In the past twenty years, geometric flows have emerged as versatile tools for describing geometric structures in Riemannian geometry. A specific class of solutions on which the metric evolves by dilation and diffeomorphisms plays a vital part in the study of singularities of the flows as they appear as possible singularity models. They are often called soliton solutions.

The theory of Yamabe flow was popularized by Hamilton in his prime research work [12] as a tool for constructing metrics of constant scalar curvature on an n-dimensional Riemannian manifold (Mⁿ,g), n ≥ 3. The Yamabe flow is an evolution equation for metrics on Riemannian manifolds. It is given by

where r is the scalar curvature corresponding to Riemannian metric g and t is time. It is used to deform a metric by smoothing out its singularities.

A Yamabe soliton is a special solution of the Yamabe flow that moves by one parameter a family of diffeomorphisms generated by a fixed vector field E on Mⁿ with a real constant λ satisfying the following equation

Here LEg is the Lie derivative of the metric g along the vector field E, called the soliton vector field of the Yamabe soliton [12]. If λ<0, λ>0, or λ=0, then the (Mⁿ,g) is called a Yamabe shrinker, Yamabe expander, or Yamabe steady soliton, respectively.

When the vector field E is the gradient of a smooth function ψ:Mn&#_10230;ℜ, the manifold will be called a gradient Yamabe soliton. The function ψ is called the potential function of the gradient Yamabe soliton. In this case equation (1.2) becomes

where Hess ψ stands for the Hessian of the potential function ψ. The gradient Yamabe soliton equation (1.3) links geometric information about the curvature of the manifold to the scalar curvature tensor and the geometry of the level sets of the potential function by means of their second fundamental form. This makes gradient Yamabe solitons under some curvature conditions an interesting topic of study.

An Einstein manifold [2] with a constant potential function is called a trivial gradient Ricci soliton. Gradient Yamabe solitons [14] play an important role in Yamabe flow as they correspond to self-similar solutions, and often arise as singularity models [18].

Introduced by Chen and Desahmukh in [4], a Riemannian manifold (Mⁿ, g) is called a quasi-Yamabe solitonif it admits a vector field E such that

for some real constant λ and smooth function µ, where E^∖ is the dual 1-form of E. The vector field E is also called a soliton vector field for the quasi-Yamabe soliton. We denote the quasi-Yamabe soliton satisfying (1.4) by (Mn,g,E,λ,μ). If E=∇ψ, then (1.4) becomes

which is the gradient generalized quasi-Yamabe soliton studied by Huang et al. [10] and Leandro et al. [13], where ∇2ψ denotes Hessian of ψ. This class of closely related Yamabe solitons hase be extensively studied; for further details see ([1], [5], [8], [19], [20], [22], [23], [25]).

According to Pigola et al. [17], if we replace the constant λ in (1.4) and (1.5) with a smooth function λ∈C∞(M), called soliton function, then we can say that (Mⁿ, g) is an almost quasi-Yamabe and gradient generalized almost quasi-Yamabe soliton, respectively.

On one hand, in 1985, Oubina [15] introduced a new class of almost contact metric manifolds, known as trans-Sasakian manifolds. This class consists the Sasakian, the Kenmotsu and the cosymplectic structures. The properties of trans-Sasakian manifolds have been studied by several authors, like Blair [3] and Marrero [14]. %The trans-Sasakian manifolds arose in a natural way from the classification of almost contact metric structures and they appear as a natural generalization of both Sasakian and Kenmotsu manifolds The main goal of this paper is to characterize the three-dimensional trans-Sasakian manifolds equipped with gradient generalized quasi-Yamabe solitons, quasi-Yamabe metrics, and gradient generalized almost quasi-Yamabe metrics.

Let M be a connected almost contact metric manifold equipped with almost contact metric structure (φ,ζ,η,g) consisting of a (1,1) tensor field ⊎, a vector field ζ, a 1-form η and a positive definite metric g such that

for all E,F∈χ(M), where χ(M) denotes the collection of all smooth vector fields of M and dimM=2m+1.

In the Grey and Harvella [9] classification of almost Hermitian manifolds, there appears a class W₄ of Hermitian manifolds which are closely related to the conformal Kaehler manifolds. In their classification, the class C6⊕C5 (see [3], [6], [15], [16]) coincides with the class of trans-Sasakian structure of type (α,β). In fact, the local nature of two sub classes, namely C₆ and C₅ of trans-Sasakian structures are characterized completely. An almost contact metric structure (ϕ, ξ, η, g) on M is called a trans-Sasakian [21] if (M×ℜ,J,G) belongs to the class W₄, where J is an almost complex structure on M×ℜ defined by

for all vector fields X on M and smooth functions f on M×ℜ. Here G is the product metric on M×ℜ and ℜ denotes the set of real numbers. This may be expressed by the condition

where α and β are some scalars functions on M and ∇ denotes the Levi-Civita connection with respect to g. We note that the trans-Sasakian structures of type (0,0), (α,0) and (0,β) are the cosymplectic, α-Sasakian and β-Kenmotsu structures, respectively. In particular, if α=1,β=0, α=0,β=1 and α=0,β=0, then the trans-Sasakian manifold reduces to Sasakian, Kenmotsu and cosymplectic manifolds, respectively. From 1.3, it follows that

equivalent to

In a 3-dimensional trans-Sasakian manifold M, we have the following relations [7]

where R, S and Q denote the curvature tensor, Ricci tensor and Ricci operator of g, respectively. Also grad stands for gradient. Further, in a three-dimensional trans-Sasakian manifold we have

and

Using (1.9) and (1.10), for constants α and β, we have

For a smooth function ψ on M, the gradient and Hessian of ψ are, respectively, defined by

For E∈Γ(TM), we define E♯∈Γ(T¯M) by

The generalized quasi-Yamabe soliton equation [4] in a Riemannian manifold M is defined by

Equation (1.3) is a generalization of Einstein manifold [10]. Note that if E=gradψ, where ψ∈C∞(M), the gradient generalized quasi-Yamabe soliton equation is given by [10]:

Main Result:

Theorem 3.1. Let M be a three-dimensional trans-Sasakian manifold satisfy the gradient generalized quasi-Yamabe soliton equation (1.4) with condition μ[λ+6(α2−β2)]=0, then ψ is a constant function. Furthermore, if µ≠ 0, then λ=−6(α2−β2) is negative, that is, a three-dimensional trans-Sasakian manifold admits a shrinking gradient generalized quasi-Yamabe soliton.

From Theorem 3.1, we get the following remarks:

Remark. Let a three-dimensional trans-Sasakian manifold M satisfy the gradient generalized quasi-Yamabe soliton equation Hessψ=(r−λ)g, then ψ is constant and M is η-Einstein.

Remark. In a three-dimensional trans-Sasakian manifold M, there is no non-constant smooth function ψ such that Hessψ=λg for some constant λ.

To prove the Theorem 3.1, we have to demonstrate the following lemmas.

Lemma 3.2. Let M be a three-dimensional trans-Sasakian manifold. Then we have

where E,F∈Γ(TM).

Proof. From the property of Lie-derivative we note that

Since LζF=[ζ,F] and Lζζ=[ζ,ζ], therefore the above equation can be written as

From (1.4) we get ∇ζζ=0, so the last equation gives

which gives

From (1.12), we lead

The Lemma 3.2 follows from the last two equations. Particularly, if Y is orthogonal to ζ then equation (1.5) assumes the form

for all E∈χ(M) and F orthogonal to ζ.

Lemma 3.3. Let M be a Riemannian manifold, and let ψ∈C∞(M). Then we have

Proof. We calculate:

Since [ζ,F](ψ)=ζ(F(ψ))−F(ζ(ψ)), therefore the above equation becomes

Hence the statement of Lemma 3.3 is proved.

Lemma 3.4. Let a three-dimensional trans-Sasakian manifold M satisfy the gradient generalized quasi-Yamabe soliton equation 1.4. Then we have

Proof. Let F∈Γ(TM), then form the definition of Ricci tensor S, scalar curvature r and the curvature condition 1.12, we have

where e1,e2,e3 is an orthonormal frame on M. From the above equations, we infer

From (1.4) and (1.9), we obtain

The Lemma 3.3 follows from equation (1.10) and the definition of Hessian (see (1.1)).

Now, we are going to prove our main Theorem 3.1 by using Lemma 3.2, Lemma 3.3 and Lemma 3.4.

Proof of Theorem 3.1. Let us suppose that the three-dimensional trans-Sasakian manifold satisfying the gradient generalized quasi-Yamabe soliton equation (1.10) and λ,μ∈ℜ. Let Y∈Γ(TM), then Lemma together with E=grad ψ leads to

From Lemma 3.4 and equations (1.1), (1.2), (1.4), (1.11), we get

for all F∈Γ(TM). Taking F orthogonal to ζ and therefore the above equation becomes

Next, the Lie derivative of the gradient generalized quasi-Yamabe soliton equation (1.4) along the vector field ζ yields

The last two equations together with Lemma infer

which is equivalent to

According to Lemma 4.3, we have

by equations (1.15) and (1.16), we obtain

since [λ+6(α2−β2)]≠0, we find that F(ψ)=0, i.e., gradψ is parallel to ζ. Hence grad ψ=0 as D=kerη is not integrable any where, which means ψ is a constant function.

Now, for particular values of α and β we turn up the following cases:

Case: For α=0, (β=1) and (α=β=0) we can state the following results:

Corollary 3.5. Let M be a 3-dimensional β-Kenmotsu (or Kenmotsu) manifold satisfies the gradient generalized quasi-Yamabe soliton 1.4 condition μ[λ−6β2)]≠0, then ψ is a constant function. Furthermore, if µ≠ 0, implies λ=6β2), then M is expanding.

Case: For β=0, or (α=1) we can state:

Corollary 3.6. Let M be a 3-dimensional α-Sasakian (or Sasakian) manifold satisfies the gradient generalized quasi-Yamabe soliton1.4 condition μ[λ+6α2)]≠0, then ψ is a constant function. Furthermore, if µ≠ 0, implies λ=−6α2), then M is shrinking.

Case: For α=β=0, we can state:

Corollary 3.7. Let M be a 3-dimensional cosymplectic manifold satisfies the gradient generalized quasi-Yamabe soliton 1.4 condition μ[λ]≠0, then ψ is a constant function. Furthermore, if µ≠ 0, implies λ=0, then M is steady.

Again, assume the equation

where g is a Riemannian metric and R is the scalar curvature, ζ is vector field, E^∖ is a 1-form and λ and µ are real constant. The data (g,ζ,λ,µ) satisfies the equation (4.1) is called the quasi-Yamabe soliton. In particular, if µ=0, (g,ζ,λ) is a Yamabe soliton.

Using the definition of Lie derivative and (4.1), we obtain

for any F,G∈χ(M).

Contracting (4.2) we get

Let (M,g,⊎,η,ζ) be a 3-dimensional trans-Sasakian manifold and (g,ζ,λ,µ) be a quasi-Yamabe soliton on M. Writing 4.2) for F=G=ζ, we obtain

Therefore

Using (4.5) we can state the following results.

Theorem 4.1. Let (M,η,φ,ζ,g) be a 3-dimensional trans-Sasakian manifold and E♯ be the g-dual 1-form of the gradient vector field ζ=grad(ψ). If (4.1) define a quasi-Yamabe soliton with non vanishing µ in M, then the Poisson equation satisfied by ψ becomes

Once again, considering the equation (4.5) we can also obtain

Remark.([24]) A C∞ function f:M&#_10230;ℝ is said to be harmonic if Δf=0 , where Δ is the Laplacian operator in M.

Now, from equation (4.7) and using above remark, we obtain the following results:

Theorem 4.2. Let (M,η,φ,ζ,g) be a 3-dimensional trans-Sasakian manifold and E♯ be the g-dual 1-form of the gradient potential vector field ζ=grad(ψ) . If the potential function ψ is harmonic, then quasi-Yamabe soliton is shrinking for the value of λ=−3(α2−β2).

Corollary 4.3. Let (M,η,φ,ζ,g) be a 3-dimensional α-Sasakian (or Sasakian) manifold and E♯ be the g-dual 1-form of the gradient potential vector field ζ=grad(ψ) . If the potential function ψ is harmonic, then quasi-Yamabe soliton is shrinking for the value of λ=−3α2.

Corollary 4.4. Let (M,η,φ,ζ,g) be a 3-dimensional β-Kenmotsu (or Kenmotsu) manifold and E♯ be the g-dual 1-form of the gradient potential vector field ζ=grad(ψ). If the potential function ψ is harmonic, then quasi-Yamabe soliton is expanding for the value of λ=3β2.

Corollary 4.5. Let (M,η,φ,ζ,g) be a 3-dimensional cosymplectic manifold and E♯ be the g-dual 1-form of the gradient potential vector field ζ=grad(ψ) . If the potential function ψ is harmonic, then quasi-Yamabe soliton is steady for the value of λ=0.

Example 5.1. Let M=(x,y,z)∈ℝ3:z≠0, where (x,y,z) is the standard coordinates of ℝ3.

The vector fields are

Let g be the Riemannian metric defined by

that is, the form of the metric becomes Let η be the 1-form defiend by η(Z)=g(Z,e3) for any Z∈χ(M).

Also, let ⊎ be the (1,1) tensor field defined by

Thus, using the linearity of ⊎ and g, we have

for any Z,W∈χ(M).

Then for e3=ξ, the structure (φ,ξ,η,g) defines an almost contact metric structure on M.

Let ∇ be the Levi-Civita connection with respect to the metric g, then we have

which is known as Koszul's formula.

Using Koszul's formula we have

From (5.1) we find that the structure (φ,ξ,η,g) satisfies the formula (4.5) for α=14 and ξ=e3. Hence the manifold is a 3-dimensional trans-Sasakian manifold of type (α,0) with the constant structure function α=14 and β=0.

Then the Riemannian and Ricci curvature tensor fields are given by:

From the above expressions of the curvature tensor we obtain

similarly we have

Now, we have constant scalar curvature as follows,

By the definition of quasi-Yamabe soliton and using (1.4), we obtain

for all i∈1,2,3, and we have

for all i∈1,2,3.

Therefore λ=−12 and μ=38 the data (g,ξ,λ,μ) admitting the shrinking quasi-Yamabe soliton on 3-dimensional trans-Sasakian manifolds with λ<0.

In [7] De and Sarkar proved that if a 3-dimensional trans-Sasakian

manifold is of constant curvature is compact and connected.

On the other hand, The classical theorem of de-Rham-Hodge asserts that the cohomology of an oriented closed Riemannian manifold can be represented by harmonic forms. The similar one still holds for an oriented compact Riemannian manifold with boundary by imposing certain boundary conditions, such as absolute and relative ones.

We consider M as a compact orientable trans-Sasakian manifold and X∈χ(M). Then Hodge-de Rham decomposition theorem [11] implies that E can be expressed as

where h∈C∞(M) and div(F)=0. The function h is called the Hodge-de Rham potential [11].

Theorem 6.1. If (g,E,λ,μ) is a compact gradient almost quasi-Yamabe soliton on trans-Sasakian manifold M. If M is also a gradient almost quasi-Yamabe soliton with potential function ψ, then up to a constant, f equals to the Hodge-de Rham potential.

Proof. Since (g,E,λ,μ) is a compact almost quasi-Yamabe soliton, now taking the trace of (1.4), we find

Hodge-de Rham decomposition implies that div(E)=Δh, hence the above equation, we get

Again since M is generalized gradient almost quasi-Yamabe soliton with Perelman potential f, hence taking trace of (1.5), we have

Now, equating the equations (5.3) and (5.4), we find 13Δ(ψ−h)=0. Hence ψ-h is a harmonic function in compact trans-Sasakian manifold. Hence f=h+c, for some constant c.

Let M=(x,y,z)∈ℝ3:z≠0 where (x,y,z) are the standard coordinates of ℝ3.

The vector fields are

Let g be the Riemannian metric defined by

that is, the form of the metric becomes

Let η be the 1-form defined by η(Z)=g(Z,e3) for any Z∈χ(M).

Also, let φ be the (1,1) tensor field defined by

Thus, using the linearity of φ and g, we have

for any Z,W∈χ(M).

Then for e3=ξ, the structure (φ,ξ,η,g) defines an almost contact metric structure on M.

Let ∇ be the Levi-Civita connection with respect to the metric g, then we have

which is known as Koszul's formula.

Using Koszul's formula we have

From (5.1) we find that the manifold satisfies (1.4) for α=0 and β=-1 and ξ=e₃. Hence the manifold is a 3-dimensional trans-Sasakian manifold of type (0,β) with the constant structure function α=0 and β=-1 [7].

Then the Riemannian and Ricci curvature tensor fields are given by:

From the above expressions of the curvature tensor we obtain

similarly, we have

Now, the scalar curvature

Because of scalar curvature r = 6, from Theorem (), we can conclude that M is an Einstein manifold.

By the definition of quasi-Yamabe soliton and using (1.4), we obtain

for all i∈1,2,3, and we have

for all i∈1,2,3.

Therefore λ =-1 and μ=38 the data (g,ξ,λ,µ) admitting the shrinking quasi-Yamabe soliton on 3-dimensional trans-Sasakian manifolds with λ<0.

The authors express their sincere thanks to the anonymous referee’s for the valuable suggestions and comments for the improvement of the paper.