The study of almost Ricci soliton was introduced by Pigola et. al. [18], where essentially they modified the definition of Ricci soliton by adding the condition on the parameter λ to be a variable function, more precisely, we say that a Riemannian manifold (Mⁿ, g) admits an almost Ricci soliton, if there exists a complete vector field V, called potential vector field and a smooth soliton function λ : Mⁿ → ℝ satisfying

where Ric and £ stand, respectively, for the Ricci tensor and Lie derivative. We shall refer to this equation as the fundamental equation of an almost Ricci soliton (Mⁿ, g, V, λ). It will be called expanding, steady or shrinking, respectively, if λ < 0, λ = 0 or λ > 0. Otherwise it will be called indefinite. When the vector field V is gradient of a smooth function f : Mⁿ → ℝ the metric will be called gradient almost Ricci soliton. In this case the preceding equation becomes

where ∇²f stands for the Hessian of f. Sometimes classical theory of tensorial calculus is more convenient to make computations. Then, we can write the fundamental equation in this language as follows:

Moreover, if the vector field X is trivial, or the potential f is constant, the almost Ricci soliton will be called trivial, otherwise it will be a non-trivial almost Ricci soliton. We notice that when n ≥ 3 and X is a Killing vector field an almost Ricci soliton will be a Ricci soliton, since in this case we have an Einstein manifold, from which we can apply Schur’s lemma to deduce that λ is constant. Taking into account that the soliton function λ is not necessarily constant, certainly comparison with soliton theory will be modified. In particular the rigidity result contained in Theorem 1.3 of [18] indicates that almost Ricci solitons should reveal a reasonably broad generalization of the fruitful concept of classical soliton. In fact, we refer the reader to [18] to see some of this changes.

In the direction to understand the geometry of almost Ricci soliton, Barros and Ribeiro Jr. proved in [2] that a compact gradient almost Ricci soliton with nontrivial conformal vector field is isometric to a Euclidean sphere. In the same paper they proved an integral formula for compact case, which was used to prove several rigidity results, for more details see [2].

The existence of Ricci almost soliton has been confirmed by Pigola et. al. [18] on some certain class of warped product manifolds. Some characterization of Ricci almost soliton on a compact Riemannian manifold can be found in ([1], [2], [3]). It is interesting to note that if the potential vector field V of the Ricci almost soliton (M,g, V, λ) is Killing then the soliton becomes trivial, provided the dimension of M > 2. Moreover, if V is conformal then Mⁿ is isometric to Euclidean sphere Sⁿ. Thus the Ricci almost soliton can be considered as a generalization of Einstein metric as well as Ricci soliton.

In [6], authors studied Ricci solitons and gradient Ricci solitons on 3-dimensional normal almost contact metric manifolds. In [10] authors studied compact Ricci soliton. Beside these, A. Ghosh [12] studied K-contact and Sasakian manifolds whose metric is gradient almost Ricci solitons. Conditions of K-contact and Sasakian manifolds are more stronger than normal almost contact metric manifolds in the sense that the 1-form η of normal almost contact metric manifolds are not contact form. The Ricci soliton and gradient Ricci soliton have been studied by several authors such as ([5], [7], [9]) and many others.

The present paper is organized as follows:

After preliminaries, in section 3 we study almost Ricci soliton in 3-dimensional f-Kenmotsu manifolds. Finally, we consider gradient almost Ricci solitons in 3-dimensional f-Kenmotsu manifolds.

Let M be an almost contact manifold, i.e., M is a connected (2n+1)-dimensional differentiable manifold endowed with an almost contact metric structure (φ, ξ, η, g) [4]. As usually, denote by Φ the fundamental 2-form of M, Φ(X, Y) = g(X, φY ), X, Y ∈ χ(M), χ(M) being the Lie algebra of differentiable vector fields on M.

For further use, we recall the following definitions ([4], [11], [19]). The manifold M and its structure (φ, ξ, η, g) is said to be:

• normal if the almost complex structure defined on the product manifold M×ℝ is integrable (equivalently [φ, φ] + 2dη ⊗ ξ = 0),

• almost cosymplectic if dη = 0 and dΦ = 0,

• cosymplectic if it is normal and almost cosymplectic (equivalently, ∇φ = 0, ∇ being covariant differentiation with respect to the Levi-Civita connection).

The manifold M is called locally conformal cosymplectic (respectively, almost cosymplectic) if M has an open covering {U_t} endowed with differentiable functions σ_t : U_t → ℝ such that over each U_t the almost contact metric structure (φ_t, ξ_t, η_t, g_t) defined by

is cosymplectic (respectively, almost cosymplectic).

Olszak and Rosca [16] studied normal locally conformal almost cosymplectic manifold. They gave a geometric interpretation of f-Kenmotsu manifolds and studied some curvature properties. Among others they proved that a Ricci symmetric f-Kenmotsu manifold is an Einstein manifold.

By an f-Kenmotsu manifold we mean an almost contact metric manifold which is normal and locally conformal almost cosymplectic.

Let M be a real (2n + 1)-dimensional differentiable manifold endowed with an almost contact structure (φ, ξ, η, g) satisfying

for any vector fields X, Y ∈ χ(M), where I is the identity of the tangent bundle TM, φ is a tensor field of (1, 1)-type, η is a 1-form, ξ is a vector field and g is a metric tensor field. We say that (M,φ, ξ, η, g) is an f-Kenmotsu manifold if the covariant differentiation of φ satisfies [15]:

where f ∈ C^∞(M) such that df ∧ η = 0. If f = α = constant ≠ 0, then the manifold is a α-Kenmotsu manifold [13]. 1-Kenmotsu manifold is a Kenmotsu manifold ([14], [17]). If f = 0, then the manifold is cosymplectic [13]. An f-Kenmotsu manifold is said to be regular if f² + f′ ≠ 0, where f′ = ξf.

For an f-Kenmotsu manifold from (2.2) it follows that

The condition df ∧ η = 0 holds if dim M ≥ 5. In general this does not hold if dim M = 3 [16].

In a 3-dimensional Riemannian manifold, we always have

In a 3-dimensional f-Kenmotsu manifold we have [16]

where r is the scalar curvature of M and f′ = ξ(f).

From (2.5), we obtain

and (2.6) yields

Example.([8])

We consider the three-dimensional manifold M = {(x, y, z) ∈ ℝ³, z ≠ 0}, where (x, y, z) are the standard coordinates in ℝ³. The vector fields

are linearly independent at each point of M. Let g be the Riemannian metric defined by

Let η be the 1-form defined by η(Z) = g(Z, e₃) for any Z ∈ χ(M). Let φ be the (1, 1) tensor field defined by φ(e₁) = −e₂, φ(e₂) = e₁, φ(e₃) = 0.

Then using linearity of φ and g we have

for any Z,W ∈ χ(M). Now, by direct computations we obtain

The Riemannian connection ∇ of the metric tensor g is given by the Koszul’s formula which is

Using (2.9) we have

Hence ∇e1e3=-2ze1. Similarly, ∇e2e3=-2ze2 and ∇_e3e₃ = 0. (2.9) further yields

From the above it follows that the manifold satisfies ∇_Xξ = f{X – η(X)ξ} for ξ = e₃, where f=-2z. Hence we conclude that M is an f-Kenmotsu manifold. Also f² + f′ ≠ 0. Hence M is a regular f-Kenmotsu manifold.

In this section we consider almost Ricci solitons on 3-dimensional f-Kenmotsu manifolds. In particular, let the potential vector field V be pointwise collinear with ξ i.e., V = bξ, where b is a function on M. Then from (1.1) we have

Using (2.3) in (3.1), we get

Putting Y = ξ in (3.2) and using (2.8) yields

Putting X = ξ in (3.3) we obtain

Putting the value of ξb in (3.3) yields

Applying d on (3.5) and using d² = 0, we get

Taking wedge product of (3.6) with η, we have

Since η ∧ dη ≠ 0 in a 3-dimensional f-Kenmotsu manifold, therefore

Using (3.8) in (3.5) gives db = 0 i.e., b =constant. Therefore from (3.2) we have

In view of (3.9) we can state the following:

Theorem 3.1

If in a 3-dimensional f-Kenmotsu manifold the metric g admits almost Ricci soliton and V is pointwise collinear with ξ, then V is constant multiple of ξ and the manifold is η-Einstein of the form (3.9).

The converse of the above theorem is not true, in general. However if we take f = constant, i.e., if we consider a 3-dimensional η-Einstein f-Kenmotsu manifold, then it admits a Ricci soliton. This can be proved as follows:

Let M be a 3-dimensional η-Einstein f-Kenmotsu manifold and V = ξ. Then

S(X, Y ) = γg(X, Y ) + δη(X)η(Y ), where γ and δ are certain scalars.

Now using (2.3)

Therefore

From equation (3.11) it follows that M admits a Ricci soliton (g, ξ, λ) if f+γ–λ = 0 and δ = f = constant. From (3.10) we have using (2.8), −2f² = γ + δ. Hence γ = −2f² – f = constant. Therefore λ = (γ + δ) = constant. So we have the following:

Theorem 3.2

If a 3-dimensional f-Kenmotsu manifold is η-Einstein of the form S = γg + δη ⊗ η, then a Ricci almost soliton (M,g, ξ, λ) reduces to a Ricci soliton (g, ξ, (γ + δ)).

Now let V = ξ. Then (3.1) reduces to

Now, in view of (2.6) we have

Putting X = Y = ξ in (3.14) yields

Assuming that f = constant, we get f′ = ξf = 0. This implies λ = 4f² = constant. Thus we can state the following:

Theorem 3.3

If a 3-dimensional f-Kenmotsu manifold with f =constant admits almost Ricci soliton then it reduces to a Ricci soliton.

This section is devoted to study 3-dimensional f-Kenmotsu manifolds admitting gradient almost Ricci soliton. For a gradient almost Ricci soliton, we have

where D denotes the gradient operator of g.

Differentiating (4.1) covariantly in the direction of X yields

Similarly, we get

and

In view of (4.2),(4.3) and (4.4), we have

We get from (2.6)

Differentiating (4.6) covariantly in the direction of X and using (2.3), we get

In view of (4.7), we get from (4.5)

This implies

Also, we have from (2.5)

In view of (4.9) and (4.10) we obtain

Assuming that the scalar curvature r and f are constants. Then it follows from (4.11) that

Applying d both sides of (12), we get

Using (13) in (12), we have

This implies λ = constant. Thus we can state the following:

Theorem 4.1

If a 3-dimensional f-Kenmotsu manifold admits gradient almost Ricci soliton then it reduces to a Ricci soliton provided the scalar curvature r and f are constants.

The author is thankful to the referee for hisher valuable comments towards the improvement of this article.