Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators

Abstract In this paper, it is proved that the Ricci operator of an almost Kenmotsu 3-h-manifold M is of transversely Killing-type if and only if M is locally isometric to the hyperbolic 3-space ℍ 3 ( − 1 ) {{\mathbb{H}}}^{3}(-1) or a non-unimodular Lie group endowed with a left invariant non-Kenmotsu almost Kenmotsu structure. This result extends those results obtained by Cho [Local symmetry on almost Kenmotsu three-manifolds, Hokkaido Math. J. 45 (2016), no. 3, 435–442] and Wang [Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Polon. Math. 116 (2016), no. 1, 79–86; Three-dimensional almost Kenmotsu manifolds with η \eta -parallel Ricci tensor, J. Korean Math. Soc. 54 (2017), no. 3, 793–805].


Introduction
The classifications of symmetric and homogeneous almost contact manifolds are one of the most important problems in differential geometry of almost contact manifolds. With regard to fruitful symmetry classification results in the framework of contact Riemannian manifolds, we refer the reader to the study of D. E. Blair [1]. In this paper, we try to study the symmetry classifications on the other kind of almost contact manifolds which are named (almost) Kenmotsu manifolds. Note that a Riemannian manifold is locally symmetric if and only if the curvature tensor is parallel with respect to the Levi-Civita connection. The study of locally symmetric Kenmotsu manifolds was initiated by K. Kenmotsu in [2], who proved that a locally symmetric Kenmotsu manifold is of constant sectional curvature −1. This result has attracted many researcher's interests. After K. Kenmotsu's result, many results concerning symmetry classification problems on Kenmotsu manifolds have emerged (see, for example, the studies of U. C. De [3] and G. Pitis [4]).
G. Dileo and A. M. Pastore in [5] initiated the symmetry classification problem on almost Kenmotsu manifolds. Locally symmetric almost Kenmotsu manifolds under ( ) = R X Y ξ , 0 for any contact vector fields X Y , (see Section 2 for the notation), some nullity conditions and CR-integrability have been classified in [5][6][7], respectively. In particular, J. T. Cho [8] and Y. Wang [9] independently completed the classification problem for almost  They proved that an almost  . Some generalizations of their results were obtained in [10,11]. Very recently, Y. Wang in [12] studied locally symmetric almost Kenmotsu manifolds for general dimensions under some addition conditions. As far as we know, the problem of "classify completely locally symmetric almost Kenmotsu manifolds of dimension >3" is still open.
In this paper, we aim to give some local classification results for almost Kenmotsu manifolds in terms of the Ricci tensor. D. E. Blair in [13] introduced the so-called Killing tensor (on a Riemannian manifold) which is defined by where ∇ is the Levi-Civita connection, T is a tensor field of type ( ) 1,1 and X denotes an arbitrary vector field. By employing the Killing tensor, D. E. Blair in [13] presented a new characterization for an almost contact metric manifold to be a cosymplectic manifold. In this paper, we say that the Ricci operator Q on an almost Kenmotsu manifold is transversely Killing if it satisfies for any vector field X orthogonal to the Reeb vector field ξ . It is known that on a Riemannian 3-manifold, the Ricci tensor is parallel if and only if the curvature tensor is parallel. Therefore, relation (1.1) is much weaker than local symmetry condition. Y. Wang in [10] considered ∇ = h 0 ξ on an almost Kenmotsu 3-manifold and constructed many examples satisfying this condition (see Section 2 for the notion of h). For simplicity, we say that an almost Kenmotsu 3-manifold is an almost Applying the aforementioned two notions, we obtain the following. there exists a Kenmotsu structure (see [14]) and on any nonunimodular Lie group there exists a left invariant non-Kenmotsu almost Kenmotsu structure (see [6,Theorem 5.2]). Remark 1.1. Since those conditions we have employed are much weaker than local symmetry, our results are generalizations of Cho and Wang's results (see [8,9]). Remark 1.2. Replacing X by + X Y in (1.1), we see that this relation is also equivalent to for any vector fields X and Y orthogonal to the Reeb vector field ξ . Such a condition is much weaker than Ricci η-parallel (i.e., ((∇ ) ) = g Q Y Z , 0 X for any vector fields X Y Z , , orthogonal to the Reeb vector field ξ ). Therefore, our Theorem 1.1 can also be viewed as an extension of results in [10].

Almost Kenmotsu manifolds
By an almost contact metric manifold, we mean a Riemannian manifold ( ) M g , of dimension + n 2 1 on which there exists a quadruple ( ) ϕ ξ η g , , , satisfying

2)
for any vector fields X Y , , where ϕ is a ( ) 1,1 -type tensor field, ξ is a vector field called the Reeb vector field, η is a global 1-form called the almost contact 1-form and g is a Riemannian metric (see the study of Blair [1]).
A vector field orthogonal to ξ is called a contact vector field. By an almost Kenmotsu manifold we mean an almost contact metric manifold on which there holds , (see [5,6]). We consider the product × of an almost contact metric manifold + M n 2 1 and and define on it an almost complex structure J by where X denotes a vector field tangent to . The almost contact metric manifold is said to be normal if J is integrable, or equivalently, denotes the Nijenhuis tensor of ϕ. A normal almost Kenmotsu manifold is said to be a Kenmotsu manifold (cf. [2,5,6]). An almost Kenmotsu manifold is a Kenmotsu manifold if and only if for any vector fields X Y , .
Let + M n 2 1 be an almost Kenmotsu manifold. We consider three tensor fields = (⋅ ) where R is the Riemannian curvature tensor of g and is the Lie differentiation. From [5,6], we know that the three ( ) 1,1 -type tensor fields l, ′ h and h are symmetric and satisfy

Main results and proofs
It is well known that an almost Kenmotsu 3-manifold becomes a Kenmotsu 3-manifold if and only if = h 0 (see [5]). Therefore, we discuss the proof of     From (3.2), we see that the scalar curvature of M 3 is given by (3.12) Applying (3.4) in the last term of (3.11), we obtain that Taking the covariant derivative of (3.13) along the Reeb flow we obtain  (3.14) The addition of (3.14) to (3.13) multiplied by 2 gives  Taking the covariant derivative of (3.16) along the Reeb flow we obtain The addition of (3.17) to (3.16) multiplied by 2 gives Consequently, the subtraction of (3.15) multiplied by b from (3.18) multiplied by c implies that (3.20) Applying again the first term of (3.6), according to the first two terms of (3.20) we get Next the proof is divided into the following two subcases. If ≠ λ 1, it follows that ( + ) ( ) + ( − ) = λ e λ λ λ b 3 1 2 1 0, which is compared with (3.28) implying ( ) = e λ 0. Therefore, in terms of (3.6) and ( ) = ( ) e λ ϕe λ we see that λ is a positive constant ≠ 1. Using this in (3.28) we obtain = = b c 0 because of (3.23) and ≠ λ 1. In this context, (3.20) becomes e  e λϕe ξ ϕe  λe ϕe  e ϕe  , , , a n d , 0 . (3.30) The so-called adjoint operator ad X is defined by for any vector fields X Y , on tangent space. Applying (3.30), it is easily seen that Thus, according to J. Milnor [16] we see that the manifold is locally isometric to a non-unimodular Lie group endowed with a left invariant non-Kenmotsu almost Kenmotsu structure. For the constructions of almost Kenmotsu structures on this kind of Lie groups we refer the reader to [6,Theorem 5.2].
Otherwise, let us consider the case = λ 1. Applying (3.25), (3.7) and (3.10) in (3.26), with the aid of (3.11) and (3.12) we observe that the scalar curvature is a constant. Therefore, using this and = λ 1 in (3.5) we know that A is also a constant. Moreover, applying = λ 1 in relation (3.14) or (3.17) we obtain = b c. Finally, applying = λ 1, = b c and (3.4), according to (3.7)-(3.10) we see that the Ricci tensor is η-parallel, i.e., ((∇ ) ) = g Q Y Z , 0 X for any vector fields X Y Z , , orthogonal to ξ . Following Y. Wang [10], we know that the manifold is locally isometric to the Riemannian product (− ) × 4 2 . Note that such a product corresponds to the non-unimodular Lie group whose Lie algebra is given by , , a n d , 0 , and this is in fact a special case of (3.30) for = λ 1. , , a n d , 0 .
As seen before, now the manifold is locally isometric to the product (− ) × 4 2 which can be realized as a non-unimodular Lie group whose Lie algebra is given by Recall that λ is assumed to be the positive eigenfunction of the operator h, it follows that ( ) = e λ 0 and hence together this with (3.6) and ( ) + ( ) = e λ ϕe λ 0 we see that λ is a constant. Applying this in (3.32) we obtain = = b c 0. Therefore, it is easily seen that (3.30) holds in this context and the manifold is locally isometric to a non-unimodular Lie group endowed with a left invariant almost Kenmotsu structure.
Conversely, as the product (− ) × 4 2 is locally symmetric, the Ricci operator of this product is necessarily parallel and hence transversely Killing. On a left invariant almost Kenmotsu structure defined on a Lie group whose Lie algebra is given by (3.30), following Wang [17,18] for any vector field X. By a direct calculation, from the aforementioned equation we get for any vector field X. And hence from (3.33) we obtain In view of ∇ = ξ 0 ξ , it follows from the aforementioned relation that (∇ ) = Q ξ 0 ξ . Therefore, if the Ricci operator is of transversely Killing-type (that is, (∇ ) = Q X 0 X for any vector field X orthogonal to ξ ), it is easily seen that the scalar curvature is a constant because of (3.26).
It is well known that on a Riemannian 3-manifold, the curvature tensor R is given by for any vector field X. On the other hand, because the scalar curvature r is a constant, applying (3.26) on (3.35) we have = − r 6. Putting this in (3.35) we see that = − QX X 2 , which is used in (3.34) giving that for any vector fields X Y Z , , . Therefore, the manifold is of constant sectional curvature − 1. The converse is also true as  It was proved by Y. Wang in [19, Theorem 3.7] that if an almost Kenmotsu 3-manifold is a Ricci soliton with the potential vector field orthogonal to the Reeb vector field ξ and ξ is an eigenvector field for the Ricci operator, then the manifold is locally isometric to either (− ) 1 3 or a non-unimodular Lie group equipped with a left invariant almost Kenmotsu structure whose Lie algebra is given by (3.30 . According to this result and our Theorem 1.1, we have the following.