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.

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 Section 2, 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 be η-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 manifold M 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.

M3 is a Kenmotsu manifold.

h=0.

∇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.

Let ξ 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 operator l 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, since ϕ 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:

M3 is an η-Einstein Kenmotsu manifold.

Qϕ=ϕQ.

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 that M3 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 have Qϕ=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 define 1-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 Corollary 5 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 structure W-=(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).

Acknowledgments

The authors would like to thank the referee for his or her valuable suggestions and comments improving of this paper. This work is supported by the NSFC (no. 10931005) and the Natural Science Foundation of Guangdong Province of China (no. S2011010000471).

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