Closed Conformal Vector Fields in Gradient Einstein Solitons

Abstract

We investigate gradient Einstein solitons admitting closed conformal vector fields. Under constant scalar curvature, we establish rigidity results showing that complete solitons with non-parallel closed conformal vector fields are isometric to Euclidean space. In the non-compact case, we prove that solitons with homothetic closed conformal vector fields are locally conformally flat in dimension four and possess a harmonic Weyl tensor in higher dimensions. Furthermore, we obtain a classification result, showing that such solitons admit a warped product structure with a one-dimensional base and space form fibers.

1. Introduction and Motivations

Conformal vectors are characterized by a distinct kind of invariance and a unique character under conformal metric transformations. Let $\mathcal{U}$ be a smooth vector field on a Riemannian manifold $({\mathfrak{B}}^n,g)$ satisfying:

$$\begin{array}{c}\hfill {\mathrm{L}}_{\mathcal{U}}g=2\phi g\end{array}$$

(1)

for any smooth function $\phi \in C^{\infty }\left(\mathfrak{B}\right)$. Moreover, $\mathrm{L}$ denotes the Lie derivative and g denotes the Riemannian metric. Thus, $\mathcal{U}$ is a conformal vector field with a conformal factor $\phi$. It is called closed if it satisfies:

$$\begin{array}{c}\hfill {\mathcal{D}}_{\mathrm{X}}\mathcal{U}=\phi \mathrm{X}\end{array}$$

(2)

for some $\mathrm{X}\in \mathfrak{X}\left(\mathfrak{B}\right)$. $\mathcal{D}$ denotes a Levi-Civita connection on $\mathfrak{B}$. It should be noted that a smooth vector $\mathcal{U}$ is called “parallel” if it satisfies the condition:

$$\begin{array}{c}\hfill {\mathcal{D}}_{\mathrm{X}}\mathcal{U}=0.\end{array}$$

(3)

for all $\mathrm{X}\in \mathfrak{X}\left(\mathfrak{B}\right)$. For more motivation, we refer to [[1,2] references therein].

On the other hand, let $\lambda \in C^{\infty }\left(\mathfrak{B}\right)$ and $\beta \in \mathbb{R}$ satisfy:

$$\begin{array}{c}\hfill \mathrm{Ric}+{\mathcal{D}}^2\lambda =\left({\displaystyle \frac{\mathrm{R}}{n}}+\beta \right)g.\end{array}$$

(4)

Then $({\mathfrak{B}}^n,g)$ is called a gradient Einstein soliton. More generally, for $\theta \in \mathbb{R}$, $({\mathfrak{B}}^n,g,\lambda ,\beta )$ is a gradient $\theta$-Einstein soliton if:

$$\begin{array}{c}\hfill \mathrm{Ric}+{\mathcal{D}}^2\lambda =(\theta \mathrm{R}+\beta )g.\end{array}$$

(5)

Thus, Einstein solitons correspond to $\theta ={\displaystyle \frac{1}{n}}$. The function $\lambda$ is called the potential function, and the soliton is said to be shrinking, steady, or expanding according to $\beta >0$, $\beta =0$, or $\beta <0$. Moreover, it was shown in [3] that if $\theta >0$ and $\beta \le 0$, then any such soliton with positive sectional curvature is rotationally symmetric. In particular, the case $\theta =0$ reduces to a gradient Ricci soliton. A generalized class of Einstein solitons as Ricci–Bourguignon solitons is discovered in [4]. Bourguignon [4] introduced a modification of the Ricci soliton equation, known as the Ricci–Bourguignon soliton. Following this development, several authors have investigated the concept (see [5,6,7,8]).

Additionally, gradient Ricci solitons admitting closed conformal vector fields were studied in [9,10,11]. Related results for quasi-Einstein manifolds with parallel vector fields can be found in [12,13,14], while Güler and Demirbag [15] treated the locally quasi-Einstein case. Recently, Diógenes et al. [9] proved the following result via the above tensor.

Theorem 1 

([9]). Let $({\mathfrak{B}}^n,g,\lambda )$, $n\ge 4$ be a complete gradient Ricci soliton possessing a non-parallel closed conformal vector field, then the tensor ${\mathcal{T}}_{abc}=0$. If $n=3\mathrm{or}4$ then it is a locally conformal flat and harmonic Weyl tensor for $n\ge 5$.

This work is devoted to the study of the conformal geometry of gradient Einstein solitons and their interplay with closed conformal vector fields in the same direction as Theorem 1. We demonstrate that gradient Einstein solitons equipped with a non-parallel closed conformal vector field can undergo a conformal transformation into Riemannian manifolds with constant scalar curvature almost everywhere. In addition, we establish a characterization for this class of manifolds when the closed conformal vector field is assumed to be of gradient type. Further, we present results concerning gradient Einstein solitons that admit non-parallel homothetic conformal vector fields, thereby shedding light on the structural implications of such vector fields. Particular emphasis is placed on the relationship between Ricci soliton structures and homothetic vector fields, a topic that has received significant attention in recent investigations. Moreover, we observe that several rigidity results concerning closed conformal vector fields on various classes of Riemannian manifolds such as gradient Ricci solitons, quasi-Einstein manifolds, solutions of the critical point equation, and generalized m-quasi Einstein manifolds established in [9,10,11,12,13,14,16,17], can be extended and refined into corresponding classification theorems.

2. Preliminaries

Moreover, we recall some background definitions and formulas that will be used in the subsequent analysis. To prove our results, we consider a special $(0,3)$-tensor ${\mathcal{T}}_{abc}$ introduced in [18], which is defined as follows:

$$\begin{array}{cc}\hfill {\mathcal{T}}_{abc}=& {\displaystyle \frac{1}{n-2}}\left\{{\mathrm{R}}_{bc}{\mathcal{D}}_a\lambda -{\mathrm{R}}_{ac}{\mathcal{D}}_b\lambda \right\}+{\displaystyle \frac{1}{2(n-1)(n-2)}}\left\{g_{bc}{\mathcal{D}}_a\mathrm{R}-g_{ac}{\mathcal{D}}_b\mathrm{R}\right\}\hfill \\ \hfill & -{\displaystyle \frac{\mathrm{R}}{(n-1)(n-2)}}\left\{g_{bc}{\mathcal{D}}_a\lambda -g_{ac}{\mathcal{D}}_b\lambda \right\}.\hfill \end{array}$$

(6)

One can readily verify that ${\mathcal{T}}_{abc}$ is trace-free with respect to any two indices and skew-symmetric in the first two indices, namely:

$$\begin{array}{c}\hfill g^{ab}{\mathcal{T}}_{abc}=g^{ac}{\mathcal{T}}_{abc}=0,\mathrm{and}{\mathcal{T}}_{abc}=-{\mathcal{T}}_{bac}.\end{array}$$

(7)

Proceeding, Catino et al. [19] derived the various characterizations of gradient Einstein-type manifolds, which generalized several classes of solitons, including the gradient Einstein soliton, and proved the following results:

Theorem 2 

([19]). Let $({\mathfrak{B}}^n,g)$, $n\ge 3$ be a complete, non-compact, non-degenerate gradient Einstein soliton. If $(0,3)$-tensor $\mathcal{T}=0$, then Cotton tensor $\mathcal{C}=0$, unless the potential function is locally constant.

In dimension four [19], they established the following:

Theorem 3 

([19]). Let $({\mathfrak{B}}^4,g)$ be a complete, non-compact, non-degenerate, gradient Einstein soliton. If $(0,3)$-tensor $\mathcal{T}=0$, then the Weyl tensor $\mathcal{W}=0$, unless the potential function is locally constant.

For Riemannian manifolds with conformal vector fields, several results have been proved. The following lemma is presented in this sequel:

Lemma 1 

([20,21]). Assume that $({\mathfrak{B}}^n,g)$ is a Riemannian manifold and $\mathcal{U}$ is a closed conformal vector field, then we have:

(i) 

The set $\mathbb{Z}\left(\mathcal{U}\right)$ of zeros of $\mathcal{U}$ is a discrete set.

(ii) 

${\mathcal{D}\left|\mathcal{U}\right|}^2=2\phi \mathcal{U}$.

(iii) 

${\mathcal{D}}^2{\left|\mathcal{U}\right|}^2=2{\phi }^{2}g+2d\phi \otimes {\mathcal{U}}^{♭}$.

(iv) 

${\left|\mathcal{U}\right|}^2\mathcal{D}\phi =\mathcal{U}\left(\phi \right)\mathcal{U}.$

(v) 

The Ricci tensor is satisfied. $\mathrm{Ric}\left(\mathcal{U}\right)=-(n-1)\mathcal{D}\phi .$

We begin by rewriting the (4) in local coordinates, which yields:

$$\begin{array}{c}\hfill {\mathrm{R}}_{ab}+{\mathcal{D}}_a{\mathcal{D}}_b\lambda =\left(\beta +{\displaystyle \frac{1}{n}}\mathrm{R}\right)g_{ab}.\end{array}$$

(8)

The equation above is reduced to the following form by taking the trace:

$$\begin{array}{c}\hfill \Delta \lambda =n\beta +\left({\displaystyle \frac{n-2}{2}}\right)\mathrm{R}.\end{array}$$

(9)

On the other hand, the covariant derivative of the curvature tensor satisfies the following well-known relation with the Ricci tensor [22].

$$\begin{array}{c}\hfill (n-2)\mathcal{D}\mathrm{R}=2\mathrm{Ric}\left(\mathcal{D}\lambda \right).\end{array}$$

(10)

Lemma 2. 

Let $\left({\mathfrak{B}}^n,g,\lambda ,\beta \right)$ be a complete gradient Einstein soliton with a closed conformal vector field $\mathcal{U}\in \mathfrak{X}\left({\mathfrak{B}}^n\right)$. Then we have:

$$\begin{array}{cc}\hfill \mathcal{D}\left(\mathcal{U}\lambda \right)=& \phi \mathcal{D}\lambda +(n-1)\mathcal{D}\phi +\left(\beta +{\displaystyle \frac{1}{n}}\mathrm{R}\right)\mathcal{U}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill d\lambda \otimes d\phi =& {\mathcal{D}}^2\mathcal{U}\left(\lambda \right)-(n-1){\mathcal{D}}^2\phi -\left(\beta +{\displaystyle \frac{1}{n}}\mathrm{R}\right)\phi g-{\displaystyle \frac{1}{n}}{\mathcal{U}}^{♭}\otimes \mathcal{D}\mathrm{R}-\phi {\mathcal{D}}^2\lambda .\hfill \end{array}$$

(12)

Proof. 

Let $\mathcal{U}\in \mathfrak{X}\left({\mathfrak{B}}^n\right)$ be a closed conformal vector field, we derive:

$$\begin{array}{cc}\hfill \mathrm{X}\left(\mathcal{U}\left(\lambda \right)\right)& =\mathrm{X}g(\mathcal{D}\lambda ,\mathcal{U})=g({\mathcal{D}}_{\mathrm{X}}\mathcal{D}\lambda ,\mathcal{U})+g({\mathcal{D}}_{\mathrm{X}}\mathcal{U},\mathcal{D}\lambda ),\hfill \\ \hfill \mathrm{X}\left(\mathcal{U}\lambda \right)& =\left({\mathcal{D}}_X\mathcal{U}\right)\lambda +\mathcal{U}\left(\mathrm{X}\lambda \right)\hfill \end{array}$$

for any $\mathrm{X}\in \mathfrak{X}\left({\mathfrak{B}}^n\right)$. Then using (2), we obtain:

$$\begin{array}{cc}\hfill \mathrm{X}\left(\mathcal{U}\left(\lambda \right)\right)& ={\lambda }^{2}g(\mathrm{X},\mathcal{U})+\phi g(\mathcal{D}\lambda ,\mathrm{X})\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \mathrm{X}\left(\mathcal{U}\lambda \right)& =\phi g(\mathcal{D}\lambda ,\mathrm{X})+\mathcal{U}\left(\mathrm{X}\lambda \right).\hfill \end{array}$$

(14)

From (4), we have:

$$\begin{array}{c}\hfill \mathrm{R}ic(\mathcal{U},\mathrm{X})+\mathcal{D}{\lambda }^2(\mathcal{U},\mathrm{X})=\left(\beta +{\displaystyle \frac{1}{n}}\mathrm{R}\right)g(\mathcal{U},\mathrm{X}).\end{array}$$

(15)

From (13) and (15), we derive the following:

$$\begin{array}{c}\hfill \mathrm{X}\left(\mathcal{U}\lambda \right)=\left(\beta +{\displaystyle \frac{1}{2}}\mathrm{R}\right)g(\mathcal{U},\mathrm{X})-\mathrm{Ric}(\mathcal{U},\mathrm{X})+\phi g(\mathcal{D}\lambda ,\mathrm{X})\end{array}$$

(16)

Taking account of (v) in Lemma 1, we have:

$$\begin{array}{cc}\hfill \mathrm{X}\left(\mathcal{U}\lambda \right)=& \left(\beta +{\displaystyle \frac{1}{2}}\mathrm{R}\right)g(\mathcal{U},\mathrm{X})+(n-1)g(\mathcal{D}\phi ,\mathrm{X})+\phi g(\mathcal{D}\lambda ,\mathrm{X}),\hfill \end{array}$$

which is equivalent to:

$$\begin{array}{cc}\hfill g(\mathcal{D}\left(\mathcal{U}\lambda \right),\mathrm{X})=& g(\phi \mathcal{D}\lambda +(n-1)\mathcal{D}\phi +\left(\beta +{\displaystyle \frac{1}{n}}\mathrm{R}\right)\mathcal{U},\mathrm{X})\hfill \end{array}$$

which we get in (11). Taking a derivative of (11), we have:

$$\begin{array}{cc}\hfill {\mathcal{D}}^2\left(\mathcal{U}\lambda \right)=& (n-1){\mathcal{D}}^2\phi +\beta \mathcal{D}\mathcal{U}+{\displaystyle \frac{1}{n}}\mathcal{D}\mathrm{R}\mathcal{U}+{\displaystyle \frac{1}{n}}\mathrm{R}\mathcal{D}\mathcal{U}+\phi {\mathcal{D}}^2\lambda +d\lambda \otimes d\phi \hfill \end{array}$$

which implies that:

$$\begin{array}{cc}\hfill {\mathcal{D}}^2\left(\mathcal{U}\lambda \right)=& (n-1){\mathcal{D}}^2\phi +\left(\beta +{\displaystyle \frac{1}{n}}\mathrm{R}\right)\phi g+{\displaystyle \frac{1}{n}}\mathcal{U}\mathcal{D}\mathrm{R}+\phi {\mathcal{D}}^2\lambda +d\lambda \otimes d\phi .\hfill \end{array}$$

(17)

We achieve our target. □

We have the first result for the non-homothetic conformal factor:

Theorem 4. 

Let $({\mathfrak{B}}^n,g,\lambda ,\beta )$ be a complete gradient Einstein soliton with a non-parallel, non-homothetic, closed conformal vector field $\mathcal{U}\in \mathfrak{X}\left({\mathfrak{B}}^n\right)$ and constant scalar curvature. Then $({\mathfrak{B}}^n,g)$ is isometric to Euclidean space ${\mathbb{R}}^n$.

Proof. 

First, we claim that:

$$T={\left|\mathcal{U}\right|}^{2}d\phi \otimes d\lambda$$

is a symmetric 2-tensor. Indeed, it is immediately shown that T is a symmetric 2-tensor from the second equation of Lemma 2 by considering constant scalar curvature. By applying the fourth assertion of Lemma 1, one can rewrite T as follows:

$$\begin{array}{c}\hfill T=\mathcal{U}\left(\phi \right){\mathcal{U}}^{♭}\otimes d\lambda \end{array}$$

Moreover, observing the symmetry of T, we conclude:

$$\begin{array}{c}\hfill \mathcal{U}\left(\phi \right)\left({\left|\mathcal{U}\right|}^{2}g(\mathcal{D}\lambda ,\mathcal{Y})-\mathcal{U}\left(\lambda \right)g(\mathcal{U},\mathcal{Y})\right)=0,\end{array}$$

which reduces to:

$$\begin{array}{c}\hfill \mathcal{U}\left(\phi \right)\left({\left|\mathcal{U}\right|}^2\mathcal{D}\lambda -\mathcal{U}\left(\lambda \right)\mathcal{U}\right)=0.\end{array}$$

We have two cases as follows:

$$\begin{array}{cc}\hfill \mathcal{U}\left(\phi \right)=& 0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\left|\mathcal{U}\right|}^2\mathcal{D}\lambda =& \mathcal{U}\left(\lambda \right)\mathcal{U}.\hfill \end{array}$$

(19)

Next, we consider (18). Combined with Lemma 1(iv), it implies that:

$$\begin{array}{c}\hfill {\left|\mathcal{U}\right|}^2\mathcal{D}\phi =0,\end{array}$$

and consequently:

$$\begin{array}{c}\hfill \mathcal{D}\phi =0,\end{array}$$

for all non-singular points of $\mathcal{U}$. Accordingly, $\phi$ applies everywhere due to Lemma 2(i) that the zeros of $\mathcal{U}$ are discrete. As a final point, due to the non-parallel nature of $\mathcal{U}$, we define a function h:${\mathfrak{B}}^n\to \mathbb{R}$ by:

$$\begin{array}{c}\hfill h={\displaystyle \frac{1}{2\phi }}{\left|\mathcal{U}\right|}^2.\end{array}$$

By combining this function with Lemma 1(iii), this result can be deduced as follows:

$$\begin{array}{c}\hfill {\mathcal{D}}^{2}h=\phi g.\end{array}$$

Hence, $\left({\mathfrak{B}}^n,g\right)$ is isometric to Euclidean space by Theorem 2 of [23].

For the second case (19): A gradient Einstein soliton possessing a non-parallel closed conformal vector field $\mathcal{U}$ and constant scalar curvature. Then, it follows by (19) that:

$$\begin{array}{c}\hfill {\left|\mathcal{U}\right|}^2\mathcal{D}\lambda =\mathcal{U}\left(\lambda \right)\mathcal{U},\end{array}$$

(20)

or $\left({\mathfrak{B}}^n,g\right)$ is isometric to the Euclidean space ${\mathbb{R}}^n$. The following identity for the Einstein soliton follows directly from (10), we have

$$\begin{array}{c}\hfill \mathrm{Ric}\left(\mathcal{D}\lambda \right)=0.\end{array}$$

The above analysis leads to the following conclusion:

$$\begin{array}{c}\hfill \mathrm{Ric}\left({\left|\mathcal{U}\right|}^2\mathcal{D}\lambda \right)={\left|\mathcal{U}\right|}^2\mathrm{Ric}\left(\mathcal{D}\lambda \right)={\mathcal{U}}^2.0=0.\end{array}$$

When used with (20) provides $\mathcal{U}\left(\lambda \right)\mathrm{Ric}\left(\mathcal{U}\right)=0$. Consequently:

$$\begin{array}{c}\hfill {\left|\mathcal{U}\right|}^2\mathcal{U}\left(\lambda \right)\mathrm{Ric}\left(\mathcal{U}\right)=0.\end{array}$$

Now, using Lemma 1(v) we obtain: ${\left|\mathcal{U}\right|}^2\mathcal{U}\left(\lambda \right)\mathcal{D}\phi =0$ and then by Lemma 1(iv), it follows that $\mathcal{U}\left(\lambda \right)\mathcal{U}\left(\phi \right)\mathcal{U}=0$. So, it follows from (20) that:

$$\begin{array}{c}\hfill {\mathcal{U}\left(\phi \right)\left|\mathcal{U}\right|}^2\mathcal{D}\lambda =0.\end{array}$$

This implies that $\mathcal{U}\left(\phi \right)\mathcal{D}\lambda =0$ in $\mathfrak{B}\setminus \mathcal{U}\left(\mathcal{U}\right)$, where $\mathcal{U}\left(\mathcal{U}\right)$ is the set of zeros of $\mathcal{U}$. Taking into account that $\mathcal{U}\left(\mathcal{U}\right)$ is a discrete set, we conclude:

$$\begin{array}{c}\hfill \mathcal{U}\left(\phi \right)\mathcal{D}\lambda =0on{\mathfrak{B}}^n.\end{array}$$

(21)

Indeed, (21) is trivially satisfied at the zeros of $\mathcal{U}$. Moreover, since any gradient Einstein soliton is analytic in harmonic coordinates, $\mathcal{D}\lambda$ cannot vanish on any nonempty open dense subset of ${\mathfrak{B}}^n$; otherwise, the manifold would be Einstein and $\lambda$ would be constant. Consequently, from (21) we obtain $\mathcal{U}\left(\phi \right)=0$. Applying Lemma 1(iv) once again, we deduce that $\mathcal{D}\phi =0$, which implies that $\phi$ must be constant. Now, we define the function ${\phi }_1:{\mathfrak{B}}^n\to \mathbb{R}$ by:

$${\phi }_1={\displaystyle \frac{1}{2\phi }}{\left|\mathcal{U}\right|}^2$$

Computing its Hessian, and using Lemma 1 (3) we obtain:

$${\mathcal{D}}^2{\phi }_1=\phi g$$

Again Theorem 2 of [23] gives $\left({\mathfrak{B}}^n,g\right)$, which is isometric to Euclidean space ${\mathbb{R}}^n$. It completes the proof of the theorem. □

Restricting to $\mathcal{U}\ne 0$ (as $\mathbb{Z}\left(\mathcal{U}\right)$ is discrete), we obtain the following result.

Proposition 1. 

Let $\left({\mathfrak{B}}^n,g,\lambda ,\beta \right)$ be a gradient Einstein soliton with a non-parallel closed conformal vector field $\mathcal{U}$ with constant scalar curvature. Then the scalar curvature is as follows:

$$\begin{array}{c}\hfill \mathrm{R}=(n-1)\left(\beta +{\displaystyle \frac{\mathrm{R}}{2}}-f\phi -\mathrm{\Psi }\right),\end{array}$$

(22)

where $\mathrm{\Psi }={\displaystyle \frac{\mathcal{U}\left(\phi \right)}{{\left|\mathcal{U}\right|}^2}}$ and $f={\displaystyle \frac{\mathcal{U}\left(\lambda \right)}{{\left|\mathcal{U}\right|}^2}}$.

Proof. 

Assuming that the scalar curvature is constant, the right-hand side of (12) becomes symmetric. It follows that $\mathrm{d}\lambda \otimes \mathrm{d}\phi$ is symmetric, and thus we obtain:

$$\begin{array}{c}\hfill g(\mathcal{D}\phi ,\mathcal{U})\mathcal{D}\lambda =g(\mathcal{D}\lambda ,\mathcal{U})\mathcal{D}\phi .\end{array}$$

(23)

Taking account of (iv) of Lemma 1, there exists a function $\mathrm{\Psi }$ such that $\mathcal{D}\phi =\mathrm{\Psi }\mathcal{U}$ by setting $\mathrm{\Psi }={\displaystyle \frac{\mathcal{U}\left(\phi \right)}{{\left|\mathcal{U}\right|}^2}}$. Next, the expression $d\phi \otimes {\mathcal{U}}^{♭}$ is symmetric due to (iii), From (23), we get:

$$\begin{array}{c}\hfill \mathrm{\Psi }(f\mathcal{U}-\mathcal{D}\lambda )=0\end{array}$$

(24)

by setting $f={\displaystyle \frac{\mathcal{U}\left(\lambda \right)}{{\left|\mathcal{U}\right|}^2}}$. Analyticity in harmonic coordinates together with the unique continuation property yields either $\mathrm{\Psi }=0$ or $\mathcal{D}\lambda =f\mathcal{U}$. If $\mathrm{\Psi }=0$, then $\phi$ is constant along $\mathcal{U}$. Hence, by Tashiro [23], ${\mathfrak{B}}^n$ is Euclidean and Ricci-flat, which implies (22).

For the case that $\mathcal{D}\lambda =f\mathcal{U}$, we have:

$$\begin{array}{c}\hfill {\mathcal{D}}^2\lambda =f\phi g+df\otimes {\mathcal{U}}^{♭}.\end{array}$$

(25)

The trace of the equation above can be calculated as follows:

$$\begin{array}{c}\hfill g(\mathcal{D}f,\mathcal{U})=\Delta \lambda -n\phi f.\end{array}$$

(26)

From (9), we obtain:

$$\begin{array}{c}\hfill g(\mathcal{D}f,\mathcal{U})=\left(\beta +{\displaystyle \frac{1}{2}}\mathrm{R}\right)n-\mathrm{R}-n\phi f.\end{array}$$

(27)

Putting (4) in (25), we get:

$$\begin{array}{c}\hfill \mathrm{Ric}+df\otimes {\mathcal{U}}^{♭}+f\phi g=(\beta +{\displaystyle \frac{1}{n}}\mathrm{R})g.\end{array}$$

(28)

Applying the conformal vector field $\mathcal{U}$ to the above equation, we have:

$$\begin{array}{c}\hfill \mathrm{Ric}(\mathcal{U},\mathcal{U})+{g(\mathcal{D}f,\mathcal{U})\left|\mathcal{U}\right|}^2+{f\phi \left|\mathcal{U}\right|}^2=(\beta +{\displaystyle \frac{1}{2}}\mathrm{R}){\mathcal{U}|}^2.\end{array}$$

(29)

Inserting (v) of Lemma 1 into the above equation:

$$\begin{array}{c}\hfill -(n-1)g\left(\mathcal{D}\phi ,\mathcal{U}\right)+{g(\mathcal{D}f,\mathcal{U})\left|\mathcal{U}\right|}^2+{f\phi \left|\mathcal{U}\right|}^2=(\beta +{\displaystyle \frac{1}{n}}\mathrm{R}){\mathcal{U}|}^2.\end{array}$$

(30)

Recall that a closed conformal vector field is characterized by the property that: its dual 1-form ${\mathcal{U}}^{♭}$ is closed. By assumption, the gradient conformal vector field $(\beta +\frac{1}{2}\mathrm{R})=\mathcal{D}\lambda$ is also closed. Hence, we obtain $\mathcal{D}\phi =\mathrm{\Psi }\mathcal{U}$. Using this relation together with (30), we conclude that:

$$\begin{array}{c}\hfill {g(\mathcal{D}f,\mathcal{U})\left|\mathcal{U}\right|}^2+{f\phi \left|\mathcal{U}\right|}^2=(\beta +{\displaystyle \frac{1}{n}}\mathrm{R}){\left|\mathcal{U}\right|}^2+(n-1)\mathrm{\Psi }{\left|\mathcal{U}\right|}^2.\end{array}$$

By direct computation with (27), we obtain:

$$\begin{array}{cc}\hfill n\left(\beta +{\displaystyle \frac{1}{2}}\mathrm{R}\right){\left|\mathcal{U}\right|}^2-{\mathrm{R}\left|\mathcal{U}\right|}^2-(n-1)\phi f{\left|\mathcal{U}\right|}^2=& (\beta +{\displaystyle \frac{1}{n}}\mathrm{R}){\left|\mathcal{U}\right|}^2+(n-1)\mathrm{\Psi }{\left|\mathcal{U}\right|}^2.\hfill \end{array}$$

(31)

Thus, our result follows from the above equation since the conformal vector field $\mathcal{U}$ is nontrivial. □

Theorem 5. 

Let $({\mathfrak{B}}^n,g,\lambda )$ be a complete gradient Einstein soliton with constant scalar curvature admitting a non-parallel closed conformal vector field $\mathcal{U}$. For $x\in \mathfrak{B}$ with $\mathcal{D}\lambda \left(x\right)\ne 0$, choose an orthonormal frame $\{E_1={\displaystyle \frac{\mathcal{D}\lambda }{\left|\mathcal{D}\lambda \right|}},E_2,\dots ,E_n\}$. Then:

$$\begin{array}{cc}\hfill {\mathrm{R}}_{11}& =-(n-1)\mathrm{\Psi },\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\mathrm{R}}_{ab}& ={\displaystyle \frac{\mathrm{R}+(n-1)\mathrm{\Psi }}{n-1}}{g}_{ab},2\le a,b\le n\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\mathrm{R}}_{1b}& =0,2\le b\le n.\hfill \end{array}$$

(34)

Proof. 

We note that if $x\in {\mathfrak{B}}^n$ satisfies $\mathcal{D}\lambda \left(x\right)\ne 0$, then necessarily $\mathcal{U}\left(x\right)\ne 0$. Consequently, from the properties of the Ricci curvature, we deduce that from virtue (v) of Lemma 1, and the fact that $\mathcal{D}\lambda =f\mathcal{U}$, which implies the first part that:

$$\begin{array}{cc}\hfill {\mathrm{R}}_{11}& =g\left(\mathrm{Ric}\left(E_1\right),E_1\right)=g\left({\displaystyle \frac{1}{\left|\mathcal{D}\lambda \right|}}\mathrm{R}ic\left(\mathcal{D}\lambda \right),E_1\right)=g\left({\displaystyle \frac{1}{\left|\mathcal{D}\lambda \right|}}\mathrm{R}ic\left(f\mathcal{U}\right),E_1\right)\hfill \\ \hfill & =g\left({\displaystyle \frac{f}{\left|\mathcal{D}\lambda \right|}}\mathrm{Ric}\left(\mathcal{U}\right),E_1\right)=-(n-1)g\left({\displaystyle \frac{f\mathcal{D}\phi }{\left|\mathcal{D}\lambda \right|}},E_1\right)=-(n-1)g\left({\displaystyle \frac{\mathrm{\Psi }\mathcal{D}\lambda }{\left|\mathcal{D}\lambda \right|}},E_1\right)\hfill \\ \hfill & =-(n-1)\mathrm{\Psi }.\hfill \end{array}$$

Since ${\mathcal{U}}_b=0$ for all $2\le b\le n$, from (28), we have:

$$\begin{array}{c}\hfill {\mathrm{R}}_{ab}=\left(\beta +{\displaystyle \frac{1}{n}}\mathrm{R}\right)g_{ab}-{\mathcal{D}}_a{\mathcal{D}}_b\lambda =((\beta +{\displaystyle \frac{1}{2}}\mathrm{R})-f\phi )g_{ab}-{\mathcal{D}}_{a}f{\mathcal{U}}_b.\end{array}$$

(35)

$$\begin{array}{c}\hfill {\mathrm{R}}_{ab}=((\beta +{\displaystyle \frac{1}{2}}\mathrm{R})-f\phi )g_{ab}-\left({\mathcal{D}}_{a}f\right){\mathcal{U}}_b=((\beta +{\displaystyle \frac{1}{n}}\mathrm{R})-f\phi )g_{ab}.\end{array}$$

Since, $f{\mathcal{U}}_b=g(f\mathcal{U},e_b)=g(\mathcal{D}\lambda ,e_b)=0.$ For $f\left(x\right)\ne 0$ we infer that ${\mathcal{U}}_b=0$ for all $2\le b\le n$. This substitutes in (35) and using Theorem 2, we get the second result. Again from (35) with $a=1$, that is:

$$\begin{array}{c}\hfill {\mathrm{R}}_{1b}=(\beta +{\displaystyle \frac{1}{2}}\mathrm{R})g_{1b}-{\mathcal{D}}_1{\mathcal{D}}_b\lambda =-f\mathrm{\Psi }{g}_{1b}-{\mathcal{D}}_{1}f{\mathcal{U}}_b=0.\end{array}$$

(36)

This completes the proof of the theorem. □

Theorem 6. 

Let $({\mathfrak{B}}^n,g,\lambda )$ be a complete non-compact gradient Einstein soliton possessing a non-parallel homothetic closed conformal vector field $\mathcal{U}$, then the tensor $\mathcal{T}=0$ for all indices.

Proof. 

Let $\{E_1={\displaystyle \frac{\mathcal{D}\lambda }{\left|\mathcal{D}\lambda \right|}},E_2,\dots E_n\}$ in $T_x\mathfrak{B}$ be an orthonormal frame at point $x\in \mathfrak{B}$ for $({\mathfrak{B}}^n,g,\lambda )$ such that $\mathcal{D}\lambda \left(x\right)\ne 0$, from identity (10), we have:

$$\begin{array}{c}\hfill (n-2)\mathcal{D}\mathrm{R}=2\mathrm{R}ic\left(\mathcal{D}\lambda \right)=2(n-1)\mathrm{\Psi }\mathcal{D}\lambda .\end{array}$$

By substituting the preceding relation into (6), we get:

$$\begin{array}{cc}\hfill {\mathcal{T}}_{abc}=& {\displaystyle \frac{1}{n-2}}\left\{{\mathrm{R}}_{bc}{\mathcal{D}}_a\lambda -{\mathrm{R}}_{ac}{\mathcal{D}}_b\lambda \right\}+{\displaystyle \frac{2(n-1)\mathrm{\Psi }}{2(n-1){(n-2)}^2}}\left\{g_{bc}{\mathcal{D}}_a\lambda -g_{ac}{\mathcal{D}}_b\lambda \right\}\hfill \\ \hfill & -{\displaystyle \frac{\mathrm{R}}{(n-1)(n-2)}}\left\{g_{bc}{\mathcal{D}}_a\lambda -g_{ac}{\mathcal{D}}_b\lambda \right\}.\hfill \end{array}$$

which equivalently:

$$\begin{array}{cc}\hfill {\mathcal{T}}_{abc}=& {\displaystyle \frac{1}{n-2}}\left\{{\mathrm{R}}_{bc}{\mathcal{D}}_a\lambda -{\mathrm{R}}_{ac}{\mathcal{D}}_b\lambda \right\}+\left\{{\displaystyle \frac{(n-1)\mathrm{\Psi }-(n-2)\mathrm{R}}{(n-1){(n-2)}^2}}\right\}\left\{g_{bc}{\mathcal{D}}_a\lambda -g_{ac}{\mathcal{D}}_b\lambda \right\}.\hfill \end{array}$$

(37)

From (33) of Theorem 5, we infer that:

$$\begin{array}{cc}\hfill {\mathrm{R}}_{ab}& ={\displaystyle \frac{\mathrm{R}+(n-1)\mathrm{\Psi }}{n-1}}{g}_{ab}\hfill \end{array}$$

(38)

for all fixed $2\le a\le n$ and $1\le b\le n$. Now from (37) and (38), we find that:

$$\begin{array}{cc}\hfill {\mathcal{T}}_{abc}=& {\displaystyle \frac{(n-1)\mathrm{\Psi }+\mathrm{R}}{(n-1)(n-2)}}\left\{g_{bc}{\mathcal{D}}_a\lambda -g_{ac}{\mathcal{D}}_b\lambda \right\}-\left\{{\displaystyle \frac{(n-1)\mathrm{\Psi }-(n-2)\mathrm{R}}{(n-1){(n-2)}^2}}\right\}\left\{g_{bc}{\mathcal{D}}_a\lambda -g_{ac}{\mathcal{D}}_b\lambda \right\}.\hfill \end{array}$$

Finally, we get:

$$\begin{array}{c}\hfill {\mathcal{T}}_{abc}={\displaystyle \frac{(n-1)\mathrm{\Psi }}{{(n-2)}^2}}\left\{g_{bc}{\mathcal{D}}_a\beta -g_{ac}{\mathcal{D}}_b\beta \right\}.\end{array}$$

(39)

As we assumed that $\mathrm{\Psi }$ is homothetic, which means $\mathrm{\Psi }$ is constant. Therefore, $\mathcal{D}\mathrm{\Psi }=0$, then we have from Theorem 1:

$$\begin{array}{c}\hfill \mathrm{\Psi }={\displaystyle \frac{\mathcal{U}\left(\phi \right)}{{\parallel \mathcal{U}\parallel }^2}}={\displaystyle \frac{g(\mathcal{D}\phi ,\mathcal{U})}{{\parallel \mathcal{U}\parallel }^2}}=0.\end{array}$$

(40)

Substituting (40) into (39), we get ${\mathcal{T}}_{abc}=0$ for all $2\le c\le n$. Next, we get the following by replacing $c=1$:

$$\begin{array}{cc}\hfill {\mathcal{T}}_{ab1}=& {\displaystyle \frac{1}{n-2}}\left\{{\mathrm{R}}_{b1}{\mathcal{D}}_a\lambda -{\mathrm{R}}_{a1}{\mathcal{D}}_b\lambda \right\}\hfill \\ \hfill & -\left\{{\displaystyle \frac{(n-1)\mathrm{\Psi }-(n-2)\mathrm{R}}{(n-1){(n-2)}^2}}\right\}\left\{g_{b1}{\mathcal{D}}_a\lambda -g_{a1}{\mathcal{D}}_b\lambda \right\},\hfill \end{array}$$

(41)

for all $1\le a,b\le n$. On the other hand, (33) and (34) imply that:

$$\begin{array}{c}\hfill {\mathrm{R}}_{a1}=-(n-1)\mathrm{\Psi }{g}_{a1}\end{array}$$

(42)

Inserting (42) into (41), we arrive at:

$$\begin{array}{c}\hfill {\mathcal{T}}_{ab1}=-\left\{{\displaystyle \frac{(n-1)\mathrm{\Psi }-(n-2)\mathrm{R}}{(n-1){(n-2)}^2}}\right\}\left\{g_{b1}{\mathcal{D}}_a\lambda -g_{a1}{\mathcal{D}}_b\lambda \right\}.\end{array}$$

(43)

Hence, ${\mathcal{T}}_{1b1}=0$. By symmetry (7), this implies ${\mathcal{T}}_{ab1}={\mathcal{T}}_{b11}=0$, and therefore ${\mathcal{T}}_{abc}=0$ at $x\in {\mathfrak{B}}^n$. Since we work in harmonic coordinates, the metric g is analytic (see DeTurck–Kazdan [24]). Hence, any tensor constructed from g and the smooth function $\lambda$ is analytic. Let $\mathcal{A}=\{x\in \mathfrak{B};\mathcal{D}\lambda (x)\ne 0\}$ and $\mathcal{A}$ is dense in ${\mathfrak{B}}^n$. Since the tensor ${\mathcal{T}}_{abc}$ is analytic and vanishes on the dense set $\mathcal{A}$, by the unique continuation property of analytic tensors, we conclude that ${\mathcal{T}}_{abc}=0\mathrm{on}{\mathfrak{B}}^n.$

This is the squared norm of the tensor ${\mathcal{T}}_{abc}$ as follows:

$$\begin{array}{c}\hfill {\left|\mathcal{T}\right|}^2={\mathcal{T}}_{abc}{\mathcal{T}}_{abc}={\displaystyle \frac{1}{n-2}}{\mathcal{T}}_{abc}\left\{{\lambda }_c{\mathrm{R}}_{ab}-{\lambda }_b{\mathrm{R}}_{ac}\right\}\end{array}$$

by using the fact that $\mathcal{T}$ is totally trace-free and skew-symmetric in the first two indices, we deduce that:

$$\begin{array}{c}\hfill {\left|\mathcal{T}\right|}^2={\displaystyle \frac{1}{n-2}}\left\{{\lambda }_c{\mathrm{R}}_{ab}{\mathcal{T}}_{abc}+{\lambda }_b{\mathrm{R}}_{ac}{\mathcal{T}}_{acb}\right\}={\displaystyle \frac{2}{(n-2)}}{\lambda }_c{\mathrm{R}}_{ab}{\mathcal{T}}_{abc}.\end{array}$$

(44)

This implies that $\mathcal{T}=0$ as ${\mathcal{T}}_{abc}=0$. It completes the proof of the theorem. □

A generalized version of a gradient Einstein-type manifold is studied in [19]. As an application of Theorem 6.3 [19], accordingly, we state the following result:

Theorem 7. 

Let $({\mathfrak{B}}^n,g)$, be a complete, non-compact gradient Einstein soliton possessing a nonparallel, homothetic, closed conformal vector field with a constant scalar curvature. Let c be a regular value of λ and let ${\mathcal{A}}_c=\{x\in \mathfrak{B}|\lambda \left(x\right)=c\}$ be the corresponding level hypersurface. Choose any local orthonormal frame such that $\{e_1\cdots e_{n-1}\}$ is tangent to ${\mathcal{A}}_c$ and $e_n={\displaystyle \frac{\mathcal{D}\lambda }{\left|\mathcal{D}\lambda \right|}}$. Then:

(i) 

${\parallel \mathcal{D}\lambda \parallel }^2$ is constant on ${\mathfrak{B}}_c;$

(ii) 

${\mathrm{R}}_{an}={\mathrm{R}}_{na}=0$ for every $a=1\cdots n-1$ and $e_n$ is eigenvector of $\mathrm{R}ic$;

(iii) 

The mean curvature $\mathcal{H}$ is constant on ${\mathcal{A}}_c;$

(iv) 

The β are constant on ${\mathcal{A}}_c;$

(v) 

${\mathcal{A}}_c$ is an Einstein with respect induced metric;

(vi) 

Ricci tensor of $\mathfrak{B}$ can be written as ${\mathrm{R}}_{ab}={\displaystyle \frac{\mathrm{R}-{\mathfrak{B}}_1}{n-1}}{\delta }_{ab}$, where ${\mathfrak{B}}_1\in \mathbb{R}$ is eigenvalue of multiplicity one or n(in this case $\mathrm{R}={\mathfrak{B}}_1$): in either case $e_n$ is an eigenvector associated to ${\mathfrak{B}}_1$.

Proof. 

The proof of Theorem 7 follows immediately from Theorem 6.3 of [19] and Theorem 8. □

Influenced by research on gradient Einstein solitons and results motivated by Theorems 2 and 1. We obtain the following result for the homothetic conformal factor.

Theorem 8. 

Let $({\mathfrak{B}}^n,g)$, be a complete, non-compact gradient Einstein soliton possessing a nonparallel, homothetic closed conformal vector field with constant scalar curvature. Then $({\mathfrak{B}}^n,g)$ is locally conformally flat for $n=4$ and has a harmonic Weyl tensor for $\ge 5$.

Proof. 

From Theorem 6, we have ${\mathcal{T}}_{abc}=0$ for all $n=4$, then Theorem 3 concludes that the Wely tensor $\mathcal{W}=0$ has vanished. That is, $({\mathfrak{B}}^4,g)$ is locally conformally flat since Einstein-type manifold is generalized to the gradient Einstein soliton. On the other hand, for ≥5, we have Theorem 2, which states that the cotton tensor $\mathcal{C}=0$ for all indices is $\mathcal{T}=0$. Hence, $({\mathfrak{B}}^n,g)$ has a harmonic Wely tensor. □

A further motivation for our study comes from several results concerning the structure of conformally flat Ricci-type manifolds. In [25], it was shown that conformally flat Ricci solitons with non-null gradient of the potential function are isometric to Robertson–Walker warped products, while in the null case they correspond to plane wave geometries. Similar phenomena were obtained in [26] for locally conformally flat Lorentzian gradient quasi-Einstein manifolds. In addition, ref. [27] proved that every locally conformally flat gradient Ricci soliton is locally isometric to a warped product whose base is an interval and whose fiber is a space form, which in particular implies rotational symmetry for complete shrinking solitons with a vanishing Weyl tensor. Motivated by these developments, we investigate alternative geometric conditions ensuring local conformal flatness. In particular, we replace the conformal flatness assumption with the existence of a homothetic closed conformal vector field and obtain the following result:

Theorem 9. 

Let $({\mathfrak{B}}^4,g)$ be a complete, non-compact gradient Einstein soliton with constant scalar curvature, equipped with a non-parallel homothetic closed conformal vector field. Then $({\mathfrak{B}}^4,g)$ admits a warped product structure with a real interval as the base and a 3-dimensional space form of constant sectional curvature c as the fiber.

Proof. 

Let $({\mathfrak{B}}^4,g,\lambda )$ be a complete gradient Einstein soliton with $n=4$. At $x\in \mathfrak{B}$, take $\{E_1={\displaystyle \frac{\mathcal{D}\lambda }{\left|\mathcal{D}\lambda \right|}},E_2,\dots ,E_n\}$ orthonormal. Then the Weyl tensor is:

$$\begin{array}{cc}\hfill \mathcal{W}(E_a,E_b,E_k,E_1)=& {\displaystyle \frac{1}{n-1}}\mathrm{Ric}(E_a,E_1){\delta }_{bk}-{\displaystyle \frac{1}{n-1}}\mathrm{Ric}(E_b,E_1){\delta }_{ak}\hfill \\ \hfill & -{\displaystyle \frac{1}{n-1}}\mathrm{Ric}(E_a,E_1){\delta }_{bk}-{\displaystyle \frac{1}{n-2}}\mathrm{Ric}(E_b,E_k)g(E_a,E_1)\hfill \\ \hfill & +{\displaystyle \frac{1}{n-2}}\mathrm{Ric}(E_a,E_k)g(E_b,E_1)+{\displaystyle \frac{1}{n-2}}\mathrm{Ric}(E_b,E_k){\delta }_{ak}\hfill \\ \hfill & +{\displaystyle \frac{\mathrm{R}}{(n-1)(n-2)}}g(E_a,E_1){\delta }_{bk}-{\displaystyle \frac{\mathrm{R}}{(n-1)(n-2)}}g(E_b,E_1){\delta }_{ak}\hfill \\ \hfill =& -{\displaystyle \frac{1}{(n-1)(n-2)}}\mathrm{Ric}(E_a,E_1){\delta }_{bk}+{\displaystyle \frac{1}{(n-1)(n-2)}}\mathrm{Ric}(E_b,E_1){\delta }_{ak}\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}\mathrm{R}ic(E_b,E_k)g(E_a,E_1)+{\displaystyle \frac{1}{n-2}}\mathrm{Ric}(E_a,E_k)g(E_b,E_1)\hfill \\ \hfill & +{\displaystyle \frac{\mathrm{R}}{(n-1)(n-2)}}g(E_a,E_1){\delta }_{bk}-{\displaystyle \frac{\mathrm{R}}{(n-1)(n-2)}}g(E_b,E_1){\delta }_{ak}.\hfill \end{array}$$

For $i=2,\cdots n,$ if we replace $E_a=E_1$ preceding equation, we have:

$$\begin{array}{cc}\hfill \mathcal{W}(E_1,E_b,E_k,E_1)=& -{\displaystyle \frac{1}{(n-1)(n-2)}}\mathrm{Ric}(E_1,E_1)-{\displaystyle \frac{1}{(n-2)}}\mathrm{Ric}(E_a,E_a)\hfill \\ \hfill & +{\displaystyle \frac{\mathrm{R}}{(n-1)(n-2)}}.\hfill \end{array}$$

(45)

Since, the second item of Theorem 8 implies $\mathcal{W}=0$ for every index, then we get the following from (45):

$$\begin{array}{c}\hfill (n-1)\mathrm{Ric}(E_a,E_a)=\mathrm{R}-\mathrm{R}ic(E_1,E_1).\end{array}$$

(46)

Taking account of (4), we get:

$$\begin{array}{cc}\hfill {\mathcal{D}}^2\lambda (E_a,E_a)=& \beta +{\displaystyle \frac{1}{n}}\mathrm{R}-{\displaystyle \frac{1}{(n-1)}}\left\{\mathrm{R}-\left((\beta +{\displaystyle \frac{1}{2}}\mathrm{R}-{\mathcal{D}}^2\lambda (E_1,E_1)\right)\right\}\hfill \\ \hfill =& {\displaystyle \frac{n\beta }{(n-1)}}+{\displaystyle \frac{(n\frac{1}{n}-1)}{(n-1)}}\mathrm{R}-{\displaystyle \frac{1}{(n-1)}}{\mathcal{D}}^2\lambda (E_1,E_1).\hfill \end{array}$$

(47)

From (9), we get:

$$\begin{array}{c}\hfill {\mathcal{D}}^2\lambda (E_a,E_a)={\displaystyle \frac{n\beta }{(n-1)}}+{\displaystyle \frac{(n\frac{1}{2}-1)}{(n-1)}}{\displaystyle \frac{n\beta -\Delta \lambda }{(1-n\frac{1}{2})}}-{\displaystyle \frac{1}{(n-1)}}{\mathcal{D}}^2\lambda (E_1,E_1).\end{array}$$

which is equivalent to:

$$\begin{array}{c}\hfill {\mathcal{D}}^2\lambda (E_a,E_a)={\displaystyle \frac{1}{(n-1)}}\left(\Delta \lambda -{\mathcal{D}}^2\lambda (E_1,E_1)\right).\end{array}$$

(48)

From the above identity, it follows that the hypersurface $\mathcal{N}={\lambda }^{-1}\left(c\right)$ is totally umbilical in $(\mathfrak{B},g)$. As a result, $\mathfrak{B}$ can be locally written as a twisted product $(t_1,t_2)\times \mathcal{N}$ with metric $d{t}^2+f_1^2{g}_{\mathcal{N}}$, where $f_1$ is a smooth function on $(t_1,t_2)$ (cf. [28], Theorem 1). Since the scalar curvature is assumed to be constant, we obtain:

$$\begin{array}{c}\hfill \mathrm{Ric}(X,\mathcal{D}\lambda )=0\end{array}$$

Therefore, $\nabla \lambda$ is an eigenvector of the Ricci tensor, which implies that: $\mathcal{R}(E_1,E_{\alpha })=0$ for all $\alpha =2,\dots ,n$. By Theorem 1 of [29], this condition ensures that the twisted product is in fact a warped product. □