# Gradient Ricci–Yamabe Soliton on Twisted Product Manifolds

Authors: Byung Hak Kim; Jin Hyuk Choi; Sang Deok Lee; Chang Yong Han
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we study the twisted product manifolds with gradient Ricci–Yamabe solitons. Then, we classify and characterize the warped product and twisted product spaces with gradient Ricci–Yamabe solitons. We also study the construction of the model space of gradient Ricci–Yamabe solitons in the Riemannian product manifolds and the warped product manifolds. Finally, we study the geometric characterization of the conformally flat twisted product manifolds with gradient Ricci–Yamabe solitons.


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## Body

## 1. Introduction

The concept of a Ricci flow was introduced by Hamilton [1], which is both generalization of Einstein metric and a special solution of the Ricci flow. Since G. Perelman solved the Poincare conjecture using Ricci flow, the study of many geometric flows and solitons has been actively conducted and have increased considerably to obtain a complete geometric classification of the manifolds and construction of model spaces.A Riemannian metricg on a complete Riemannian manifold M is called a Ricci soliton if there exists a smooth vector field X such that the Ricci tensor satisfies the following equation:(1)Ric+12LXg=ρg,for some constant ρ, where LX is the Lie derivative with respect to X [2–4]. It is said that M,g or M is a Ricci soliton if the metric g on M is a Ricci soliton. The Ricci soliton is called shrinking if ρ>0, steady if ρ=0, and expanding if ρ<0. The metric of a Ricci soliton is useful in not only physics but also mathematics and is often referred to as quasi-Einstein metric [5]. If X=∇h for some function h on M, then M is called a gradient Ricci soliton with a potential function h or h,ρ [6]. In this case, equation (1) can be rewritten as(2)Ric+Hessh=ρg,where Hessh is Hessian of h.Ifρ is a function, then M is called an almost gradient Ricci soliton with h,ρ.A Riemannian metricg on a Riemannian manifold M is called a Yamabe soliton [7–10] if there exist a smooth vector field X and a constant ρ such that(3)r−ρg=12LXg,where r is the scalar curvature of M. The gradient 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 or h,ρ. In this case, equation (3) becomes(4)r−ρg=Hessh.In (4), if ρ is a function on M, then M is called an almost gradient Yamabe soliton with h,ρ [7, 8, 10]. When the function h is constant, we say that the corresponding Yamabe soliton is a trivial Yamabe soliton. It is well known that every compact Yamabe soliton is of constant curvature and hence trivial because h turns out to be constant in this case [11]. In 2019, Guler and Crasmareanu [12] introduced a scalar combination of the Ricci flow and the Yamabe flow under the name of Ricci–Yamabe map. A Riemannian metric g on a Riemannian manifold M with dim M>2 is said to admit a Ricci–Yamabe soliton g,X,λ,α,β if there exist a smooth vector field X on M and constants λ,α,β such that(5)LXg+2αS=2λ−βrg,where S is the Ricci curvature [13, 14]. In particular, if X=∇h for some smooth function h on M, we say that M is a gradient Ricci–Yamabe soliton with g,h,λ,α,β. In this case, equation (5) becomes(6)Hessh+αS=λ−12βrg.The Ricci–Yamabe soliton (or gradient Ricci–Yamabe soliton) is said to be a(a)
Ricci soliton (or gradient Ricci soliton) ifα=1,β=0 [4, 12].(b)
Yamabe soliton (or gradient Yamabe soliton) ifα=0,β=1 [7, 9, 10].(c)
Einstein soliton (or gradient Einstein soliton) ifα=1,β=−1 [3].(d)
ρ-Einstein soliton (or gradient ρ-Einstein soliton) if α=1,β=−2ρ [14].The Ricci–Yamabe soliton or gradient Ricci–Yamabe soliton is said to proper ifα≠0,1 [13, 14] .This paper is organized as follows. In Section2, we introduce the results of geometric characterization of gradient Ricci–Yamabe soliton in the product manifold. In Section 3, we study the geometric characterization of the warped product manifold Rn×fF with gradient Ricci–Yamabe soliton. In Sections 4 and 5, we introduce the geometric results obtained under more generalized assumptions than the assumptions used in previous studies of the warped product manifold B×fF and the twisted product manifold B×fF, respectively.


## 2. Gradient Ricci–Yamabe Soliton in the Product Manifold

LetM be the product manifold B×F of the n-dimensional Riemannian manifold B,g and p-dimensional Riemannian manifold F,g¯. If M=B×F is a generalized gradient Ricci–Yamabe soliton withg˜,h˜,λ˜,α˜,β˜,w˜, then(7)∇bha˜+α˜Sba=λ˜−12β˜r˜gba+w˜h˜bh˜a,∂bh˜x=w˜h˜bh˜x,∇¯yh˜x+α˜S¯yx=λ˜−12β˜r˜g¯yx+w˜h˜yh˜x,r˜=r+r¯,where ∇ and ∇¯ are Levi-Civita connections for g and g¯, respectively, S and S¯ are Ricci curvatures on B and F, respectively, and w˜ is a constant. The ranges of the indices a,b,…, and x,y,… are 1,2,…,n, and n+1,n+2,…,n+p, respectively.From the second equation of (7), we see that the potential function h˜ is repressed by h˜=k+l for some functions k on B and l on F, respectively. Using this fact and the first and third equations of (7), we get λ˜−1/2β˜r˜=1/nΔk+α˜r=1/pΔ˜l+α˜r¯. Since Δk+α˜r and Δ˜l+α˜r¯ depend only B and F, respectively, the quantity λ˜−1/2β˜r˜ becomes constant, so we can put λ˜−1/2β˜r˜=K for a constant K. From these facts and (7), the first and third equations reduced to ∇∇k+α˜S=Kg and ∇¯∇¯l+α˜S¯=Kg¯, that is, B and F become gradient Ricci soliton with k/α˜ and l/α˜ as potential functions, respectively.Thus, we have the following.Theorem 1.
LetM=B×F be a gradient Ricci–Yamabe soliton with g˜,h˜,λ˜,α˜,β˜. Then, we have(1)
h˜=k+lfor some functionskandlonBandF, respectively.(2)
r˜ is constant.(3)
B is gradient Ricci soliton with a potential function k/α˜.(4)
F is gradient Ricci soliton with a potential function l/α˜.Conversely, ifB and F are the gradient Ricci solitons with potential functions h and h¯, respectively, then we obtain(8)S=ρg−∇∇h,S¯=ρ¯g¯−∇¯∇¯h,for constants ρ and ρ¯.If we take a functionh˜ on B×F by h˜=αh+h¯ for a constant α and let ρ=ρ¯, then we see that(9)αS˜+∇˜∇˜h˜=αS+∇∇h∂bhx∂bhxS¯+∇¯∇¯h¯=αρg00ρg¯=αρg˜.From (8), we get r=nρ−Δh and r¯=pρ−Δ¯h¯. So,(10)r˜=mρ−Δ˜h˜α,where m=n+p the dimension of M and Δ˜h˜=α△h+Δ¯h¯. From (10), we see that ρ=r˜/m+Δ˜h˜/mα and r˜ is constant if and only if Δ˜h˜ is constant. Hence, equation (9) is reduced to αS˜+∇˜∇˜h˜=αρg˜=Δ˜h˜/m+αr˜/mg˜.Thus, we have the following.Theorem 2.
LetBandFbe the gradient Ricci solitons withh,ρandh¯,ρ¯, respectively. Assume thatρ=ρ¯andr+r¯are constants; then, the product spaceM=B×Fbecomes a gradient Ricci–Yamabe soliton withg+g¯,αh+αh¯,αr+r¯/n+p,α,−2α/n+p.If we apply Theorem2, we can construct the model spaces of gradient Ricci–Yamabe solitons on the Riemannian product manifold.


## 3. Gradient Ricci–Yamabe Soliton in the Warped Product Manifold

Let the warped product manifoldℝ×fF be a gradient Ricci–Yamabe soliton with g˜,h˜,λ˜,α˜,β˜ and f be a warping function depending only on ℝ. Then, we get(11)h˜11−α˜pff11=λ˜−12β˜r˜,∂1h˜x−f1fh˜x=0,λ˜−12β˜r˜f2g¯yx=∇¯yh˜x+ff1h˜1g¯yx+α˜S¯yx−fΔfg¯yx−p−1f12g¯yx,r˜=r¯f2−2pΔff−pp−1f2f12,where f1=∂f/∂t and f1=g11f1 for the coordinate t in ℝ.From (11), we have the following.Theorem 3.
If the warped product manifoldM=ℝ×fFis a proper gradient Ricci–Yamabe soliton withg˜,h˜,λ˜,α˜,β˜andp>2, then we have the following:(a)
Ifh˜x=0for allx=2,3,…,n+1, then we see thata1F becomes Einstein manifold.a2M becomes an almost gradient Ricci soliton with a potential function h˜/α˜ and a function ρ˜.(b)
Ifh˜1=0, then eitherMis the Riemann product ofℝand steady gradient Ricci soliton orFis Einstein.Proof.
a Assume that h˜x=0 for all x, that is, h˜ depends only on ℝ. Then, we see that λ˜−1/2β˜r˜ becomes a function on ℝ from the first equation of (11), so that r˜=r˜t and r¯ become a constant. On the other hand, from the third equation of (11), we obtain(12)α˜S¯yx=Kf2−ff1h˜1+α˜fΔf+α˜p−1f12g¯yx,where K=λ˜−1/2β˜r˜. Then, equation (12) is reduced to the form α˜S¯yx=α˜r¯/pg¯yx for Kf2−ff1h˜1+α˜fΔf+α˜p−1f12α˜/p=r¯, that is, F is an Einstein when α˜≠0 and p>2 because p=2 does not guarantee that r¯ is constant.
SinceK=Kt depends only on ℝ, we get an equation ∇˜∇˜h˜/α˜+S˜=ρ˜g˜, where we have put ρ˜=K/α˜. Therefore, M becomes an almost gradient Ricci soliton with a potential function h˜/ρ˜.
b If h˜1=0, then we see that r¯ becomes constant from the first and fourth equation of (11). From the second equation of (11), we get f1h˜x=0 and that f1=0 or h˜x=0 because h˜ is a function only on F. If we consider the case f1=0, then we see that r˜ becomes constant from the fourth equation of (11), that is, M becomes a trivial Yamabe soliton. Moreover, we see that λ˜−1/2β˜r˜=0 from the first equation of (11), so ∇¯yh˜x+α˜S¯yx=0 from the third equation of (11). Hence, we can state that M and F become a steady gradient Ricci soliton, that is, M is the Riemannian product of ℝ and steady gradient Ricci soliton. Moreover, M itself becomes a steady gradient Ricci soliton. For the second case h˜x=0, h˜ becomes constant. Then, we get α¯S¯yx=αfΔf+p−1f12−pff11g¯yx from the first and third equations of (11). Hence, F becomes Einstein.Theorem 4.
Let the warped product manifoldM=ℝ×fFbe a proper gradient Ricci–Yamabe soliton withg˜,h˜,λ˜,α˜,β˜andh˜=k+lfor some functionskandlinℝandF, respectively. Ifp≥3, then M is one of the following manifolds:(a)
M is the Riemann product ofℝand a gradient Ricci soliton manifold and thatr˜andr¯become constant.(b)
F becomes an Einstein manifold.(c)
M is the Riemann product ofℝand an Einstein manifold.Proof.
Sinceh=k+l, we get f1lx=0 from the second equation of (11). From this fact, we can consider the following three cases:(i)
fis constant.(ii)
lis constant, sayw, and thath˜=l+w.(iii)
fis constant and l is constant.
In case (i),M becomes a Riemann product ℝ and F because f is constant. Moreover, we obtain(13)k11=λ˜−12β˜r˜,λ˜−12β˜r˜f2g¯yx=∇¯ylx+α˜S¯yx,r˜=r¯f2.
from the first, third, and fourth equations of (11), respectively. Moreover, from the first equation of (13), we see that r˜ depends only on ℝ and that r¯=f2r˜ depend only on ℝ, that is, r¯ is constant and that r˜=r¯/f2 becomes constant. Hence, the second equation of (13) is reduced to ∇¯∇¯l/α˜+S¯=λ˜−1/2β˜r˜f2g¯yx. This means that F is a gradient Ricci soliton with l/α˜,λ˜−1/2β˜r˜f2/α˜ so that we can get result (i). For case (ii), we see that h˜x=0. Hence, we see that the fibre F becomes Einstein from the first case of Theorem 3. Finally, if we combine cases (i) and (ii), then result (iii) can be obtained.
Let us consider the case that the warped product manifoldM=ℝ×fF is a proper gradient Ricci–Yamabe soliton with f=e4t and h˜x=0. Then, we get(14)h˜11−16pα˜=K,4e8th1g¯yx+α˜S¯yx−16pe8tg¯yx=Ke8tg¯yx,where K=λ˜−1/2β˜e−8tr¯−16p2−16p. From equation (14), we see that r¯ is constant and S¯=A/α˜g¯, where A=K−4h1+16pα˜e8t. Hence, we see that F is Einstein. Moreover, we get(15)h˜11−4h˜1=α˜r˜pe−8t.
The general solution of (15) is h˜=c1+c2e4t+α˜r¯/96pe−8t for some constants c1 and c2. Thus, we have the following.Theorem 5.
If the warped product manifoldM=ℝ×e4tFis a gradient Ricci–Yamabe soliton withg˜,h˜,λ˜,α˜,β˜andh˜x=0for allx, then the potential functionh˜becomesh˜=c1+c2e4t+α˜r¯/96pe−8tand the fibreFbecomes Einstein ifp>2.If we consider the converse of Theorem5 and the definition of the gradient Ricci–Yamabe soliton in (6), then we obtain Theorem 6, and we can construct gradient Ricci–Yamabe soliton on the warped product manifold of ℝ and an Einstein manifold F with warping function e4t if we use Theorem 6.Theorem 6.
LetFbe an Einstein manifold withp=dimF>2andh˜=c+αr¯/96pe−8tfor an arbitrary constantc. Then, the warped product manifoldM=ℝ×e4tFbecomes a gradient Ricci–Yamabe soliton withg˜=100e8tg¯,h˜,−16/3pα+32/3α,α,−4α/3p.Proof.
SinceF is an Einstein manifold, S¯=r¯/pg¯ where r¯ is constant. It is sufficient to show the following for proof: ∇˜∇˜h+αS˜=λ−1/2βr˜g˜, which is equivalent to(16)2αr¯3pe−8t−16αp00−αr¯3pg¯yx+αr¯pg¯yx−16pαe8tg¯yx=λ−12βr¯e−8t−16p2−16p100e8tg¯yx.
Hence, if we takeλ=−16/3pα+32/3α, β=−4α/3p for arbitrary constant α, then we can show relation (16).Theorem 7.
If the warped product manifoldℝn×fFis a gradient Ricci–Yamabe soliton withg˜,h˜,λ˜,α˜,β˜, then we have the following:(a)
Ifh˜x=0for allx, then we see thata1M becomes an Einstein.a2M becomes an almost gradient Ricci soliton with a potential functionh˜/αand a functionρ˜=K/α˜.(b)
Ifh˜1=0, then eitherMis the Riemann product ofRnand steady gradient Ricci soliton orFis Einstein.Proof.
Now consider the case of the warped product manifoldM of ℝn and F with warping function f being a gradient Ricci–Yamabe soliton with g˜,h˜,λ˜,α˜,β˜, that is, M=ℝn×fF; then, we have(17)∇bh˜a−α˜pf∇bfa=Kδab,∂bh˜x−fbfh˜x=0,∇¯yh˜x+ffch˜cg¯yx+α˜S¯yx−fΔfg¯yx−p−1fe2g¯yx=Kf2g¯yx,where K=λ˜−1/2β˜r˜ and r˜=r¯/f2−2pΔf/f−pp−1/f2fe2.
Ifh˜x=0 for all x, then α˜S¯yx=−ffch˜c+Kf2+αfΔf+αp−1fe2g¯yx and r¯ becomes constant from the first and third equations of (17). Hence, K becomes a function on ℝn. Therefore, F becomes an Einstein when α˜≠0 and p>2, and that M becomes an almost gradient Ricci soliton with a potential function h˜ and a function ρ˜.
Ifha=0 for all a, then we get fbh˜x=0 from the second equation of (17), that is, f is constant or h˜ is constant. By the analogous way of the proof of Theorem 3 (b), we see that either M is the Riemann product of ℝn and F or F is Einstein.


## 4. Gradient Ricci–Yamabe Soliton in the Warped Product ManifoldB×fF

In this section, we consider the warped product manifoldB×fF of Riemannian manifolds B,g and F,g¯. If M is a gradient Ricci–Yamabe soliton with g˜,h˜,λ˜,α˜,β˜, then we have(18)∇bh˜a+α˜Sab−pf∇bfa=Kgab,∂bh˜x−fbfh˜x=0,∇¯yh˜x+ffch˜cg¯yx+α˜S¯yx−fΔfg¯yx−p−1fe2g¯yx=Kf2g¯yx,where K=λ˜−1/2β˜r˜ and r˜=r+r¯/f2−2pΔf/f−pp−1/f2fe2.From (18), we have the following.Theorem 8.
If the warped product spaceM=B×fFis a gradient Ricci–Yamabe soliton withg˜,h˜,λ˜,α˜,β˜, then we have the following:(a)
Ifh˜a=0for alla=1,2,…,n, thenMis either a Riemann product ofBandForFbecomes Einstein.(b)
Ifh˜x=0for allx=n+1,…,n+p, thenr¯is constant andFbecomes Einstein.Proof.
a Since h˜a=0 for all a, we see that fbh˜x=0 from the second equation of (18). The relation fbh˜x=0 means that f is constant or h is constant. In the case of h˜ being a constant, we can reduce the relation S¯=Ag¯ from the third equation of (18) and A depends only on the base space B because the quantity A depends only on B with the help of the first equation of (18). Hence, A is constant on F and F becomes Einstein.
(b) Ifh˜x=0, then we see that K depends only on the base space B from the first equation of (18). Moreover, we obtain pffch˜c+α˜r¯−pfΔf−pp−1fe2=pKf2, and ∂xr¯=0 is reduced, and hence r¯ is constant. From this fact and the third equation of (18), we see that F becomes an Einstein manifold.Theorem 9.
Let the warped product spaceM=B×fFbe a gradient Ricci–Yamabe soliton withg˜,h˜,λ˜,α˜,β˜andh˜=k+lfor some functionskandlinRandF, respectively, thenMis one of the following manifolds:(a)
Mis the Riemann product of the almost Ricci soliton and Ricci soliton.(b)
Fbecomes an Einstein manifold.(c)
Mis the Riemann product of the almost Ricci soliton and Einstein manifold.Proof.
If the potential functionh˜=k+l for some functions k and l on B and F, respectively, in (18), then we get fblx=0 from the second equation of (18). This means that f is constant or l is constant. If f is constant, then we get(19)∇∇kα˜+S=Kα˜g,∇¯∇¯lα˜+S¯=Kα˜f2g¯,from which we see that K depends only on B from the first equation of (19) and that the base space B becomes almost Ricci soliton. From the second equation of (19), we see that K/α˜f2 depends only on B, and hence F becomes Ricci soliton. Therefore, in this case, M becomes the Riemann product of the almost Ricci soliton and Ricci soliton.
Ifl is constant, then we obtain(20)ffch˜cg¯yx+α˜S¯yx−fΔfg¯yx−p−1fe2g¯yx=Kf2g¯yx,pffch˜c+α˜r¯−pfΔf−pp−1fe2=pKf2,and hence we see that r¯ is constant from the second equation of (20). If we substitute the second equation of (20) into the first equation of (20), then we get S¯=r¯/p¯g¯, that is, F becomes Einstein.


## 5. Gradient Ricci–Yamabe Soliton in the Twisted Product Manifold

The twisted product ofn-dimension Riemannian manifold B,g and p-dimension F,g¯ is the product manifold B×F with g˜=g00f2g¯, where f:B×F⟶R+. In [15], the authors proved the following.Theorem 10.
If the twisted product manifoldM=B×fFof the Riemannian manifoldsB,gandF,g¯is conformally flat andp≠1,n≠1, thenMis the warped product spaceB×f∗F∗ofBwith andF∗, whereF∗isFwith the metricg∗=f¯2g¯.In the process of proving Theorem10, the authors knew that the warping function f can be expressed as a product of two functions f∗ and f¯ in B and F, respectively, that is, f=f∗f¯. Now, let us study the geometric characterization of the conformally flat twisted product manifold with gradient Ricci–Yamabe soliton. If the twisted product manifold M=B×fF is a gradient Ricci–Yamabe soliton with g˜,h˜,λ˜,α˜,β˜, then(21)∇bh˜a+α˜Sab−pf∇bfa=Kgab,∂bh˜x−fbfh˜x−α˜p−11f∂bfx−1f2fbfx=0,∇¯yh˜x−1ffyh˜x+fxh˜y−fzh˜zg¯yx+ffch˜cg¯yx+α˜S¯yx−fΔfg¯yx−p−1fe2g¯yx−∇¯ffg¯yx−p−2f∇¯yfx+2p−2f2fyfx−p−3f2fw2g¯yx=Kf2g¯yx,where K=λ˜−1/2β˜r˜ and r˜=r+r¯/f2−2pΔf/f−2p−1/f3Δ¯f−pp−1/f2fe2−p−1p−4/f4fx2.If the potential functionh˜=k+l for some functions k and l on B and F, respectively, and M is conformally flat, then the warping function f is expressed by f=f∗f¯ according to the proof process of Theorem 10 and that the first equation of (21) becomes ∇bka+α˜Sab−p/f∗∇bfa∗=Kgba.Hence, we see that the quantityK depends only on B. Moreover, we get fb∗lx=0 from the second equation of (21), that is, f∗ is constant or l is constant. Considering that f is a constant as the first case, then the warping function f=f∗f¯ depends only on F. Hence, the Riemannian metric g˜ becomes a product metric and the first equation of (21) is reduced to ∇bka+αSba=Kgba and ∇b∇ak/α+Sba=K/αgba, that is, B becomes an almost gradient Ricci soliton with k/α,K/α. Considering that the second case, l is a constant, we see that h˜=k+l depends only on B. Thus, we have the following.Theorem 11.
Let the twisted product manifoldM=B×fFbe a gradient Ricci–Yamabe soliton withg˜,h˜,λ˜,α˜,β˜and conformally flat. Ifh˜=k+lfor some functionskandlonBandF, respectively, then the quantityK=λ˜−1/2β˜r˜depends only onBandMis either Riemann product of the almost gradient Ricci soliton andFor the potential functionh˜depends only onB.It can be seen from the structure equation of the twisted product manifold that it is established even under the conditionS˜A,X=0 for vector fields A and X on B and F, respectively. Thus, the condition of conformally flat in Theorem 11 can be replaced by condition S˜A,X=0.



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