstein manifold. Also the equation (4.10) reduces to (Fψ)η(E)(Eψ)η(F)=0, that is gradψ=(ξψ)ξ. Now applying g(φEgradψ,φF)=g(φFgradψ,φE), we obtain (ξψ)g(E,φF)=0. This implies ξψ=0. Therefore gradψ=0, that is, ψ is constant. Thus, we can state that:

Corollary 4.2. A Sasakian manifold whose metric satisfies gradient Ricci soliton equation is an Einstein manifold and the potential function is constant.

The corollary 4.2 has been proved by He and Zhu [17].

If α=0 and βC(M), from the equation (4.12) we have S=2ng. Then the equation becomes (Fψ)η(E)(Eψ)η(F)=(Eβ)η(F)(Fβ)η(E). Thus for any Eξ, we have g(gradψ,E)=(Eβ). Since β is a non-zero function, ψ is non-constant. Also gradψ is not perpendicular to Eξ. Thus we get the following:

Corollary 4.3. A Sasakian manifold whose metric satisfies gradient almost Ricci soliton equation is an Einstein manifold. Moreover, neither ψ is a constant function nor gradψ is perpendicular to the vector field Eξ.

The second part of the Corollary 4.3 is also proved in the paper [3].

If α=1m,m and βC(M), from (4.12), we see that S=2ng. Thus, we can state that:

Corollary 4.4. A Sasakian manifold with m-quasi-Einstein metric is an Einstein manifold.

The above result has also been obtained in [16].

Now we consider GQE metric on Sasakian 3-manifold. Using (2.15) in (4.11), it follows that

0=(r6)g(E,φF)+(ξψ){g(φE,gradα)g(φF,gradψ)  g(φF,gradα)g(φE,gradψ)}.

If the potential function ψ remains invariant under the characteristic vector field ξ, from the above equation we have r=6. Thus, we can state that

Corollary 4.4. A Sasakian 3-manifold with GQE metric is a manifold of constant sectional curvature 1, provided the potential function ψ remains invariant under the Reeb vector field ξ.

Remark 1. It can be easily shown that in a 3-dimensional Sasakian manifold the φ-sectional curvature is equal to r42. Under the hypothesis of Corollary 4.5, we can prove that the scalar curvature of a 3-dimensional Sasakian manifold is constant. Therefore the φ-sectional curvature is constant and the manifold becomes a 3-dimensional Sasakian space form [5], provided the potential function remains invariant under the Reeb vector field ξ.

The authors would like to thank the referees and editor for reviewing the paper carefully and their valuable comments to improve the quality of the paper.

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