# A Study on Ricci Solitons in Kenmotsu Manifolds

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

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## Abstract

We study and obtain results on Ricci solitons in Kenmotsu manifolds satisfyingR(ξ,X)·B=0, B(ξ,X)·S=0, S(ξ,X)·R=0, R(ξ,X)·P¯=0, and P¯(ξ,X)·S=0, where B and P¯ are C-Bochner and pseudo-projective curvature tensor.


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## Body

## 1. Introduction

A Ricci soliton is a natural generalization of an Einstein metric and is defined on a Riemannian manifold(M,g). A Ricci soliton is a triple (g,V,λ) with g a Riemannian metric, V a vector field, and λ a real scalar such that
(1)ℒVg+2S+2λg=0,
where S is a Ricci tensor of M and ℒV denotes the Lie derivative operator along the vector field V. The Ricci soliton is said to be shrinking, steady, and expanding accordingly as λ is negative, zero, and positive, respectively [1]. In this paper, we prove conditions for Ricci solitons in Kenmotsu manifolds to be shrinking, steady, and expanding.In 1972, Kenmotsu [2] studied a class of contact Riemannian manifolds satisfying some special conditions and this manifold is known as Kenmotsu manifolds. Kenmotsu proved that a locally Kenmotsu manifold is a warped product I×fN of an interval I and a Kaehler manifold N with warping function f(t)=set, where s is a nonzero constant. Kenmotsu proved that if in a Kenmotsu manifold the condition R(X,Y)·R=0 holds, then the manifold is of negative curvature -1, where R is the curvature tensor of type (1,3) and R(X,Y) denotes the derivation of the tensor algebra at each point of the tangent space.The authors in [3–7] have studied Ricci solitons in contact and Lorentzian manifolds. The authors in [8] have obtained some results on Ricci solitons satisfying R(ξ,X)·C~=0, P(ξ,X)·C~=0, H(ξ,X)·S=0 and C~(ξ,X)·S=0 and now we extend the work to R(ξ,X)·B=0, B(ξ,X)·S=0, S(ξ,X)·R=0, R(ξ,X)·P-=0 and P-(ξ,X)·S=0.


## 2. Preliminaries

Ann-dimensional differential manifold M is said to be an almost contact metric manifold [9] if it admits an almost contact metric structure (ϕ,ξ,η,g) consisting of a tensor field ϕ of type (1,1), a vector field ξ, a 1-form η, and a Riemannian metric g compatible with (ϕ,ξ,η,g) satisfying
(2)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 Kenmotsu manifold [2] if
(3)(∇Xϕ)Y=g(ϕX,Y)ξ-η(Y)ϕX.
From (3), we have
(4)∇Xξ=X-η(X)ξ,
where ∇ denotes the Riemannian connection of g.In ann-dimensional Kenmotsu manifold, we have
(5)η(R(X,Y)Z)=g(X,Z)η(Y)-g(Y,Z)η(X),(6)R(X,Y)ξ=η(X)Y-η(Y)X,(7)R(ξ,X)Y=η(Y)X-g(X,Y)ξ,(8)R(ξ,X)ξ=X-η(X)ξ,
where R is the Riemannian curvature tensor.Let(g,V,λ) be a Ricci soliton in an n-dimensional Kenmotsu manifold M. From (4) we have
(9)(ℒξg)(X,Y)=2[g(X,Y)-η(X)η(Y)].
From (1) and (9) we get
(10)S(X,Y)=-(λ+1)g(X,Y)+η(X)η(Y).The above equation yields that(11)QX=-(λ+1)X+η(X)ξ,(12)S(X,ξ)=-λη(X),(13)r=-λn-(n-1),
where S is the Ricci tensor, Q is the Ricci operator, and r is the scalar curvature on M.


### 2.1. Example for 3-Dimensional Kenmotsu Manifolds

We consider3-dimensional manifold M={(x,y,z)∈R3;z≠0}, where (x,y,z) are the standard coordinates in R3. Let {E1,E2,E3} be linearly independent given by
(14)E1=z∂∂x,E2=z∂∂y,E3=-z∂∂z.Letg 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
(15)g=1z2(dx⊗dx+dy⊗dy+dz⊗dz).The(ϕ,ξ,η) structure is given by
(16)η=-1zdz,ξ=E3=-z∂∂z,ϕE1=-E2,ϕE2=E1,ϕE3=0.The linearity property ofϕ 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 definition of Lie bracket, we have
(17)[E1,E2]=0,[E1,E3]=E1,[E2,E3]=E2.Let∇ be the Levi-Civita connection; with respect to above metric g is given by Koszula formula
(18)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]),
and by virtue of it we have
(19)∇E1E3=E1,∇E2E3=E2,∇E3E3=0,∇E1E2=0,∇E2E2=-E3,∇E3E2=0,∇E1E1=-E3,∇E2E1=0,∇E3E1=0.Clearly (19) shows that (ϕ,ξ,η,g) satisfies (2), (3), and (4). Thus M is a Kenmotsu manifold.It is known that(20)R(X,Y)Z=∇X∇YZ-∇Y∇XZ-∇[X,Y]Z.
With the help of (19) and (20), it can be easily verified that
(21)R(E1,E2)E2=-E1,R(E1,E3)E3=-E1,R(E1,E1)E1=0,R(E2,E1)E1=-E2,R(E2,E3)E3=-E2,R(E2,E2)E2=0,R(E3,E1)E1=-E3,R(E3,E2)E2=-E3,R(E3,E3)E3=0.From the above expression of the curvature tensor we obtain(22)S(E1,E1)=g(R(E1,E2)E2,E1)+g(R(E1,E3)E3,E1)=-2.Similarly we have(23)S(E2,E2)=S(E3,E3)=-2,(ℒξg)(Ei,Ei)=2[g(Ei,Ei)-η(Ei)η(Ei)].Now byX=Y=Ei, in (1), where i=1,2,3 and by virtue of above equations we get the value of λ which is strictly greater than 0. Thus this is an example of expanding Ricci solitons in Kenmotsu manifolds.


## 2.1. Example for 3-Dimensional Kenmotsu Manifolds

We consider3-dimensional manifold M={(x,y,z)∈R3;z≠0}, where (x,y,z) are the standard coordinates in R3. Let {E1,E2,E3} be linearly independent given by
(14)E1=z∂∂x,E2=z∂∂y,E3=-z∂∂z.Letg 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
(15)g=1z2(dx⊗dx+dy⊗dy+dz⊗dz).The(ϕ,ξ,η) structure is given by
(16)η=-1zdz,ξ=E3=-z∂∂z,ϕE1=-E2,ϕE2=E1,ϕE3=0.The linearity property ofϕ 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 definition of Lie bracket, we have
(17)[E1,E2]=0,[E1,E3]=E1,[E2,E3]=E2.Let∇ be the Levi-Civita connection; with respect to above metric g is given by Koszula formula
(18)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]),
and by virtue of it we have
(19)∇E1E3=E1,∇E2E3=E2,∇E3E3=0,∇E1E2=0,∇E2E2=-E3,∇E3E2=0,∇E1E1=-E3,∇E2E1=0,∇E3E1=0.Clearly (19) shows that (ϕ,ξ,η,g) satisfies (2), (3), and (4). Thus M is a Kenmotsu manifold.It is known that(20)R(X,Y)Z=∇X∇YZ-∇Y∇XZ-∇[X,Y]Z.
With the help of (19) and (20), it can be easily verified that
(21)R(E1,E2)E2=-E1,R(E1,E3)E3=-E1,R(E1,E1)E1=0,R(E2,E1)E1=-E2,R(E2,E3)E3=-E2,R(E2,E2)E2=0,R(E3,E1)E1=-E3,R(E3,E2)E2=-E3,R(E3,E3)E3=0.From the above expression of the curvature tensor we obtain(22)S(E1,E1)=g(R(E1,E2)E2,E1)+g(R(E1,E3)E3,E1)=-2.Similarly we have(23)S(E2,E2)=S(E3,E3)=-2,(ℒξg)(Ei,Ei)=2[g(Ei,Ei)-η(Ei)η(Ei)].Now byX=Y=Ei, in (1), where i=1,2,3 and by virtue of above equations we get the value of λ which is strictly greater than 0. Thus this is an example of expanding Ricci solitons in Kenmotsu manifolds.


## 3. Ricci Soliton in a Kenmotsu Manifold SatisfyingR(ξ,X)·B=0

Bochner introduced a Kähler analogue of the Weyl conformal curvature tensor by purely formal considerations, which is now well known as the Bochner curvature tensor [10]. A geometric meaning of the Bochner curvature tensor is given by Blair in [11] by using the Boothby-Wang's fibration. In 1969, Matsumoto and Chūman [12] constructed the notion of C-Bochner curvature tensor in a Sasakian manifold and studied its several properties.The C-Bochner curvature tensor [13] B in M is defined by
(24)B(X,Y)Z=R(X,Y)Z+1n+3×[g(X,Z)QY-S(Y,Z)X-g(Y,Z)QX+S(X,Z)Y+g(ϕX,Z)QϕY-S(ϕY,Z)ϕX-g(ϕY,Z)QϕX+S(ϕX,Z)ϕY+2S(ϕX,Y)ϕZ+2g(ϕX,Y)QϕZ+η(Y)η(Z)QX-η(Y)S(X,Z)ξ+η(X)S(Y,Z)ξ-η(X)η(Z)QY]-D+n-1n+3[g(ϕX,Z)ϕY-g(ϕY,Z)ϕXmmmmmm,+2g(ϕX,Y)ϕZ]+Dn+3[η(Y)g(X,Z)ξ-η(Y)η(Z)Xmmmm,+η(X)η(Z)Y-η(X)g(Y,Z)ξ]-D-4n+3[g(X,Z)Y-g(Y,Z)X],
where D=(r+n-1)/(n+1).TakingZ=ξ in (24) and using (6), (10), (11), we get
(25)B(X,Y)ξ=[1-λn+3+4n+3][η(X)Y-η(Y)X].Similarly using (5), (10), (11), (12) in (24), we get
(26)η(B(X,Y)Z)=[1-λn+3+4n+3]mmmmmmmm×[g(X,Z)η(Y)-g(Y,Z)η(X)].We assume that the conditionR(ξ,X)·B=0, then we have
(27)R(ξ,X)B(Y,Z)W-B(R(ξ,X)Y,Z)W-B(Y,R(ξ,X)Z)W-B(Y,Z)R(ξ,X)W=0.
Using (7) in (27), we get
(28)η(B(Y,Z)W)X-g(B(Y,Z)W,X)ξ+g(X,Y)B(ξ,Z)W-η(Y)B(X,Z)W+g(X,Z)B(Y,ξ)W-η(Z)B(Y,X)W+g(X,W)B(Y,Z)ξ-η(W)B(Y,Z)X=0.By taking an inner product withξ, we have
(29)η(B(Y,Z)W)η(X)-g(B(Y,Z)W,X)+g(X,Y)η(B(ξ,Z)W)-η(Y)η(B(X,Z)W)+g(X,Z)η(B(Y,ξ)W)-η(Z)η(B(Y,X)W)+g(X,W)η(B(Y,Z)ξ)-η(W)η(B(Y,Z)X)=0.By using (25), (26) in (29), we have
(30)[1-λn+3+4n+3][g(Y,W)g(Z,X)-g(Z,W)g(Y,X)]-g(B(Y,Z)W,X)=0.In view of (24) in (30), then we have
(31)[1-λn+3+4n+3][g(Y,W)g(Z,X)-g(Z,W)g(Y,X)]-g(R(Y,Z)W,X)-1n+3[g(Y,W)S(Z,X)-S(Z,W)g(Y,X)-1n+3m-g(Z,W)S(Y,X)+S(Y,W)g(Z,X)-1n+3m+g(ϕY,W)S(ϕZ,X)-S(ϕZ,W)g(ϕY,X)-1n+3m-g(ϕZ,W)S(ϕY,X)+S(ϕY,W)g(ϕZ,X)-1n+3m+2S(ϕY,Z)g(X,ϕW)+2g(ϕY,Z)S(X,ϕW)-1n+3m+η(W)η(Z)S(Y,X)-η(X)η(Z)S(Y,W)-1n+3m+η(Y)η(X)S(Z,W)-η(W)η(Y)S(Z,X)]-Dn+3[η(X)η(Z)g(Y,W)-η(W)η(Z)g(Y,X)-Dn+3m+η(W)η(Y)g(Z,X)-η(Y)η(X)g(Z,W)]+D+n-1n+3[g(ϕY,W)g(ϕZ,X)+D+n-1n+3,-g(ϕZ,W)g(ϕY,X)+D+n-1n+3,m+2g(ϕY,Z)g(X,ϕW)]+D-4n+3[g(Y,W)g(Z,X)-g(Z,W)g(Y,X)]=0.TakingX=Y=ei in (31) and summing over i=1,2,…,n. By virtue of (10), (11), (12), and on simplification, we get
(32)S(Z,W)=[-(n+4)λ-2n-3n+3+-n2-6n+8n+3-rn+3]g(W,Z)m+[(λ+1)(n+4)-2+nn+3+r+4(n-1)n+3]η(W)η(Z).PuttingZ=W=ξ in (32) and by virtue of (10) and (13), we have
(33)λ=(n-1).Therefore,λ positive that is, the Ricci soliton in Kenmotsu manifold is expanding.Hence we state the following theorem:Theorem 1.
A Ricci soliton in a Kenmotsu manifold satisfyingR(ξ,X)·B=0 is expanding.


## 4. Ricci Soliton in a Kenmotsu Manifolds SatisfyingB(ξ,X)·S=0

The conditionB(ξ,X)·S=0 implies that
(34)S(B(ξ,X)Y,Z)+S(Y,B(ξ,X)Z)=0.By using (10) in (34), we have
(35)η(Z)η(B(ξ,X)Y)-(λ+1)g(B(ξ,X)Y,Z)-(λ+1)g(Y,B(ξ,X)Z)+η(Y)η(B(ξ,X)Z)=0,
the above equation implies that
(36)[η(Z)η(B(ξ,X)Y)+η(Y)η(B(ξ,X)Z)]=(λ+1)[g(B(ξ,X)Y,Z)+g(Y,B(ξ,X)Z)].By using (24) and (26) in (36), we have
(37)2η(X)η(Y)η(Z)[1-λ(n+3)+4(n+3)]-[1-λ(n+3)+4(n+3)]×[g(X,Z)η(Y)+g(X,Y)η(Z)]=0.PutX=Y=ξ in (37) then the equation is identically satisfied and we do not get the value for λ. So, we proceed as follows: Taking X=Y=ei in (37) and summing over i=1,2,…,n and by virtue of (13) and η(Z)≠0 conditions, we obtain
(38)λ=n+7.Therefore,λ is positive that is Ricci soliton in Kenmotsu manifolds satisfying B(ξ,X)·S=0 is expanding.Hence we can state the following theorem.Theorem 2.
A Ricci soliton in a Kenmotsu manifold satisfyingB(ξ,X)·S=0 is expanding.


## 5. Ricci Soliton in a Kenmotsu Manifold SatisfyingS(ξ,X)·R=0

Using the following equations:(39)S((X,ξ)·R)(U,V)W=((X∧Sξ)·R)(U,V)W=(X∧Sξ)R(U,V)Wm+R((X∧Sξ)U,V)W+R(U,(X∧Sξ)V)Wm+R(U,V)(X∧Sξ)W,
where the endomorphism X∧SY is defined by
(40)(X∧SY)Z=S(Y,Z)X-S(X,Z)Y,we have(41)S((X,ξ)·R)(U,V)W=S(ξ,R(U,V)W)X-S(X,R(U,V)W)ξm+S(ξ,U)R(X,V)W-S(X,U)R(ξ,V)Wm+S(ξ,V)R(U,X)W-S(X,V)R(U,ξ)Wm+S(ξ,W)R(U,V)X-S(X,W)R(U,V)ξ.By using the conditionS(ξ,X)·R=0, and by virtue of (10), (12), we have
(42)-λη(R(U,V)W)X-[-(λ+1)g(X,R(U,V)W)+η(X)η(R(U,V)W)]ξ-λη(U)R(X,V)W-[-(λ+1)g(X,U)+η(X)η(U)]R(ξ,V)W-λη(V)R(U,X)W-[-(λ+1)g(X,V)+η(X)η(V)]R(U,ξ)W-λη(W)R(U,V)X-[-(λ+1)g(X,W)+η(X)η(W)]R(U,V)ξ=0.By taking an inner product withξ and by virtue of (5), (6), (7), and (8), we have
(43)-(λ+1)η(X)[g(U,W)η(V)-g(V,W)η(U)]+λ[g(V,W)η(X)η(U)-g(U,W)η(X)η(V)mm-g(U,X)η(V)η(W)+g(V,X)η(U)η(W)]+(λ+1)g(X,R(U,V)W)+(λ+1)g(X,U)[η(W)η(V)-g(V,W)]+(λ+1)g(X,V)[g(U,W)-η(W)η(U)]+g(V,W)η(X)η(U)-g(U,W)η(X)η(V)=0.TakingX=U=ei and summing over i=1,2,…,n, we obtain
(44)2(λ+1)[g(V,W)-η(W)η(V)]+(λ+1)S(V,W)-(λ+1)(n-1)g(V,W)+(n-1)η(V)η(W)=0.TakingV=W=ξ in (44) and by virtue of (12), (13), we obtain
(45)-λ(λ+n)=0.This implies either(46)λ=0orλ=-n.Therefore for anyλ=0 or λ=-n the Ricci soliton in Kenmotsu manifolds satisfying S(ξ,X)·R=0 is either steady or shrinking.Hence we can state the following theorem.Theorem 3.
A Ricci soliton in a Kenmotsu manifold satisfyingS(ξ,X)·R=0 is either steady or shrinking.


## 6. Ricci Soliton in a Kenmotsu Manifolds SatisfyingR(ξ,X)·P-=0

The Pseudo-projective curvature tensorP- is defined by
(47)P-(X,Y)Z=aR(X,Y)Z+b[S(Y,Z)X-S(X,Z)Y]-rn(an-1+b)[g(Y,Z)X-g(X,Z)Y],
where a,b≠0 are constants. Taking Z=ξ in (47) and using (6), (10), (11), we get
(48)P-(X,Y)ξ=[a+bλ+rn(an-1+b)][η(X)Y-η(Y)X].Similarly using (5), (10), (11), (12) in (47), we get
(49)η(P-(X,Y)Z)=[a+b(λ+1)+rn(an-1+b)]×[g(X,Z)η(Y)-g(Y,Z)η(X)].We assume that the conditionR(ξ,X)·P-=0, then we have
(50)R(ξ,X)P-(U,V)W-P-(R(ξ,X)U,V)W-P-(U,R(ξ,X)V)W-P-(U,V)R(ξ,X)W=0.Using (7) in (50), we find
(51)η(P-(U,V)W)X-g(X,P-(U,V)W)ξ-η(U)P-(X,V)W+g(X,U)P-(ξ,V)W-η(V)P-(U,X)W+g(X,V)P-(U,ξ)W-η(W)P-(U,V)X+g(X,W)P-(U,V)ξ=0.By taking an inner product withξ then we get
(52)η(P-(U,V)W)η(X)-g(X,P-(U,V)W)-η(U)η(P-(X,V)W)+g(X,U)η(P-(ξ,V)W)-η(V)η(P-(U,X)W)+g(X,V)η(P-(U,ξ)W)-η(W)η(P-(U,V)X)+g(X,W)η(P-(U,V)ξ)=0.By using (48), (49) in (52), we have
(53)-g(X,P-(U,V)W)+[a+b(λ+1)+rn[an+1+b]]×[g(X,V)g(U,W)-g(X,U)g(V,W)]=0.In view of (47) in (53), we have
(54)-ag(X,R(U,V)W)-b[(λ+1){g(V,X)g(U,W)-g(V,W)g(U,X)}mmmmm+η(V)η(W)g(U,X)mmmmm-g(V,X)η(U)η(W)]+[a+b(λ+1)]×[g(X,V)g(U,W)-g(X,U)g(V,W)]=0.TakingX=U=ei in (54) and summing over i=1,2,…,n, and on simplification, we get
(55)aS(V,W)=-a(n-1)g(V,W)-b(n-1)η(V)η(W).PuttingV=W=ξ in (55) and by virtue of (12), (13), we get the following equation:
(56)λ=(n-1)(a+b)a.Since(a+b)/a≠0 implies that λ>0, that is, the Ricci soliton in Kenmotsu manifold satisfying R(ξ,X)·P-=0 is expanding,hence we state the following theorem.Theorem 4.
A Ricci soliton in a Kenmotsu manifold satisfyingR(ξ,X)·P-=0 is expanding.


## 7. Ricci Soliton in a Kenmotsu Manifolds SatisfyingP-(ξ,X)·S=0

The conditionP-(ξ,X)·S=0 implies that
(57)S(P-(ξ,X)Y,Z)+S(Y,P-(ξ,X)Z)=0.By using (10) in (57), we have
(58)η(Z)η(P-(ξ,X)Y)-(λ+1)g(P-(ξ,X)Y,Z)-(λ+1)g(Y,P-(ξ,X)Z)+η(Y)η(P-(ξ,X)Z)=0,
that is,
(59)[η(Z)η(P-(ξ,X)Y)+η(Y)η(P-(ξ,X)Z)]=(λ+1)[g(P-(ξ,X)Y,Z)+g(Y,P-(ξ,X)Z)].By using (47) and (48) in (59), we have
(60)[a+rn[an-1+b]]×[2η(X)η(Y)η(Z)m-g(X,Z)η(Y)-g(X,Y)η(Z)]=0.PutX=Y=ξ in (60); then the equation is identically satisfied and we do not get the value for λ. So, we proceed as follows: taking X=Y=ei, summing over i=1,2,…,n, and by virtue of (13) and η(Z)≠0 conditions we obtain
(61)λ=(n-1)2(a-b)n[a+b(n-1)].Therefore, ifa=b in (61) then λ=0; that is, Ricci soliton in Kenmotsu manifolds satisfying P-(ξ,X)·S=0 is steady. If a≠b then either λ>0 for a>b or λ<0 for a<b, that is, the Ricci soliton in Kenmotsu manifold satisfying P-(ξ,X)·S=0 is expanding or shrinking.Hence we can state the following theorem.Theorem 5.
A Ricci soliton in a Kenmotsu manifolds satisfyingP-(ξ,X)·S=0 is steady for a=b, expanding for a>b and shrinking for a<b.



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