# On the Classification of Almost Kenmotsu Manifolds of Dimension 3

Authors: Yaning Wang; Ximin Liu
Journal: ISRN Geometry
 (2013)

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## Abstract

This paper deals with the classification of a 3-dimensional almost Kenmotsu manifold satisfying certain geometric conditions. Moreover, by applying our main classification theorem, we obtain some suffcient conditions for an almost Kenmotsu manifold of dimension 3 to be an Einstein-Weyl manifold.


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## Body

## 1. Introduction

Contact metric manifolds known as a special class of almost contact metric manifolds are objects of increasing interest both from geometers and physicists [1] recently. We refer the reader to the recent monograph [2] for a wide and detailed overview of the results in this field. From (14) (see Section 3) we know that a normal almost contact metric manifold (which includes Sasakian and Kenmotsu manifolds as its special cases) of dimension 3 satisfies Qϕ=ϕQ. But the above property need not be true in an almost contact metric manifold. Blair et al. [3] obtained a classification of 3-dimensional contact metric manifold with Qϕ=ϕQ. However, in higher dimensions the classification of contact metric manifold with Qϕ=ϕQ is still open. It is worthy to point out that Ghosh [4] recently proved that a contact metric manifold admitting the Einstein-Weyl structures W±=(g,±ω) (see Section 4) and Qϕ=ϕQ is either a K-contact or an Einstein manifold.On the other hand, in 1972, Kenmotsu [5] introduced a class of almost contact metric manifolds which are known as Kenmotsu manifolds nowadays. Recently, almost Kenmotsu manifolds satisfying η-parallelism and locally symmetries are studied by Dileo and Pastore [6] and [7], respectively. We notice that Dileo and Pastore [8] complete the classification of 3-dimensional almost Kenmotsu manifold with the assumption that ξ belongs to the (k,μ)′-nullity distribution. However, to the best of our knowledge the study of 3-dimensional almost Kenmotsu manifolds is still lacking so far. The object of this paper is to classify the 3-dimensional almost Kenmotsu manifolds satisfying Qϕ=ϕQ and other geometric conditions, providing some results which show the differences between almost Kenmotsu manifolds and the contact metric manifolds of dimension 3 [3, 9]. Moreover, by applying our main classification theorem, we obtain some suffcient conditions for an almost Kenmotsu manifold of dimension 3 to be an Einstein-Weyl manifold.This paper is organized as the following way. In Section2, we provide some basic formulas and properties of almost Kenmotsu manifolds. Section 3 is devoted to present our main theorems and their proofs. Finally, in Section 4, we prove that if an almost Kenmotsu manifold of dimension 3 is η-Einstein with certain condition then it admits both Einstein-Weyl structures W+=(g,+ω) and W-=(g,-ω).


## 2. Almost Kenmotsu Manifolds

First of all, we give some basic notions of almost Kenmotsu manifolds which follow from [5, 7]. An almost contact structure on a (2n+1)-dimensional smooth manifold M2n+1 is a triplet (ϕ,ξ,η), where ϕ is a (1,1)-tensor, ξ a global vector field, and η a 1-form, such that
(1)ϕ2=-Id+η⊗ξ,η(ξ)=1,
which implies that ϕ(ξ)=0,  η∘ϕ=0 and rank(ϕ)=2n. It follows from [2, 10] that a Riemannian metric g on M2n+1 is said to be compatible with the almost contact structure (ϕ,ξ,η) if
(2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y).
An almost contact structure endowed with a compatible Riemannian metric is said to be an almost contact metric structure. The fundamental 2-form Φ is defined by Φ(X,Y)=g(X,ϕY) for any vector fields X and Y on M2n+1. An almost Kenmotsu manifold is defined as an almost contact metric manifold together with dη=0 and dΦ=2η∧Φ. It is well known that the normality of almost contact structure is expressed by the vanishing of the tensor Nϕ=[ϕ,ϕ]+2dη⊗ξ, where [ϕ,ϕ] is the Nijenhuis tensor of ϕ. A normal almost Kenmotsu manifold is said to be a Kenmotsu manifold.Now let(M2n+1,ϕ,ξ,η,g) be an almost Kenmotsu manifold. We denote by l=R(·,ξ)ξ and h=(1/2)ℒξϕ on M2n+1, where R is the curvature tensor and ℒ is the Lie differentiation, respectively. Thus, the two (1,1)-type tensors l and h are symmetric and satisfy
(3)hξ=0,lξ=0,trh=0,tr(hϕ)=0,hϕ+ϕh=0.
We also have the following formulas following from [5, 7, 8]:
(4)∇Xξ=-ϕ2X-ϕhX(⇒∇ξξ=0),(5)ϕlϕ-l=2(h2-ϕ2),(6)tr(l)=S(ξ,ξ)=g(Qξ,ξ)=-2n-trh2,(7)R(X,Y)ξ=η(X)(Y-ϕhY)-η(Y)(X-ϕhX)+(∇Yϕh)X-(∇Xϕh)Y,(8)∇ξh=-ϕ-2h-ϕh2-ϕl,
for any X,Y∈Γ(TM2n+1), where S,  Q,  ∇, and Γ(TM) denote the Ricci curvature tensor, Ricci operator, the Levi-Civita connection of g, and the Lie algebra of vector fields in M2n+1, respectively. From the above formulas we also have ∇ξϕ=0.An almost contact manifold is said to beη-Einstein if
(9)Q=αid+βη⊗ξ,
where α and β are both smooth functions on M2n+1. It is easy to see that an η-Einstein almost Kenmotsu manifold satisfies Qϕ=ϕQ because of ϕξ=0. We also recall that the k-nullity distribution [11] is defined by
(10)Np(k)={Z∈TpM:R(X,Y)Z=k(g(Y,Z)XZ∈TpM:R(X,Y)Z=Np(k)=-g(X,Z)Y)Tp},
where k is a real number. When k in (10) is a smooth functions, then the nullity distributions are called the generalized nullity distributions [12]. Also, the sectional curvature K(ξ,X) of a plane section spanned by ξ and a vector X orthogonal to ξ is called a ξ-sectional curvature and the sectional curvature K(X,ϕX) of a plane section spanned by vectors X and ϕX with X orthogonal to ξ is called a ϕ-sectional curvature [3].A Riemanian manifoldM of dimension m is conformally flat if and only if the Weyl tensor C defined by
(11)C(X,Y)Z=R(X,Y)Z-1m-2C(X,Y)Z=×{g(X,Z)QYS(Y,Z)X-S(X,Z)YC(X,Y)Z=+g(Y,Z)QX-g(X,Z)QY}C(X,Y)Z=+r(m-1)(m-2){g(Y,Z)X-g(X,Z)Y}
vanishes for m>3 and
(12)(∇XS)(Y,Z)-(∇YS)(X,Z)=14{X(r)g(Y,Z)-Y(r)g(X,Z)},
for m=3, where S,  Q, and r denote the Ricci tensor, Ricci operator, and the scalar curvature, respectively.


## 3. A Classification Theorem

Let(M3,ϕ,ξ,η,g) be a normal almost contact metric manifold of dimension 3, then we have Qϕ=ϕQ. In fact, it follows from [13, 14] that M3 satisfies
(13)S(X,Y)=(r2+α2-β2)g(ϕX,ϕY)-2(α2-β2)η(X)η(Y),∀X,Y∈Γ(TM),
where α=div(ξ) and β=tr(ϕ∇ξ) are both constants and r denotes the scalar curvature of M3. Then from the above equation it is easy to get
(14)Q=-(r2+α2-β2)ϕ2-2(α2-β2)η⊗ξ.
Noticing that ϕξ=0, then from (14) we know that Qϕ=ϕQ. If β=1 (resp., α=1) then M3 is just the 3-dimensional Sasakian (resp., Kenmotsu) manifold [13]. However, the condition Qϕ=ϕQ need not be true in a contact metric manifold as well as an almost Kenmotsu manifold. In a contact metric manifold M2n+1 if the characteristic vector ξ is a Killing vector field, then the manifold is said to be a K-contact manifold. A Sasakian manifold is always a K-contact one, but the converse need not hold only if M is of dimension 3. We present the following result to characterize Kenmotsu manifold of dimension 3 which is analogous to the contact metric manifold of dimension 3.Lemma 1.
Let(M3,ϕ,ξ,η,g) be an almost Kenmotsu manifold of dimension 3, then the following conditions are equivalent.(1)
M3 is a Kenmotsu manifold.(2)
h=0.(3)
∇ξ=-ϕ2.Proof.
It follows from [7, 8] that an almost Kenmotsu is a Kenmotsu manifold if and only if
(15)(∇Xϕ)Y=-g(X,ϕY)-η(Y)ϕX,∀X,Y∈Γ(TM).
Noticing that ∇ξϕ=0 and (∇Xϕ)Y+(∇ϕX)(ϕY)=-η(Y)ϕX-2g(X,ϕY)-η(Y)h(X) (see [7]) for any X,Y∈Γ(TM), thus M3 is a Kenmotsu manifold if and only if every nonvanishing vector field X in contact distribution 𝒟 (defined by 𝒟=ker(η)=Im(ϕ)) satisfies the following equations:
(16)(∇Xϕ)X=0,(∇Xϕ)ϕX=g(X,X)ξ,(∇Xϕ)ξ=-ϕX,(∇ϕXϕ)ξ=X.
On the other hand, on an almost Kenmotsu manifold by using (1)–(4) we obtain
(17)g((∇Xϕ)X,X)=g((∇Xϕ)X,ϕX)=0,g((∇Xϕ)X,ξ)=g(X,hX),g((∇Xϕ)ϕX,X)=g((∇Xϕ)ϕX,ϕX)=0,g((∇Xϕ)ϕX,ξ)=g(X,X)-g(X,ϕhX),(∇Xϕ)ξ=-ϕX-hX,(18)(∇ϕXϕ)ξ=X-hϕX.
Comparing the above three equations with (16), we complete the proof of equivalence between (1) and (2). The equivalence between (2) and (3) follows from the fact that ∇Xξ=-ϕ2X-ϕhX for any X∈Γ(TM).Theorem 2.
Let(M3,ϕ,ξ,η,g) be an almost Kenmotsu manifold of dimension 3 for which the characteristic vector field ξ is an eigenvector field of the Ricci operator. If M3 is conformally flat, then it satisfies
(19)h′Q=Qh′,(20)X(tr(l))=14X(r),
for any X∈Γ(TM), where r denotes the scalar curvature of M3.Proof.
Letξ be an eigenvector field of the Ricci operator corresponding to the eigenvalue α, that is, Qξ=αξ. Substituting the above equation into (6) implies that α=tr(l). Then we obtain
(21)Qξ=tr(l)ξ.
Differentiating (21) along an arbitrary vector field X and using (4) gives
(22)(∇XQ)ξ=X(tr(l))ξ+Q(ϕ2X+ϕhX)-tr(l)(ϕ2X+ϕhX).
Since M3 is conformally flat, then substituting Z=ξ into (12) and using g((∇XQ)Y,Z)=(∇XS)(Y,Z) gives
(23)g((∇XQ)Y-(∇YQ)X,ξ)=14{X(r)η(Y)-Y(r)η(X)},
for any X,Y∈Γ(TM). Using the symmetry g((∇XQ)Y,Z)=g((∇XQ)Z,Y), then from (23) we have
(24)g((∇XQ)ξ,Y)-g((∇YQ)ξ,X)=14{X(r)η(Y)-Y(r)η(X)},
for any X,Y∈Γ(TM). Substituting (22) into (24) gives
(25)X(tr(l))η(Y)-Y(tr(l))η(X)+g(Y,QϕhX)-g(X,QϕhY)=14{X(r)η(Y)-Y(r)η(X)},
for any X,Y∈Γ(TM). Replacing ϕX and ϕY by X and Y, respectively, in (25) gives
(26)g(ϕY,QϕhϕX)=g(ϕX,QϕhϕY),
for any X,Y∈Γ(TM). Thus, it follows from (26) that h′Q=Qh′ and hence by substituting Y=ξ in (25) we obtain
(27)X(tr(l))-14X(r)=[ξ(tr(l))-14ξ(r)]η(X),
for any X∈Γ(TM). Applying the exterior derivation d on both sides of (27) and using the well-known Poincaré lemma d2=0, and then replacing X,Y, respectively, by ϕX,ϕY in the resulting equation, we obtain
(28)ξ(tr(l))-14ξ(r)=0.
Thus, substituting (28) into (27) gives (20).Theorem 3.
Let(M3,ϕ,ξ,η,g) be a 3-dimensional almost Kenmotsu manifold satisfying Qϕ=ϕQ. Then one of the following cases occurs.
Case  1. ξ(tr(l))=0 and hence M3 is a Kenmotsu manifold.
Case  2. ξ(tr(l))≠0 and hence the eigenvalues of h are locally given by {0,ce-2t,-ce-2t} with coordinate t on R and c a nonzero number.Proof.
For an almost Kenmotsu manifold the operatorl never vanishes. In fact, if l=0, it follows from (6) that tr(h2)=-2, there is a contradiction. Now we suppose that M3 is an almost Kenmotsu manifold of dimension 3 with Qϕ=ϕQ. Noticing that ϕξ=0 then we have ϕQξ=0. Denoting by P the projective component of Qξ on contact distribution 𝒟, then from (6) we have Qξ=tr(l)ξ+P. By using the hypothesis Qϕ=ϕQ on the above equation we get P=0, then we have
(29)Qξ=tr(l)ξ.
It is well known that the curvature tensor of a 3-dimensional Riemannian manifold (M3,g) is given by
(30)R(X,Y)Z=g(Y,Z)QX-g(X,Z)QY+g(QY,Z)X-g(QX,Z)Y-r2[g(Y,Z)X-g(X,Z)Y],
for any X,Y,Z∈Γ(TM), where r denotes the scalar curvature of M3. Noticing (29) and replacing Y,Z by ξ in (30) yields
(31)QX=lX-(tr(l)-r2)X-(r2-2tr(l))η(X)ξ,∀X∈Γ(TM).
Using ϕξ=0 and ϕQ=Qϕ, it follows from (31) that ϕlX=ϕQ+(tr(l)-r/2)ϕX=lϕX for any X∈Γ(TM), that is,
(32)ϕl=lϕ.
On the other hand, substituting (32) into (5) gives that
(33)l=ϕ2-h2.
From (8) and (33) we obtain that
(34)∇ξh=-2h,∇ξl=4h2.
On the other hand, sinceϕ is antisymmetric on 𝒟 then from (32) we have g(ϕX,lX)=0 for any X∈𝒟. Thus, noticing that g(ξ,lX)=0, we obtain lX=(tr(l)/2)X for any X∈𝒟, that is, lX=-(tr(l)/2)ϕ2X for any X∈Γ(TM). Substituting the above equation into (31) gives
(35)QX=r-tr(l)2X+3tr(l)-r2η(X)ξ,∀X∈Γ(TM).
Differentiating (35) along Y∈Γ(TM) we obtain
(36)(∇YQ)X=Y(r-tr(l)2)X+3tr(l)-r2η(X)∇Yξ+12{Y(3tr(l)-r)η(X)+(3tr(l)-r)g(X,∇Yξ)}ξ,∀X,Y∈Γ(TM).
Also, it is well known that
(37)12grad(r)=(∇XQ)X+(∇ϕXQ)ϕX+(∇ξQ)ξ,
for any unit vector filed X in contact distribution 𝒟, where grad(r) denotes the gradient of scalar curvature of M3. Now letting Y=X in (36) be unit vector fields in 𝒟 gives
(38)(∇XQ)X=X(r-tr(l)2)X+3tr(l)-r2g(X,X+hϕX)ξ.
Similarly, we have
(39)(∇ϕXQ)ϕX=ϕX(r-tr(l)2)ϕX+3tr(l)-r2g(X,X-hϕX)ξ.
Substituting (38) and (39) and (∇ξQ)ξ=ξ(tr(l))ξ into (37) implies
(40)12grad(r)=X(r-tr(l)2)X+ϕX(r-tr(l)2)ϕX+{(3tr(l)-r)+ξ(tr(l))}ξ.
By using (40) and taking an inner product with X and ϕX, respectively, we obtain X(tr(l))=0 and ϕX(tr(l))=0 for any X∈𝒟, that is, tr(l) is a constant on 𝒟.
Case  1. If ξ(tr(l))=0, then we find that tr(l) is a constant on M and hence from (6) we see that tr(h2) is also a constant on M. Let X∈𝒟 be a unit eigenvector filed of h with eigenvalue λ, that is, hX=λX (and hence hϕX=-λϕX), then λ is a constant since that tr(h2) is a constant. Using the first term of (34) gives g((∇ξh)X,X)=g(ξ(λ)X,X)=-2g(hX,X)=-2λ; thus, noting that λ is a constant then we obtain λ=0, that is, h=0 and hence from Lemma 1 we see that M3 is a Kenmotsu manifold.
Case  2. If ξ(tr(l))≠0, it follows from Case  1 that ξ(λ)=-2λ≠0, where ±λ denotes the eigenvalues of h on 𝒟. Locally, we can write ξ=∂/∂t and hence λ=ce-2t with coordinate t on R and c a nonzero number following the fact that λ is a constant on contact distribution 𝒟, which completes the proof.Corollary 4.
Let(M3,ϕ,ξ,η,g) be an almost Kenmotsu manifold of dimension 3. Then the following assertions are equivalent:(a)
M3 is an η-Einstein Kenmotsu manifold.(b)
Qϕ=ϕQ.(c)
ξ belongs to the generalized k-nullity distribution.Moreover, if one of the above conditions holds, then the ξ-sectional curvature of M3 is -1 and the ϕ-sectional curvature is 2+r/2, where r denotes the scalar curvature of M3.Proof.
Suppose thatM3 is an η-Einstein almost Kenmotsu manifold of dimension 3; from (9) it is easy to see (a)⇒(b). If Qϕ=ϕQ, then replacing Z by ξ in (30) and using (35) we see that ξ belongs to the generalized k-nullity distribution, which means that (b)⇒(c). Now letting ξ belongs to the generalized k-nullity distribution, then by a straightforward calculation we know that ξ is an eigenvector field of Ricci operator. Replacing Y=Z by ξ in (30) implies that (b)⇒(a).
Finally, if one of the above conditions holds, then from the above statements we haveQϕ=ϕQ. We choose a unit nonvanishing vector field X in contact distribution 𝒟. Replacing Y=Z=ϕX in (30) gives an equation, taking an inner product with X on the resulting equation and taking into account Qϕ=ϕQ we obtain
(41)R(X,ϕX,X,ϕX)=g(R(X,ϕX)ϕX,X)=2S(X,X)-r2.
On the other hand, by the definition of Ricci curvature tensor we have
(42)S(X,X)=R(X,ξ,X,ξ)+R(X,ϕX,X,ϕX)=g(lX,X)+R(X,ϕX,X,ϕX).
Using lX=(tr(l)/2)X for any X∈𝒟 (see Theorem 3), then it follows from (41) into (42) that S(X,X)=1+r/2. Then from (41) we know that the ϕ-sectional curvature of M3 is 2+r/2, which completes the proof.Kenmotsu [5] proved that if a Kenmotsu manifold M2n+1 is a space of constant ϕ-holomorphic sectional curvature H then M2n+1 is a space of constant curvature and H=-1. Thus, the following result follows from Corollary 4 and Theorem 3.Corollary 5.
Let(M3,ϕ,ξ,η,g) be an η-Einstein almost Kenmotsu manifold of dimension 3 with ξ(tr(l))=0. If the scalar curvature of M3 is a constant, then M3 is locally isometric to a hyperbolic space ℍ3(-1) with constant scalar curvature -6.


## 4. Einstein-Weyl Structures

Recall that a Weyl structure [4, 15] on a Riemannian manifold (M,g) of dimension ≥3 is defined by the pair W+=(g,ω) satisfying
(43)Dg=ω⊗g,
where D is a unique torsion-free connection which preserves the conformal class [g]={λg:λ∈C∞(M)} on M and ω is an 1-form on M. It follows from (43) that
(44)DXY=∇XY-12ω(X)Y-12ω(Y)X+12g(X,Y)E,
where ∇ and E denote the Levi-Civita connection of g and the dual vector field of ω with respect to g, respectively. The Weyl structure is said to be Einstein-Weyl if the symmetrized Ricci tensor associated with the Weyl connection is proportional to the Riemannian metric g, that is,
(45)SD(X,Y)+SD(Y,X)=Λg(X,Y),∀X,Y∈Γ(TM),
where SD denotes the Ricci tensor associated with D and Λ is a smooth function on M. Notice that Narita [15] proved that an η-Einstein almost contact metric manifold satisfying ∇Xξ=-ϕX admits an Einstein-Weyl structure W+=(g,ω). However, (4) implies that an almost Kenmotsu manifold never satisfies Narita’s condition even if h=0. Since then, we present the following sufficient conditions to characterize Einstein-Weyl structure on an almost Kenmotsu manifold of dimension 3.Theorem 6.
Let(M3,ϕ,ξ,η,g) be an η-Einstein almost Kenmotsu manifold of dimension 3 with ξ(tr(l))=0, that is, S(X,Y)=αg(X,Y)+βη(X)η(Y). If β is a constant ≤1/4, then M3 admits an Einstein-Weyl structure W+=(g,ω).Proof.
We define1-form ω by ω=λη, where λ is a nonvanishing function on M3. Then the dual vector field of ω respective to g is λξ. We also define a connection D on M3 by
(46)DXY=∇XY-12γη(X)Y-12γη(Y)X+12γg(X,Y)ξ,
then from (46) it is easy to verify that D is torsion free and Dg=λη⊗g, that is, (g,ω) is a Weyl structure on M3. It follows from [15, 16] that
(47)SD(X,Y)=S(X,Y)+(∇Xω)Y-12(∇Yω)X+14ω(X)ω(Y)+(12div(γξ)-14γ2)g(X,Y),
for any X,Y∈Γ(TM3). By using (4) then a simple computation gives that
(48)(∇Xω)Y+(∇Yω)X=X(λ)η(Y)+Y(λ)η(X)+2f[g(X,Y)-η(X)η(Y)+g(hϕX,Y)],
for any X,Y∈Γ(TM3) and
(49)div(λξ)=g(∇ξ(λξ),ξ)+g(∇e(λξ),e)+g(∇ϕe(λξ),ϕe)=ξ(λ)+2λ,
where e∈𝒟 is a unit vector field on M3. Thus, substituting (48) and (49) into (47) yields that
(50)SD(X,Y)+SD(Y,X)=2S(X,Y)+12((∇Xω)Y+(∇Yω)X)+12ω(X)ω(Y)+(div(λξ)-12λ2)g(X,Y)=(2α+3λ+ξ(λ)-12λ2)g(X,Y)+(2β-λ+12λ2)η(X)η(Y)+12(X(λ)η(Y)+Y(λ)η(X))+λg(hϕX,Y),∀X,Y∈Γ(TM3).
On the other hand, noticing Theorem 3 and Corollary 5, we know that h=0 and M3 is a Kenmotsu manifold. We set λ=1+1-4β or λ=1-1-4β, then it follows from (50) that
(51)SD(X,Y)+SD(Y,X)=2(α+λ+β)g(X,Y),∀X,Y∈Γ(M3),
which completes the proof.Remark 7.
From Corollary5 we know that an η-Einstein almost Kenmotsu manifold of dimension 3 with ξ(tr(l))=0 is a Kenmotsu manifold and h=0; however, this property need not be true in higher dimensions more than 3. Thus, our result cannot be generalized to the case of higher dimensions.
A Weyl structureW-=(g,-ω) on a Riemannian manifold M of dimension (2n+1) is defined by Dg=-ω⊗g for a unique torsion-free connection D and a 1-form ω. The Weyl structure W-=(g,-ω) is said to be an Einstein-Weyl structure if (45) holds for a smooth function Λ on M. For an Einstein-Weyl structure W-=(g,-ω) it follows from [4] that
(52)DXY=∇XY+12γη(X)Y+12γη(Y)X-12γg(X,Y)ξ,SD(X,Y)=S(X,Y)-n(∇Xω)Y+12(∇Yω)X+14ω(X)ω(Y)+(12div(E)-14ω2)g(X,Y),
for any X,Y∈Γ(TM2n+1), where E denotes the dual vector field of ω with respect to g. Thus, a straightforward calculation which is similar to the proof of Theorem 6 gives the following.Theorem 8.
Let(M3,ϕ,ξ,η,g) be an η-Einstein almost Kenmotsu manifold of dimension 3 with ξ(tr(l))=0, that is, S(X,Y)=αg(X,Y)+βη(X)η(Y). If β is a constant ≤1/4, then M3 admits both Einstein-Weyl structures W±=(g,±ω).



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