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Geometry of para-Sasakian metric as an almost conformal η\eta-Ricci soliton

Sumanjit Sarkar Department of Mathematics
Jadavpur University
Kolkata-700032, India. [email protected]    Santu Dey Department of Mathematics
Bidhan Chandra College
Asansol, Burdwan, West Bengal-713304, India. [email protected]    Arindam Bhattacharyya Department of Mathematics
Jadavpur University
Kolkata-700032, India [email protected]
Abstract.

In this paper, we initiate the study of conformal η\eta-Ricci soliton and almost conformal η\eta-Ricci soliton within the framework of para-Sasakian manifold. We prove that if para-Sasakian mteric admits conformal η\eta-Ricci soliton, then the manifold is η\eta-Einstein and either the soliton vector field VV is Killing or it leaves ϕ\phi invariant. Here, we have shown the characteristics of the soliton vector field VV and scalar curvature when the manifold admitting conformal η\eta-Ricci soliton and vector field is pointwise collinear with the characteristic vector field ξ\xi. Next, we show that a para-Sasakian metric endowed an almost conformal η\eta-Ricci soliton is η\eta-Einstein metric if the soliton vector field VV is an infnitesimal contact transformation. We have also displayed that the manifold is Einstein if it represents a gradient almost conformal η\eta-Ricci soliton. We have developed an example to display the alive of conformal η\eta-Ricci soliton on 3-dimensional para-Sasakian manifold.

Key words and phrases:
Ricci flow, conformal η\eta-Ricci soliton, almost conformal η\eta-Ricci soliton, gradient almost conformal η\eta-Ricci soliton, para-Sasakian manifold.
1991 Mathematics Subject Classification:
53C15, 53C21, 53C25, 53C44

1. Introduction

In modern mathematics, the methods of paracontact geometry play an important role. The notion of almost paracontact manifold was first introduced by Sato [19]. After that he and Matsumoto [20] defined and studied a para-Sasakian manifold as special case of an almost paracontact manifold. Adati et al. [1] deduced some fundamental properties of para-Sasakian manifold. Later Kaneyuki and Williams [10] associated pseudo-Riemannian metric with an almost paracontact manifold after Takahashi [22] intoduced pseudo- Riemannian metric in contact manifold, in particular, in Sasakian manifold. Zamkovoy in [25] proved that any almost paracontact structure admits a pseudo-Riemannian metric with signature (n+1,n)(n+1,n). Para-Sasakian manifold (in short p-Sasakian manifold) was studied by many authors, namely: Calvaruso [4], Cappelletti et al. [6], Tripathi et al. [23] and many others.

A pseudo-Riemannian manifold (M,g)(M,g) admits a Ricci soliton which is a generalization of Einstein metric (i.e, S=a​gS=ag for some constant aa) if there exists a smooth non-zero vector field VV and a constant λ\lambda such that,

12​ℒV​g+S+λ​g=0,\frac{1}{2}\mathcal{L}_{V}g+S+\lambda g=0,

where ℒV\mathcal{L}_{V} denotes Lie derivative along the direction VV and SS denotes the Ricci curvature tensor of the manifold. The vector field VV is called potential vector field and λ\lambda is called soliton constant.

The Ricci soliton is a self-similar solution of the Hamilton’s Ricci flow [9] which is defined by the equation ∂g​(t)∂t=−2​S​(g​(t))\frac{\partial g(t)}{\partial t}=-2S(g(t)) with initial condition g​(0)=gg(0)=g, where g​(t)g(t) is a one-parameter family of metrices on MM. The potential vector field VV and soliton constant λ\lambda play vital roles while determining the nature of the soliton. A soliton is said to be shrinking, steady or expanding according as λ<0\lambda<0, λ=0\lambda=0 or λ>0\lambda>0. Now if VV is zero or Killing then the Ricci soliton reduces to Einstein manifold and the soliton is called trivial soliton.

If the potential vector field VV is the gradient of a smooth function ff, denoted by D​fDf then the soliton equation reduces to,

H​e​s​s​f+S+λ​g=0,Hessf+S+\lambda g=0,

where H​e​s​s​fHessf is Hessian of the smooth function ff. Perelman [13] proved that a Ricci soliton on a compact manifold is a gradient Ricci soliton.

In 2005, Fischer [8] has introduced conformal Ricci flow which is a mere generalisation of the classical Ricci flow equation that modifies the unit volume constraint to a scalar curvature constraint. The conformal Ricci flow equation was given by,

∂g∂t+2​(S+gn)\displaystyle\frac{\partial g}{\partial t}+2(S+\frac{g}{n}) =\displaystyle= −p​g,\displaystyle-pg,
r​(g)\displaystyle r(g) =\displaystyle= −1,\displaystyle-1,

where r​(g)r(g) is the scalar curvature of the manifold, pp is scalar non-dynamical field and nn is the dimension of the manifold. Corresponding to the conformal Ricci flow equation in 2015, Basu and Bhattacharyya [2] introduced the notion of conformal Ricci soliton equation as a generalization of Ricci soliton equation given by,

ℒV​g+2​S+[2​λ−(p+2n)]​g=0.\mathcal{L}_{V}g+2S+[2\lambda-(p+\frac{2}{n})]g=0.

In 2009, Cho and Kimura [7] introduced the concept of η\eta-Ricci soliton which is another generalization of classical Ricci soliton and is given by,

ℒξ​g+2​S+2​λ​g+2​μ​η⊗η=0,\mathcal{L}_{\xi}g+2S+2\lambda g+2\mu\eta\otimes\eta=0,

where μ\mu is a real constant, η\eta is a 1-form defined as η​(X)=g​(X,ξ)\eta(X)=g(X,\xi) for any X∈χ​(M)X\in\chi(M). Clearly it can be noted that if μ=0\mu=0 then the η\eta-Ricci soliton reduces to Ricci soliton.

Recently Siddiqi [21] established the notion of conformal η\eta-Ricci soliton which generalizes both conformal Ricci soliton and η\eta-Ricci soliton. The equation for conformal η\eta-Ricci soliton is given by,

ℒV​g+2​S+[2​λ−(p+2n)]​g+2​μ​η⊗η=0.\mathcal{L}_{V}g+2S+[2\lambda-(p+\frac{2}{n})]g+2\mu\eta\otimes\eta=0. (1.1)

In the foregoing equation if we consider the soliton vector field as a gradient of a smooth function ff and μ\mu as a smooth function then the soliton equation changes to

H​e​s​s​f+S+[λ−(p2+1(2​n+1))]​g+μ​η⊗η=0,Hessf+S+[\lambda-(\frac{p}{2}+\frac{1}{(2n+1)})]g+\mu\eta\otimes\eta=0, (1.2)

and the soliton is called gradient almost conformal η\eta-Ricci soliton.
As follows in the literature, Ricci soliton on paracontact geometry studied by many authors ([3], [5], [15]). In particular, Calvaruso and Perrone [5] explicitly studied Ricci soliton on 3-dimensional almost paracontact manifolds. In 2019, Patra [12] studied Ricci soliton on paracontact metric manifolds and proved that if a para-Sasakian manifold satisfy Ricci soliton equation then the manifold is either Einstein or η\eta-Einstein. The case of η\eta-Ricci soliton in para-Sasakian manifold was treated by Naik and Venkatesha in [11] and showed that if the metric of a para-Sasakian manifold represents a η\eta-Ricci soliton then the manifold is either Einstein or D-homothetically fixed η\eta-Einstein manifold. Very recently conformal η\eta-Ricci soliton and its generalizations have been studied by [16, 17, 18] and they have obtained some beautiful results.

Motivated by above mentioned works, in this paper, we consider conformal η\eta-Ricci soliton and gradient almost conformal η\eta-Ricci soliton in the framework of para-Sasakian manifold. We have organized this paper as follows: in first section we look back on some elementary properties of para-Sasakian manifolds; in later section first we prove that if a para-Sasakian manifold satisfies conformal η\eta-Ricci soliton then the manifold is η\eta-Einstein and either the soliton vector field VV is Killing or it leaves ϕ\phi invariant, secondly we prove that if VV is pointwise collinear with the characteristic vector field ξ\xi then VV is a constant multiple of ξ\xi. In the next section, we examine para-Sasakian manifold of dimension greater than 3 with an almost conformal η\eta Ricci soliton and show that the manifold is η\eta-Einstein. Then, we think about a gradient almost conformal η\eta Ricci soliton and deduce that the manifold is Einstein and finally, we provide some examples to verify our results.

2. Notes on para-Sasakian manifold

A (2​n+1)(2n+1)-dimensional smooth manifold MM is said to have an almost paracontact structure if it admits a vector field ξ\xi, (1,1)-tensor field ϕ\phi and a 1-form η\eta satisfying the following conditions

i)ϕ2=I−η⊗ξ,\displaystyle i)\phi^{2}=I-\eta\otimes\xi, (2.1)
ii)η(ξ)=1,\displaystyle ii)\eta(\xi)=1, (2.2)

iii)iii) ϕ\phi induces on the 2n-dimensional distribution 𝒟≡k​e​r​(η)\mathcal{D}\equiv ker(\eta), an almost paracomplex structure 𝒫\mathcal{P} i.e., 𝒫2≡Iχ​(M)\mathcal{P}^{2}\equiv I_{\chi(M)} and the eigensubbundles 𝒟+\mathcal{D}^{+} and 𝒟−\mathcal{D}^{-}, corresponding to the eigenvalues 11, −1-1 of 𝒫\mathcal{P} respectively, have equal dimension nn; hence 𝒟=𝒟+⊕𝒟−\mathcal{D}=\mathcal{D}^{+}\oplus\mathcal{D}^{-}.
The vector field ξ\xi is called characteristic vector field or Reeb vector field. An immediate consequence of those relations are

ϕ​ξ\displaystyle\phi\xi =\displaystyle= 0\displaystyle 0 (2.3)
η∘ϕ\displaystyle\eta\circ\phi =\displaystyle= 0.\displaystyle 0. (2.4)

The tensor field ϕ\phi induces an almost paracomplex structure on each fibre of K​e​r​(η)Ker(\eta) i.e., the eigendistributions corresponding to eigenvalues 11 and −1-1 have same dimension nn. A pseudo-Riemannian metric gg is said to be compatible with the almost paracontact structure if

g​(ϕ​X,ϕ​Y)=−g​(X,Y)+η​(X)​η​(Y)g(\phi X,\phi Y)=-g(X,Y)+\eta(X)\eta(Y) (2.5)

holds for arbitrary vector fields XX and YY and (M,ϕ,ξ,η,g)(M,\phi,\xi,\eta,g) is called an almost paracontact metric manifold.

On an almost paracontact metric manifold fundamental 2-form Φ\Phi is defined by Φ​(X,Y)=g​(X,ϕ​Y)\Phi(X,Y)=g(X,\phi Y) for all vector fields XX and YY on MM. An almost paracontact metric manifold for which

Φ​(X,Y)=d​η​(X,Y)=g​(X,ϕ​Y)\Phi(X,Y)=d\eta(X,Y)=g(X,\phi Y) (2.6)

is said to be paracontact metric manifold. In this case, η\eta becomes a contact form i.e., η∧(d​η)n≠0\eta\wedge(d\eta)^{n}\neq 0 and the manifold becomes a contact manifold. On a paracontact metric manifold M2​n+1​(ϕ,ξ,η,g)M^{2n+1}(\phi,\xi,\eta,g) we consider a self-adjoint operator h=12​ℒξ​ϕh=\frac{1}{2}\mathcal{L}_{\xi}\phi, where ℒξ\mathcal{L}_{\xi} denotes the Lie derivative along ξ\xi. This operator hh is symmetric and satisfies

h​ϕ\displaystyle h\phi =−ϕ​h,\displaystyle=-\phi h, h​ξ\displaystyle h\xi =0,\displaystyle=0, ∇Xξ\displaystyle\nabla_{X}\xi =−ϕ​X+ϕ​h​X,\displaystyle=-\phi X+\phi hX,

where ∇\nabla is the operator of covariant differentiation w.r.t. the metric gg. The normality of a paracontact metric manifold (M,ϕ,ξ,η,g)(M,\phi,\xi,\eta,g) is equivalent to vanishing of the (1,2)(1,2)-torsion tensor defined by Nϕ​(X,Y)=[ϕ,ϕ]​(X,Y)−2​d​η​(X,Y)​ξN_{\phi}(X,Y)=[\phi,\phi](X,Y)-2d\eta(X,Y)\xi, where [ϕ,ϕ]​(X,Y)=ϕ2​[X,Y]+[ϕ​X,ϕ​Y]−ϕ​[X,ϕ​Y]−ϕ​[ϕ​X,Y][\phi,\phi](X,Y)=\phi^{2}[X,Y]+[\phi X,\phi Y]-\phi[X,\phi Y]-\phi[\phi X,Y] for any X,Y∈χ​(M)X,Y\in\chi(M). A normal paracontact metric manifold is called a para-Sasakian metric manifold. It is equivalent to say, an almost paracontact metric manifold is called a para-Sasakian manifold if it satisfies

(∇Xϕ)​Y=−g​(X,Y)​ξ+η​(Y)​X(\nabla_{X}\phi)Y=-g(X,Y)\xi+\eta(Y)X (2.7)

for arbitrary X,Y∈χ​(M)X,Y\in\chi(M). In a para-Sasakian manifold the operator hh vanishes and the manifold satisfies,

∇Xξ\displaystyle\nabla_{X}\xi =\displaystyle= −ϕ​X,\displaystyle-\phi X, (2.8)
R​(X,Y)​ξ\displaystyle R(X,Y)\xi =\displaystyle= η​(X)​Y−η​(Y)​X,\displaystyle\eta(X)Y-\eta(Y)X, (2.9)
R​(X,ξ)​Y\displaystyle R(X,\xi)Y =\displaystyle= g​(X,Y)​ξ−η​(Y)​X,\displaystyle g(X,Y)\xi-\eta(Y)X, (2.10)
Q​ξ\displaystyle Q\xi =\displaystyle= −2​n​ξ,\displaystyle-2n\xi, (2.11)

for all vector fields XX and YY on MM and RR, QQ denote Riemannian curvature tensor and Ricci operator associated with the Ricci tensor SS defined by S​(X,Y)=g​(Q​X,Y)S(X,Y)=g(QX,Y).

In [11], authors have prove that in a para-Sasakian manifold the following result holds (for proof see lemma-4)

Q​ϕ=ϕ​Q.Q\phi=\phi Q. (2.12)

Prakasha and Veeresha in [14] established another beautiful result on para-Sasakian manifold (see lemma-1) which using (2.12) can be restated as

(∇ξQ)​X\displaystyle(\nabla_{\xi}Q)X =\displaystyle= 0,\displaystyle 0, (2.13)
(∇XQ)​ξ\displaystyle(\nabla_{X}Q)\xi =\displaystyle= Q​ϕ​X+2​n​ϕ​X.\displaystyle Q\phi X+2n\phi X. (2.14)

3. On Conformal η\eta-Ricci soliton

In this section, we have studied conformal η\eta-Ricci soliton on Para-Sasakian manifold. First we prove the following lemma which has been used to prove the next theorems.

Lemma 3.1.

If the metric gg of a para-Sasakian manifold represents a conformal η\eta-Ricci soliton, then

η​(ℒV​ξ)=−(ℒV​η)​ξ=λ−p2−12​n+1−2​n+μ.\eta(\mathcal{L}_{V}\xi)=-(\mathcal{L}_{V}\eta)\xi=\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu. (3.1)
Proof.

As the metric gg satisfies conformal η\eta-Ricci soliton equation (1.1), using (2.11), we can easily obtain

(ℒV​g)​(X,ξ)+2​(λ−p2−12​n+1−2​n+μ)​η​(X)=0(\mathcal{L}_{V}g)(X,\xi)+2(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)\eta(X)=0

for arbitrary vector field XX. Lie differentiation of the relation η​(X)=g​(X,ξ)\eta(X)=g(X,\xi) along the soliton vector field VV yields (ℒV​g)​(X,ξ)=(ℒV​η)​X−g​(X,ℒV​ξ)(\mathcal{L}_{V}g)(X,\xi)=(\mathcal{L}_{V}\eta)X-g(X,\mathcal{L}_{V}\xi). Using this in the foregoing equation, we have

(ℒV​η)​X−g​(X,ℒV​ξ)=−2​(λ−p2−12​n+1−2​n+μ)​η​(X).(\mathcal{L}_{V}\eta)X-g(X,\mathcal{L}_{V}\xi)=-2(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)\eta(X). (3.2)

Finally taking Lie derivative of (2.2) along VV into account, we can easily obtain our desired result (3.1). ∎

Theorem 3.2.

Let M2​n+1​(ϕ,ξ,η,g)M^{2n+1}(\phi,\xi,\eta,g) be a para-Sasakian manifold. If the metric gg represents a conformal η\eta-Ricci soliton then the manifold is η\eta-Einstein and either the soliton vector field VV is Killing or it leaves ϕ\phi invariant.

Proof.

Taking covariant derivative of (2.2) along arbitrary vector field YY and using (2.8) we can easily have (∇Yη)​X=g​(ϕ​X,Y)(\nabla_{Y}\eta)X=g(\phi X,Y).

Since the metric gg of the manifold represents a conformal η\eta-Ricci soliton, taking covariant derivative of (1.1) along arbitrary vector field ZZ, we obtain

(∇ZℒV​g)​(X,Y)=−2​(∇ZS)​(X,Y)−2​μ​[g​(ϕ​X,Z)​η​(Y)+g​(ϕ​Y,Z)​η​(X)](\nabla_{Z}\mathcal{L}_{V}g)(X,Y)=-2(\nabla_{Z}S)(X,Y)-2\mu[g(\phi X,Z)\eta(Y)+g(\phi Y,Z)\eta(X)] (3.3)

for all vector fields XX, YY and ZZ on MM. Again from Yano[24], we have the following commutation formula

(ℒV​∇Xg−∇XℒV​g−∇[V,X]g)​(Y,Z)\displaystyle(\mathcal{L}_{V}\nabla_{X}g-\nabla_{X}\mathcal{L}_{V}g-\nabla_{[V,X]}g)(Y,Z) =\displaystyle= −g​((ℒV​∇)​(X,Y),Z)\displaystyle-g((\mathcal{L}_{V}\nabla)(X,Y),Z)
−g​((ℒV​∇)​(X,Z),Y),\displaystyle-g((\mathcal{L}_{V}\nabla)(X,Z),Y),

where gg is the metric connection i.e., ∇g=0\nabla g=0. So the above equation reduces to

(∇XℒV​g)​(Y,Z)=g​((ℒV​∇)​(X,Y),Z)+g​((ℒV​∇)​(X,Z),Y)(\nabla_{X}\mathcal{L}_{V}g)(Y,Z)=g((\mathcal{L}_{V}\nabla)(X,Y),Z)+g((\mathcal{L}_{V}\nabla)(X,Z),Y) (3.4)

for all vector fields XX, YY, ZZ on MM. Combining (3.3) and (3.4), we have

g​((ℒV​∇)​(X,Z),Y)+\displaystyle g((\mathcal{L}_{V}\nabla)(X,Z),Y)+ g​((ℒV​∇)​(Y,Z),X)=−2​(∇ZS)​(X,Y)\displaystyle g((\mathcal{L}_{V}\nabla)(Y,Z),X)=-2(\nabla_{Z}S)(X,Y)
−2​μ​[g​(ϕ​X,Z)​η​(Y)+g​(ϕ​Y,Z)​η​(X)].\displaystyle-2\mu[g(\phi X,Z)\eta(Y)+g(\phi Y,Z)\eta(X)].

By a straightforward combinatorial computation, the foregoing equation yields

g​((ℒV​∇)​(X,Y),Z)\displaystyle g((\mathcal{L}_{V}\nabla)(X,Y),Z) =\displaystyle= (∇ZS)​(X,Y)−(∇XS)​(Y,Z)−(∇YS)​(Z,X)\displaystyle(\nabla_{Z}S)(X,Y)-(\nabla_{X}S)(Y,Z)-(\nabla_{Y}S)(Z,X) (3.5)
+2​μ​[g​(ϕ​X,Z)​η​(Y)+g​(ϕ​Y,Z)​η​(X)]\displaystyle+2\mu[g(\phi X,Z)\eta(Y)+g(\phi Y,Z)\eta(X)]

for all X,Y,Z∈χ​(M)X,Y,Z\in\chi(M). Setting Y=ξY=\xi and making use of (∇ZS)​(X,Y)=g​((∇ZQ)​X,Y)(\nabla_{Z}S)(X,Y)=g((\nabla_{Z}Q)X,Y), (2.12), (2.13) and (2.14), we obtain

(ℒV​∇)​(X,ξ)=2​(μ−2​n)​(ϕ​X)−2​Q​ϕ​X.(\mathcal{L}_{V}\nabla)(X,\xi)=2(\mu-2n)(\phi X)-2Q\phi X. (3.6)

Differentiating the last equation covariantly with respect to arbitrary vector field YY and using (2.7) and (2.8), we acquire

(∇YℒV​∇)​(X,ξ)\displaystyle(\nabla_{Y}\mathcal{L}_{V}\nabla)(X,\xi) =\displaystyle= (ℒV​∇)​(X,ϕ​Y)−2​(∇YQ)​(ϕ​X)−2​η​(X)​(Q​Y)\displaystyle(\mathcal{L}_{V}\nabla)(X,\phi Y)-2(\nabla_{Y}Q)(\phi X)-2\eta(X)(QY) (3.7)
−2​μ​g​(X,Y)​ξ+2​(μ−2​n)​η​(X)​Y.\displaystyle-2\mu g(X,Y)\xi+2(\mu-2n)\eta(X)Y.

From Yano [24], we know (ℒV​R)​(X,Y)​ξ=(∇XℒV​∇)​(Y,ξ)−(∇YℒV​∇)​(X,ξ)(\mathcal{L}_{V}R)(X,Y)\xi=(\nabla_{X}\mathcal{L}_{V}\nabla)(Y,\xi)-(\nabla_{Y}\mathcal{L}_{V}\nabla)(X,\xi). Substituting the values from (3.7) and using (2.13), (3.6), we get

(ℒV​R)​(X,ξ)​ξ=4​(μ−2​n)​X−4​Q​X−4​μ​η​(X)​ξ,(\mathcal{L}_{V}R)(X,\xi)\xi=4(\mu-2n)X-4QX-4\mu\eta(X)\xi, (3.8)

which holds for an arbitrary vector field XX. Again, from (2.9) we get R​(X,ξ)​ξ=η​(X)​ξ−XR(X,\xi)\xi=\eta(X)\xi-X. By virtue of (3.1) and (3.2), Lie differentiation of the last relation along VV yields

(ℒV​R)​(X,ξ)​ξ=2​(λ−p2−12​n+1−2​n+μ)​[X−η​(X)​ξ].(\mathcal{L}_{V}R)(X,\xi)\xi=2(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)[X-\eta(X)\xi]. (3.9)

Substituting the value of (ℒV​R)​(X,ξ)​ξ(\mathcal{L}_{V}R)(X,\xi)\xi from (3.8) in the foregoing equation and taking inner product with arbitrary vector field YY, we obtain

S​(X,Y)\displaystyle S(X,Y) =\displaystyle= 12​(λ−p2−12​n+1−2​n−μ)​η​(X)​η​(Y)\displaystyle\frac{1}{2}(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n-\mu)\eta(X)\eta(Y) (3.10)
−12​(λ−p2−12​n+1+2​n−μ)​g​(X,Y)\displaystyle-\frac{1}{2}(\lambda-\frac{p}{2}-\frac{1}{2n+1}+2n-\mu)g(X,Y)

for all X,Y∈χ​(M)X,Y\in\chi(M). This transforms the soliton equation (1.1) to

(ℒV​g)​(X,Y)=(p2+12​n+1+2​n−λ−μ)​[g​(X,Y)+η​(X)​η​(Y)].(\mathcal{L}_{V}g)(X,Y)=(\frac{p}{2}+\frac{1}{2n+1}+2n-\lambda-\mu)[g(X,Y)+\eta(X)\eta(Y)]. (3.11)

Differentiating (3.10) covariantly along arbitrary vector field ZZ, we get
(∇ZS)​(X,Y)=12​(λ−p2−12​n+1−2​n−μ)​[g​(ϕ​X,Z)​η​(Y)+g​(ϕ​Y,Z)​η​(X)](\nabla_{Z}S)(X,Y)=\frac{1}{2}(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n-\mu)[g(\phi X,Z)\eta(Y)+g(\phi Y,Z)\eta(X)]. Repeated use of this in (3.3), gives rise to

(ℒV​∇)​(X,Y)=(λ−p2−12​n+1−2​n+μ)​[η​(Y)​(ϕ​X)+η​(X)​(ϕ​Y)](\mathcal{L}_{V}\nabla)(X,Y)=(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)[\eta(Y)(\phi X)+\eta(X)(\phi Y)] (3.12)

for arbitrary vector fields XX and YY on MM. Covariant differentiation of the aforementioned equation along arbitrary vector field ZZ and use of (2.7), yields

(∇ZℒV​∇)​(X,Y)=\displaystyle(\nabla_{Z}\mathcal{L}_{V}\nabla)(X,Y)= (λ−p2−12​n+1−2n+μ)[g(ϕY,Z)(ϕX)+g(ϕX,Z)\displaystyle(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)[g(\phi Y,Z)(\phi X)+g(\phi X,Z)
(ϕY)−g(X,Z)η(Y)ξ−g(Y,Z)η(X)ξ+2η(X)η(Y)Z].\displaystyle(\phi Y)-g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi+2\eta(X)\eta(Y)Z].

Using this relation in (ℒV​R)​(X,Y)​Z=(∇XℒV​∇)​(Y,Z)−(∇YℒV​∇)​(X,Z)(\mathcal{L}_{V}R)(X,Y)Z=(\nabla_{X}\mathcal{L}_{V}\nabla)(Y,Z)-(\nabla_{Y}\mathcal{L}_{V}\nabla)(X,Z) (for details see Yano [24]) and contracting XX, we get

(ℒV​S)​(Y,Z)=2​(λ−p2−12​n+1−2​n+μ)​[(2​n+1)​η​(Y)​η​(Z)−g​(Y,Z)].(\mathcal{L}_{V}S)(Y,Z)=2(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)[(2n+1)\eta(Y)\eta(Z)-g(Y,Z)]. (3.13)

Taking Lie derivative of (3.10) along VV and using (3.11), we achieve

(ℒV​S)​(Y,Z)=\displaystyle(\mathcal{L}_{V}S)(Y,Z)= 12(λ−p2−12​n+1−2n−μ)[((ℒVη)Y)η(Z)+\displaystyle\frac{1}{2}(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n-\mu)[((\mathcal{L}_{V}\eta)Y)\eta(Z)+
η(Y)((ℒVη)Z)]+12(λ−p2−12​n+1+2n−μ)\displaystyle\eta(Y)((\mathcal{L}_{V}\eta)Z)]+\frac{1}{2}(\lambda-\frac{p}{2}-\frac{1}{2n+1}+2n-\mu)
(λ−p2−12​n+1−2​n+μ)​[g​(Y,Z)+η​(Y)​η​(Z)].\displaystyle(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)[g(Y,Z)+\eta(Y)\eta(Z)]. (3.14)

Comparisons of (3.13) and (3.2) gives

2​(λ−p2−12​n+1−2​n+μ)​[(2​n+1)​η​(Y)​η​(Z)−g​(Y,Z)]=\displaystyle 2(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)[(2n+1)\eta(Y)\eta(Z)-g(Y,Z)]=
12​(λ−p2−12​n+1−2​n−μ)​[((ℒV​η)​Y)​η​(Z)+η​(Y)​((ℒV​η)​Z)]\displaystyle\frac{1}{2}(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n-\mu)[((\mathcal{L}_{V}\eta)Y)\eta(Z)+\eta(Y)((\mathcal{L}_{V}\eta)Z)]
+12​(λ−p2−12​n+1+2​n−μ)​(λ−p2−12​n+1−2​n+μ)\displaystyle+\frac{1}{2}(\lambda-\frac{p}{2}-\frac{1}{2n+1}+2n-\mu)(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)
[g​(Y,Z)+η​(Y)​η​(Z)].\displaystyle[g(Y,Z)+\eta(Y)\eta(Z)]. (3.15)

Substituting YY and ZZ by ϕ2​Y\phi^{2}Y and ϕ​Z\phi Z respectively and using (2.1), (2.4) and (2.6), we obtain

(λ−p2−12​n+1−2​n+μ)​(λ−p2−12​n+1+2​n−μ+4)​d​η​(Y,Z)=0(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)(\lambda-\frac{p}{2}-\frac{1}{2n+1}+2n-\mu+4)d\eta(Y,Z)=0 (3.16)

∀Y,Z∈χ​(M)\forall Y,Z\in\chi(M). As we know, in para-Sasakian manifold d​η≠0d\eta\neq 0, we have (λ−p2−12​n+1−2​n+μ)​(λ−p2−12​n+1+2​n−μ+4)=0(\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu)(\lambda-\frac{p}{2}-\frac{1}{2n+1}+2n-\mu+4)=0. This gives either λ=p2+12​n+1+2​n−μ\lambda=\frac{p}{2}+\frac{1}{2n+1}+2n-\mu or λ=p2+12​n+1−2​n+μ−4\lambda=\frac{p}{2}+\frac{1}{2n+1}-2n+\mu-4.

Case-I: If λ=p2+12​n+1+2​n−μ\lambda=\frac{p}{2}+\frac{1}{2n+1}+2n-\mu, then (3.10) reduces to S​(X,Y)=(μ−2​n)​g​(X,Y)−μ​η​(X)​η​(Y)S(X,Y)=(\mu-2n)g(X,Y)-\mu\eta(X)\eta(Y) i.e., the manifold is η\eta-Einstein. Also (3.11) gives ℒV​g=0\mathcal{L}_{V}g=0. So, VV is Killing vector field.

Case-II: Using λ=p2+12​n+1−2​n+μ−4\lambda=\frac{p}{2}+\frac{1}{2n+1}-2n+\mu-4 in (3.10), we get S​(X,Y)=2​g​(X,Y)−2​(n+1)​η​(X)​η​(Y)S(X,Y)=2g(X,Y)-2(n+1)\eta(X)\eta(Y). So, the manifold is η\eta-Einstein. Substituting YY by ϕ​Y\phi Y and setting Z=ξZ=\xi in (3.2), we obtain (ℒV​η)​(ϕ​Y)=0(\mathcal{L}_{V}\eta)(\phi Y)=0. Further, replacing YY by ϕ​Y\phi Y and using (2.1), (3.1) and λ=p2+12​n+1−2​n+μ−4\lambda=\frac{p}{2}+\frac{1}{2n+1}-2n+\mu-4, we get

ℒV​η=2​(2​n−μ+2)​η\mathcal{L}_{V}\eta=2(2n-\mu+2)\eta

Exterior differentiation of the foregoing equation and use of well-known relation d​(ℒV​η)=ℒV​d​ηd(\mathcal{L}_{V}\eta)=\mathcal{L}_{V}d\eta and (2.6), yields

(ℒV​d​η)​(X,Y)=2​(2​n−μ+2)​g​(X,ϕ​Y)(\mathcal{L}_{V}d\eta)(X,Y)=2(2n-\mu+2)g(X,\phi Y) (3.17)

for arbitrary vector fields XX and YY on MM. Lie differentiation of (2.6) along VV, infers

(ℒV​d​η)​(X,Y)=2​(2​n−μ+2)​g​(X,ϕ​Y)+g​(X,(ℒV​ϕ)​Y)(\mathcal{L}_{V}d\eta)(X,Y)=2(2n-\mu+2)g(X,\phi Y)+g(X,(\mathcal{L}_{V}\phi)Y) (3.18)

∀X,Y∈χ​(M)\forall X,Y\in\chi(M). Comparing this with (3.17) gives ℒV​ϕ=0\mathcal{L}_{V}\phi=0, as XX and YY are arbitrary vector fields. So, VV leaves ϕ\phi invariant. ∎

Theorem 3.3.

If the metric gg of a para-Sasakian manifold MM represents a conformal η\eta-Ricci soliton and if the soliton vector field VV is pointwise collinear with the characteristic vector field ξ\xi then VV is a constant multiple of ξ\xi and the scalar curvature of the manifold is constant.

Proof.

Since the soliton vector field VV is pointwise collinear with the characteristic vector field ξ\xi, so, V=f​ξV=f\xi where, ff is a smooth function on M2​n+1M^{2n+1}. Substituting V=f​ξV=f\xi in (ℒV​g)​(X,Y)=g​(∇XV,Y)+g​(X,∇YV)(\mathcal{L}_{V}g)(X,Y)=g(\nabla_{X}V,Y)+g(X,\nabla_{Y}V) and using (2.8), we get

(ℒV​g)​(X,Y)=(X​f)​η​(Y)+(Y​f)​η​(X)(\mathcal{L}_{V}g)(X,Y)=(Xf)\eta(Y)+(Yf)\eta(X) (3.19)

for arbitrary vector field XX and YY on MM. Using (3.19) in the soliton equation (1.1), we obtain

(X​f)​η​(Y)+(Y​f)​η​(X)+2​S​(X,Y)+2​μ​η​(X)​η​(Y)\displaystyle(Xf)\eta(Y)+(Yf)\eta(X)+2S(X,Y)+2\mu\eta(X)\eta(Y)
+2​(λ−p2−12​n+1)​g​(X,Y)=0.\displaystyle+2(\lambda-\frac{p}{2}-\frac{1}{2n+1})g(X,Y)=0. (3.20)

Setting Y=ξY=\xi and using (2.2) and (2.11), the above equation becomes

D​f=2​[2​n+p2+12​n+1−12​(ξ​f)−λ−μ]​ξ.Df=2[2n+\frac{p}{2}+\frac{1}{2n+1}-\frac{1}{2}(\xi f)-\lambda-\mu]\xi. (3.21)

Taking inner product with respect to Reeb vector field ξ\xi, we acquire

ξ​f=2​n+p2+12​n+1−λ−μ.\xi f=2n+\frac{p}{2}+\frac{1}{2n+1}-\lambda-\mu. (3.22)

From previous theorem we obtained either λ=p2+12​n+1+2​n−μ\lambda=\frac{p}{2}+\frac{1}{2n+1}+2n-\mu or λ=p2+12​n+1−2​n+μ−4\lambda=\frac{p}{2}+\frac{1}{2n+1}-2n+\mu-4. If we consider λ=p2+12​n+1+2​n−μ\lambda=\frac{p}{2}+\frac{1}{2n+1}+2n-\mu, then from (3.22) we get ξ​f=0\xi f=0. Substituting these values in (3.21), yields

D​f=0.Df=0. (3.23)

Now, if we consider λ=p2+12​n+1−2​n+μ−4\lambda=\frac{p}{2}+\frac{1}{2n+1}-2n+\mu-4, then from (3.22) we obtain ξ​f=2​(2​n−μ+2)\xi f=2(2n-\mu+2) and from (3.21), we get

D​f=(ξ​f)​ξ.Df=(\xi f)\xi. (3.24)

Taking (2.8) into account, differentiating the foregoing equation along XX and then taking scalar product with arbitrary vector field YY, leads to

g​(∇XD​f,Y)=(X​(ξ​f))​η​(Y)−(ξ​f)​g​(ϕ​X,Y).g(\nabla_{X}Df,Y)=(X(\xi f))\eta(Y)-(\xi f)g(\phi X,Y). (3.25)

Anti-symmetrizing the last equation and using g​(∇XD​f,Y)=g​(X,∇YD​f)g(\nabla_{X}Df,Y)=g(X,\nabla_{Y}Df), we have

(X​(ξ​f))​η​(Y)−(Y​(ξ​f))​η​(X)−2​(ξ​f)​g​(ϕ​X,Y)=0(X(\xi f))\eta(Y)-(Y(\xi f))\eta(X)-2(\xi f)g(\phi X,Y)=0 (3.26)

for arbitrary vector fields XX and YY. If we let XX to be unit vector (i.e., g​(X,X)=1g(X,X)=1) in K​e​r​(η)Ker(\eta), then ϕ​X\phi X also becomes a unit vector with g​(ϕ​X,ϕ​Y)=−1g(\phi X,\phi Y)=-1. Now replacing YY by ϕ​X\phi X in (3.26), we get ξ​f=0\xi f=0. Substituting this value in (3.24) leads to

D​f=0.Df=0. (3.27)

Combining (3.23) and (3.27) we can conclude that D​f=0Df=0 in entire manifold. Therefore ff is constant. So, VV is a constant multiple of ξ\xi.
The equation (3.3) reduces to Q​X+2​(λ−p2−12​n+1)​X+2​μ​η​(X)​ξ=0QX+2(\lambda-\frac{p}{2}-\frac{1}{2n+1})X+2\mu\eta(X)\xi=0. Tracing of this equation leads to r=2−2​μ+(2​n+1)​(p−2​λ)r=2-2\mu+(2n+1)(p-2\lambda), where rr denotes the scalar curvature of the manifold. This completes the proof. ∎

4. On almost conformal η\eta-Ricci soliton

In this section, we consider almost conformal η\eta-Ricci soliton on para-Sasakian manifold. It follows from (1.1) that almost conformal η\eta-Ricci soliton is the generalization of almost η\eta-Ricci soliton because it involve two smooth functions λ\lambda and μ\mu.

Definition 4.1.

A vector field VV is said to be an infinitesimal contact transformation if there exists a certain a∈C∞​(M)a\in C^{\infty}(M) such that

ℒV​η=a​η.\mathcal{L}_{V}\eta=a\eta. (4.1)

If a=0a=0, then VV is called strictly infinitesimal contact transformation.

Theorem 4.2.

Let M2​n+1M^{2n+1} be a para-Sasakian manifold with n>1n>1. If gg represents an almost conformal η\eta-Ricci soliton with the soliton vector field VV as infinitesimal contact transformation, then the manifold is η\eta-Einstein and either the soliton vector field VV is Killing or it leaves ϕ\phi invariant.

Proof.

Taking d​(ℒV​η)=ℒV​d​ηd(\mathcal{L}_{V}\eta)=\mathcal{L}_{V}d\eta into account, exterior derivative of (4.1) gives

ℒV​d​η=(d​a)∧η+a​d​η.\mathcal{L}_{V}d\eta=(da)\wedge\eta+ad\eta.

Using (2.6), the foregoing equation can be rewritten as

(ℒV​d​η)​(X,Y)=12​[(X​a)​η​(Y)−η​(X)​(Y​a)]+a​g​(X,ϕ​Y)(\mathcal{L}_{V}d\eta)(X,Y)=\frac{1}{2}[(Xa)\eta(Y)-\eta(X)(Ya)]+ag(X,\phi Y) (4.2)

for arbitrary vector fields XX and YY on MM. Lie differentiation of (2.6) along the soliton vector field VV, yields

(ℒV​d​η)​(X,Y)=g​(X,(ℒV​ϕ)​Y)−2​g​((λ−p2−12​n+1)​X+Q​X,ϕ​Y)(\mathcal{L}_{V}d\eta)(X,Y)=g(X,(\mathcal{L}_{V}\phi)Y)-2g((\lambda-\frac{p}{2}-\frac{1}{2n+1})X+QX,\phi Y) (4.3)

∀X,Y∈χ​(M)\forall X,Y\in\chi(M). Comparing the aforementioned equation with (4.2), we obtain

2​(ℒV​ϕ)​Y=η​(Y)​(D​a)−(Y​a)​ξ+(2​a+4​λ−2​p−42​n+1)​(ϕ​Y)+4​Q​ϕ​Y2(\mathcal{L}_{V}\phi)Y=\eta(Y)(Da)-(Ya)\xi+(2a+4\lambda-2p-\frac{4}{2n+1})(\phi Y)+4Q\phi Y (4.4)

for any vector YY in χ​(M)\chi(M). Setting Y=ξY=\xi and using (2.2) and (2.4), we get

2​(ℒV​ϕ)​ξ=D​a−(ξ​a)​ξ.2(\mathcal{L}_{V}\phi)\xi=Da-(\xi a)\xi. (4.5)

Taking (1.1) and (4.1) into consideration, Lie differentiating g​(ξ,ξ)=1g(\xi,\xi)=1 and g​(X,ξ)=η​(X)g(X,\xi)=\eta(X) along VV, we achieve

η​(ℒV​ξ)\displaystyle\eta(\mathcal{L}_{V}\xi) =\displaystyle= λ−p2−12​n+1−2​n+μ\displaystyle\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n+\mu (4.6)
ℒV​ξ\displaystyle\mathcal{L}_{V}\xi =\displaystyle= (a+2​λ−p−22​n+1−4​n+2​μ)​ξ.\displaystyle(a+2\lambda-p-\frac{2}{2n+1}-4n+2\mu)\xi. (4.7)

Combining above two relations, we get

a=2​n−λ−μ+p2+12​n+1.a=2n-\lambda-\mu+\frac{p}{2}+\frac{1}{2n+1}. (4.8)

Lie differentiating (2.4) along the soliton vector field VV and using (4.8), we obtain (ℒV​ϕ)​ξ=0(\mathcal{L}_{V}\phi)\xi=0. Combining this with (4.5) we have D​a=(ξ​a)​ξDa=(\xi a)\xi, which further implies

d​a=(ξ​a)​η.da=(\xi a)\eta. (4.9)

Operating the foregoing equation by exterior derivative operator dd and using d2=0d^{2}=0, yields

d​(ξ​a)∧η+(ξ​a)​d​η=0.d(\xi a)\wedge\eta+(\xi a)d\eta=0.

Taking η∧η=0\eta\wedge\eta=0 and η∧d​η≠0\eta\wedge d\eta\neq 0 into account, wedge product of the above equation with respect to the 1-form η\eta gives ξ​a=0\xi a=0. Substituting this in (4.9) we get d​a=0da=0 and so aa is constant. Then the equation (4.4) reduces to

(ℒV​ϕ)​Y=(a+2​λ−p−22​n+1)​(ϕ​Y)+2​Q​ϕ​Y(\mathcal{L}_{V}\phi)Y=(a+2\lambda-p-\frac{2}{2n+1})(\phi Y)+2Q\phi Y (4.10)

for any vector field YY on MM. Operating (2.1) by ℒV\mathcal{L}_{V} and using (4.1) and (4.7) we get ℒV​ϕ2=0\mathcal{L}_{V}\phi^{2}=0. Consequently, we get (ℒV​ϕ)​(ϕ​X)+ϕ​(ℒV​ϕ)​X=0(\mathcal{L}_{V}\phi)(\phi X)+\phi(\mathcal{L}_{V}\phi)X=0 for an arbitrary vector field XX. After repeated application of (4.10) and use of (2.1), (2.11) and (2.12), the aforementioned relation leads to

Q​X=−(a2+λ−p2−12​n+1)​X+(a2+λ−p2−12​n+1−2​n)​η​(X)​ξQX=-(\frac{a}{2}+\lambda-\frac{p}{2}-\frac{1}{2n+1})X+(\frac{a}{2}+\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n)\eta(X)\xi (4.11)

for all field XX of MM. Covariant derivative of (4.11) along an aritrary vector field YY yields

(∇YQ)​X=\displaystyle(\nabla_{Y}Q)X= (a2+λ−p2−12​n+1−2​n)​[g​(Y,ϕ​X)​ξ−η​(X)​(ϕ​Y)]\displaystyle(\frac{a}{2}+\lambda-\frac{p}{2}-\frac{1}{2n+1}-2n)[g(Y,\phi X)\xi-\eta(X)(\phi Y)]
−(Y​λ)​[X−η​(X)​ξ].\displaystyle-(Y\lambda)[X-\eta(X)\xi]. (4.12)

It is well-known that X​r=g​((∇XQ)​ei,ei)Xr=g((\nabla_{X}Q)e_{i},e_{i}) and 12​X​r=g​((∇eiQ)​X,ei)\frac{1}{2}Xr=g((\nabla_{e_{i}}Q)X,e_{i}), where {ei}i=12​n+1\{e_{i}\}_{i=1}^{2n+1} is an orthonormal basis of the manifold. Successive application of (4.2) in these two relations infers

X​r\displaystyle Xr =\displaystyle= −2​n​(X​λ),\displaystyle-2n(X\lambda), (4.13)
X​r\displaystyle Xr =\displaystyle= −2​(X​λ)+2​η​(X)​(ξ​λ)\displaystyle-2(X\lambda)+2\eta(X)(\xi\lambda) (4.14)

for an arbitrary vector field XX. Since the characteristic vector field ξ\xi is a Killing vector field in a para-Sasakian manifold, it follows that ξ​r=0\xi r=0. Plugging this in (4.13), we obtain ξ​λ=0\xi\lambda=0 as we assume n≠0n\neq 0. Consequently, (4.14) reduces to X​r=−2​(X​λ),∀X∈χ​(M)Xr=-2(X\lambda),\forall X\in\chi(M). Setting this in (4.13), we get

(n−1)​(X​λ)=0.(n-1)(X\lambda)=0.

Since n>1n>1 and XX is an arbitrary vector field, we conclude that λ\lambda is a constant. Thus it follows from (4.8) that μ\mu is also constant. Then the soliton reduces to conformal η\eta-Ricci soliton and the result follows from theorem-3.2. ∎

Theorem 4.3.

Let M2​n+1M^{2n+1} be a para-Sasakian manifold of dimension >3>3. If gg represents a gradient almost conformal η\eta-Ricci soliton, then the soliton reduces to gradient almost conformal Ricci soliton and the manifold is Einstein.

Proof.

From the gradient almost conformal η\eta-Ricci soliton equation (1.2), we easily obtain

∇XD​f=−Q​X+(p2+12​n+1−λ)​X−μ​η​(X)​ξ\nabla_{X}Df=-QX+(\frac{p}{2}+\frac{1}{2n+1}-\lambda)X-\mu\eta(X)\xi (4.15)

for an arbitrary vector field XX of MM. Taking covariant derivative along an arbitrary vector field YY, we acquire

∇Y∇X⁡D​f=\displaystyle\nabla_{Y}\nabla_{X}Df= −(∇YQ)​X−Q​(∇YX)+(p2+12​n+1−λ)​(∇YX)−(Y​λ)​X\displaystyle-(\nabla_{Y}Q)X-Q(\nabla_{Y}X)+(\frac{p}{2}+\frac{1}{2n+1}-\lambda)(\nabla_{Y}X)-(Y\lambda)X
−(Y​μ)​η​(X)​ξ−μ​[g​(ϕ​X,Y)​ξ+η​(∇YX)​ξ−η​(X)​(ϕ​Y)].\displaystyle-(Y\mu)\eta(X)\xi-\mu[g(\phi X,Y)\xi+\eta(\nabla_{Y}X)\xi-\eta(X)(\phi Y)].

From Yano [24], we know R​(X,Y)​Z=∇X∇Y⁡Z−∇Y∇X⁡Z−∇[X,Y]ZR(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z for all X,Y,Z∈χ​(M)X,Y,Z\in\chi(M). Plugging the above equation along with (4.15) in this curvature property, we infer

R​(X,Y)​D​f\displaystyle R(X,Y)Df =(∇YQ)​X−(∇XQ)​Y+(Y​λ)​X−(X​λ)​Y+(Y​μ)​η​(X)​ξ\displaystyle=(\nabla_{Y}Q)X-(\nabla_{X}Q)Y+(Y\lambda)X-(X\lambda)Y+(Y\mu)\eta(X)\xi
−(X​μ)​η​(Y)​ξ+μ​[2​g​(ϕ​X,Y)​ξ−η​(X)​(ϕ​Y)+η​(Y)​(ϕ​X)]\displaystyle-(X\mu)\eta(Y)\xi+\mu[2g(\phi X,Y)\xi-\eta(X)(\phi Y)+\eta(Y)(\phi X)] (4.16)

for all X,Y∈χ​(M)X,Y\in\chi(M). Setting Y=ξY=\xi in the foregoing equation and using (2.13) and (2.14), we get

R​(X,ξ)​D​f=−Q​ϕ​X+(μ−2​n)​(ϕ​X)+(ξ​λ)​X−(X​λ)​ξ+(ξ​μ)​η​(X)​ξ−(X​μ)​ξ.R(X,\xi)Df=-Q\phi X+(\mu-2n)(\phi X)+(\xi\lambda)X-(X\lambda)\xi+(\xi\mu)\eta(X)\xi-(X\mu)\xi.

Combining the last equation with (2.7) and (2.10), we obtain

g​((∇Xϕ)​Y,D​f)=\displaystyle g((\nabla_{X}\phi)Y,Df)= −g​(Q​ϕ​X,Y)+(μ−2​n)​g​(ϕ​X,Y)+(ξ​λ)​g​(X,Y)\displaystyle-g(Q\phi X,Y)+(\mu-2n)g(\phi X,Y)+(\xi\lambda)g(X,Y)
−(X​λ)​η​(Y)+(ξ​μ)​η​(X)​η​(Y)−(X​μ)​η​(Y)\displaystyle-(X\lambda)\eta(Y)+(\xi\mu)\eta(X)\eta(Y)-(X\mu)\eta(Y) (4.17)

for arbitrary vector fields XX and YY on MM. Replacing XX and YY by ϕ​X\phi X and ϕ​Y\phi Y in the foregoing equation, we achieve

g​((∇ϕ​Xϕ)​ϕ​Y,D​f)=\displaystyle g((\nabla_{\phi X}\phi)\phi Y,Df)= g​(Q​ϕ​X,Y)−(μ−2​n)​g​(ϕ​X,Y)\displaystyle g(Q\phi X,Y)-(\mu-2n)g(\phi X,Y)
−(ξ​λ)​[g​(X,Y)−η​(X)​η​(Y)]\displaystyle-(\xi\lambda)[g(X,Y)-\eta(X)\eta(Y)] (4.18)

where we have used (2.1), (2.2), (2.5), (2.11) and (2.12). Subtraction of (4.3) from (4.3) yields

g​((∇ϕ​Xϕ)​ϕ​Y−(∇Xϕ)​Y,D​f)=2​g​(Q​ϕ​X,Y)−2​(μ−2​n)​g​(ϕ​X,Y)−\displaystyle g((\nabla_{\phi X}\phi)\phi Y-(\nabla_{X}\phi)Y,Df)=2g(Q\phi X,Y)-2(\mu-2n)g(\phi X,Y)-
2​(ξ​λ)​g​(X,Y)+(ξ​λ)​η​(X)​η​(Y)+(X​λ)​η​(Y)−(ξ​μ)​η​(X)​η​(Y)+(X​μ)​η​(Y).\displaystyle 2(\xi\lambda)g(X,Y)+(\xi\lambda)\eta(X)\eta(Y)+(X\lambda)\eta(Y)-(\xi\mu)\eta(X)\eta(Y)+(X\mu)\eta(Y). (4.19)

From Zamkovoy [25], we know

(∇ϕ​Xϕ)​ϕ​Y−(∇Xϕ)​Y=2​g​(X,Y)​ξ−η​(Y)​[X+η​(X)​ξ](\nabla_{\phi X}\phi)\phi Y-(\nabla_{X}\phi)Y=2g(X,Y)\xi-\eta(Y)[X+\eta(X)\xi] (4.20)

holds for arbitrary vector fields XX and YY in a para-Sasakian manifold (for proof see lemma-2.7 of [25] and here we have used h=0h=0 which holds in para-Sasakian manifold). Taking (4.20) into account, (4.3) can be rewritten as

2​g​(X,Y)​(ξ​f)−η​(Y)​(X​f)−η​(X)​η​(Y)​(ξ​f)=2​g​(Q​ϕ​X,Y)\displaystyle 2g(X,Y)(\xi f)-\eta(Y)(Xf)-\eta(X)\eta(Y)(\xi f)=2g(Q\phi X,Y)
−2​(μ−2​n)​g​(ϕ​X,Y)−2​(ξ​λ)​g​(X,Y)+(ξ​λ)​η​(X)​η​(Y)+\displaystyle-2(\mu-2n)g(\phi X,Y)-2(\xi\lambda)g(X,Y)+(\xi\lambda)\eta(X)\eta(Y)+
(X​λ)​η​(Y)−(ξ​μ)​η​(X)​η​(Y)+(X​μ)​η​(Y).\displaystyle(X\lambda)\eta(Y)-(\xi\mu)\eta(X)\eta(Y)+(X\mu)\eta(Y). (4.21)

Anti-symmetrizing the last equation and then replacing XX and YY by ϕ​X\phi X and ϕ​Y\phi Y, respectively we get Q​ϕ​X=(μ−2​n)​(ϕ​X)Q\phi X=(\mu-2n)(\phi X). Further, substitution of XX by ϕ​X\phi X in the last relation yields

Q​X=(μ−2​n)​X−μ​η​(X)​ξ.QX=(\mu-2n)X-\mu\eta(X)\xi. (4.22)

Covariant differentiation of (4.22) along an arbitrary vector field YY and using that expression of (∇YQ)​X(\nabla_{Y}Q)X in (4.3), we obtain

R​(X,Y)​D​f=(Y​μ)​X−(X​μ)​Y+(Y​λ)​X−(X​λ)​YR(X,Y)Df=(Y\mu)X-(X\mu)Y+(Y\lambda)X-(X\lambda)Y (4.23)

∀X,Y∈χ​(M)\forall X,Y\in\chi(M). Contraction of (4.3) and (4.23) over XX yields

Q​(D​f)\displaystyle Q(Df) =\displaystyle= 12​(D​r)+2​n​(D​λ)+(D​μ)−(ξ​μ)​ξ\displaystyle\frac{1}{2}(Dr)+2n(D\lambda)+(D\mu)-(\xi\mu)\xi (4.24)
Q​(D​f)\displaystyle Q(Df) =\displaystyle= 2​n​(D​μ+D​λ)\displaystyle 2n(D\mu+D\lambda) (4.25)

Comparing the last two relations we get

(2​n−1)​D​μ=12​(D​r)−(ξ​μ)​ξ.(2n-1)D\mu=\frac{1}{2}(Dr)-(\xi\mu)\xi. (4.26)

Tracing (4.22), we find r=2​n​μ−2​n​(2​n+1)r=2n\mu-2n(2n+1). So, D​r=2​n​D​μDr=2nD\mu. Plugging this relation in (4.26), we obtain

(n−1)​D​μ+(ξ​μ)​ξ=0.(n-1)D\mu+(\xi\mu)\xi=0. (4.27)

As we know g​((∇eiQ)​ξ,ei)=12​(ξ​r)g((\nabla_{e_{i}}Q)\xi,e_{i})=\frac{1}{2}(\xi r), we can easily obtain ξ​r=0\xi r=0. Combining this result with D​r=2​n​D​μDr=2nD\mu, we get ξ​μ=0\xi\mu=0. Using this in (4.27), we have D​μ=0D\mu=0 (since n≠1n\neq 1), so μ\mu is constant. Scalar product of (4.25) with respect to the characteristic vector field ξ\xi and use of (2.11), yields

ξ​(f+λ)=0\xi(f+\lambda)=0 (4.28)

as n>1n>1. Now, setting Y=ξY=\xi in (4.23) and using (2.10), we obtain

(X​(f+λ))​ξ=(ξ​(f+λ))​X(X(f+\lambda))\xi=(\xi(f+\lambda))X (4.29)

for an arbitrary vector field XX. Plugging equation (4.28) in the above equation, we get that f+λf+\lambda is constant. As we have μ\mu and f+λf+\lambda are constant, replacing XX by D​fDf in (4.22), we get

μ​[(D​f)−(ξ​f)​ξ]=0.\mu[(Df)-(\xi f)\xi]=0. (4.30)

We suppose, μ\mu is a non-zero constant. Then, from last equation, we get D​f=(ξ​f)​ξDf=(\xi f)\xi. Substitution of D​fDf by (ξ​f)​ξ(\xi f)\xi in (4.15) and use of (2.8), yields

(X​(ξ​f))​ξ−(ξ​f)​(ϕ​X)=(p2+12​n+1−λ−μ+2​n)​X(X(\xi f))\xi-(\xi f)(\phi X)=(\frac{p}{2}+\frac{1}{2n+1}-\lambda-\mu+2n)X (4.31)

for any vector field XX of χ​(M)\chi(M). Taking inner product of the foregoing equation with respect to ξ\xi and substituting the resultant relation in (4.31), we get

(ξ​f)​(ϕ​X)+(p2+12​n+1−λ−μ+2​n)​[X−η​(X)​ξ]=0.(\xi f)(\phi X)+(\frac{p}{2}+\frac{1}{2n+1}-\lambda-\mu+2n)[X-\eta(X)\xi]=0.

Contracting the last equation over XX and using n≠0n\neq 0, we obtain p2+12​n+1−λ−μ+2​n=0\frac{p}{2}+\frac{1}{2n+1}-\lambda-\mu+2n=0. Since μ\mu is a constant, we have λ\lambda is also constant. As we know f+λf+\lambda is constant, this yields ff is constant. This contradicts the fact that the soliton vector field VV is non-zero as we get V=D​f=0V=Df=0.
So, μ\mu must be identically equal to zero and the soliton reduces to gradient almost conformal Ricci soliton. Finally, the equation (4.22) reduces to Q​X=−2​n​X​∀X∈χ​(M)QX=-2nX~{}\forall X\in\chi(M). So, the manifold becomes Einstein with Einstein constant −2​n-2n. Furthermore, the scalar curvature of the manifold can be expressed as r=−2​n​(2​n+1)r=-2n(2n+1). ∎

Example 4.1.

We consider the example of the paper [11]. In this paper, authors considered the Euclidean space M=ℝ3M=\mathbb{R}^{3} with Cartesian coordinates (x,y,z)(x,y,z) and defined the normal almost paracontact metric structure (φ,ξ,η,g)(\varphi,\xi,\eta,g) on MM as follows:

φ​(∂∂x)\displaystyle\varphi(\frac{\partial}{\partial x}) =∂∂y,\displaystyle=\frac{\partial}{\partial y}, φ​(∂∂y)\displaystyle\varphi(\frac{\partial}{\partial y}) =∂∂x−y​∂∂z,\displaystyle=\frac{\partial}{\partial x}-y\frac{\partial}{\partial z}, φ​(∂∂z)\displaystyle\varphi(\frac{\partial}{\partial z}) =0,\displaystyle=0,
ξ\displaystyle\xi =2​∂∂z,\displaystyle=2\frac{\partial}{\partial z}, η\displaystyle\eta =12​(d​z+y​d​x),\displaystyle=\frac{1}{2}(dz+ydx),
(gi​j)=(y2−120y40140y4014)(g_{ij})=\begin{pmatrix}\frac{y^{2}-1}{2}&0&\frac{y}{4}\\ 0&\frac{1}{4}&0\\ \frac{y}{4}&0&\frac{1}{4}\end{pmatrix}

and authors has shown that the manifold MM is para-Sasakian. Authors has taken pseudo-orthonormal φ\varphi-basis e1=2​∂ye_{1}=2\partial y, e2=2​∂x−2​y​∂ze_{2}=2\partial x-2y\partial z and e3=ξ=2​∂ze_{3}=\xi=2\partial z and also obtained the expressions of the curvature tensor and the Ricci tensor respectively as follows:

R​(e1,e2)​e1\displaystyle R(e_{1},e_{2})e_{1} =−3​e2,\displaystyle=-3e_{2}, R​(e1,e2)​e2\displaystyle R(e_{1},e_{2})e_{2} =−3​e1,\displaystyle=-3e_{1}, R​(e1,e2)​e3\displaystyle R(e_{1},e_{2})e_{3} =0,\displaystyle=0,
R​(e1,e3)​e1\displaystyle R(e_{1},e_{3})e_{1} =ξ,\displaystyle=\xi, R​(e1,e3)​e2\displaystyle R(e_{1},e_{3})e_{2} =0,\displaystyle=0, R​(e1,e3)​e3\displaystyle R(e_{1},e_{3})e_{3} =−e1,\displaystyle=-e_{1},
R​(e2,e3)​e1\displaystyle R(e_{2},e_{3})e_{1} =0,\displaystyle=0, R​(e2,e3)​e2\displaystyle R(e_{2},e_{3})e_{2} =−ξ,\displaystyle=-\xi, R​(e2,e3)​e3\displaystyle R(e_{2},e_{3})e_{3} =−e2,\displaystyle=-e_{2},

and

S​(e1,e1)\displaystyle S(e_{1},e_{1}) =2,\displaystyle=2, S​(e2,e2)\displaystyle S(e_{2},e_{2}) =−2,\displaystyle=-2, S​(e3,e3)\displaystyle S(e_{3},e_{3}) =−2.\displaystyle=-2.

Also the scalar curvature rr=2. Thus

S​(X,Y)=2​g​(X,Y)−4​η​(X)​η​(Y)​∀X,Y∈χ​(M).S(X,Y)=2g(X,Y)-4\eta(X)\eta(Y)~{}~{}\forall X,Y\in\chi(M). (4.32)

Let f:M→ℝf:M\rightarrow\mathbb{R} be a smooth function defined by,

f​(x,y,z)=x22+y22+z2.f(x,y,z)=\frac{x^{2}}{2}+\frac{y^{2}}{2}+z^{2}. (4.33)

Then the gradient of ff, D​fDf is given by,

D​f=(x​∂∂x+y​∂∂y+2​z​∂∂z)Df=(x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+2z\frac{\partial}{\partial z}) (4.34)

Now (ℒD​f​g)​(e1,e1)=−2​g​(e1,e1)=2(\mathcal{L}_{Df}g)(e_{1},e_{1})=-2g(e_{1},e_{1})=2, (ℒD​f​g)​(e2,e2)=2​g​(e2,e2)=−2(\mathcal{L}_{Df}g)(e_{2},e_{2})=2g(e_{2},e_{2})=-2 and (ℒD​f​g)​(e3,e3)=4(\mathcal{L}_{Df}g)(e_{3},e_{3})=4. Then from the above results we can verify that,

(ℒD​f​g)​(X,Y)=2​{g​(X,Y)+η​(X)​η​(Y)}(\mathcal{L}_{Df}g)(X,Y)=2\{g(X,Y)+\eta(X)\eta(Y)\} (4.35)

for all X,Y∈χ​(M).X,Y\in\chi(M). From (4.32) and (4.35) we obtain that gg represents a gradient almost conformal η\eta-Ricci soliton i.e., it satisfies (1.2) for V=D​fV=Df, where ff is defined by (4.33), λ=p2−83\lambda=\frac{p}{2}-\frac{8}{3} and μ=3\mu=3.

5. Conclusion and Remarks

The study of conformal Ricci solitons and conformal η\eta Ricci solitons on Riemannian manifolds and pseudo-Riemannian manifolds is a great importance in the area of differential geometry, specially in Riemannian geometry and in special relativistic physics as well. Basically, conformal Ricci flow is the most prominent flagship of modern physics. The conformal η\eta-Ricci soliton is a new notion not only in the area of differentiable manifold but in the area of mathematical physics, general relativity and quantum cosmology, quantum gravity, Black hole as well. It deals a geometric and physical applications with relativistic viscous fluid spacetime admitting heat flux and stress, dark and dust fluid general relativistic spacetime, radiation era in general relativistic spacetime. The application of this solitons and our results will not only play an important and significant role in paracontact geometry, but also it has a motivational contribution in mathematical fluid dynamics, thermodynamics etc. We can also find the application of such solitons on some Einstein space time like hyper-generalized quasi-Einstein spacetime, mixed super quasi-Einstein spacetime with the connection of general relativity. There are some questions arise from our article to study further research.
(i) Is theorem 3.3 true if we assume soliton vector field VV is pointwise collinear with the characteristic vector field ξ\xi ?
(ii) Is theorem 4.2 true without assuming soliton vector field VV as an infnitesimal contact transformation?
(iii) Are theorem 4.2 and theorem 4.3 true if we consider the dimension of the manifold is 3?

6. Acknowledgements

The first author is the corresponding author and this work was financially supported by UGC Senior Research Fellowship of India, Sr. No. 2061540940. Ref. No:21/06/2015(i)EU-V.

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