Some Properties of Noncompact Ricci Solitons

Abstract

In this article, we are interested in finding sufficient conditions for a noncompact Ricci soliton (M, h, v, λ) to be trivial. The major role is played by a skew-symmetric operator φ associated to the potential field v of the connected Ricci soliton (M, h, v, λ). In the firstresult, it is shown that if the potential field v of the Ricci soliton (M, h, v, λ) is an eigenvector of the Ricci operator S with eigenvalue τ/n, where τ is the scalar curvature and n = dim M, with additional conditions v(τ) ≤ 0 and squared norm of the covariant derivative ∇v having a suitable upper bound is necessarily trivial. In another result, it is shown that a connected Ricci soliton (M, h, v, λ) with potential field v an eigenvector of the Ricci operator S with eigenvalue λ with an additional condition v(τ) ≤ 0 is trivial. Similarly, it is shown that for a complete and connected Ricci soliton (M, h, v, λ) with potential field v annihilating the Ricci operator S and incompressible vector field φv and v(τ) having a suitable upper bound is either trivial, or else is isometric to the Euclidean space. Also, we show that if the Ricci operator S of a connected Ricci soliton (M, h, v, λ) is invariant under the potential field v with additional two suitable conditions, then (M, h, v, λ) is trivial. Finally, it has been observed that the function ψ related to the squared length of the potential field v of a connected Ricci soliton (M, h, v, λ) has a role, namely it is shown that if ψ is a superharmonic function and the Ricci curvature Ric(v, v) has a suitable upper bound then (M, h, v, λ) is necessarily trivial.

1. Introduction

In Hamilton’s quest to resolve the Poincare conjecture, he introduced the Ricci flow on an n-dimensional Riemannian manifold $\left(M,h\right)$ as the heat equation (cf. [1,2])

$${\displaystyle \frac{\partial h}{\partial s}}=-2Ric,$$

(1)

where $Ric$ is the Ricci tensor with respect to the evolving metric $h\left(s\right)$. The Ricci flow is not only significant in geometry (as ultimately it became the basis for the resolution of the long standing Poincare conjecture (cf. [2,3])), but also has shown immense applications in medical imaging [4], economics [5], biology, chemistry, and physics [6,7]. Apart from this importance of the Ricci flow, its stable solution namely, the Ricci soliton is the quadruple $\left(M,h,\mathbf{v},\lambda \right)$, where $\mathbf{v}$ is the vector field on $\left(M,h\right)$ called potential field satisfying

$${\pounds }_{\mathbf{v}}h+2Ric=2\lambda h$$

(2)

$\lambda$ being a constant, has tremendous significance and is a widely studied geometric structure. The Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ has very interesting geometry as well as topology (cf. [1,8]) and has attracted many mathematicians. In [7,9,10], it has been exhibited that a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ has very rich topology. In [11] the impact of potential field $\mathbf{v}$ on the geometry of the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is studied. If the potential field $\mathbf{v}=\nabla f$ is a gradient of a function f, then the Ricci soliton Equation (2) takes the form

$$Hess\left(f\right)+Ric=\lambda h,$$

(3)

and the Ricci soliton $\left(M,h,\nabla f,\lambda \right)$ is called a gradient Ricci soliton and the function f is called the potential function. It is worth noting that the main step in the resolution of the Poincare conjecture was establishing that a compact Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is necessarily a gradient Ricci soliton (cf. [1,2,8]). Therefore, the importance of studying the geometry of Ricci solitons has shifted to studying noncompact Ricci solitons. A noncompact Ricci soliton can be a gradient Ricci soliton as well as a nongradient Ricci soliton. In [3], important properties of complete gradient solitons are derived.

Note that through the defining Equation (2), it follows that if the potential field $\mathbf{v}$ is a Killing vector field, then a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is an Einstein manifold and in this case a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is said to be a trivial Ricci soliton. Thus, a Ricci soliton can be considered as a generalization of an Einstein manifold. If a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is compact and has scalar curvature $\tau$, then the potential field $\mathbf{v}=\nabla f$ is a gradient and in this case the function f satisfies

$$2\lambda f=\tau +{∥\nabla f∥}^2,$$

(4)

up to normalization. This interesting property is used to derive geometric as well as topological information of the compact Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ (cf. [3,7,9,10,12]). In an interesting article [3], authors obtained optimal lower and upper estimates for the growth of the potential function f on complete noncompact gradient Ricci solitons $\left(M,h,\nabla f,\lambda \right)$ for $\lambda >0$. Using these estimates on f, they proved that any complete noncompact gradient Ricci soliton $\left(M,h,\nabla f,\lambda \right)$, $\lambda >0$, has at most Euclidean volume growth. An interesting part of this result is that it is seen as a soliton analog of the classical Bishop volume comparison theorem for Riemannian manifolds of nonnegative Ricci curvature. In [10], the authors proved that any complete gradient Ricci soliton $\left(M,h,\nabla f,\lambda \right)$, $\lambda >0$, with strictly positive sectional curvature must necessarily be compact. Note that in noncompact nongradient Ricci solitons $\left(M,h,\mathbf{v},\lambda \right)$ the tool provided by Equation (4) is missing. It is for this reason, studying the geometry of noncompact Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is interesting as well as challenging. In [11], the author has shown that a complete and connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ of positive Ricci curvature with potential field $\mathbf{v}$ a Jacobi-type vector field is necessarily a trivial Ricci soliton. In [13], the authors considered an n-dimensional connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with energy function $\psi$ defined by

$$\psi ={\displaystyle \frac{1}{2}}{∥\mathbf{v}∥}^2,$$

and have shown that the inequality

$$\Delta \psi \le {∥\phi ∥}^2-Ric\left(\mathbf{v},\mathbf{v}\right),$$

is necessary and sufficient for $\left(M,h,\mathbf{v},\lambda \right)$ to be trivial. Also, in [14], the authors have shown that an n-dimensional compact Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, $n\ge 3$, having Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature orthogonal is a trivial Ricci soliton. In this article, we focus on finding sufficient conditions for a connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ to be trivial.

In Section 2, we recall necessary information on curvature tensor, Ricci tensor and the scalar curvature of a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$. It is worth noting that an important operator associated to the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is the skew-symmetric operator $\phi$ defined by

$$d\alpha \left(F_1,F_2\right)=2h\left(\phi F_1,F_2\right),$$

where $\alpha$ is the 1-form dual to the potential field $\mathbf{v}$ and $F_1$ and $F_2$ are smooth vector fields on M. This operator $\phi$ plays a crucial role in this article, in particular in the basic Lemmas in Section 2. In Section 3, connected Ricci solitons $\left(M,h,\mathbf{v},\lambda \right)$ of dimension $n>2$ are considered with the emphasis on the fact that the potential field $\mathbf{v}$ is an eigenvector of the Ricci operator S. In the first result, the potential field $\mathbf{v}$ is considered to be an eigenvector of S corresponding to eigenvalue ${\displaystyle \frac{\tau }{n}}$, ($\tau$ is the scalar curvature) and together with other suitable conditions, it is proved that in this case the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ becomes trivial (see Theorem 1). Then in the next result of this section, the potential field $\mathbf{v}$ is considered to be eigenvector of the Ricci operator S corresponding to eigenvalue $\lambda$ and with another suitable condition it is proved that the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ becomes trivial (see Theorem 2). In the last result of this section, we consider a complete and connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with potential field $\mathbf{v}$ eigenvector of the Ricci operator S corresponding to eigenvalue 0 (that is, $\mathbf{v}$ annihilates S) and with other suitable conditions, it is proved that either the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is trivial, or else it is isometric to the Euclidean space (see Theorem 3).

In Section 4, we consider the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with Ricci operator S invariant under $\mathbf{v}$, that is, S commutes with the differential of the local flow of the potential field $\mathbf{v}$. We prove that under the condition that the Ricci operator S is invariant under the potential field $\mathbf{v}$ together with suitable restrictions on the covariant derivative ${\nabla }_{\mathbf{v}}\mathbf{v}$ and the Ricci curvature $Ric\left(\mathbf{v},\mathbf{v}\right)$, the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is trivial (see Theorem 4).

In Section 5, we prove two triviality results for a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$. It is clear that if the potential field $\mathbf{v}$ is Killing then the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is trivial. There is a notion of a 2-Killing vector field, which is weaker than the notion of a Killing vector field. In the first result of this section, we show that if the potential field $\mathbf{v}$ of the connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is 2-Killing and additionally the condition $\mathbf{v}\left(\tau \right)\le 0$ holds, then this implies that Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is trivial (see Theorem 5). In the last result of this section, we consider the energy function

$$\psi ={\displaystyle \frac{1}{2}}{∥\mathbf{v}∥}^2,$$

on a connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ and show that if the function $\psi$ is superharmonic and the Ricci curvature $Ric\left(\mathbf{v},\mathbf{v}\right)$ has a suitable upper bound, then the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is trivial (see Theorem 6).

2. Preliminaries

Here, we recall the notions of curvature tensor, Ricci tensor and scalar curvature as well as their properties on a Riemannian manifold $\left(M,h\right)$. We denote by ∇ the Riemannian connection. We have the following expression for the curvature tensor of $\left(M,h\right)$ (cf. [2,15])

$$R(F_1,F_2)F_3={\nabla }_{F_1}{\nabla }_{F_2}{F}_3-{\nabla }_{F_2}{\nabla }_{F_1}{F}_3-{\nabla }_{\left[F_1,F_2\right]}{F}_3,\hspace{1em}{F}_i\in \mathfrak{X}\left(M\right),$$

(5)

where $i=1,2,3$ and $\mathfrak{X}\left(M\right)$ denotes the set of smooth vector fields on M. The Ricci tensor $Ric$ of $\left(M,h\right)$ is obtained by taking the trace in the above equation, that is, we have

$$Ric\left(F_1,F_2\right)=\sum _{j}h\left(R\left(f_j,F_1\right)F_2,f_j\right),\hspace{1em}{F}_1,F_2\in \mathfrak{X}\left(M\right),$$

(6)

$\left\{f_1,\dots ,f_n\right\}$ being a local frame on $\left(M,h\right)$. The Ricci tensor is symmetric and the Ricci operator S of $\left(M,h\right)$ is defined by

$$Ric\left(F_1,F_2\right)=h\left(S{F}_1,F_2\right),\hspace{1em}{F}_1,F_2\in \mathfrak{X}\left(M\right),$$

(7)

which is also a symmetric operator. Now, taking again the trace in the above equation, gives the scalar curvature $\tau$ of $\left(M,h\right)$, namely

$$\tau =\sum _{k}Ric\left(f_k,f_k\right).$$

(8)

It is quite interesting to see that the gradient $\nabla \tau$ of the scalar curvature $\tau$ is related to the derivative of the Ricci operator, namely (cf. [2,15])

$${\displaystyle \frac{1}{2}}\nabla \tau =\sum _k\left({\nabla }_{f_k}S\right)\left(f_k\right),$$

(9)

where

$$\left({\nabla }_{F_1}S\right)\left(F_2\right)={\nabla }_{F_1}S{F}_2-S\left({\nabla }_{F_1}{F}_2\right).$$

We will see that this identity is very useful as we progress in our work on Ricci solitons.

Now, considering a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, $\dim M=n$, we denote by $\alpha$ the 1-form dual to the potential field $\mathbf{v}$ and use the exterior derivative $d\alpha$ to introduce a skew-symmetric operator $\phi$, which we call the associated operator of the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, defined by

$$d\alpha \left(F_1,F_2\right)=2h\left(\phi F_1,F_2\right),\hspace{1em}{F}_1,F_2\in \mathfrak{X}\left(M\right).$$

Using the following outcome from the definition of Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, namely Equation (2) and the above equation and employing these in Koszul’s formula, yields the following equation for the covariant derivative of the potential field

$${\nabla }_F\mathbf{v}=\lambda F-SF+\phi F,\hspace{1em}F\in \mathfrak{X}\left(M\right).$$

(10)

Noting that the associated operator $\phi$ being skew-symmetric, its trace is zero and thus, from the above equation, we derive the divergence of the potential field as follows:

$$\mathrm{d}\mathrm{i}\mathrm{v}\mathbf{v}=n\lambda -\tau .$$

(11)

Differentiating Equation (10) and using the expression for the curvature tensor field given in (5) for a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, we see that

$$R\left(F_1,F_2\right)\mathbf{v}=-\left({\nabla }_{F_1}S\right)\left(F_2\right)+\left({\nabla }_{F_2}S\right)\left(F_1\right)+\left({\nabla }_{F_1}\phi \right)\left(F_1\right)-\left({\nabla }_{F_2}\phi \right)\left(F_1\right).$$

(12)

Tracing the above equation as in Equation (6) and using Equation (9) while using symmetry of the Ricci operator S and skew-symmetry of the associated operator $\phi$, leads to

$$Ric\left(F_2,\mathbf{v}\right)={\displaystyle \frac{1}{2}}h\left(F_2,\nabla \tau \right)-\sum _{k}h\left(F_2,\left({\nabla }_{f_k}\phi \right)\left(f_k\right)\right),\hspace{1em}{F}_2\in \mathfrak{X}\left(M\right),$$

(13)

that is,

$$S\mathbf{v}={\displaystyle \frac{1}{2}}\nabla \tau -\sum _k\left({\nabla }_{f_k}\phi \right)\left(f_k\right).$$

(14)

Lemma 1. 

On a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with associated operator φ, the Lie derivative of the Ricci tensor with respect to the potential field $\mathbf{v}$ is given by

$$\begin{array}{ccc}\hfill \left({\pounds }_{\mathbf{v}}Ric\right)\left(F_1,F_2\right)& =& h\left(\left({\nabla }_{\mathbf{v}}S\right)\left(F_1\right),F_2\right)+2\lambda Ric\left(F_1,F_2\right)-2h\left(S{F}_1,S{F}_2\right)\hfill \\ & & +h\left(S{F}_1,\phi F_2\right)+h\left(S{F}_2,\phi F_1\right),\hfill \end{array}$$

for $F_1,F_2\in \mathfrak{X}\left(M\right)$.

Proof. 

Using Equation (10) in

$$\left({\pounds }_{\mathbf{v}}Ric\right)\left(F_1,F_2\right)=\mathbf{v}Ric\left(F_1,F_2\right)-Ric\left(\left[\mathbf{v},F_1\right],F_2\right)-Ric\left(F_1,\left[\mathbf{v},F_2\right]\right)$$

we get the desired result. □

In the next Lemma, we compute the second Lie derivative of the metric h with respect to the potential field $\mathbf{v}$ following the techniques described in [16].

Lemma 2. 

On a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with Ricci operator S, we have

$$\begin{array}{ccc}\hfill \left({\pounds }_{\mathbf{v}}{\pounds }_{\mathbf{v}}h\right)\left(F_1,F_2\right)& =& 4{\lambda }^{2}h\left(F_1,F_2\right)-8\lambda Ric\left(F_1,F_2\right)-2h\left(\left({\nabla }_{\mathbf{v}}S\right)\left(F_1\right),F_2\right)\hfill \\ & & +4h\left(S{F}_1,S{F}_2\right)-2h\left(S{F}_1,\phi F_2\right)-2h\left(S{F}_2,\phi F_1\right),\hfill \end{array}$$

for $F_1,F_2\in \mathfrak{X}\left(M\right)$.

Proof. 

Using Equation (2) in the form

$${\pounds }_{\mathbf{v}}h=2\lambda h-2Ric,$$

and taking the Lie derivative in the above equation, we conclude

$${\pounds }_{\mathbf{v}}{\pounds }_{\mathbf{v}}h=2\lambda {\pounds }_{\mathbf{v}}h-2{\pounds }_{\mathbf{v}}Ric=2\lambda \left(2\lambda h-2Ric\right)-2{\pounds }_{\mathbf{v}}Ric.$$

Combining above equation with Lemma 1, we get the desired result. □

Lemma 3. 

On a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with Ricci operator S, we have

$$\begin{array}{ccc}\hfill \left({\pounds }_{{\nabla }_{\mathbf{v}}\mathbf{v}}h\right)\left(F_1,F_2\right)& =& 2{\lambda }^{2}h\left(F_1,F_2\right)-4\lambda Ric\left(F_1,F_2\right)-h\left(\left({\nabla }_{F_1}S\right)\left(\mathbf{v}\right),F_2\right)\hfill \\ & & -h\left(\left({\nabla }_{F_2}S\right)\left(\mathbf{v}\right),F_1\right)+2h\left(S{F}_1,S{F}_2\right)+h\left(\left({\nabla }_{F_1}\phi \right)\left(\mathbf{v}\right),F_2\right)\hfill \\ & & +h\left(\left({\nabla }_{F_2}\phi \right)\left(\mathbf{v}\right),F_1\right)-2h\left(\phi F_1,\phi F_2\right),\hfill \end{array}$$

for $F_1,F_2\in \mathfrak{X}\left(M\right)$.

Proof. 

Using Equation (10), we have

$${\nabla }_{\mathbf{v}}\mathbf{v}=\lambda \mathbf{v}-S\mathbf{v}+\phi \mathbf{v}$$

and from Equation (2), we have

$${\pounds }_{{\nabla }_{\mathbf{v}}\mathbf{v}}h=2\lambda \left(\lambda h-Ric\right)-{\pounds }_{S\mathbf{v}}h+{\pounds }_{\phi \mathbf{v}}h.$$

(15)

Now, taking $F_1,F_2\in \mathfrak{X}\left(M\right)$ and using Equation (10), we compute

$$\left({\pounds }_{S\mathbf{v}}h\right)\left(F_1,F_2\right)=h\left({\nabla }_{F_1}S\mathbf{v},F_2\right)+h\left({\nabla }_{F_2}S\mathbf{v},F_1\right),$$

that is,

$$\begin{array}{cc}\hfill \left({\pounds }_{S\mathbf{v}}h\right)\left(F_1,F_2\right)& =h\left(\left({\nabla }_{F_1}S\right)\left(\mathbf{v}\right),F_2\right)+h\left(\left({\nabla }_{F_2}S\right)\left(\mathbf{v}\right),F_1\right)+2\lambda Ric\left(F_1,F_2\right)\hfill \\ \hfill \hspace{1em}& -2h\left(S{F}_1,S{F}_2\right)+h\left(\phi F_1,S{F}_2\right)+h\left(\phi F_2,S{F}_1\right).\hfill \end{array}$$

(16)

Similarly, we have

$$\begin{array}{cc}\hfill \left({\pounds }_{\phi \mathbf{v}}h\right)\left(F_1,F_2\right)& =h\left(\left({\nabla }_{F_1}\phi \right)\left(\mathbf{v}\right),F_2\right)+h\left(\left({\nabla }_{F_2}\phi \right)\left(\mathbf{v}\right),F_1\right)+h\left(S{F}_1,\phi F_2\right)\hfill \\ \hfill \hspace{1em}& +h\left(S{F}_2,\phi F_1\right)-2h\left(\phi F_1,\phi F_2\right).\hfill \end{array}$$

(17)

Inserting Equations (16) and (17) in Equation (15) gives the desired result. □

On a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with associated operator $\phi$, we define the squared lengths of the covariant derivative of the potential field $\mathbf{v}$ and $\phi$ by

$${∥\nabla \mathbf{v}∥}^2=\sum _{k}h\left({\nabla }_{f_k}\mathbf{v},{\nabla }_{f_k}\mathbf{v}\right)\hspace{1em}\mathrm{and}\hspace{1em}{∥\phi ∥}^2=\sum _{k}h\left(\phi f_k,\phi f_k\right),$$

(18)

where $\left\{f_1,\dots ,f_n\right\}$ is an orthonormal frame on $\left(M,h,\mathbf{v},\lambda \right)$.

Recall that a vector field $\xi$ on a Riemannian manifold $\left(M,h\right)$ is said to be incompressible if $\mathrm{div}\xi =0$. Denoting by $L^1\left(M\right)$ the space of Lebesgue integrable functions on M, we have the following result from [17]:

Proposition 1. 

Let ξ be a smooth vector field on the complete, noncompact, oriented Riemannian manifold $\left(M,h\right)$, such that $\mathit{div}\xi$ does not change sign on M. If the length $∥\xi ∥\in L^1\left(M\right)$, then $\mathit{div}\xi =0$.

3. Ricci Solitons with Potential Field Eigenvector of S

Note that on the Euclidean space $\left(R^n,\overline{h}\right)$, where $\overline{h}$ is the flat Euclidean metric. Taking $\mathbf{v}$ as the position vector of $R^n$, we see that

$${\displaystyle \frac{1}{2}}{\pounds }_{\mathbf{v}}\overline{h}+\overline{Ric}=\overline{h},$$

where $\overline{Ric}$ is the Ricci tensor of $\left(R^n,\overline{h}\right)$, which is zero. Thus, $\left(R^n,\overline{h},\mathbf{v},\lambda \right)$ is a Ricci soliton, with $\lambda =1$. Moreover, as the Ricci operator $\overline{S}=0$ and the scalar curvature $\overline{\tau }=0$ for the Ricci soliton $\left(R^n,\overline{h},\mathbf{v},\lambda \right)$, it trivially satisfies

$$\overline{S}\mathbf{v}={\displaystyle \frac{\overline{\tau }}{n}}\mathbf{v}.$$

In this article, we are interested in finding conditions under which a connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with $\dim M=n$, is trivial. The above example hints at trying a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, $\dim M=n$ with the Ricci operator S satisfying

$$S\mathbf{v}={\displaystyle \frac{\tau }{n}}\mathbf{v}.$$

(19)

That is, the potential field $\mathbf{v}$ is an eigenvector of the Ricci operator S corresponding to the eigenvalue $n^{-1}\tau$, for triviality of the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$. First we consider the connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with $\dim M=n>2$ and having nonzero scalar curvature. We prove the following:

Theorem 1. 

A connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, $\dim M=n>2$ with Ricci operator S, scalar curvature τ and associated operator φ satisfying

$$\mathbf{v}\left(\tau \right)\le 0,\hspace{1em}S\mathbf{v}={\displaystyle \frac{\tau }{n}}\mathbf{v}\hspace{1em}\mathit{and}\hspace{1em}{∥\nabla \mathbf{v}∥}^2\le {∥\phi ∥}^2,$$

is a trivial Ricci soliton.

Proof. 

Choosing a local frame $\left\{f_1,\dots ,f_n\right\}$ on the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, we proceed to compute $\mathrm{div}S\mathbf{v}$ and we have

$$\mathrm{div}S\mathbf{v}=\sum _{k}h\left({\nabla }_{f_k}S\mathbf{v},f_k\right)=\sum _{k}h\left(\left({\nabla }_{f_k}S\right)\left(\mathbf{v}\right)+S{\nabla }_{f_k}\mathbf{v},f_k\right).$$

Using symmetry of the operator S, skew-symmetry of the associated operator $\phi$ and Equations (9) and (10) in the above equation, it leads to

$$\begin{array}{cc}\hfill \mathrm{div}S\mathbf{v}& ={\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)+\sum _{k}h\left(\lambda f_k-S{f}_k+\phi f_k,S{f}_k\right)\hfill \\ \hfill \hspace{1em}& ={\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)+\lambda \tau -{∥S∥}^2.\hfill \end{array}$$

(20)

Also, using Equation (11), we have

$$\mathrm{div}\left(\tau \mathbf{v}\right)=\mathbf{v}\left(\tau \right)+\tau \left(n\lambda -\tau \right).$$

Combining the above equation with Equations (19) and (20), yields

$${\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)+\lambda \tau -{∥S∥}^2={\displaystyle \frac{1}{n}}\mathbf{v}\left(\tau \right)+{\displaystyle \frac{\tau }{n}}\left(n\lambda -\tau \right).$$

Thus, we have

$${\displaystyle \frac{n-2}{2n}}\mathbf{v}\left(\tau \right)={∥S∥}^2-{\displaystyle \frac{1}{n}}{\tau }^2.$$

(21)

Now, using the restrictions $\mathbf{v}\left(\tau \right)\le 0$ and $n>2$, we conclude

$${∥S∥}^2-{\displaystyle \frac{1}{n}}{\tau }^2\le 0.$$

(22)

However, the Schwartz’s inequality conveys

$${∥S∥}^2\ge {\displaystyle \frac{1}{n}}{\tau }^2,$$

and by virtue of inequality (22), we see the equality

$${∥S∥}^2={\displaystyle \frac{1}{n}}{\tau }^2,$$

holds, if and only if

$$S={\displaystyle \frac{\tau }{n}}I.$$

(23)

Since, $n>2$ the above equation implies $\tau$ is a constant. Note that using Equation (23) in Equation (21), we conclude $\mathbf{v}\left(\tau \right)=0$. Also, the Equation (10) transforms to

$${\nabla }_F\mathbf{v}=\left(\lambda -{\displaystyle \frac{\tau }{n}}\right)F+\phi F,\hspace{1em}F\in \mathfrak{X}\left(M\right).$$

Using skew-symmetry of $\phi$ and Equation (18) with the above equation to find the squared length of the covariant derivative of the potential field $\mathbf{v}$, we have

$${∥\nabla \mathbf{v}∥}^2=n{\left(\lambda -{\displaystyle \frac{\tau }{n}}\right)}^2+{∥\phi ∥}^2.$$

(24)

Using the inequality

$${∥\nabla \mathbf{v}∥}^2\le {∥\phi ∥}^2,$$

in Equation (24), yields

$${\left(\lambda -{\displaystyle \frac{\tau }{n}}\right)}^2=0.$$

Consequently, the associated operator $\tau =n\lambda$ and Equation (23) now takes the form $Ric=\lambda h$. □

Consider a $(2n-1)$-dimensional sphere $S^{2n-1}\left(c\right)$ of constant curvature c, which is an embedded hypersurface of the Euclidean space $\left(R^{2n},\overline{h},J\right)$, with complex structure J and Hermitian Euclidean metric $\overline{h}$. Let N be the unit normal to $S^{2n-1}\left(c\right)$. The shape operator of the sphere $S^{2n-1}\left(c\right)$ is $A=-\sqrt{c}I$. As the vector $JN$ is orthogonal to N, we get the unit vector field $\mathbf{v}=-JN$ on the sphere $S^{2n-1}\left(c\right)$. For a vector field F on $S^{2n-1}\left(c\right)$, we decompose $JF$ as

$$JF=\varphi F+\alpha \left(F\right)N,$$

where $\varphi F$ is tangential to $S^{2n-1}\left(c\right)$ and $\alpha$ is the 1-form dual to $\mathbf{v}$ and as J is skew symmetric, the operator $\varphi$ is also skew symmetric. Differentiating $\mathbf{v}=-JN$ with respect to F and noticing that J is parallel, while using fundamental equations of hypersurfaces, we get

$${\nabla }_F\mathbf{v}-\sqrt{c}g\left(F,\mathbf{v}\right)N=-\sqrt{c}JF=-\sqrt{c}\varphi F-\sqrt{c}\alpha \left(F\right)N,$$

which gives

$${\nabla }_F\mathbf{v}=-\sqrt{c}\varphi F.$$

Thus, using this equation and the expression $Ric=(2n-2)ch$, where h is the induced metric on the sphere $S^{2n-1}\left(c\right)$, we conclude that

$${\pounds }_{\mathbf{v}}h+2Ric=4(n-1)ch,$$

showing that $\left(S^{2n-1}\left(c\right),h,\mathbf{v},\lambda \right)$ is a Ricci soliton with $\lambda =2(n-1)c$ and associated operator $\phi =-\sqrt{c}\varphi$. Moreover, the equation ${\nabla }_F\mathbf{v}=-\sqrt{c}\varphi F$ implies

$${∥\nabla \mathbf{v}∥}^2=c{∥\varphi ∥}^2={∥\phi ∥}^2.$$

We see that the Ricci soliton $\left(S^{2n-1}\left(c\right),h,\mathbf{v},\lambda \right)$ satisfies all the conditions of the Theorem 1.

If $\left(M,h,\mathbf{v},\lambda \right)$ is a trivial Ricci soliton, then the scalar curvature $\tau$ is a constant and $S\mathbf{v}=\lambda \mathbf{v}$ holds. It implies that $\mathbf{v}\left(\tau \right)=0$. It naturally raises a question: Do the conditions with $\mathbf{v}\left(\tau \right)=0$ and $S\mathbf{v}=\lambda \mathbf{v}$ necessarily imply that a connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is a trivial Ricci soliton? We answer this question in affirmative and indeed prove the following:

Theorem 2. 

A connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, with scalar curvature τ and Ricci operator S satisfying

$$\mathbf{v}\left(\tau \right)\le 0 \mathit{and}\hspace{1em}S\mathbf{v}=\lambda \mathbf{v},$$

is a trivial Ricci soliton.

Proof. 

Differentiating the relation $S\mathbf{v}=\lambda \mathbf{v}$ with respect to $F\in \mathfrak{X}\left(M\right)$, and using Equation (10), we obtain

$$\left({\nabla }_{F}S\right)\left(\mathbf{v}\right)+S\left(\lambda F-SF+\phi F\right)=\lambda \left(\lambda F-SF+\phi F\right).$$

That is,

$$\left({\nabla }_{F}S\right)\left(\mathbf{v}\right)={\left(S-\lambda I\right)}^{2}F+\lambda \phi F-S\phi F,\hspace{1em}F\in \mathfrak{X}\left(M\right).$$

Choosing $F=f_k$ in the above equation and then taking the inner product with $f_k$ and summing the resulting equation for a frame $\left\{f_1,\dots ,f_n\right\}$ on the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, we conclude

$${\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)={∥S-\lambda I∥}^2.$$

(25)

where we have used

$$\sum _{k}h\left(S{f}_k,\phi f_k\right)=0.$$

Using the inequality $\mathbf{v}\left(\tau \right)\le 0$ in Equation (25), yields $S=\lambda I$ proving the Theorem. □

We have seen that the Euclidean space $\left(R^n,\overline{h},\mathbf{v},\lambda \right)$ is a nontrivial Ricci soliton, where $\lambda =1$. It is Ricci flat, and therefore, its Ricci operator $\overline{S}$ being zero satisfies $\overline{S}\mathbf{v}=0$. This observation leads to a natural question, namely what additional conditions on a complete and connected n-dimensional Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with Ricci operator S satisfying $S\mathbf{v}=0$ is necessarily isometric to the Euclidean space $\left(R^n,\overline{h}\right)$. As a partial answer to this question, we prove the following:

Theorem 3. 

An n-dimensional complete and connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, $n>2$ and $\lambda \ne 0$ with potential field $\mathbf{v}$ and Ricci operator S satisfying

$$S\mathbf{v}=0,\hspace{1em}{\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)\le {\displaystyle \frac{\tau }{n}}\left(\tau -n\lambda \right),$$

and the vector $\phi \left(\mathbf{v}\right)$ being incompressible, is either trivial or else isometric to the Euclidean space $\left(R^n,\overline{h}\right)$.

Proof. 

Let $\left(M,h,\mathbf{v},\lambda \right)$ be an n-dimensional complete and connected Ricci soliton. Differentiating the equation $S\mathbf{v}=0$, we get

$$\left({\nabla }_{F}S\right)\left(\mathbf{v}\right)+S\left(\lambda F-SF+\phi F\right)=0,\hspace{1em}F\in \mathfrak{X}\left(M\right),$$

where we used Equation (10). Choosing $F=f_k$ in the above equation and taking the inner product with $f_k$ and summing the equation over a frame $\left\{f_1,\dots ,f_n\right\}$ on the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ results into

$${\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)+\lambda \tau -{∥S∥}^2=0,$$

(26)

where we have used the symmetry of S and skew-symmetry of $\phi$ and

$$\sum _{k}h\left(S{f}_k,\phi f_k\right)=0.$$

Rearranging Equation (26), we have

$${\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)-{\displaystyle \frac{\tau }{n}}\left(\tau -n\lambda \right)={∥S∥}^2-{\displaystyle \frac{1}{n}}{\tau }^2,$$

which on treating with the inequality in the statement yields

$${∥S∥}^2-{\displaystyle \frac{1}{n}}{\tau }^2\le 0.$$

(27)

However, the Schwartz’s inequality is

$${∥S∥}^2\ge {\displaystyle \frac{1}{n}}{\tau }^2,$$

which together with the inequality (27) implies the equality

$${∥S∥}^2={\displaystyle \frac{1}{n}}{\tau }^2.$$

Hence, we have

$$S={\displaystyle \frac{\tau }{n}}I,$$

(28)

and combining it with $n>2$ yields $\tau$ is a constant. Now using $S\mathbf{v}=0$ in Equation (28) implies $\tau \mathbf{v}=0$. If $\mathbf{v}=0$, then we get that the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is trivial. If $\mathbf{v}\ne 0$, we get $\tau =0$ and consequently, Equation (28) now yields $S=0$. With these implications, we see that the Equation (10) takes the form

$${\nabla }_F\mathbf{v}=\lambda F+\phi F,\hspace{1em}F\in \mathfrak{X}\left(M\right).$$

(29)

Now, as both $\tau =0$ and $S=0$, Equation (14) reduces to

$$\sum _k\left({\nabla }_{f_k}\phi \right)\left(f_k\right)=0.$$

(30)

As the vector $\phi \left(\mathbf{v}\right)$ is incompressible, we have $\mathrm{div}\phi \left(\mathbf{v}\right)=0$, which implies

$$\sum _{k}h\left({\nabla }_{f_k}\phi \left(\mathbf{v}\right),f_k\right)=0.$$

(31)

That is

$$\sum _{k}h\left(\left({\nabla }_{f_k}\phi \right)\left(\mathbf{v}\right)+\phi \left(\lambda f_k+\phi f_k\right),f_k\right)=0,$$

where we used Equation (29). Now, using Equation (30) and the fact that $\phi$ is skew-symmetric in the above equation, we obtain

$${∥\phi ∥}^2=0.$$

Consequently, the Equation (29) further reduces to

$${\nabla }_F\mathbf{v}=\lambda F,\hspace{1em}F\in \mathfrak{X}\left(M\right).$$

(32)

Defining a smooth function

$$\psi ={\displaystyle \frac{1}{2}}{∥\mathbf{v}∥}^2$$

and using Equation (32) we find the gradient $\nabla \psi =\lambda \mathbf{v}$. Note that as both $\lambda \ne 0$ and $\mathbf{v}\ne 0$, the function $\psi$ is not a constant. Using equations $\nabla \psi =\lambda \mathbf{v}$ and Equation (32), we find

$$Hess\left(\psi \right)={\lambda }^{2}h,$$

where the constant ${\lambda }^2\ne 0$. Hence, $\left(M,h,\mathbf{v},\lambda \right)$ is isometric to the Euclidean space $\left(R^n,\overline{h}\right)$ (cf. [18]). □

Note that the conditions (i) M is complete and connected with $\dim M>2$, $\lambda \ne 0$, (ii) $S\mathbf{v}=0$ and (iii) ${\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)\le {\displaystyle \frac{\tau }{n}}\left(\tau -n\lambda \right)$ in Theorem 3 are ideal combinations yielding the desired outcome. For instance, conditions (ii) and (iii) are responsible in arriving to the conclusion

$$S={\displaystyle \frac{\tau }{n}}I,$$

and then the dimensional restriction and connectedness of M is utilized to achieving that the scalar curvature $\tau$ is a constant. The combination of the above equation with condition (ii) gives us two alternatives, either the potential field $\mathbf{v}=0$ or else the constant $\tau =0$. The first alternative makes the Ricci soliton trivial and the second alternative together with the completeness of M and the condition $\lambda \ne 0$, makes $\left(M,h,\mathbf{v},\lambda \right)$ isometric to the Euclidean space $\left(R^n,\overline{h}\right)$.

4. Ricci Solitons with Invariant Ricci Operator S

On a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, the Ricci operator S is said to be invariant under the potential field $\mathbf{v}$ if

$${\pounds }_{\mathbf{v}}S=0.$$

(33)

We are interested in getting conditions on a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ including the Ricci operator being invariant so that the Ricci soliton is trivial. We prove the following:

Theorem 4. 

A connected noncompact Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, with the Ricci operator S invariant under the potential field $\mathbf{v}$ and $\mathit{div}\left({\nabla }_{\mathbf{v}}\mathbf{v}\right)$ does not change sign on M with $∥{\nabla }_{\mathbf{v}}\mathbf{v}∥\in L^1\left(M\right)$, and the Ricci curvature $Ric\left(\mathbf{v},\mathbf{v}\right)$ satisfying

$$Ric\left(\mathbf{v},\mathbf{v}\right)\ge {∥\phi ∥}^2,$$

is a trivial Ricci soliton.

Proof. 

As S is invariant under the potential field $\mathbf{v}$, using Equation (33) we have

$$\left[\mathbf{v},SF\right]=S\left[\mathbf{v},F\right],\hspace{1em}F\in \mathfrak{X}\left(M\right).$$

Treating it with Equation (10), we arrive at

$$\left({\nabla }_{\mathbf{v}}S\right)\left(F\right)=\phi SF-S\phi F,\hspace{1em}F\in \mathfrak{X}\left(M\right).$$

(34)

Taking trace in the above equation yields

$$\mathbf{v}\left(\tau \right)=0,$$

(35)

where we have used the symmetry and skew-symmetry of S and $\phi$ respectively. Taking inner product in Equation (14) with potential field $\mathbf{v}$ and using Equation (35) yields

$$Ric\left(\mathbf{v},\mathbf{v}\right)=-\sum _{k}h\left(\mathbf{v},\left({\nabla }_{f_k}\phi \right)\left(f_k\right)\right).$$

(36)

Note that

$$\sum _k\left({\pounds }_{{\nabla }_{\mathbf{v}}\mathbf{v}}h\right)\left(f_k,f_k\right)=2\mathrm{div}\left({\nabla }_{\mathbf{v}}\mathbf{v}\right),$$

and thus, taking trace in the expression of Lemma 3, yields

$$\mathrm{div}\left({\nabla }_{\mathbf{v}}\mathbf{v}\right)=n{\lambda }^2-2\lambda \tau -{\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)+{∥S∥}^2-\sum _{k}h\left(\mathbf{v},\left({\nabla }_{f_k}\phi \right)\left(f_k\right)\right)-{∥\phi ∥}^2.$$

Using Equations (35) and (36) in the above equation, we have

$$\mathrm{div}\left({\nabla }_{\mathbf{v}}\mathbf{v}\right)=\left({∥S∥}^2-{\displaystyle \frac{1}{n}}{\tau }^2\right)+{\displaystyle \frac{1}{n}}{\left(n\lambda -\tau \right)}^2+\left(Ric\left(\mathbf{v},\mathbf{v}\right)-{∥\phi ∥}^2\right).$$

As the conditions in the statement are imposed on the vector ${\nabla }_{\mathbf{v}}\mathbf{v}$, Proposition 1 conveys $\mathrm{div}\left({\nabla }_{\mathbf{v}}\mathbf{v}\right)=0$. Thus, we have

$$\left({∥S∥}^2-{\displaystyle \frac{1}{n}}{\tau }^2\right)+{\displaystyle \frac{1}{n}}{\left(n\lambda -\tau \right)}^2+\left(Ric\left(\mathbf{v},\mathbf{v}\right)-{∥\phi ∥}^2\right)=0,$$

with all three terms in the sum being non-negative, we conclude

$${∥S∥}^2={\displaystyle \frac{1}{n}}{\tau }^2\hspace{1em}\mathrm{and}\hspace{1em}\tau =n\lambda .$$

Consequently, we have

$$S=\lambda I.$$

Therefore, $\left(M,h,\mathbf{v},\lambda \right)$ is trivial. □

Consider, the Euclidean space $\left(R^{2n},\overline{h},J\right)$, with complex structure J and Hermitian Euclidean metric $\overline{h}$ and the vector field $\mathbf{v}$ defined by

$$\mathbf{v}=\chi +J\chi ,$$

where

$$\chi =\sum _k{y}^k{\displaystyle \frac{\partial }{\partial y^k}},$$

$y^1,\dots ,y^{2n}$ being Euclidean coordinates on $R^{2n}$. For a vector field F on $R^{2n}$, as the complex structure J is parallel, we have

$${\overline{\nabla }}_F\mathbf{v}=F+JF,$$

that is, using the Ricci tensor $\overline{Ric}=0$, we have

$${\pounds }_{\mathbf{v}}\overline{h}+2\overline{Ric}=2\overline{h}.$$

Thus, $\left(R^{2n},\overline{h},\mathbf{v},\lambda \right)$ is a Ricci soliton with $\lambda =1$ and it is a nontrivial Ricci soliton with associated operator $\phi =J$. Moreover, we have

$${\overline{\nabla }}_{\mathbf{v}}\mathbf{v}=\mathbf{v}+J\mathbf{v}=2J\chi ,$$

that is, $div\left({\overline{\nabla }}_{\mathbf{v}}\mathbf{v}\right)=0$ and

$$∥{\overline{\nabla }}_{\mathbf{v}}\mathbf{v}∥=2∥\chi ∥\in L^1\left(M\right).$$

As the Ricci operator $\overline{S}=0$, $\overline{S}$ is invariant under $\mathbf{v}$, we observe that the Ricci soliton $\left(R^{2n},\overline{h},\mathbf{v},\lambda \right)$ satisfies the first three conditions in the statement of Theorem 4 and contradicts the last condition

$$\overline{Ric}\left(\mathbf{v},\mathbf{v}\right)\ge {∥\phi ∥}^2.$$

5. Ricci Solitons with Generic Potential Field

As the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with the Killing potential field $\mathbf{v}$ is trivial. It is of interest to seek conditions on potential fields which are weaker than Killing. One of such notions is 2-Killing. In [18], the author introduces a 2-Killing vector field $\xi$ on a Riemannian manifold $\left(M,h\right)$ satisfying

$${\pounds }_{\xi }{\pounds }_{\xi }h=0.$$

(37)

It is known that a Killing vector field is 2-Killing but a 2-Killing vector need not be Killing. In the first result of this section, we use this restriction on the potential field $\mathbf{v}$ of the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$.

Theorem 5. 

A connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with potential field $\mathbf{v}$ a 2-Killing vector field and scalar curvature τ satisfying $\mathbf{v}\left(\tau \right)\le 0$ is trivial.

Proof. 

Using Equation (37) in Lemma 2, we have

$$\begin{array}{ccc}\hfill 0& =& 4{\lambda }^{2}h\left(F_1,F_2\right)-8\lambda Ric\left(F_1,F_2\right)-2h\left(\left({\nabla }_{\mathbf{v}}S\right)\left(F_1\right),F_2\right)\hfill \\ & & +4h\left(S{F}_1,S{F}_2\right)-2h\left(S{F}_1,\phi F_2\right)-2h\left(S{F}_2,\phi F_1\right),\hfill \end{array}$$

for $F_1,F_2\in \mathfrak{X}\left(M\right)$. Taking trace in the above equation yields

$$n{\lambda }^2-2\lambda \tau -{\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)+{∥S∥}^2=0.$$

That is,

$${\displaystyle \frac{1}{n}}{\left(n\lambda -\tau \right)}^2+\left({∥S∥}^2-{\displaystyle \frac{1}{n}}{\tau }^2\right)={\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right).$$

Using the Schwartz’s inequality and $\mathbf{v}\left(\tau \right)\le 0$ in the above equation yields $\tau =n\lambda$ and

$${∥S∥}^2={\displaystyle \frac{1}{n}}{\tau }^2.$$

As the above equality is an equality in the Schwarz’s inequality, which holds if and only if

$$S={\displaystyle \frac{\tau }{n}}I,$$

and we have $S=\lambda I$, proving that $\left(M,h,\mathbf{v},\lambda \right)$ is trivial. □

Finally, we deal with the length of the potential field of the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$. We define the energy function $\psi$ by

$$\psi ={\displaystyle \frac{1}{2}}{∥\mathbf{v}∥}^2,$$

which when using Equation (10) gives the following expression of gradient $\nabla \psi$,

$$\nabla \psi =\lambda \mathbf{v}-S\mathbf{v}-\phi \mathbf{v}.$$

(38)

Recall that a function f on a Riemannian manifold $\left(M,h\right)$ is said to be superharmonic if $\Delta f\le 0$ (cf. [19]). In the next result we use the energy function $\psi$ to be superharmonic to get the triviality of the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$.

Theorem 6. 

A connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with energy function ψ a superharmonic function and Ricci curvature $Ric\left(\mathbf{v},\mathbf{v}\right)$ satisfying

$$Ric\left(\mathbf{v},\mathbf{v}\right)\le {∥\phi ∥}^2,$$

is trivial.

Proof. 

Differentiating Equation (38) with respect to $F\in X\left(M\right)$, we have

$${\nabla }_F\nabla \psi =\lambda {\nabla }_F\mathbf{v}-\left({\nabla }_{F}S\right)\left(\mathbf{v}\right)-S\left({\nabla }_F\mathbf{v}\right)-\left({\nabla }_F\phi \right)\left(\mathbf{v}\right)-\phi \left({\nabla }_F\mathbf{v}\right).$$

Choosing $F=f_k$ in the above equation and taking the inner product with $f_k$ for a frame $\left\{f_1,\dots ,f_n\right\}$ and summing the resulting equation while using Equation (10), we conclude

$$\begin{array}{ccc}\hfill \Delta \psi & =& \lambda \left(n\lambda -\tau \right)-{\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)-\sum _{k}h\left({\nabla }_{f_k}\mathbf{v},S{f}_k\right)+\sum _{k}h\left(\mathbf{v},\left({\nabla }_{f_k}\phi \right)\left(f_k\right)\right)\hfill \\ & & +\sum _{k}h\left({\nabla }_{f_k}\mathbf{v},\phi f_k\right).\hfill \end{array}$$

Again using Equation (10) in the above equation leads to

$$\begin{array}{cc}\hfill \Delta \psi & =\lambda \left(n\lambda -\tau \right)-{\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)-\lambda \tau +{∥S∥}^2+\sum _{k}h\left(\mathbf{v},\left({\nabla }_{f_k}\phi \right)\left(f_k\right)\right)\hfill \\ \hfill \hspace{1em}& +{∥\phi ∥}^2.\hfill \end{array}$$

(39)

Now, using Equation (14), we have

$$Ric\left(\mathbf{v},\mathbf{v}\right)={\displaystyle \frac{1}{2}}\mathbf{v}\left(\tau \right)-\sum _{k}h\left(\mathbf{v},\left({\nabla }_{f_k}\phi \right)\left(f_k\right)\right),$$

and inserting it in Equation (39) leads to

$$\Delta \psi =n{\lambda }^2-2\lambda \tau -Ric\left(\mathbf{v},\mathbf{v}\right)+{∥S∥}^2+{∥\phi ∥}^2.$$

Rearranging the above equation we arrive at

$$\Delta \psi ={\displaystyle \frac{1}{n}}{\left(n\lambda -\tau \right)}^2+\left({∥S∥}^2-{\displaystyle \frac{1}{n}}{\tau }^2\right)+\left({∥\phi ∥}^2-Ric\left(\mathbf{v},\mathbf{v}\right)\right).$$

(40)

As $\psi$ is superharmonic, we get

$${\displaystyle \frac{1}{n}}{\left(n\lambda -\tau \right)}^2+\left({∥S∥}^2-{\displaystyle \frac{1}{n}}{\tau }^2\right)+\left({∥\phi ∥}^2-Ric\left(\mathbf{v},\mathbf{v}\right)\right)\le 0.$$

From the condition in the statement and the Schwartz’s inequality, each of the three terms above are non-negative. We therefore conclude

$$S={\displaystyle \frac{\tau }{n}}I=\lambda I,$$

proving that the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is trivial. □

If the length of the potential field $\mathbf{v}$ of the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is a constant, then as a trivial consequence of the above result we have the following:

Corollary 1. 

A connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with potential field $\mathbf{v}$ of constant length and the Ricci curvature $Ric\left(\mathbf{v},\mathbf{v}\right)$ satisfying

$$Ric\left(\mathbf{v},\mathbf{v}\right)\le {∥\phi ∥}^2,$$

is trivial.

6. A Concluding Remark

We have seen in Section 5 that a Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ with potential field $\mathbf{v}$ having a constant length or the energy function $\psi ={\displaystyle \frac{1}{2}}{∥\mathbf{v}∥}^2$ superharmonic leads to triviality of the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$. There are many nontrivial Ricci solitons which do not have constant lengths nor the energy function $\psi ={\displaystyle \frac{1}{2}}{∥\mathbf{v}∥}^2$ is superharmonic. For instance, on the Euclidean space $\left(R^n,\overline{h}\right)$ with Euclidean metric $\overline{h}$, consider the vector field $\mathbf{v}$ defined by

$$\mathbf{v}=\chi +y^3{\displaystyle \frac{\partial }{\partial y^2}}-y^2{\displaystyle \frac{\partial }{\partial y^3}},$$

(41)

where

$$\chi =\sum _k{y}^k{\displaystyle \frac{\partial }{\partial y^k}},$$

and $y^1,\dots ,y^n$ are the Euclidean coordinates on $R^n$. Denoting by $\overline{\nabla }$ the Euclidean connection on $\left(R^n,\overline{h}\right)$, the covariant derivative of $\mathbf{v}$ with respect to a vector field F on $R^n$ is given by

$${\overline{\nabla }}_F\mathbf{v}=F+\phi F,$$

(42)

where the operator $\phi$ is given by

$$\phi F=F\left(y^3\right){\displaystyle \frac{\partial }{\partial y^2}}-F\left(y^2\right){\displaystyle \frac{\partial }{\partial y^3}}.$$

(43)

It is straight forward to see that the operator $\phi$ satisfies

$$\overline{h}\left(\phi F_1,F_2\right)=-\overline{h}\left(\phi F_2,F_1\right),$$

that is, it is a skew-symmetric operator. Using Equation (42) we have

$$\left({\pounds }_{\mathbf{v}}\overline{h}\right)=2\overline{h}-\overline{Ric},$$

where $\overline{Ric}=0$ is the Ricci tensor of the Euclidean space $\left(R^n,\overline{h}\right)$. Hence, $\left(R^n,\overline{h},\mathbf{v},\lambda \right)$ is a nontrivial Ricci soliton with $\lambda =1$. Note that the Ricci curvature $\overline{Ric}\left(\mathbf{v},\mathbf{v}\right)\le {∥\phi ∥}^2$ but the energy function

$$\psi ={\displaystyle \frac{1}{2}}{∥\mathbf{v}∥}^2,$$

is neither constant nor superharmonic as $\left(R^n,\overline{h},\mathbf{v},\lambda \right)$ is a nontrivial Ricci soliton.

Therefore, it will be an interesting question to study the geometry of nontrivial Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ for which the energy function $\psi$ is non-constant and in particular study the geometry of level sets $M_c$ namely

$$M_c={\psi }^{-1}\left\{c\right\},$$

which will be a hypersurface of the Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$.

Note that examples of trivial solitons are in abundance, for instance all Einstein manifolds $\left(M,h\right)$ with potential field zero, that is, $\left(M,h,0,\lambda \right)$ are trivial Ricci solitons, or an Einstein manifold $\left(M,h\right)$ admitting a Killing vector field $\mathbf{v}$ is a trivial Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$. It is easy to check that these two sets of examples of trivial Ricci solitons satisfy the statements of the triviality results in this article.

We have pointed out in the introduction that an n-dimensional compact Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ is a gradient Ricci soliton and it considerably simplifies the geometry of $\left(M,h,\mathbf{v},\lambda \right)$ as in this case the associated operator $\phi =0$. Moreover, tools such as Stokes’ Theorem as well as critical point theory is available on a compact Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$. However, on noncompact Ricci solitons there are fewer number of tools and associated operator $\phi$ may not be zero. Therefore addressing geometric issues of a noncompact Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$ becomes harder and more challenging. Therefore, when finding triviality results on noncompact Ricci solitons, it appears we are imposing more conditions to achieve the results. For instance, in Theorem 1, in order for an n-dimensional connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, $n>2$, to be trivial, we needed three conditions

$$\left(\mathrm{i}\right)\hspace{1em}\mathbf{v}\left(\tau \right)\le 0,\hspace{1em}\left(\mathrm{ii}\right)\hspace{1em}S\mathbf{v}={\displaystyle \frac{\tau }{n}}\mathbf{v},\hspace{1em}\left(\mathrm{iii}\right)\hspace{1em}{∥\nabla \mathbf{v}∥}^2\le {∥\phi ∥}^2.$$

For those readers, who are seasoned with elegant results on compact Ricci solitons, may find results on noncompact Ricci solitons require too many conditions, especially due to the presence of associated operator $\phi$. It will be an interesting question, whether we can reduce the number of conditions in Theorem 1, to get the triviality result on an n-dimensional connected Ricci soliton $\left(M,h,\mathbf{v},\lambda \right)$, $n>2$. This can be said about other triviality results obtained in this article.