Let $(M, \phi, \eta, \xi)$ be a $(2n+1)$-dimensional almost contact manifold. Then the product $\overline{M}=M\times\mathbb{R}$ is a almost Hermitian manifold with almost complex structure $J$ and product metric $G$ being Hermitian metric. In [10], Gray and Harvella gave sixteen different structures of the almost Hermitian manifold $(\overline{M}, J, G)$. Using the structure in the class $\mathcal{W}_4$ on $(\overline{M}, J, G)$, the trans-Sasakian structure $(\phi, \eta, \xi, \alpha, \beta)$ on $M$, was defined (see [15]) that is the generalization of Sasakian and Kenmotsu structure on a contact metric manifold (see [1, 12]), where $\alpha, \beta$ are smooth functions on $M$. In general, we denote $(M, \phi, \eta, \xi, \alpha, \beta)$ by a trans-Sasakian manifold of type $(\alpha, \beta)$. Note that trans-Sasakian manifolds of type $(0, 0)$, $(\alpha, 0)$ and $(0, \beta)$ are called cosymplectic, $\alpha$-Sasakian and $\beta$-Kenmotsu manifolds respectively.

Recall that a Ricci soliton is the generalization of Einstein metric and defined on a Riemannian manifold $(M, g)$ by

$ \begin{equation}\label{eq:1.1} {\rm Ric}+\frac{1}{2}\mathcal{L}_V g=\lambda g, \end{equation} $

(1.1)

where $V$ is a smooth vector field, $\lambda$ a constant on $M$. It is called gradient Ricci soliton if $V=\nabla f$ for some smooth function $f$ on $M$. The Ricci soliton became important not only for studying topology of manifold but in study of string theory. Compact Ricci solitons are the fixed point of Ricci flow

$ \frac{\partial}{\partial t}g=-2{\rm Ric} $

projected from the space of metrics onto its quotient modulo diffeomorphisms and scalings, and often arise as blow-up limits for Ricci flow on compact manifolds. The Ricci soliton is said to be shrinking, steady and expanding according as $\lambda$ is negative, zero and positive respectively. More details about Ricci soliton can refer to [2, 4].

Recently in [3], Calin and Carasmareanu started to study Ricci solitons in $f$-Koenmotsu manifolds. Later Nagaraja and Premalatha [13] also considered Ricci soliton $(g, V, \lambda)$ in $f$-Koenmotsu manifolds and Ricci soliton in 3-dimensional trans-Sasakian manifolds when $V$ is a conformal killing vector field, and gave the conditions for Ricci solitons to be shrinking, steady and expanding. Otherwise, De [9] studied Ricci solitons on normal almost contact metric manifolds.

Concerning the Ricci solitons in contact manifolds, Sharama [16] began to study the Ricci solitons in $K$-contact manifolds, where the contact structure $\xi$ is a killing vector field, i.e., $\mathcal{L}_\xi g=0$, which is not in general in a trans-Sasakian manifold. Recently, He and Zhu [11] proved that a Sasakian manifold satisfying the gradient Ricci soliton equation is necessarily Einstein. Also, Cho [5, 6] considered contact Ricci solitons and transversal Ricci solitons in 3-contact manifolds, and proved that a compact contact Ricci soliton is Sasakian-Einstein and a 3-contact manifold admitting a transversal Ricci soliton is either Sasakian or locally isometric to one of the following Lie group with a left invariant metric: $SU(2)$, $SL(2, \mathbb{R})$, $E(2)$, respectively.

Motivated by the above work, in this paper, we study the Ricci soliton in a 3-dimensional trans-Sasakian manifold $(M, \phi, \eta, \xi, \alpha, \beta)$ of type $(\alpha, \beta)$ in case of $V=\xi$ in Ricci soliton equation (1.1) and the gradient Ricci solitons in trans-Sasakian manifolds.

An almost contact manifold $(M, \phi, \xi, \eta)$ is a $(2n+1)$-dimensional Riemannian manifold $M$ equipped with an almost contact structure $(\phi, \xi, \eta)$, where $\phi$ is a $(1, 1)$-tensor field, $\xi$ a unit vector field, $\eta$ a one-form dual to $\xi$ satisfying

$ \begin{equation} \phi^2=-I+\eta\otimes\xi, \, \eta\circ \phi=0, \, \phi\circ\xi=0. \end{equation} $

(2.1)

It is well-known that there exists a Riemannian metric $g$ such that

$ \begin{equation} g(\phi X, \phi Y)=g(X, Y)-\eta(X)\eta(Y), \end{equation} $

(2.2)

$ \begin{equation} g(\phi X, Y)=-g(X, \phi Y), \, g(X, \xi)=\eta(X), \end{equation} $

(2.3)

where $X, Y\in\mathfrak{X}(M)$. If there are two smooth functions $\alpha, \beta$ on $(M, \phi, \xi, \eta)$ such that

$ \begin{equation} (\nabla_X\phi)Y=-\alpha(g(X, Y)\xi-\eta(Y)X)+\beta(g(\phi X, Y)\xi-\eta(Y)\phi X), \end{equation} $

(2.4)

then $M$ is called a trans-Sasakian manifold of type $(\alpha, \beta)$, denote by $(M, \phi, \xi, \eta, \alpha, \beta)$, where $\nabla$ is the Levi-Civita connection with respect to metric $g$. It is clear that a trans-Sasakian manifold of type $(1, 0)$ is a Sasakian manifold and a trans-Sasakian manifold of type $(0, 1)$ is a Kenmotsu manifold. A trans-Sasakian manifold of type $(0, 0)$ is called cosymplectic manifold.

Using (2.4), it follows that for any $X, Y\in \mathfrak{X}(M)$

$ \begin{equation} \nabla_X\xi=-\alpha\phi(X)+\beta(X-\eta(X)\xi), \, (\nabla_X\eta)Y=-\alpha g(\phi X, Y)+\beta g(\phi X, \phi Y). \end{equation} $

(2.5)

Then it is easy to get the divergence $\hbox{div}\xi={\rm tr}(X\rightarrow\nabla_X\xi)=2n\beta$ and $\nabla_\xi\xi=0$.

Let Ric be the Ricci tensor on a Riemmaian manifold $(M, g)$, then the Ricci operator $Q:\mathfrak{X}(M)\to\mathfrak{X}(M)$ is defined by $Ric(X, Y)=g(QX, Y), \, X, Y\in\mathfrak{X}(M)$. It is well known that for any vector field $X, Y\in\mathfrak{X}(M)$, the following results were hold [7, Theorem 3.2, Proposition 3.4]:

$ \begin{array}{l} R(X, Y)\xi&= (\alpha^2-\beta^2)(\eta(Y)X-\eta(X)Y)+2\alpha\beta(\eta(Y)\phi(X)-\eta(X)\phi(Y))\\&+(Y\alpha)\phi X-(X\alpha)\phi Y+(Y\beta)\phi^{2}X-(X\beta)\phi^{2}Y, \end{array} $

(2.6)

$ \begin{eqnarray} 2\alpha\beta+\xi\alpha&=&0, \end{eqnarray} $

(2.7)

$ \begin{eqnarray} {\rm Ric}(X, \xi)&=&(2n(\alpha^2-\beta^2)-\xi\beta)\eta(X)-(2n-1)X\beta-(\phi X)\alpha. \end{eqnarray} $

(2.8)

Lemma 2.1    For any Riemannain manifold $(M, g)$ and a local orthogonal frame $\{e_j\}$ on $M$, $j=1, \cdots, \hbox{dim} M$, the gradient of scalar curvature $r$ satisfies

$ \frac{1}{2}\nabla r=\sum\limits_{j}(\nabla Q)(e_j, e_j), $

where $(\nabla Q)(X, Y)=\nabla_{X}Q(Y)-Q(\nabla_{X}Y), \, X, Y\in\mathfrak{X}(M)$.

Proof   For any $X\in\mathfrak{X}(M)$,

$ \begin{align*} X(r)&=\sum\limits_{j}\nabla_{X}{\rm Ric}(e_j, e_j)=\sum\limits_{j}\nabla_{X}g(Qe_j, e_j)\\ &=\sum\limits_{j}\biggl\{g(\nabla_{X}(Qe_j), e_j)+g(Qe_j, \nabla_{X}e_j)\biggl\}\\ &=\sum\limits_{j}g((\nabla Q)(e_j, X), e_j) =2\sum\limits_{j}g((\nabla Q)(e_j, e_j), X). \end{align*} $

Note that the last equation is held because of the second Bianchi identity.

In this section we consider Ricci soliton $(g, \xi, \lambda)$ in 3-dimensional trans-Sasakian manifolds $(M, \phi, \xi, \eta, \alpha, \beta)$, i.e., there exists some constant $\lambda$ satisfies

$ \begin{equation} {\rm Ric}+\frac{1}{2}\mathcal{L}_\xi g=\lambda g. \end{equation} $

(3.1)

The next lemma play important role in proving our results.

Lemma 3.1 For any $(2n+1)$-dimensional manifold with trans-Sasakian structure $(\phi, \xi, \eta, \alpha, \beta)$, we have

$ \frac{1}{2}\xi r=2n\beta^2, $

where $r$ is the scalar curvature.

Proof   In term of (3.1) for any vector field $X$,

$ \begin{equation} Q(X)=\lambda X+\beta\phi^2X. \end{equation} $

(3.2)

We compute the differentiation of (3.2) with respect to any vector field $Y$,

$ \begin{array}{l} (\nabla_YQ)X&= \nabla_Y(Q(X))-Q(\nabla_YX)\nonumber\\ &=\nabla_Y(\lambda X+\beta\phi^2X)-\lambda\nabla_YX-\beta\phi^2(\nabla_YX)\nonumber\\ &= Y(\beta)\phi^2X-\alpha\beta g(X, \phi Y)\xi+\beta^2g(\phi X, \phi Y)\xi\nonumber\\& -\alpha\beta\eta(X)\phi(Y)-\beta^2\eta(X)\phi^2(Y). \end{array} $

(3.3)

Since there is a canonical splitting of tangent bundle $\ker\eta\oplus {\rm span}{\xi}$ as the case of a contact structure, we can choose an orthogonal frame $\{e_1, \cdots, e_{2n+1}\}$ such that $e_{j+n}=\phi e_j, $ $e_{2n+1}=\xi, $ $j=1, \cdots, n$. It reduces from Lemma 2.1 and (3.3) that

$ \begin{eqnarray*} \frac{1}{2}\xi r&=&\frac{1}{2}g(\nabla r, \xi)=\sum\limits_{j=1}^{2n+1}g((\nabla Q)(e_j, e_j), \xi) =\sum\limits_{j=1}^{2n+1}g((\nabla_{e_j}Q)e_j, \xi)\\ &=&\beta^2\sum\limits_{j=1}^{2n}g(\phi e_j, \phi e_j)=2n\beta^2. \end{eqnarray*} $

For the 3-dimensional trans-Sasakian manifolds, the Ricci tensor Ric may express as follows (see[7]):

$ \begin{array}{l} {\rm Ric}(X, Y)&=(\frac{1}{2}r+\xi\beta-(\alpha^2-\beta^2))g(X, Y)\nonumber\\ & -(\frac{1}{2}r+\xi\beta-3(\alpha^2-\beta^2))\eta(X)\eta(Y)\nonumber\\ &-(Y\beta+\phi(Y)\alpha)\eta(X)-(X\beta+\phi(X)\alpha)\eta(Y), \end{array} $

(3.4)

where $r$ is the scalar curvature.

Thus

$ \begin{eqnarray} {\rm Ric}(\phi X, \phi Y)=(\frac{1}{2}r+\xi\beta-(\alpha^2-\beta^2))g(\phi X, \phi Y), \end{eqnarray} $

(3.5)

$ \begin{eqnarray} {\rm Ric}(\xi, \xi)=2(\alpha^2-\beta^2)-2\xi\beta. \end{eqnarray} $

(3.6)

By the first equation of (2.5), a straightforward calculation implies that

$ (\mathcal{L}_\xi g)(X, Y)=g(\nabla_X\xi, Y)+g(X, \nabla_Y\xi)=2\beta g(\phi X, \phi Y). $

Therefore

$ \begin{equation} (\mathcal{L}_\xi g)(\phi X, \phi Y)=2\beta g(\phi^2 X, \phi^2 Y)=2\beta g(\phi X, \phi Y). \end{equation} $

(3.7)

Applying (3.7), (3.5) in Ricci soliton equation (3.1), we have

$ \begin{equation} \frac{1}{2}r+\xi\beta-(\alpha^2-\beta^2)+\beta=\lambda. \end{equation} $

(3.8)

Obviously, since $\phi\xi=0$,

$ \begin{equation} (\mathcal{L}_\xi g)(\xi, \xi)=0. \end{equation} $

(3.9)

On the other hand, it implies from equations(3.6), (3.6) and Ricci soliton equation (3.1) that

$ \begin{equation} -2\xi\beta+2(\alpha^2-\beta^2)=\lambda. \end{equation} $

(3.10)

Then from (3.8) and (3.10) we obtain

$ \begin{equation} r+2\beta=3\lambda. \end{equation} $

(3.11)

Differentiating (3.11) w.r.t. $\xi$ and together with Lemma 3.1 when $n=1$, we get

$ \begin{equation} \xi\beta=-2\beta^2. \end{equation} $

(3.12)

It implies immediately from (3.12) and (3.11) that $\lambda=2(\alpha^2+\beta^2)$, then we have the following result.

Proposition 3.2    A Ricci soliton $(g, \lambda, \xi)$ in a 3-dimensional trans-Sasakian manifold is shrinking.

Moreover, we get from equation (3.12) the following.

Theorem 3.3   If $(M, \phi, \xi, \eta, \alpha, \beta)$ is a 3-dimensional compact and connected trans-Sasakian manifold admitting Ricci soliton $(g, \lambda, \xi)$, then $M$ is homothetic to a Sasakian manifold.

Proof   Using (3.12) and $\hbox{div}\xi=2\beta$, we get $\beta=0$ and $\alpha$ is a non-zero constant. It deduces that for any $X, Y\in\mathfrak{X}(M)$,

$ \alpha^{-2}(\nabla_{X}\nabla_Y\xi-\nabla_{\nabla_X Y}\xi)=g(Y, \xi)X-g(X, Y)\xi, $

and $(\mathcal{L}_\xi g)(X, Y)=0$, i.e., $\xi$ is a killing vector field. Thus it completes the proof of theorem by [14, Theorem 1.1]. The detail of proof can be seen in [8, Theorem 3.1].

Corollary 3.4   A 3-dimensional compact and connected trans-Sasakian manifold $M$ of type$(\alpha, \beta)$ admitting Ricci soliton $(g, \lambda, \xi)$ is an Einstein manifold.

Proof   From the proof of Theorem 3.3, we know $\beta=0$. Thus the scalar curvature $r=3\lambda$ is constant via (3.11). Moreover, Sharama [16] proved that a compact Ricci soliton of constant scalar curvature is Einstein, then we obtain immediately the result.

In this section we consider gradient Ricci solitons in trans-Sasakian manifolds. We assume that $(M, \phi, \xi, \alpha, \beta)$ is a $(2n+1)$-dimensional trans-Sasakian manifold.

First, we note that the following conclusion has been proved by taking Lie derivative of $\mathcal{L}_V g$ with respect to $\xi$.

Lemma 4.1[11]  For any manifold with a almost contact metric structure $(\phi, \xi, \eta, g)$,

$ \mathcal{L}_\xi(\mathcal{L}_{V}g)(Y, \xi)=R(V, \xi, \xi, Y)+g(\nabla_\xi\nabla_\xi V, Y) +\nabla_Y g(\nabla_\xi V, \xi) $

for any vector field $Y$.

Using(2.7), equation (2.6) implies

$ \begin{eqnarray} R(X, \xi, \xi, Y)&=& g(R(X, \xi)\xi, Y)=g\biggl((\alpha^2-\beta^2)(X-\eta(X)\xi)+(\xi\beta)\phi^{2}X, Y\biggl)\nonumber\\ &=& -(\alpha^2-\beta^2-\xi\beta)g(\phi X, \phi Y). \end{eqnarray} $

(4.1)

When $\alpha, \beta=$constant, it implies immediately from (2.8) that

$ {\rm Ric}(X, \xi)=2n(\alpha^2-\beta^2)\eta(X). $

Then

$ \begin{eqnarray} (\mathcal{L}_\xi {\rm Ric})(Y, \xi)&=&\nabla_\xi({\rm Ric}(\xi, Y))-{\rm Ric}([\xi, Y], \xi)\nonumber\\ &=&\nabla_\xi(2n(\alpha^2-\beta^2)\eta(Y)))-{\rm Ric}(\nabla_{\xi}Y-\nabla_Y\xi, \xi)\nonumber\\ &=& 2n(\alpha^2-\beta^2)g(\nabla_\xi Y, \xi)-{\rm Ric}(\nabla_{\xi}Y, \xi)\nonumber\\ &=& 2n(\alpha^2-\beta^2)\eta(\nabla_\xi Y)-{\rm Ric}(\nabla_{\xi}Y, \xi)=0, \end{eqnarray} $

(4.2)

and

$ \begin{eqnarray*} & (\mathcal{L}_\xi g)(Y, \xi)=g(\nabla_Y\xi, \xi)=0, \\ & R(V, \xi, \xi, Y)=-(\alpha^2-\beta^2)g(\phi V, \phi Y)=-(\alpha^2-\beta^2)g(V, Y). \end{eqnarray*} $

On the other hand,

$ \begin{eqnarray*} 2(\lambda-2n(\alpha^2-\beta^2))g(X, \xi)&=&2(\lambda g(X, \xi)-{\rm Ric}(X, \xi))=(\mathcal{L}_V g)(X, \xi)\\ &=&g(\nabla_X V, \xi)+g(\nabla_\xi V, X). \end{eqnarray*} $

Replacing $X$ by $\xi$ in above equation, we get

$ \lambda-2n(\alpha^2-\beta^2)=g(\nabla_\xi V, \xi). $

This implies $\nabla_Y g(\nabla_\xi V, \xi)=0$ since $\alpha, \beta, \lambda$ are constant. Therefore, from Lemma 4.1, taking the Lie derivative $\mathcal{L}_\xi$ to the Ricci soliton equation (1.1) yields

$ \begin{equation}\label{eq:3.22} -(\alpha^2-\beta^2)g(V, Y)+g(\nabla_\xi\nabla_\xi V, Y)=0. \end{equation} $

(4.3)

In case of where $V=\nabla f$ for some smooth function $f$, since for any $X\in\mathfrak{X}(M)$ Ricci soliton equation (1.1) yields $\nabla_X\nabla f+QX=\lambda X$,

$ \nabla_\xi\nabla_\xi\nabla f=\nabla_\xi(\lambda\xi-Q\xi)=-\nabla_\xi(2n(\alpha^2-\beta^2)\xi)=0. $

Using (4.3), therefore we have

$ \begin{equation}\label{eq:3.23} (\alpha^2+\beta^2)g(V, Y)=0. \end{equation} $

(4.4)

Next we consider the following cases:

(ⅰ) If $\alpha=0$ then $\beta\neq 0$ since $\alpha^2\neq\beta^2$. So we have $g(V, Y)=0$ via (4.4), i.e., $\nabla f=0$ for any $Y\bot\xi$. It follows that $f=$constant.

(ⅱ) If $\alpha\neq 0$ then $g(V, Y)=0$ by (4.4), i.e., $f=$constant.

Summarizing the above discussion, we obtain the following conclusions:

Theorem 4.2    Any trans-Sasakian manifold $(M, \phi, \xi, \eta, \alpha, \beta)$ admitting a gradient Ricci soliton is an Einstein manifold provided $\alpha$ and $\beta$ are constants.

Remark 4.3   In fact, our result can be regarded as the generalization of [11, Theorem 1.1].