On the Classification of Almost Kenmotsu Manifolds of Dimension 3

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.

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 𝑀^2𝑛+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 𝜙²=−Id+𝜂⊗𝜉,𝜂(𝜉)=1, (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 𝑀^2𝑛+1 is said to be compatible with the almost contact structure (𝜙,𝜉,𝜂) if 𝑔𝜙𝑋,𝜙𝑌=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌). (2) 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 𝑀^2𝑛+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 𝑁_𝜙=[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 (𝑀^2𝑛+1,𝜙,𝜉,𝜂,𝑔) be an almost Kenmotsu manifold. We denote by 𝑙=𝑅(⋅,𝜉)𝜉 and ℎ=(1/2)ℒ_𝜉𝜙 on 𝑀^2𝑛+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 ℎ𝜉=0,𝑙𝜉=0,trℎ=0,trℎ𝜙=0,ℎ𝜙+𝜙ℎ=0. (3) We also have the following formulas following from [5, 7, 8]: ∇_𝑋𝜉=−𝜙²𝑋−𝜙ℎ𝑋⇒∇_𝜉ℎ𝜉=0,𝜙𝑙𝜙−𝑙=2²−𝜙²(𝑙)()(),tr=𝑆𝜉,𝜉=𝑔𝑄𝜉,𝜉=−2𝑛−trℎ²,𝑅()(𝑋)(𝑌)+∇𝑋,𝑌𝜉=𝜂𝑌−𝜙ℎ𝑌−𝜂𝑋−𝜙ℎ𝑋_𝑌𝜙ℎ𝑋−∇_𝑋∇𝜙ℎ𝑌,_𝜉ℎ=−𝜙−2ℎ−𝜙ℎ²−𝜙𝑙, (4)(5)(6)(7)(8) for any 𝑋,𝑌∈Γ(𝑇𝑀^2𝑛+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 𝑀^2𝑛+1, respectively. From the above formulas we also have ∇_𝜉𝜙=0.

An almost contact manifold is said to be 𝜂-Einstein if 𝑄=𝛼id+𝛽𝜂⊗𝜉, (9) where 𝛼 and 𝛽 are both smooth functions on 𝑀^2𝑛+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 𝑁_𝑝(𝑘)=𝑍∈𝑇_𝑝()𝑔()𝑋()𝑌,𝑀∶𝑅𝑋,𝑌𝑍=𝑘𝑌,𝑍−𝑔𝑋,𝑍 (10) 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 𝐶()()1𝑋,𝑌𝑍=𝑅𝑋,𝑌𝑍−×𝑆()()𝑌()()+𝑟𝑚−2𝑌,𝑍𝑋−𝑆𝑋,𝑍+𝑔𝑌,𝑍𝑄𝑋−𝑔𝑋,𝑍𝑄𝑌()𝑔()()𝑌𝑚−1)(𝑚−2𝑌,𝑍𝑋−𝑔𝑋,𝑍 (11) vanishes for 𝑚>3 and ∇_𝑋𝑆(𝑌,𝑍)−∇_𝑌𝑆()=1𝑋,𝑍4𝑋(𝑟)𝑔(𝑌,𝑍)−𝑌(𝑟)𝑔(𝑋,𝑍), (12) 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 𝑆()=𝑟𝑋,𝑌2+𝛼²−𝛽²𝑔𝛼𝜙𝑋,𝜙𝑌−2²−𝛽²𝜂(𝑋)𝜂(𝑌),∀𝑋,𝑌∈Γ(𝑇𝑀), (13) 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 𝑟𝑄=−2+𝛼²−𝛽²𝜙²−2𝛼²−𝛽²𝜂⊗𝜉. (14) 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 𝑀^2𝑛+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 ∇_𝑋𝜙𝑌=−𝑔𝑋,𝜙𝑌−𝜂(𝑌)𝜙𝑋,∀𝑋,𝑌∈Γ(𝑇𝑀). (15) 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: ∇_𝑋𝜙∇𝑋=0,_𝑋𝜙()∇𝜙𝑋=𝑔𝑋,𝑋𝜉,_𝑋𝜙∇𝜉=−𝜙𝑋,_𝜙𝑋𝜙𝜉=𝑋. (16) On the other hand, on an almost Kenmotsu manifold by using (1)–(4) we obtain 𝑔∇_𝑋𝜙𝑋,𝑋=𝑔∇_𝑋𝜙𝑔∇𝑋,𝜙𝑋=0,_𝑋𝜙(),𝑔∇𝑋,𝜉=𝑔𝑋,ℎ𝑋_𝑋𝜙𝜙𝑋,𝑋=𝑔∇_𝑋𝜙𝑔∇𝜙𝑋,𝜙𝑋=0,_𝑋𝜙(),∇𝜙𝑋,𝜉=𝑔𝑋,𝑋−𝑔𝑋,𝜙ℎ𝑋_𝑋𝜙∇𝜉=−𝜙𝑋−ℎ𝑋,_𝜙𝑋𝜙𝜉=𝑋−ℎ𝜙𝑋. (17)(18) 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 ℎ^′𝑄=𝑄ℎ^′,𝑋((𝑙=1tr))4𝑋(𝑟), (19)(20) 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 𝑄𝜉=tr(𝑙)𝜉. (21) Differentiating (21) along an arbitrary vector field 𝑋 and using (4) gives ∇_𝑋𝑄𝜉=𝑋(tr(𝑙))𝜉+𝑄𝜙²(𝑙)𝜙𝑋+𝜙ℎ𝑋−tr²𝑋+𝜙ℎ𝑋. (22) Since 𝑀³ is conformally flat, then substituting 𝑍=𝜉 into (12) and using 𝑔((∇_𝑋𝑄)𝑌,𝑍)=(∇_𝑋𝑆)(𝑌,𝑍) gives 𝑔∇_𝑋𝑄𝑌−∇_𝑌𝑄=1𝑋,𝜉4𝑋(𝑟)𝜂(𝑌)−𝑌(𝑟)𝜂(𝑋), (23) for any 𝑋,𝑌∈Γ(𝑇𝑀). Using the symmetry 𝑔((∇_𝑋𝑄)𝑌,𝑍)=𝑔((∇_𝑋𝑄)𝑍,𝑌), then from (23) we have 𝑔∇_𝑋𝑄𝜉,𝑌−𝑔∇_𝑌𝑄=1𝜉,𝑋4𝑋(𝑟)𝜂(𝑌)−𝑌(𝑟)𝜂(𝑋), (24) for any 𝑋,𝑌∈Γ(𝑇𝑀). Substituting (22) into (24) gives 𝑋((𝑙𝜂(𝑌)((𝑙𝜂(𝑋)=1tr))−𝑌tr))+𝑔𝑌,𝑄𝜙ℎ𝑋−𝑔𝑋,𝑄𝜙ℎ𝑌4𝑋(𝑟)𝜂(𝑌)−𝑌(𝑟)𝜂(𝑋), (25) for any 𝑋,𝑌∈Γ(𝑇𝑀). Replacing 𝜙𝑋 and 𝜙𝑌 by 𝑋 and 𝑌, respectively, in (25) gives 𝑔𝜙𝑌,𝑄𝜙ℎ𝜙𝑋=𝑔𝜙𝑋,𝑄𝜙ℎ𝜙𝑌, (26) 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 𝑋((𝑙−1tr))4𝑋(𝑟)=𝜉((𝑙−1tr))4𝜉(𝑟)𝜂(𝑋), (27) 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 𝜉((𝑙−1tr))4𝜉(𝑟)=0. (28) 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 𝑄𝜉=tr(𝑙)𝜉. (29) It is well known that the curvature tensor of a 3-dimensional Riemannian manifold (𝑀³,𝑔) is given by 𝑅()()()()𝑋()𝑟𝑋,𝑌𝑍=𝑔𝑌,𝑍𝑄𝑋−𝑔𝑋,𝑍𝑄𝑌+𝑔𝑄𝑌,𝑍−𝑔𝑄𝑋,𝑍𝑌−2𝑔(𝑌,𝑍)𝑋−𝑔(𝑋,𝑍)𝑌, (30) 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 (𝑙)−𝑟𝑄𝑋=𝑙𝑋−tr2𝑋−𝑟2−2tr(𝑙)𝜂(𝑋)𝜉,∀𝑋∈Γ(𝑇𝑀). (31) 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, 𝜙𝑙=𝑙𝜙. (32) On the other hand, substituting (32) into (5) gives that 𝑙=𝜙²−ℎ². (33) From (8) and (33) we obtain that ∇_𝜉ℎ=−2ℎ,∇_𝜉𝑙=4ℎ². (34)
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 (𝑙)𝑄𝑋=𝑟−tr2(𝑙)X+3tr−𝑟2𝜂(𝑋)𝜉,∀𝑋∈Γ(𝑇𝑀). (35) Differentiating (35) along 𝑌∈Γ(𝑇𝑀) we obtain ∇_𝑌𝑄(𝑙)𝑋=Y𝑟−tr2(𝑙)𝑋+3tr−𝑟2𝜂(𝑋)∇_𝑌𝜉+12𝑌((𝑙))𝜂(𝑋)+((𝑙))𝑔3tr−𝑟3tr−𝑟𝑋,∇_𝑌𝜉().𝜉,∀𝑋,𝑌∈Γ𝑇𝑀 (36) Also, it is well known that 12grad(𝑟)=∇_𝑋𝑄𝑋+∇_𝜙𝑋𝑄𝜙𝑋+∇_𝜉𝑄𝜉, (37) 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 ∇_𝑋𝑄(𝑙)𝑋=𝑋𝑟−tr2(𝑙)𝑋+3tr−𝑟2𝑔𝑋,𝑋+ℎ𝜙𝑋𝜉. (38) Similarly, we have ∇_𝜙(𝑙)𝑋𝑄𝜙𝑋=𝜙𝑋𝑟−tr2+(𝑙)𝜙𝑋3tr−𝑟2𝑔𝑋,𝑋−ℎ𝜙𝑋𝜉. (39) Substituting (38) and (39) and (∇_𝜉𝑄)𝜉=𝜉(tr(𝑙))𝜉 into (37) implies 12(𝑟)(𝑙)grad=𝑋𝑟−tr2𝑋(𝑙)+𝜙𝑋𝑟−tr2+(𝑙))((𝑙𝜙𝑋{(3tr−𝑟+𝜉tr))}𝜉. (40) 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 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 𝑅𝑅()−𝑟𝑋,𝜙𝑋,𝑋,𝜙𝑋=𝑔𝑋,𝜙𝑋𝜙𝑋,𝑋=2𝑆𝑋,𝑋2. (41) On the other hand, by the definition of Ricci curvature tensor we have 𝑆()()().𝑋,𝑋=𝑅𝑋,𝜉,𝑋,𝜉+𝑅𝑋,𝜙𝑋,𝑋,𝜙𝑋=𝑔𝑙𝑋,𝑋+𝑅𝑋,𝜙𝑋,𝑋,𝜙𝑋 (42) 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 𝑀^2𝑛+1 is a space of constant 𝜙-holomorphic sectional curvature 𝐻 then 𝑀^2𝑛+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 𝐷𝑔=𝜔⊗𝑔, (43) 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 𝐷_𝑋𝑌=∇_𝑋1𝑌−2𝜔(𝑋)1𝑌−2𝜔(𝑌)1𝑋+2𝑔(𝑋,𝑌)𝐸, (44) 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, 𝑆^𝐷(𝑋,𝑌)+𝑆^𝐷(𝑌,𝑋)=Λ𝑔(𝑋,𝑌),∀𝑋,𝑌∈Γ(𝑇𝑀), (45) where 𝑆^𝐷 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 𝐷_𝑋𝑌=∇_𝑋1𝑌−2(𝑋)1𝛾𝜂𝑌−2(𝑌)1𝛾𝜂𝑋+2𝛾𝑔(𝑋,𝑌)𝜉, (46) 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 𝑆^𝐷(𝑋,𝑌)=𝑆(𝑋,𝑌)+∇_𝑋𝜔𝑌−12∇_𝑌𝜔1𝑋+4𝜔(𝑋)𝜔(𝑌)+12−1div𝛾𝜉4𝛾²𝑔(𝑋,𝑌), (47) for any 𝑋,𝑌∈Γ(𝑇𝑀³). By using (4) then a simple computation gives that ∇_𝑋𝜔𝑌+∇_𝑌𝜔𝑋(𝜆)𝜂(𝑌)(𝜆)𝜂(𝑋)𝑔()(𝑋)𝜂(𝑌),=𝑋+𝑌+2𝑓𝑋,𝑌−𝜂+𝑔ℎ𝜙𝑋,𝑌 (48) for any 𝑋,𝑌∈Γ(𝑇𝑀³) and div(𝜆𝜉)=𝑔∇_𝜉(𝜆𝜉),𝜉+𝑔∇_𝑒(𝜆𝜉),𝑒+𝑔∇_𝜙𝑒()(𝜆)𝜆𝜉,𝜙𝑒=𝜉+2𝜆, (49) where 𝑒∈𝒟 is a unit vector field on 𝑀³. Thus, substituting (48) and (49) into (47) yields that 𝑆^𝐷(𝑋,𝑌)+𝑆^𝐷()()+1𝑌,𝑋=2𝑆𝑋,𝑌2∇_𝑋𝜔𝑌+∇_𝑌𝜔𝑋+12𝜔(𝑋)𝜔(𝑌)+()−1div𝜆𝜉2𝜆²𝑔()=(𝜆)−1𝑋,𝑌2𝛼+3𝜆+𝜉2𝜆²𝑔()+1𝑋,𝑌2𝛽−𝜆+2𝜆²𝜂(𝑋)𝜂(𝑌)+12𝑋(𝜆)𝜂(𝑌)+𝑌(𝜆)𝜂(𝑋)+𝜆𝑔ℎ𝜙𝑋,𝑌,∀𝑋,𝑌∈Γ𝑇𝑀³. (50) 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 𝑆^𝐷(𝑋,𝑌)+𝑆^𝐷()𝑔()𝑀𝑌,𝑋=2𝛼+𝜆+𝛽𝑋,𝑌,∀𝑋,𝑌∈Γ³, (51) 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 𝐷_𝑋𝑌=∇_𝑋1𝑌+2(𝑋)1𝛾𝜂𝑌+2(𝑌)1𝛾𝜂𝑋−2()𝑆𝛾𝑔𝑋,𝑌𝜉,_𝐷(𝑋,𝑌)=𝑆(𝑋,𝑌)−𝑛∇_𝑋𝜔𝑌+12∇_𝑌𝜔1𝑋+4𝜔(𝑋)𝜔(𝑌)+12(𝐸)−1div4𝜔²𝑔(𝑋,𝑌), (52) for any 𝑋,𝑌∈Γ(𝑇𝑀^2𝑛+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).