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

Abstract

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

1. Introduction

A trans-Sasakian manifold is usually denoted by $(M,\varphi ,\xi ,\eta ,g,\alpha ,\beta )$, where both $\alpha$ and $\beta$ are smooth functions and $(\varphi ,\xi ,\eta ,g)$ is an almost contact metric structure. M is said to be proper if either $\alpha =0$ or $\beta =0$. When $\beta =0$, $\alpha$ is a constant if $\dim M\ge 5$ (see [1]) and in this case M becomes an $\alpha$-Sasakian manifold if $\alpha \in {\mathbb{R}}^{*}$ or a cosymplectic manifold if $\alpha =0$. This conclusion is not necessarily true for dimension three. However, unlike the above case, when $\alpha =0$, $\beta$ is not necessarily a constant even if $\dim M\ge 5$ or M is compact for dimension three (see [2]). The set of all trans-Sasakian manifolds of type $(0,\beta )$ coincides with that of all f-cosymplectic manifolds (see [3]) or f-Kenmotsu manifolds (see [4,5,6]). A trans-Sasakian manifold of dimension $\ge 5$ must be proper (see [1]). In the geometry of trans-Sasakian 3-manifolds, there exists a basic interesting problem, that is:

Under what condition is a trans-Sasakian 3-manifold proper?

De [7,8,9,10,11,12], Deshmukh [13,14,15], Wang and Liu [16] and Wang [2,17] answered this question from various points of view. In this paper, we study this question under a new geometric condition. Before stating our main results, we recall some results related with such a condition.

On an almost contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$, the Ricci operator of M is said to be Reeb flow invariant if it satisfies

$${\mathcal{L}}_{\xi }Q=0,$$

(1)

where $\mathcal{L}$, $\xi$ 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 $SO(3)$), $SL(2,R)$ (or $O(1,2)$), 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., $\nabla Q=0$) 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.

2. Trans-Sasakian Manifolds

On a smooth Riemannian manifold $(M,g)$ of dimension $2n+1$, we assume that $\varphi$, $\xi$ and $\eta$ are $(1,1)$-type, $(1,0)$-type and $(0,1)$-type tensor fields, respectively. According to [23], M is called an almost contact metric manifold if

$$\begin{array}{c}{\varphi }^{2}X=-X+\eta (X)\xi ,\eta (\xi )=1,\eta (\varphi X)=0,\\ g(\varphi X,\varphi Y)=g(X,Y)-\eta (X)\eta (Y),\eta (X)=g(X,\xi )\end{array}$$

(2)

for any vector fields X and Y. An almost contact metric manifold is said to be normal if $[\varphi ,\varphi ]=-2\mathrm{d}\eta \otimes \xi$, where $[\varphi ,\varphi ]$ denotes the Nijenhuis tensor of $\varphi$.

A normal almost contact metric manifold is called a trans-Sasakian manifold (see [1]) if

$$({\nabla }_X\varphi )Y=\alpha (g(X,Y)\xi -\eta (Y)X)+\beta (g(\varphi X,Y)\xi -\eta (Y)\varphi X)$$

(3)

for any vector fields $X,Y$ and two smooth functions $\alpha$, $\beta$. In particular, a three-dimensional almost contact metric manifold is trans-Sasakian if and only if it is normal (see [24,25]).

A normal almost contact metric manifold is called an α-Sasakian manifold if $\mathrm{d}\eta =\alpha \Phi$ and $\mathrm{d}\Phi =0$, where $\alpha \in {\mathbb{R}}^{*}$ (see [26]). An $\alpha$-Sasakian manifold reduces to a Sasakian manifold (see [23]) when $\alpha =1$. A normal almost contact metric manifold is called a β-Kenmotsu manifold if it satisfies $\mathrm{d}\eta =0$ and $\mathrm{d}\Phi =2\beta \eta \wedge \Phi$, where $\beta \in {\mathbb{R}}^{*}$ (see [26]). A $\beta$-Kenmotsu manifold becomes a Kenmotsu manifold when $\beta =1$. A normal almost contact metric manifold is called a cosymplectic manifold if it satisfies $\mathrm{d}\eta =0$ and $\mathrm{d}\Phi =0$.

Putting $Y=\xi$ into (3) and using (2), we have

$${\nabla }_X\xi =-\alpha \varphi X+\beta (X-\eta (X)\xi )$$

(4)

for any vector field X. In this paper, all manifolds are assumed to be connected.

3. 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 $(\alpha ,\beta )$ we have

$$\xi (\alpha )+2\alpha \beta =0.$$

(5)

Lemma 2.

On a trans-Sasakian 3-manifold of type $(\alpha ,\beta )$, the Ricci operator is given by

$$\begin{array}{cc}\hfill Q=& \left(\frac{r}{2}+\xi (\beta )-{\alpha }^2+{\beta }^2\right)\mathrm{id}-\left(\frac{r}{2}+\xi (\beta )-3{\alpha }^2+3{\beta }^2\right)\eta \otimes \xi \hfill \\ & +\eta \otimes (\varphi (\nabla \alpha )-\nabla \beta )+g(\varphi (\nabla \alpha )-\nabla \beta ,·)\otimes \xi ,\hfill \end{array}$$

(6)

where by $\nabla f$ 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 $(\alpha ,\beta )$, the following three conditions are equivalent:

(1) 

The Reeb vector field is minimal or harmonic.

(2) 

The following equation holds: $\varphi \nabla \alpha -\nabla \beta +\xi (\beta )\xi =0$(⇔$\nabla \alpha +\varphi \nabla \beta +2\alpha \beta \xi =0$).

(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 $\alpha$-Sasakian 3-manifold, the Ricci operator is given by

$$\begin{array}{c}\hfill QX=\left(\frac{r}{2}-{\alpha }^2\right)X-\left(\frac{r}{2}-3{\alpha }^2\right)\eta (X)\xi ,\end{array}$$

(7)

for any vector field X and certain nonzero constant $\alpha$. Moreover, according to [16] (Corollary 3.10), we observe that the scalar curvature r is invariant along the Reeb vector field $\xi$, i.e., $\xi (r)=0$. In fact, such an equation can be deduced directly by using the formula $\mathrm{div}Q=\frac{1}{2}\nabla r$ and (7). Applying $\xi (r)=0$, it follows directly from (7) that ${\mathcal{L}}_{\xi }Q=0$. □

Proof of Theorem 1.

Let M be a trans-Sasakian 3-manifold and e be a unit vector field orthogonal to $\xi$. Then, $\{\xi ,e,\varphi e\}$ 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])

$$\begin{array}{cc}\hfill {\nabla }_{\xi }\xi =& 0,{\nabla }_{\xi }e=\lambda \varphi e,{\nabla }_{\xi }\varphi e=-\lambda e,\hfill \\ \hfill {\nabla }_e\xi =& \beta e-\alpha \varphi e,{\nabla }_{e}e=-\beta \xi +\gamma \varphi e,{\nabla }_e\varphi e=\alpha \xi -\gamma e,\hfill \\ \hfill {\nabla }_{\varphi e}\xi =& \alpha e+\beta \varphi e,{\nabla }_{\varphi e}e=-\alpha \xi -\delta \varphi e,{\nabla }_{\varphi e}=-\beta \xi +\delta e,\hfill \end{array}$$

(8)

where $\lambda$, $\gamma$ and $\delta$ 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

$$0=({\mathcal{L}}_{\xi }Q)X=({\nabla }_{\xi }Q)X+\alpha \varphi QX-\alpha Q\varphi X+\beta \eta (QX)\xi -\beta \eta (X)Q\xi$$

(9)

for any vector field X.

By using the local basis $\{\xi ,e,\varphi e\}$ and Lemma 2, the Ricci operator can be rewritten as the following:

$$\begin{array}{cc}\hfill Q\xi =& \varphi \nabla \alpha -\nabla \beta +(2{\alpha }^2-2{\beta }^2-\xi (\beta ))\xi ,\hfill \\ \hfill Qe=& \left(\frac{r}{2}+\xi (\beta )-{\alpha }^2+{\beta }^2\right)e-(\varphi e(\alpha )+e(\beta ))\xi ,\hfill \\ \hfill Q\varphi e=& \left(\frac{r}{2}+\xi (\beta )-{\alpha }^2+{\beta }^2\right)\varphi e+(e(\alpha )-\varphi e(\beta ))\xi .\hfill \end{array}$$

(10)

Replacing X in (9) by $\xi$, we obtain

$$\begin{array}{c}{\nabla }_{\xi }(\varphi \nabla \alpha -\nabla \beta )+\xi (2{\alpha }^2-2{\beta }^2-\xi (\beta ))\xi +\alpha (-\nabla \alpha +\xi (\alpha )\xi -\varphi \nabla \beta )\\ +2\beta ({\alpha }^2-{\beta }^2-\xi (\beta ))\xi -\beta (\varphi \nabla \alpha -\nabla \beta )-\beta (2{\alpha }^2-2{\beta }^2-\xi (\beta ))\xi =0.\end{array}$$

(11)

Taking the inner product of the above equation with $\xi$, e and $\varphi e$, respectively, we obtain

$$\begin{array}{cc}\hfill \xi (\xi (\beta ))+2\beta \xi (\beta )+4{\alpha }^2\beta =& 0,\hfill \\ \hfill \alpha e(\alpha )-\beta \varphi e(\alpha )-\beta e(\beta )-\alpha \varphi e(\beta )=& 0,\hfill \\ \hfill \beta e(\alpha )+\alpha \varphi e(\alpha )+\alpha e(\beta )-\beta \varphi e(\beta )=& 0,\hfill \end{array}$$

(12)

where we have employed Lemma 1. The addition of the second term of (12) multiplied by $\alpha$ to the third term of (12) multiplied by $\beta$ gives

$$({\alpha }^2+{\beta }^2)(e(\alpha )-\varphi e(\beta ))=0.$$

(13)

Following (13), we consider the following several cases.

Case i: ${\alpha }^2+{\beta }^2=0$, or equivalently, $\alpha =\beta =0$. In this case, the manifold becomes a cosymplectic 3-manifold. The proof for this case is completed because of Lemma 4.

Case ii: ${\alpha }^2+{\beta }^2\ne 0$. It follows immediately from (13) that $e(\alpha )-\varphi e(\beta )=0$, or equivalently, $g(\nabla \alpha +\varphi \nabla \beta ,e)=0$. Because e is assumed to be an arbitrary vector field, it follows that $\nabla \alpha +\varphi \nabla \beta =\eta (\nabla \alpha +\varphi \nabla \beta )\xi$, i.e.,

$$\nabla \alpha +\varphi \nabla \beta +2\alpha \beta \xi =0,$$

(14)

or equivalently, $\varphi \nabla \alpha -\nabla \beta +\xi (\beta )\xi =0$, where we have used Lemma 1. When $\beta =0$, it follows from (14) that $\alpha$ is a nonzero constant. Thus, the proof can be done by applying Lemma 5. In what follows, we consider the last case.

Case iii: ${\alpha }^2+{\beta }^2\ne 0$ and $\beta \ne 0$. In this context, (10) becomes

$$\begin{array}{cc}\hfill Q\xi =& 2({\alpha }^2-{\beta }^2-\xi (\beta ))\xi ,\hfill \\ \hfill Qe=& \left(\frac{r}{2}+\xi (\beta )-{\alpha }^2+{\beta }^2\right)e,\hfill \\ \hfill Q\varphi e=& \left(\frac{r}{2}+\xi (\beta )-{\alpha }^2+{\beta }^2\right)\varphi e.\hfill \end{array}$$

(15)

Replacing X by e in (9) and using (8), (15), we acquire

$$0=({\mathcal{L}}_{\xi }Q)e=\xi \left(\frac{r}{2}+\xi (\beta )-{\alpha }^2+{\beta }^2\right)e.$$

With the aid of Lemma 1 and the first term of (12), from the previous relation, we have

$$\xi (r)=0.$$

(16)

From (15), we calculate the derivative of the Ricci operator as the following:

$$\begin{array}{cc}\hfill ({\nabla }_{\xi }Q)\xi =& 0,\hfill \\ \hfill ({\nabla }_{e}Q)e=& e(A)e-\beta A\xi +2\beta ({\alpha }^2-{\beta }^2-\xi (\beta ))\xi ,\hfill \\ \hfill ({\nabla }_{\varphi e}Q)\varphi e=& \varphi e(A)\varphi e-\beta A\xi +2\beta ({\alpha }^2-{\beta }^2-\xi (\beta ))\xi ,\hfill \end{array}$$

(17)

where we have used the first term of (8) and (12) and, for simplicity, we put

$$A=\frac{r}{2}+\xi (\beta )-{\alpha }^2+{\beta }^2.$$

(18)

On a Riemannian manifold, we have $\mathrm{div}Q=\frac{1}{2}\nabla r$. In this context, it is equivalent to

$$g(({\nabla }_{\xi }Q)\xi +({\nabla }_{e}Q)e+({\nabla }_{\varphi e}Q)\varphi e,X)=\frac{1}{2}X(r)$$

(19)

for any vector field X. Replacing X in (19) by $\xi$ and recalling (16) and the first term of (12), we obtain $2\beta (A-2{\alpha }^2+2{\beta }^2+2\xi (\beta ))=0$, or equivalently,

$$\xi (\beta )-{\alpha }^2+{\beta }^2=-\frac{r}{6},$$

(20)

where we have used the assumption $\beta \ne 0$ and (18). According to (15), it is clear to see that the manifold is Einstein, i.e, $Q=\frac{r}{3}\mathrm{id}$. 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 $\nabla R=0$ and this is equivalent to $\nabla Q=0$ 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 ${\mathbb{S}}^3(c^2)$, the hyperbolic space ${\mathbb{H}}^3(-c^2)$, the Euclidean space ${\mathbb{R}}^3$, product space $\mathbb{R}\times {\mathbb{S}}^2(c^2)$ or $\mathbb{R}\times {\mathbb{H}}^2(-c^2)$, 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 $\alpha$-Sasakian manifold, a cosymplectic manifold or a space of constant sectional curvature. It is well known that an $\alpha$-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 $\xi$ of a compact trans-Sasakian 3-manifold is minimal or harmonic, then $\alpha$ 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 $\xi$ gives

$$\xi (\xi (\beta ))+2\beta \xi (\beta )=0.$$

(21)

Adding the above equation to the first term of (12) implies that $\alpha =0$ because of $\beta \ne 0$. Using this in (14), we have $\nabla \beta =\xi (\beta )\xi$. The following proof follows directly from [2]. For sake of completeness, we present the detailed proof.

Applying $\nabla \beta =\xi (\beta )\xi$ and (7), we obtain

$${\nabla }_X\nabla \beta =X(\xi (\beta ))\xi +\xi (\beta )(\beta X-\beta \eta (X)\xi )=0$$

for any vector field X. Contracting X in the previous relation and using (21), we obtain $\Delta \beta =\xi (\xi (\beta ))+2\beta \xi (\beta )=0$. Because the manifold is assumed to be compact, the application of the divergence theorem gives that $\beta$ is a non-zero constant. Next, we show that this is impossible. In fact, the application of (4) gives that $\mathrm{div}\xi =2\beta$. Since the manifold is assumed to be compact, it follows that $\beta =0$, 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])

$$\begin{array}{cc}& R(X,Y)Z\hfill \\ \hfill =& B(g(Y,Z)X-g(X,Z)Y)-Cg(Y,Z)\eta (X)\xi \hfill \\ & +g(Y,Z)(\eta (X)(\varphi \nabla \alpha -\nabla \beta )-g(\nabla \beta -\varphi \nabla \alpha ,X)\xi )\hfill \\ & +Cg(X,Z)\eta (Y)\xi -g(X,Z)(\eta (Y)(\varphi \nabla \alpha -\nabla \beta )-g(\nabla \beta -\varphi \nabla \alpha ,Y)\xi )\hfill \\ & -(g(\nabla \beta -\varphi \nabla \alpha ,Z)\eta (Y)+g(\nabla \beta -\varphi \nabla \alpha ,Y)\eta (Z))X-C\eta (Y)\eta (Z)X\hfill \\ & +(g(\nabla \beta -\varphi \nabla \alpha ,Z)\eta (X)+g(\nabla \beta -\varphi \nabla \alpha ,X)\eta (Z))X+C\eta (X)\eta (Z)Y\hfill \end{array}$$

(22)

for any vector fields $X,Y,Z$, where, for simplicity, we set

$$B=\frac{r}{2}+2\xi (\beta )-2{\alpha }^2+2{\beta }^2,C=\frac{r}{2}+\xi (\beta )-3{\alpha }^2+3{\beta }^2.$$

(23)

Substituting (14) and (20) into (22), with the aid of (23), we get

$$R(X,Y)Z=\frac{r}{6}(g(Y,Z)X-g(X,Z)Y)$$

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 $\alpha =0$ or not. In view of this, we introduce an interesting question:

Problem 1.

Is there a non-proper and non-compact trans-Sasakian 3-manifold of constant sectional curvature?

Remark 2.

According to De and Sarkar [10] (Theorem 5.1), we observe that a compact trans-Sasakian 3-manifold of constant sectional curvature is either α-Sasakian or β-Kenmotsu.

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 $\alpha =0$ and the manifold is compact (see [2]).