Soliton-Type Equations on a Riemannian Manifold

Abstract

We study some particular cases of soliton-type equations on a Riemannian manifold. We give an estimation of the first nonzero eigenvalue of the Laplace operator and provide necessary and sufficient conditions for the manifold to be isometric to a sphere. Finally, we characterize trivial generalized gradient Ricci solitons.

1. Introduction

Regarded as stationary solutions to the Ricci flow [1], Ricci solitons have been intensively studied in various frameworks and from different points of view. Properties of Ricci solitons deal with aspects concerning the curvature of the manifold on which they are defined as well as provide information on the behavior of the flow. Recently, generalizations of this notion have been used and soliton-type equations have been studied. Such a generalization was considered in [2]. Indeed, a generalized gradient Ricci soliton on a smooth manifold M is given by the data $\left(g,\nabla f,\alpha ,\beta \right)$ fulfilling

$$Hess\left(f\right)+\alpha Ric=\beta g,$$

(1)

where g and Ric are the Riemannian metric and the Ricci curvature with respect to g and f, $\alpha$ and $\beta$ are smooth functions on M. If $(\alpha ,\beta )=\left(-f,-\frac{1}{n-1}(1+rf)\right)$, then the metric is said to satisfy the Miao–Tam equation [3], and if $(\alpha ,\beta )=(-f,\mathsf{\Delta }(f\left)\right)$, then g is said to satisfy the Fischer–Marsden equation [4], where r denotes the scalar curvature of $(M,g)$.

It is interesting to note that an n-sphere $S^n\left(c\right)$ is a generalized gradient Ricci soliton $\left(g,\nabla f,\alpha ,\beta \right)$, where g is the canonical metric on $S^n\left(c\right)$, f is an eigenfunction of the Laplace operator corresponding to the first nonzero eigenvalue and $\alpha =-f$ and $\beta =-nfc$. This example initiates the question of finding conditions under which a generalized gradient Ricci soliton $\left(g,\nabla f,\alpha ,\beta \right)$ on an n-dimensional compact smooth manifold M is isometric to $S^n\left(c\right)$.

In the present paper, we treat this kind of soliton, finding necessary and sufficient conditions for the manifold to be isometric to a sphere and also characterizing the so-called trivial solitons, i.e., solitons with Killing potential vector fields.

2. Generalized Gradient Ricci Solitons

Let $\left(g,\nabla f,\alpha ,\beta \right)$ be a generalized gradient Ricci soliton on an n-dimensional smooth manifold M. From the soliton in Equation (1), we have

$$H_f+\alpha Q=\beta I,$$

(2)

where $H_f$ is the Hessian operator defined by $g\left(H_{f}X,Y\right):=Hess\left(f\right)(X,Y)$ and Q is the Ricci operator defined by $g\left(QX,Y\right):=Ric(X,Y)$. Taking the trace in Equation (2), we have

$$\mathsf{\Delta }\left(f\right)=n\beta -r\alpha .$$

(3)

On taking the inner product with $H_f$ in (2) and using (3), one obtains

$$\parallel H_f{\parallel }^2+\alpha ⟨H_f,Q⟩=\beta \mathsf{\Delta }\left(f\right)=\beta (n\beta -r\alpha )$$

and taking the inner product with Q, we have

$$⟨H_f,Q⟩+\alpha {\parallel Q\parallel }^2=r\beta .$$

Comparing the above relations, we obtain

$$\parallel H_f{\parallel }^2={\alpha }^2{\parallel Q\parallel }^2+\beta (n\beta -2r\alpha )={\alpha }^2\left({\parallel Q\parallel }^2-\frac{r^2}{n}\right)+\frac{1}{n}{(n\beta -r\alpha )}^2$$

(4)

and we can state:

Proposition 1.

If $(g,\nabla f,\alpha ,\beta )$ is a generalized gradient Ricci soliton on an n-dimensional smooth manifold M and $\parallel H_f{\parallel }^2\le {\alpha }^2\left({\parallel Q\parallel }^2-\frac{r^2}{n}\right)$, then f is a harmonic function; hence, $\beta =\frac{r\alpha }{n}$.

Proof. 

The hypothesis implies ${(n\beta -r\alpha )}^2\le 0$; therefore, $\beta =\frac{r\alpha }{n}$. □

As a consequence, we obtain:

Corollary 1.

Let f be a smooth non-constant function on an n-dimensional Riemannian manifold $(M,g)$ and assume that $\parallel H_f{\parallel }^2\le f^2\left({\parallel Q\parallel }^2-\frac{r^2}{n}\right)$. If the Riemannian metric g satisfies the Fischer–Marsden equation, then the scalar curvature is zero and $H_f=0$; hence, M is a Ricci flat manifold.

Next, we prove the following lemmas, which will be useful for our main results.

Lemma 1.

Let $\left(g,\nabla f,\alpha ,\beta \right)$ be a generalized gradient Ricci soliton on an n-dimensional smooth manifold M. Then

$$Q\left(\nabla f+\nabla \alpha \right)=-(n-1)\nabla \beta +r\nabla \alpha +\frac{1}{2}\alpha \nabla r.$$

Proof. 

We have

$$H_{f}X=\beta X-\alpha QX,$$

and differentiating the above equation, we obtain

$$\left({\nabla }_X{H}_f\right)Y=X\left(\beta \right)Y-X\left(\alpha \right)QY-\alpha \left({\nabla }_{X}Q\right)Y.$$

Now, using the above equation in

$$R(X,Y)\nabla f=\left({\nabla }_X{H}_f\right)Y-\left({\nabla }_Y{H}_f\right)X,$$

we conclude

$$R(X,Y)\nabla f=X\left(\beta \right)Y-Y\left(\beta \right)X-X\left(\alpha \right)QY+Y\left(\alpha \right)QX-\alpha \left(\left({\nabla }_{X}Q\right)Y-\left({\nabla }_{Y}Q\right)X\right).$$

Taking the trace in the above equation, while using the symmetry of the Ricci operator Q and the formula

$$\frac{1}{2}\nabla r=\sum _{i=1}^n\left({\nabla }_{e_i}Q\right)e_i,$$

we obtain

$$Ric\left(Y,\nabla f\right)=-(n-1)Y\left(\beta \right)-Ric\left(\nabla \alpha ,Y\right)+rY\left(\alpha \right)-\frac{1}{2}\alpha Y\left(r\right)+\alpha Y\left(r\right),$$

that is,

$$Ric\left(\nabla f+\nabla \alpha ,Y\right)=-(n-1)Y\left(\beta \right)+rY\left(\alpha \right)+\frac{1}{2}\alpha Y\left(r\right),$$

which implies the conclusion. □

For $\alpha =-f$, we obtain

$$(n-1)\nabla \beta =-r\nabla f-\frac{1}{2}f\nabla r,$$

from Lemma 1. Now, for the Miao–Tam equation, we obtain

$$f\nabla r=0,$$

by means of (3). Thus, we have the following:

Corollary 2.

If f is a non-trivial solution of the Miao–Tam equation on a complete Riemannian manifold $(M,g)$, then the scalar curvature r is a constant.

In order to give an estimation for the first nonzero eigenvalue of the Laplace operator, we prove the following result.

Lemma 2.

Let $\left(g,\nabla f,\alpha ,\beta \right)$ be a generalized gradient Ricci soliton on an n-dimensional smooth manifold M. If $\nabla f+\nabla \alpha$ is an eigenvector of the Ricci operator Q corresponding to the eigenvalue $\frac{r}{n}$, then

$$\nabla \left(\mathsf{\Delta }\left(f\right)+\frac{r}{n-1}f\right)=\frac{1}{n-1}\left(f-\left(\frac{n-2}{2}\right)\alpha \right)\nabla r.$$

Proof. 

Assume $Q\left(\nabla f+\nabla \alpha \right)=\frac{r}{n}\left(\nabla f+\nabla \alpha \right)$. Using Lemma 1, we have

$$\frac{r}{n}\left(\nabla f+\nabla \alpha \right)=-(n-1)\nabla \beta +r\nabla \alpha +\frac{1}{2}\alpha \nabla r,$$

that is,

$$\begin{array}{ccc}\hfill \frac{r}{n}\nabla f& =& -(n-1)\nabla \beta +\frac{n-1}{n}r\nabla \alpha +\frac{1}{2}\alpha \nabla r\hfill \\ & =& -(n-1)\nabla \beta +\frac{n-1}{n}\nabla \left(r\alpha \right)-\frac{n-1}{n}\alpha \nabla r+\frac{1}{2}\alpha \nabla r\hfill \\ & =& -\frac{n-1}{n}\nabla \left(n\beta -r\alpha \right)-\frac{n-2}{2n}\alpha \nabla r.\hfill \end{array}$$

Now, using Equation (3), we obtain

$$\frac{n-1}{n}\nabla \left(\mathsf{\Delta }\left(f\right)\right)+\frac{1}{n}\nabla \left(rf\right)-\frac{1}{n}f\nabla r=-\frac{n-2}{2n}\alpha \nabla r,$$

which gives

$$\frac{n-1}{n}\nabla \left(\mathsf{\Delta }\left(f\right)+\frac{r}{n-1}f\right)=\frac{1}{n}\left(f-\frac{n-2}{2}\alpha \right)\nabla r,$$

and implies the conclusion. □

As a consequence, we have the following result, which gives an estimate of the first nonzero eigenvalue of the Laplace operator.

Proposition 2.

Let $\left(g,\nabla f,\alpha ,\beta \right)$ be a generalized gradient Ricci soliton on an n-dimensional smooth compact and connected manifold M of constant scalar curvature r. If $\nabla f+\nabla \alpha$ is an eigenvector of the Ricci operator Q corresponding to the eigenvalue $\frac{r}{n}$, then the first nonzero eigenvalue ${\lambda }_1$ of the Laplace operator satisfies ${\lambda }_1\le \frac{r}{n-1}$.

Proof. 

Since the scalar curvature is constant, Lemma 2 implies

$$\nabla \left(\mathsf{\Delta }\left(f\right)+\frac{r}{n-1}f\right)=0,$$

that is, $\mathsf{\Delta }\left(f\right)+\frac{r}{n-1}f$ is a constant and we have

$$\mathsf{\Delta }\left(f\right)=-\frac{r}{n-1}\left(f-\overline{c}\right)$$

for a constant $\overline{c}$. Denoting $\overline{f}:=f-\overline{c}$, the above equation becomes

$$\mathsf{\Delta }\left(\overline{f}\right)=-\frac{r}{n-1}\overline{f}.$$

(5)

Note that as f is non-constant, the function $\overline{f}$ is also non-constant and it is an eigenfunction of the Laplace operator corresponding to the eigenvalue $\frac{r}{n-1}$. Since M is compact, we conclude

$${\lambda }_1\le \frac{r}{n-1}.\square$$

3. Characterization of Spheres

We prove the following result for further use.

Lemma 3.

Let $\left(g,\nabla f,\alpha ,\beta \right)$ be a generalized gradient Ricci soliton on an n-dimensional smooth compact manifold M. Then

$${\int }_M{\alpha }^2\left({∥Q∥}^2-\frac{r^2}{n}\right)={\int }_M\left({∥H_{\overline{f}}∥}^2-\frac{1}{n}{\left(\mathsf{\Delta }\left(\overline{f}\right)\right)}^2\right),$$

for $\overline{f}=f-\overline{c}$, with $\overline{c}$ a constant.

Proof. 

Using Equation (2) and the fact that $H_f=H_{\overline{f}}$, we have

$${∥H_{\overline{f}}∥}^2=n{\beta }^2+{\alpha }^2{∥Q∥}^2-2\alpha \beta r$$

(6)

which gives

$${\alpha }^2\left({∥Q∥}^2-\frac{r^2}{n}\right)={∥H_{\overline{f}}∥}^2-\frac{1}{n}\left(n^2{\beta }^2-2n\alpha \beta r+{\alpha }^2{r}^2\right)={∥H_{\overline{f}}∥}^2-\frac{1}{n}{\left(n\beta -r\alpha \right)}^2.$$

Integrating the above equation and using Equation (3), we obtain

$${\int }_M{\alpha }^2\left({∥Q∥}^2-\frac{r^2}{n}\right)={\int }_M\left({∥H_{\overline{f}}∥}^2-\frac{1}{n}{\left(\mathsf{\Delta }\left(\overline{f}\right)\right)}^2\right).\square$$

Next, we see that the tools developed above give us the following characterization of a sphere ${\mathbf{S}}^n\left(c\right)$.

Theorem 1.

Let $\left(g,\nabla f,\alpha ,\beta \right)$ be a generalized gradient Ricci soliton on an n-dimensional smooth compact and connected manifold M of constant scalar curvature r. If $\nabla f+\nabla \alpha$ is an eigenvector of the Ricci operator Q corresponding to the eigenvalue $\frac{r}{n}$, then

$$Ric\left(\nabla f,\nabla f\right)\ge \frac{r}{n}{∥\nabla f∥}^2$$

if and only if $r>0$ and M is isometric to the sphere ${\mathbf{S}}^n\left(c\right)$ with $r=n(n-1)c$.

Proof. 

Assume that the conditions in the statement hold. Then using Equation (5), we have

$$\overline{f}\mathsf{\Delta }\left(\overline{f}\right)=-\frac{r}{n-1}{\overline{f}}^2,$$

where $\overline{f}=f-\overline{c}$. Integrating the above equation yields

$${\int }_M{∥\nabla \overline{f}∥}^2=\frac{r}{n-1}{\int }_M{\overline{f}}^2,$$

that is,

$${\int }_M{∥\nabla f∥}^2=\frac{r}{n-1}{\int }_M{\overline{f}}^2,$$

(7)

and as $\overline{f}$ is non-constant, it implies that $r>0$. Note that $\nabla f=\nabla \overline{f}$, $H_f=H_{\overline{f}}$ and $\mathsf{\Delta }\left(f\right)=\mathsf{\Delta }\left(\overline{f}\right)$. Thus, from Bochner’s formula [5]

$${\int }_M\left(Ric\left(\nabla \overline{f},\nabla \overline{f}\right)+{∥H_{\overline{f}}∥}^2-{\left(\mathsf{\Delta }\left(\overline{f}\right)\right)}^2\right)=0$$

we have

$${\int }_M\left(Ric\left(\nabla f,\nabla f\right)+n{\beta }^2+{\alpha }^2{∥Q∥}^2-2\alpha \beta r-\frac{r^2}{{(n-1)}^2}{\overline{f}}^2\right)=0,$$

by means of (6). Using $n{\beta }^2-2\alpha \beta r=\frac{1}{n}\left[{\left(n\beta -r\alpha \right)}^2-{\alpha }^2{r}^2\right]$ and Equation (3), from the above relation, we obtain

$${\int }_M\left(Ric\left(\nabla f,\nabla f\right)+\frac{1}{n}{\left(\mathsf{\Delta }\left(f\right)\right)}^2+{\alpha }^2\left({∥Q∥}^2-\frac{r^2}{n}\right)-\frac{r^2}{{(n-1)}^2}{\overline{f}}^2\right)=0.$$

By Equation (5), we have

$${\int }_M\left(Ric\left(\nabla f,\nabla f\right)+\frac{r^2}{n{(n-1)}^2}{\overline{f}}^2+{\alpha }^2\left({∥Q∥}^2-\frac{r^2}{n}\right)-\frac{r^2}{{(n-1)}^2}{\overline{f}}^2\right)=0,$$

that is,

$${\int }_M\left(Ric\left(\nabla f,\nabla f\right)-\frac{r^2}{n(n-1)}{\overline{f}}^2+{\alpha }^2\left({∥Q∥}^2-\frac{r^2}{n}\right)\right)=0.$$

Now, using Equation (7), we have

$${\int }_M\left(\left(Ric\left(\nabla f,\nabla f\right)-\frac{r}{n}{∥\nabla f∥}^2\right)+{\alpha }^2\left({∥Q∥}^2-\frac{r^2}{n}\right)\right)=0.$$

From the hypothesis $Ric\left(\nabla f,\nabla f\right)\ge \frac{r}{n}{∥\nabla f∥}^2$ and Schwartz’s inequality ${∥Q∥}^2\ge \frac{r^2}{n}$, the above equation implies

$${\alpha }^2\left({∥Q∥}^2-\frac{r^2}{n}\right)=0.$$

Inserting the above equation into Lemma 3, we have

$${\int }_M\left({∥H_{\overline{f}}∥}^2-\frac{1}{n}{\left(\mathsf{\Delta }\left(\overline{f}\right)\right)}^2\right)=0.$$

Using Schwartz’s inequality, we obtain that the equality ${∥H_{\overline{f}}∥}^2=\frac{1}{n}{\left(\mathsf{\Delta }(\overline{f})\right)}^2$ holds if and only if $H_{\overline{f}}=\frac{\mathsf{\Delta }(\overline{f})}{n}I$. Now, from Equation (5), we arrive at

$$H_{\overline{f}}=-\frac{r}{n(n-1)}\overline{f}I=-c\overline{f}I,$$

where c is a positive constant. Hence, by the Theorem of Obata, M is isometric to ${\mathbf{S}}^n\left(c\right)$.

Conversely, for the sphere ${\mathbf{S}}^n\left(c\right)$, its Ricci tensor and scalar curvature are given by

$$Ric=(n-1)cg,r=n(n-1)c.$$

Moreover, there exists a smooth function f (the eigenfunction corresponding to the first nonzero eigenvalue ${\lambda }_1=nc$) on ${\mathbf{S}}^n\left(c\right)$ that satisfies

$$H_f=-cfI,\mathsf{\Delta }\left(f\right)=-ncf.$$

Thus, we see that

$$Hess\left(f\right)+(-f)Ric=(-ncf)g,$$

that is, $\left(g,\nabla f,\alpha ,\beta \right)$ is a generalized gradient Ricci soliton on ${\mathbf{S}}^n\left(c\right)$, with $\alpha =-f$ and $\beta =-ncf$. We see that all the conditions in the hypothesis are satisfied by this generalized gradient Ricci soliton on the sphere ${\mathbf{S}}^n\left(c\right)$. □

Finally, we prove the following characterization of the sphere ${\mathbf{S}}^n\left(c\right)$.

Theorem 2.

A generalized gradient Ricci soliton $\left(g,\nabla f,\alpha ,\beta \right)$ on an n-dimensional smooth compact and connected manifold M is isometric to the sphere ${\mathbf{S}}^n\left(c\right)$ if and only if the positive constant c satisfies

$$Ric\left(\nabla f,\nabla f\right)\ge (n-1)c{∥\nabla f∥}^2$$

and

$${\int }_M\left(n\beta -r\alpha +cf\right)\left(n\beta -r\alpha +ncf\right)\le 0.$$

Proof. 

Using Equation (3), we have

$$\begin{array}{ccc}\hfill {\int }_M\left(n\beta -r\alpha +cf\right)\left(n\beta -r\alpha +ncf\right)& =& {\int }_M\left({\left(\mathsf{\Delta }\left(f\right)\right)}^2+(n+1)cf\mathsf{\Delta }\left(f\right)+n{c}^2{f}^2\right)\hfill \\ & =& {\int }_M\left(Ric\left(\nabla f,\nabla f\right)+{∥H_f∥}^2+(n+1)cf\mathsf{\Delta }\left(f\right)+n{c}^2{f}^2\right).\hfill \end{array}$$

Now, using $Ric\left(\nabla f,\nabla f\right)\ge (n-1)c{∥\nabla f∥}^2$ and ${\int }_M\left(n\beta -r\alpha +cf\right)\left(n\beta -r\alpha +ncf\right)\le 0$ in the above equation, we conclude

$${\int }_M\left((n-1)c{∥\nabla f∥}^2+{\parallel H_f\parallel }^2+(n+1)cf\mathsf{\Delta }\left(f\right)+n{c}^2{f}^2\right)\le 0.$$

Inserting ${\int }_{M}f\mathsf{\Delta }\left(f\right)=-{\int }_M{∥\nabla f∥}^2$ into the above inequality, we have

$${\int }_M\left(-(n-1)cf\mathsf{\Delta }\left(f\right)+{\parallel H_f\parallel }^2+(n+1)cf\mathsf{\Delta }\left(f\right)+n{c}^2{f}^2\right)\le 0,$$

that is,

$${\int }_M\left({\parallel H_f\parallel }^2+2cf\mathsf{\Delta }\left(f\right)+n{c}^2{f}^2\right)\le 0$$

and we conclude

$${\int }_M{\parallel H_f+cfI\parallel }^2\le 0.$$

This proves that $H_f=-cfI$, that is, M is isometric to the sphere ${\mathbf{S}}^n\left(c\right)$.

Conversely, as seen in the proof of Theorem 1, we know that $\left(g,\nabla f,\alpha ,\beta \right)$ is a generalized gradient Ricci soliton on the sphere ${\mathbf{S}}^n\left(c\right)$, where f is the eigenfunction of the Laplace operator $\mathsf{\Delta }$ corresponding to the first nonzero eigenvalue $nc$ and $\alpha =-f$ and $\beta =-ncf$. In addition, we have

$${\int }_M\left(n\beta -r\alpha +cf\right)\left(n\beta -r\alpha +ncf\right)={\int }_M\left(-ncf+cf\right)\left(-ncf+ncf\right)=0.$$

Hence, all the conditions in the hypothesis are satisfied. □

4. Trivial Solitons

Following the ideas from [2,6,7], we shall further provide some characterizations for trivial generalized gradient Ricci solitons $(g,\nabla f,\alpha ,\beta )$ with unit geodesic potential vector fields, i.e., ${\nabla }_{\nabla f}\nabla f=0$. Note that it is not a unit vector field, but to distinguish between a geodesic vector field (whose integral curves are conformal geodesics) and those whose integral curves are geodesics, we use the term unit geodesic vector field.

Theorem 3.

Let $(g,\nabla f,\alpha ,\beta )$ be a generalized gradient Ricci soliton on an n-dimensional compact and connected smooth manifold M ($n>2$) with a unit geodesic potential vector field and nonzero scalar curvature. Assume that α and β are constant, $\alpha \ne 0$. Then $\nabla f$ is an eigenvector of the Ricci operator with constant eigenvalue $\frac{\beta }{\alpha }$ satisfying $(n\beta -r\alpha )r\alpha \ge 0$ if and only if the soliton is trivial.

Proof. 

The proof follows the same steps as [6,7]. The converse implication is trivial. For the direct implication, if we assume that $Q(\nabla f)=\sigma \nabla f$, $\sigma \in {\mathbb{R}}^{*}$, then taking the inner product with $\nabla f$ implies $\sigma =\frac{\beta }{\alpha }$, so $\left(g,\nabla (\frac{f}{\alpha }),\frac{\beta }{\alpha }\right)$ is a gradient Ricci soliton. Then from (1), (3) and Lemma 1, we obtain

$$\nabla r=\frac{2\beta }{{\alpha }^2}\nabla f,$$

$$Hess\left(r\right)=\frac{2\beta }{{\alpha }^2}Hess\left(f\right)=\frac{2\beta }{{\alpha }^2}(\beta g-\alpha Ric),$$

$$\mathsf{\Delta }\left(r\right)=\frac{2\beta }{{\alpha }^2}\mathsf{\Delta }\left(f\right)=\frac{2\beta }{{\alpha }^2}(n\beta -r\alpha ),$$

$$Ric(\nabla r,\nabla r)=\frac{\beta }{\alpha }{\parallel \nabla r\parallel }^2.$$

In this case, Bochner’s formula

$${\int }_M\left(Ric(\nabla r,\nabla r)+{\parallel Hess\left(r\right)\parallel }^2-{\left(\mathsf{\Delta }\left(r\right)\right)}^2\right)=0$$

becomes

$${\int }_M{\alpha }^2\left({\parallel Q\parallel }^2-\frac{r^2}{n}\right)={\int }_M\left(\frac{n-1}{n}{(n\beta -r\alpha )}^2-\frac{{\alpha }^3}{4\beta }{\parallel \nabla r\parallel }^2\right).$$

However, $\mathsf{\Delta }\left(r\right)=\frac{2\beta }{{\alpha }^2}(n\beta -r\alpha )$ and $div(r\nabla r)=r\mathsf{\Delta }\left(r\right)+{\parallel \nabla r\parallel }^2$ imply

$${\int }_M(n\beta -r\alpha )=0,{\int }_M{\parallel \nabla r\parallel }^2=-\frac{2\beta }{{\alpha }^2}{\int }_M\left(n\beta -r\alpha \right)r,$$

which, replaced in the previous relation, gives

$${\int }_M\left({\parallel Q\parallel }^2-\frac{r^2}{n}\right)=-\frac{n-2}{2n\alpha }{\int }_M(n\beta -r\alpha )r.$$

Using Schwartz’s inequality, we deduce that ${\parallel Q\parallel }^2=\frac{r^2}{n}$; hence, $Q=\frac{r}{n}I$. Moreover, since r is nonzero, we obtain $n\beta =r\alpha$; therefore, $Hess\left(f\right)=0$ by (4), i.e., the soliton is trivial. □

Theorem 4.

Let $(g,\nabla f,\alpha ,\beta )$ be a generalized gradient Ricci soliton on an n-dimensional compact and connected smooth manifold M ($n>2$) with a unit geodesic potential vector field. Then

$$Ric(\nabla f,\nabla f)\ge \frac{n-1}{n}{(n\beta -r\alpha )}^2$$

if and only if the soliton is trivial.

Proof. 

The converse implication is trivial. For the direct implication, from (2), we obtain

$${\parallel Hess\left(f\right)\parallel }^2={\alpha }^2\left({\parallel Q\parallel }^2-\frac{r^2}{n}\right)+\frac{{(n\beta -r\alpha )}^2}{n}.$$

Using (3) and Bochner’s formula

$${\int }_M\left(Ric(\nabla f,\nabla f)+{\parallel Hess\left(f\right)\parallel }^2-{\left(\mathsf{\Delta }\left(f\right)\right)}^2\right)=0$$

we obtain

$${\int }_M{\alpha }^2\left({\parallel Q\parallel }^2-\frac{r^2}{n}\right)={\int }_M\left(\frac{n-1}{n}{(n\beta -r\alpha )}^2-Ric(\nabla f,\nabla f)\right).$$

By using Schwartz’s inequality, we deduce ${\parallel Q\parallel }^2=\frac{r^2}{n}$; hence, $Q=\frac{r}{n}I$. Therefore,

$$\frac{r}{n}\nabla f=Q(\nabla f)=\frac{\beta }{\alpha }\nabla f-\frac{1}{\alpha }{\nabla }_{\nabla f}\nabla f=\frac{\beta }{\alpha }\nabla f$$

which implies $n\beta =r\alpha$, and we deduce that $Hess\left(f\right)=0$, i.e., the soliton is trivial. □

For particular cases, we can state:

Corollary 3.

Let $(M,g)$ be an n-dimensional compact and connected Riemannian manifold M ($n>2$) and $\nabla f$ a unit geodesic vector field.

(i) If g satisfies the Miao–Tam equation and $Ric(\nabla f,\nabla f)\ge \frac{1}{n(n-1)}{(n+rf)}^2$, then M is an Einstein manifold.

(ii) If g satisfies the Fischer–Marsden equation and $Ric(\nabla f,\nabla f)\ge \frac{1}{n(n-1)}{\left(rf\right)}^2$, then M is a Ricci flat manifold.

If $\alpha$ is nowhere zero and $\nabla f$ is a conformal vector field with ${\pounds }_{\nabla f}g=2\beta g$, then $Hess\left(f\right)=\beta g$ and M is an Einstein manifold, provided $n\ge 3$. If $\alpha =0$, Equation (1) becomes

$$Hess\left(f\right)=\beta g,$$

(8)

hence, ${\parallel Hess\left(f\right)\parallel }^2=n{\beta }^2$ and $\mathsf{\Delta }\left(f\right)=n\beta$, which implies the equality case in Schwartz’s inequality. Note that in [8,9,10,11,12], the authors proved that a non-constant function f on a complete n-dimensional Riemannian manifold $(M,g)$ satisfies Equation (8) for $\beta$ a negative constant if and only if M is isometric to the n-dimensional Euclidean space. In [13], the authors proved that if Equation (8) holds with $\beta$ a function, then $(M,g)$ is locally a warped product $(a,b){\times }_h{N}^{n-1}$. If $\beta$ is a non-constant function on M, we prove the following result.

Proposition 3.

Let $(M,g)$ be an n-dimensional compact Riemannian manifold and let f be a smooth function on M satisfying Equation (8). If $Ric(\nabla f,\nabla f)\le 0$, then $\nabla f\in kerQ$ and $Hess\left(f\right)=0$.

Proof. 

From (8), we obtain $\mathsf{\Delta }\left(f\right)=n\beta$. Hence,

$$d\beta =div(Hess\left(f\right))=d\left(\mathsf{\Delta }\left(f\right)\right)+i_{Q(\nabla f)}g=nd\beta +i_{Q(\nabla f)}g$$

which implies

$$\nabla \beta =-\frac{1}{n-1}Q(\nabla f).$$

However, $\nabla \beta =\frac{1}{n}\nabla \left(\mathsf{\Delta }\left(f\right)\right)$. Therefore,

$$\nabla \left(\mathsf{\Delta }\left(f\right)\right)=-\frac{n}{n-1}Q(\nabla f).$$

Replacing these relations in Bochner’s formula

$$\frac{1}{2}{\mathsf{\Delta }(\parallel \nabla f\parallel }^2{)=Ric(\nabla f,\nabla f)+\parallel Hess\left(f\right)\parallel }^2+g(\nabla \left(\mathsf{\Delta }\left(f\right)\right),\nabla f)$$

we obtain

$$\frac{1}{2}{\mathsf{\Delta }(\parallel \nabla f\parallel }^2)=-\frac{1}{n-1}Ric(\nabla f,\nabla f)+n{\beta }^2=-\frac{1}{n-1}Ric(\nabla f,\nabla f)+\frac{1}{n}{\left(\mathsf{\Delta }\left(f\right)\right)}^2,$$

which, by integration, in the compact case, gives

$${\int }_{M}Ric(\nabla f,\nabla f)=\frac{n-1}{n}{\int }_M{\left(\mathsf{\Delta }\left(f\right)\right)}^2$$

and using the hypothesis, we deduce that $Ric(\nabla f,\nabla f)=0$, $\mathsf{\Delta }\left(f\right)=0$, $\beta =0$, $Q(\nabla f)=0$ and $Hess\left(f\right)=0$. □