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
1 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
G. Dileo and A. M. Pastore in [5] initiated the symmetry classification problem on almost Kenmotsu manifolds. Locally symmetric almost Kenmotsu manifolds under
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
for any vector field X orthogonal to the Reeb vector field
Theorem 1.1
The Ricci operator of an almost Kenmotsu 3-h-manifold M is transversely Killing if and only if M is locally isometric to either the hyperbolic 3-space
On the hyperbolic 3-space
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
2 Almost Kenmotsu manifolds
By an almost contact metric manifold, we mean a Riemannian manifold
for any vector fields
where X denotes a vector field tangent to
where
for any vector fields
Let
and
3 Main results and proofs
It is well known that an almost Kenmotsu 3-manifold becomes a Kenmotsu 3-manifold if and only if
Applying (2.3), the following lemma was obtained by Cho and Kimura in [15, Lemma 6].
Lemma 3.1
On a non-Kenmotsu almost Kenmotsu 3-manifold we have
where
Applying Lemma 3.1, the Ricci operator Q of
with respect to the local basis
and
From (3.2), we see that the scalar curvature of
Theorem 3.1
The Ricci operator of a non-Kenmotsu almost Kenmotsu 3-h-manifold is transversely Killing if and only if the manifold is locally isometric to a non-unimodular Lie group endowed with a left invariant almost Kenmotsu structure.
Proof
Applying Lemma 3.1, by a direct calculation we obtain
By the aforementioned two relations, it is easily seen that
where we have used the assumption
Suppose that the Ricci operator is transversely Killing, setting
and
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
The addition of (3.14) to (3.13) multiplied by 2 gives
Similarly, applying (3.4) in the second term of (3.12) we obtain that
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
where we have used
In view of (3.6), from Lemma 3.1, by a direct calculation we have
Applying again the first term of (3.6), according to the first two terms of (3.20) we get
Moreover, applying again (3.20), with the help of (3.6), the well-known Jacobi identity for tangent vector fields
Taking the covariant derivative of (3.19) gives
Case 1
Now with the help of the second term of (3.4), the last term of (3.12) becomes
where we have used
Recall that on any Riemannian manifold
where
Now putting (3.27) into (3.24) we obtain
Taking the covariant derivative of (3.28) along the Reeb flow, with the help of (3.6), (3.21), (3.22), we obtain
Next the proof is divided into the following two subcases.
If
The so-called adjoint operator
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
and this is in fact a special case of (3.30) for
Case 2
If
As seen before, now the manifold is locally isometric to the product
If
Similarly, applying
Finally, multiplying (3.31) by
Conversely, as the product
with
From the aforementioned relations, it is easily seen that (1.1) is true for
Theorem 3.1
in fact is the non-Kenmotsu version of Theorem 1.1 and the corresponding Kenmotsu version of Theorem 1.1 is given as follows.
Theorem 3.2
The Ricci operator of a Kenmotsu 3-manifold is transversely Killing if and only if it is locally isometric to the hyperbolic 3-space
Proof
On a Kenmotsu 3-manifold,
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
It is well known that on a Riemannian 3-manifold, the curvature tensor R is given by
for any vector fields
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
for any vector fields
Proof of Theorem 1.1
The proof of Theorem 1.1 follows immediately from Theorems 3.1 and 3.2.□
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
Remark 3.1
If the Ricci operator of an almost Kenmotsu 3-h-manifold M is transversely Killing, then M is a Ricci soliton.
Acknowledgements
Quanxiang Pan was supported by the Doctoral Foundation of Henan Institute of Technology (No. KQ1828). Hui Wu was supported by the National Natural Science Foundation of China (No. 11801306) and the Project Funded by China Postdoctoral Science Foundation (No. 2020M672023). The authors would like to thank the reviewers for their useful comments and careful reading.
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