Hamilton introduced Yamabe flow in 1982 [8] at the same time he introduced Ricci flow. Ricci solitons and Yamabe solitons are the limit of the solutions of Ricci flow and Yamabe flow respectively. In dimension n = 2 the Yamabe soliton is equivalent to the Ricci soliton. However, in dimension n>2, the Yamabe and Ricci solitons do not agree as the Yamabe soliton preserves the conformal class of the metric but the Ricci soliton does not in general.

Over the past twenty years the theory of geometric flows, such as Ricci flow and Yamabe flow has been the focus of attention of many geometers. Recently, in 2019, Guler and Crasmareanu [7] introduced the study of a new geometric flow which is a scalar combination of Ricci and Yamabe flows under the name Ricci-Yamabe map. This is also called the Ricci-Yamabe flow of type (α,β). The Ricci-Yamabe flow is an evolution equation for the metrics on the Riemannian or semi-Riemannian manifolds defined as [7]

A solution to the Ricci-Yamabe flow is called Ricci-Yamabe soliton if it depends only on one parameter group of diffeomorphism and scaling. A Ricci-Yamabe soliton on a Riemannian manifold (M,g) consists of data (g,V,λ,α,β) satisfying

where £V is the Lie-derivative, S is the Ricci tensor, r is the scalar curvature and λ,α,β∈ℝ. If V is the gradient of a smooth function f on M, then the above soliton is called the gradient Ricci-Yamabe soliton and then equation (1.2) reduces to

where ∇2f is the Hessian of f.

The Ricci-Yamabe soliton is said to be expanding, steady, or shrinking according to whether λ is negative, zero, or positive. It is called an almost Ricci-Yamabe soliton if α,β and λ are smooth functions on M. A Ricci-Yamabe soliton is said to be a [5]

• • Ricci soliton [8] if α =1, β=0.

• • Yamabe soliton[9] if α =0, β = 1.

• • Einstein soliton [2] if α =1, β = -1.

• • ρ-Einstein soliton [3] if α =1, β = -2ρ.

When α≠0,1, a Ricci-Yamabe soliton is proper. All of these classifications apply to gradient Ricci-Yamabe solitons as well.

The paper is organized as follows. After preliminaries in Section 2, we consider Ricci-Yamabe solitons on Kenmotsu 3-manifolds in Section 3. In Section 4 we study Ricci-Yamabe solitons on Kenmotsu 3-manifolds with η-parallel Ricci tensors. Section 5 is devoted to studying gradient Ricci-Yamabe solitons on Kenmotsu 3-manifolds. Finally, in Section 6 we construct an example of a 3-dimensional Kenmotsu manifold admitting a Ricci-Yamabe soliton.

An almost contact structure [1] on a (2n+1)-dimensional smooth manifold M2n+1 is a triplet (ϕ,ξ,η), where ϕ is a (1,1)-type tensor, ξ a global vector field and η a 1-form, such that

where 'id' denotes the identity mapping and relation (2.1) implies that ϕ(ξ)=0, η∘ϕ=0 and rank(ϕ)=2n. The almost contact structure induces a natural almost complex structure J on the product manifold M×ℝ defined by J(U,λd/dt)=(ϕU−λξ,η(U)d/dt), where U is tangent to M, t the coordinate of ℝ and λ a smooth function on M×ℝ. The almost contact structure is said to be normal [12] if the almost complex structure J is integrable or equivalently [ϕ,ϕ]+2dη⊗ξ vanishes, where [ϕ,ϕ] is the Nijenhuis torsion of ϕ. Let g be a compatible Riemannian metric with (ϕ,ξ,η), that is, let

or equivalently, Φ(U,V)=g(U,ϕV) along with g(U,ξ)=η(U) for all U,V∈χ(M). With this, M is an almost contact metric manifold equipped with an almost contact metric structure (ϕ,ξ,η,g). An almost contact metric manifold is called a Kenmotsu manifold if it satisfies

for all U,V∈χ(M), where ∇ is Levi-Civita connection of the Riemannian metric. A Kenmotsu manifold is normal but not Sasakian. Moreover, it is also not compact since from the formula (2.3) we get

which gives divξ=2n. A conformal change g^* of a Riemannian metric g is called a concircular transformation [14] if geodesic circles of g are transformed into geodesic circles of g^*. Here a geodesic circle means a curve whose first curvature is constant and whose second curvature is identically zero. A cosymplectic structure is defined to be a normal almost contact metric structure (ϕ,ξ,η,g) with both the fundamental 2-form 𝚽 and the 1-form η is closed. An almost contact metric structure is cosymplectic if and only if ∇ϕ=0. In [11], Kirichenko obtained the class of Kenmotsu manifolds from cosymplectic manifolds by the canonical concircular transformations. A Kenmotsu manifold is of constant curvature -1 if and only if it is canonically concircular to Cn×ℝ [11].

For a (2n+1)-dimensional Kenmotsu manifold, the following formulas hold:

for any U,V∈χ(M), where S is the Ricci tensor and Q is the Ricci operator.

From [4], we know that for a 3-dimensional Kenmotsu manifold

where R is the curvature tensor and r is the scalar curvature of the manifold M.

An almost contact metric manifold is said to be η-Einstein if the Ricci tensor S satisfies

for any vector field V,W on M and arbitrary functions a, b on M. An η-Einstein manifold with b vanishing and a constant is obviously an Einstein manifold. An η-Einstein manifold is said to be proper η-Einstein if b≠0.

Three-dimensional Kenmotsu manifold have been studied in the papers ([4], [13]).

Lemma 2.1.([13]) On any three-dimensional Kenmotsu manifold (M3,ϕ,ξ,η,g) we have

Lemma 2.2.([4]) A 3-dimensional Riemannian manifold is a manifold of constant sectional curvature -1 if and only if the scalar curvature r=-6.

Lemma 2.3.(Proposition 8 of ([10]) Let M be an η-Einstein Kenmotsu manifold S=ag+bη⊗η for scalars a and b. If either a or b is constant then the manifold becomes an Einstein manifold.

Assume that the Kenmotsu 3-manifold admits a proper Ricci-Yamabe soliton (g,ξ,λ,α,β). Then the relation (1.2) yields

In Kenmotsu 3-manifolds

Using (3.2) in (3.1), we get

which implies

Hence we can state the following theorem using Lemma 2.3 :

Theorem 3.1. A proper Ricci-Yamabe soliton on a 3-dimensional Kenmotsu manifold is an Einstein manifold.

In this section we study Ricci-Yamabe solitons on Kenmotsu 3-manifolds with η-parallel Ricci tensor. A Kenmotsu manifold is said to have η-parallel Ricci tensor if [6]

for all smooth vector fields U,V,W.

In Kenmotsu 3-manifolds

Using (4.2) in (4.1), we get

Putting V=ξ in (4.3) and using Lemma 2.1 we obtain

If α=1, then (4.4) implies either β = 0 or, r=-6.

Case I: If β = 0 and α=1, then Ricci-Yamabe soliton becomes Ricci soliton.

Case II: If r=-6, then from Lemma 2.2, the manifold becomes a manifold of constant sectional curvature -1.

Hence we conclude the following:

Theorem 4.1. If a 3-dimensional Kenmotsu manifold admits a Ricci-Yamabe soliton with η-parallel Ricci tensor, then either it is a Ricci soliton or it is a manifold of constant sectional curvature -1, provided α=1.

Suppose a Kenmotsu 3-manifold admits the gradient Ricci-Yamabe soliton. Then from equation (1.3), we get

Differentiating (5.1) covariantly along any vector field V, we get

Interchanging U and V in the above equation, we infer

Hence from the above equations, we get

Now, in 3-dimension Kenmotsu manifolds

Using (5.5) in (5.4), we get

Contracting (5.6) and using Lemma 2.1, we get

Again, replacing U by Df in (2.8), we get

In view of (5.7) and (5.8) we infer

Putting V = ξ in the above equation, we get

Taking inner product of (5.7) with the vector field ξ, we get

Putting U = ξ in (5.11) and using Lemma 2.1, we get

Using (5.10) and (5.12) in (5.9), we obtain

Hence above equation implies either, β2r+3β+α=0 or, Vr+2(r+6)η(V)=0.

Case I: If β2r+3β+α=0, then r is constant.

Case II: If Vr=−2(r+6)η(V). Using this in (5.12), we get

which implies

Using the above equation in (5.1), we get

which implies r is constant.

Thus, we state the following:

Theorem 5.1. If the metric of a Kenmotsu 3-manifold M is a gradient Ricci-Yamabe soliton, then the scalar curvature is constant.

If we take β =1 and α=0, then both cases implies r=-6. Then from Lemma 2.2, we say that it is a manifold of constant sectional curvature -1.

We consider the 3-dimensional manifold M={(x,y,z)∈ℝ3,z≠0}, where (x,y,z) are the standard coordinate of ℝ3. Let {e1,e2,e3} be a linearly independent global frame on M given by

Let g be the Riemannian metric defined by

Let η be the 1-form defined by η(W)=g(W,e3), for all W∈χ(M) and ϕ be the (1,1)-tensor defined by

Then using the linearity of ϕ and g, we get

for any V,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

The Riemannian connection ∇ of the metric g and using Koszul's formula , we have

For the above we see that ∇Wξ=W−η(W)ξ for all W∈χ(M). Hence the manifold is a Kenmotsu manifold.

Now, we have

With the help of the above results and using (6.1), we obtain

From the above, we can easily calculate the non-vanishing components of the Ricci tensor S as follows:

Hence from the above, we get

where r is the scalar curvature.

From (3.3) we obtain

Therefore λ=2α−3β. Hence it is Ricci-Yamabe soliton on Kenmotsu 3-manifolds.

The authors are thankful to the referee for his valuable suggestions towards the improvement of the paper.