Journal of Mathematics
Volume 2021 (2021), Article ID 9958573, 7 pages
https://doi.org/10.1155/2021/9958573

Research Article

Study on Twisted Product Almost Gradient Yamabe Solitons

Byung Hak Kim,¹ Jin Hyuk Choi,² and Sang Deok Lee³

¹Department of Applied Mathematics and Institute of Natural Sciences, Kyung Hee University, Yongin 17104, Republic of Korea
²Humanitas College, Kyung Hee University, Yongin 17104, Republic of Korea
³Department of Mathematics, Dankook University, Cheonan 31116, Republic of Korea

Correspondence should be addressed to Jin Hyuk Choi; jinhchoi@khu.ac.kr

Received 5 March 2021; Accepted 26 April 2021; Published 13 May 2021

Academic Editor: Antonio Masiello

Copyright © 2021 Byung Hak Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we first study gradient Yamabe solitons on the twisted product spaces. Then, we classify and characterize the warped product and twisted product spaces with almost gradient Yamabe solitons. We also study the construction of almost gradient Yamabe solitons in the Riemannian product spaces.

1. Introduction

The notion of Yamabe flow was introduced by Hamilton [1] in 1989, which is defined on a Riemannian manifold as , where is the Riemannian metric on and is the scalar curvature of . The significance of Yamabe flow lies in the fact that it is a natural geometric deformation to metrics of constant scalar curvature. A Riemannian manifold is called a Yamabe soliton if there exist a smooth vector field and constant such that

where is the Lie derivative with respect to the vector field . The Yamabe soliton is called shrinking if , steady if , and expanding if . When for some function on , we say that is a gradient Yamabe soliton with a potential function . In this case, equation (1) becomes

[2–7]. In (2), if is a function on , then is called an almost gradient Yamabe soliton with [8]. The Yamabe soliton (resp., gradient Yamabe soliton) is said to be trivial if is killing (resp., is a constant).

The warped product or twisted product metric on the product manifold of two Riemannian manifolds and is given by , where is a positive function with for a warped product metric and for a twisted product metric.

It was known [6] that the metric of any compact Yamabe soliton is a metric of constant scalar curvature when the dimension of the manifold . In 2011, Cao and two coauthors [3] studied classification theorems for complete nontrivial locally conformally flat gradient Yamabe soliton. In [9], they found sufficient conditions on the soliton vector field under which the metric of a Yamabe soliton is a Yamabe metric, that is, a metric of constant scalar curvature. Moreover, in [8], we can see the various examples of compact and noncompact almost gradient Yamabe soliton. The present authors [4] studied gradient Yamabe soliton in the warped product manifolds and admittance of gradient Yamabe solitons and geometric structures for some model spaces. In 2019, Karaca [9] obtained a classification theorem regarding a gradient Yamabe soliton on multiplying warped product space with splitting potential function. With respect to the result, we obtain similar classification theorems for the warped product or the twisted product space with almost gradient Yamabe soliton. Especially considering the condition of the conformal flatness, the classification and characterizations of the space were greatly helped. In addition to these, there are many works on solitons on the twisted product spaces [10–14] and Yamabe solitons [2, 5–9].

From this point of view, the purpose of this paper is to get a more generalized classification theorem of the results which is already published on the warped and twisted product space for a gradient Yamabe soliton and almost gradient Yamabe soliton. This paper is organized as follows. In Section 2, we discuss gradient Yamabe soliton in the twisted product space. Sections 3–5 are devoted to studying almost gradient Yamabe soliton in the Riemannian, warped, and twisted product spaces.

2. Gradient Yamabe Solitons in the Twisted Product Spaces

In this section, we consider the case that the twisted product space of n-dimensional Riemannian manifold and p-dimensional Riemannian manifold is a gradient Yamabe soliton with . Let and are Riemannian connections in and , respectively. Then, we have

where and are local coordinate systems in and , respectively. Moreover, we have put , , and .

Assume that the potential function is decomposed by for some functions and on and , respectively. Then, from equation (3), we see that , so becomes a function on because is a constant. Moreover, we obtain

and since and are functions on , the quantity is a function on . Hence, by using the first equation of (3), we get

where we have put ; that is, becomes an almost gradient Yamabe soliton. Thus, we have the following theorem.

Theorem 1. If the twisted product manifold is gradient Yamabe soliton with and , for some functions and on and , respectively, then the base space becomes an almost gradient Yamabe soliton with and depends only on .
In [15], the authors proved the following theorem.

Theorem 2. If the twisted product manifold is conformally flat and and , then is the warped product space of and .
In [4], the present authors proved the following theorem.

Theorem 3. If the product manifold is a gradient Yamabe soliton, then , and become trivial gradient Yamabe soliton. This means that there is a nontrivial gradient Yamabe soliton in the Riemannian product manifold.

In the proof process of Theorem 2, the authors derived that is the product of certain functions on and on , respectively. In this sense, if we consider the conformally flat twisted product space with gradient Yamabe soliton and , then we get

where and . From the first and second equations of (6), we see that is a function only on and is constant or . Since is a positive function, we see that is constant or is constant. Let us consider the first case, that is, is constant. Then, depends only on and that becomes a Riemannian product metric. Hence, , , and become trivial gradient Yamabe soliton due to Theorem 3. Moreover, from the third equation of (6), we get

and that depends only on because the quantities of right-hand side of (7) depend only on . Therefore, becomes a constant and that becomes a constant due to the first equation of (6). On the other hand, if is constant, then the potential function becomes a function on so that becomes an almost gradient Yamabe soliton by Theorem 1 and the first and third equations of (6). Thus, we have the following theorem.

Theorem 4. Let the twisted product manifold be a gradient Yamabe soliton and conformally flat. If for some functions and on and , respectively, and and , then depends only on , and is one of the following two cases:

(1) is a Riemannian product of and , and that , , and become trivial gradient Yamabe soliton, and and become constant(2)The potential function depends only on , and becomes an almost gradient Yamabe soliton

In 1965, Tashiro [16] proved that the following theorem.

Theorem 5. Let be an -dimensional complete Riemannian manifold of dimension and suppose it admits a special concircular field satisfying the equation

for constants and . Then, becomes either direct product of an -dimensional complete Riemannian manifold with a straight line when or a Euclidean space when but .

Hence, if we combine Theorems 4 and 5, then we can state the following theorem.

Theorem 6. Let the twisted product manifold be a gradient Yamabe soliton and conformally flat. If and and , then we have two cases as follows:

(i) becomes a Riemannian product space of an -dimension complete Riemannian manifold with a straight line when or becomes a Euclidean space when . In any case, , and become trivial gradient Yamabe soliton.(ii)The potential function depends only on , and becomes an almost gradient Yamabe soliton.

3. Almost Gradient Yamabe Solitons in the Riemannian Product Spaces

Tokura and other coauthors [7] introduced various examples of an almost gradient Yamabe soliton in . In this section, we consider the relation between the structure of an almost gradient Yamabe soliton with in the Riemannian product manifold of and and the structure of an almost gradient Yamabe soliton in and . If the potential function is expressed by for some functions and on and , respectively, then we have

Theorem 7. Let the Riemannian product manifold be an almost gradient Yamabe soliton with with for some functions and in and , respectively. Then, is a constant on , becomes an almost gradient Yamabe soliton with , and becomes an almost gradient Yamabe soliton with and . In this case, and become constants on and , respectively.

Proof. From the first and third equations of equation (9), we can see that is a quantity on and , respectively. Hence, becomes constant. Since is an almost gradient Yamabe soliton with for some function and on and , respectively, we get and from equation (9). If we put and by and , then and . Therefore, and become functions on and , respectively. Hence, we can see that and become an almost gradient Yamabe soliton with and , respectively. Moreover, we obtain . Hence, we obtain and that and are also constants.
For the converse case of Theorem 7, if we assume that and are almost gradient Yamabe solitons with and , respectively, and , then we get and . If we take and put , then we can derive and ; that is, becomes an almost gradient soliton with , where . Thus, we have the following theorem.

Theorem 8. Let be an almost Yamabe soliton with and be an almost Yamabe soliton with . If , then becomes an almost Yamabe soliton with .

If we combine Theorems 7 and 8, we can state the following theorem.

Theorem 9. The Riemannian product space with is almost gradient Yamabe soliton for some functions and on and , respectively, if and only if and are almost gradient Yamabe soliton with and , respectively, and .

If we combine Theorems 5 and 7, then we can state the following theorem.

Theorem 10. If the complete Riemannian product manifold is an almost gradient Yamabe soliton with , then is a constant on and that becomes one of the following:

(a)If , then of an -dimensional complete Riemannian manifold with a straight line (b)If , then a Euclidean space

Proof. We see that is a constant by Theorem 7. Then, from the first equation of (7), for some constant . Hence, we can prove Theorem 10 by using Theorem 1.
From Theorems 7 and 10, we see that if the constant , then the product manifold becomes of an -dimension complete Riemannian manifold with the first equation of (9), . Moreover, becomes an almost gradient Yamabe soliton with and because is constant and . Hence, the -dimension Riemannian manifold becomes of an -dimensional Riemannian manifold with from Theorem 7, that is, . Hence, becomes , where we put . Hence these facts, Theorems 5 and 7 give the followinf theorem

Theorem 11. Let the complete Riemannian product manifold be an almost gradient Yamabe soliton with . Then, becomes a constant. If the constant , then becomes of an -dimensional complete Riemannian manifold with a 2-dimensional Euclidean space .

4. Almost Gradient Yamabe Solitons in the Warped Product Spaces

In this section, we study that the warped product space of is an almost gradient Yamabe soliton with . Then, we have

and that

Theorem 12. If the warped product space is an almost gradient Yamabe soliton with , then we have the following:

(a)If for all , then , , and is either a Riemannian product of and or the potential function is a constant(b)If for all , then is an almost gradient Yamabe soliton

Proof.

(a)If , then becomes a function on , , and from the first and third equations of (11). Moreover, we get from the second equation of (11); that is, is either a Riemannian product of and or the potential function is a constant.(b)From the first equation of (11), we see that becomes a function on . Moreover, we have from the fourth equation of (3) and the first equation of (11). If we put , then we obtain and that is a function on . Hence, becomes an almost gradient Yamabe soliton.

Consider the case of for some functions and on and , respectively. Then, from equations (3) and (11), we obtain

Hence, we can see that the function or is constant from the second equation of (12). If the function is constant, then becomes a product space; henceforth, we can apply Theorem 9.
On the other hand, if the function is a constant, then for all . Hence, if we apply this fact to Theorem 12 (b), then we have.

Theorem 13. If the warped product space is an almost gradient Yamabe soliton with for some functions and in and , respectively, then we have two cases as follows:

(i) is either of an -dimensional complete Riemannian manifold with a 2-dimensional Euclidean space if the constant or a Euclidean space if the constant (ii) is an almost gradient Yamabe soliton with

5. Almost Gradient Yamabe Solitons in the Twisted Product Spaces

Let the twisted product space be an almost gradient Yamabe soliton with . Then, we have

Assume that the potential function is decomposed by for some functions and on and , respectively. Then, from the first equation of (13), we see that ; that is, is a function on . Moreover, we obtain

and since and are functions on , the quantity is a function on . Hence, by use of the first equation of (11), we get

and that becomes an almost gradient Yamabe soliton. Thus, we have the following theorem.

Theorem 14. If the twisted product manifold is an almost gradient Yamabe soliton with and , then the base space becomes an almost gradient Yamabe soliton and is a function on .
If the conformally flat twisted product space is an almost gradient Yamabe soliton with and for some functions and on and , respectively, then we get

and and are functions on and , respectively, which come from the conformally flatness and the Proof of Theorem 2. From the first and second equations of (16), we see that is a function only on , and is constant or , respectively,. Since is a positive function, we see that is a constant or is a constant.
Let us consider the first case, that is, is a constant. Then, depends only on and that becomes a Riemannian product metric. Hence, , , and become trivial gradient Yamabe solitons due to Theorem 3. Moreover, from the third equation of (16), we get

and that depends only on because the quantities of right-hand side of (17) depend only on . Therefore, becomes a constant and that becomes a constant due to the first equation of (16). Moreover, we see that where we have put . Then, we can apply this fact to Theorem 5, and we see that becomes either direct product of an (n-1)-dimensional complete Riemannian manifold with a straight line when or a Euclidean space when .
On the other hand, if is constant, then the potential function becomes a function on and becomes an almost gradient Yamabe soliton by Theorem 14. Thus, we have the following theorem.

Theorem 15. Let the twisted product manifold be an almost gradient Yamabe soliton with and conformally flat. If and and , then depends only on , and is one of the following two cases:

(1) is either direct product of an (n−1)-dimensional complete Riemannian manifold with a straight line when or a Euclidean space when moreover and become constants(2)The potential function depends only on , and becomes an almost gradient Yamabe soliton.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts interest.

Acknowledgments

The first author was supported by the National Research Foundation of Korea (NRF-2017 R1E1A1A 03071005), and the third author was supported by the 2020 Research Fund of Dankook University.

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