Geometry of conformal η - Ricci solitons and conformal η - Ricci almost solitons on paracontact geometry

: We prove that if an η - Einstein para - Kenmotsu manifold admits a conformal η - Ricci soliton then it is Einstein. Next, we proved that a para - Kenmotsu metric as a conformal η - Ricci soliton is Einstein if its potential vector ﬁ eld V is in ﬁ nitesimal paracontact transformation or collinear with the Reeb vector ﬁ eld. Furthermore, we prove that if a para - Kenmotsu manifold admits a gradient conformal η - Ricci almost soliton and the Reeb vector ﬁ eld leaves the scalar curvature invariant then it is Einstein. We also construct an example of para - Kenmotsu manifold that admits conformal η - Ricci soliton and satisfy our results. We also have studied conformal η - Ricci soliton in three - dimensional para - cosymplectic manifolds.


Introduction
In recent years, geometric flows, in particular, the Ricci flow, have been an interesting research topic in differential geometry. The concept of Ricci flow was first introduced by Hamilton and developed to answer Thurston's geometric conjecture. A Ricci soliton can be considered as a fixed point of Hamilton's Ricci flow (see details in [1]) and a natural generalization of the Einstein metric (i.e., the Ricci tensor [Ric] is a constant multiple of the pseudo-Riemannian metric g), defined on a pseudo-Riemannian manifold M g , ( ) by £ g λg 1 2 Ric where £ V denotes the Lie derivative in the direction of V χ M ( ) ∈ , Ric is the Ricci tensor of g, and λ is a constant. The Ricci soliton is said to be shrinking, steady, and expanding accordingly as λ is negative, zero, and positive, respectively. Otherwise, it will be called indefinite. A Ricci soliton is trivial if V is either zero or Killing on M. First, Pigola et al. [2] assumed the soliton constant λ to be a smooth function on M and named as Ricci almost soliton. After that, Barros et al. studied Ricci almost soliton detailed in [3,4]. Recently, Cho and Kimura [5] generalized the notion of Ricci soliton to η-Ricci soliton, and Calin and Crasmareanu [6] studied this in Hopf hypersurfaces of complex space forms.
A Riemannian or pseudo-Riemannian metric g, defined on a smooth manifold M n of dimension n, is said to be an η-Ricci soliton if there exists a vector field V and constants λ, μ such that £ g λg μη η 1 2 Ric 0. V + + + ⊗ = (1.1) If λ μ M , : → are smooth functions, then M g , ( ) is called η-Ricci almost soliton. If the potential vector field V is a gradient of a smooth function f on M, then the manifold is called a gradient η-Ricci almost soliton. In this case, equation (1.1) can be exhibited as f λ g μ η η Hess Ric 0, where f Hess denotes the Hessian of f . The function f is known as the potential function. In 2005, Fischer [7] has introduced conformal Ricci flow which is a mere generalization of the classical Ricci flow equation that modifies the unit volume constraint to a scalar curvature constraint. The conformal Ricci flow equation was given by where r g ( ) is the scalar curvature of the manifold, p is the scalar nondynamical field, and n is the dimension of the manifold. Corresponding to the conformal Ricci flow equation in 2015, Basu and Bhattacharyya [8] introduced the notion of conformal Ricci soliton equation as a generalization of Ricci soliton equation given by £ g λ p n g 1 2 Ric 1 2 Recently, Siddiqi [9] established the definition of conformal η-Ricci soliton which generalizes the conformal Ricci soliton and η-Ricci soliton. The definition of conformal η-Ricci soliton is given by If the potential vector field V is a gradient of a smooth function f on M, then the manifold is called a gradient conformal η-Ricci almost soliton. In this case, equation ( where f Hess denotes the Hessian of f . The function f is known as the potential function. In the literature, many authors studied Ricci soliton and η-Ricci soliton in the framework of contact metric manifolds. For instance, Sharma [10] considered a K -contact and κ μ , ( )-contact metric as Ricci soliton; Cho and Sharma considered a contact metric as Ricci soliton [11]; η-Einstein almost Kenmotsu metric as Ricci soliton by Wang and Liu [12]. Furthermore, Ghosh considered a noncompact almost contact metric, in particular, a Kenmotsu metric as Ricci soliton (see [13,14]). The interest in Ricci solitons and η-Ricci solitons has risen among theoretical physicists in relation with string theory and connection to general relativity and therefore these have been extensively studied in pseudo-Riemannian settings (see [15,16]). So, several authors studied Ricci soliton and η-Ricci soliton on paracontact metric manifolds, for instance, Patra et al. [17,16] considered a paracontact metric as a Ricci soliton and Naik et al. [18] considered a para-Sasakian metric as η-Ricci soliton. In [19], Welyczko introduced notion of para-Kenmotsu manifold, which is the analogous of Kenmotsu manifold [20] in paracontact geometry and studied in detail by Zamkovoy [21]. Furthermore, Balaga studied some aspects of η-Ricci solitons on para-Kenmotsu and Lorentzian para-Sasakian manifolds (see [22][23][24]).
Are there paracontact metric almost manifolds, whose metrics are conformal η-Ricci soliton?
We will give the answer of the above question very affirmatively in the following sections. This paper is organized as follows. After collecting some of the basic definitions and formulas on para-Kenmotsu manifold in Section 2, we prove in Section 3 that a para-Kenmotsu metric as a conformal η-Ricci soliton is Einstein if it is η-Einstein or the potential vector field V is infinitesimal paracontact transformation or V is collinear with the Reeb vector field ξ . In Section 4, we consider conformal η-Ricci almost solitons on para-Kenmotsu manifold and find some η-Einstein and Einstein manifolds using conformal η-Ricci almost solitons. We draw an example of para-Kenmotsu manifold that admits conformal η-Ricci soliton. In the last section, we consider three-dimensional para-coysmplectic manifold as a conformal η-Ricci soliton and deduce some relations on the scalar curvature of the manifold.

Preliminaries
In this section, we give a brief review of several fundamental notions and formulas which we will need later on. We refer to [15,21,44,45] for more details as well as some examples. ) admits a pseudo-Riemannian metric g such that , then we say that M has an almost paracontact metric structure and g is called compatible metric. The fundamental 2-form Φ of an almost paracontact metric structure φ ξ η g , , , is called a paracontact metric manifold. In this case, η is a contact form, i.e., η η d 0 n ( ) ∧ ≠ , ξ is its Reeb vector field, and M is a contact manifold (see [46]). An almost paracontact metric manifold is said to be para-Kenmotsu manifold (see [21] . On para-Kenmotsu manifold the following formulas hold [21]: , where ∇, R, and Q denote, respectively, the Riemannian connection, the curvature tensor, and the Ricci operator of g associated with the Ricci tensor given by . Now, we prove the following lemma on para-Kenmotsu manifold. In [47], Sarkar et al. have proved the following lemma in different techniques. Here we have used some short techniques and different methods to prove the lemma. which gives proof of the first part of the lemma. Now differentiating (2.4) along Z, we obtain From second Bianchi identity, we infer Putting equation (2.12) in (2.11), we obtain which gives our complete proof. □

On conformal η-Ricci soliton
In this section, we study the conformal η-Ricci soliton on para-Kenmotsu manifold and find some important conditions so that a para-Kenmotsu metric as a conformal η-Ricci soliton is Einstein. First, we recall a definition: a contact metric manifold M n 2 1 + is said to be η-Einstein, if the Ricci tensor Ric can be written as where α, β are smooth functions on M. For an η-Einstein K -contact manifold (see Yano and Kon [48]) and para-Sasakian manifold [45] of dimension 3 > , it is well known that the functions α, β are constants, but for a η-Einstein para-Kenmotsu manifold this is not true. So, we continue α, β as functions. In [13], Ghosh studied three-dimensional Kenmotsu metric as a Ricci soliton and for higher dimension in [14]. Recently, Patra [16] considered Ricci soliton on para-Kenmotsu manifold and proved that an η-Einstein para-Kenmotsu metric as a Ricci soliton is Einstein and therefore here we consider η-Einstein para-Kenmotsu metric as a conformal η-Ricci soliton. Before obtaining our main results first we derive the following lemma.
Furthermore, from (2.3) we obtain R X ξ ξ X η X ξ , ( ) ( ) = − + and the Lie derivative of this along V yields for any X χ M ( ) ∈ . If g represents a conformal η-Ricci soliton with potential vector field V , then Lemma (3.1) holds, i.e., £ R X ξ ξ , 0 Plugging it into (3.15) and using (2.4) provides for any X χ M ( ) ∈ . Again, applying (1.3) and (3.14) in (3.16) yields n λ p μ φ X 2 0 Next we consider a para-Kenmotsu metric as a conformal η-Ricci soliton with nonzero potential vector field V is collinear with ξ and prove the following result. Proof. Since the potential vector field V is collinear with ξ , i.e., V νξ = for some smooth function ν on M.
. By virtue of this, the soliton equation and therefore the smooth function ν reduces to a constant and it equals to μ and hence V μξ = .
In particular, we can also say that if a para-Kenmotsu manifold admits a conformal η-Ricci soliton with the nonzero potential vector field V is ξ , then it is Einstein with constant scalar curvature r n n 2 2 1 ( ) = − + . On paracontact metric manifold M, a vector field X is said to be infinitesimal paracontact transformation if it preserves the paracontact form η, i.e., there exists a smooth function ρ on M that satisfies for any Y χ M ( ) ∈ and if ρ 0 = then X is said to be strict. Here we consider that a para-Kenmotsu metric as a conformal η-Ricci soliton with potential vector field V is infinitesimal paracontact transformation and prove the following. , n 1 > , be a para-Kenmotsu manifold. If g represents a conformal η-Ricci soliton with the potential vector field V is infinitesimal paracontact transformation, then V is strict and g is Einstein with constant scalar curvature r n n 2 2 Proof. First, recalling the well-known formula (see page no. 23 of [49]): along V , we acquire £ ξ ρξ V = . Thus, equations (3.14) and (3.17) entail that ρ 0 = and therefore £ ξ . Thus, from (3.8) we conclude the rest part of this theorem. Hence the proof.

Proof. Equation (1.4) can be exhibited as
Df QX λ p n X μη X ξ 1 2 for any X χ M ( ) ∈ . Using this in R X Y , , [ ] = ∇ ∇ − ∇ , we can easily obtain the curvature tensor expression in the following form: . Taking contraction of (4.2) over X with respect to an orthonormal basis e i : 1,2, , for any Y χ M ( ) ∈ . Now, substituting ξ for Y in (4.2) and using Lemma 2.1 provides for any X χ M ( ) ∈ . Next, taking inner product of (4.4) with ξ and using (2.6), we obtain g R X ξ Df ξ , , . By virtue of (2.5), the preceding equation reduces to for any X χ M ( ) ∈ . Furthermore, using (2.5) in (4.4), we infer for any X χ M ( ) ∈ . By virtue of (4.5), equation (4.6) reduces to for any X χ M ( ) ∈ .
for any X on χ M ( ) ∈ . By hypothesis: ξr 0 = and therefore, the trace of (2.8) gives r n n 2 2 1 ( ) = − + . It follows from (4.8) that QX nX 2 = − , as required. So, we complete the proof. □ Note that, Theorem 4.1 is a more general version, where λ, μ are smooth functions on M, and therefore it also holds for gradient conformal η-Ricci soliton, where λ, μ are constants. Next, considering a para-Kenmotsu metric as a conformal η-Ricci almost soliton with the potential vector field V is pointwise collinear with the Reeb vector field ξ , we extend Theorem 4.1 from gradient conformal η-Ricci almost soliton to conformal η-Ricci almost soliton and prove the following. Proof. By hypothesis: V σξ = for some smooth function σ on M. It follows that . By virtue of this, the soliton equation . Now, putting X Y ξ = = in (4.9) and using (2.6) yields ξσ n λ p μ 2 Using this in (4.9) entails that X Y σ λ p n g X Y n λ p n σ η X η Y Ric , 1 2 . Hence, M is η-Einstein. Moreover, if the Reeb vector field ξ leaves the scalar curvature r invariant, i.e., ξr 0 = . Now, tracing (2.7) yields ξr r n n 2 2 2 1 ( ) { ( )} = − + + and therefore, r n n 2 2 1 ( ) = − + . Using this in the trace of (4.10) gives λ σ n p 2   The pseudo-Riemannian metric is given by . Thenη e φ X η X e 1, 3 2 3 . Thus, φ ξ η g , , , By virtue of this we can verify (2.2) and therefore M φ ξ η g , , , 3 ( )is a para-Kenmotsu manifold. Using the well-known expression of curvature tensor R X Y , , . If we choose the potential function f x y z z , , 5 Three-dimensional para-cosymplectic metric as conformal η-Ricci soliton Dacko [50] introduced the notion of para-cosymplectic manifold. The fundamental 2-form Φ is defined on an almost paracontact metric manifold M ϕ ξ η g , , , , for any vector fields X and Y on M. Clearly, the skew-symmetricness of the 2-form Φ inherits from ϕ.
An almost paracontact metric manifold is said to be almost para-cosymplectic if the forms η and Φ are closed, i.e., η d 0 = and dΦ 0 = , respectively. In addition, if the normality of almost para-cosymplectic manifold is fulfilled, then it is called para-cosymplectic manifold. Equivalently, we can say that an almost paracontact metric manifold is para-cosymplectic if the forms η and ϕ are parallel with respect to the corresponding Levi-Civita connection ∇ of the metric g, i.e., η 0 ∇ = and Φ 0 ∇ = , respectively. We recall some useful relations which are satisfied by any para-cosymplectic manifold where X is the arbitrary vector field and R, ∇, S, and Q are the usual notations. For three-dimensional case we have additional Riemannian curvature property for arbitrary vector fields X Y , , Z. Using this result we deduce that three-dimensional para-cosymplectic manifold satisfies . A vector field V is said to be conformal Killing vector field or simply conformal vector field if there is a smooth function ρ such that ρ is called the conformal coefficient. If we consider the conformal coefficient ρ to be zero, then the conformal vector field reduces to Killing vector field. Now we first prove some lemmas whose results are used to deduce our main result.
Lemma 5.1. (See [49]) If a n-dimensional Riemannian manifold admits a conformal vector field V, then we obtain for any vector fields X and Y , where D and Δ denote the gradient and Laplacian operator of g, respectively.
Lemma 5.2. If the metric g of a three-dimensional para-cosymplectic manifold represents a conformal η-Ricci soliton, then the following properties hold Proof. As the vector field ξ is a unit vector field we have g ξ ξ , 1 ( ) = . Taking Lie derivative of the previous relation w.r.t. vector field V we have £ g ξ ξ η £ ξ Using Proof. For proof, we refer to [36]. A vector field V on an n-dimensional semi-Riemannian manifold M g , ( ) is said to be conformal vector field if where ρ is called the conformal coefficient. □

Lemma 5.4. [49]
On an n-dimensional semi-Riemannian manifold M g , ( ) endowed with conformal vector field V, we obtain Now, we prove the following lemma.
Hence, if r 0 ≠ , then the manifold is an Einstein manifold, and if r 0 = , then the manifold is Ricciflat. □