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

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.


Introduction
On a Riemannian manifold (M, g) if there exists a vector eld V and a constant λ satisfying L V g + Ric + λg = , (1.1) then it is said that the triple (g, V , λ), for simplicity, g, de nes a Ricci soliton (see Hamilton [1,2]), where Ric denotes the Ricci tensor. Usually, V and λ are said to be the potential vector eld and the soliton constant, respectively. If the potential vector eld 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 > ). The Ricci ow is an evolution equation for metrics on a Riemannian manifold de ned by ∂ ∂ t g ij (t) = − Ric ij (t).
Ricci solitons are self-similar solutions to the Ricci ow. 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 classi cation 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 de ned by L V g + Ric * + λg = , (1.2) where V and λ still denote a vector eld (called the potential vector eld) and a constant (called the soliton constant). On an almost contact metric manifold (M, ϕ, ξ , η, g), the *-Ricci tensor Ric * is de ned by for any vector elds 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 non at 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 -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 -manifold represents a *-Ricci soliton, then the manifold is locally isometric to the hyperbolic space H (− ).
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 eld 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 H n+ (− ) × R n provided that the potential vector eld is not in nitesimal contact transformation. We also construct two concrete examples to illustrate our main results.

(κ, µ) -almost Kenmotsu manifolds
Let (M n+ , g) be a smooth Riemannian manifold of dimension n + . On this manifold if there exist a ( , )type tensor eld ϕ, a global vector eld ξ and a -form η satisfying for any vector elds X, Y, then (ϕ, ξ , η, g) is called an almost contact metric structure and M n+ is called an almost contact metric manifold (see Blair [24]). Usually, ξ and η are called the Reeb or characteristic vector eld and an almost contact -form respectively. The fundamental 2-form Φ on an almost contact metric manifold M n+ is de ned by Φ(X, Y) = g(X, ϕY) for any vector elds X and Y. Let (M n+ , ϕ, ξ , η, g) be an almost contact manifold. We de ne on the product M n+ × R an almost complex structure J by where X denotes a vector eld tangent to M n+ , t is the coordinate of R and f is a C ∞ -function on M n+ × R. We denote by [ϕ, ϕ] the Nijenhuis tensor of ϕ (see [24]), if [ϕ, ϕ] = − dη ⊗ 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 = and dΦ = η ∧ 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 elds X, Y, where ∇ denotes the Levi-Civita connection of the metric g. On an almost Kenmotsu manifold, we set h = L ξ ϕ 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 eld ξ of an almost Kenmotsu manifold M n+ belongs to the so called (κ, µ) -nullity distribution, i.e., for some constants κ and µ, then M n+ is said to be a (κ, µ) -almost Kenmotsu manifold (see [10,23]). It follows from (2.6) that for any vector elds X, Y.

*-Ricci solitons on (κ, µ) -almost Kenmotsu manifolds
Given a (κ, µ) -almost Kenmotsu manifold M n+ , it has been proved in [23,Proposition 4 In view of symmetry of h and h ξ = , we denote by X an eigenvector eld of h orthogonal to ξ with corresponding eigenvalue θ. By (2.1) and (3.1), it follows that θ = −(κ + ) and hence we have κ ≤ − . The equality holds if and only if h = 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 < − , i.e., h ≠ everywhere.
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 for any vector elds 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 for any vector elds X, Y , Z.
Proof. Let M n+ be a (κ, µ) -almost Kenmotsu manifold with κ < − whose metric g is a *-Ricci soliton for certain potential vector eld V and constant λ. From (1.1) and (3.3) we have for any vector elds X, Y. Taking the covariant derivative of the above relation gives From Yano [31, pp. 23] we have the following formula On Riemannian manifold (M, g), because the metric g is parallel, it follows that (3.9) Putting (3.7) into (3.9) we obtain Taking the covariant derivative of (3.10) we get (3.11) With the aid of (3.1), it has been proved by Dileo Substituting (3.12) into (3.11) yields (3.13) Notice that the following formula was given by Yano in [31, pp. 23] for any vector elds X, Y , Z. Putting (3.13) into the above relation we complete the proof.

Following Blair [24, pp.72] and Tanno [32] we give
De nition 3.1. On an almost contact metric manifold (M, ϕ, ξ , η, g) a vector eld X is said to be in nitesimal contact transformation if L V η = fη for some function f . In particular, X is said to be strict in nitesimal contact transformation if L X η = .
Our main result in this paper is the following. From (3.15) and (3.16), contracting X in (3.5) or (3.14) we have From (3.2), the Ricci tensor can be written as Taking the Lie derivative of this relation along the potential vector eld V we obtain for any vector elds Y , Z, where we have applied (3.6). Comparing (3.18) with (3.17) gives Note that by setting X = Y = ξ in (3.6) we obtain η(∇ ξ V) + λ = . Applying this equation and replacing Y = Z = ξ in the previous relation we obtain λ = because of κ < − and hence for any vector eld Y. With the aid of λ = , replacing Y in (3.19) by ξ we get The action of the operator h on the above relation gives h L V ξ = L V ξ because of (3.1), λ = and κ < − . Comparing this equation with the previous one gives either L V ξ = or κ = − . It has been proved by Dileo and Pastore in [23, pp. 56] that if κ = − , then a (κ, µ) -almost Kenmotsu manifold is locally isometric to the Riemannian product H n+ (− ) × R n . In this case, from (3.6) we see that the potential vector eld V is Killing. Note that the almost Kenmotsu structure on the product H n+ (− ) × R n was constructed in [23]. If κ ≠ − , then we get L V ξ = , and with the aid of λ = and (L V g)(ξ , X) = we have for any vector eld X. Then, V is strict in nitesimal contact transformation. This completes the proof.

Remark 3.1.
In [15], it was proved that if a -dimensional cosymplectic manifold admits a Ricci soliton, then either the manifold is locally at or the potential vector eld is an in nitesimal contact transformation. In view of this result and our Theorem 3.1, it is interesting to investigate the existence and properties of in nitesimal contact transformations on almost Kenmotsu and cosymplectic manifolds in future. [20,Theorem 16] proved that a non-Sasakian (κ, µ)-contact manifold of dimension > admitting a non-trivial *-Ricci soliton is locally isometric to R n+ × S n ( ). Our Theorem 3.1 is in analogy with Ghosh-Patra's result in almost Kenmotsu geometry.

Examples
Before closing this paper, we present two concrete examples of (κ, µ) -almost Kenmotsu manifolds of dimension three admitting a *-Ricci soliton.
Let G be a Lie group whose Lie algebra g is given by We denote e by e and ϕe by e , respectively. According to the Koszul formula, the Levi-Civita connection on G can be written as follows: ∇ ξ ξ = , ∇ ξ e = , ∇ ξ ϕe = , One can check that on G there exists a left invariant almost Kenmotsu structure (ϕ, ξ , η, g) satisfying the (− − θ , − ) -nullity condition. In particular, Lie group G is locally isometric to the product H (− ) × R (see also [23]) when θ = . Next, we show that there exist two kinds of *-Ricci solitons on G.