Pseudosymmetric normal paracontact metric space forms admitting ( α, β ) − type almost η − Ricci-Yamabe solitons

: In this paper, we have considered normal paracontact metric space forms ad-mitting ( α, β ) − type almost η − Ricci-Yamabe solitons by means of some curvature tensors. Ricci pseudosymmetry concepts of normal paracontact metric space forms admit-ting ( α, β ) − type almost η − Ricci-Yamabe soliton have introduced according to choosing of some special curvature tensors such as Riemann, concircular, projective, W 1 curvature tensor . After that, according to choosing of the curvature tensors, necessary conditions are given for normal paracontact metric space form admitting ( α, β ) − type almost η − Ricci-Yamabe soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications are made under the some conditions.


Introduction
In differential geometry, an interesting problem is whether a compact connected Riemannian manifold is conformally equivalent to a manifold of constant scalar curvature.This problem was formulated by Yamabe in 1960.Yamabe himself gave the affirmative answer, though there were some lacuna in his arguments.
In the past twenty years, the theory of geometric flows has been the most significant geometrical tool to explain the geometric structures in Riemannian geometry.A certain section of solutions on which the metric evolves by dilations and diffeomorphisms plays an important part in the study of singularities of the flows as they appear as possible singularity models.They are often called soliton solutions.
Another important topic of differential geometry is Ricci flow which was developed by Richerd Hamilton in order to solve the century long open problem "Poincare conjecture".The notion of Yamabe flow also arose parallelly from the work of Hamilton [1].
Hamilton first time introduced the concept of Ricci flow and Yamabe flow simultaneously in 1988.Ricci soliton and Yamabe soliton emerge as the limit of the solutions of the Ricci flow and Yamabe flow, respectively.The notion of Yamabe flow was introduced by Hamilton as a tool for constructing metrics of constant scalar curvature in a given conformal class of Riemannian metrics on Riemannian manifold (Φ, g) , n ≥ 3. The Yamabe flow is an evolution equation for metrics on a Riemannian manifolds as follows: ∂ ∂t g (t) = −r (t) g (t) , (1) where r (t) denotes the scalar curvature of the metric g (t) .Yamabe solitons correspond to self-similar solutions of the Yamabe flow.In dimension n = 2 the Yamabe soliton is equivalent to Ricci soliton.However, in dimension n > 2, the Yamabe and Ricci solitons do not agree as the first preserves the conformal class of the metric but the Ricci soliton does not in general.
Over the past twenty years, the theory of geometric flows, such as Ricci flow and Yamabe flow has been the focus of attraction of many geometers.Recently, in 2019, Güler and Crasmareanu introduced the study of a new geometric flow which is a scalar combination of Ricci and Yamabe flow under the name Ricci-Yamabe map in [2].This is also called the Ricci-Yamabe flow of the type (α, β) .The Ricci-Yamabe flow is an evolution for the metrics on the Riemannian or semi-Riemannian manifolds defined as Due to the sign of involved scalars and the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow.This kind of multiple choice can be useful in some geometrical or physical models for example relativistic theories.Therefore naturally Ricci-Yamabe soliton emerges as the limit of the soliton of Ricci-Yamabe flow.This is a strong inspiration for initiating the study of Ricci-Yamabe solitons is the fact that although Ricci solitons and Yamabe soliton are same in two dimensional spaces, there are essentially differences in higher dimensions.An interpolation soliton between Ricci and Yamabe soliton is considered, where the name Ricci-Bourguignon soliton corresponds to Ricci-Bourguignon flow but it depends on a single scalar in [3].A soliton to the Ricci-Yamabe flow is called Ricci-Yamabe solitons if it moves only by one parameter group of diffeomorphism and scaling.To be precise a Ricci-Yamabe soliton on Riemannian manifold (Φ, g) in [4] is a data (g, V, λ, α, β) satisfying where S is the Ricci tensor, r is the scalar curvature and L V is the Lie-derivative along the vector field.If λ > 0, λ < 0, or λ = 0, then the (Φ, g) is called Ricci-Yamabe expander, Ricci-Yamabe shrinker, or Ricci-Yamabe steady soliton, respectively.Therefore, equation (2) is called Ricci-Yamabe soliton of (α, β) −type, which is a generalization of Ricci and Yamabe solitons.We note that Ricci-Yamabe soliton of type (α, 0) , (0, β) −type are α−Ricci soliton and β−Yamabe soliton respectively.An advenced extension of Ricci soliton is the concept of η−Ricci soliton defined by Siddiqi and Akyol in [5] and by Cho and Kimura in [6].Therefore analogously we can define the new notion by perturbing the Equation (2) that defines the type of soliton by a multiple of a certain (0, 2) −tensor field η ⊗ η, we obtain a slightly more general notion, namely, η−Ricci-Yamabe soliton of type (α, β) defined as: Again, let us remark that η−Ricci-Yamabe soliton of type (α, 0) or (1, 0) , (0, β) or (0, 1) −type are α−η−Ricci soliton (or η−Ricci soliton) and β −η−Yamabe soliton (or η−Yamabe soliton) respectively for more details about these particular cases see in [7][8][9][10][11].
According to Pigola et al. if we replace the constant λ in (3) with a smooth function λ ∈ C ∞ (Φ) , called soliton function, then we say (Φ, g) is an almost Ricci soliton.
In this paper, we have considered normal paracontact metric space forms admitting (α, β) −type almost η−Ricci-Yamabe solitons by means of some curvature tensors.Ricci pseudosymmetry concepts of normal paracontact metric space forms admitting (α, β) −type almost η−Ricci-Yamabe soliton have introduced according to choosing of some special curvature tensors such as Riemann, concircular, projective, W 1 curvature tensor.After that, according to choosing of the curvature tensors, necessary conditions are given for normal paracontact metric space form admitting (α, β) −type almost η−Ricci-Yamabe soliton to be Ricci semisymmetric.Then some characterizations are obtained and some classifications are made under the some conditions.
For simplicity's sake, the normal paracontact metric space form expression will be expressed as N P M S-form after this part of the article.Similarly, for brevity, after this part of the article, η−Ricci-Yamabe soliton expressions will be shown as η − RY S, Ricci pseudosymmetric as Ricci − P , and Ricci semisymetric as Ricci − S.

Preliminaries
Let's take an n−dimensional differentiable Φ manifold.If it admits a tensor field ϕ of type (1, 1), a contravariant vector field ξ, and a 1-form η satisfying the following conditions; and then, Φ is called a normal paracontact metric manifold, where ∇ is Levi-Civita connection.From (7) , we can easily see that for any ϵ 1 ∈ χ (Φ) [12].
Moreover, if such a manifold has constant sectional curvature equal to c, then is the Riemannian curvature tensor is R given by for any vector fields ϵ 1 , ϵ 2 , ϵ 3 ∈ χ (Φ) [16].In a N P M S−form by direct calculations, we can easily see that from which for any ϵ 1 , ϵ 2 ∈ χ (Φ) , where Q is the Ricci operator and S is the Ricci tensor of Φ.
Lemma 1.Let Φ be an n-dimensional N P M S−forms.In this case, the following equations are obtained.
where R, S, and Q are Riemann curvature tensor, Ricci curvature tensor, and Ricci operator, respectively.Let Φ be a Riemannian manifold, T is (0, k) −type tensor field and A is (0, 2) −type tensor field.In this case, Tachibana tensor field Q (A, T ) is defined as where,
For an n−dimensional semi-Riemann manifold Φ, the concircular curvature tensor is defined as For an n−dimensional N P M S−form, if we choose ϵ 3 = ξ in (28) , we can write For an n−dimensional N P M S−form, if we choose ϵ 3 = ξ in (28) , we can write Definition 2. Let Φ be an n−dimensional N P M S−form.If there exists a function , Theorem 3. Let Φ be a N P M S−form and (g, ξ, λ, µ, α, β) be (α, β) −type almost Proof.Let's assume that N P M S−form Φ be concircular Ricci−P and (g, ξ, λ, µ, α, β) be (α, β) −type almost η − RY S on Φ.That's mean for all ϵ 1 , ϵ 2 , ϵ 4 , ϵ 5 ∈ Γ (T Φ) .From the last equation, we can easily write If we choose ϵ 5 = ξ in (33) , we get By using of ( 22) and (31) in (34) , we have where A = 1 − r n(n−1) .Substituting (32) into (35), we have If we use (21) in the (36), we can write If we write ϕϵ 4 instead of ϵ 4 in (37) and make use of (1) , we obtain By means of (37) and (38) , we conclude and so we get This completes the proof of Theorem.
For an n−dimensional semi-Riemann manifold Φ, the projective curvature tensor is defined as For an n−dimensional N P M S−form, if we choose ϵ 3 = ξ in (37) , we can write and similarly, if we take the inner product of both sides of (37) by ξ, we get Definition 3. Let Φ be an n−dimensional N P M S−form.If there exists a function then the Φ is said to be projective Ricci − P .Also, if Theorem 4. Let Φ be a N P M S−form and (g, ξ, λ, µ, α, β) be (α, Proof.Let's assume that N P M S−form Φ be projective Ricci−P and (g, ξ, λ, µ, α, β) be (α, β) −type almost η − RY S on Φ.That's mean for all ϵ 1 , ϵ 2 , ϵ 4 , ϵ 5 ∈ Γ (T Φ) .From the last equation, we can easily write If we choose ϵ 5 = ξ in (42) , we get If we make use of ( 22) and (40) in (43) , we have If we use (41) in the (44), we get If we use (21) in the (45), we can write If we write ϕϵ 4 instead of ϵ 4 in (46) and make use of (1) , we obtain It is clear from (46) and (47) , we obtain This completes the proof of Theorem.
For an n−dimensional semi-Riemann manifold Φ, the W 1 −curvature tensor is defined as For an n−dimensional N P M S−form, if we choose ϵ 3 = ξ in (48) , we can write and similarly, if we take the inner product of both of sides of (48) by ξ, we get , Proof.Let's assume that N P M S−form Φ be W 1 − Ricci − P and (g, ξ, λ, µ, α, β) be (α, β) −type almost η − RY S on Φ.That's mean for all ϵ 1 , ϵ 2 , ϵ 4 , ϵ 5 ∈ Γ (T Φ) .From the last equation, we can easily write If we choose ϵ 5 = ξ in (51) , we get By means of (22) and (49) in (52) , we have If we use (50) in the (53), we get Funding: No support was received from any institution or organization for the fee of the article.

Corollary 5 .
Let Φ be a N P M S−form and (g, ξ, λ, µ, α, β) be (α, β) −type almost η − RY S on Φ.If Φ is a projective Ricci − P , then Φ is a projective Ricci − S or the following results are observed depending on the state of α

) Definition 4 .
Let Φ be an n−dimensional N P M S−form.If there exists a function H 4 on Φ such that