Abstract
In this note, we characterize Sasakian manifolds endowed with --Ricci-Yamabe solitons. Also, the existence of --Ricci-Yamabe solitons in a 5-dimensional Sasakian manifold has been proved through a concrete example.
1. Introduction
In 1982 (resp., 1988), Hamilton introduced the idea of Ricci flow [1] (resp., Yamabe flow [2]). On a smooth Riemannian (or semi-Riemannian manifold), the Yamabe flow is determined as the evolution of the Riemannian (or semi-Riemannian) metric at time to using the following equation: where refers to the scalar curvature of the metric . In case , the Yamabe and Ricci flows are related as in the following equation: where defines the Ricci tensor. Thus, for the case , there is not such an equivalence, since the Yamabe flow preserves the conformal class of metric but generally this is not true.
The solutions of both Ricci and Yamabe flows are presented as Ricci and Yamabe solitons, respectively. On a Riemannian manifold , the Ricci and Yamabe solitons are defined by respectively, where is the Lie derivative operator along vector field (called soliton vector field) at and , where is the set of real numbers. Recently in 2018, Deshmukh and Chen ([3, 4]) briefly studied Yamabe solitons to find sufficient conditions at the soliton vector field so that the metric of the Yamabe soliton is of constant scalar curvature. Yamabe solitons have also been studied in ([5–8]) and many others.
In 2019, Ricci-Yamabe flow, as a new class of geometric flows of the type , was presented by Güler and Crasmareanu [9] and defined as
After Güler and Crasmareanu, Dey [10] proposed the concept of Ricci-Yamabe solitons; according to him, the Ricci-Yamabe soliton of the type is a Riemannian manifold that admits where . In addition, it is noted that Ricci-Yamabe solitons of types and are known as -Ricci solitons and -Yamabe solitons, respectively.
The concept of -Ricci soliton was investigated by Kaimakamis and Panagiotidou [11] in case of real hypersurfaces at complex space forms. More specifically, it is noted that the concept of -Ricci tensor was presented firstly by Tachibana [12] in almost Hermitian manifolds, and later by Hamada [13] to consider different case which is the real hypersurfaces of nonflat complex space forms. The Riemannian metric on the smooth manifold is named the -Ricci soliton in case , a smooth vector field and obeying: where for every vector fields on , as well as and are the -Ricci operator and the -Ricci tensor, respectively. In this connection, we recommend the papers ([14–21]) for the specific contents regarding Ricci, -Ricci, and -Ricci solitons in case of contact Riemannian geometry. In [22], the authors studied gradient Yamabe, gradient Einstein, and quasi-Yamabe solitons on almost co-Kähler manifolds.
Recently, Dey and Roy [23] presented the concept of --Ricci soliton in Sasakian manifolds. The Riemannian manifold is named --Ricci soliton in case
Motivated by previous studies, we introduce the notion of --Ricci-Yamabe soliton of type which is a Riemannian manifold satisfying for , . The --Ricci-Yamabe soliton is described as shrinking, steady or expanding if it admits the soliton vector for which or , respectively. Particularly, if , then this concept of --Ricci-Yamabe soliton reduces to a concept of -Ricci-Yamabe soliton .
Throughout the paper, we denote a -dimensional Sasakian manifold by , -Ricci-Yamabe soliton by -RYS, and --Ricci-Yamabe soliton by --RYS. We present our work as follows: Section 2 includes essential results and some basic definitions of Sasakian manifolds. Section 3 covers the study of --RYS on leading to several significant characterizations of the manifold. Section 4 deals with the study of pseudo-Ricci-symmetric and Ricci recurrent admitting --RYS. The --RYS on satisfying the curvature conditions and have been studied in Sections 5 and 6, respectively.
2. Preliminaries
A -dimensional differentiable manifold is said to admit an almost contact structure, sometimes called a -structure, in case it admits a (1,1) type tensor field , a structure vector field , and a 1-form satisfying [24]
The almost contact structure is called normal in case , where is the Nijenhuis tensor of Considering the Riemannian metric tensor that is defined on and satisfies for any where refers to the set of all smooth vector fields of . The structure is named the almost contact metric structure. Next, considering , the tensor field of type as . In case , then the structure is named as normal metric structure. The normal contact metric structure is named Sasakian structure satisfying ([25–27]): for any where stands for the Levi-Civita connection.
In case of , we have
for any ; and refers to the curvature tensor and the Ricci operator.
Definition 1. A Sasakian manifold is called an -Einstein in case the non-vanishing Ricci tensor is expressed as where . In particular, if , then is named as an Einstein manifold.
Definition 2. The vector field is named as an affine conformal vector field in case it satisfies [28] where . In case , then is called an affine vector field.
Lemma 3. The -Ricci tensor of is given by [14] for any
3. --Ricci-Yamabe Solitons on Sasakian Manifolds
First, we prove the following:
Theorem 4. An admitting --RYS is an -Einstein manifold of the constant scalar curvature. Moreover, the scalars , related to each other by .
Proof. Let the metric of an be --RYS , then Equation (9) turns to
for all vector fields as well on . Using (16), Equation (21) leads to
Using (20), (22) takes the form
where and .
By putting at (23) as well the use of (10) and (11), we have
where .
In view of (15), from (24), it follows that
On contracting (23), we find , which by using the values of , and (25) leads to
where and are constants. Thus, (23) together with (25) and (26) leads to the statement of Theorem 4.
Particularly, taking in (23) as well in (25) resulted in and , respectively, being . Thus, we have the following.
Corollary 5. An admitting -RYS is an -Einstein manifold, and the soliton is shrinking, steady or expanding according to or , respectively.
Next, we prove the following.
Theorem 6. If an admits --RYS such that the vector field represents an affine conformal vector field. Then, is an -Einstein manifold, and is an affine vector field.
Proof. The use of (20) in (9) gives Referencing Yano [29], the expression is well-known for all at . As is parallel respecting to , the previous equation turns to as a result of (19), it leads to Taking the covariant derivative of (27) respecting to and using (17), we have Putting in (31) and using (10), (11), (15), and (30), we get From (30)–(32), we find which by replacing gives Now, the covariant differentiation of (15) yields From (34) and (35), it follows that By replacing by in (36) and using (10), we get The contraction of (37) gives Therefore, from (32), it follows that . This implies that ; therefore, is an affine vector field. This completes the proof.
Furthermore, we prove the following.
Lemma 7. An satisfies the following equations: where refers to the Ricci operator.
Proof. Differentiating along and using (16), we get (38). Next, differentiating (14) along and using (16), we find Taking a frame field and then contracting (40), we get From Bianchi’s second identity, we can easily obtain that By equating (41) and (42), then using (38), Equation (39) follows.
Now, we prove the next theorem:
Theorem 8. If an admits --RYS such that the vector field represents the gradient of defined by (9), then either is a pointwise collinear with the structure vector field or .
Proof. Suppose an admits --RYS such that the vector field represents the gradient of , i.e., . Then, from (9), we find
for any on .
The covariant differentiation of (43) respecting to and the use of (16) and (17) leads to
Interchanging and in (44), we have
In view of (43), we also have
From (44)–(46), we get
By replacing by in (47) and using (10), (13), (38), and (39), we get
The inner product of (48) with leads to
Therefore, we have either or , that is, is pointwise collinear with . The proof is completed.
4. Pseudo-Ricci-Symmetric and Ricci-Recurrent Sasakian Manifolds Admitting --Ricci-Yamabe Solitons
Definition 9. The non-flat is named pseudo-Ricci-symmetric and is represented by , in case the Ricci tensor of the manifold satisfies the condition [30] where the non-zero 1-form is given by , vector fields being the vector field that corresponds to the associated 1-form . In particular, if , then is called Ricci-symmetric.
The covariant derivative of (23) leads to
Now, using (23) and (51), (50) becomes
Choosing , (52) reduces to which by putting gives . This implies that . Thus, we have te following.
Theorem 10. A pseudo-Ricci-symmetric admitting --RYS is Ricci-symmetric.
Definition 11 [31]. An is named as Ricci-recurrent in case there exists a 1-form holds: for all and on and 1-form
By the use of (51) in (53), we find which by putting then using (10) and (15) reduces to
By taking , (55) takes the form
Now, replacing by in (56) and using (10), we find
Since , therefore, we obtain . This leads to Hence, by the use of (25), we have Therefore, we give the next theorem.
Theorem 12. If a Ricci-recurrent admits --RYS, then as well .
Hence, by using these values of and in (23), we obtain
Thus, we state:
Corollary 13. A Ricci-recurrent admitting a --RYS defines an Einstein manifold.
5. Sasakian Manifolds Admitting --Ricci-Yamabe Solitons Satisfying
Considering an admitting --RYS which satisfies , this implies that for all on In view of (23) and the symmetries of , (59) takes the form which by taking then using (10) and (11) turns to
From (61), it follows that , which leads to ; hence, (25) gives This helps us to state:
Theorem 14. For an admitting --RYS that satisfies , we have and .
Now by using and , (23) takes the form
Thus, we have:
Corollary 15. In case an satisfies and admits --RYS, then it defines an Einstein manifold.
6. Sasakian Manifolds Admitting --Ricci-Yamabe Solitons Satisfying
Let an admitting --RYS satisfies where
This can be expressed as where being used.
From (63), (65), and (23), we get
From the preceeding equation, it follows that . This implies that . Hence, from (25), we get Thus, we have
Theorem 16. If an admits --RYS and the manifold satisfies , then and .
Now, by using these values of as well , (23) yields
Thus, we give the next corollary:
Corollary 17. In case an admitting --RYS satisfies , then it is an Einstein manifold.
Example 1. Let a manifold of dimension 5, where refer to the usual coordinates at . Suppose , , , and are the vector fields at defined as and these are linearly independent at each point of .
Suppose is the Riemannian metric defined as
Considering , a 1-form on determined as of all . Let be a tensor field on defined by
The linearity of and leads to for all . Therefore and for others and . By using well-known Koszul’s formula, we can easily calculate
It can be easily verified that the manifold satisfies
It is clear that this manifold is a Sasakian manifold.
It is easy to have the following non-vanishing components:
Utilizing the previous results we calculate the following:
Using (23), we have . By equating both the values of , we obtain
Hence, as well insures Equation (25), and so, is the --RYS on the given -dimensional Sasakian manifold.
Data Availability
No data were used to support this study.
Conflicts of Interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Acknowledgments
The third author (Fatemah Mofarreh) expresses her gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.