*-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds

Xinxin Dai; Yan Zhao; Uday Chand De

doi:10.1515/math-2019-0056

Article Open Access

*-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds

Xinxin Dai , Yan Zhao and Uday Chand De

Published/Copyright: July 31, 2019

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From the journal Open Mathematics Volume 17 Issue 1

Abstract

Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍⁿ⁺¹(−4)×ℝⁿ or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

Keywords: Almost Kenmotsu manifold; *-Ricci soliton; nullity distribution; Lie group

MSC 2010: Primary 53D15; Secondary 53C25

1 Introduction

On a Riemannian manifold (M, g) if there exists a vector field V and a constant λ satisfying

12LVg+Ric+λg=0, (1.1)

then it is said that the triple (g, V, λ), for simplicity, g, defines a Ricci soliton (see Hamilton [1, 2]), where Ric denotes the Ricci tensor. Usually, V and λ are said to be the potential vector field and the soliton constant, respectively. If the potential vector field V is Killing, (1.1) reduces to an Einstein metric (that is, the Ricci tensor is a constant multiple of the Riemannian metric when dimM > 2). The Ricci flow is an evolution equation for metrics on a Riemannian manifold defined by

∂∂tgij(t)=−2Ricij(t).

Ricci solitons are self-similar solutions to the Ricci flow.

The studying of Ricci solitons on almost contact metric manifolds was introduced by R. Sharma in [3]. In the last decade, a large number of papers were published regarding classification of Ricci solitons on almost contact manifolds. Among others, we refer the readers to [4, 5, 6, 7], [8, 9, 10, 11, 12] and [13, 14, 15, 16] for fruitful results on (almost) Ricci solitons on contact metric, (almost) Kenmotsu and (almost) cosymplectic manifolds, respectively. Recently, a new research interest has appeared regarding the so called *-Ricci soliton which is defined by

12LVg+Ric∗+λg=0, (1.2)

where V and λ still denote a vector field (called the potential vector field) and a constant (called the soliton constant). On an almost contact metric manifold (M, ϕ, ξ, η, g), the *-Ricci tensor Ric^* is defined by

Ric∗(X,Y)=12trace{Z→R(X,ϕY)ϕZ} (1.3)

for any vector fields X, Y. The *-Ricci tensor Ric^* in almost contact geometry can be regarded in analogy with the usual Ricci tensor in Riemannian geometry. As usual, for simplicity, we say that the metric g of an almost contact metric manifold is a *-Ricci soliton if (1.2) is true.

The notion of *-Ricci tensor was introduced on an almost Hermitian manifold by Tachibana in [17]. Later, such notion was considered on real hypersurfaces of a nonflat complex space form by Kaimakamis and Panagiotidou [18] (see also [19]). Recently, *-Ricci solitons on an almost contact metric manifold (M, ϕ, ξ, η, g) were started to be considered by some authors. More precisely, *-Ricci solitons on Sasakian 3-manifolds and (κ, μ)-contact manifolds were investigated in [20] and [21] respectively. Y. Wang in [22] proved that if the metric of a Kenmotsu 3-manifold represents a *-Ricci soliton, then the manifold is locally isometric to the hyperbolic space ℍ³(−1).

In this paper, we start to investigate the *-Ricci solitons on almost Kenmotsu manifolds. Because the class of almost Kenmotsu manifolds is rather large, then we have to consider some other special almost Kenmotsu manifolds. By a (κ, μ)′-almost Kenmotsu manifold, we mean that the Reeb vector field of the manifold belongs to the (κ, μ)′-nullity distribution (see [23]). We prove that if the metric of a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold is a *-Ricci soliton, then the manifold is locally isometric to the product ℍⁿ⁺¹(−4) × ℝⁿ provided that the potential vector field is not infinitesimal contact transformation. We also construct two concrete examples to illustrate our main results.

2 (κ, μ)′-almost Kenmotsu manifolds

Let (M²ⁿ⁺¹, g) be a smooth Riemannian manifold of dimension 2n + 1. On this manifold if there exist a (1, 1)-type tensor field ϕ, a global vector field ξ and a 1-form η satisfying

ϕ2=−id+η⊗ξ,η(ξ)=1, (2.1)

g(ϕX,ϕY)=g(X,Y)−η(X)η(Y) (2.2)

for any vector fields X, Y, then (ϕ, ξ, η, g) is called an almost contact metric structure and M²ⁿ⁺¹ is called an almost contact metric manifold (see Blair [24]). Usually, ξ and η are called the Reeb or characteristic vector field and an almost contact 1-form respectively.

The fundamental 2-form Φ on an almost contact metric manifold M²ⁿ⁺¹ is defined by Φ(X, Y) = g(X, ϕ Y) for any vector fields X and Y. Let (M²ⁿ⁺¹,ϕ, ξ, η, g) be an almost contact manifold. We define on the product M²ⁿ⁺¹ × ℝ an almost complex structure J by

JX,fddt=ϕX−fξ,η(X)ddt,

where X denotes a vector field tangent to M²ⁿ⁺¹, t is the coordinate of ℝ and f is a 𝓒^∞-function on M²ⁿ⁺¹ × ℝ. We denote by [ϕ, ϕ] the Nijenhuis tensor of ϕ (see [24]), if [ϕ, ϕ] = −2dη ⊗ ξ (or equivalently, the almost complex structure J is integrable), then the almost contact metric structure is said to be normal.

On an almost contact metric manifold if there hold dη = 0 and dΦ = 2η ∧ Φ, then the manifold is said to be an almost Kenmotsu manifold (see [25]). A normal almost Kenmotsu manifold is said to be a Kenmotsu manifold (see [26]) and this is also equivalent to

(∇Xϕ)Y=g(ϕX,Y)ξ−η(Y)ϕX (2.3)

for any vector fields X, Y, where ∇ denotes the Levi-Civita connection of the metric g. On an almost Kenmotsu manifold, we set h = 12 𝓛_ξϕ and h′ = h ∘ ϕ, where 𝓛 is the Lie derivative. It is easily seen that both the above two operators are symmetric. The following formulas can be seen in [23, 27]:

hξ=0,trh=tr(h′)=0,hϕ+ϕh=0, (2.4)

∇ξ=h′+id−η⊗ξ. (2.5)

If the Reeb vector field ξ of an almost Kenmotsu manifold M²ⁿ⁺¹ belongs to the so called (κ, μ)′-nullity distribution, i.e.,

R(X,Y)ξ=κ(η(Y)X−η(X)Y)+μ(η(Y)h′X−η(X)h′Y) (2.6)

for some constants κ and μ, then M²ⁿ⁺¹ is said to be a (κ, μ)′-almost Kenmotsu manifold (see [10, 23]). It follows from (2.6) that

R(ξ,X)Y=κ(g(X,Y)ξ−η(Y)X)+μ(g(h′X,Y)ξ−η(Y)h′X) (2.7)

for any vector fields X, Y.

3 *-Ricci solitons on (κ, μ)′-almost Kenmotsu manifolds

Given a (κ, μ)′-almost Kenmotsu manifold M²ⁿ⁺¹, it has been proved in [23, Proposition 4.1] that μ = −2. Also, we have

h2=h′2=(κ+1)ϕ2. (3.1)

In view of symmetry of h′ and h′ξ = 0, we denote by X an eigenvector field of h′ orthogonal to ξ with corresponding eigenvalue θ. By (2.1) and (3.1), it follows that θ² = −(κ + 1) and hence we have κ ≤ −1. The equality holds if and only if h = 0 and in this case the manifold is called a C-almost Kenmotsu manifold because in this context the Reeb foliation is conformal (see [28]). Throughout this paper, we consider those (κ, μ)′-almost Kenmotsu manifolds with κ < −1, i.e., h ≠ 0 everywhere.

Lemma 3.1

([29]). On a (κ, μ)′-almost Kenmotsu manifold with κ < −1 the Ricci operator is given by

Q=−2nid+2n(κ+1)η⊗ξ−2nh′, (3.2)

where the Ricci operator is defined by Ric(X, Y) = g(QX, Y).

Lemma 3.2

On a (κ, μ)′-almost Kenmotsu manifold with κ < −1 the *-Ricci tensor is given by

Ric∗(X,Y)=−(κ+2)(g(X,Y)−η(X)η(Y)) (3.3)

for any vector fields X, Y.

Proof

On a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold, using [23, Proposition 4.2.] we compute the curvature tensor of the manifold as the following

R(X,Y)Z=κη(Z)(η(Y)X−η(X)Y)+κ(g(Y,Z)η(X)−g(X,Z)η(Y))ξ+(g(Y−h′Y,Z)η(X)−g(X−h′X,Z)η(Y))ξ+η(Z)(η(Y)(X−h′X)−η(X)(Y−h′Y))−(g(Y+h′Y,Z)(X+h′X)−g(X+h′X,Z)(Y+h′Y)) (3.4)

for any vector fields X, Y, Z. We remark that expression of curvature tensor R of non-Kenmotsu α-almost Kenmotsu manifolds satisfying (κ, μ)′-nullity condition was obtain by D. Dileo [30, Corollary 2]. However, her formula contains a little error. The application of (3.4) together with the definition of *-Ricci tensor (1.3) and (2.4) give (3.3).□

Notice that (3.2) can be deduced directly from (3.4).

Lemma 3.3

If a triple (g, V, λ) on a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold is a *-Ricci soliton, then we have

(LVR)(X,Y,Z)=2(κ+2)(g(X+h′X,Z)(Y+h′Y)−g(Y+h′Y,Z)(X+h′X))+2(κ+2)(η(Y)η(Z)(X+h′X)−η(X)η(Z)(Y+h′Y))+2(κ+2)(g(κY−2h′Y,Z)η(X)−g(κX−2h′X,Z)η(Y))+2(κ+2)(g(Y+h′Y,Z)η(X)−g(X+h′X,Z)η(Y))ξ (3.5)

for any vector fields X, Y, Z.

Proof

Let M²ⁿ⁺¹ be a (κ, μ)′-almost Kenmotsu manifold with κ < −1 whose metric g is a *-Ricci soliton for certain potential vector field V and constant λ. From (1.1) and (3.3) we have

(LVg)(X,Y)=2(κ+2−λ)g(X,Y)−2(κ+2)η(X)η(Y) (3.6)

for any vector fields X, Y. Taking the covariant derivative of the above relation gives

(∇XLVg)(Y,Z)=−2(κ+2)(g(X+h′X,Y)η(Z)+g(X+h′X,Z)η(Y)−2η(X)η(Y)η(Z)). (3.7)

From Yano [31, pp. 23] we have the following formula

(∇ZLVg−LV∇Zg+∇[V,Z]g)(X,Y)=g((LV∇)(Z,X),Y)+g((LV∇)(Z,Y),X).

On Riemannian manifold (M, g), because the metric g is parallel, it follows that

(∇ZLVg)(X,Y)=g((LV∇)(Z,X),Y)+g((LV∇)(Z,Y),X). (3.8)

In view of the symmetry of the (1, 2)-type tensor filed 𝓛_V∇, i.e., (𝓛_V∇)(X, Y) = (𝓛_V∇)(Y, X), interchanging cyclicly the roles of X, Y, Z in (3.8) we obtain

g((LV∇)(X,Y),Z)=12(∇XLVg)(Y,Z)+12(∇YLVg)(Z,X)−12(∇ZLVg)(X,Y). (3.9)

Putting (3.7) into (3.9) we obtain

(LV∇)(X,Y)=−2(κ+2)g(X+h′X,Y)ξ+2(κ+2)η(X)η(Y)ξ. (3.10)

Taking the covariant derivative of (3.10) we get

(∇XLV∇)(Y,Z)=−2(κ+2)g((∇Xh′)Y,Z)ξ−2(κ+2)g(Y+h′Y,Z)(X−η(X)ξ+h′X)+2(κ+2)g(X−η(X)ξ+h′X,Y)η(Z)ξ+2(κ+2)g(X−η(X)ξ+h′X,Z)η(Y)ξ+2(κ+2)η(Y)η(Z)(X−η(X)ξ+h′X). (3.11)

With the aid of (3.1), it has been proved by Dileo and Pastore in [23, Lemma 4.1] that on M²ⁿ⁺¹ there holds

(∇Xh′)Y=g((κ+1)X−h′X,Y)ξ+η(Y)((κ+1)X−h′X)−2(κ+1)η(X)η(Y)ξ. (3.12)

Substituting (3.12) into (3.11) yields

(∇XLV∇)(Y,Z)=−2(κ+2)(g(Y+h′Y,Z)−η(Y)η(Z))(X+h′X)−2(κ+2)g(κX−2h′X+(1−2κ)ξ,Y)η(Z)ξ−2(κ+2)g(κX−2h′X+(1−2κ)ξ,Z)η(Y)ξ. (3.13)

Notice that the following formula was given by Yano in [31, pp. 23]

(LVR)(X,Y)Z=(∇XLV∇)(Y,Z)−(∇YLV∇)(X,Z) (3.14)

for any vector fields X, Y, Z. Putting (3.13) into the above relation we complete the proof.□

Following Blair [24, pp.72] and Tanno [32] we give

Definition 3.1

On an almost contact metric manifold (M, ϕ, ξ, η, g) a vector field X is said to be infinitesimal contact transformation if 𝓛_Vη = fη for some function f. In particular, X is said to be strict infinitesimal contact transformation if 𝓛_Xη = 0.

Our main result in this paper is the following.

Theorem 3.1

If a triple (g, V, λ) on a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold is a *-Ricci soliton, then the soliton constant λ = 0, and either the manifold is locally isometric to the product ℍⁿ⁺¹(−4) × ℝⁿ and the potential vector field is Killing, or the potential vector field is strict infinitesimal contact transformation.

Proof

Let {e_i : i = 0, 1, 2, …2n} be an orthonormal basis for tangent space at each point of the manifold. Taking the inner product of (3.13) with X, setting X = e_i in (3.13) and summing over i ∈ {0, 1, 2 …2n} we obtain

∑i=02ng((∇eiLV∇)(Y,Z),ei)=−4n(κ+2)g(Y−η(Y)ξ+h′Y,Z), (3.15)

where we have employed trh′ = 0 and h′² = (κ + 1)ϕ². Similarly, taking the inner product of (3.13) with Y, replacing Y by e_i in (3.13) and summing over i ∈ {0, 1, 2 …2n} we obtain

∑i=02ng((∇XLV∇)(ei,Z),ei)=0. (3.16)

From (3.15) and (3.16), contracting X in (3.5) or (3.14) we have

(LVRic)(Y,Z)=−4n(κ+2)g(Y−η(Y)ξ+h′Y,Z). (3.17)

From (3.2), the Ricci tensor can be written as

Ric(Y,Z)=−2ng(Y,Z)+2n(κ+1)η(Y)η(Z)+2ng(h′Y,Z).

Taking the Lie derivative of this relation along the potential vector field V we obtain

(LVRic)(Y,Z)=−4n(κ+2−λ)g(Y,Z)+4n(κ+2−2λ(κ+1))η(Y)η(Z)+2n(κ+1)(η(Y)g(LVξ,Z)+η(Z)g(LVξ,Y))−2ng((LVh′)Y,Z)−4n(κ+2−λ)g(h′Y,Z) (3.18)

for any vector fields Y, Z, where we have applied (3.6). Comparing (3.18) with (3.17) gives

g((LVh′)Y,Z)=2λg(Y,Z)+2λg(h′Y,Z)−4λ(κ+1)η(Y)η(Z)+(κ+1)(η(Y)g(LVξ,Z)+η(Z)g(LVξ,Y)).

Note that by setting X = Y = ξ in (3.6) we obtain η(∇_ξ V) + λ = 0. Applying this equation and replacing Y = Z = ξ in the previous relation we obtain λ = 0 because of κ < −1 and hence

(LVh′)Y=(κ+1)(η(Y)LVξ+g(LVξ,Y)ξ) (3.19)

for any vector field Y. With the aid of λ = 0, replacing Y in (3.19) by ξ we get

h′LVξ=−(κ+1)LVξ.

The action of the operator h′ on the above relation gives h′𝓛_Vξ = 𝓛_Vξ because of (3.1), λ = 0 and κ < −1. Comparing this equation with the previous one gives either 𝓛_Vξ = 0 or κ = −2.

It has been proved by Dileo and Pastore in [23, pp. 56] that if κ = −2, then a (κ, μ)′-almost Kenmotsu manifold is locally isometric to the Riemannian product ℍⁿ⁺¹(−4) × ℝⁿ. In this case, from (3.6) we see that the potential vector field V is Killing. Note that the almost Kenmotsu structure on the product ℍⁿ⁺¹(−4) × ℝⁿ was constructed in [23]. If κ ≠ −2, then we get 𝓛_Vξ = 0, and with the aid of λ = 0 and (𝓛_Vg)(ξ, X) = 0 we have

(LVη)X=LVη(X)−η(∇VX)+η(∇XV)=0

for any vector field X. Then, V is strict infinitesimal contact transformation. This completes the proof.□

Remark 3.1

In [15], it was proved that if a 3-dimensional cosymplectic manifold admits a Ricci soliton, then either the manifold is locally flat or the potential vector field is an infinitesimal contact transformation. In view of this result and our Theorem 3.1, it is interesting to investigate the existence and properties of infinitesimal contact transformations on almost Kenmotsu and cosymplectic manifolds in future.

Remark 3.2

Ghosh and Patra in [20, Theorem 16] proved that a non-Sasakian (κ, μ)-contact manifold of dimension > 3 admitting a non-trivial *-Ricci soliton is locally isometric to ℝⁿ⁺¹ × 𝕊ⁿ(4). Our Theorem 3.1 is in analogy with Ghosh-Patra′s result in almost Kenmotsu geometry.

4 Examples

Before closing this paper, we present two concrete examples of (κ, μ)′-almost Kenmotsu manifolds of dimension three admitting a *-Ricci soliton.

It was shown in [22, Theorem 3.2] that any 3-dimensional non-unimodular Lie group admits a left invariant almost Kenmotsu structure satisfying the (κ, μ, ν)-nullity condition. In particular, (κ, μ)′-almost Kenmotsu manifolds of dimension three were completely classified in [23, Theorem 5.1].

Let G be a Lie group whose Lie algebra 𝔤 is given by

[e1,e2]=(1+θ)e2,[e2,e3]=0,[e3,e1]=(θ−1)e3,

where θ is a positive constant. Let g be the metric defined on G by g(e_i, e_j) = δ_ij for 1 ≤ i, j ≤ 3. We denote by ξ = -e₁ and denote by η the dual 1-form of ξ with respect to g. We define a (1, 1)-type tensor field ϕ by

ϕ(ξ)=0,ϕ(e2)=e3,ϕ(e3)=−e2.

We denote e by e₂ and ϕ e by e₃, respectively. According to the Koszul formula, the Levi-Civita connection on G can be written as follows:

∇ξξ=0,∇ξe=0,∇ξϕe=0, (4.1)

∇eξ=(1+θ)e,∇ee=−(θ+1)ξ,∇eϕe=0, (4.2)

∇ϕeξ=(1−θ)ϕe,∇ϕee=0,∇ϕeϕe=(θ−1)ξ, (4.3)

One can check that on G there exists a left invariant almost Kenmotsu structure (ϕ, ξ, η, g) satisfying the (−1 − θ², −2)′-nullity condition. In particular, Lie group G is locally isometric to the product ℍ²(−4) × ℝ (see also [23]) when θ = 1. Next, we show that there exist two kinds of *-Ricci solitons on G.

θ = 1. We suppose that V := α ξ+βe + γϕ e is a Killing vector field defined on G, i.e., g(∇_XV, Y) + g(∇_YV, X) = 0 for any vector fields X, Y, where α, β and γ are three smooth functions varying along ξ, e and ϕ e. Locally, we observe that V is a Killing vector field if and only if the following six equations hold:

g(∇ξV,e)+g(∇eV,ξ)=0,g(∇ξV,ϕe)+g(∇ϕeV,ξ)=0,g(∇eV,ϕe)+g(∇ϕeV,e)=0,g(∇ξV,ξ)=g(∇eV,e)=g(∇ϕeV,ϕe)=0.

The above six equations can be rewritten as the following

ξ(β)+e(α)=2β,ξ(γ)+ϕe(α)=0,e(γ)+ϕe(β)=0, (4.4)

ξ(α)=0,2α+e(β)=0,ϕe(γ)=0. (4.5)

One can find many solutions of the system of partial differential equations (4.4)-(4.5). For instance, if we set α = 0 and γ = 1, from (4.4)-(4.5) we obtain

ξ(β)=2β,e(β)=0,ϕe(β)=0. (4.6)

On the (−2, −2)′-almost Kenmotsu 3-manifold G mentioned above, βe + ϕ e is a Killing vector field if β satisfies (4.6). Next we give some local expressions of these solutions.

Example 4.1

Let M³ := {(x, y, z) ∈ ℝ³} be an open submanifold of the Euclidean space ℝ³. On M³, we consider three linear independently vector fields

e1=∂∂x−2y∂∂y+∂∂z,e2=∂∂y,e3=∂∂z.

One can check that

[e1,e2]=2e2,[e2,e3]=0,[e3,e1]=0.

On M³ we define a (1, 1)-type tensor field ϕ by ϕ(e₁) = 0, ϕ e₂ = e₃ and ϕ e₃ = -e₂, and we define a Riemannian metric g such that g(e₁, e₁) = g(e₂, e₂) = g(e₃, e₃) = 1. We denote by ξ = −e₁ and η its dual 1-form with respect to the metric g. One can check that (M³,ϕ, ξ, η, g) is a (−2, −2)′-almost Kenmotsu 3-manifold. On M³, from (4.6) we have β = ce^−2x, where c is a non-zero constant. Thus, ce−2x∂∂y+∂∂z is a Killing vector field and hence (ce−2x∂∂y+∂∂z,0,g) is a *-Ricci soliton on M³.

θ ≠ 1. Following the second case of Theorem 3.1, if there exists a *-Ricci soliton (V, λ, g) on G, then it becomes (3.6) with κ = −θ² − 1 ≠ −2, and we also have λ = 0 and 𝓛_Vξ = 0. Also, in this case, the potential vector field V is strict infinitesimal contact transformation. Suppose that V is the vector field satisfying the above conditions.

Replacing Y by ξ in (3.6), with the aid of λ = 0 and 𝓛_Vξ = 0, we see that η(V) is a constant, say α. Then, we may write V = αξ+βe + γϕ e with β, γ smooth functions. In this case, with the aid of (4.1)-(4.3), 𝓛_Vξ = 0 is equivalent to

ξ(β)=β(1+θ),ξ(γ)=γ(1−θ). (4.7)

Putting X = Y = e in (3.6), using λ = 0 and (4.1)-(4.3), we have

e(β)=−α(θ+1)−θ2+1. (4.8)

Similarly, putting X = Y = ϕ e in (3.6), using λ = 0 and (4.1)-(4.3), we have

ϕe(γ)=α(θ−1)−θ2+1. (4.9)

Finally, putting X = e and Y = ϕ e in (3.6), using λ = 0 and (4.1)-(4.3), we have

ϕe(β)+e(γ)=0. (4.10)

For simplicity, we set γ = 0, then V = αξ + βe + γϕ e on G with θ = 2 is a *-Ricci soliton if and only if

ξ(β)=3β,e(β)=−12,ϕe(β)=0. (4.11)

Example 4.2

Let M³ := {(x, y, z) ∈ ℝ³} be an open submanifold of the Euclidean space ℝ³. On M³, we consider three linear independently vector fields

e1=∂∂x−3y∂∂y+z∂∂z,e2=∂∂y,e3=∂∂z.

One can check that

[e1,e2]=3e2,[e2,e3]=0,[e3,e1]=e3.

On M³ we define a (1, 1)-type tensor field ϕ by ϕ(e₁) = 0, ϕ e₂ = e₃ and ϕ e₃ = −e₂, and we define a Riemannian metric g such that g(e₁, e₁) = g(e₂, e₂) = g(e₃, e₃) = 1. We denote by ξ = −e₁ and η its dual 1-form with respect to the metric g. One can check that (M³,ϕ, ξ, η, g) is a (−5, −2)′-almost Kenmotsu 3-manifold. On M³, from (4.11) we have β = ce^−3x −12y, where c is a non-zero constant. Thus, (3∂∂x+(ce−3x−12y)∂∂y,0,g) is a *-Ricci soliton on M³.

Acknowledgement

The authors would like to thank anonymous referees for their careful reading and many valuable suggestions. The second author was supported by the Doctoral Foundation of Henan University of Technology (No. 2018BS061).

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Received: 2018-12-25

Accepted: 2019-04-25

Published Online: 2019-07-31

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

Articles in the same Issue

https://doi.org/10.1515/math-2019-0056

Keywords for this article

Almost Kenmotsu manifold; *-Ricci soliton; nullity distribution; Lie group

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