Study on Twisted Product Almost Gradient Yamabe Solitons

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.


Introduction
e notion of Yamabe flow was introduced by Hamilton [1] in 1989, which is defined on a Riemannian manifold (M, g) as (zg/zt) � − rg, where g is the Riemannian metric on M and r is the scalar curvature of M. e significance of Yamabe flow lies in the fact that it is a natural geometric deformation to metrics of constant scalar curvature. A Riemannian manifold (M, g) is called a Yamabe soliton if there exist a smooth vector field X and constant ρ such that where L X is the Lie derivative with respect to the vector field X. e Yamabe soliton is called shrinking if ρ > 0, steady if ρ � 0, and expanding if ρ < 0. When X � ∇h for some function h on M, we say that M is a gradient Yamabe soliton with a potential function h. In this case, equation (1) becomes [2 -7]. In (2), if ρ is a function on M, then M is called an almost gradient Yamabe soliton with (h, ρ) [8]. e Yamabe soliton (resp., gradient Yamabe soliton) is said to be trivial if X is killing (resp., h is a constant). e warped product or twisted product metric g on the product manifold of two Riemannian manifolds (B, g) and (F, g) is given by g � g 0 0 f 2 g , where f is a positive function with f: B ⟶ R + for a warped product metric and f: B × F ⟶ R + 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 n ≥ 3. 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. e 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][11][12][13][14] and Yamabe solitons [2,[5][6][7][8][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. is 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.

Gradient Yamabe Solitons in the Twisted Product Spaces
In this section, we consider the case that the twisted product space M � B× f F of n-dimensional Riemannian manifold (B, g) and p-dimensional Riemannian manifold (F, g) is a gradient Yamabe soliton with (h, ρ). Let ∇ and ∇ are Riemannian connections in B and F, respectively. en, we have where u a : a � 1, 2, . . . , n and u x : x � n + 1, 2, . . . , n + p are local coordinate systems in B and F, respectively. Moreover, we have put h a � (zh/zu a ), f c � (zf/zu c ), and f x � (zf/zu x ).
Assume that the potential function h is decomposed by h � k + l for some functions k and l on B and F, respectively. en, from equation (3), we see that (r − ρ)g ab � ∇ b k a , so r − ρ becomes a function on B because ρ is a constant. Moreover, we obtain and since r − ρ and r are functions on B, the quantity − (r/ Hence, by using the first equation of (3), we get where we have put In [15], the authors proved the following theorem.

Theorem 2.
If the twisted product manifold M � B× f F is conformally flat and p ≠ 1 and n ≠ 1, then M is the warped product space B× f * F * of B and F * .
In [4], the present authors proved the following theorem.

Theorem 3. If the product manifold M � B × F is a gradient Yamabe soliton, then B, F, and M become trivial gradient Yamabe soliton. is means that there is a nontrivial gradient Yamabe soliton in the Riemannian product manifold.
In the proof process of eorem 2, the authors derived that f is the product of certain functions f * on B and f on F, respectively. In this sense, if we consider the conformally flat twisted product space with gradient Yamabe soliton and h � k + l, then we get where f � f * f and l x � (zl/zu x ). From the first and second equations of (6), we see that r − ρ is a function only on B and depends only on F and that g becomes a Riemannian product metric. Hence, B, F, and M become trivial gradient Yamabe soliton due to eorem 3. Moreover, from the third equation of (6), we get and that (r − ρ) depends only on F because the quantities of right-hand side of (7) depend only on F. erefore, (r − ρ) becomes a constant and that Δk becomes a constant due to the first equation of (6). On the other hand, if l is constant, then the potential function h becomes a function on B so that B becomes an almost gradient Yamabe soliton by eorem 1 and the first and third equations of (6). us, we have the following theorem. In 1965, Tashiro [16] proved that the following theorem.
Theorem 5. Let M be an n-dimensional complete Riemannian manifold of dimension n ≥ 2 and suppose it admits a special concircular field ρ satisfying the equation for constants k and b. en, M becomes either direct product Hence, if we combine eorems 4 and 5, then we can state the following theorem.

Almost Gradient Yamabe Solitons in the Riemannian Product Spaces
Tokura and other coauthors [7] introduced various examples of an almost gradient Yamabe soliton in R 3 . In this section, we consider the relation between the structure of an almost gradient Yamabe soliton with (h, ρ) in the Riemannian product manifold (B × F, g) of (B, g) and (F, g) and the structure of an almost gradient Yamabe soliton in B and F. If the potential function h is expressed by h � k + l for some functions k and l on B and F, respectively, then we have Theorem 7. Let the Riemannian product manifold M � B × F be an almost gradient Yamabe soliton with (h, ρ) with h � k + l for some functions k and l in B and F, respectively. en, r − ρ is a constant on M, B becomes an almost gradient Yamabe soliton with (k, ρ � ρ − r), and F becomes an almost gradient Yamabe soliton with (l, ρ � ρ − r) and ρ + r � ρ + r. In this case, r − ρ and r − ρ become constants on B and F, respectively.
Proof. From the first and third equations of equation (9), we can see that r − ρ is a quantity on B and F, respectively. Hence, r − ρ becomes constant. Since M � B × F is an almost gradient Yamabe soliton with (h � k + l, ρ) for some function k and l on B and F, respectively, we get (r − (ρ − r))g ab � ∇ b k a and (r − (ρ − r))g yx � ∇ y l x from equation (9). If we put ρ and ρ by ρ � ρ − r and ρ � ρ − r, then ρ � r − (∇ b k a )g ba and ρ � r − (∇ y l x )g yx . erefore, ρ and ρ become functions on B and F, respectively. Hence, we can see that B and F become an almost gradient Yamabe soliton with (ρ, k) and (ρ, l), respectively. Moreover, we obtain ρ + r � ρ � ρ + r. Hence, we obtain r − ρ � r − ρ � r − ρ and that r − ρ and r − ρ are also constants.
For the converse case of eorem 7, if we assume that B and F are almost gradient Yamabe solitons with (ρ, k) and (ρ, l), respectively, and ρ + r � ρ + r, then we get (r − ρ)g ba � ∇ b k a and (r − ρ)g yx � ∇ y l x . If we take h � k + l and put ρ + r � ρ + r � ρ, then we can derive (r us, we have the following theorem.
If we combine eorems 7 and 8, we can state the following theorem.

Theorem 9.
e Riemannian product space M � B × F with (h � k + l, ρ) is almost gradient Yamabe soliton for some functions k and l on B and F, respectively, if and only if B and F are almost gradient Yamabe soliton with (ρ, k) and (ρ, l), respectively, and ρ + r � ρ � ρ + r.
If we combine eorems 5 and 7, then we can state the following theorem. Proof. We see that r − ρ is a constant by eorem 7. en, from the first equation of (7), ∇ ∇ h � βg for some constant β � r − ρ. Hence, we can prove eorem 10 by using eorem 1.
Hence, M becomes M � W n− 2 × I 2 , where we put I 2 � I 2 × I. Hence these facts, eorems 5 and 7 give the followinf theorem □ Theorem 11. Let the complete Riemannian product manifold M be an almost gradient Yamabe soliton with (h � k + l, ρ). en, r − ρ becomes a constant. If the constant r − ρ � 0, then M becomes M � W × I 2 of an (n − 2)-dimensional complete Riemannian manifold W with a 2-dimensional Euclidean space I 2 .

Almost Gradient Yamabe Solitons in the Warped Product Spaces
In this section, we study that the warped product space of M � B× f F is an almost gradient Yamabe soliton with (h, ρ). en, we have and that (a) If h a � 0, then h becomes a function on F, r � ρ, and ∇ y h x � 0 from the first and third equations of (11). Moreover, we get f b h x � 0 from the second equation of (11); that is, M is either a Riemannian product of B and F or the potential function h is a constant. (b) From the first equation of (11), we see that r − ρ becomes a function on B. Moreover, we have ∇ b h a � (r − ρ)g ab � (r − (− (r/f 2 ) + (2p△f/f) + (p(p − 1) ‖f e ‖ 2 /f 2 ) + ρ))g ab from the fourth equation of (3) and the first equation of (11). If we put then we obtain ρ � r − r − ρ � r − (△h/n) and that ρ is a function on B. Hence, B becomes an almost gradient Yamabe soliton.
Consider the case of h � k + l for some functions k and l on B and F, respectively. en, from equations (3) and (11), we obtain Hence, we can see that the function f or l is constant from the second equation of (12). If the function f is constant, then M becomes a product space; henceforth, we can apply eorem 9.
On the other hand, if the function l is a constant, then l x � 0 for all x. Hence, if we apply this fact to eorem 12 (b), then we have. □ Theorem 13. If the warped product space M � B× f F is an almost gradient Yamabe soliton with (h � k + l, ρ) for some functions k and l in B and F, respectively, then we have two cases as follows:

Almost Gradient Yamabe Solitons in the Twisted Product Spaces
and that B becomes an almost gradient Yamabe soliton. us, we have the following theorem.

Theorem 14.
If the twisted product manifold M � B× f F is an almost gradient Yamabe soliton with (h, ρ) and h � k + l, then the base space B becomes an almost gradient Yamabe soliton and r − ρ is a function on B.
If the conformally flat twisted product space M � B× f F is an almost gradient Yamabe soliton with (h, ρ) and h � k + l for some functions k and l on B and F, respectively, then we get and f � f * f, where f * and f are functions on B and F, respectively, which come from the conformally flatness and the Proof of eorem 2. From the first and second equations of (16), we see that r − ρ is a function only on B, and f * is constant or fl x � 0, respectively,. Since f � f * f is a positive function, we see that f * is a constant or l is a constant. Let us consider the first case, that is, f * is a constant. en, f � f * f depends only on F and that g becomes a Riemannian product metric. Hence, B, F, and M become trivial gradient Yamabe solitons due to eorem 3. Moreover, from the third equation of (16), we get and that (r − ρ) depends only on F because the quantities of right-hand side of (17) depend only on F. erefore, (r − ρ) becomes a constant and that Δk becomes a constant due to the first equation of (16). Moreover, we see that Hess k � αg where we have put α � r − ρ. en, we can apply this fact to eorem 5, and we see that M becomes either direct product V × I of an (n-1)-dimensional complete Riemannian manifold V with a straight line I when α � 0 or a Euclidean space when α ≠ 0.
On the other hand, if l is constant, then the potential function h becomes a function on B and B becomes an almost gradient Yamabe soliton by eorem 14. us, we have the following theorem.
Theorem 15. Let the twisted product manifold M � B× f F be an almost gradient Yamabe soliton with (h, ρ) and conformally flat. If h � k + l and p ≠ 1 and n ≠ 1, then (r − ρ) depends only on B, and M is one of the following two cases: (1) M is either direct product V × I of an (n− 1)-dimensional complete Riemannian manifold V with a straight line I when α � 0 or a Euclidean space when α ≠ 0 moreover (r − ρ) and Δk become constants (2) e potential function h depends only on B, and B becomes an almost gradient Yamabe soliton.

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

Conflicts of Interest
e authors declare that they have no conflicts interest.