Almost Kenmotsu 3- h -metric as a cotton soliton

Purpose – Cotton soliton is a newly introduced notion in the field of Riemannian manifolds. The object of this article is to study the properties of this soliton on certain contact metric manifolds. Design/methodology/approach – The authors consider the notion of Cotton soliton on almost Kenmotsu 3-manifolds. The authors use a local basis of the manifold that helps to study this notion in terms of partial differential equations. Findings – First the authorsconsiderthatthe potentialvector fieldispointwisecollinearwiththe Reeb vector fieldandproveanon-existenceofsuchCottonsoliton.NexttheauthorsassumethatthepotentialvectorfieldisorthogonaltotheReebvectorfield.ItisprovedthatsuchaCottonsolitononanon-KenmotsualmostKenmotsu3-h-manifoldsuchthattheReebvectorfieldisaneigenvectoroftheRiccioperatorissteadyandthemanifoldislocallyisometricto. Originality/value – Theresultsofthispaperarenew andinteresting.Also,the Proposition3.2willbehelpful in further study of this space.


Introduction
An almost contact metric manifold is an odd dimensional differentiable manifold M 2nþ1 together with a structure (w, ξ, η, g) satisfying ( [1,2]) w 2 X ¼ −X þ ηðXÞξ; ηðξÞ ¼ 1; (1.1) for all vector fields X, Y on M 2nþ1 , where g is the Riemannian metric, w is a (1, 1)-tensor field, ξ is a unit vector field called the Reeb vector field and η is a 1-form defined by η(X) 5 g(X, ξ).
Here also fξ 5 0 and η 8 f 5 0; both can be derived from (1.1) easily. The fundamental 2-form Φ on an almost contact metric manifold is defined by Φ(X, Y) 5 g(X, wY) for all vector fields X, Y on M 2nþ1 . The condition for an almost contact metric manifold being normal is equivalent to vanishing of the (1, 2)-type torsion tensor N w , defined by by (∇ X w)Y 5 g(wX, Y)ξ À η(Y)wX, for any vector fields X, Y on M 2nþ1 . For further details on Kenmotsu manifolds we refer the reader to go through the references ( [5,6]). The Weyl tensor on an n-dimensional Riemannian manifold is defined as.
CðX ; Y ÞZ ¼ RðX ; Y ÞZ þ r ðn À 1Þðn À 2Þ ½gðY ; Z ÞX À gðX ; Z ÞY À 1 n À 2 ½SðY ; Z ÞX À SðX ; Z ÞY þ gðY ; Z ÞQX À gðX ; Z ÞQY ; where R is the curvature tensor, S denotes the Ricci tensor, Q stands for Ricci operator and r is the scalar curvature. A (0, 3)-Cotton tensor of a 3-dimensional Riemannian manifold (M 3 , g) is defined as (see [7]) where S is the Ricci tensor and r is the scalar curvature of M 3 . The Cotton tensor is skew-symmetric in first two indices and totally trace free. It is well known that for n ≥ 4, an n-dimensional Riemannian manifold is conformally flat if the Weyl tensor vanishes. For n 5 3, the Weyl tensor always vanishes but the Cotton tensor does not vanish in general.
In 2008, Kicisel, Sario glu and Tekin [8] introduced the notion of Cotton flow as an analogy of the Ricci flow. The Cotton flow is based on the conformally invariant Cotton tensor and defined exclusively for 3-dimension as vg vt ¼ C; where C is the (0, 2)-Cotton tensor of g. From the Cotton flow, they defined the notion of Cotton soliton as follows: Definition 1.1. A Cotton soliton is a metric g defined on 3-dimensional smooth manifold M 3 such that the following equation holds for a constant σ and a vector field V, called the potential vector field, where L V denotes the Lie derivative along V and C is the (0, 2)-Cotton tensor defined by in a local frame of M 3 , where g 5 det(g ij ), C ijk is the (0, 3)-Cotton tensor and e is a tensor density.
In an orthonormal frame, e 123 5 1. Also exchange of any two indices will give rise to minus sign and it will be zero if there has two same indices. For example, e 231 5 À e 213 and e 112 5 e 122 5 e 223 5 0. Cotton solitons are fixed points of the Cotton flow up to diffeomorphisms and rescaling. The Cotton soliton is said to be shrinking, steady or expanding according as σ is positive, zero or negative respectively. As far as we know, the Cotton soliton was studied by Chen [9] on certain almost contact metric manifold, precisely on almost coK€ ahler 3-manifolds. Motivated by the study of Chen [9], we consider the notion of Cotton soliton on an almost Kenmotsu 3-h-manifold and prove some related results.

Almost Kenmotsu 3-h-manifolds
Let (M 3 , w, ξ, η, g) be a 3-dimensional almost Kenmotsu manifold. We denote by l 5 R($, ξ)ξ, where R is the Riemannian curvature tensor. The tensor fields l and h are symmetric operators and satisfy the following relations ( [3,4]): where a, b and c are smooth functions.
Lemma 2.3. On U 1 , the coefficients of the Riemannian connection ∇ of an almost Kenmotsu 3-h-manifold with respect to a local orthonormal basis {ξ, e, we} is given by where b and c are smooth functions.
From Lemma 2.3, the Lie brackets can be calculated as follows: In [12], Wang obtained the components of the Ricci operator Q for an almost Kenmotsu 3manifold on U 1 as follows: . Now, we write the components of the Ricci operator Q for an almost Kenmotsu 3-h-manifold as follows: Lemma 2.4. On U 1 , the Ricci operator of an almost Kenmotsu 3-h-manifold with respect to a local orthonormal basis {ξ, e, we} is given by The scalar curvature r of an almost Kenmotsu 3-h-manifold is given by Using Lemma 2.4, we obtain It is well known that an almost Kenmotsu 3-manifold is Kenmotsu if and only if h 5 0. Thus a Kenmotsu metric always admits an almost Kenmotsu 3-h-metric structure. We now close this section by providing an example of a non-Kenmotsu almost Kenmotsu 3-h-manifold. We now close this section by recalling an important result of Cho [14].
Theorem 2.6. A non-Kenmotsu almost Kenmotsu 3-manifold M 3 is locally symmetric if and only if M 3 is locally isometric to the product space H 2 ð−4Þ 3 R.

Cotton soliton
In this section, we consider the notion of Cotton soliton within the framework of almost Kenmotsu 3-h-manifolds. To study the notion of Cotton soliton, we need to compute the components of the (0, 2)-Cotton tensor. In this regard, we prove the following Lemma: Lemma 3.1. The components of the (0, 2)-Cotton tensor C with respect to an orthonormal frame {ξ, e, we} of a non-Kenmotsu almost Kenmotsu 3-h-manifold M 3 can be expressed as follows: C 11 ¼ Cðξ; ξÞ ¼ b½weðλÞ þ 2λb À c½eðλÞ þ 2λc ÀeðeðλÞ þ 2λcÞ þ weðweðλÞ þ 2λbÞ; (3.1) þξðeðλÞ þ 2λcÞ À 1 4 weðrÞ; (3.2) Proof. The components of the metric tensor g with respect to an orthonormal frame {ξ, e, we} of a non-Kenmotsu almost Kenmotsu 3-h-manifold M 3 is given by and hence det(g ij ) 5 1. Therefore, Eqn (1.5) reduces to where C ijk 5 C(e i , e j , e k ). Also, C ijk 5 À C jik and C iik 5 0 for all i, j, k 5 1, 2, 3. It can be easily obtained that (see [9]) Making use of (1.3), we get the following: Using (3.19) and ξ(λ) 5 0, we get from (2.5) Since λ is a positive function, then the second and third equations of (3.24) and (3.25) implies b 5 c 5 0. From Lemma 2.4, we get f 5 2. Also from (3.19), we get e(λ) 5 we(λ) 5 0 and therefore λ is a constant. Now, the Lie brackets given in (2.4) reduces to ½e; ξ ¼ e À λwe; ½e; we ¼ 0 and ½we; ξ ¼ −λe þ we: is an eigen vector of the Ricci operator, then the components of the (0, 2)-Cotton tensor C with respect to an orthonormal frame {ξ, e, we} on M 3 can be expressed as follows: We first consider the Cotton soliton with potential vector field V pointwise collinear with the Reeb vector field. In this regard, we prove the following non-existing result.  It is now quite tempting to consider the potential vector field V as orthogonal to the Reeb vector field. In this setting, we prove the following: Theorem 3.6. Let (M 3 , g) be a non-Kenmotsu almost Kenmotsu 3-h-manifold such that the Reeb vector field is an eigen vector of the Ricci operator. If g is a Cotton soliton with potential vector field orthogonal to the Reeb vector field, then M 3 is locally isometric to H 2 ð−4Þ 3 R and the Cotton soliton is steady.
Consider e 1 5 ξ. We define the 1-form η be by η(Z) 5 g(Z, e 1 ) for any smooth vector field Z on M. Let us define the (1, 1)-tensor fields w and h by wðe 1 Þ ¼ 0; wðe 2 Þ ¼ e 3 and wðe 3 Þ ¼ −e 2 : Using the linearity of w and g, we have for any smooth vector field Z, U on M.
Here e 1 5 ξ, e 2 5 e and e 3 5 we. Comparing the obtained components of ∇ e i e j with Lemma 2.3, we get a 5 b 5 c 5 0, λ 5 1, f 5 2 and r 5 À 6. Then from Lemma 2.4, we can see that ξ is an eigenvector of the Ricci operator Q.