Para-Ricci-like Solitons on Almost Paracontact Almost Paracomplex Riemannian Manifolds

It is introduced and studied para-Ricci-like solitons with potential Reeb vector field on almost paracontact almost paracomplex Riemannian manifolds. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field have been considered. It is proved a necessary and sufficient condition the manifold to admit a para-Ricci-like soliton which is the structure to be para-Einstein-like. Explicit examples are provided in support of the proven statements.


Introduction
The concept of Ricci solitons is introduced in 1982 by R. S. Hamilton [13] as a special self-similar solution of the Ricci flow equation, and it plays an important role in understanding its singularities. In other words, Ricci solitons represent a generalization of Einstein metrics on a Riemannian manifold, being generalized fixed points of the Ricci flow, considered as a dynamical system. A detailed study on Riemannian Ricci solitons may be found in [9].
The study of Ricci solitons in contact Riemannian geometry began in 2008 by R. Sharma [23]. After that the investigation of the geometric characteristics of the Ricci solitons continues on different types of almost contact metric manifolds (see [14], [3], [20], [1], [12]). The notion of η-Ricci soliton, introduced by J. T. Cho and M. Kimura [10], is a generalization of the concept of Ricci soliton. Investigations on η-Ricci solitons have been made in [4], [5], [21] but in the context of paracontact geometry.
Recently, various geometers have been extensively studying the pseudo-Riemannian properties of Ricci solitons as its generalizations. This interest is based on the growing interest of theoretical physicists in Ricci solitons and their relation with string theory. For further references on pseudo-Riemannian Ricci solitons, see [8], [7], [2], [6], [16].
The object of our focus is the geometry of the almost paracontact almost paracomplex Riemannian manifolds. In these manifolds, the induced almost product structure on the paracontact distribution is traceless and the restriction on the paracontact distribution of the almost paracontact structure is an almost paracomplex structure. In [17], these manifolds are introduced and classified. Their investigation continues in [18,19].
In the present paper, due to the presence of two associated metrics each other, we introduce and study our generalization of the Ricci soliton compatible with the manifold structure, called para-Ricci-like soliton. We characterize it in terms of some important types of manifolds under considerationpara-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field. We comment two examples in support of the proven statements.
In particular, we have:

Almost paracontact almost paracomplex Riemannian manifolds
Let (M, φ, ξ, η, g) be an almost paracontact almost paracomplex Riemannian manifold (briefly, an apapR manifold). Namely, M is a differentiable odd-dimensional manifold, equipped with a Riemannian metric g and an almost paracontact structure (φ, ξ, η), i.e. φ is a (1,1)-tensor field, ξ is the characteristic vector field and η is its dual 1-form, which satisfy the following: where I is the identity transformation on T M ( [22], [18]). Consequently, we get the following equations: where ∇ denotes the Levi-Civita connection of g. Here and further x, y, z, w stand for arbitrary vector fields from X(M) or vectors in T M at a fixed point of M.
The apapR manifolds are classified in [17]. This classification contains eleven basic classes F 1 , F 2 , . . . , F 11 and it is made with respect to the (0,3)-tensor F determined by The following basic properties of F are valid: The intersection of the basic classes is denoted by F 0 , which is the special class determined by the condition F = 0. The 1-forms associated with F (also known as Lee forms) are defined by: where g ij is the inverse matrix of (g ij ) of g with respect to a basis {ξ; 3.1. Para-Sasaki-like manifolds. In [15], the class of para-Sasaki-like spaces is defined in the set of almost paracontact almost paracomplex Riemannian manifolds obtained from a specific cone construction. The considered para-Sasaki-like spaces are determined by Remark 3.1. The apapR manifolds of para-Sasaki-like type form a subclass of the basic class F 4 from [17] and their Lee forms are θ = −2n η and θ * = ω = 0.
In [15], the truthfulness of the following identities is proved where R and ρ denote the curvature tensor and the Ricci tensor, respectively.

Para-Einstein-like manifolds.
Definition 3.2. An apapR manifold (M, φ, ξ, η, g) is said to be para-Einstein-like with constants (a, b, c) if its Ricci tensor ρ satisfies In particular, the manifold is known to be called an η-Einstein manifold and an Einstein manifold when b = 0 and b = c = 0, respectively.
, where div and div * stand for the divergence with respect to g andg, respectively.

ApapR manifolds with a torse-forming Reeb vector field.
It is known that a vector field ξ is called torse-forming if ∇ x ξ = f x + α(x)ξ is valid for a smooth function f and an 1-form α on the manifold. The 1-form α on an apapR manifold (M, φ, ξ, η, g) is determined by α = −f η, taking into account the last equality in (4). Then, we have the following equivalent equalities (17) ∇ In the further investigations we omit the trivial case when f = 0. Proof. Taking into account (6) and (7), equalities (17) imply θ * (ξ) = −2n f and θ(ξ) = ω = 0. Therefore, bearing in mind the components of F in the basic classes F i , given in [18], we get the statement.

Proofs of Theorem A and Theorem B
A Ricci soliton is a pseudo-Riemannian manifold (M, g) which admits a smooth non-zero vector field v on M such that [13] ρ = − 1 2 L v g − λ g, where L stands for the Lie derivative and λ is a constant. In particular, a Ricci soliton with negative, zero or positive λ is called shrinking, steady or expanding, respectively [11].
The authors of [10] made a generalization of the Ricci soliton on an almost contact metric manifold (M, ϕ, ξ, η, g) called an η-Ricci soliton determined by ρ = − 1 2 L v g − λ g − ν η ⊗ η, where ν is also a constant. Due to the presence of two associated metrics on almost contact B-metric manifolds, a subsequent generalization of the Ricci soliton and the η-Ricci soliton is presented in [16], called a Ricci-like soliton. Here, we introduce our generalization in terms of apapR manifolds as follows.  An apapR manifold (M, φ, ξ, η, g) is called a para-Riccilike soliton with potential vector field ξ and constants (λ, µ, ν) if its Ricci tensor ρ satisfies: Taking into account (19) and (20) ( we obtain the scalar curvature (M, φ, ξ, η, g) be a para-Einstein-like manifold with constants (a, b, c) and let (M, φ, ξ, η, g) admit a para-Ricci-like soliton with potential ξ and constants (λ, µ, ν). Then:
Proof. The characterization of the basic classes F i by the components of F , given in [18], and the assertions (iii) and (iv) in Proposition 4.2, complete the proof of the corollary.
Using the expression of ∇ x ξ from (9) in (20) and (19), we obtain that (19) takes the form which coincides with (10) under conditions (1) in Theorem A. The equality a + b + c = −2n comes from (14), whereas λ + µ + ν = 2n is a result of (21) for ρ(x, ξ) and the corresponding formula from (9). Therefore, the main assertion in Theorem A is proved. We get the truthfulness of the assertions (i), (ii), (iii) and (iv) in Theorem A as corollaries of the main statement for the cases µ = 0, µ = ν = 0, b = 0 and b = c = 0, respectively. This completes the proof of Theorem A. Now we focus our considerations on Theorem B. Let ξ on an apapR manifold (M, φ, ξ, η, g) be torse-forming with function f and let (M, φ, ξ, η, g) admit a para-Ricci-like soliton with potential ξ and constants (λ, µ, ν). Taking into account (17), (19) and (20), we get the form of the Ricci tensor of (M, φ, ξ, η, g) as follows Bearing in mind (10), the latter identity shows that a necessary and sufficient condition (M, φ, ξ, η, g) to be para-Einstein-like is f to be a constant. Then, the main statements (2) in Theorem B are valid.
We get the truthfulness of the assertions (i), (ii) and (iii) in Theorem B as corollaries of the main statement for the cases b = µ = 0, b = µ = ν = 0 and b = c = µ = 0, respectively. This completes the proof of Theorem B.  Let (M, φ, ξ, η, g) be a para-Einstein-like manifold with constants (a, b, c) and let (M, φ, ξ, η, g) admits a para-Ricci-like soliton with potential ξ and constants (λ, µ, ν). Then f is determined by:

Some consequences of the main theorems
Proof. Using the form of ∇ξ in (17), we get We get that −2nf 2 = a + b + c from (23)  Proof. Taking into account (13), (17), (18) and (2), we obtain Using the latter result and following the proof for (ii) and (iii) in Corollary 3.6, we establish the truthfulness of the statements.
6. Examples In the same paper, G is equipped with an invariant apapR structure (φ, ξ, η, g) as follows: It is proved that the constructed manifold (G, φ, ξ, η, g) is a para-Sasakilike manifold. Taking into account (24) and (25), we compute the components R ijkl = R(e i , e j , e k , e l ) of the curvature tensor and the components ρ ij = ρ(e i , e j ) of the Ricci tensor. The non-zero of them are determined by the following equalities and the well-known their symmetries and antisymmetries: Therefore, using (10), we establish that the constructed manifold is η-Einstein with constants Now, we check whether (G, φ, ξ, η, g) admits a Ricci-like soliton with potential ξ. By virtue of (25) and (20), we obtain the components (L ξ g) ij = (L ξ g) (e i , e j ) and the non-zero of them are (28) (L ξ g) 13 = (L ξ g) 24 = (L ξ g) 31 = (L ξ g) 42 = 2.
Moreover, it is easy to check that ξ is a torse-forming vector field with constant By virtue of (30) and (20), we obtain the components (L ξ g) ij and the non-zero of them are (33) (L ξ g) 11 = (L ξ g) 22 = −2p.