The Ricci soliton is a natural generalization of Einstein metrics, which is a self-similar solution to Hamiltion's Ricci flow. In [1], Pigola, Rigoli, Rimoldi and Setti introduced the gradient Ricci almost soliton. That is, if there exist two smooth functions $f, \lambda$ such that

$ \begin{equation}\label{Int1} R_{ij}+f_{ij}=\lambda g_{ij}, \end{equation} $

(1.1)

then $(M^n, g)$ is called a gradient Ricci almost soliton, where $R_{ij}+f_{ij}$ is called the $\infty$-dimensional Bakry-Emery Ricci tensor. Clearly, a gradient Ricci soliton is a special case of the gradient Ricci almost soliton when $\lambda$ is a constant. In particular, if $\lambda=\rho R+\mu$, where $R$ is the scalar curvature and $\rho, \mu$ are two constants, then (1.1) is called the gradient $\rho$-Einstein soliton defined in [2] which is a self-similar solution to the following geometric flow first considered by Bourguignon in [3]

$ \begin{equation}\label{H-Int2} \frac{\partial}{\partial t}g_{ij}=-2(R_{ij}-\rho Rg_{ij}). \end{equation} $

(1.2)

For some study with respect to the gradient Ricci almost soliton, the interested reader can refer to [1, 4-7] for more details.

Note that if $f$ given in (1.1) satisfies $f_{ij}=0$, then (1.1) becomes

$ \begin{equation}\label{Int2} R_{ij}=\lambda g_{ij} \end{equation} $

(1.3)

and the classical Schur's lemma states that the scalar curvature $R=n\lambda$ must be a constant when $n\geq3$. However, there exist gradient Ricci almost solitons with nontrivial function $f$ such that $\lambda$ is not a constant. A natural question is to consider whether can one find manifolds satisfying (1.1) with nontrivial function $f$ on which $\lambda$ is constant. In this paper, we consider this problem on Kähler manifolds and prove the following results.

Theorem 1.1 Let $(M^n, g_{i\bar{j}})$ be an $n$-dimensional Kähler manifold with $n\geq2$. If there exist two smooth real-valued functions $f, \lambda$ satisfying the equation

$ \begin{equation}\label{Proof1} R_{i\bar{j}}+f_{i\bar{j}}=\lambda g_{i\bar{j}}, \end{equation} $

(1.4)

then $\lambda$ must be a constant.

Therefore, by virtue of Theorem 1.4 of Chen and Zhu in [8], we obtain the following.

Corollary 1.2 Let $(M^n, g_{i\bar{j}})$ be an $n$-dimensional ($n\geq2$) complete Kähler manifold with harmonic Bochner tensor. If there exist two smooth real-valued functions $f, \lambda$ satisfying (1.4) with $f_{ij}=0$(that is, $\nabla f$ is a holomorphic vector field), then we have

1) if the function $\lambda>0$, then $(M^n, g_{i\bar{j}})$ is isometric to the quotient of $N^k\times \mathbb{C}^{n-k}$, where $N^k$ is a $k$-dimensional Kähler-Einstein manifold with positive scalar curvature;

2) if the function $\lambda<0$, then $(M^n, g_{i\bar{j}})$ is isometric to the quotient of $N^k\times \mathbb{C}^{n-k}$, where $N^k$ is a $k$-dimensional Kähler-Einstein manifold with negative scalar curvature.

Remark If $\lambda$ defined in (1.4) is a constant, then it is called a Kähler-Ricci soliton. For the classification of the Kähler-Ricci soliton, we refer to [8, 9].

Using the concepts as in [8], under the Kähler metric $g=(g_{i\bar{j}})$, the Ricci curvature and the scalar curvature defined by

$ R_{i\bar{j}}=R_{i\bar{j}k\bar{k}}, \ \ \ R=R_{i\bar{i}}=R_{i\bar{i}j\bar{j}}, $

respectively. By the first Bianchi identity, we have

$ \begin{equation} R_{i\bar{j}, j}=R_{i\bar{j}l\bar{l}, j} =-(R_{\bar{j}jl\bar{l}, i}+R_{jil\bar{l}, \bar{j}}) =-R_{\bar{j}jl\bar{l}, i} =R_{j\bar{j}l\bar{l}, i} =R_{, i}. \end{equation} $

(2.1)

By virtue of (1.4), we obtain

$ \begin{equation}\label{Proof3} R= {\rm tr}_{g}(R_{i\bar{j}})=n\lambda-\Delta f. \end{equation} $

(2.2)

Therefore, from (2.3), we obtain

$ (n\lambda)_{, i} = R_{, i}+(\Delta f)_{, i} =R_{, i}+f_{j\bar{j}, i} =R_{, i}+f_{i\bar{j}, j} \nonumber\\ \ \ \ \ \ \ \ \ \ = R_{, i}+[\lambda g_{i\bar{j}}-R_{i\bar{j}}]_{, j} =R_{, i}+\lambda_{, i}-R_{i\bar{j}, j} =\lambda_{, i}, $

(2.3)

where in the third equality we used

$ \begin{equation} \label{Proof4} f_{j\bar{j}, i}=f_{\bar{j}j, i} =f_{\bar{j}i, j}-f_{l}R_{\bar{l}\, \bar{j}j, i} =f_{\bar{j}i, j} =f_{i\bar{j}, j} \end{equation} $

(2.4)

and in the last equality we used (2.1). Therefore, from (2.4), it is easy to see that

$ \begin{equation}\label{Proof5} (n-1)\lambda_{, i}=0~~ {\rm for \ all}\ i, \end{equation} $

(2.5)

which shows that $\lambda$ is a constant.

We complete the proof of Theorem 1.1.