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

1 Introduction

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

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

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

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

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

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

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

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]:

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

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

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

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.

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

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

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:

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.

1. 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.

1. 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)