CURVATURE PROPERTIES OF ALMOST RICCI-LIKE SOLITONS WITH TORSE-FORMING VERTICAL POTENTIAL ON ALMOST CONTACT B-METRIC MANIFOLDS

A generalization of η-Ricci solitons is considered involving an additional metric and functions as soliton coefficients. The soliton potential is torseforming and orthogonal to the contact distribution of the almost contact B-metric manifold. Then such a manifold can also be considered as an almost Einstein-like manifold, a generalization of an η-Einstein manifold with respect to both B-metrics and functions as coefficients. Necessary and sufficient conditions are found for a number of properties of the curvature tensor and its Ricci tensor of the studied manifolds. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.


Introduction
A notion of a Ricci soliton is a natural extension of an Einstein metric on a smooth manifold. Recently, Hamilton's concept of the Ricci flow introduced on Riemannian manifolds in [14], has been developed by many authors for (pseudo-)Riemannian manifolds with various additional tensor structures (e.g. [1], [2], [3], [4], [5], [6], [16], [25], [26], [27], [28], [29]). A significant part of these works are for η-Ricci solitons on manifolds with a horizontal distribution ker η for some 1-form η. Moreover, the manifold is of Kenmotsu type, where the soliton potentials can be torse-forming and orthogonal to ker η. Torse-forming vector fields are defined by a certain recurrent condition for their covariant derivative with respect to the Levi-Civita connection of the basic metric [30]. This research has been expanded to study various properties of the curvature tensor and its Ricci tensor compatible with the additional tensor structures.
In [18], the author begins the study of Ricci-like solitons on almost contact Bmetric manifolds, whose Reeb vector field ξ is the soliton potential. These solitons are a generalization of η-Ricci solitons because use also the associated B-metric of the manifold. In [20], the author generalizes the Ricci-like solitons to almost Ricci-like solitons on the considered manifolds, considering the soliton coefficients as functions. Then, he studies the case when the potential is torse-forming and vertical, i.e. collinear to ξ.
Following [18], the only basic class of almost contact B-metric manifolds that allows a torse-forming Reeb vector field is F 5 , denoted according to the Ganchev-Mihova-Gribachev classification [11]. This class is considered in the present work and F 5 is the counterpart of the class of β-Kenmotsu manifolds among the classes of almost contact metric manifolds. Ricci solitons and η-Ricci solitons on manifolds with almost contact or almost paracontact structures, both of Kenmotsu type, have been studied in a number of works (e.g. [1], [2], [5], [15], [25], [27]).
In the present paper, our aim is to study curvature properties of almost Riccilike solitons whose potential is torse-forming and vertical on almost contact Bmetric manifolds. The paper is organized as follows. In Section 2, we recall basic definitions and facts for the considered manifolds and the tensor structures on 1 them. In Section 3, we investigate a series of properties of the curvature tensor and its Ricci tensor on the manifolds under study. In Section 4, we give an explicit example of a smooth manifold of the considered type and comment on the results for this example supporting the relevant assertions in the previous section.

Preliminaries
2.1. Torse-forming vertical vector fields on almost contact B-metric manifolds. We consider almost contact B-metric manifolds denoted by (M, φ, ξ, η, ). This means that any M is a (2n + 1)-dimensional smooth manifold equipped with an almost contact structure and a B-metric, i.e. a pseudo-Riemannian metric of signature (n + 1, n), satisfying the following conditions [11] (1) where ι is the identity transformation on Γ(TM).
In the latter equality and further, x, y, z, w will stand for arbitrary elements of Γ(TM) or vectors in the tangent space T p M of M at an arbitrary point p in M.
An important characteristic of these manifolds is the existence of an associated B-metric˜ of defined bỹ (x, y) = (x, φy) + η(x)η(y).
A classification of almost contact B-metric manifolds is given in [11]. It is made with respect to the (0, . Each basic class F i , i ∈ {1, 2, . . . , 11} contains the special class F 0 , determined by F = 0, and known as the class of the cosymplectic B-metric manifolds. A vector field ϑ on a (pseudo-)Riemannian manifold (M, ) is called torse-forming vector field if it satisfies the following condition for an arbitrary vector field x, a differentiable function f and a 1-form γ where ∇ is the Levi-Civita connection of [30]. The 1-form γ is called the generating form and the function f is called the conformal scalar of ϑ [22]. In [18], manifolds (M, φ, ξ, η, ) with torse-forming Reeb vector field ξ are studied. Then, γ has the form γ = − f η. There, it is deduced that the class of the considered manifolds is F 1 ⊕ F 2 ⊕ F 3 ⊕ F 5 ⊕ F 6 ⊕ F 10 and only F 5 can contain such manifolds among the basic classes. If (M, φ, ξ, η, ) is an F 5 -manifold with torse-forming ξ, then we have In this case, it is found that the condition for Ricci-symmetry is equivalent to the Einstein condition on this Einstein-like manifold. We further exclude from our consideration the trivial case when f = 0, since it implies the parallelism of ϑ.
In the rest part of our work, we consider a torse-forming vector field ϑ, i.e. (2) is valid. Moreover, ϑ is pointwise collinear with ξ, i.e. ϑ = k ξ, where k is a nonzero pointwise differentiable function on M. Therefore ϑ belongs to the vertical distribution H ⊥ = span ξ, which is orthogonal to the contact distribution H = ker η with respect to . For that reason, we call such a vector field ϑ vertical.
(ii) ξ is a torse-forming vector field with conformal scalar f /k and generating form (iv) the following equalities for the curvature tensor R, the Ricci tensor ρ and the sectional curvature K are valid: According to Theorem 2.1(ii) and (3) for (M, φ, ξ, η, ) ∈ F 5 , we have the following 2.2. Einstein-like almost contact B-metric manifolds. In [18], it is said that (M, φ, ξ, η, ) is Einstein-like if its Ricci tensor ρ has the following form for some triplet of constants (a, b, c): In particular, when b = 0 and b = c = 0, the manifold is called an η-Einstein manifold and an Einstein manifold, respectively. If a, b, c are functions on M, then the manifold is called almost Einstein-like, almost η-Einstein and almost Einstein, respectively. Other particular cases of Einstein-like types of the considered manifolds are given in [21] and [17]. For example, when a = b = 0, the manifold is called v-Einstein for short from vertical-Einstein since η ⊗ η is the vertical component of and˜ .
Now we introduce the notion of an almost v-Einstein manifold when the condition (11) is satisfied for a = b = 0 and a function c on M.
As a consequence of (11) we obtain that the corresponding scalar curvature and its associated quantity on an almost Einstein-like manifold have the form:
A Ricci soliton is called shrinking, steady or expanding depending on whether λ is negative, zero or positive, respectively [8].
Similar to the case of a Ricci-like soliton with potential ϑ = ξ, studied in [18], we recall the following more general result for almost Ricci-like solitons with torse-forming vertical potential.

Theorem 2.3 ([20]
). Let (M, φ, ξ, η, ) be (2n + 1)-dimensional and ϑ be a vector field on M, which is vertical with a constant k = η(ϑ) as well as torse-forming with a function f as a conformal scalar. Moreover, let a, b, c, λ, µ, ν be functions on M that satisfy the following equalities: Then, M admits an almost Ricci-like soliton with potential ϑ and functions (λ, µ, ν) if and only if M is almost Einstein-like with functions (a, b, c).

manifold is almost Einstein with function a if and only if it admits an almost
η-Ricci soliton with potential ϑ and functions (λ, µ, ν) = (−a − f, 0, f ), which is not almost Ricci soliton with potential ϑ.

Corollary 2.6 ([20]). Under the hypothesis of Theorem 2.3 we have that M is almost
Einstein-like with functions (a, b, c) and scalar curvature τ. Then the conformal scalar f satisfies the following equation where f ′ is the derivative of the function f = f (t) and t is a coordinate on H ⊥ and the sectional curvature of an arbitrary ξ-section is 3. Some curvature properties of almost Ricci-like solitons with torse-forming vertical potential on almost contact B-metric manifolds Throughout this section, we assume that (M, φ, ξ, η, ) is a (2n + 1)-dimensional manifold belonging to the class F 5 \ F 0 and admitting an almost Ricci-like soliton with potential ϑ, which is torse-forming with a non-zero conformal scalar f and a vertical one with a non-zero constant coefficient k.
3.1. Locally Ricci symmetric manifolds. If consider the condition for locally Ricci symmetry, i.e. ∇ρ vanishes, we immediately obtain the following and the scalar curvature τ of M is the following constant (iii) It is Einstein with constant a and τ = (2n + 1)a.
Proof. Because of Theorem 2.1(ii)-(iii) and (15), we get Using the latter equality, Theorem 2.1(ii)-(iii), (9), (15) and (16), we compute the covariant derivative of the Ricci tensor with respect to ∇ of an F 5 -manifold admitting an almost Ricci-like soliton with torse-forming vertical potential as follows (20) ( As consequences of the latter formula we have Therefore, taking into account (20), we establish that ρ is ∇-parallel (i.e. (i) holds) if and only if the soliton functions are: Bearing in mind (21), we deduce that λ and ν (just like f ) are horizontal constants, i.e. constants on H, and thus they are vertical functions, i.e. on H ⊥ . By (14) we obtain that the constant in (21) is −a and b = c = 0 is true. Hence, the manifold under consideration is Einstein with the constant a, which is equal to τ/(2n + 1) due to (12). Therefore, we obtain (19) using (15), and this completes the proof.

Manifolds of Ricci semi-symmetric type. A manifold is called
Ricci semisymmetric if the following condition is satisfied R(x, y) · ρ = 0, where R(x, y) is considered as a field of linear operators, acting on the Ricci (0, 2)-tensor ρ.
Proof. By using of (4) and (15), we get the following Then, applying (16) and (23), we obtain the following equality Since ϑ is regular, the later equality is valid for arbitrary x and y if and only if a = b = 0. Therefore, we have ρ = c η ⊗ η and τ = c, i.e. the manifold is v-Einstein. Because of (17), we get (22) and τ 0 due to the regularity of ϑ. Then, taking into account (12), we establish that (i) is equivalent to (iii).
It is known that (24) is equivalent to requiring that the Ricci tensor is a Killing tensor, i.e. (∇ x ρ)(x, x) = 0.
(iii) The manifold is almost Einstein-like with vertical functions (a, b, c) satisfying the following equations Proof. Applying (20) to (24), we obtain that the condition ρ to be cyclic parallel is equivalent to the following condition for the cyclic sum of x, y and z which by substituting x, y and z consecutively with ξ yields d(λ + µ + ν) = 0 and Taking into account (27), then (26) is valid for arbitrary x, y and z if and only if Bearing in mind (28) and (15), we obtain that the soliton functions λ, µ and ν satisfy (25). Therefore they are horizontal constants, i.e. they depend only on the coordinate of H ⊥ just as f . This means that the equivalence between (i) and (ii) holds true. Finally, the equivalence between (ii) and (iii) follows from Theorem 2.3.
Here, we remark that the latter equation is equivalent to requiring that the curvature tensor is harmonic, i.e. div R = 0. In a similar way as in Theorem 3.3, we establish the truthfulness of the following (iii) The manifold is almost Einstein-like with vertical functions (a, b, c) satisfying the following equations
In [12], if the latter condition is satisfied for arbitrary vector fields on the manifold, it is called globally Ricci φ-symmetric, and when these vector fields are orthogonal to ξ, then the manifold is called locally Ricci φ-symmetric.
Using (20), we obtain the following expression It is not difficult to verify the validity of the following two statements.  -like with functions (a, b, c), where a and b are vertical functions.
3.6. Manifolds with vanishing Q · R. Let us consider the curvature condition Q · R = 0, which can be written in the form Proof. Taking the trace of (29) for x = e i , w = e j and due to the symmetry of Q, we obtain ρ(Qy, z) + ρ(y, Qz) = 0.

3.7.
Almost pseudo Ricci symmetric manifolds. In [7], the notion of almost pseudo Ricci symmetric manifolds is introduced by the following condition for its nonvanishing Ricci tensor where α and β are non-vanishing 1-forms.
Proof. Combining (20) with (30) and (14), we obtain the following equality Substituting sequentially ξ for x, y, z and applying the result equalities in (32), we get the following condition which holds true for arbitrary x, y and z if and only if the following system of equalities satisfied: Solving the obtained system, we obtain the three solutions implying (i), (ii) and (iii), using (12) and (17).
Proof. The statement follows from Theorem 3.8 with equalities (14).
where α is a non-vanishing 1-form. It is obvious that any special weakly Ricci symmetric manifold with 1-form α is almost pseudo Ricci symmetric with 1-forms α = β. Thus Theorem 3.8 implies the following where a and c are given in (31).
As a consequence of (31) and (iii) in Corollary 3.11 we get the following differential condition for the scalar curvature and the conformal scalar .

Example
Let us consider the example from [20] of a manifoldM := (R 2n+1 , φ, ξ, η,¯ ) belonging to F 5 \ F 0 , where the real space and the almost contact B-metric structure are defined as follows , φ Initially, a B-metric is introduced by the equality where z = z i ∂ ∂x i + z n+i ∂ ∂x n+i + z 2n+1 ∂ ∂t , i ∈ {1, 2, . . . , n} and δ ij are the Kronecker's symbols. Then, the B-metric¯ is obtained by the following contact conformal transformation of determined by the functions u and v on R 2n+1 : = e 2u cos 2v + e 2u sin 2v˜ + ( 1 − e 2u cos 2v − e 2u sin 2v where ℓ(t) is an arbitrary twice differentiable function on R such that ℓ ′ 0. A torse-forming vertical vector field ϑ is considered, i.e. ϑ = kξ and (10) is valid. Then, the conformal scalar f of ϑ is determined by f = kℓ ′ .
(8)M is not special weakly Ricci symmetric for an arbitrary function f with the condition k f ′ + f 2 0.