Yamabe Solitons and τ-Quasi Yamabe Gradient Solitons on Riemannian Manifolds Admitting Concurrent-Recurrent Vector Fields

Devaraja Mallesha Naik; Ghodratallah Fasihi-Ramandi; H. Aruna Kumara; Venkatesha Venkatesha

doi:10.1515/ms-2023-0037

Article Open Access

Yamabe Solitons and τ-Quasi Yamabe Gradient Solitons on Riemannian Manifolds Admitting Concurrent-Recurrent Vector Fields

Devaraja Mallesha Naik , Ghodratallah Fasihi-Ramandi , H. Aruna Kumara and Venkatesha Venkatesha

Published/Copyright: March 31, 2023

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From the journal Mathematica Slovaca Volume 73 Issue 2

Abstract

We consider a Riemannian manifold (M,g) admitting a concurrent-recurrent vector field for which the metric g is a Yamabe soliton or a τ-quasi Yamabe gradient soliton. We show that if the metric of a Riemannian three-manifold (M, g) admitting a concurrent-recurrent vector field is a Yamabe soliton, then M is of constant negative curvature –α². In this case, we see that the potential vector field is Killing. Next, we show that if the metric of a Riemannian manifold M admitting concurrent-recurrent vector field is a non-trivial r-quasi Yamabe gradient soliton with potential function f, then M has constant scalar curvature and is equal to –n(n – 1)α². Finally, an illustrative example is presented.

2020 Mathematics Subject Classification: Primary 06B05; 06D35; Secondary 35R30; 39A10

Keywords: Yamabe soliton; τ-quasi Yamabe gradient soliton; conformal vector field

1 Introduction

Let (M, g) be a compact Riemannian manifold with s as its scalar curvature. Then, the Yamabe problem concerns the existence of a Riemannian metric g′ conformal to g, for which the scalar curvature s′ of the metric g′ is constant. As an effort to solve the Yamabe problem, Hamilton in [4] came up with the concept of Yamabe flow. Given a conformal class of Riemannian metrics, Yamabe flow can be used for constructing metrics whose scalar curvature s is constant. The Yamabe flow is an evolving metric family (g(t)) satisfying

1.1 ∂∂tg(t)=−s(t)g(t)

with the initial data g(0) = g. Geometric flows like Yamabe flows, Ricci flows, mean and the inverse mean curvature flows, and Kaehler-Ricci flows have been applied to a variety of topological, geometric, and physical problems. It is interesting to note that in dimension two the Yamabe flow and Ricci flow are equivalent, but in higher dimension they are non-identical.

The Yamabe solitons are special solutions of the Yamabe flows, that is, there exist scalars σ(t) and diffeomorphisms ϕ_t in such manner that g(t)=σ(t)ϕt∗(g0) is the solution of the Yamabe flow (1.1), with σ(0) = 1 and ϕ₀ = I. In other words, a Riemannian metric g is called a Yamabe soliton if there exist a smooth vector field V∈ 𝔛 (M) (called a potential vector field) and a scalar λ∈ ℝ such that

1.2 12LVg=(s−λ)g,

where V stands for the Lie derivative operator along V. In the case which V is Killing, we say that the Yamabe soliton is trivial. In particular, if the potential vector field V= Df, where D is the gradient operator and f is a smooth scalar function, then we say that the metric g is Yamabe gradient soliton and in this case (1.2) turns into

Hessf=(s−λ)g.

where Hess_f is Hessian of f. If f is constant, then the above soliton called trivial Yamabe gradient soliton.

It is interesting to notice that there is a nice connection between warped product structure and Yamabe soliton. In [6], Ma and Cheng showed that a non-compact complete Riemannian manifold admitting a Yamabe gradient soliton has a warped product structure. In [10], Tokura et al. studied Yamabe gradient soliton on warped product manifold with compact Riemannian base and in that case it has been shown that the soliton is trivial. Yamabe solitons have been studied by many geometers in many different contexts (see [1‒3,8,9,11,13]). Recently, in[7], the first author introduced a special vector field ν which satisfies the relation

1.3 ∇Xν=α{X−ν♭(X)ν},

where α∈ ℝ and ν^♭ is the 1-form equivalent to ν in a Reimannian manifold (M, g). A unit non-parallel (i.e., α ≠ 0) vector field ν satisfying the preceding equation is called a concurrent-recurrent vector field. It is shown that an n-dimensional connected Riemannian manifold (M, g) admits concurrent-recurrent vector field ν, if and only if, (M, g) is the warped product I × _{f_(t)} F, where I is an open interval and f(t) = e^αt (see [7: Theorem 3]). In this paper, we consider a Yamabe soliton on a Riemannian manifold admitting concurrent-recurrent vector field and prove:

Theorem 1.1

Let (M, g) be Riemannian three-manifold admitting concurrent-recurrent vector field. If the metric g is a Yamabe soliton with potential vector field V, then the manifold is of constant negative curvature −α² and V is Killing.

Due to the Theorem 3 of [7], we have:

Corollary 1.1.1

Let M = I × _f(t)) F with the warping function f(t) = e^αt, where α∈ ℝ, I is an open interval in ℝ and F is a Riemannian 2-manifold. If the metric of M is a Yamabe soliton, then the manifold is of constant negative curvature –α².

In [5], Huang and Li introduced the notion of a τ-quasi Yamabe gradient soliton which naturally extends the concept of Yamabe gradient soliton. According to Huang and Li[5], a τ-quasi Yamabe gradient soliton is a Riemannian metric g satisfying

1.4 Hessf=1τdf⊗df+(s−λ)g,

where f is a smooth scalar function and τ > 0 is a constant. Notice that a Yamabe gradient soliton is nothing but an ∞-quasi Yamabe gradient soliton. For λ < 0 the Yamabe soliton (or τ-quasi Yamabe gradient soliton) is said to be shrinking, for λ > 0 is said to be expanding, and for λ = 0 is said to be steady. In [12], Wang showed that a non-compact complete Riemannian manifold admitting a τ-quasi Yamabe gradient soliton has warped product structure. In this direction, we consider a τ-quasi Yamabe gradient soliton on a Riemannian manifold admitting concurrent-recurrent vector field and prove the following theorem.

Theorem 1.2

Let M be Riemannian manifold admitting concurrent-recurrent vector field. If the metric of M is non-trivial τ-quasi Yamabe gradient soliton with potential function f, then the scalar curvature of M is constant and is equal to −n(n−1)α².

From Theorem 3 of [7], we immediately have the following.

Corollary 1.2.1

Let M = I × _f(t) F with the warping function f(t) = e^αt, where α∈ ℝ, I is an open interval in ℝ and F is a Riemannian n-manifold. If the metric of M is a non-trivial τ-quasi Yamabe gradient soliton, then the scalar curvature of M is constant and is equal to −n(n− 1)α².

2 Background and key lemmas

A unit vector field ν on a Reimannian manifold (M, g) is said to be a concurrent-recurrent vector field if it satisfies

2.1 ∇Xν=α{X−ν♭(X)ν},

where ∇ is the Levi-Civita connection of g and α is a non-zero constant. In [7], the author constructed certain examples of n-dimensional Riemannian manifolds admitting such vector fields. An interesting property of this vector field is that it is an eigenvector of the Ricci operator of the Riemannian manifold (M, g) on which this vector field is defined. Moreover, the defining equation (2.1) dictates that integral curves of ν are geodesics. The following result has been proved in [7].

Theorem 2.1

A Riemannian n-manifold admitting a concurrent-recurrent vector field is locally isometric to the warped product I × _f(t)F, where I⊆ ℝ is an open interval and F is a Riemannian (n−1)-manifold. Conversely, the warped product I × _f(t) F with the warping function f(t) =e^αt admits a concurrent-recurrent vector field .

2.1 Key lemmas

In this subsection, we give some lemmas that are needed to prove our main results.

Lemma 2.1

A Riemannian manifold equipped with a concurrent-recurrent vector field ν satisfies

2.2 ν(s)=−2α(s+n(n−1)α2).

Proof. Using (1.3) in the definition of Riemann curvature tensor, we obtain

2.3 R(X,Y)ν=−α2{ν♭(Y)X−ν♭(X)Y}.

Contracting the above equation gives

2.4 Ric⁡(X,ν)=−(n−1)α2ν♭(X),

which yields Qν= −(n−1)α²ν, where Q is Ricci operator defined by g(QX,Y) = Ric(X,Y). Differentiating Qν=−(n−1)α²ν along X implies that

2.5 (∇XQ)ν=−(n−1)α3X−αQX.

Taking the g-trace of the above equation gives (2.2).

Let (M, g) be a Riemannian manifold. If there exists ρ ∈ C^∞(M), called the potential function, such that

£Vg=2ρg

then we say that the vector field V is a conformal vector field (see Yano [14]). Moreover, V is homothetic when ρ is constant, whereas Killing when ρ= 0. Now, we recall the following result from Yano [14].

Lemma 2.2

A conformal vector field V on a Riemannian n-manifold (Mⁿ,g) satisfies

(£VRic)(X,Y)=−(n−2)g(∇XDρ,Y)−(Δρ)g(Y,X),£Vs=−2ρs−2(n−1)Δρ,

where Δ = div D is the Laplacian operator of g.

Lemma 2.3

If the metric of Riemannian three-manifold M equipped with a concurrent-recurrent vector field is a Yamabe soliton, then the scalar curvature of M is harmonic and the Yamabe soliton is shrinking with λ = –6α².

Proof. First, we take Lie-derivative to g (ν,ν) = 1 along V, and employ the equations [1.2] and [1.3] to get

2.6 (LVν♭)ν=−ν♭(LVν)=(s−λ).

Now, taking n= 3 and ρ= s− λ in Lemma 2.2, we find

2.7 (ℒVRic)(X,Y)=−g(∇XDs,Y)−(Δs)g(X,Y),

2.8 ℒVs=−2s(s−λ)−4Δs.

In Riemannian three-manifolds, the curvature tensor is given by

2.9 R(X,Y)Z=g(Z,Y)QX−g(Z,X)QY+g(Z,QY)X−g(Z,QX)Y−s2{g(Z,Y)X−g(Z,X)Y}.

Taking Z = ν in the above equation and using (2.3), we easily deduce

2.10 Ric⁡(X,Y)=s2+α2g(Y,X)−3α2+s2ν♭(X)ν♭(Y).

Lie-differentiating of (2.10) along V and employing the equations (2.8) and(1.2), we find

(LVRic)(X,Y)=(2α2(s−λ)−2Δs)g(X,Y)+(s(s−λ)+2Δs)ν♭(X)ν♭(Y)−s2+3α2{(LVν♭)(X)ν♭(Y)+ν♭(X)(LVν♭)(Y)}.

Comparing the above equation with (2.7),we obtain

2.11 g(∇XDs,Y)=(Δs−2α2(s−λ))g(X,Y)−(s(s−λ)+2Δs)ν♭(X)ν♭(Y)+s2+3α2{(LVν♭)(X)ν♭(Y)+ν♭(X)(LVν♭)(Y)}.

Replacing X and Y in the above equation by ν and utilizing(2.6), we see that

ν(νs)=−Δs+4α2(s−λ.

Now, we use (2.2) in the above equation in order to obtain

2.12 Δs=−4α2(λ+6α2).

We replace Y by ν in (2.11) and use the equations (2.6) and (2.12) to deduce

g(∇XDs,ν)=(4α2(λ+6α2)+((α2−s2)s−λ))ν♭(X)+(s2+3α2)(LVν♭)X.

On the other hand, differentiating (2.2) along X and utilizing (1.3) we obtain

g(∇XDs,ν)=−3α(Xs)−2α2(s+6α2)ν♭(X).

Using the preceding equation in (2.11), we get

s2+3α2(LVν♭)(X)={(s−λ)(s2−α2)−2α2(s+6α2)−4α2(λ+6α2)}ν♭(X)−3α(Xs).

Substituting the above equation in (2.11) and using(2.12), we find

g(∇XDs,Y)=−2α2(λ+s+12α2)g(X,Y)+2α2(λ−3s−12α2)ν♭(X)ν♭(Y)−3α(Xs)ν♭(Y)−3α(Ys)ν♭(X),

which further leads to

2.13 ∇XDs=−2α2(λ+s+12α2)X+2α2(λ−3s−12α2)ν♭(X)ν−3α(Xs)ν−3αν♭(X)Ds.

Replacing X in the previous equation by ν and applying(2.2), we get

∇νDs=−2α2(s+6α2)ν−3αDs.

Operating the above equation by ∇_X gives us

2.14 ∇X∇νDs=−2α2(Xs)ν−3α∇XDs−2α2(s+6α2)∇Xν.

On the other hand, differentiating (2.13) along ν, we get

∇ν∇XDs=−2α2(νs)X−2α2(λ+s+12α2)∇νX−6α2(νs)ν♭(X)ν+2α2(λ−3s−12α2)ν♭(∇νX)ν−3αν(Xs)ν−3αν♭(∇νX)Ds−3αν♭(X)∇νDs.

Again from (2.13)we immediately get

2.15 ∇[X,ν]Ds=−2α2(λ+s+12α2)(∇Xν−∇νX)−3αg(∇Xν−∇νX,Ds)ν+2α2(λ−3s−12α2)ν♭(∇Xν−∇νX)ν−3αν♭(∇Xν−∇νX)Ds.

Now, employing the equations (2.14)–(2.15), one can easily get

R(X,ν)Ds=−2α2(Xs)ν−3α∇XDs−2α2(s+6α2)∇Xν+2α2(νs)X+2α2(λ+s+12α2)∇νX+6α2(νs)ν♭(X)ν+3αν♭(X)∇νDs−2α2(λ−3s−12α2)ν♭(∇νX)ν+3αν(Xs)ν+3αν♭(∇νX)Ds+2α2(λ+s+12α2)(∇Xν−∇νX)+3αg(∇Xν−∇νX,Ds)ν+3αν♭(∇Xν−∇νX)Ds−2α2(λ−3s−12α2)ν♭(∇Xν−∇νX)ν.

Contracting the above equation, we find

Ric⁡(ν,Ds)=10α2(νs)−3αΔs−4α3(s+6α2)+6αν(νs)+4α3(λ+s+12α2).

Making use of (2.2) together with (2.12)in the preceding equation, we obtain

Ric⁡(ν,Ds)=4α3(3λ+s+30α2).

Now, we employ (2.4) and (2.2) in the above equation to deduce the value of soliton constant λ=−6α², which means Yamabe soliton is expanding. Using this in (2.12), we have Δs= 0, that is, the scalar curvature is harmonic. □

3 Proof of the main results

Proof of Theorem 1.1. first, differentiate (1.2) along Z to achieve

3.1 (∇ZℒVg)=2(Zs)g(X,Y).

In [14], Yano reveals the following relation:

(LV∇Xg−∇XLVg−∇[V,X]g)(Y,Z)=−g((LV∇)(X,Z),Y)−g((LV∇)(X,Y),Z).

Due to ∇_g = 0, it appears from the preceding equation that

(∇XLVg)(X,Y)=g((LV∇)(X,Z),Y)+g((LV∇)(X,Y),Z).

Utilizing the symmetric property of L_V∇, the foregoing equation brings into view

2g((LV∇)(X,Y),Z)=(∇YLVg)(Z,X)+(∇XLVg)(Y,Z)−(∇ZLVg)(X,Y).

Utilizing (3.1) in the previous equation, we find

3.2 (LV∇)(X,Y)=(Xs)Y+(Ys)X−g(Y,X)Ds.

Taking differentiation of (3.2) covariantly along Z yields

(∇ZL∇)(X,Y)=g(∇ZDs,X)Y+g(∇ZDs,Y)X−g(Y,X)∇ZDs.

Applying the above relation on the following well known formula

(LVR)(X,Y)Z=(∇XLV∇)(Y,Z)−(∇YLV∇)(X,Z),

appears that

(LVR)(X,Y)Z=g(∇XDs,Z)Y−g(Z,Y)∇XDs−g(∇YDs,Z)X+g(X,Z)∇YDs.

Replacing Z in the previous equation by ν and calling back λ = –6α² and the equation (2.13), we infer

3.3 (LVR)(X,Y)ν=(4α2(s+6α2)ν♭(Y)+3α(Ys))X−3α(Ys)ν♭(X)ν−(4α2(s+6α2)ν♭(X)+3α(Xs))Y+3α(Xs)ν♭(Y)ν.

At this point, we differentiate (2.3)and apply (1.2) to ensure

3.4 (LVR)(X,Y)ν=−α2{(g(Y,LVν)+2(s+6α2)ν♭(Y))X−(g(X,LVν)+2(s+6α2)ν♭(X))Y}−R(X,Y)LVν.

Subtracting (3.3) from (3.4), we see that

R(X,Y)LVν=(6α2(s+6α2)ν♭(X)+3α(Xs)+α2g(LVν,X))Y+3α(Ys)ν♭(X)ν−(6α2(s+6α2)ν♭(Y)+3α(Ys)+α2g(Y,LVν))X−3α(Xs)ν♭(Y)ν.

Contracting the above equation, we may obtain that

Ric⁡(Y,LVν)=−6α2(s+6α2)−3α(Ys)−2α2g(Y,LVν).

By the support of (2.10) and (2.6), the above equation shows that

3.5 (s+6α2)LVν=−(s+18α2)(s+6α2)ν−6αDs.

If possible, we suppose that on an open subset O of M there holds s ≠ –6α². Then it appears from equation (3.5) that

3.6 LVν=−(s+18α2)ν−6αs+6α2Ds.

Replacing Y with ν in the well-known formula (see [14]):

(LV∇)(X,Y)=LV∇XY−∇XLVY−∇[V,X]Y,

and then utilizing λ = –6α²,(1.3) and (3.6)we obtain the following equality

3.7 (LV∇)(X,ν)=−α(s+30α2)ν♭(X)ν+s−6α2s+6α2(Xs)ν−12α2s+6α2ν♭(X)Ds+α(s+6α2)X−6α(s+6α2)2(Xs)Ds.

On the other hand, replacing Y in (3.2) by ν and using (2.2) we have

3.8 (LV∇)(X,ν)=(Xs)ν−2α(s+6α2)X−ν♭(X)Ds.

Comparing (3.7) with (3.8) implies that

3.9 1−s−6α2s+6α2(Xs)ν−3α(s+6α2)X+12α2s+6α2ν♭(X)Ds+α(s+30α2)ν♭(X)ν+6α(s+6α2)2(Xs)Ds=0.

Finally, replacing X by ν in (3.9)and applying (2.2) we get

3.10 Ds=−2α(s+6α2)ν.

Using the previous equation in (3.6) gives us L_Vν = –(s + 6α²)ν. This relation together with (1.2) gives

3.11 (LVν♭)(X)=(LVg)(X,ν)+g(X,LVν)=(s+6α2)ν♭(X).

With the help of Lemma (2.3), one can easily derived from (2.8) that

3.12 LVs=−2s(s+6α2).

Now we write (3.10) as: ds = –2α(s + 6α²)ν^♭. Now, we take Lie derivative to this equation in order to deduce

LVds=−2α(LVs)ν♭−2α(s+6α2)LVν♭.

From (3.12), the preceding equation transforms into

3.13 LVds=4αs(s+6α2)ν♭−2α(s+6α2)LVν♭.

Operating the equation (3.12) by d and using ds = –2α(s+6α²)ν^♭, we have

3.14 LVds=4α(s+6α2)2ν♭+4αs(s+6α2)ν♭,

where used the fact that Lie-derivative commutes with the exterior derivative. Now, comparing (3.13) with (3.14), leads us to the following formula

(s+6α2){LVν♭+2(s+6α2)}ν♭=0.

As we know s ≠ –6α² on O, we must have

LVν♭=−2(s+6α2)ν♭.

Comparing the previous equation with (3.11) shows that the scalar curvature s = –6α² on O and this is a contradiction. So, we must have s = –6α² on M. Substituting this in (2.10)we see that Ric = (1–n)α²g which along with(2.9) shows that the manifold is of constant negative curvature –α², and consequently, V is Killing. This concludes the proof of the theorem.

Proof of Theorem 1.2. We may write the equation (1.4) as

3.15 ∇XDf=1τg(X,Df)Df+(s−λ)Df.

Using the previous equation in the definition of curvature tensor, we find

3.16 R(X,Y)Df=(s−λ)τ{(Yf)X−(Xf)Y}+(Xs)Y−(Ys)X.

Replacing X by ν in the previous equation and comparing the obtained equation with R(ν, X)Df = –α²{g(X, Df)ν – (νf)X} (which follows from(2.3)), we have

3.17 −α2(Yf)ν+α2(νf)Y=s−λτ(Yf)ν−s−λτ(νf)Y+(νs)Y−(Ys)ν.

On the other hand, we contract the equation (3.16) over X to find the expression of Ricci tensor as

3.18 Ric⁡(Y,Df)=(n−1)(s−λ)τ(Yf)−(n−1)(Ys).

Replacing Y in the above equation by ν and utilizing (2.4) imply

3.19 (νs)−α2(νf)=s−λτ(νf),

which is a relation involving the scalar curvature and the potential function of the τ-quasi Yamabe gradient soliton. Feeding (3.19) in (3.17), we obtain

3.20 τDs=(s−λ+τα2)Df.

Differentiating (3.20) with respect to X and using (3.15), we reach at

3.21 τ∇XDs=(Xs)Df+(Xf)Df+(s−λ+τα2)(s−λ)X.

Taking scalar product of (3.16) with D f, we have (Xs)Df= (Xf) Ds. Employing this in (3.21), we find

3.22 τ∇XDs=2g(X,Df)Ds+(s−λ+τα2)(s−λ)X.

At this stage, we use (3.20) in (3.28) in order to ensure

QDf=−α2(n−1)Df.

Differentiating the previous equation with respect to X and calling back (3.15) give

3.23 (∇XQ)Df+(s−λ)QX+α2(n−1)(s−λ)X=0.

On the other hand, from second Bianchi identity one can find

traceg⁡{X→(∇XQ)Y}=(div⁡ Q)(Y)=12Y(s).

Now, we contract (3.23) over X and utilize the above identity to deduce

3.24 g(Ds,Df)+2(s−λ)(s+nα2(n−2))=0.

Combining the equations (3.15),(3.20), and (3.22) one can easily find

3τg(X,Df)g(Ds,Df)+2(s−λ+τα2)(s−λ)τg(X,Df)=−2(2s−2λ+τα2)(s−λ+τα2)τg(X,Df)=0.

Setting X = Df in the above equation and using the fact that |Df| ≠ 0 (as the soliton is non-trivial), we have

3τg(Ds,Df)+2(s−λ+τα2)(s−λ)τ+2(2s−2λ+τα2)(s−λ+τα2)τ=0.

Now, we employ (3.24) in the preceding equation to deduce

(s−λ+τα2)(3s−3λ+τα2)−3(s−λ)(s+nα2(n−1))=0.

The above equation shows that s is constant. So that we have ν(s)= 0, which together tracing of (2.5) gives that s = –nα²(n–1). This completes the proof.

4 Example

In this section, we construct a Riemannian manifold (M, g) admitting a concurrent-recurrent vector field for which the metric g is a Yamabe soliton.

Consider a manifold M = {(u, v, w) ∈ ℝ³:v > 0,w ≠ 0} with a coordinate system (u, v, w). Let us define a Riemannian metric on M as

g=2α2w2(du)2+α2w22v(du⊗ dv+dv⊗du)+α2w24v2(dv)2+1α2w2(dw)2,

where α( ≠ 0) ∈ ℝ. From Koszul’s formula, one can easily compute

∇∂∂u∂∂u =−2α4w3∂∂w, ∇∂∂u∂∂v=−α4w32v∂∂w, ∇∂∂u∂∂w=1w∂∂u, ∇∂∂v∂∂u=−α4w32v∂∂w, ∇∂∂v∂∂v=−1v∂∂v−α4w34v2∂∂w,∇∂∂v∂∂w=1w∂∂v,∇∂∂w∂∂u=1w∂∂u,∇∂∂w∂∂v=1w∂∂v, ∇∂∂w∂∂w=−1w∂∂w.

Let us take ν=αw∂∂w. Then, from above we can verify that

∇Xiν=α{Xi−ν♭(Xi)ν},

for all 1≤ i≤ 3, where X1=∂∂u, X2=∂∂v and X3=∂∂w. Thus, the vector field ν=αw∂∂w is concurrent-recurrent vector field. Now we use Levi-Civita connection to find the non-zero components of curvature tensor as given below

R(∂∂u,∂∂v)∂∂u=−α4w22v∂∂u+2α4w2∂∂v, R(∂∂v,∂∂w)∂∂v =α4w24v2∂∂w,R(∂∂u,∂∂w)∂∂w=−1w2∂∂u, R(∂∂u,∂∂w)∂∂v=α4w22v∂∂w,R(∂∂u,∂∂v)∂∂v=α4w22v∂∂v−α4w24v2∂∂u, R(∂∂v,∂∂w)∂∂w=−1w2∂∂v,R(∂∂u,∂∂w)∂∂u=2α4w2∂∂w, R(∂∂v,∂∂w)∂∂u=α4w22v∂∂w,

From the curvature tensor we find the scalar curvature as s = –6α². Also, it is not hard to verify that

R(Xi,Xj)Xk=−α2(g(Xj,Xk)Xi−g(Xi,Xk)Xj),

for all 1≤ i, j, k ≤ 3. This shows that (M, g) is of constant curvature –α².

Now we shall show that the metric g is a Yamabe soliton on M. Let

V=ln(v)2∂∂u−2v(ln(v)+2u)∂∂v+w∂∂w

be a vector field on M. It is not hard to see that L_Vg = 0, and so V is Killing. Thus, we see that

LVg=(s−λ)g

for λ = –6α². Hence, g is a Yamabe soliton having the potential vector field V=ln(v)2∂∂u−2v(ln(v)+2u)∂∂v+w∂∂w and the soliton constant λ = –6α². Also, we see that this example verifies our Theorem 1.1.

(Communicated by Július Korbaš)

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Received: 2022-01-11

Accepted: 2022-03-19

Published Online: 2023-03-31

Published in Print: 2023-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ms-2023-0037

Keywords for this article

Yamabe soliton; τ-quasi Yamabe gradient soliton; conformal vector field

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