Characterizations of Trivial Ricci Solitons

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.


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 n-dimensional Ricci soliton ðM, g, u, λÞ is a Riemannian manifold ðM, gÞ on which there is a smooth vector field u (called potential field) satisfying (cf. [1]), where Ric is the Ricci tensor, £ u g is the Lie derivative of the metric g with respect to u, and λ is a constant. A Ricci soliton ðM, g, u, λÞ is said to be expanding, stable, or shrinking depending on λ < 0, λ = 0, or λ > 0, respectively. If the potential field u = ∇f is a gradient of a smooth function f , then ðM, g,∇f , λÞ is called a gradient Ricci soliton, and in this case, equation (1) takes the form where H f is the Hessian of the function f . 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 impor-tance. One of the important findings on Ricci solitons is that if it is compact, the potential field u is a gradient of a smooth function f , that is, a compact Ricci soliton is a gradient Ricci soliton (cf. [1]). A Ricci soliton ðM, g, u, λÞ is said to be trivial if £ u g = 0, and in this case, the metric g becomes an Einstein metric with λ becoming the Einstein constant. Several authors have studied the geometry of Ricci solitons (cf. [2][3][4]); in [5][6][7], Myers-type theorems have been proved for Ricci soliton; similarly in [8], it has been observed that a complete shrinking Ricci soliton ðM, g, u, λÞ 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 ðM, g, u, λÞ 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 S of a compact Ricci soliton ðM, g, u, λÞ satisfying a differential inequality involving the first nonzero eigenvalue λ 1 of the Laplace operator gives a characterization of a trivial Ricci soliton (cf. Theorem 1).
We also show that for a connected Ricci soliton ðM, g, u, λÞ the flow of potential field u being geodesic flow with its length kuk satisfying certain inequality gives a characterization of a trivial Ricci soliton (cf. Theorem 2). Finally, it is observed that potential field u being of constant length satisfying certain inequality on a connected Ricci soliton ðM, g, u, λÞ also gives a characterization of a trivial Ricci soliton (cf. Theorem 4).

Preliminaries
Let ðM, g, u, λÞ be an n-dimensional Ricci soliton and α be smooth 1-form dual to the potential field u. We define a skew symmetric tensor field ψ on the Ricci soliton ðM, g, u, λÞ by where XðMÞ is the Lie algebra of smooth vector fields on M. We call this tensor field ψ the associated tensor field of the Ricci soliton ðM, g, u, λÞ. The Ricci operator Q on the Ricci soliton ðM, g, u, λÞ is a symmetric operator defined by RicðX, YÞ = gðQX, YÞ, X, Y ∈ XðMÞ. The gradient ∇S of the scalar curvature S = TrQ satisfies where fe 1 , ⋯, e n g is a local orthonormal frame and the covariant derivative ð∇QÞðX, YÞ = ∇ X QY − Qð∇ X YÞ.
Using equations (1) and (3) and Koszul's formula, the covariant derivative of the potential field u is given by Now, using equation (5), we get the following expression for Riemannian curvature tensor of the Ricci soliton ðM, g, u, λÞ: As the operator Q is symmetric and ψ is skew-symmetric, using equations (4) and (6), we obtain which leads to We denote by λ 1 the first nonzero eigenvalue of the Laplace operator Δ acting on smooth functions on compact ðM, g, u, λÞ.
then by minimum principle, we have ð

A Characterization of Compact Trivial Ricci Solitons
Now, we prove the first result of this paper.

Theorem 1.
An n-dimensional complete shrinking Ricci soliton ðM, g, u, λÞ with Ricci curvature bounded below by a constant c > 0 and first nonzero eigenvalue λ 1 of the Laplacian operator is trivial if and only if the scalar curvature S satisfies the inequality Proof. Suppose ðM, g, u, λÞ is a complete shrinking Ricci soliton with Ricci curvature satisfying Ric ≥ c > 0 and the scalar curvature S satisfies the inequality Note that the assumption on the Ricci curvature in view of Myers' theorem implies that M is compact. Thus, ðM, g, u, λÞ is a compact Ricci soliton, and therefore, it is a gradient Ricci soliton (cf. [1]). Consequently, u is a closed vector field, that is, ψ = 0. Equation (8) takes the form which gives Moreover equation (5) becomes which we use to compute the divergence of Qu and obtain Now, using equation (13) in the above equation leads to which on integrating gives

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Using equation (15), we have div u = nλ − S, which gives ð and consequently, we conclude ð Thus, equation (18) takes the form Now, equations (13) and (16) imply which together with div Su = uðSÞ + Sðnλ − SÞ gives Integrating the above equation, we conclude ð that is, Now, using equation (20) in the above equation yields Thus, equations (21) and (27) imply Also, we have Bochner's formula ð where A S ðXÞ = ∇ X ∇S is the Hessian operator of the scalar curvature S. Note that equation (19) implies Ð M ðS − nλÞ = 0, 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 λ > 0 (for a shrinking Ricci soliton) and the inequality (30) gives Thus, Since the Ricci curvature satisfies Ric ≥ c for a constant c > 0, the above inequality takes the form 3 Advances in Mathematical Physics Using the Schwarz inequality kA S k 2 ≥ ð1/nÞðΔSÞ 2 , and the inequality (12) in the above inequality, we conclude Also, the equality in the Schwarz inequality holds if and only if A S = ðΔS/nÞI. Moreover, the equation ∇S = 2λu in view of equation (15) implies Consequently, using A S = ðΔS/nÞI, we get that is, QðXÞ = ðS/nÞX. Now, using QðXÞ = ðS/nÞX with equation (13) and first equation in equation (37), we get S = nλ, that is, Q = λI. Hence, Ric = λg and the Ricci soliton ðM, g, u, λÞ is trivial. Conversely, if ðM, g, u, λÞ is a trivial soliton, then Ric = λg, λ > 0 gives S = nλ which implies ΔS = 0, and consequently, the equality (12) holds.
It is well known that the odd-dimensional unit sphere S 2n+1 with induced metric g as a hypersurface of the Euclidean space ðC n+1 , h,iÞ admits a unit Killing vector field u, and consequently, we have the trivial Ricci soliton ðS 2n+1 , g, u, λÞ, λ = 2n, satisfying the hypothesis of Theorem 1.

Characterizations of Connected Trivial Ricci Solitons
In this section, we consider a connected Ricci soliton ðM, g, u, λÞ and find necessary and sufficient conditions under which it is a trivial Ricci soliton. Recall that the local flow fϕ t g of a smooth vector field u on a Riemannian manifold ðM, gÞ is said to be geodesic flow if the orbits of fϕ t g are geodesics on ðM, gÞ. 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 S 3 which has a Sasakian structure ðϕ, u, α, gÞ (cf. [14]). Then for a positive function f on S 3 , deform the metric g by Then, u is still a unit vector field on the Riemannian manifold ðS 3 , gÞ. However, u is no more a Killing vector field on ðS 3 , gÞ but instead ðψ, u, α, gÞ is a trans-Sasakian structure [15], and the flow of u on the Riemannian manifold ðS 3 : gÞ is a geodesic flow.
In the next result, we use this notion of geodesic flow for the potential field u of the Ricci soliton ðM, g, u, λÞ to characterize trivial Ricci solitons. Theorem 2. Let ðM, g, u, λÞ be an n-dimensional connected shrinking Ricci soliton with the local flow of potential field u be the geodesic flow. Then, ðM, g, u, λÞ is trivial Ricci soliton if and only if the scalar curvature S is a constant along the integral curves of u and the associated tensor ψ satisfies the inequality Proof. Suppose ðM, g, u, λÞ is connected with local flow of u a geodesic flow and the scalar curvature S is a constant along the integral curves of u and the associated tensor ψ satisfies As the local flow of u is a geodesic flow, equation (5) gives As the scalar curvature S is a constant along the integral curves of u, using equations (4) and (8), we conclude Now, using equations (5) and (44), we find the divergence of the vector field Qu. After some straight forward computations, we get Similarly, using equations (5) and (45), we get Equation (43) gives Ricðu, uÞ = λkuk 2 , which on inserting in the above equation yields Note that equation (5) gives div u = ðnλ − SÞ. Consequently, on taking divergence in equation (43) and using equations (46) and (48), we conclude Advances in Mathematical Physics which gives Using the Schwarz inequality kQk 2 ≥ ð1/nÞS 2 and inequality (42), in the above equation, we conclude Since the equality in the Schwarz inequality holds if and only if Q = ðS/nÞI, we get Ric = λg, that is, ðM, g, u, λÞ is trivial.
Conversely, if ðM, g, u, λÞ is a trivial Ricci soliton with local flow of u a geodesic flow, then it follows that S is a constant and equation (5) takes the form ∇ X u = ψX and ψu = 0: ð52Þ Then finding the divergence of ψu using above equation, gives the equality Remark 3.
(1) It is clear that an odd-dimensional unit sphere ðS 2n+1 , g, u, λÞ is a trivial Ricci soliton, where λ = 2n, the potential field u = −JN, J being the complex structure on C n+1 and N is the unit normal to the hypersurface S 2n+1 . The associated tensor ψ is given by ψX = ðJXÞ T , the tangential component of JX. It follows that kψk 2 = 2n = λkuk 2 holds. Naturally, u being the Killing vector field, its flow consists of isometries of S 2n+1 , and therefore, it is a geodesic flow.
(2) Next, we give an example of a nontrivial Ricci soliton with the flow of potential field u not a geodesic field. Consider the open subset of the Euclidean space ðR n , gÞ, where g is the Euclidean metric. Consider the vector field u ∈ XðMÞ defined by where is the position vector field and x 1 , ⋯, x n are the Euclidean coordinates on M. It follows that Hence, we have that is, ðM, g, u, λÞ, λ = 1 is a nontrivial Ricci soliton with associated tensor field ψ, given by The flow fφ t g of u is given by which is not a geodesic flow. Moreover, we have kψk 2 = 2 and kuk 2 = kΨk 2 + ðx 1 Þ 2 + ðx 2 Þ 2 , that is, kψk 2 < λkuk 2 holds.
Next, we consider Ricci solitons ðM, g, u, λÞ, with potential field u of constant length. Note that if M is compact and kuk is a constant, then ðM, g, u, λÞ is trivial, the argument goes as follows: in this case, u = ∇h for a smooth function h, and as M is compact, there is point p ∈ M (the critical point of h), where u p = 0. As kuk = c, a constant, that will give u = 0, that is, ðM, g, u, λÞ is trivial.
We get the following characterization of noncompact trivial Ricci solitons with potential field u having constant length.
Theorem 4. Let ðM, g, u, λÞ be an n-dimensional connected noncompact Ricci soliton with a constant length of potential field. Then, ðM, g, u, λÞ is trivial if and only if the associated tensor ψ satisfies the inequality Proof. Suppose ðM, g, u, λÞ is an n-dimensional Ricci soliton with kuk a constant and As kuk 2 is a constant, using equation (5), we conclude Now, div u = nλ − S and using equations (5) and (8), we get

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Taking divergence in equation (63), and using the above equations, we conclude − ψ k k 2 + Ric u, u ð Þ= nλ 2 − 2λS + Q k k 2 : Also, the inner product with u in equation (63) gives Ricðu, uÞ = λkuk 2 , and consequently, the above equation becomes Using the Schwarz inequality and the inequality (62), in the above equation, we conclude that Q k k 2 = 1 n S 2 , which, as in the proof of Theorem 2, implies that ðM, g, u, λÞ 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 M be the unit open ball M = x ∈ C n : x k k < 1 f g ð68Þ in the Euclidean space ðC n , J, gÞ, where J is the complex structure and g is the Euclidean metric. Consider the smooth vector field u ∈ XðMÞ defined by where is the position vector field. Then, it follows that that is, Hence, ðM, g, u, λÞ is a nontrivial Ricci soliton with λ = 1 and associated tensor ψ = J. We get kψk 2 = 2n and kuk 2 = 2 kΨk 2 < 2, that is, λkuk 2 < kψk 2 .

Data Availability
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Conflicts of Interest
The authors declare no conflict of interest.