# Certain Results on Ricci Solitons inα-Sasakian Manifolds

Authors: S. R. Ashoka; C. S. Bagewadi; Gurupadavva Ingalahalli
Journal: Geometry
 (2013)

---

## Abstract

We study Ricci solitons inα-Sasakian manifolds and show that it is a shrinking or expanding soliton and the manifold is Einstein with Killing vector field. Further, we prove that if V is conformal Killilng vector field, then the Ricci soliton in 3-dimensional α-Sasakian manifolds is shrinking or expanding but cannot be steady.


---

## Body

## 1. Introduction

A Ricci soliton(g,V,λ) is a generalization of an Einstein metric and is defined on a Riemannian manifold (M,g) by
(1)ℒVg+2S+2λg=0,
where V is a complete vector field on M and λ is a constant. The Ricci soliton is said to be shrinking, steady, or expanding according as λ is negative, zero, and positive, respectively. Long-existing solutions, that is, solutions which exist on an infinite time interval, are the self-similar solutions, which in Ricci flow are called Ricci soliton.Compact Ricci solitons are the fixed points of the Ricci flow∂g/∂t=-2Ric(g) projected from the space of metrics onto its quotient modulo diffeomorphisms and scalings and often arise as blow-up limits for the Ricci flow on compact manifolds. If the vector field V is the gradient of a potential function -f, then g is called a gradient Ricci soliton and (1) assumes the form
(2)∇∇f=S+λg.
A Ricci soliton on a compact manifold is a gradient Ricci soliton. A Ricci soliton on a compact manifold has constant curvature in dimension 2 [1] and also in dimension 3 [2]. In [3], Perelman proved that a Ricci soliton on a compact n-manifold is a gradient Ricci soliton. In [4], Sharma studied Ricci solitons in K-contact manifolds, where the structure field ξ is Killing, and he proved that a complete K-contact gradient soliton is compact Einstein and Sasakian. In [5], Tripathi studied Ricci solitons in N(k)-contact metric and (k,μ) manifolds. In [6], Ghosh and Sharma studied K-contact metrics as Ricci solitons. In [7], Nagaraja and Premalatha studied Ricci solitons in f-Kenmotsu manifolds and 3-dimensional trans-Sasakian manifolds. Recently, Bagewadi and Ingalahalli [8] studied Ricci solitons in Lorentzian α-Sasakian manifolds. Motivated by the previous studies on Ricci solitons, in this paper, we study Ricci solitons in an α-Sasakian manifolds, where α is some constant.


## 2. Preliminaries

LetM be an almost contact metric manifold of dimension n equipped with an almost contact metric structure (ϕ,ξ,η,g) consisting of a (1,1) tensor field ϕ, a vector field ξ, a 1-form η, and a Riemannian metric g, which satisfy
(3)ϕ2=-I+η⊗ξ,η(ξ)=1,η∘ϕ=0,ϕξ=0,(4)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),η(X)=g(X,ξ),
for all X,Y∈𝔛(M). An almost contact metric manifold M(ϕ,ξ,η,g) is said to be α-Sasakian manifold if the following conditions hold:
(5)(∇Xϕ)Y=α(g(X,Y)ξ-η(Y)X),∇Xξ=-αϕX,(∇Xη)Y=αg(X,ϕY),
for some nonzero constant α on M.In anα-Sasakian manifold, we have the following relations:
(6)R(X,Y)ξ=α2[η(Y)X-η(X)Y],(7)R(ξ,X)Y=α2[g(X,Y)ξ-η(Y)X],(8)η(R(X,Y)Z)=α2[g(Y,Z)η(X)-g(X,Z)η(Y)],(9)S(X,ξ)=α2(n-1)η(X),(10)S(ξ,ξ)=α2(n-1),(11)Qξ=α2(n-1)ξ,
for all X,Y,Z∈𝔛(M), where R is the Riemannian curvature tensor, S is the Ricci tensor, and Q is the Ricci operator.


## 3. Ricci Solitons inα-Sasakian Manifold

In this section, we prove some theorems on Ricci solitons inα-Sasakian manifold.Proposition 1.
A complete Einsteinα-Sasakian manifold is compact.Proof.
LetM be a complete Einstein α-Sasakian manifold, then the general form is given by
(12)S(X,Y)=rng(X,Y)⟹Q=rnI.
Operating ξ in (12) and using (11) show r=n(n-1)α2. Hence we get Q=α2(n-1)I. So the Ricci curvatures are equal to α2(n-1) which is a positive constant. By Myers's theorem [9], we conclude that M is compact.Theorem 2.
If the metricg of an α-Sasakian manifold (M,g) is a gradient Ricci soliton, then the Ricci soliton is a shrinking soliton and (M,g) is compact Einstein.Proof.
Equation (2) can be written as
(13)∇YDf=QY+λY,
where D denotes the gradient operator of g and Y denotes an arbitrary vector field on M. Using this we derive
(14)R(X,Y)Df=(∇XQ)Y-(∇YQ)X.
Taking its inner product with ξ, substituting X=ξ, and using (7) and (11), we have
(15)Df=(ξf)ξ.
Substituting (15) in (13), we get
(16)Y(ξf)η(X)-α(ξf)g(ϕY,X)=S(X,Y)+λg(X,Y).
Interchanging X and Y in (16), we have
(17)X(ξf)η(Y)-α(ξf)g(ϕX,Y)=S(Y,X)+λg(Y,X).
Adding (16) and (17), we have
(18)X(ξf)η(Y)+Y(ξf)η(X)=2S(Y,X)+2λg(Y,X).
Putting Y=ξ in (18), we have
(19)X(ξf)=[α2(n-1)+λ]η(X).
The use of the previous two equations provides
(20)S(X,Y)=[α2(n-1)+λ]η(X)η(Y)-λg(Y,X).
Consequently, (13) assumes the form
(21)∇YDf=[α2(n-1)+λ]η(Y)ξ.
Using this, we compute R(X,Y)Df, and taking inner product with ξ (bearing in mind that Df=(ξf)ξ), we obtain [α2(n-1)+λ]=0. Therefore, from (19), we have X(ξf)=0; that is, ξf is constant or ξf=c. Hence (15) can be written as df=cη. Its exterior derivative implies cdη=0. Hence c=0. Thus f is constant.
Consequently, (13) reduces to S=α2(n-1)g; that is, an α-Sasakian manifold is an Einstein. Also, as λ=-α2(n-1) is negative for α>0 or α<0; that is, Ricci soliton in α-Sasakian manifolds is shrinking.From above-mentioned theorem, we state the following corollary.Corollary 3.
If a metricg of a compact α-Sasakian manifold (M,g) is a Ricci soliton, then g is a shrinking soliton and the manifold is Einstein.Theorem 4.
If a metricg in an α-Sasakian manifold is a Ricci soliton with V=ξ, then it is Einstein.Proof.
PuttingV=ξ in (1), then we have
(22)(ℒξg+2S+2λg)(X,Y)=0,
where
(23)(ℒξg)(X,Y)=g(∇Xξ,Y)+g(X,∇Yξ)=0.
Substituting (23) in (22), then we get the result.Proposition 5.
If anα-Sasakian manifold is a Ricci soliton with V point-wise collinear with ξ, then V is a constant multiple of ξ and the manifold is Einstein.Proof.
From (1), we have
(24)(ℒVg)(X,Y)+2S(X,Y)+2λg(X,Y)=0,
where
(25)(ℒVg)(X,Y)=g(∇XV,Y)+g(X,∇YV).
Substituting (25) in (24), then we obtain
(26)g(∇XV,Y)+g(X,∇YV)+2S(X,Y)+2λg(X,Y)=0.
Putting V=aξ in (26), we get
(27)(Xa)η(Y)+(Ya)η(X)+2S(X,Y)+2λg(X,Y)=0.
Putting X=Y=ξ in (27), we have
(28)(ξa)+α2(n-1)+λ=0.
Again putting X=ξ in (27), we obtain
(29)(Ya)=[-α2(n-1)-λ]η(Y).
Equation (30) implies that
(30)da=[-α2(n-1)-λ]η.
Applying d on both sides,
(31)d2a=[-α2(n-1)-λ]dη.
Equation (31) implies that d2a=0, but dη is nowhere vanishing. Therefore, -λ-α2(n-1)=0 which implies da=0; that is, a is constant. As ξ is Killing, we conclude that the manifold is Einstein which completes the proof.Definition 6.
A vector fieldV is said to be conformal Killing vector field if it satisfies
(32)ℒVg=2ρg
for some scalar function ρ.Theorem 7.
Let(g,V,λ) be a Ricci soliton in an α-Sasakian manifolds (M,g). Then (M,g) is Ricci-semisymmetric if and only if V is conformal Killing.Proof.
Suppose thatV is a conformal Killing vector field, and from (1), we have
(33)2ρg(X,Y)+2S(X,Y)+2λg(X,Y)=0.
The previous equation implies that
(34)S(X,Y)=[-ρ-λ]g(X,Y).
This shows that the Ricci soliton is Einstein as follows:
(35)QX=[-ρ-λ]X.
LetM be an α-Sasakian manifolds; then we have [10]  (1)
Einstein,(2)
locally Ricci symmetric,(3)
Ricci semisymmetric; that is,R·S=0.
The implication(1)→(2)→(3) is trivial. Now, we prove the implication (3)→(1).
Now,(36)(R(X,Y)·S)(U,V)=-S(R(X,Y)U,V)-S(U,R(X,Y)V).
Considering R·S=0 and putting X=ξ in (36), we have
(37)S(R(ξ,Y)U,V)+S(U,R(ξ,Y)V)=0.
By using (7) in (37), we obtain
(38)α2[g(Y,U)S(ξ,V)-η(U)S(Y,V)]+α2[g(Y,V)S(U,ξ)-η(V)S(U,Y)]=0.
Putting U=ξ in (38) and by using (3), (9), and (10) on simplification, we obtain
(39)S(Y,V)=(n-1)α2g(Y,V).
Substituting (39) in (1), we have
(40)(ℒVg)(X,Y)=ρg(X,Y),
where ρ=2[-α2(n-1)-λ]; that is, V is conformal Killing.Now we study Ricci solitons in 3-dimensionalα-Sasakian manifolds.Theorem 8.
In a 3-dimensionalα-Sasakian manifolds, a Ricci soliton (g,V,λ), where V is conformal Killing, is (i)
shrinking forρ=2α2,(ii)
expanding forρ<2α2 and ρ>2α2.Proof.
Suppose that(M,g) is a 3-dimensional α-Sasakian manifolds and (g,V,λ) is a Ricci soliton in (M,g). If V is a conformal Killing vector field, then
(41)ℒVg=2ρg
for some scalar function ρ.
In a 3-dimensionalα-Sasakian manifolds and from (1), we have
(42)S(X,Y)=[-ρ-λ]g(X,Y),(43)QX=[-ρ-λ]X,(44)r=3[-ρ-λ].
In a 3-dimensional α-Sasakian manifolds, the curvature tensor R is given by
(45)R(X,Y)Z=g(Y,Z)QX-g(X,Z)QY+S(Y,Z)X-S(X,Z)Y-r2[g(Y,Z)X-g(X,Z)Y].
Using (42), (43), and (44) in (45), we get
(46)R(X,Y)Z={2[-ρ-λ]-r2}[g(Y,Z)X-g(X,Z)Y].
Putting X=Z=ξ in (46), we get
(47)R(ξ,Y)ξ={2[-ρ-λ]-r2}[η(Y)ξ-Y].
In an α-Sasakian manifolds R(ξ,Y)ξ is given by
(48)R(ξ,Y)ξ=α2[η(Y)ξ-Y].
From (47) and (48), we have
(49){2[-ρ-λ]-r2-α2}[η(Y)η(W)-g(Y,W)]=0.
The previous equation implies that
(50){2[-ρ-λ]-r2-α2}=0.
From (44) and (50), we have
(51)λ=-[ρ+2α2].(i)
Ifρ=2α2 implies λ=-4α2, that is, λ<0. Hence Ricci soliton is shrinking.(ii)
Letρ<2α2, suppose ρ=(-2α2-1)<2α2 implies ρ+2α2=-1, that is, -λ=1>0. Hence Ricci soliton is expanding.(iii)
Letρ>2α2 implies -2α2<-ρ. If -2α2-2α2<-ρ-2α2 then -4α2<-λ implies λ>4α2. Hence Ricci soliton is expanding.Ifρ=0 in (41), then ℒVg=0; that is, conformal vector field does not exist, and then the Ricci soliton is generalization of Einstein metric, then V is a Killing vector field. On base of this condition, we state the following.Remark 9.
A Ricci soliton inα-Sasakian manifolds is shrinking, if V is a Killing vector field.



---

*Source: 573925-2013-07-15.xml*