Abstract

Finding characterizations of trivial solitons is an important problem in geometry of Ricci solitons. In this paper, we find several characterizations of a trivial Ricci soliton. First, on a complete shrinking Ricci soliton, we show that the scalar curvature satisfying a certain inequality gives a characterization of a trivial Ricci soliton. Then, it is shown that the potential field having geodesic flow and length of potential field satisfying certain inequality gives another characterization of a trivial Ricci soliton. Finally, we show that the potential field of constant length satisfying an inequality gives a characterization of a trivial Ricci soliton.

1. Introduction

Recall that Ricci solitons, being self-similar solutions of the Ricci flow (cf. [1]), are a topic of current interest. Moreover, they are models for some singularities which make their geometry very interesting. An -dimensional Ricci soliton is a Riemannian manifold on which there is a smooth vector field (called potential field) satisfying (cf. [1]), where is the Ricci tensor, is the Lie derivative of the metric with respect to , and is a constant. A Ricci soliton is said to be expanding, stable, or shrinking depending on , , or , respectively. If the potential field is a gradient of a smooth function , then is called a gradient Ricci soliton, and in this case, equation (1) takes the form where is the Hessian of the function . Ricci solitons are stable solutions of the Ricci flow (cf. [1]) and have been used in settling Poincare conjecture, and since then, the study of Ricci solitons has picked up immense importance. One of the important findings on Ricci solitons is that if it is compact, the potential field is a gradient of a smooth function , that is, a compact Ricci soliton is a gradient Ricci soliton (cf. [1]). A Ricci soliton is said to be trivial if , and in this case, the metric becomes an Einstein metric with becoming the Einstein constant. Several authors have studied the geometry of Ricci solitons (cf. [24]); in [57], Myers-type theorems have been proved for Ricci soliton; similarly in [8], it has been observed that a complete shrinking Ricci soliton has a finite fundamental group. In [9, 10], Bishop-type volume comparison theorems have been proved for noncompact shrinking Ricci solitons.

As Ricci solitons generalize Einstein metrics, a natural open problem is the existence of triviality results (i.e., conditions under which a Ricci soliton becomes an Einstein manifold). Thus, an important question in the geometry of a Ricci soliton is to find conditions under which it becomes trivial. Recently in [11, 12], authors have found necessary and sufficient conditions for a compact Ricci soliton to be a trivial Ricci soliton. In this paper, we find necessary and sufficient conditions for compact Ricci solitons as well as noncompact Ricci solitons to be trivial. In our first result, we show that the scalar curvature of a compact Ricci soliton satisfying a differential inequality involving the first nonzero eigenvalue of the Laplace operator gives a characterization of a trivial Ricci soliton (cf. Theorem 1). We also show that for a connected Ricci soliton the flow of potential field being geodesic flow with its length satisfying certain inequality gives a characterization of a trivial Ricci soliton (cf. Theorem 2). Finally, it is observed that potential field being of constant length satisfying certain inequality on a connected Ricci soliton also gives a characterization of a trivial Ricci soliton (cf. Theorem 4).

2. Preliminaries

Let be an -dimensional Ricci soliton and be smooth 1-form dual to the potential field . We define a skew symmetric tensor field on the Ricci soliton by where is the Lie algebra of smooth vector fields on . We call this tensor field the associated tensor field of the Ricci soliton . The Ricci operator on the Ricci soliton is a symmetric operator defined by , . The gradient of the scalar curvature satisfies where is a local orthonormal frame and the covariant derivative .

Using equations (1) and (3) and Koszul’s formula, the covariant derivative of the potential field is given by

Now, using equation (5), we get the following expression for Riemannian curvature tensor of the Ricci soliton :

As the operator is symmetric and is skew-symmetric, using equations (4) and (6), we obtain which leads to

We denote by the first nonzero eigenvalue of the Laplace operator acting on smooth functions on compact . If is a smooth function satisfying then by minimum principle, we have

3. A Characterization of Compact Trivial Ricci Solitons

Now, we prove the first result of this paper.

Theorem 1. An -dimensional complete shrinking Ricci soliton with Ricci curvature bounded below by a constant and first nonzero eigenvalue of the Laplacian operator is trivial if and only if the scalar curvature satisfies the inequality

Proof. Suppose is a complete shrinking Ricci soliton with Ricci curvature satisfying and the scalar curvature satisfies the inequality Note that the assumption on the Ricci curvature in view of Myers’ theorem implies that is compact. Thus, is a compact Ricci soliton, and therefore, it is a gradient Ricci soliton (cf. [1]). Consequently, is a closed vector field, that is, . Equation (8) takes the form which gives Moreover equation (5) becomes which we use to compute the divergence of and obtain Now, using equation (13) in the above equation leads to which on integrating gives Using equation (15), we have , which gives and consequently, we conclude Thus, equation (18) takes the form Now, equations (13) and (16) imply which together with gives Integrating the above equation, we conclude that is, which gives Now, using equation (20) in the above equation yields Thus, equations (21) and (27) imply Also, we have Bochner’s formula where is the Hessian operator of the scalar curvature . Note that equation (19) implies , which in view of equation (10) gives Now, we use equation (14) to compute Integrating the above equation and using equations (28) and (29), we get which on using (for a shrinking Ricci soliton) and the inequality (30) gives or Thus, Since the Ricci curvature satisfies for a constant , the above inequality takes the form Using the Schwarz inequality , and the inequality (12) in the above inequality, we conclude Also, the equality in the Schwarz inequality holds if and only if . Moreover, the equation in view of equation (15) implies Consequently, using , we get that is, . Now, using with equation (13) and first equation in equation (37), we get , that is, . Hence, and the Ricci soliton is trivial.
Conversely, if is a trivial soliton, then , gives which implies , and consequently, the equality (12) holds.

It is well known that the odd-dimensional unit sphere with induced metric as a hypersurface of the Euclidean space admits a unit Killing vector field , and consequently, we have the trivial Ricci soliton , , satisfying the hypothesis of Theorem 1.

4. Characterizations of Connected Trivial Ricci Solitons

In this section, we consider a connected Ricci soliton and find necessary and sufficient conditions under which it is a trivial Ricci soliton. Recall that the local flow of a smooth vector field on a Riemannian manifold is said to be geodesic flow if the orbits of are geodesics on . Geodesic flows have been used in studying geometry of foliations on a Riemannian manifold (cf. [7, 13]). Note that a flow consisting of isometries is a geodesic flow and the converse is not true. For example, consider the 3-dimensional unit sphere which has a Sasakian structure (cf. [14]). Then for a positive function on , deform the metric by

Then, is still a unit vector field on the Riemannian manifold . However, is no more a Killing vector field on but instead is a trans-Sasakian structure [15], and the flow of on the Riemannian manifold is a geodesic flow.

In the next result, we use this notion of geodesic flow for the potential field of the Ricci soliton to characterize trivial Ricci solitons.

Theorem 2. Let be an -dimensional connected shrinking Ricci soliton with the local flow of potential field be the geodesic flow. Then, is trivial Ricci soliton if and only if the scalar curvature is a constant along the integral curves of and the associated tensor satisfies the inequality

Proof. Suppose is connected with local flow of a geodesic flow and the scalar curvature is a constant along the integral curves of and the associated tensor satisfies As the local flow of is a geodesic flow, equation (5) gives As the scalar curvature is a constant along the integral curves of , using equations (4) and (8), we conclude Now, using equations (5) and (44), we find the divergence of the vector field . After some straight forward computations, we get Similarly, using equations (5) and (45), we get Equation (43) gives , which on inserting in the above equation yields Note that equation (5) gives . Consequently, on taking divergence in equation (43) and using equations (46) and (48), we conclude which gives Using the Schwarz inequality and inequality (42), in the above equation, we conclude Since the equality in the Schwarz inequality holds if and only if , we get , that is, is trivial.
Conversely, if is a trivial Ricci soliton with local flow of a geodesic flow, then it follows that is a constant and equation (5) takes the form Then finding the divergence of using above equation, gives the equality

Remark 3. (1)It is clear that an odd-dimensional unit sphere is a trivial Ricci soliton, where , the potential field , being the complex structure on and is the unit normal to the hypersurface . The associated tensor is given by , the tangential component of . It follows that holds. Naturally, being the Killing vector field, its flow consists of isometries of , and therefore, it is a geodesic flow.(2)Next, we give an example of a nontrivial Ricci soliton with the flow of potential field not a geodesic field. Consider the open subset of the Euclidean space , where is the Euclidean metric. Consider the vector field defined by where is the position vector field and are the Euclidean coordinates on . It follows that

Hence, we have that is, , is a nontrivial Ricci soliton with associated tensor field , given by

The flow of is given by which is not a geodesic flow. Moreover, we have and , that is, holds.

Next, we consider Ricci solitons , with potential field of constant length. Note that if is compact and is a constant, then is trivial, the argument goes as follows: in this case, for a smooth function , and as is compact, there is point (the critical point of ), where . As , a constant, that will give , that is, is trivial.

We get the following characterization of noncompact trivial Ricci solitons with potential field having constant length.

Theorem 4. Let be an -dimensional connected noncompact Ricci soliton with a constant length of potential field. Then, is trivial if and only if the associated tensor satisfies the inequality

Proof. Suppose is an -dimensional Ricci soliton with a constant and As is a constant, using equation (5), we conclude Now, and using equations (5) and (8), we get Taking divergence in equation (63), and using the above equations, we conclude Also, the inner product with in equation (63) gives , and consequently, the above equation becomes Using the Schwarz inequality and the inequality (62), in the above equation, we conclude that which, as in the proof of Theorem 2, implies that is trivial.
Converse follows on the similar lines as in Theorem 2.

We construct an example of a nontrivial Ricci soliton with a nonconstant length of potential. Let be the unit open ball in the Euclidean space , where is the complex structure and is the Euclidean metric. Consider the smooth vector field defined by where is the position vector field. Then, it follows that that is,

Hence, is a nontrivial Ricci soliton with and associated tensor . We get and , that is, .

Data Availability

No data have been used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Acknowledgments

The authors extend their appreciations to the Deanship of Scientific Research at King Saud University for funding this work through research group no. (RG-1440-142).