Sufficient conditions for triviality of Ricci solitons

1.   Introduction

In first decade of the nineteenth century, Poincare made the following conjecture: " A compact simply connected three-manifold without boundary is diffeomorphic to the three sphere $S^{3}$." A more general conjecture than Poincare conjecture is Thurston's geometrization conjecture, which says that any closed three-manifold can be decomposed into pieces such that each piece has a locally homogeneous metric and are $S^{3}$, $R^{3}$, $H^{3}$, $S^{2}\times R$, $H^{2}\times R$, $SL\left(2, R\right)$, $nil^{3}$ and $sol^{3}$. With the aim of proving the geometrization conjecture, Hamilton ^[1] initiated a program in 1982 called Ricci flow that starts with a given Riemannian metric $g_{0}$ on a smooth $n$-dimensional manifold $M$ and evolves it as a one-parameter family of metrics $g\left(s\right)$ satisfying

where $Ric$ is the Ricci tensor of the evolving metric $g\left(s\right)$. The generalized fixed points of the Ricci flow are those manifolds that change by a diffeomorphism and a rescaling under the Ricci flow. More precisely, let $M$\ be an $n$-dimensional smooth manifold and $\left(M, g\left(s\right) \right)$ be a solution of the Ricci flow such that $g\left(0\right) = g_{0}$. Let $f_{s}:M\rightarrow M$ be a one-parameter family of diffeomorphisms generated by the family of vector fields $X(s)$ and let $\rho (s)$ be a time-dependent scale factor, then a solution of the Ricci flow of the form $g\left(s\right) = \rho \left(s\right) f_{s}^{\ast }\left(g\right)$ is called Ricci soliton. Thus, a Ricci soliton is a generalized fixed point of the Ricci flow, viewed as a dynamical system on the space of Riemannian metrics modulo diffeomorphisms and scalings. Taking the derivative of the above equation with respect to $s$, substituting $s = 0$ and assuming $\overset{.}{\rho }\left(0\right) = -2\lambda$, $\rho \left(0\right) = 1$, $f_{0} = id$, $X(0) = \mathbf{u}$, we get

where $\lambda$ is a constant, $£_{\mathbf{u}}$ is the Lie-derivative of $g$ with respect to $\mathbf{u}$ and $\mathrm{Ric}$ is the Ricci tensor of $\left(M, g \right)$. We shall denote a Ricci soliton by $\left(M, g, \mathbf{u}, \lambda \right)$. The topic Ricci soliton is important in geometry as well as global analysis, especially since it was deployed in settling the famous Poincarè conjecture. The vector field $\mathbf{u}$ appearing in Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is called the potential field of the Ricci soliton. If the potential field $\mathbf{u}$ is Killing; that is, $£_{\mathbf{u}} g = 0$, then the definition of Ricci soliton implies

that is, the Ricci soliton is an Einstein manifold. In this case the Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is called a trivial Ricci soliton.

In ^{[1,2,3,4,5,6,7]}, authors found different conditions under which a Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is a trivial Ricci soliton. For compact gradient Ricci solitons in ^[5], the author derived several identities, and later in ^[6] these identities were used to prove that a compact gradient shrinking Ricci soliton, which is locally conformally flat, must be trivial. In ^[8], authors used the Ricci mean value $\delta$ of an $n$-dimensional compact gradient Ricci soliton $\left(M, g, \nabla f, \lambda \right)$ defined by

where $V$ is the volume of $M$ and $Ric\left(\nabla f, \nabla f\right)$ is the Ricci curvature in the direction of $\nabla f$, to prove for $n\geq 2$ that $\delta \geq 0$; the equality holds if, and only if, the Ricci soliton is trivial. Similarly, in ^[9] the author has considered Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ of positive Ricci curvature and has shown that if the potential field $\mathbf{u}$ is a Jacobi-type vector field, then the Ricci soliton is trivial.

Apart from finding conditions under which a Ricci soliton is trivial, there are several important aspects of the geometry of Ricci solitons. For instance, on space times such as spherically symmetric static space times Lorentzian plane-symmetric static space times and Kantowski Sachs space times, treated as Ricci solitons, the role of potential field on these respective space times is studied in ^{[3,10,11,12,13]}, respectively.

We denote by $\eta$ the smooth one-form dual to $\mathbf{u}$ that is

for smooth vector field $X$ on $M$, then we obtain a skew-symmetric tensor field $F$ defined on $M$ by

for smooth vector fields $X$ and $Y$ on $M$.

One of the important questions on Ricci solitons is to find conditions under which a Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is trivial. Note that the squared length of the covariant derivative of potential field $\mathbf{u}$ is given by

where $\{ u_1, u_2, \ldots, u_n \}$ is a local orthonormal frame on $M$, $n = \mathrm{dim} M$. Also, we define

Our first result is the following:

Theorem 1. If the covariant derivative of the potential field $\mathbf{u}$ of a connected Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ satisfies

then the Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is trivial.

Recall that if potential field $\mathbf{u}$ is Killing makes $\left(M, g, \mathbf{u}, \lambda \right)$ a trivial Ricci soliton. In ^[14], authors introduced the notion of the geodesic vector field. Note that a Killing vector field of constant length is a geodesic vector field, and there are many examples of geodesic vector fields that are not Killing. Recall that a vector field $\xi$ on a Riemannian $\left(M, g \right)$ is said to be a geodesic vector field if

that is, the integral curves of $\xi$ are geodesics. An interesting example is provided by the vector field $\xi$ of a proper tran-Sasakian manifold $\left(M, g, \phi, \xi, \eta, \alpha, \beta \right)$, which is a geodesic vector field that is not killing ^[15,16]. A similar example is provided by the vector field $\xi$ of a Kenmotsu manifold $\left(M, g, \phi, \xi, \eta \right)$ ^[17].

In our next result on an $n$-dimensonal Ricci soliton $\left(M, g, \mathbf{u}, \lambda\right)$, we use the condition that the potential field $\mathbf{u}$ is a geodesic vector field to prove the following:

Theorem 2. If the Ricci curvature $Ric\left(\mathbf{u}, \mathbf{u} \right)$ of an $n$-dimensional compact Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ with scalar curvature $\tau$ satisfies

and the potential field $\mathbf{u}$ is a geodesic vector field, then the Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is trivial.

For an $n$-dimensional Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$, we let $\left\{ \varphi _{t}\right\}$ be the local flow of the potential field $\mathbf{u}$, then the Ricci operator $S$ of $\left(M, g, \mathbf{u}, \lambda \right)$ is said to be invariant under $\mathbf{u}$ if

or, equivalently,

If the Ricci operator $S$ of the Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is invariant under the potential field $\mathbf{u}$, then we have the following characterization of a trivial Ricci soliton.

Theorem 3. If the Ricci operator $S$ an $n$-dimensional compact Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is invariant under the potential field $\mathbf{u}$ and satisfies

for each vector field $U$ on $M$, then $\left(M, g, \mathbf{u}, \lambda\right)$ is a trivial Ricci soliton and the converse also holds.

A Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is said to be shrinking if the constant is $\lambda > 0$ ^[1]; it is an important question to find geometric conditions under which a Ricci soliton is shrinking. One of important classical vector fields is concurrent vector field $\xi$ on a Riemannian manifold $\left(M, g\right)$, which obeys

for any smooth vector field $U$ on $M$. This means that the holonomy group of $M$ leaves a point of $M$ invariant ^[18,19]. In our final result, we use the condition that the potential field $\mathbf{u}$ of the Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is a concurrent vector field to prove the following:

Theorem 4. Let $\left(M, g, \mathbf{u}, \lambda \right)$ be an $n$-dimensional connected Ricci soliton with $\mathbf{u}$ as a concurrent vector field, then $\left(M, g, \mathbf{u}, \lambda\right)$ is a shrinking Ricci soliton.

2.   Preliminaries

Suppose $\left(M, g, \mathbf{u}, \lambda\right)$ is an $n$-dimensional Ricci soliton, then by Eq 1.1 we have

We denote by $S$ the Ricci operator of $\left(M, g, \mathbf{u}, \lambda\right)$ satisfying

where $\mathfrak{X}(M)$ is the Lie-algebra of vector field on $M$. Using the following expressions

and

we derive

that is,

Equation 2.1 implies

Recall that the scalar curvature $\tau$ of the Ricci soliton $\left(M, g, \mathbf{u}, \lambda\right)$ is given by

where $\{ u_1, ..., u_n \}$ is a local orthonormal frame on $M$, $n = \text{dim}M$.

Lemma 2.1. Let $\left(M, g, \mathbf{u}, \lambda\right)$ be an $n$-dimnesional Ricci soliton, then

(i) $\mathrm{div} \mathbf{u} = n \lambda - \tau$;

(ii) $\| S- \frac{\tau}{n} I \|^2 = -n \left(\lambda - \frac{\tau}{n} \right)^2 + \| \nabla \mathbf{u} \|^2 - \|F\|^2$.

Proof. Using Eq 2.2, we have

where we used $\mathrm{tr}.F = 0$. This proves (i). Also, Eq 2.2 implies

that is,

Thus, we have

where $\{ u_1, ..., u_n \}$ is a local orthonormal frame on $M$.

Now, using Eq 2.2 and (i) in Lemma 2.1, we obtain the result in (ii). □

3.   Proof of Theorem 1

Using the definition of Ricci soliton 1.1, we have

Thus,

Also, note that

Using 3.2 in (ii) of Lemma 2.1, we obtain

Combining it with Eq 3.1, we conclude

Hence, using the condition in the statement, we conclude $£_{\mathbf{u}} g = 0$; that is, $\left(M, g, \mathbf{u}, \lambda\right)$ is a trivial soliton.

Remark 3.1. As Theorem 1 suggests, the Ricci soliton $(M, g, \mathbf{u}, \lambda)$ with potential field $\mathbf{u}$ satisfying

is trivial. It is natural to expect to see through an example of a nontrivial Ricci soliton $(M, g, \mathbf{u}, \lambda)$ that the potential field $\mathbf{u}$ does not satisfy the above condition. We consider the $n$-dimensional Euclidean space $\left(\mathbf{E}^{n}, g\right)$ with Euclidean metric $g$ and the vector field $\mathbf{u}$ defined by

where $u^{1}, ..., u^{n}$ are the Euclidean coordinates, then we have

and

This shows that $\left(\mathbf{E}^{n}, g, \mathbf{u}, 1\right)$ is an $n$-dimensional nontrivial Ricci soliton. It follows that $\mathbf{u}$ is a closed field and, therefore, $F = 0$. Moreover, we have

that is, we have

4.   Proof of Theorem 2

Assume that the potential field $\mathbf{u}$ of the Ricci soliton $\left(M, g, \mathbf{u}, \lambda\right)$ is a geodesic vector field; that is,

By virtue of Eqs 2.2 and 4.4, we have

that is,

where we used $g \left(F(\mathbf{u}), \mathbf{u} \right) = 0$, owing to skew-symmetry of $F$. Also, on using (i) in Lemma 2.1, we have that

Integrating the above equation and using Eq 4.3, we conclude

Using the condition in the statement, we conclude

Comparing Eqs 4.3 and 4.4, we have

using it in (ii) of Lemma 2.1, we have

in (i) of Lemma 2.1, we have

Finally, the integral formula of Yano ^[20] and Eq 4.7 implies

Using Eqs 4.6 and 4.8, we conclude

Note that in view of Eq 4.9, Eq 3.1 takes the form

and using Eq 3.2, we conclude

Using the above equation in integral 4.9, we have

Now, using the lower bound on $\mathrm{Ric} (\mathbf{u}, \mathbf{u})$ in the statement, we conclude

that is, in view of Eq 4.5,

Hence, $\left(M, g, \mathbf{u}, \lambda\right)$ is trivial.

5.   Proof of Theorem 3

Suppose $\left(M, g, \mathbf{u}, \lambda \right)$ is an $n$-dimensional compact Ricci soliton such that the Ricci operator $S$ satisfies

and is invariant under the potential field $\mathbf{u}$; that is,

Using Eq 5.2, we have

which in view of Eq 2.2 gives

Now, define a function $\psi$ on $M$ by

then using Eq 2.2 and symmetry of the Ricci operator and skew symmetry of the operator $F$, we find the gradient $\nabla \psi$ of $\psi$ as

Note that by using Lemma 2.1, we have

Next, we compute the divergence of $S\left(\mathbf{u}\right)$ while using Eqs 2.2, 5.1 and 5.3 through a local orthonormal frame $\left\{u_{1}, .., u_{n}\right\}$, and we have

Since $S$ is symmetric and $F$ is skew symmetric, it follows that

Thus, we confirm

Thus, using Eq 5.4, we have

which in view of Eqs 5.4 and 5.7, we have

integrating the above equation, we reach

We rearrange the above integral as

and as by Schwartz's inequality $\left\Vert S\right\Vert ^{2}\geq \frac{1}{n}\tau ^{2}$, both integrands in the above equation are nonnegative and we can confirm

Thus, we conclude that

and

Now, the Eq 5.12 is equal in the Schwartz's inequality and it holds if, and only if,

and combining it with Eq 5.11, we confirm

Hence, $\left(M, g, \mathbf{u}, \lambda \right)$ is trivial Ricci soliton.

Conversely, if $\left(M, g, \mathbf{u}, \lambda \right)$ is trivial Ricci soliton, then the potential field $\mathbf{u}$ is Killing and, therefore, the flow $\left\{ \varphi _{t}\right\}$ consists of isometries. Hence, the Ricci operator $S = \lambda I$ is invariant under $\mathbf{u}$. Moreover, the condition

holds. This completes the proof.

6.   Proof of Theorem 4

Suppose the potential field $\mathbf{u}$ is a concurrent vector field, then we have ^[18,19]

and, consequently, we have

Using a local frame $\left\{ u_{1}, ...u_{n}\right\}$, we have

On the basis of it, we conclude

Now, using Eq 2.2, we have

which in view of Eqs 6.1 and 6.2 yields

Taking the inner product with $\mathbf{u}$ in the above equation and paying attention to skew-symmetry of $F$, we reach

Since $\mathbf{u} = 0$ together with Eq 6.1 gives a contradiction, the above equation on connected $M$ implies $\lambda = 1$; that is, the Ricci soliton is shrinking.

7.   Conclusions

In Theorems 1 and 2, we discussed conditions under which a Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is trivial. In Theorem 2, we used that the potential field $\mathbf{u}$ is a geodesic vector field, and in Theorem 4, it was shown that if the potential field $\mathbf{u}$ is a concurrent vector field, then the Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ is shrinking. There is yet another important vector field defined on a Riemannian manifold called the incompressible vector field ^[21]; this notion was taken from fluid dynamics, where the velocity field of an incompressible fluid satisfies the equation of continuity. A smooth vector field $\xi$ on a Riemannian manifold $(M, g)$ is said to be incompressible if $\mathrm{div}\xi = 0$. In view of Theorems 2 and 4, it will be interesting to study the behavior of the Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ under the condition that the potential field $\mathbf{u}$ is incompressible. It is worth finding a condition under which a Ricci soliton $\left(M, g, \mathbf{u}, \lambda \right)$ with the potential field $\mathbf{u}$ is incompressible and trivial.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This project was supported by the Researchers Supporting Project number (RSP2023R413), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.