On warped product gradient $\eta$-Ricci solitons

TL;DR

This paper investigates gradient $ ext{η}$-Ricci solitons on warped product manifolds, deriving key equations using Bochner formula, and explores conditions under which these solitons are uniquely determined or influence scalar curvature bounds.

Contribution

It provides a new method to construct gradient $ ext{η}$-Ricci solitons on warped products and links soliton properties to scalar curvature in specific geometric settings.

Findings

Derived Laplacian equation for the potential function $f$

Proved soliton is determined by $f$ in irrotational case

Established scalar curvature bounds for compact base manifolds

Abstract

If the potential vector field of an $\eta$-Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a Laplacian equation satisfied by the potential function $f$. In a particular case of irrotational potential vector field we prove that the soliton is completely determined by $f$. We give a way to construct a gradient $\eta$-Ricci soliton on a warped product manifold and show that if the base manifold is oriented, compact and of constant scalar curvature, the soliton on the product manifold gives a lower bound for its scalar curvature.

Equations

$$g({\nabla }_X\xi ,Y)=g({\nabla }_Y\xi ,X),$$

$$div(\xi )=\Delta (f)$$

$$div(\eta ):=trace(Z\mapsto ♯((\nabla \eta )(Z,\cdot )))=\sum _{i=1}^m({\nabla }_{E_i}\eta )E_i=\sum _{i=1}^{m}g(E_i,{\nabla }_{E_i}\xi ):=div(\xi ),$$

$$\frac{1}{2}\Delta (|\xi {|}^2)=|\nabla \xi {|}^2+S(\xi ,\xi )+\xi (div(\xi )),$$

$$(div({\mathcal{L}}_{\xi }g))(X):=trace(Z\mapsto ♯((\nabla ({\mathcal{L}}_{\xi }g))(Z,\cdot ,X)))=\sum _{i=1}^m({\nabla }_{E_i}({\mathcal{L}}_{\xi }g))(E_i,X)=$$

$$=\sum _{i=1}^m\{E_i(({\mathcal{L}}_{\xi }g)(E_i,X))-({\mathcal{L}}_{\xi }g)(E_i,{\nabla }_{E_i}X)\}=2\sum _{i=1}^{m}g({\nabla }_{E_i}{\nabla }_X\xi -{\nabla }_{{\nabla }_{E_i}X}\xi ,E_i):=$$

$$:=2\sum _{i=1}^{m}g({\nabla }_{E_i,X}^2\xi ,E_i)=2\sum _{i=1}^{m}g({\nabla }_{X,E_i}^2\xi +R(E_i,X)\xi ,E_i):=$$

$$:=2\sum _{i=1}^{m}g({\nabla }_{X,E_i}^2\xi ,E_i)+2trace(Z\mapsto R(Z,X)\xi ):=2\sum _{i=1}^{m}g({\nabla }_X{\nabla }_{E_i}\xi -{\nabla }_{{\nabla }_X{E}_i}\xi ,E_i)+2S(X,\xi )=$$

$$=2\sum _{i=1}^{m}g({\nabla }_X{\nabla }_{E_i}\xi ,E_i)+2S(X,\xi )=2\sum _{i=1}^{m}X(g({\nabla }_{E_i}\xi ,E_i))+2S(X,\xi )=2X(div(\xi ))+2S(X,\xi ),$$

$$(div({\mathcal{L}}_{\xi }g))(\xi )=2\xi (div(\xi ))+2S(\xi ,\xi ).$$

$$(div({\mathcal{L}}_{X}g))(X)=\frac{1}{2}\Delta (|X{|}^2)-|\nabla X{|}^2+S(X,X)+X(div(X))$$

$$\Delta (|\xi {|}^2)-2|\nabla \xi {|}^2=2S(\xi ,\xi )+2\xi (div(\xi )).$$

$$div(L_{\xi }g)=2d(div(\xi ))+2i_{Q\xi }g,$$

$${\mathcal{L}}_{\xi }g+2S+2\lambda g+2\mu \eta \otimes \eta =0,$$

$$({\nabla }_{X}Q)Y-({\nabla }_{Y}Q)X=-{\nabla }_{X,Y}^2\xi +{\nabla }_{Y,X}^2\xi +\mu (df\otimes \nabla \xi -\nabla \xi \otimes df)(X,Y),$$

$$\nabla \xi +Q+\lambda I_{\chi (M)}+\mu df\otimes \xi =0.$$

$$({\nabla }_{X}Q)Y=-({\nabla }_X{\nabla }_Y\xi -{\nabla }_{{\nabla }_{X}Y}\xi )-\mu \{g(Y,{\nabla }_X\xi )\xi +df(Y){\nabla }_X\xi \}:=$$

$$:=-{\nabla }_{X,Y}^2\xi -\mu \{g(Y,{\nabla }_X\xi )\xi +df(Y){\nabla }_X\xi \}$$

$$\frac{1}{2}(\Delta -{\nabla }_{\xi })(|\xi {|}^2)=|Hess(f){|}^2+\lambda |\xi {|}^2+\mu |\xi {|}^2\{|\xi {|}^2-2\Delta (f)\}.$$

$$trace(\eta \otimes \eta )=\sum _{i=1}^m[df(E_i){]}^2=\sum _{1\le i,j,k\le m}{\xi }^j{\xi }^{k}g(E_i,E_j)g(E_i,E_k)=\sum _{i=1}^m({\xi }^i{)}^2=$$

$$=\sum _{1\le i,j\le m}{\xi }^i{\xi }^{j}g(E_i,E_j)=|\xi {|}^2.$$

$$div(\xi )+scal+m\lambda +\mu |\xi {|}^2=0$$

$$d(div(\xi ))+d(scal)+\mu d(|\xi {|}^2)=0.$$

$$div({\mathcal{L}}_{\xi }g)+2div(S)+2\mu \cdot div(df\otimes df)=0.$$

$$d(scal)=2div(S),$$

$$\frac{1}{2}\Delta (|\xi {|}^2)-|\nabla \xi {|}^2+S(\xi ,\xi )+\mu \{2(div(df\otimes df))(\xi )-\xi (|\xi {|}^2)\}=0.$$

$$(div(df\otimes df))(\xi ):=\sum _{i=1}^m\{E_i(df(E_i)df(\xi ))-df(E_i)df({\nabla }_{E_i}\xi )\}=$$

$$=\sum _{i=1}^m\{g(E_i,\xi )g({\nabla }_{E_i}\xi ,\xi )+g(\xi ,\xi )g(E_i,{\nabla }_{E_i}\xi )\}=g({\nabla }_{\xi }\xi ,\xi )+|\xi {|}^2\sum _{i=1}^{m}g({\nabla }_{E_i}\xi ,E_i):=$$

$$:=\frac{1}{2}\xi (|\xi {|}^2)+|\xi {|}^{2}div(\xi ),$$

$$\frac{1}{2}\Delta (|\xi {|}^2)-|\nabla \xi {|}^2+S(\xi ,\xi )+2\mu |\xi {|}^{2}div(\xi )=0.$$

Full text

†† 2010 Mathematics Subject Classification. Primary 53C21, 53C44; Secondary 53C25. †† Key words and phrases. gradient $\eta$-Ricci solitons, scalar curvature, Laplacian equation.

1 Introduction

Ricci flow, introduced by R. S. Hamilton [16], deforms a Riemannian metric $g$ by the evolution equation $\frac{\partial}{\partial t}g=-2S$, called the ”heat equation” for Riemannian metrics, towards a canonical metric. Modeling the behavior of the Ricci flow near a singularity, Ricci solitons [15] have been studied in the contexts of complex, contact and paracontact geometries [2].

The more general notion of $\eta$-Ricci soliton was introduced by J. T. Cho and M. Kimura [11] and was treated by C. Călin and M. Crasmareanu on Hopf hypersurfaces in complex space forms [10]. We also discussed some aspects of $\eta$-Ricci solitons in paracontact [6], [7] and Lorentzian para-Sasakian geometry [5].

A particular case of soliton arises when the potential vector field is the gradient of a smooth function. The gradient vector fields play a central rôle in the Morse-Smale theory [22]. G. Y. Perelman showed that if the manifold is compact, then the Ricci soliton is gradient [18]. In [14], R. S. Hamilton conjectured that a compact gradient Ricci soliton on a manifold $M$ with positive curvature operator implies that $M$ is Einstein manifold. In [12], S. Deshmukh proved that a Ricci soliton of positive Ricci curvature and whose potential vector field is of Jacobi-type, is compact and therefore, a gradient Ricci soliton. Different aspects of gradient Ricci solitons were studied in various papers. In [1], N. Basu and A. Bhattacharyya treated gradient Ricci solitons in Kenmotsu manifolds having Killing potential vector field. P. Petersen and W. Wylie discussed the rigidity of gradient Ricci solitons [20] and gave a classification imposing different curvature conditions [19].

The aim of our paper is to investigate some properties of gradient $\eta$-Ricci solitons. After deducing some results derived from the Bochner formula, we construct a gradient $\eta$-Ricci soliton on a warped product manifold and for the particular case of product manifolds, we show that if the base is oriented, compact and of constant scalar curvature, then we obtain a lower bound for the scalar curvature of the product manifold.

2 Bochner formula revisited

Let $(M,g)$ be an $m$-dimensional Riemannian manifold and consider $\xi$ a gradient vector field on $M$. If $\xi:=grad{(f)}$, for $f$ a smooth function on $M$, then the $g$-dual $1$-form $\eta$ of $\xi$ is closed, as $\eta(X):=g(X,\xi)=df(X)$. Then $0=(d\eta)(X,Y):=X(\eta(Y))-Y(\eta(X))-\eta([X,Y])=g(\nabla_{X}\xi,Y)-g(\nabla_{Y}\xi,X)$, hence:

[TABLE]

for any $X$, $Y\in\chi(M)$, where $\nabla$ is the Levi-Civita connection of $g$.

Also:

[TABLE]

and

[TABLE]

for $\{E_{i}\}_{1\leq i\leq m}$ a local orthonormal frame field with $\nabla_{E_{i}}E_{j}=0$ in a point. From now on, whenever we make a local computation, we will consider this frame.

In this case, the Bochner formula becomes:

[TABLE]

where $S$ is the Ricci curvature of $g$. Indeed:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $R$ is the Riemann curvature and $S$ is the Ricci curvature tensor fields of the metric $g$ and the relation (5), for $X:=\xi$, becomes:

[TABLE]

But the Bochner formula states that for any vector field $X$ [20]:

[TABLE]

and from (6) and (7) we deduce that:

[TABLE]

Remark that (5) can be written in terms of $(1,1)$-tensor fields:

[TABLE]

where $Q$ is the Ricci operator defined by $g(QX,Y):=S(X,Y)$.

3 Gradient $\eta$-Ricci solitons

Consider now the equation:

[TABLE]

where $g$ is a Riemannian metric, $S$ its Ricci curvature, $\eta$ a $1$-form and $\lambda$ and $\mu$ are real constants. The data $(g,\xi,\lambda,\mu)$ which satisfy the equation (10) is said to be an $\eta$-Ricci soliton on $M$ [11]; in particular, if $\mu=0$, $(g,\xi,\lambda)$ is a Ricci soliton [15]. If the potential vector field $\xi$ is of gradient type, $\xi=grad(f)$, for $f$ a smooth function on $M$, then $(g,\xi,\lambda,\mu)$ is called gradient $\eta$-Ricci soliton.

Proposition 3.1**.**

Let $(M,g)$ be a Riemannian manifold. If (10) defines a gradient $\eta$-Ricci soliton on $M$ with the potential vector field $\xi:=grad(f)$ and $\eta$ is the $g$-dual $1$-form of $\xi$, then:

[TABLE]

for any $X$, $Y\in\chi(M)$, where $Q$ stands for the Ricci operator.

Proof..

As $g(QX,Y):=S(X,Y)$, follows:

[TABLE]

Then:

[TABLE]

[TABLE]

and using (1) we get the required relation. ∎

Theorem 3.2**.**

If (10) defines a gradient $\eta$-Ricci soliton on the $m$-dimensional Riemannian manifold $(M,g)$ and $\eta$ is the $g$-dual $1$-form of the gradient vector field $\xi:=grad(f)$, then:

[TABLE]

Proof..

First remark that if $\xi=\sum_{i=1}^{m}\xi^{i}E_{i}$, for $\{E_{i}\}_{1\leq i\leq m}$ a local orthonormal frame field with $\nabla_{E_{i}}E_{j}=0$ in a point, then:

[TABLE]

[TABLE]

Taking the trace of the equation (10), we obtain:

[TABLE]

and differentiating it:

[TABLE]

Then taking the divergence of the same equation, we get:

[TABLE]

Substracting the relations (18) and (17) computed in $\xi$, considering (6), (8) and using the fact that the scalar and the Ricci curvatures satisfy [20]:

[TABLE]

we obtain:

[TABLE]

As

[TABLE]

[TABLE]

[TABLE]

the equation (20) becomes:

[TABLE]

From the $\eta$-soliton equation (10), we get:

[TABLE]

and the equation (22) becomes:

[TABLE]

As $\xi:=grad(f)$ follows $Hess(f)=\nabla(df)$ and $|\nabla\xi|^{2}=|Hess(f)|^{2}$. ∎

Remark 3.3**.**

For $\mu=0$ in Theorem 3.2, we obtain the relation for the particular case of gradient Ricci soliton [20].

Remark 3.4**.**

i) Assume that $\mu\neq 0$. Denoting by $\Delta_{\xi}:=\Delta-\nabla_{\xi}$, the equation (14) can be written:

[TABLE]

where $\xi:=grad(f)$. If $\lambda\geq\mu[2\Delta(f)-|\xi|^{2}]$, then $\Delta_{\xi}(|\xi|^{2})\geq 0$ and from the maximum principle follows that $|\xi|^{2}$ is constant in a neighborhood of any local maximum. If $|\xi|$ achieve its maximum, then $M$ is quasi-Einstein. Indeed, since $Hess(f)=0$, from (10) we have $S=-\lambda g-\mu df\otimes df$. Moreover, in this case, $|\xi|^{2}\{\lambda+\mu[|\xi|^{2}-2\Delta(f)]=0$, which implies either $\xi=0$, so $M$ is Einstein, or $|\xi|^{2}=2\Delta(f)-\frac{\lambda}{\mu}\geq 0$. Since $\Delta(f)=-scal-m\lambda-\mu|\xi|^{2}$ we get $\mu(2\mu+1)|\xi|^{2}=-(2\mu\cdot scal+2m\lambda\mu+\lambda)$. If $\mu=-\frac{1}{2}$, the scalar curvature equals to $\lambda(1-m)$ and if $\mu\neq-\frac{1}{2}$, it is either locally upper (or lower) bounded by $-\frac{\lambda(1+2m\mu)}{2\mu}$, for $\mu<-\frac{1}{2}$ ($\mu>-\frac{1}{2}$, respectively). On the other hand, if the potential vector field is of constant length, then $2\mu\Delta(f)\geq\lambda+\mu|\xi|^{2}$ equivalent to $\mu(2\mu+1)|\xi|^{2}+(2\mu\cdot scal+2m\lambda\mu+\lambda)\leq 0$ with equality for $\Delta(f)=\frac{\lambda}{2\mu}+\frac{|\xi|^{2}}{2}\geq\frac{\lambda}{2\mu}$ and $Hess(f)=0$ which yields the quasi-Einstein case.

ii) For $\mu=0$, we get the Ricci soliton case [20].

Proposition 3.5**.**

Let $(M,g)$ be an $m$-dimensional Riemannian manifold and $\eta$ be the $g$-dual $1$-form of the gradient vector field $\xi:=grad(f)$. If $\xi$ satisfies $\nabla\xi=I_{\chi(M)}-\eta\otimes\xi$, where $\nabla$ is the Levi-Civita connection associated to $g$, then:

$Hess(f)=g-\eta\otimes\eta$; 2. 2.

$R(X,Y)\xi=\eta(X)Y-\eta(Y)X$, for any $X$, $Y\in\chi(M)$; 3. 3.

$S(\xi,\xi)=(1-m)|\xi|^{2}$.

Proof..

Express the Lie derivative along $\xi$ as follows:

[TABLE]

[TABLE]

[TABLE] 2. 2.

Replacing now the expression of $\nabla\xi$ in $R(X,Y)\xi:=\nabla_{X}\nabla_{Y}\xi-\nabla_{Y}\nabla_{X}\xi-\nabla_{[X,Y]}\xi$, from a direct computation we get $R(X,Y)\xi=\eta(X)Y-\eta(Y)X$. 3. 3.

[TABLE]

∎

The condition satisfied by the potential vector field $\xi$, namely, $\nabla\xi=I_{\chi(M)}-\eta\otimes\xi$, naturally arises if $(M,\varphi,\xi,\eta,g)$ is for example, Kenmotsu manifold [17]. In this case, $M$ is a quasi-Einstein manifold.

Example 3.6**.**

Let $M=\{(x,y,z)\in\mathbb{R}^{3},z>0\}$, where $(x,y,z)$ are the standard coordinates in $\mathbb{R}^{3}$. Set

[TABLE]

[TABLE]

Then $(\varphi,\xi,\eta,g)$ is a Kenmotsu structure on $M$.

Consider the linearly independent system of vector fields:

[TABLE]

Follows

[TABLE]

[TABLE]

[TABLE]

and the Levi-Civita connection $\nabla$ is deduced from Koszul’s formula

[TABLE]

[TABLE]

precisely

[TABLE]

[TABLE]

[TABLE]

Then the Riemann and the Ricci curvature tensor fields are given by:

[TABLE]

[TABLE]

[TABLE]

From (10) computed in $(E_{i},E_{i})$:

[TABLE]

for all $i\in\{1,2,3\}$, we have:

[TABLE]

for all $i\in\{1,2,3\}$. Therefore, $\lambda=\mu=1$ define an $\eta$-Ricci soliton on $(M,\varphi,\xi,\eta,g)$. Moreover, it is a gradient $\eta$-Ricci soliton, as the potential vector field $\xi$ is of gradient type, $\xi=grad(f)$, where $f(x,y,z):=-\ln z$.

Assume now that (10) defines a gradient $\eta$-Ricci soliton on $(M,g)$ with $\mu\neq 0$. Under the hypotheses of the Proposition 3.5, the equation (24) simplifies a lot. Compute:

[TABLE]

for $\{E_{i}\}_{1\leq i\leq m}$ a local orthonormal frame field with $\nabla_{E_{i}}E_{j}=0$ in a point,

[TABLE]

[TABLE]

From the equation (10) we obtain:

[TABLE]

Using Proposition 3.5 and the relation (28), we get:

[TABLE]

so $|\xi|^{2}(\mu-1)=m-2-\lambda$ i.e. $\xi$ is of constant length. Using (26) we get $|\xi|=1$. It follows $\lambda+\mu=m-1$ and we deduce:

Theorem 3.7**.**

Under the hypotheses of the Proposition 3.5, if (10) defines a gradient $\eta$-Ricci soliton on $(M,g)$ with $\mu\neq 0$, then the Laplacian equation (24) becomes:

[TABLE]

Therefore, the existence of a gradient $\eta$-Ricci soliton defined by (10) with the potential vector field $\xi:=grad(f)$, yields the Laplacian equation (30), and the soliton is completely determined by $f$.

4 Warped product $\eta$-Ricci solitons

Consider $(B,g_{B})$ and $(F,g_{F})$ two Riemannian manifolds of dimensions $n$ and $m$, respectively. Denote by $\pi$ and $\sigma$ the projection maps from the product manifold $B\times F$ to $B$ and $F$ and by $\widetilde{\varphi}:=\varphi\circ\pi$ the lift to $B\times F$ of a smooth function $\varphi$ on $B$. In this context, we shall call $B$ the base and $F$ the fiber of $B\times F$, the unique element $\widetilde{X}$ of $\chi(B\times F)$ that is $\pi$-related to $X\in\chi(B)$ and to the zero vector field on $F$, the horizontal lift of $X$ and the unique element $\widetilde{V}$ of $\chi(B\times F)$ that is $\sigma$-related to $V\in\chi(F)$ and to the zero vector field on $B$, the vertical lift of $V$. Also denote by $\mathcal{L}(B)$ the set of all horizontal lifts of vector fields on $B$, by $\mathcal{L}(F)$ the set of all vertical lifts of vector fields on $F$, by $\mathcal{H}$ the orthogonal projection of $T_{(p,q)}(B\times F)$ onto its horizontal subspace $T_{(p,q)}(B\times\{q\})$ and by $\mathcal{V}$ the orthogonal projection of $T_{(p,q)}(B\times F)$ onto its vertical subspace $T_{(p,q)}(\{p\}\times F)$.

Let $\varphi>0$ be a smooth function on $B$ and

[TABLE]

be a Riemannian metric on $B\times F$.

Definition 4.1**.**

[3] The product manifold of $B$ and $F$ together with the Riemannian metric $g$ defined by (31) is called the warped product of $B$ and $F$ by the warping function $\varphi$ (and is denoted by $(M:=B\times_{\varphi}F,g)$).

Remark that if $\varphi$ is constant equal to 1, the warped product becomes the usual product of the Riemannian manifolds.

For simplification, in the rest of the paper we shall simply denote by $X$ the horizontal lift of $X\in\chi(B)$ and by $V$ the vertical lift of $V\in\chi(F)$.

Due to a result of J. Case, Y.-J. Shu and G. Wei [8], we know that for a gradient $\eta$-Ricci soliton $(g,\xi:=grad(f),\lambda,\mu)$ with $\mu\in(-\infty,0)$ and $\eta=df$ the $g$-dual of $\xi$, on a connected $n$-dimensional Riemannian manifold $(M,g)$, the function

[TABLE]

is constant.

Choosing properly an Einstein manifold, a smooth function and considering the warped product manifold, we can characterize the gradient $\eta$-Ricci soliton on the base manifold as follows [8]. Let $(B,g_{B})$ be an $n$-dimensional connected Riemannian manifold, $\lambda$ and $\mu$ real constants such that $-\frac{1}{\mu}$ is a natural number, $f$ a smooth function on $B$, $k:=\mu e^{2\mu f}[\Delta(f)-|\xi|^{2}-\frac{\lambda}{\mu}]$ and $(F,g_{F})$ an $m$-dimensional Riemannian manifold with $m=-\frac{1}{\mu}$ and $S_{F}=kg_{F}$. Then $(g,\xi:=grad(f),\lambda,\mu)$ is a gradient $\eta$-Ricci soliton on $(B,g_{B})$ with $\eta=df$ the $g$-dual of $\xi$, if and only if the warped product manifold $(M:=B\times_{\varphi}F,g)$ with the warping function $\varphi=e^{-\frac{f}{m}}$ is Einstein manifold with $S=\lambda g$.

Let $S$, $S_{B}$, $S_{F}$ the Ricci tensors on $M$, $B$ and $F$ and $\widetilde{S_{B}}$, $\widetilde{S_{F}}$ the lift on $M$ of $S_{B}$ and $S_{F}$, which satisfy the following properties:

Lemma 4.2**.**

[3]** If $(M:=B\times_{\varphi}F,g)$ is the warped product of $B$ and $F$ by the warping function $\varphi$ and $m>1$, then for any $X$, $Y\in\mathcal{L}(B)$ and any $V$, $W\in\mathcal{L}(F)$, we have:

$S(X,Y)=\widetilde{S_{B}}(X,Y)-\frac{m}{\widetilde{\varphi}}H^{\varphi}(X,Y)$, where $H^{\varphi}$ is the lift on $M$ of $Hess(\varphi)$; 2. 2.

$S(X,V)=0$; 3. 3.

$S(V,W)=\widetilde{S_{F}}(V,W)-\pi^{*}[\frac{\Delta(\varphi)}{\varphi}+(m-1)\frac{|grad(\varphi)|^{2}}{\varphi^{2}}]|_{F}g(V,W)$.

Notice that the lift on $M$ of the gradient and the Hessian satisfy:

[TABLE]

[TABLE]

for any smooth function $f$ on $B$.

We shall construct a gradient $\eta$-Ricci soliton on a warped product manifold following [13].

Let $(B,g_{B})$ be a Riemannian manifold, $\varphi>0$ a smooth function on $B$ and $f$ a smooth function on $B$ such that:

[TABLE]

where $\lambda$, $\mu$ and $m>1$ are real constants.

Take $(F,g_{F})$ an $m$-dimensional manifold with $S_{F}=kg_{F}$, for $k:=\pi^{*}[-\lambda\varphi^{2}+\varphi\Delta(\varphi)+(m-1)|grad(\varphi)|^{2}-\varphi(grad(f))(\varphi)]|_{F}$, where $\pi$ and $\sigma$ be the projection maps from the product manifold $B\times F$ to $B$ and $F$, respectively, and $g:=\pi^{*}g_{B}+(\varphi\circ\pi)^{2}\sigma^{*}g_{F}$ a Riemannian metric on $B\times F$. Then, for $\xi:=grad(f\circ\pi)$, if consider $\mu=0$ in (35), $(g,\xi,\lambda)$ is a gradient Ricci soliton on $B\times_{\varphi}F$ called the warped product Ricci soliton [13].

With the above notations, we prove that:

Theorem 4.3**.**

Let $(B,g_{B})$ be a Riemannian manifold, $\varphi>0$, $f$ two smooth functions on $B$, let $m>1$, $\lambda$, $\mu$ be real constants satisfying (35) and $(F,g_{F})$ an $m$-dimensional Riemannian manifold. Then $(g,\xi,\lambda,\mu)$ is a gradient $\eta$-Ricci soliton on the warped product manifold $(B\times_{\varphi}F,g)$, where $\xi=grad(\widetilde{f})$ and the $1$-form $\eta$ is the $g$-dual of $\xi$, if and only if:

[TABLE]

and

[TABLE]

where $k:=\pi^{*}[-\lambda\varphi^{2}+\varphi\Delta(\varphi)+(m-1)|grad(\varphi)|^{2}-\varphi(grad(f))(\varphi)]|_{F}$.

Proof..

The gradient $\eta$-Ricci soliton $(g,\xi,\lambda,\mu)$ on $(B\times_{\varphi}F,g)$ is given by:

[TABLE]

Then for any $X$, $Y\in\mathcal{L}(B)$ and for any $V$, $W\in\mathcal{L}(F)$, from Lemma 4.2 we get:

[TABLE]

[TABLE]

and using the fact that

[TABLE]

we obtain:

[TABLE]

Conversely, notice that the left-hand side term in (38) computed in $(X,V)$, for $X\in\mathcal{L}(B)$ and $V\in\mathcal{L}(F)$ vanishes identically and using again Lemma 4.2, for each situation $(X,Y)$ and $(V,W)$, we can recover the equation (38) from (36) and (37). ∎

Remark 4.4**.**

In the case of product manifold (for $\varphi=1$), notice that the equation (35) defines a gradient $\eta$-Ricci soliton on $B$ and the chosen manifold $(F,g_{F})$ is Einstein ($S_{F}=-\lambda g_{F}$), so a gradient $\eta$-Ricci soliton on the product manifold $B\times F$ can be naturally obtained by ”lifting” a gradient $\eta$-Ricci soliton on $B$.

Remark 4.5**.**

If for the function $\varphi$ and $f$ on $B$ there exists two constants $a$ and $b$ such that $\nabla(grad(\varphi))=\varphi[aI_{\chi(B)}+bdf\otimes grad(f)]$, then $Hess(\varphi)=\varphi(ag_{B}+bdf\otimes df)$ and $(g_{B},grad(f),\lambda-ma,\mu-mb)$ is a gradient $\eta$-Ricci soliton on $B$.

Let us make some remark on the class of manifolds that satisfy the condition (35):

[TABLE]

for $\varphi>0$, $f$ smooth functions on the oriented and compact Riemannian manifold $(B,g_{B})$, $\lambda$, $\mu$ and $m>1$ real constants. Denote by $\xi:=grad(f)$.

Taking the trace of (39), we obtain:

[TABLE]

Remark that:

[TABLE]

[TABLE]

Also:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

therefore:

[TABLE]

Applying the divergence to (39), computing it in $\xi$ and considering (3), we get:

[TABLE]

[TABLE]

[TABLE]

From (40), (41), (42) and (43), we obtain:

[TABLE]

[TABLE]

Integrating with respect to the canonical measure on $B$, we have:

[TABLE]

and similarly:

[TABLE]

Using:

[TABLE]

and integrating (44) on $B$, from the above relations and the divergence theorem, we obtain:

[TABLE]

[TABLE]

Proposition 4.6**.**

Let $(B,g_{B})$ be an oriented and compact Riemannian manifold, $f$ a smooth function on $B$, let $m>1$, $\lambda$, $\mu$ be real constants satisfying (35) (for $\varphi=1$) and $(F,g_{F})$ be an $m$-dimensional Einstein manifold with $S_{F}=-\lambda g_{F}$. If $(g,\xi,\lambda,\mu)$ is a gradient $\eta$-Ricci soliton on the product manifold $(B\times F,g)$, where $\xi=grad(\widetilde{f})$ and the $1$-form $\eta$ is the $g$-dual of $\xi$, then:

[TABLE]

Let now consider the product manifold $B\times F$, in which case (40) (for $\varphi=1$) becomes:

[TABLE]

and integrating it on $B$, we get:

[TABLE]

Replacing it in (46), we obtain:

[TABLE]

Proposition 4.7**.**

Let $(B,g_{B})$ be an oriented, compact and complete $n$-dimensional ($n>1$) Riemannian manifold of constant scalar curvature, $\varphi>0$, $f$ two smooth functions on $B$, let $m>1$, $\lambda$, $\mu$ be real constants satisfying (39). If one of the following two conditions hold:

$\varphi=1$* and $\lambda=-\frac{scal_{B}}{n}$;* 2. 2.

there exists a positive function $h$ on $B$ such that $Hess(f)=-h\cdot Hess(\varphi)$ and $\mu\geq 0$,

then $B$ is conformal to a sphere in the $(n+1)$-dimensional Euclidean space.

Proof..

From (49) we obtain:

[TABLE]

so $Hess(f)=\frac{\Delta(f)}{n}g_{B}$ which implies by [23] that $B$ is conformal to a sphere in the $(n+1)$-dimensional Euclidean space. 2. 2.

From the condition $Hess(f)=-h\cdot Hess(\varphi)$ we obtain $\Delta(f)=-h\Delta(\varphi)$ and replacing them in (45), we get:

[TABLE]

From $\mu\geq 0$ we deduce that $Hess(f)=\frac{\Delta(f)}{n}g_{B}$ and according to [23], we get the conclusion.

∎

Finally, we state a result on the scalar curvature of a product manifold admitting an $\eta$-Ricci soliton:

Proposition 4.8**.**

Let $(B,g_{B})$ be an oriented and compact Riemannian manifold of constant scalar curvature, $f$ a smooth function on $B$, let $m>1$, $\lambda$, $\mu$ be real constants satisfying (35) (for $\varphi=1$) and $(F,g_{F})$ be an $m$-dimensional Einstein manifold with $S_{F}=-\lambda g_{F}$. If $(g,\xi,\lambda,\mu)$ is a gradient $\eta$-Ricci soliton on the product manifold $(B\times F,g)$, where $\xi=grad(\widetilde{f})$ and the $1$-form $\eta$ is the $g$-dual of $\xi$, then the scalar curvature of $B\times F$ is $\geq-(n+m)\lambda$.

Proof..

From (49) we deduce that $\frac{n+2}{2}(\frac{scal_{B}}{n}+\lambda)\cdot vol(B)=\int_{B}|Hess(f)-\frac{\Delta(f)}{n}g_{B}|^{2}\geq 0$ and since $scal_{F}=-m\lambda$, we get the conclusion. ∎

We end these considerations by giving an example of gradient $\eta$-Ricci soliton on a product manifold.

Example 4.9**.**

Let $(g_{M},\xi_{M},1,1)$ be the gradient $\eta$-Ricci soliton on the Riemannian manifold $M=\{(x,y,z)\in\mathbb{R}^{3},z>0\}$, where $(x,y,z)$ are the standard coordinates in $\mathbb{R}^{3}$, with the metric $g_{M}:=\frac{1}{z^{2}}(dx\otimes dx+dy\otimes dy+dz\otimes dz)$ (given by Example 3.6) and let $S^{3}$ be the $3$-sphere with the round metric $g_{S}$ (which is Einstein with the Ricci tensor equals to $2g_{S}$). By Remark 4.4 we obtain the gradient $\eta$-Ricci soliton $(g,\xi,1,1)$ on the ”generalized cylinder” $M\times S^{3}$, where $g=g_{M}+g_{S}$ and $\xi$ is the lift on $M\times S^{3}$ of the gradient vector field $\xi_{M}=grad(f)$, where $f(x,y,z):=-\ln z$.