Trans-Sasakian 3-Manifolds with Reeb Flow Invariant Ricci Operator

Let M be a three-dimensional trans-Sasakian manifold of type (α, β). In this paper, we obtain that the Ricci operator of M is invariant along Reeb flow if and only if M is an α-Sasakian manifold, cosymplectic manifold or a space of constant sectional curvature. Applying this, we give a new characterization of proper trans-Sasakian 3-manifolds.

On an almost contact metric manifold (M, φ, ξ, η, g), the Ricci operator of M is said to be Reeb flow invariant if it satisfies where L, ξ and Q are the Lie derivative, Reeb vector field and the Ricci operator, respectively.Cho in [18] proved that a contact metric 3-manifold satisfies Equation (1) if and only if it is Sasakian or locally isometric to SU(2) (or )), the group E(2) of rigid motions of Euclidean 2-plane.Cho in [19] proved that an almost cosymplectic 3-manifold satisfies (1) if and only if it is either cosymplectic or locally isometric to the group E(1, 1) of rigid motions of Minkowski 2-space.
In addition, Cho and Kimura in [20] proved that an almost Kenmotsu 3-manifold satisfies (1) if and only if it is of constant sectional curvature −1 or a non-unimodular Lie group.Reeb flow invariant Ricci operators were also investigated on the unit tangent sphere bundle of a Riemannian manifold (see [21]), even on real hypersurfaces in complex two-plane Grassmannians (see [22]).In this paper, we obtain a new characterization of proper trans-Sasakian 3-manifolds by employing (1) and proving Theorem 1.The Ricci operator of a trans-Sasakian 3-manifold is invariant along Reeb flow if and only if the manifold is an α-Sasakian manifold, cosymplectic manifold or a space of constant sectional curvature.
According to calculations shown in Section 3, we observe that Ricci parallelism with respect to the Levi-Civita connection (i.e., ∇Q = 0) is stronger than a Reeb flow invariant Ricci operator.Thus, we have Remark 1. Theorem 1 is an extension of Wang and Liu [16] (Theorem 3.12).Some corollaries induced from Theorem 1 are also given in the last section.
A normal almost contact metric manifold is called a trans-Sasakian manifold (see [1]) if for any vector fields X, Y and two smooth functions α, β.In particular, a three-dimensional almost contact metric manifold is trans-Sasakian if and only if it is normal (see [24,25]).
Putting Y = ξ into (3) and using (2), we have for any vector field X.In this paper, all manifolds are assumed to be connected.

Reeb Flow Invariant Ricci Operator on Trans-Sasakian 3-Manifolds
In this section, we give a proof of our main result Theorem 1. First, we introduce the following two important lemmas (see [12]) which are useful for our proof.
Lemma 1.On a trans-Sasakian 3-manifold of type (α, β) we have Lemma 2. On a trans-Sasakian 3-manifold of type (α, β), the Ricci operator is given by where by ∇ f we mean the gradient of a function f .
We also need the following lemma (see [17]) Lemma 3. On a trans-Sasakian 3-manifold of type (α, β), the following three conditions are equivalent: (1) The Reeb vector field is minimal or harmonic.
(2) The following equation holds: (3) The Reeb vector field is an eigenvector field of the Ricci operator.
Lemma 4. The Ricci operator on a cosymplectic 3-manifold is invariant along the Reeb flow.
The above lemma can be seen in [19] Lemma 5.The Ricci operator on an α-Sasakian 3-manifold is invariant along the Reeb flow.
Proof.According to Lemma 2 and the definition of an α-Sasakian 3-manifold, the Ricci operator is given by for any vector field X and certain nonzero constant α.Moreover, according to [16] (Corollary 3.10), we observe that the scalar curvature r is invariant along the Reeb vector field ξ, i.e., ξ(r) = 0.In fact, such an equation can be deduced directly by using the formula divQ = 1 2 ∇r and (7).Applying ξ(r) = 0, it follows directly from (7) that L ξ Q = 0.
Proof of Theorem 1.Let M be a trans-Sasakian 3-manifold and e be a unit vector field orthogonal to ξ.Then, {ξ, e, φe} forms a local orthonormal basis on the tangent space for each point of M. The Levi-Civita connection ∇ on M can be written as the following (see [12]) where λ, γ and δ are smooth functions on some open subset of the manifold.We assume that the Ricci operator is invariant along the Reeb flow.From ( 1) and ( 4), we have for any vector field X.By using the local basis {ξ, e, φe} and Lemma 2, the Ricci operator can be rewritten as the following: Replacing X in ( 9) by ξ, we obtain Taking the inner product of the above equation with ξ, e and φe, respectively, we obtain where we have employed Lemma 1.The addition of the second term of ( 12) multiplied by α to the third term of ( 12) multiplied by β gives Following ( 13), we consider the following several cases.Case i: α 2 + β 2 = 0, or equivalently, α = β = 0.In this case, the manifold becomes a cosymplectic 3-manifold.The proof for this case is completed because of Lemma 4.
Case iii: α 2 + β 2 = 0 and β = 0.In this context, (10) becomes Replacing X by e in (9) and using ( 8), (15), we acquire With the aid of Lemma 1 and the first term of ( 12), from the previous relation, we have From ( 15), we calculate the derivative of the Ricci operator as the following: where we have used the first term of ( 8) and ( 12) and, for simplicity, we put On a Riemannian manifold, we have divQ = 1 2 ∇r.In this context, it is equivalent to for any vector field X. Replacing X in ( 19) by ξ and recalling (16) and the first term of ( 12), we obtain 2β(A − 2α 2 + 2β 2 + 2ξ(β)) = 0, or equivalently, where we have used the assumption β = 0 and (18).According to (15), it is clear to see that the manifold is Einstein, i.e, Q = r 3 id.Because the manifold is of dimension three, then it must be of constant sectional curvature.
A Riemannian manifold is said to be locally symmetric if ∇R = 0 and this is equivalent to ∇Q = 0 for dimension three.Wang and Liu in [16] proved that a trans-Sasakian 3-manifold is locally symmetric if and only if it is locally isometric to the sphere space S 3 (c 2 ), the hyperbolic space , where c is a nonzero constant.According to [16], on a locally symmetric trans-Sasakian 3-manifold, the Reeb vector field is an eigenvector field of the Ricci operator.Thus, following Lemma 3 and relations ( 9) and ( 10), we observe that Ricci parallelism is stronger than the Reeb flow invariant Ricci operator.Hence, our main result in this paper extends [16] (Theorem 3.12).
From Theorem 1, we obtain a new characterization of proper trans-Sasakian 3-manifolds.
Theorem 2. A compact trans-Sasakian 3-manifold with Reeb flow invariant Ricci operator is homothetic to either a Sasakian manifold or a cosymplectic manifold.
Proof.As seen in the proof of Theorem 1, a trans-Sasakian 3-manifold with Reeb flow invariant Ricci operator is a α-Sasakian manifold, a cosymplectic manifold or a space of constant sectional curvature.It is well known that an α-Sasakian manifold is homothetic to a Sasakian manifold.Moreover, there do exist compact Sasakian and cosymplectic manifolds.To complete the proof, we need only to prove that Case iii in the proof of Theorem 1 cannot occur.
Let M be a trans-Sasakian 3-manifold satisfying Case iii.According to (14) and Lemma 5, we know that the Reeb vector field is minimal or harmonic.It has been proved in [17] (Lemma 5.1) that when ξ of a compact trans-Sasakian 3-manifold is minimal or harmonic, then α is a constant.Because the manifold is of constant sectional curvature, then the scalar curvature r is also a constant.Therefore, the differentiation of (20) along ξ gives Adding the above equation to the first term of (12) implies that α = 0 because of β = 0. Using this in ( 14), we have ∇β = ξ(β)ξ.The following proof follows directly from [2].For sake of completeness, we present the detailed proof.
Applying ∇β = ξ(β)ξ and (7), we obtain for any vector field X. Contracting X in the previous relation and using (21), we obtain ∆β = ξ(ξ(β)) + 2βξ(β) = 0.Because the manifold is assumed to be compact, the application of the divergence theorem gives that β is a non-zero constant.Next, we show that this is impossible.In fact, the application of (4) gives that divξ = 2β.Since the manifold is assumed to be compact, it follows that β = 0, a contradiction.This completes the proof.
Theorem 2 can also be written as follows.
Theorem 3. A compact trans-Sasakian 3-manifold with Reeb flow invariant Ricci operator is proper.
The curvature tensor R of a trans-Sasakian 3-manifold is given by (see [10,27]) for any vector fields X, Y, Z, where, for simplicity, we set Substituting ( 14) and ( 20) into (22), with the aid of ( 23), we get R(X, Y)Z = r 6 (g(Y, Z)X − g(X, Z)Y) for any vector fields X, Y, Z.This implies that, on a trans-Sasakian 3-manifold satisfying Case iii in the proof of Theorem 1, we do not know whether α = 0 or not.In view of this, we introduce an interesting question: Remark 3. Given a trans-Sasakian 3-manifold, following proof of Theorem 1, we still do not know whether β is a constant or not even when α = 0 and the manifold is compact (see [2]).
Author Contributions: X.L. introduced the problem.Y.Z.investigated the problem.W.W. wrote the paper.