Yamabe solitons on conformal Sasaki-like almost contact B-metric manifolds

A Yamabe soliton is defined on arbitrary almost contact B-metric manifold, which is obtained by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. The cases when the given manifold is cosymplectic or Sasaki-like are studied. In this way, manifolds from one of the main classes of the studied manifolds are obtained. The same class contains the conformally equivalent manifolds of cosymplectic manifolds by the usual conformal transformation of the B-metric. An explicit 5-dimensional example of a Lie group is given, which is characterized in relation to the obtained results.


Introduction
Richard Hamilton introduced the notion of the Yamabe flow in 1988 (see [8]) as an apparatus for constructing metrics with constant scalar curvature. The significance of this problem from the point of view of mathematical physics is that the Yamabe flow corresponds to the case of fast diffusion of the porous medium equation [4].
A self-similar solution of the Yamabe flow, defined on a Riemannian manifold or a pseudo-Riemannian manifold, is called a Yamabe soliton and is determined by ( [1]) 1 2 where L v g stands for the Lie derivative of the metric g along the vector field v, τ denotes the scalar curvature of g and σ is a constant. Many authors have studied the Yamabe soliton on different types of manifolds since the introduction of this concept (see e.g. [2,3,5,7,14]).
In the present paper we begin the study of the mentioned Yamabe solitons on almost contact B-metric manifolds. The geometry of these manifolds is significantly influenced by the presence of two B-metrics, which are interconnected by the almost contact structure.
It is a well-known fact that the Yamabe flow preserves the conformal class of the metric. So this gives us a reason to study Yamabe solitons and conformal transformations together. Contact conformal transformations were studied in [11], which transform not only the metric but also the Reeb vector field and its associated contact 1-form through the pair of B-metrics. According to this work, the class of almost contact B-metric manifolds, which is closed under the action of contact conformal transformations, is the direct sum of the four main classes among the eleven basic classes of these manifolds known from the classification of Ganchev-Mihova-Gribachev in [6]. Main classes are called those whose manifolds are characterized by the fact that the covariant derivative of the structure tensors with respect to the Levi-Civita connection of some of the B-metrics is expressed only by the pair of B-metrics and the corresponding traces.
We study Yamabe solitons on two of the simplest types of manifolds among those studied, namely cosymplectic and Sasaki-like. The former have parallel structure tensors with respect to the Levi-Civita connections of the B-metrics. The latter are those whose complex cone is a Kähler manifold with a pair of Norden metrics, i.e. again with parallel structure tensors with respect to the Levi-Civita connections of the metrics. Note that the class of Sasaki-like manifolds does not contain cosymplectic manifolds, although they are in each of the eleven basic classes of the classification used.
We find that the manifolds thus constructed in both cases belong to one of the main classes, the only one which contains the conformally equivalent manifolds of the cosymplectic ones by the usual conformal transformations.
The present paper is organized as follows. Section 1 is devoted to the basic concepts of almost contact B-metric manifolds, contact conformal transformations of the structure tensors on them and the introduction of the notion of a Yamabe soliton on a transformed almost contact B-metric manifold. In Section 2 and Section 3, we study the constructed manifolds when the initial manifold is cosymplectic and Sasaki-like, respectively. In Section 4, we provide an explicit example of a Lie group as a 5-dimensional manifold equipped with the structures studied.

Almost contact B-metric manifolds, contact conformal transformations and Yamabe solitons
Let (M, ϕ, ξ, η, g) be an almost contact manifold with B-metric or an almost contact B-metric manifold, i.e. M is a (2n + 1)-dimensional differentiable manifold with an almost contact structure (ϕ, ξ, η) consisting of an endomorphism ϕ of the tangent bundle, a vector field ξ, its dual 1-form η as well as M is equipped with a pseudo-Riemannian metric g of signature (n + 1, n), such that the following algebraic relations are satisfied [6] for arbitrary X, Y of the algebra X(M ) on the smooth vector fields on M , where ι stands for the identity transformation on X(M ).
Further, X, Y , Z will stand for arbitrary elements of X(M ) or vectors in the tangent space T p M of M at an arbitrary point p in M .
The associated B-metricg of g on M is defined bỹ and it is also of signature (n + 1, n) as its counterpart g.
The fundamental tensor F of type (0,3) on (M, ϕ, ξ, η, g) is defined by where ∇ is the Levi-Civita connection of g. The following properties of F are consequences of (1) and (2): and relations of F with ∇ξ and ∇η are: The following 1-forms, known as Lee forms of the manifold, are associated with F : where g ij are the components of the inverse matrix of g with respect to a basis {E i ; ξ} (i = 1, 2, . . . , 2n) of T p M . The following general identities for the Lee forms of (M, ϕ, ξ, η, g) are known from [10] (4) A classification of the almost contact B-metric manifolds in terms of F is given in [6]. This classification includes eleven basic classes F 1 , F 2 , . . . , F 11 . Their intersection is the special class F 0 defined by condition for the vanishing of F . Hence F 0 is the class of cosymplectic B-metric manifolds, where the structures ϕ, ξ, η, g,g are ∇-parallel.
Using the pair of B-metrics g andg as well as η ⊗η, in [10], the author and K. Gribachev introduced the so-called contact conformal transformation of the B-metric g into a new B-metricḡ for (M, ϕ, ξ, η, g). Later, in [11], this transformation is generalized as a contact conformal transformation that gives an almost contact B-metric structure (ϕ,ξ,η,ḡ) as follows g = e 2u cos 2v g + e 2u sin 2vg + e 2w − e 2u cos 2v − e 2u sin 2v η ⊗ η, The corresponding tensorsF and F are related by [11], see also [13, (22)], where we use the following notations for brevity In the general case, the relations between the Lee forms of the manifolds (M, ϕ, ξ, η, g) and (M, ϕ,ξ,η,ḡ) are the following (see [11]) Definition 1.1. We say that the B-metricḡ generates a Yamabe soliton with potential the Reeb vector fieldξ and soliton constantσ on a conformal almost contact B-metric manifold (M, ϕ,ξ,η,ḡ), if the following condition is satisfied whereτ is the scalar curvature ofḡ.
It is well known the following expression of the Lie derivative in terms of the covariant derivative with respect to the Levi-Civita connection∇ ofḡ (10) Lξḡ

The case when the given manifold is cosymplectic
In this section, we consider (M, ϕ, ξ, η, g) to be a cosymplectic manifold with almost contact B-metric structure, i.e. an F 0 -manifold, defined by covariant constant structure tensor ϕ (and consequently the same condition for ξ, η,g is valid) with respect to ∇ of g. Therefore F = 0 and (6) takes the form From the latter equality, bearing in mind the general identity (3) for the manifolds under consideration, we obtain Combining the last expression with (7) and (10) gives the following Moreover, the obtained Yamabe soliton has a constant scalar curvature with valueτ =σ and the obtained almost contact B-metric manifold belongs to the subclass of the main class F 1 determined by the conditions: Proof. Is we assume thatḡ generates a Yamabe soliton with potentialξ and a soliton constantσ on (M, ϕ,ξ,η,ḡ), then due to (9) and (12)  − [sin 2v du(ξ) + cos 2v dv(ξ)] g(X, ϕY )} The substitution ξ for Y in the latter equality gives the following which is valid if and only if (17)τ =σ.
Let us remark that Lξḡ vanishes because of (17) and (9), i.e.ξ is a Killing vector field in the considered case.
Then we use (13) and the formulaη = e w η from (5) to obtain the following expression Now, we substitute the Lee forms ofF from (8) in (19) and get The last expression ofF means that the obtained manifold belongs to the main class F 1 , according to the classification in [6], and the corresponding Lee forms have the following properties: Finally, taking into account (8), (13) and the vanishing of θ, θ * and ω for any cosymplectic B-metric manifold, we find the expressions of the Lee forms of the transformed manifold as in (14).
As a result from (13), we obtain that the case in the present section is possible when the functions (u, v, w) of the contact conformal transformation in (5) satisfy the conditions: • u and v are constants on the vertical distribution V = span ξ = ker ϕ; • w is a constant on the horizontal (contact) distribution H = ker η = im ϕ.

The case when the given manifold is Sasaki-like
In the present section, we suppose that the given almost contact B-metric manifold (M, ϕ, ξ, η, g) is Sasaki-like, i.e. the complex cone M × R − is a Kähler manifold with Norden metric also known as a holomorphic complex Riemannian manifold [9].
If (M, ϕ, ξ, η, g) is Sasaki-like, then the following condition is met: The Sasaki-like condition in terms of F given in [9] is the following Then, by virtue of (3)  Moreover, the obtained Yamabe soliton has a constant scalar curvature with a valueτ =σ and the transformed almost contact B-metric manifold belongs to a subclass of the main class F 1 determined by: Proof. The expression in (21) and the assumption thatḡ generates a Yamabe soliton with potentialξ and a soliton constantσ on (M, ϕ,ξ,η,ḡ) imply the An obvious consequence for Y = ξ is the following which is satisfied if and only it the following conditions are fulfilled Due to (25) and (9), we have the vanishing of Lξḡ, which means thatξ is a Killing vector field in this case as well.

Example
We recall a known example of a Sasaki-like manifold, given in [9] as Example 2. A Lie group G of dimension 5 is considered to have a basis of left-invariant vector fields {E 0 , . . . , E 4 } defined by the following commutators for λ, µ ∈ R: An invariant almost contact B-metric structure is then defined on G by It is verified that the constructed manifold (G, ϕ, ξ, η, g) is an almost contact B-metric manifold that is Sasaki-like.
In [12], the components R ijkl = R(e i , e j , e k , e l ) of its curvature tensor are calculated and the type of the corresponding Ricci tensor ρ is found. Namely, it is ρ = 4 η ⊗ η and the scalar curvature is τ = 4.
Using the non-zero ones of R ijkl determined by the following equalities: and the properties R ijkl = −R jikl = −R ijlk , we compute that the associated quantity τ * of τ is zero, i.e.
Let us consider a contact conformal transformation defined by (5), where the functions (u, v, w) are determined as in (30).
Then, bearing in mind the last results, we obtain for our example that τ = 4 e −2w ,τ * = 0.