Para-Ricci-like Solitons with Vertical Potential on Para-Sasaki-Like Riemannian $\Pi$-Manifolds

Object of study are para-Ricci-like solitons on para-Sasaki-like almost paracontact almost paracomplex Riemannian manifolds, briefly, Riemannian $\Pi$-manifolds. Different cases when the potential of the soliton is the Reeb vector field or pointwise collinear to it are considered. Some additional geometric properties of the constructed objects are proven. Results for a parallel symmetric second-order covariant tensor on the considered manifolds are obtained. Explicit example of dimension 5 in support of the given assertions is provided.


Introduction
In 1982 R. S. Hamilton introduced the concept of Ricci solitons as a special solution of the Ricci flow equation ( [11]). In [5], the author made a detailed study on Riemannian Ricci solitons. The start of the study of Ricci solitons in contact Riemannian geometry is given with [24]. Following this work the investigation of the Ricci solitons in different types of almost contact metric manifolds are done in [8,12,21].
We investigate the noted concept of Ricci solitons in the geometry of almost paracontact almost paracomplex Riemannian manifolds, briefly, Riemannian Π-Manifolds. The induced almost product structure on the paracontact distribution of these manifolds is traceless and the restriction on the paracontact distribution of the almost paracontact structure is an almost paracomplex structure. The study of the considered manifolds starts in [16], where they are called almost paracontact Riemannian manifolds of type (n, n). Their investigation continues in [18,19,20], under the name almost paracontact almost paracomplex Riemannian manifolds.
In the present paper, we continue the investigation of the introduced in [17] generalization of the Ricci soliton called para-Ricci-like soliton. Here, the potential of the considered para-Ricci-like soliton is a vector field, which is pointwise collinear to the Reeb vector field. The paper is organized as follows. After the present introductory Section 1, in Section 2 we give some preliminary definitions and facts about para-Sasaki-like Riemannian Π-manifolds. In Section 3 we investigate para-Riccilike solitons on the considered manifolds and we prove some additional geometric properties. Section 4 is devoted to some characterization for para-Ricci-like solitons on para-Sasaki-like Riemannian Π-manifolds concerning a parallel symmetric (0, 2)-tensor. In the final Section 5 we comment an explicit example in support of some of the proven assertions.

Para-Sasaki-like Riemannian Π-Manifolds
We denote by (M, φ, ξ, η, g) a Riemannian Π-manifold, where M is a differentiable (2n + 1)-dimensional manifold, g is a Riemannian metric and (φ, ξ, η) is an almost paracontact structure, i.e. φ is a (1,1)-tensor field, ξ is a Reeb vector field and η is its dual 1-form. The following conditions are valid: where I is the identity transformation on T M ( [23,18]). Consequently, from the latter equalities we obtain the following: where ∇ denotes the Levi-Civita connection of g. Here and further, by x, y, z, w we denote arbitrary vector fields from X(M) or vectors in T M at a fixed point of M.
The associated metricg of g on (M, φ, ξ, η, g) is determined by the equality Obviously,g is compatible with (M, φ, ξ, η, g) in the same way as g and it is indefinite metric of signature (n + 1, n).
In [13], it is introduced and studied the class of para-Sasaki-like spaces in the set of Riemannian Π-manifolds which are obtained from a specific cone construction. This special subclass of the considered manifolds is determined by the following condition: In [13] is proven that the following identities are valid for any para-Sasaki-like Riemannian Π-manifold: where R and ρ stand for the curvature tensor and the Ricci tensor, respectively. It is known from [17] that a Riemannian Π-manifold (M, φ, ξ, η, g) is said to be para-Einstein-like with constants (a, b, c) if its Ricci tensor ρ satisfies: Moreover, if b = 0 or b = c = 0 the manifold is called an η-Einstein manifold or an Einstein manifold, respectively. If a, b, c are functions on M, then the manifold is called almost para-Einstein-like, almost η-Einstein manifold or an almost Einstein manifold, respectively. Let is consider a (2n + 1)-dimensional Riemannian Π-manifold (M, φ, ξ, η, g) which is para-Sasaki-like and para-Einstein-like with constants (a, b, c). Tracing (6) and using the last equalities of (5), we have: [17] where τ stands for the scalar curvature with respect to g of (M, φ, ξ, η, g). Moreover, for the scalar curvatureτ with respect tog on (M, φ, ξ, η, g) we obtain Taking into account (7) and (8), expression (6) gets the following form ρ = τ 2n + 1 g + τ 2n + 1 g + −2(n + 1) − τ +τ 2n η ⊗ η. , c), then the scalar curvatures τ andτ are constants Proof. If (M, φ, ξ, η, g) is almost para-Einstein-like then (7) and (8) are valid, where (a, b, c) are a triad of functions. Using (5) and substituting y = ξ, we can express R(x, ξ)ξ as follows After that, bearing in mind (4) and (5), we compute the covariant derivative of R(x, ξ)ξ with respect to ∇ z and we take its trace for z = e i and x = e j which gives The following consequence of the second Bianchi identity is valid For a para-Sasaki-like manifolds, according to (5), the equalities Qξ = −2n ξ and ∇ x ξ = φx hold. Using them, it follows that (∇ x Q)ξ = −Qφx + 2n φx. As a consequence of the latter equality we have that the trace in the left hand side of (10) vanishes. Then, by virtue of (9) and (10) we get which implies dτ (ξ) = 0,τ = −2n. The latter equalities together with (7) and (8) complete the proof.
For a para-Sasaki-like (M, φ, ξ, η, g) we have Then, bearing in mind the definition equality ofg, it follows that Because of (11), ρ takes the form  (15) we complete the proof.
The latter two equations for (λ, µ) = (0, −1) can be solved as a system with respect to g(φx, φy) and g(x, φy) as follows The recurrent dependence (17) of the Ricci tensor is get by substituting the latter equalities into (16). Thus, we complete the proof of (iv).

Para-Ricci-like solitons with a potential pointwise collinear with the Reeb vector field on para-Sasaki-like manifolds.
Similarly to the definition of a para-Ricci-like soliton with potential ξ, given in (11), we introduce the following notion.

Definition 3.1.
A Riemannian Π-manifold (M, φ, ξ, η, g) admits a para-Riccilike soliton with potential vector field v and constants (λ, µ, ν) if its Ricci tensor ρ satisfies the following: Let (M, φ, ξ, η, g) be a para-Sasaki-like Riemannian Π-manifold admitting a para-Ricci-like soliton whose potential vector field v is pointwise collinear with ξ, i.e. v = k ξ, where k is a differentiable function on M. The vector field v belongs to the vertical distribution H ⊥ = span ξ, which is orthogonal to the contact distribution H = ker η with respect to g. (M, φ, ξ, η, g) be a para-Sasaki-like Riemannian Π-manifold of dimension 2n + 1 and let it admits a para-Ricci-like soliton with constants (λ, µ, ν) whose potential vector field v satisfies the condition v = k ξ, i.e. it is pointwise collinear with the Reeb vector field ξ, where k is a differentiable function on M. Then:
In [7], it is defined a Ricci φ-symmetric Ricci operator Q, i.e. the non-vanishing Q satisfies φ 2 (∇ x Q)y = 0. Moreover, according to [9], if the latter property is valid for an arbitrary vector field on the manifold or for an orthogonal vector field to ξ the manifold is called globally Ricci φ-symmetric or locally Ricci φ-symmetric, respectively.
An almost pseudo Ricci symmetric manifold is a manifold whose non-vanishing Ricci tensor has the following condition [6] (27) where α and β are non-vanishing 1-forms. According to [25], a manifold is called special weakly Ricci symmetric when its non-vanishing Ricci tensor satisfies the following Proof. In a similar way as for (16), taking into account (25), we get Then, it is easy to conclude the statement (ii-a).
Substituting successively x, y and z for ξ, we get (33), which combined with (34) implies for arbitrary x, y, z and therefore α(ξ) = 0 holds. Thus, we complete the proof of assertion (ii-d).
We come to the conclusion that an almost pseudo Ricci symmetric manifold with α = β is a special weakly Ricci symmetric manifold, comparing (28) with (27). Then, from (33) we obtain that α = 0 and therefore (M, φ, ξ, η, g) has ∇ρ = 0. Taking into account (ii-d), we get the validity of the statement (ii-e).

Parallel symmetric second order covariant tensor on (M, φ, ξ, η, g)
Let h be a symmetric (0, 2)-tensor field which is parallel with respect to the Levi-Civita connection of g, i.e. ∇h = 0. The Ricci identity for h is valid, i.e.
Vice versa, the valid condition (11) can be rewritten as h = −λg. Taking into account that λ is constant and g is parallel, it follows that h is also parallel with respect to ∇ of g.