In [8], Hamilton introduced the notion of Yamabe solitons. According to the author, a Riemannian metric g of a complete Riemannian manifold (Mⁿ, g) is called a Yamabe soliton if it satisfies

where W, λ, r and £ denotes a smooth vector field, a real number, the scalar curvature and Lie-derivative respectively. The vector field W is said to be the soliton field of the Yamabe solitons. If W is the gradient of a C^∞ function f:Mn→ℝ, then the manifold will be called gradient yamabe soliton. In this occasion, the previous equation reduces to

where ∇2f indicates the Hessian of f. A Yamabe soliton is said to be shrinking, steady or expanding according to λ>0, λ=0 or λ<0, respectively. Yamabe solitons have been investigated by many researchers in different context (see, [1],[4], [5], [6]).

The concept of the Gradient Einstein solitons was initiated by Catino and Mazzieri [3]. For some smooth function f and some constant λ∈ℝ, the Gradient Einstein solitons are Riemannian manifolds obeying

where S denotes the Ricci tensor.

Many years ago in [10], Olszak investigated the 3-dimensional normal almost contact metric(briefly, acm) manifolds mentioning several examples. After the citation of [10], in recent years normal acm manifolds have been studied by many researchers in different context (see, [7],[10] and references contained in those).

The present article is constructed as follows:

In section 2, we recall a few basic facts and formulas of 3-dimensional non-cosymplectic normal acm manifolds which will be needed throughout the article. In section 3, we investigate the Yamabe, gradient Yamabe and gradient Einstein solitons. Specifically, we establish the below stated Theorems:

Theorem 1.1. If a 3-dimensional non-cosymplectic normal acm manifold admits a Yamabe soliton of the type (M3,g,ξ), then the scalar curvature is constant and the characteristic vector field ξ is Killing.

Theorem 1.2. If a 3-dimensional non-cosymplectic normal acm manifold M³ admits a Yamabe soliton of the type (M³,g, W), then either the manifold is quasi-Sasakian or the scalar curvature of the manifold is constant and the soliton vector field W is Killing provided α,β are constants and the scalar curvature r is invariant under the characteristic vector field ξ.

Theorem 1.3. Let the Riemannian metric of a 3-dimensional non-cosymplectic normal acm manifold with α,β=constant be the gradient Yamabe soliton. Then either the manifold is of constant sectional curvature −(α2−β2) or the manifold is α-Kenmotsu, provided the gradient yamabe soliton is trivial.

Theorem 1.4. Let the Riemannian metric of a 3-dimensional non-cosymplectic normal acm manifold with α, β =constant and α≠±β be the gradient Einstein metric. Then either the manifold is α-Kenmotsu or is a manifold of constant sectional curvature.

Let M³ be an acm manifold endowed with a triplet of almost contact structure(η,ξ,ϕ). In details, M³ is an odd-dimensional differentiable manifold equipped with a global 1-form η, a unique characteristic vector field ξ and a (1,1)-type tensor field ϕ, respectively, such that

An almost complex structure J on M×ℝ is defined by

where (E,λddt) denotes a tangent vector on M×ℝ, E and λddt being tangent to M and ℝ respectively. After fulfilling the condition, the structure J is integrable, M and (ϕ,ξ,η) are called normal (see, [2]).

The Nijenhuis torsion is defined by

The structure (η,ξ,ϕ) is said to be normal if and only if

A Riemannian metric g on M³ is called compatible with the structure (η,ξ,ϕ) if the condition

holds for any E,F∈χ(M). In such case, the quadruple (η,ξ,ϕ,g) is termed as an acm structure on M³ and M³ is an acm manifold. The equation

is also valid on such a manifold.

Certainly, we can define the fundamental 2-form Φ by

where F,Z∈χ(M).

For a normal acm, we can write [10]:

where 2α=divξ and 2β=tr(ϕ∇ξ), divξ is the divergent of ξ defined by divξ=trace{E→∇Eξ} and tr(ϕ∇ξ)=trace{E→ϕ∇Eξ}. Utilizing (2.8) in (2.7) we lead

Also in this manifold the subsequent relations hold [10]:

It is well admitted that the Riemann curvature tensor always satisfies

By (2.11), (2.12) and (2.15) we infer

From (2.10) it follows that if α,β =constant, then the manifold is either α-Kenmotsu [9] or cosymplectic [2]or β-Sasakian. Also, it is well known that a 3-dimensional normal almost contact manifold reduces to a quasi-Sasakian manifold if and only if α =0 (see, [10]).

Now before producing the detailed proof of our main theorems, we first prove the following results:

Lemma 2.1. For a 3-dimensional non-cosymplectic normal acm manifold with α, β =constant, we have

Proof. For α ,β =constants, we get from (2.16)

Differentiating (2.18) covariantly in the direction of E and using (2.8) and (2.14), we get

Replacing F by ξ in (2.19) and using (2.8), we obtain

Lemma 2.2. Let M3(η,ξ,ϕ,g) be a non-cosymplectic normal acm manifold with α, β =constant. Then we have

Proof. Recalling (2.19), we can write

Putting E=Z=ei (where {ei} is an orthonormal basis for the tangent space of M and taking ∑{i}, 1≤i≤3 ) in the foregoing equation and using the so called formula of Riemannian manifolds divQ=12grad r, we obtain

Superseding F=ξ in the previous equation we have the required result.

Definition 2.1. A vector field W on an n dimensional Riemannian manifold (M,g) is said to be conformal if

ρ being the conformal coefficient. If the conformal coefficient is zero then the conformal vector field is a Killing vector field.

Lemma 2.3. [12] On an n-dimensional Riemannian or, Pseudo-Riemannian manifold (Mⁿ,g) endowed with a conformal vector field W, the following relations are satisfied:

for E,F∈χ(M), D being the gradient operator and Δ=−divD being the Laplacian operator of g.

Proof of Theorem 1.1. Let a normal acm manifold M³ admits a Yamabe soliton of the type (g,ξ). Then superseding W=ξ in (1.1) yields

In view of (2.8), (3.1) becomes

Superseding E=F=ξ in (3.2) and using (2.1), we have

Therefore the scalar curvature r is constant. Putting λ =r in (3.1) yields £ξg=0 i.e., ξ is Killing vector field. Hence the theorem.

Proof of Theorem 1.2. Let M³ be a normal acm manifold with the structure which endowed the Yamabe soliton (g, W).

Taking Lie differential of g(ξ,ξ)=1 along the soliton vector field V and making use of (1.1) yields

Again, in view of (1.1) and (2.23) it is obvious that the soliton vector field W is conformal with the conformal coefficient ρ=λ−r2. As we consider the metric g of the 3-dimensional normal acm manifold M is a Yamabe soliton, using ρ=λ−r2 and n=3 in Lemma 2.3, we have

and

Here we consider α,β are constants. So from (2.16), we have

Taking Lie differentiation of (3.7) along W and making use of (1.1) and (2.10), we obtain

Making use of (3.5) in (3.8), we acquire

Replacing E=F=ξ in (3.9) and using of (2.1) and (3.4), we obtain

Let r is invariant under the characteristic vector field ξ. Then either α=0 or r=constant. Thus we conclude that either the manifold is quasi-Sasakian or using r=constant in (3.10), we have λ-r=0. Hence (1.1) immediately yields £Wg=0, i.e., the soliton vector field W is Killing. This completes the proof.

If α=0 and β=1, then the manifold reduces to a 3-dimensional Sasakian manifold. Since the characteristic vector field ξ is Killing in a Sasakian manifold, therefore ξ r=0. Hence from the above theorem we can state the following:

Corollary 3.1. If a three dimensional Sasakian manifold admits a Yamabe soliton, then the scalar curvature of the manifold is constant and the soliton vector field W is Killing.

The foregoing Corollary was established by Sharma in [11].

Proof of Theorem 1.3. Let us consider a gradient Yamabe soliton on a 3-dimensional non-cosymplectic normal acm manifold with α, β =constant. Then from (1.2) we obtain

from which we acquire

Contraction of previous equation along F yields

Now, the equation (2.16) gives

Equation (3.13) and (3.14) together reveal that

Putting E=ξ and using (2.20), we get

Hence, using (3.16) in (3.15), we have

Now, from (3.12) we infer that

Again (2.11) implies that

Combining equation (3.18) and (3.19), we acquire

Setting F=ξ in the above equation gives

Using (3.21) in (3.17) we infer that

This shows that either r=−6(α2−β2) or Df=(ξf)ξ. Next, we consider the above two cases as follows.

Case i: If r=−6(α2−β2), then from (2.16) we get S=−2(α2−β2)g , that is the manifold is an Einstein manifold and hence from (2.15) it follows that the manifold is of constant sectional curvature −(α2−β2).

Case ii: If

Taking the covariant differentiation of (3.23) along any vector field E∈χ(M) we get

Replacing E by ϕE and taking inner product with ϕF yields

Interchanging E and F in the foregoing equation, we infer

Applying Poincare's lemma, we have d2f(E,F)=0 and hence by a straightforward calculation we lead

Since ∇g=0, the above equation yields

Replacing E by ϕE and F by ϕF in (3.28) and utilizing (3.25) and (3.26), we obtain

which implies that

Since dη≠0, either β=0 or (ξ f)=0. Hence we conclude that either the manifold is α-Kenmotsu or (f= constant) the gradient yamabe soliton is trivial. This completes the proof.

Proof of Theorem 1.4. Let us suppose that the Riemannian metric of a 3-dimensional non-cosymplectic normal acm manifold with α, β =constant is a gradient Einstein metric. Then from (1.3) we obtain

From which we get

Now, from (2.19) we infer that

The contraction of above equation along E and using (2.20), gives

Equation (3.14) and (3.33) together reveal that

Putting E=ξ and utilizing (2.20), we have

Hence, using (3.35) in (3.34), we get

Now, from (3.32) we infer that

Combining equation (3.19) and (3.37) reveal that

Putting F=ξ in the above equation gives

This shows that Df=(ξf)ξ, provided α≠±β. Hence from the previous theorem, we conclude that either the manifold is α-Kenmotsu or f= constant. If f=constant, then we get from (3.30) that the manifold is an Einstein manifold. Since the manifold is under consideration of dimension 3, hence the manifold is of constant sectional curvature.

This finishes the proof.

We consider a 3-dimensional Riemannian manifold M={(x,y,z)∈ℝ3}, (x,y,z) being the standard coordinate in ℝ3. Here we take the vector fields v₁, v₂ and v₃ given by

We define the Riemannian metric g on M by g(vi,vj)=δij, i,j=1,2,3 and η, a 1-form on M by η(E)=g(E,v1), for E∈χ(M). Let ϕ be a second order mixed tensor field defined by ϕ(v1)=0,ϕ(v2)=−v3,ϕ(v3)=v2.

In this setting (ϕ,ξ,η,g) becomes an almost contact structure on M with ξ=v1.

The setting of the vector fields v1,v2,v3 gives

Using Koszul's formula, we calculate the following:

It is easy to verify that the manifold M is a 3-dimensional non-cosymplictic normal almost contact metric manifold with α =0 and β=12.

By using the well-known formula R(E,F)W=∇E∇FW−∇F∇EW−∇[E,F]W, we calculate the non-zero independent components of the curvature tensor as follows:

Therefore we get the non-zero components of Ricci tensor as S(v1,v1)=12,S(v2,v2)=S(v3,v3)=−12. Hence the scalar curvature r=−12=a constant.

Let E=a1v1+a2v2+a3v3 and F=b1v1+b2v2+b3v3. Then

Therefore if we set λ=−12 the (g,v1=ξ) becomes a Yamabe Soliton on M and also v₁ is a Killing vector field with the scalar curvature r=constant.

Hence the Theorem 1.1 is verified.

The authors are thankful to the referee and the Editor in Chief for their valuable suggestions towards the improvement of the paper.