# Certain Results on Ricci Solitons in Trans-Sasakian Manifolds

Authors: C. S. Bagewadi; Gurupadavva Ingalahalli
Journal: Journal of Mathematics
 (2013)

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## Abstract

We study and obtain results on Ricci solitons in trans-Sasakian manifolds satisfyingR(ξ,X)·C̃=0, P(ξ,X)·C̃=0, H(ξ,X)·S=0, and C̃(ξ,X)·S=0, where C̃, P, and H are quasiconformal, projective, and conharmonic curvature tensors.


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## Body

## 1. Introduction

During 1982, Hamilton [1] made the fundamental observation that Ricci flow is an excellent tool for simplifying the structure of a manifold. It is a process which deforms the metric of a Riemannian manifold analogous to the diffusion of heat there by smoothing out the regularity in the metric. It is given by (∂g/∂t)=-2Ricg.


### 1.1. Example

(i) If the manifold is an Euclidean space or more generally Ricci-flat, then Ricci flow leaves the metric unchanged. Conversely, any metric unchanged by Ricci flow is Ricci-flat.(ii) LetMn=Sn and gSn the standard metric on the unit n-sphere in the Euclidean space. If g0=r02gSn for some r0>0 (r0 is the radius), then g(t)=(r02-2(n-1)t)gSn is a solution. So the Ricci flow with g(0)=g0 collapses to a point at time t=r0/2(n-1).Ricci solitons move under the Ricci flow simply by diffeomorphisms of the initial metric, that is, they are stationary points of the Ricci flow in space of metrics onM. Here the metric g(t) is the pull back of the initial metric g(0) by a 1-parameter family of diffeomorphisms ϕt generated by a vector field on a manifold M.Letσ(t) be a smooth function of time. Since ϕt:M→M is a diffeomorphism and g(t) is a Riemannian metric on M (codomain), then by definition of pull back ϕt*g(t) is a metric on M (domain).Setg~(t)=σ(t)ϕt*(g(t)); then we have [2]
(1)∂g~∂t=σ′(t)ϕt*(g(t))+σ(t)ϕt*∂g∂t+σ(t)ϕt*(LXg).
(This follows from the definition of the Lie derivative.)Suppose we have a metricg0, a vector field Y, and λ∈R (all independent of time) such that
(2)LYg0+2Ricg0+2λg0=0.
If we choose g(t)=g0,  σ(t)=1-2λt, and X(t)=(1/σ(t))Y, then it gives a family of diffeomorphisms ϕt with ϕ0 identity; then using (2) in (1),  g~ defined above is a Ricci flow with g(0)=g0, that is,
(3)∂g~∂t=-2Ricg~.
Hence, LXg0+2Ricg0+2λg0=0 is a solution of the Ricci flow and is known as a Ricci soliton.Definition 1 (see [3]).
A Ricci soliton on a Riemannian manifold is defined by(4)LXg+2S+2λg=0.
It is said to be shrinking, steady, or expanding according to λ<0,  λ=0, and λ>0.


### 1.2. Example

(1)Hamilton Cigar Soliton [4]. Let M=R2, and ϕt:R2→R2 defined by ϕt(x,y)=(e-2tx,e-2ty) forms a family of a one-parameter group of diffeomorphisms. The vector field X generated by {ϕt} is X=-2(x(∂/∂x)+y(∂/∂y)). The metric g0 is obtained as g0=(dx2+dy2)/(1+x2+y2),  g~(t)=ϕt*(g0)=(dx2+dy2)/(e4t+x2+y2),  Ricg0=(2/(1+x2+y2))g0,  LXg0=(4/(1+x2+y2))g0. Using (4), we have λ=0. Hence, this Ricci soliton is steady and is called cigar soliton as it is asymptotic to a flat cylinder at infinity.(2)Cylinder Shrinking Soliton [4]. Consider the product of the sphere with a like {(S2×R,g(t)):t∈(-∞,0)}, where g(t)=2|t|gS2+dr2. Its Ricci tensor is given by Ric(g(t))=gS2=(1/2|t|)g(t)-(1/2|t|)dr2. If we set ϕ(θ,r,t)=(r2/4|t|), θ∈S2, r∈R, t<0, then 2Ricg(t)+2t∇ϕg(t)+(1/t)g(t)=0, λ=1/t<0. Hence g(t) is a shrinking soliton.In [3], Sharma obtained some interesting results on Ricci solitons in K-contact manifolds. Cǎlin and Crasmareanu [5] extended the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds. They studied the case of f-Kenmotsu manifolds satisfying a special condition called regular and showed that a symmetric parallel tensor field of second order is a constant multiple of the Riemannian metric. Using this result, they obtained the results on Ricci solitons concerned in f-Kenmotsu manifolds and 3-dimensional β-Kenmotsu manifolds.In [6, 7], Bagewadi and Ingalahalli studied Ricci solitons in α-Sasakian and Lorentzian α-Sasakian manifolds and proved that a symmetric parallel tensor field of second order is a constant multiple of the Riemannian metric; based on these they obtained the results on Ricci Solitons concerned in these manifolds. In [8], Tripathi obtained some results on Ricci solitons in contact metric manifolds. In [9], Nagaraja and Premalatha used semisymmetric conditions in Kenmotsu manifolds and obtained reuslts on Ricci solitons in Kenmotsu manifolds.Motivated by all this work in this paper we are studying the Ricci solitons in trans-Sasakian manifolds.Trans-Sasakian manifolds arose in a natural way from the classification of almost contact metric structures, and they appear as a natural generalization of both Sasakian and Kenmotsu manifolds. In [10], Gray-Hervella classification of almost Hermite manifolds appears as a class W4 of Hermitian manifolds which are closely related to locally conformally Kähler manifolds. An almost contact metric structure on a manifold M is called a trans-Sasakian structure [11] if the product manifold M×R belongs to the class W4. The class C6⊕C5 [12] coincides with the class of trans-Sasakian structures of type (α,β). The local nature of the two subclasses C5 and C6 of trans-Sasakian structures is characterized completely. It is known that [13] trans-Sasakian structures of type (0,0),   (α,0), and (0,β), are cosymplectic, α-Sasakian, and β-Kenmotsu, respectively, where α,β∈R. In [12], Marrero has shown that a trans-Sasakian manifold of dimension n≥5 is either cosymplectic or α-Sasakian or β-Kenmotsu manifold. Later many authors worked in this topic, trans-Sasakian manifolds, like Blair and Oubiña [14], Janssens and Vanhecke [13], Bagewadi and Venkatesha [15], U. De and K. De [16], De and Tripathi [17], Shaikh et al. [18], Nagaraja et al. [19]. Now we move on to the preliminaries.


## 1.1. Example

(i) If the manifold is an Euclidean space or more generally Ricci-flat, then Ricci flow leaves the metric unchanged. Conversely, any metric unchanged by Ricci flow is Ricci-flat.(ii) LetMn=Sn and gSn the standard metric on the unit n-sphere in the Euclidean space. If g0=r02gSn for some r0>0 (r0 is the radius), then g(t)=(r02-2(n-1)t)gSn is a solution. So the Ricci flow with g(0)=g0 collapses to a point at time t=r0/2(n-1).Ricci solitons move under the Ricci flow simply by diffeomorphisms of the initial metric, that is, they are stationary points of the Ricci flow in space of metrics onM. Here the metric g(t) is the pull back of the initial metric g(0) by a 1-parameter family of diffeomorphisms ϕt generated by a vector field on a manifold M.Letσ(t) be a smooth function of time. Since ϕt:M→M is a diffeomorphism and g(t) is a Riemannian metric on M (codomain), then by definition of pull back ϕt*g(t) is a metric on M (domain).Setg~(t)=σ(t)ϕt*(g(t)); then we have [2]
(1)∂g~∂t=σ′(t)ϕt*(g(t))+σ(t)ϕt*∂g∂t+σ(t)ϕt*(LXg).
(This follows from the definition of the Lie derivative.)Suppose we have a metricg0, a vector field Y, and λ∈R (all independent of time) such that
(2)LYg0+2Ricg0+2λg0=0.
If we choose g(t)=g0,  σ(t)=1-2λt, and X(t)=(1/σ(t))Y, then it gives a family of diffeomorphisms ϕt with ϕ0 identity; then using (2) in (1),  g~ defined above is a Ricci flow with g(0)=g0, that is,
(3)∂g~∂t=-2Ricg~.
Hence, LXg0+2Ricg0+2λg0=0 is a solution of the Ricci flow and is known as a Ricci soliton.Definition 1 (see [3]).
A Ricci soliton on a Riemannian manifold is defined by(4)LXg+2S+2λg=0.
It is said to be shrinking, steady, or expanding according to λ<0,  λ=0, and λ>0.


## 1.2. Example

(1)Hamilton Cigar Soliton [4]. Let M=R2, and ϕt:R2→R2 defined by ϕt(x,y)=(e-2tx,e-2ty) forms a family of a one-parameter group of diffeomorphisms. The vector field X generated by {ϕt} is X=-2(x(∂/∂x)+y(∂/∂y)). The metric g0 is obtained as g0=(dx2+dy2)/(1+x2+y2),  g~(t)=ϕt*(g0)=(dx2+dy2)/(e4t+x2+y2),  Ricg0=(2/(1+x2+y2))g0,  LXg0=(4/(1+x2+y2))g0. Using (4), we have λ=0. Hence, this Ricci soliton is steady and is called cigar soliton as it is asymptotic to a flat cylinder at infinity.(2)Cylinder Shrinking Soliton [4]. Consider the product of the sphere with a like {(S2×R,g(t)):t∈(-∞,0)}, where g(t)=2|t|gS2+dr2. Its Ricci tensor is given by Ric(g(t))=gS2=(1/2|t|)g(t)-(1/2|t|)dr2. If we set ϕ(θ,r,t)=(r2/4|t|), θ∈S2, r∈R, t<0, then 2Ricg(t)+2t∇ϕg(t)+(1/t)g(t)=0, λ=1/t<0. Hence g(t) is a shrinking soliton.In [3], Sharma obtained some interesting results on Ricci solitons in K-contact manifolds. Cǎlin and Crasmareanu [5] extended the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds. They studied the case of f-Kenmotsu manifolds satisfying a special condition called regular and showed that a symmetric parallel tensor field of second order is a constant multiple of the Riemannian metric. Using this result, they obtained the results on Ricci solitons concerned in f-Kenmotsu manifolds and 3-dimensional β-Kenmotsu manifolds.In [6, 7], Bagewadi and Ingalahalli studied Ricci solitons in α-Sasakian and Lorentzian α-Sasakian manifolds and proved that a symmetric parallel tensor field of second order is a constant multiple of the Riemannian metric; based on these they obtained the results on Ricci Solitons concerned in these manifolds. In [8], Tripathi obtained some results on Ricci solitons in contact metric manifolds. In [9], Nagaraja and Premalatha used semisymmetric conditions in Kenmotsu manifolds and obtained reuslts on Ricci solitons in Kenmotsu manifolds.Motivated by all this work in this paper we are studying the Ricci solitons in trans-Sasakian manifolds.Trans-Sasakian manifolds arose in a natural way from the classification of almost contact metric structures, and they appear as a natural generalization of both Sasakian and Kenmotsu manifolds. In [10], Gray-Hervella classification of almost Hermite manifolds appears as a class W4 of Hermitian manifolds which are closely related to locally conformally Kähler manifolds. An almost contact metric structure on a manifold M is called a trans-Sasakian structure [11] if the product manifold M×R belongs to the class W4. The class C6⊕C5 [12] coincides with the class of trans-Sasakian structures of type (α,β). The local nature of the two subclasses C5 and C6 of trans-Sasakian structures is characterized completely. It is known that [13] trans-Sasakian structures of type (0,0),   (α,0), and (0,β), are cosymplectic, α-Sasakian, and β-Kenmotsu, respectively, where α,β∈R. In [12], Marrero has shown that a trans-Sasakian manifold of dimension n≥5 is either cosymplectic or α-Sasakian or β-Kenmotsu manifold. Later many authors worked in this topic, trans-Sasakian manifolds, like Blair and Oubiña [14], Janssens and Vanhecke [13], Bagewadi and Venkatesha [15], U. De and K. De [16], De and Tripathi [17], Shaikh et al. [18], Nagaraja et al. [19]. Now we move on to the preliminaries.


## 2. Preliminaries

Ann-dimensional differential manifold M is said to be an almost contact metric manifold [20] if it admits a (1,1) tensor field ϕ, a vector field ξ, a 1-form η, and a Riemannian metric g, satisfying
(5)ϕ2=-I+η⊗ξ,η(ξ)=1,η∘ϕ=0,ϕξ=0,g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),g(X,ξ)=η(X),
for all vector fields X, Y on M.An almost contact metric manifoldM(ϕ,ξ,η,g) is said to be trans-Sasakian manifold if (M×R,J,G) belongs to the class W4 of the Hermitian manifolds [12], where J is the almost complex structure of M×R defined by
(6)J(Z,fddt)=(ϕZ-fξ,η(Z)ddt),
for all vector fields Z on M and smooth function f on M×R, and G is the product metric on M×R. This may be expressed by the following condition [13]:
(7)(∇Xϕ)Y=α(g(X,Y)ξ-η(Y)X)+β(g(ϕX,Y)ξ-η(Y)ϕX),
where α and β are scalar functions on M, and such a structure is said to be the trans-Sasakian structure of type (α,β). From (7), we have
(8)∇Xξ=-αϕX+β(X-η(X)ξ),(9)(∇Xη)(Y)=-αg(ϕX,Y)+βg(ϕX,ϕY).Note 1.
(1) For some smooth functionsα and β, if α≠0 and β=0,  α=0, and β≠0, then it reduces to α-Sasakian and β-Kenmotsu manifold, respectively.
(2) Ifα and β are scalars and α=1 and β=0,  α=0 and β=1, then it reduces to Sasakian and Kenmotsu manifold, respectively.
(3) Ifα=0 and β=0, then it reduces to cosymplectic manifold, respectively.In a trans-Sasakian manifold, we have [17]
(10)R(X,Y)ξ=(α2-β2)[η(Y)X-η(X)Y]-(Xα)ϕY-(Xβ)ϕ2Y+2αβ[η(Y)ϕX-η(X)ϕY]+(Yα)ϕX+(Yβ)ϕ2X,(11)R(ξ,X)ξ=(α2-β2-ξβ)(η(X)ξ-X),(12)η(R(X,Y)Z)=(α2-β2)·[g(Y,Z)η(X)-g(X,Z)η(Y)]+2αβ·[g(ϕY,Z)η(X)-g(ϕX,Z)η(Y)]+[(Xα)g(ϕY,Z)-(Yα)g(ϕX,Z)]+[(Xβ)g(ϕ2Y,Z)-(Yβ)g(ϕ2X,Z)],(13)2αβ+ξα=0,
where α,  β are functions and R is the Riemannian curvature tensor. From (8) we have
(14)(ℒξg)(X,Y)=2β[g(X,Y)-η(X)η(Y)].
From (4) and (14), we get
(15)S(X,Y)=-(λ+β)g(X,Y)+βη(X)η(Y).
The above equation yields
(16)QX=-(λ+β)X+βη(X)ξ,(17)S(X,ξ)=-λη(X),(18)r=-λn-(n-1)β,
where Q is the Ricci operator and r is the scalar curvature on M.Remark 2.
Since our study deals with Ricci solitons of trans-Sasakian manifolds and by Definition1, it is known that λ is a scalar quantity. Further, from (18) it is seen that λ is related to β. Hence in the calculations of our results which will be proved in the following it is necessary to assume α and β as simply scalar quantities.


### 2.1. Example for 3-Dimensional Trans-Sasakian Manifold

We consider the3-dimensional manifold M={(x,y,z)∈R3;z≠0}, where (x,y,z) are the standard co-ordinates in R3. Let {E1,E2,E3} be linearly independent global frame field on M given by
(19)E1=z(∂∂x+y∂∂z),E2=z∂∂y,E3=∂∂z.
Let g be the Riemannian metric defined by g(E1,E2)=g(E2,E3)=g(E1,E3)=0,   g(E1,E1)=g(E2,E2)=g(E3,E3)=1, where g is given by
(20)g=1z2[(1-y2z2)dx⊗dx+dy⊗dy+z2dz⊗dz].
The (ϕ,ξ,η) is given by η=dz-ydx, ξ=E3=∂/∂z, ϕE1=E2, ϕE2=-E1, ϕE3=0. The linearity property of ϕ and g yields that η(E3)=1, ϕ2U=-U+η(U)E3, g(ϕU,ϕW)=g(U,W)-η(U)η(W), for any vector fields U,W on M. By the definition of Lie bracket, we have
(21)[E1,E2]=yE2-z2E3,[E1,E3]=-1zE1,[E2,E3]=-1zE2.Let∇ be the Levi-Civita connection with respect to the above metric g given by the Koszula formula
(22)2g(∇XY,Z)=X(g(Y,Z))+Y(g(Z,X))-Z(g(X,Y))-g(X,[Y,Z])-g(Y,[X,Z])+g(Z,[X,Y]).
Then,
(23)∇E1E3=-1zE1+12z2E2,∇E2E3=-1zE2-12z2E1,∇E3E3=0,∇E2E2=yE1+1zE3,∇E1E2=-12z2E3,∇E2E1=12z2E3-yE2,∇E1E1=1zE3,∇E3E2=-12z2E1,∇E3E1=12z2E2.The tangent vectorsX and Y to M are expressed as linear combinations of E1,E2,E3, that is, X=Σi=13aiEi and Y=Σi=13biEi, where ai,bi (i=1,2,3) are scalars. Clearly (ϕ,ξ,η,g) and X,Y satisfy (5), (7), and (8) with α=-(1/2)z2≠0 and β=-1/z≠0. Thus M is a trans-Sasakian manifold.


### 2.2. Example for 3-Dimensionalα-Sasakian Manifolds

LetM={(x,y,z)∈R3}. Let (E1,E2,E3) be linearly independent vector fields given by
(24)E1=k∂∂y,E2=k(∂∂x+2y∂∂z),E3=∂∂z.
Let g be the Riemannian metric defined by g(E1,E2)=g(E2,E3)=g(E1,E3)=0, g(E1,E1)=g(E2,E2)=g(E3,E3)=1, where g is given by
(25)g=1k2[(1-4k2y2)dx⊗dx+dy⊗dy+k2dz⊗dz].
The (ϕ,ξ,η) is given by
(26)η=dz-2ydx,ξ=E3=∂∂z,ϕE1=E2,ϕE2=-E1,ϕE3=0.
The linearity property of ϕ and g yields that η(E3)=1, ϕ2U=-U+η(U)E3, g(ϕU,ϕW)=g(U,W)-η(U)η(W), for any vector fields U,W. By the definition of Lie bracket, we have
(27)[E1,E2]=2k2E3,[E1,E3]=[E2,E3]=0.
Let ∇ be the Levi-Civita connection; with respect to the above metric g given by Koszula formula (22) and by virtue of it we have
(28)∇E1E1=0,∇E2E2=0,∇E3E3=0,∇E1E2=k2E3,∇E2E1=-k2E3,∇E2E3=k2E1,∇E1E3=-k2E2,∇E3E1=-k2E2,∇E3E2=k2E1.
The tangent vectors X and Y to M are expressed as linear combinations of E1,E2,E3, that is, X=Σi=13aiEi and Y=Σi=13biEi, where ai and bi (i=1,2,3) are scalars. This becomes α-Sasakian manifold with α=k2.


### 2.3. Example for 3-Dimensionalβ-Kenmotsu Manifold

LetM={(x,y,z)∈R3}. Let (E1,E2,E3) be linearly independent vector fields given by
(29)E1=e-kz(∂∂x+∂∂y),E2=e-kz(-∂∂x+∂∂y),E3=∂∂z.
Let g be the Riemannian metric defined by g(E1,E2)=g(E2,E3)=g(E1,E3)=0, g(E1,E1)=g(E2,E2)=g(E3,E3)=1, where g is given by
(30)g=e2kz2(dx⊗dx+dy⊗dy)+dz⊗dz.
The (ϕ,ξ,η) is given by
(31)η=dz,ξ=E3=∂∂z,ϕE1=E2,ϕE2=-E1,ϕE3=0.
The linearity property of ϕ and g yields that η(E3)=1, ϕ2U=-U+η(U)E3, g(ϕU,ϕW)=g(U,W)-η(U)η(W), for any vector fields U,W on M. By the definition of Lie bracket, we have
(32)[E1,E2]=0,[E2,E3]=kE2,[E1,E3]=kE1.
Let ∇ be the Levi-Civita connection; with respect to above metric g given by Koszula formula (22) and by virtue of it, we have
(33)∇E1E3=kE1,∇E2E3=kE2,∇E3E3=0,∇E1E2=0,∇E2E2=-kE3,∇E3E2=0,∇E1E1=-kE3,∇E2E1=0,∇E3E1=0.
The tangent vectors X and Y to M are expressed as linear combinations of E1,E2,E3; that is, X=Σi=13aiEi and Y=Σi=13biEi, ai,bi (i=1,2,3) are scalars. Clearly M is a β-Kenmotsu manifold with β=k.


## 2.1. Example for 3-Dimensional Trans-Sasakian Manifold

We consider the3-dimensional manifold M={(x,y,z)∈R3;z≠0}, where (x,y,z) are the standard co-ordinates in R3. Let {E1,E2,E3} be linearly independent global frame field on M given by
(19)E1=z(∂∂x+y∂∂z),E2=z∂∂y,E3=∂∂z.
Let g be the Riemannian metric defined by g(E1,E2)=g(E2,E3)=g(E1,E3)=0,   g(E1,E1)=g(E2,E2)=g(E3,E3)=1, where g is given by
(20)g=1z2[(1-y2z2)dx⊗dx+dy⊗dy+z2dz⊗dz].
The (ϕ,ξ,η) is given by η=dz-ydx, ξ=E3=∂/∂z, ϕE1=E2, ϕE2=-E1, ϕE3=0. The linearity property of ϕ and g yields that η(E3)=1, ϕ2U=-U+η(U)E3, g(ϕU,ϕW)=g(U,W)-η(U)η(W), for any vector fields U,W on M. By the definition of Lie bracket, we have
(21)[E1,E2]=yE2-z2E3,[E1,E3]=-1zE1,[E2,E3]=-1zE2.Let∇ be the Levi-Civita connection with respect to the above metric g given by the Koszula formula
(22)2g(∇XY,Z)=X(g(Y,Z))+Y(g(Z,X))-Z(g(X,Y))-g(X,[Y,Z])-g(Y,[X,Z])+g(Z,[X,Y]).
Then,
(23)∇E1E3=-1zE1+12z2E2,∇E2E3=-1zE2-12z2E1,∇E3E3=0,∇E2E2=yE1+1zE3,∇E1E2=-12z2E3,∇E2E1=12z2E3-yE2,∇E1E1=1zE3,∇E3E2=-12z2E1,∇E3E1=12z2E2.The tangent vectorsX and Y to M are expressed as linear combinations of E1,E2,E3, that is, X=Σi=13aiEi and Y=Σi=13biEi, where ai,bi (i=1,2,3) are scalars. Clearly (ϕ,ξ,η,g) and X,Y satisfy (5), (7), and (8) with α=-(1/2)z2≠0 and β=-1/z≠0. Thus M is a trans-Sasakian manifold.


## 2.2. Example for 3-Dimensionalα-Sasakian Manifolds

LetM={(x,y,z)∈R3}. Let (E1,E2,E3) be linearly independent vector fields given by
(24)E1=k∂∂y,E2=k(∂∂x+2y∂∂z),E3=∂∂z.
Let g be the Riemannian metric defined by g(E1,E2)=g(E2,E3)=g(E1,E3)=0, g(E1,E1)=g(E2,E2)=g(E3,E3)=1, where g is given by
(25)g=1k2[(1-4k2y2)dx⊗dx+dy⊗dy+k2dz⊗dz].
The (ϕ,ξ,η) is given by
(26)η=dz-2ydx,ξ=E3=∂∂z,ϕE1=E2,ϕE2=-E1,ϕE3=0.
The linearity property of ϕ and g yields that η(E3)=1, ϕ2U=-U+η(U)E3, g(ϕU,ϕW)=g(U,W)-η(U)η(W), for any vector fields U,W. By the definition of Lie bracket, we have
(27)[E1,E2]=2k2E3,[E1,E3]=[E2,E3]=0.
Let ∇ be the Levi-Civita connection; with respect to the above metric g given by Koszula formula (22) and by virtue of it we have
(28)∇E1E1=0,∇E2E2=0,∇E3E3=0,∇E1E2=k2E3,∇E2E1=-k2E3,∇E2E3=k2E1,∇E1E3=-k2E2,∇E3E1=-k2E2,∇E3E2=k2E1.
The tangent vectors X and Y to M are expressed as linear combinations of E1,E2,E3, that is, X=Σi=13aiEi and Y=Σi=13biEi, where ai and bi (i=1,2,3) are scalars. This becomes α-Sasakian manifold with α=k2.


## 2.3. Example for 3-Dimensionalβ-Kenmotsu Manifold

LetM={(x,y,z)∈R3}. Let (E1,E2,E3) be linearly independent vector fields given by
(29)E1=e-kz(∂∂x+∂∂y),E2=e-kz(-∂∂x+∂∂y),E3=∂∂z.
Let g be the Riemannian metric defined by g(E1,E2)=g(E2,E3)=g(E1,E3)=0, g(E1,E1)=g(E2,E2)=g(E3,E3)=1, where g is given by
(30)g=e2kz2(dx⊗dx+dy⊗dy)+dz⊗dz.
The (ϕ,ξ,η) is given by
(31)η=dz,ξ=E3=∂∂z,ϕE1=E2,ϕE2=-E1,ϕE3=0.
The linearity property of ϕ and g yields that η(E3)=1, ϕ2U=-U+η(U)E3, g(ϕU,ϕW)=g(U,W)-η(U)η(W), for any vector fields U,W on M. By the definition of Lie bracket, we have
(32)[E1,E2]=0,[E2,E3]=kE2,[E1,E3]=kE1.
Let ∇ be the Levi-Civita connection; with respect to above metric g given by Koszula formula (22) and by virtue of it, we have
(33)∇E1E3=kE1,∇E2E3=kE2,∇E3E3=0,∇E1E2=0,∇E2E2=-kE3,∇E3E2=0,∇E1E1=-kE3,∇E2E1=0,∇E3E1=0.
The tangent vectors X and Y to M are expressed as linear combinations of E1,E2,E3; that is, X=Σi=13aiEi and Y=Σi=13biEi, ai,bi (i=1,2,3) are scalars. Clearly M is a β-Kenmotsu manifold with β=k.


## 3. Ricci Soliton in Trans-Sasakian Manifolds SatisfyingR(ξ,X)·C~=0

The quasiconformal curvature tensorC~ is defined by
(34)C~(X,Y)Z=aR(X,Y)Z+b[S(Y,Z)X-S(X,Z)YaR(X,Y)Z+b+g(Y,Z)QX-g(X,Z)QY]-rn(an-1+2b)[g(Y,Z)X-g(X,Z)Y],
where a,b are constants. Taking Z=ξ in (34) and using (10), (15), (16), and (17), we get
(35)C~(X,Y)ξ=[a(α2-β2)-b(2λ+β)-rn(an-1+2b)]·[η(Y)X-η(X)Y]+2αβa·[η(Y)ϕX-η(X)ϕY].
Similarly using (12), (15), (16), and (17) in (34), we get
(36)η(C~(X,Y)Z)=[a(α2-β2)-b(2λ+β)-rn(an-1+2b)]·[g(Y,Z)η(X)-g(X,Z)η(Y)]-2αβa·[g(ϕX,Z)η(Y)-g(ϕY,Z)η(X)].
We assume that the condition R(ξ,X)·C~=0; then we have
(37)R(ξ,X)C~(Y,Z)W-C~(R(ξ,X)Y,Z)W-C~(Y,R(ξ,X)Z)W-C~(Y,Z)R(ξ,X)W=0,
for all vector fields X, Y, Z, and W on M. Using (10) in (37), we have
(38)(α2-β2)[g(C~(Y,Z)W,X)ξ-η(C~(Y,Z)W)X]+2αβ[g(ϕC~(Y,Z)W,X)ξ+η(C~(Y,Z)W)ϕX]+(gradα)g(ϕC~(Y,Z)W,X)-(gradβ)·g(ϕC~(Y,Z)W,ϕX)-(α2-β2)·[g(X,Y)C~(ξ,Z)W-η(Y)C~(X,Z)W]-2αβ·[g(ϕY,X)C~(ξ,Z)W+η(Y)C~(ϕX,Z)W]-g(ϕY,X)C~(gradα,Z)W+g(ϕX,ϕY)·C~(gradβ,Z)W-(α2-β2)·[g(X,Z)C~(Y,ξ)W-η(Z)C~(Y,X)W]-2αβ·[g(ϕZ,X)C~(Y,ξ)W+η(Z)C~(Y,ϕX)W]-g(ϕZ,X)C~(Y,gradα)W+g(ϕX,ϕZ)·C~(Y,gradβ)W-(α2-β2)·[g(X,W)C~(Y,Z)ξ-η(W)C~(Y,Z)X]-2αβ·[g(ϕW,X)C~(Y,Z)ξ+η(W)C~(Y,Z)ϕX]-g(ϕW,X)C~(Y,Z)gradα+g(ϕX,ϕW)C~(Y,Z)gradβ=0.
By taking an inner product with ξ, we get
(39)(α2-β2)[g(C~(Y,Z)W,X)-η(C~(Y,Z)W)η(X)]-(α2-β2)[g(X,Y)η(C~(ξ,Z)W)-(α2-β2)-η(Y)η(C~(X,Z)W)]-2αβ·[g(ϕY,X)η(C~(ξ,Z)W)+η(Y)η(C~(ϕX,Z)W)]-g(ϕY,X)η(C~(gradα,Z)W)+g(ϕX,ϕY)·η(C~(gradβ,Z)W)-(α2-β2)·[g(X,Z)η(C~(Y,ξ)W)-η(Z)η(C~(Y,X)W)]-2αβ·[g(ϕZ,X)η(C~(Y,ξ)W)+η(Z)η(C~(Y,ϕX)W)]-g(ϕZ,X)η(C~(Y,gradα)W)+g(ϕX,ϕZ)·η(C~(Y,gradβ)W)-(α2-β2)·[g(X,W)η(C~(Y,Z)ξ)-η(W)η(C~(Y,Z)X)]-2αβ[g(ϕW,X)η(C~(Y,Z)ξ)+η(W)η(C~(Y,Z)ϕX)]-g(ϕW,X)η(C~(Y,Z)gradα)+g(ϕX,ϕW)η(C~(Y,Z)gradβ)=0.
In view of (34), (35), and (36) and Remark 2 in (39), then we have
(40)(α2-β2)[g(R(Y,Z)W,X)(*-β2)+b[S(Z,W)g(Y,X)-S(Y,W)g(Z,X)+b(α2-β2)+g(Z,W)S(Y,X)-g(Y,W)S(Z,X)](α2-β2)-rn(an-1+2b)(α2-β2)·[g(Z,W)g(Y,X)-g(Y,W)g(Z,X)]]+(α2-β2)[a(α2-β2)-b(2λ+β)-rn(an-1+2b)]·[g(X,Z)g(Y,W)-g(X,Y)g(Z,W)]+2αβa(α2-β2)·[g(X,Z)g(ϕY,W)-g(ϕY,X)η(W)η(Z)+g(ϕZ,X)·η(Y)η(W)]+2αβ[a(α2-β2)-b(2λ+β)-rn(an-1+2b)]·[g(X,ϕZ)g(Y,W)-g(X,ϕY)g(Z,W)]+4α2β2a·[g(ϕZ,X)g(ϕY,W)+g(ϕY,ϕX)η(W)η(Z)-g(ϕZ,ϕX)η(Y)η(W)]=0.
Taking X=Y=ei in (40) and summing over i=1,2,…,n, we get
(41)(α2-β2)[a+b(n-2)]S(Z,W)+[br(α2-β2)-r(α2-β2)(n-1)n(an-1+2b)]·g(Z,W)-(α2-β2)(n-1)·[a(α2-β2)-b(2λ+β)-rn(an-1+2b)]·g(Z,W)-2αβa(α2-β2)g(ϕZ,W)+2αβ·[a(α2-β2)-b(2λ+β)-rn(an-1+2b)]·g(ϕZ,W)+4α2β2a[-g(Z,W)+nη(Z)η(W)]=0.
Putting Z=W=ei in (41) and summing over i=1,2,…,n, we get from (18)
(42)λ[-n(a+b(n-2))-n2b+2bn(n-1)]=(n-1)β(a+b(n-2))+a(α2-β2)n(n-1).
Since by [12], a trans-Sasakian manifold of dimension n≥5 is either cosymplectic or α-Sasakian or β-Kenmotsu manifold. So, based on this we state the following.Theorem 3.
A Ricci soliton(g,ξ,λ) in a quasiconformally semisymmetric α-Sasakian manifold is shrinking.Proof.
In (42) put β=0; then we have
(43)λ[-n(a+b(n-2))-n2b+2bn(n-1)]=aα2n(n-1)⇒-aλn=aα2n(n-1).
On simplification we get λ=-α2(n-1) and λ<0. Hence by Definition 1 Ricci soliton is shrinking.Theorem 4.
A Ricci soliton(g,ξ,λ) in a quasiconformally semisymmetric β-Kenmotsu manifold is expanding.Proof.
In (42) put α=0; then we have
(44)-aλn=(n-1)β(a+b(n-2))-aβ2n(n-1).⇒-aλn=-β(n-1)[-(a+b(n-2))+aβn].
If a+b(n-2)=0, then
(45)λ=β2(n-1).
The above equation implies thatλ>0. Hence Ricci soliton is expanding.Theorem 5.
A Ricci soliton(g,ξ,λ) in a quasiconformally semisymmetric cosympletic manifold is steady.Proof.
In (42) put α=0 and β=0; then we have
(46)-aλn=0.
The above equation implies that λ=0. Hence Ricci soliton is steady.


## 4. Ricci Soliton in Trans-Sasakian Manifolds SatisfyingP(ξ,X)·C~=0

The projective curvature tensor is defined by(47)P(X,Y)Z=R(X,Y)Z-1n-1[S(Y,Z)X-S(X,Z)Y].
Putting X=ξ in (47), we get
(48)P(ξ,Y)Z=R(X,Y)Z-1n-1[S(Y,Z)X-S(X,Z)Y].
We assume that P(ξ,X)·C~=0; then we have
(49)P(ξ,X)C~(Y,Z)W-C~(P(ξ,X)Y,Z)W-C~(Y,P(ξ,X)Z)W-C~(Y,Z)P(ξ,X)W=0.
Using (48) in (49), we obtain
(50)(α2-β2)[g(X,C~(Y,Z)W)ξ-η(C~(Y,Z)W)X]+2αβ·[g(ϕC~(Y,Z)W,X)ξ+η(C~(Y,Z)W)ϕX]+(gradα)·g(ϕC~(Y,Z)W,X)-(gradβ)g(ϕC~(Y,Z)W,ϕX)-1n-1[-(λ+β)g(X,C~(Y,Z)W)ξ+βη(C~(Y,Z)W)·η(X)ξ+λη(C~(Y,Z)W)X]-(α2-β2)·[g(X,Y)C~(ξ,Z)W-η(Y)C~(X,Z)W]-2αβ·[g(ϕY,X)C~(ξ,Z)W+η(Y)C~(ϕX,Z)W]-g(ϕY,X)·C~(gradα,Z)W+g(ϕX,ϕY)C~(gradβ,Z)W+1n-1·[-(λ+β)g(X,Y)C~(ξ,Z)W+βη(Y)η(X)C~(ξ,Z)W+λη(Y)C~(X,Z)W]-(α2-β2)·[g(X,Z)C~(Y,ξ)W-η(Z)C~(Y,X)W]-2αβ·[g(ϕZ,X)C~(Y,ξ)W+η(Z)C~(Y,ϕX)W]-g(ϕZ,X)·C~(Y,gradα)W+g(ϕX,ϕZ)C~(Y,gradβ)W+1n-1·[-(λ+β)g(X,Z)C~(Y,ξ)W+βη(Z)η(X)C~(Y,ξ)W+λη(Z)C~(Y,X)W]-(α2-β2)·[g(X,W)C~(Y,Z)ξ-η(W)C~(Y,Z)X]-2αβ·[g(ϕW,X)C~(Y,Z)ξ+η(W)C~(Y,Z)ϕX]-g(ϕW,X)·C~(Y,Z)gradα+g(ϕX,ϕW)C~(Y,Z)gradβ+1n-1·[-(λ+β)g(X,W)C~(Y,Z)ξ+βη(W)η(X)C~(Y,Z)ξ+λη(W)C~(Y,Z)X]=0.
By taking an inner product with ξ and by using Remark 2 in (50), then, we obtain
(51)[(α2-β2)+λ+βn-1]·[g(X,C~(Y,Z)W)-η(C~(Y,Z)W)η(X)]-(α2-β2)·[g(X,Y)η(C~(ξ,Z)W)-η(Y)η(C~(X,Z)W)]-2αβ·[g(ϕY,X)η(C~(ξ,Z)W)+η(Y)η(C~(ϕX,Z)W)]+1n-1[-(λ+β)g(X,Y)η(C~(ξ,Z)W)+βη(Y)+1n-1·η(X)η(C~(ξ,Z)W)+λη(Y)η(C~(X,Z)W)]-(α2-β2)[g(X,Z)η(C~(Y,ξ)W)-η(Z)η(C~(Y,X)W)]-2αβ[g(ϕZ,X)η(C~(Y,ξ)W)+η(Z)η(C~(Y,ϕX)W)]+1n-1[-(λ+β)g(X,Z)η(C~(Y,ξ)W)+βη(Z)+1n-1·η(X)η(C~(Y,ξ)W)+λη(Z)η(C~(Y,X)W)]+(α2-β2)η(W)η(C~(Y,Z)X)-2αβη(W)·η(C~(Y,Z)ϕX)+λn-1η(W)η(C~(Y,Z)X)=0.
By using (34), (35), and (36) in (51) and on simplification, we obtain
(52)[(α2-β2)+λ+βn-1]ag(X,R(Y,Z)W)+b·[S(Z,W)g(Y,X)-S(Y,W)g(Z,X)+g(Z,W)S(Y,X)-g(Y,W)S(Z,X)]-rn(an-1+2b)[g(Z,W)g(Y,X)-g(Y,W)g(Z,X)]-[(α2-β2)+λ+βn-1]·[aaaa(α2-β2)η(X){g(Z,W)η(Y)-g(Y,W)η(Z)}+2αβ[g(ϕZ,W)η(Y)-g(ϕY,W)η(Z)]+b·[S(Z,W)η(Y)η(X)-S(Y,W)η(Z)η(X)+g(Z,W)·S(Y,ξ)η(X)-g(Y,W)S(Z,ξ)η(X)]-rn(an-1+2b)·[g(Z,W)η(Y)η(X)-g(Y,W)η(Z)η(X)](α2-β2)]+(α2-β2)k·[g(Z,W)η(X)η(Y)-g(X,Y)g(Z,W)+g(X,Z)g(Y,W)-g(Y,W)η(X)η(Z)]+2αβk·[g(ϕZ,X)g(Y,W)-g(ϕY,X)g(Z,W)]+λ+βn-1k·[g(X,Y)η(Z)η(W)-g(X,Y)g(Z,W)+g(Z,W)η(X)η(Y)-g(X,Z)η(Y)η(W)+g(X,Z)g(Y,W)-g(Y,W)η(X)η(Z)]+2aλαβn-1·[g(ϕZ,W)η(X)η(Y)+g(ϕZ,X)η(W)η(Y)-g(ϕY,X)η(W)η(Z)]+λn-1k·[g(X,Z)η(Y)η(W)-g(X,Y)η(Z)η(W)]+λ+βn-1·[2aαβ{g(X,Z)g(ϕY,W)-g(ϕY,W)η(X)η(Z)}]+2aλαβ(α2-β2)·[g(X,Z)g(ϕY,W)-g(ϕY,W)η(X)η(Z)+g(ϕZ,W)η(X)η(Y)-g(ϕY,X)η(W)η(Z)+g(ϕZ,X)η(Y)η(W)]+4aα2β2·[g(ϕZ,X)g(ϕY,W)+g(ϕY,ϕX)η(W)η(Z)-g(ϕZ,ϕX)η(W)η(Y)]=0,
where k=[a(α2-β2)-b(2λ+β)-(r/n)(a/(n-1)+2b)].TakingX=Y=ei in (52) and summing over i=1,2,…,n, and on simplification, we obtain
(53)[(α2-β2)+λ+βn-1](a+b(n-2))S(Z,W)+(α2-β2)·[bλ(n-2)-a(α2-β2)(n-1)]g(Z,W)+λ+βn-1·[bλ(n-2)-a(α2-β2)(n-1)]g(Z,W)+βn-1·[a(α2-β2)-b(2λ+β)-rn(an-1+2b)](n-1)η(Z)·η(W)+2αβ[a(α2-β2)-b(2λ+β)-rn(an-1+2b)]·g(ϕZ,W)+2aαβλn-1g(ϕZ,W)+2aαβ(α2-β2)·g(ϕZ,W)+4aα2β2[-g(Z,W)+nη(Z)η(W)]=0.
Taking Z=W=ei in (53) and summing over i=1,2,…,n, and on simplification, we obtain
(54)2r(a+b(n-2))[(α2-β2)+λ+βn-1]+2n(α2-β2)·[bλ(n-2)-a(α2-β2)(n-1)]+2n(λ+β)n-1·[bλ(n-2)-a(α2-β2)(n-1)]+2βn-1·[a(α2-β2)-b(2λ+β)-rn(an-1+2b)]·(n-1)=0.
If a+b(n-2)=0 and on simplification, we get
(55)nλ2+λ(n-1)[β+2n(α2-β2)]+(n-1)2(α2-β2)[n(α2-β2)+β]=0.
Again by [12], a trans-Sasakian manifold of dimension n≥5 is either cosymplectic or α-Sasakian or β-Kenmotsu manifold. So, based on this we state the following.Theorem 6.
A Ricci soliton in anα-Sasakian manifold satisfying P(ξ,X)·C~=0 is shrinking, provided a+b(n-2)=0.Proof.
In (55) put β=0; then we have
(56)nλ2+2nα2λ(n-1)+n(n-1)2α4=0⇒[λ+(n-1)α2]2=0,
that is, λ=-(n-1)α2 and λ<0. This completes the proof of the theorem.Theorem 7.
A Ricci soliton in aβ-Kenmotsu manifold satisfying P(ξ,X)·C~=0 is expanding, provided a+b(n-2)=0.Proof.
In (55) put α=0; then we have
(57)nλ2+λ(n-1)[β-2nβ2]-(n-1)2β2[-nβ2+β]=0.
On simplifying the above quadratic equation then we get
(58)λ=-(n-1)β(1-2nβ)±(n-1)β2n
that is either
(59)λ=-(n-1)β(1-2nβ)+(n-1)β2norλ=-(n-1)β(1-2nβ)-(n-1)β2n
on simplification we get
(60)λ=(n-1)β2orλ=(n-1)β(nβ-1)n.
In both cases λ>0. This completes the proof of the theorem.Theorem 8.
A Ricci soliton in a cosympletic manifolds satisfyingP(ξ,X)·C~=0 is steady, provided a+b(n-2)=0.Proof.
In (55) put α=0 and β=0; then we have
(61)nλ2=0,
since λ=0. This completes the proof of the theorem.


## 5. Ricci Soliton in Trans-Sasakian Manifolds SatisfyingH(ξ,X)·S=0

LetM be an n-dimensional trans-Sasakian manifolds admitting a Ricci soliton (g,V,λ). The conharmonic curvature tensor on M is given by
(62)H(X,Y)Z=R(X,Y)Z-1n-2·[S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY].
Putting X=ξ in (62) and by using (10), (15), (16), and (17), we get
(63)H(ξ,Y)Z=(α2-β2)[g(Y,Z)ξ-η(Z)Y]+2αβ[g(ϕY,Z)ξ+η(Z)ϕY]+g(Y,ϕZ)·(gradα)-g(ϕY,ϕZ)(gradβ)-1n-2·[-(2λ+β)g(Y,Z)ξ+(2λ+β)η(Z)Y],η(H(ξ,Y)Z)=(α2-β2)[g(Y,Z)-η(Z)η(Y)]-1n-2·[-(2λ+β)g(Y,Z)+(2λ+β)η(Z)η(Y)].
We assume that H(ξ,X)·S=0 holds. Then we have
(64)S(H(ξ,X)Y,Z)+S(Y,H(ξ,X)Z)=0.
By using (15) in (64), then we have
(65)βη(Z)η(H(ξ,X)Y)-(λ+β)g(H(ξ,X)Y,Z)-(λ+β)·g(Y,H(ξ,X)Z)+βη(Y)η(H(ξ,X)Z)=0,
that is,
(66)β[η(Z)η(H(ξ,X)Y)+η(Y)η(H(ξ,X)Z)]=(λ+β)[g(H(ξ,X)Y,Z)+g(Y,H(ξ,X)Z)].
By using (63) in (66), we have
(67)2η(X)η(Y)η(Z)[-β(α2-β2)-β(2λ+β)(n-2)]+[β(α2-β2)+β(2λ+β)(n-2)]·[g(X,Z)η(Y)+g(X,Y)η(Z)]=0.
Taking X=Y=ei and summing over i=1,2,…,n, and by virtue of (18), we obtain
(68)λ=-(α2-β2)(n-2)-β2.
Again by [12], a trans-Sasakian manifold of dimension n≥5 is either cosymplectic or α-Sasakian or β-Kenmotsu manifold. So, based on this we state the following.Theorem 9.
A Ricci soliton in anα-Sasakian manifold satisfying H(ξ,X)·S=0 is shrinking.Proof.
In (68) put β=0; then we have
(69)λ=-α2(n-2)2.
We get λ=-(α2(n-2))/2<0. This completes the proof of the theorem.Theorem 10.
A Ricci soliton in aβ-Kenmotsu manifold satisfying H(ξ,X)·S=0 is expanding if β>0 and shrinking if β<0.Proof.
In (68) put α=0; then we have
(70)λ=β[β(n-2)+1]2.
If β>0, then λ>0 and Ricci soliton is expanding. If β<0, then λ<0 and Ricci soliton is shrinking. This completes the proof of the theorem.Theorem 11.
A Ricci soliton in a cosymplectic manifolds satisfyingH(ξ,X)·S=0 is steady.Proof.
In (68) put α=0 and β=0; then we have
(71)λ=0.
This completes the proof of the theorem.


## 6. Ricci Soliton in Trans-Sasakian Manifolds SatisfyingC~(ξ,X)·S=0

We assume thatC~(ξ,X)·S=0 holds. Then we have
(72)S(C~(ξ,X)Y,Z)+S(Y,C~(ξ,X)Z)=0.
By using (15) in (72), then we have
(73)βη(Z)η(C~(ξ,X)Y)-(λ+β)g(C~(ξ,X)Y,Z)-(λ+β)·g(Y,C~(ξ,X)Z)+βη(Y)η(C~(ξ,X)Z)=0,
that is,
(74)β[η(Z)η(C~(ξ,X)Y)+η(Y)η(C~(ξ,X)Z)]=(λ+β)[g(C~(ξ,X)Y,Z)+g(Y,C~(ξ,X)Z)].
By using (63) in (74), we have
(75)[aβ(α2-β2)-bβ(2λ+β)-rβn[a(n-1)+2b]]·[g(X,Z)η(Y)+g(X,Y)η(Z)-2η(X)η(Y)η(Z)]=0.
Taking X=Y=ei and summing over i=1,2,…,n, and if a+b(n-2)=0, and by virtue of (18), we obtain
(76)λ=-(α2-β2)(n-1).
Again by [12], a trans-Sasakian manifold of dimension n≥5 is either cosymplectic or α-Sasakian or β-Kenmotsu manifold. So, based on this we state the following.Theorem 12.
A Ricci soliton in anα-Sasakian manifold satisfying C~(ξ,X)·S=0 is shrinking, provided a+b(n-2)=0.Proof.
In (76) put β=0; then we have
(77)λ=-α2(n-1).
We get λ=-α2(n-1)<0. This completes the proof of the theorem.Theorem 13.
A Ricci soliton in aβ-Kenmotsu manifold satisfying C~(ξ,X)·S=0 is expanding, provided a+b(n-2)=0.Proof.
In (76) put α=0; then we have
(78)λ=β2(n-1)
then the above equation implies that λ>0. This completes the proof of the theorem.Theorem 14.
A Ricci soliton in a cosymplectic manifolds satisfyingC~(ξ,X)·S=0 is steady, provided a+b(n-2)=0.Proof.
In (76) put α=0 and β=0, then we have
(79)λ=0.
This completes the proof of the theorem.



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*Source: 787408-2013-02-13.xml*