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

  • Yanlin Li , Santu Dey , Sampa Pahan and Akram Ali EMAIL logo
Published/Copyright: August 9, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

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 field V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η -Ricci almost soliton and the Reeb vector field 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.

Keywords: conformal η-Ricci soliton; Kenmotsu manifold; Einstein manifold; infinitesimal contact transformation; para-cosymplectic manifold
MSC 2010: 53C15; 53C25; 53D15

1 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

1 2 £ V g + Ric = λ g ,

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

(1.1) 1 2 £ V g + Ric + λ g + μ η ⊗ η = 0 .

If λ , μ : M → R 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

(1.2) Hess f + Ric + λ g + μ η ⊗ η = 0 ,

where Hess f 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

∂ g ∂ t + 2 S + g n = − p g , r ( g ) = − 1 ,

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

1 2 £ V g + Ric + λ − 1 2 p + 2 2 n + 1 g = 0 .

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

(1.3) 1 2 £ V g + Ric + λ − 1 2 p + 2 2 n + 1 g + μ η ⊗ η = 0 .

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 (1.3) can be exhibited as

(1.4) Hess f + Ric + λ − 1 2 p + 2 2 n + 1 g + μ η ⊗ η = 0 ,

where Hess f 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]).

Recently, Patra [16] considered Ricci soliton on para-Kenmotsu manifold and proved that a para-Kenmotsu metric as a Ricci soliton is Einstein if it is η -Einstein or the potential vector field V is infinitesimal paracontact transformation. Many authors studied Conformal Ricci solitons and their generalizations in the framework of almost contact and paracontact geometries, e.g., Kenmotsu manifold by [8,25, 26,27], Sasakian manifold by [28,29], generalized Sasakian space forms by [30], f -Kenmotsu manifold by [31,32], para-Kähler manifold by [33], and ( κ , μ ) -Paracontact metric manifolds by Siddiqi [34]. Also, recently Venkatesha et al. [35] considered the metric of η -Einstein para-Kenmotsu manifold as ∗ -Ricci soliton and proved that the manifold is Einstein. Erken [36] considered Yamabe solitons on three-dimensional para-cosymplectic manifold and proved some vital results like the manifold are either η -Einstein or Ricci flat. Motivated by these results we consider a para-Kenmotsu metric as conformal η -Ricci solitons and conformal η -Ricci almost solitons. Li and Ali et al. also did several relevant investigations [25,37,38, 39,40,41, 42,43]. Based on the aforementioned facts and discussions in the research of para-contact geometry, a natural question arises.

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.

2 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. A ( 2 n + 1 ) -dimensional smooth manifold M 2 n + 1 has an almost paracontact structure ( φ , ξ , η ) if it admits a ( 1 , 1 ) -tensor field φ , a vector field ξ , and a 1-form η satisfying the following conditions:

(2.1) ( i ) φ 2 = I − η o ξ , φ ( ξ ) = 0 , η o φ = 0 , η ( ξ ) = 1 ,

(ii) There exists a distribution D : p ∈ M → D p ⊂ T p M : D p = K e r ( η ) = { x ∈ T p M : η ( x ) = 0 } , called paracontact distribution generated by η . If an almost paracontact manifold M 2 n + 1 with a structure ( φ , ξ , η ) admits a pseudo-Riemannian metric g such that

g ( φ X , φ Y ) = − g ( X , Y ) + η ( X ) η ( Y ) ,

for all X , Y ∈ χ ( M ) , 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 ) defined by Φ ( X , Y ) = g ( X , φ Y ) for all vector fields X , Y on M . If Φ = d η , then the manifold M 2 n + 1 ( φ , ξ , η , g ) is called a paracontact metric manifold. In this case, η is a contact form, i.e., η ∧ ( d η ) n ≠ 0 , 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]) if

(2.2) ( ∇ X φ ) Y = g ( φ X , Y ) ξ − η ( Y ) φ X ,

for any X , Y ∈ χ ( M ) . On para-Kenmotsu manifold the following formulas hold [21]:

(2.3) ∇ X ξ = X − η ( X ) ξ ,

(2.4) R ( X , Y ) ξ = η ( X ) Y − η ( Y ) X ,

(2.5) R ( X , ξ ) Y = g ( X , Y ) ξ − η ( Y ) X ,

(2.6) Q ξ = − 2 n ξ ,

for all X , Y ∈ χ ( M ) , 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 Ric ( X , Y ) = g ( Q X , Y ) for all X , Y ∈ χ ( M ) . 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.

Lemma 2.1

On para-Kenmotsu manifold M 2 n + 1 ( φ , ξ , η , g ) the following formulas hold for any X , Y ∈ χ ( M ) ,

(2.7) ( ∇ X Q ) ξ = − Q X − 2 n X ,

(2.8) ( ∇ ξ Q ) X = − 2 Q X − 4 n X .

Proof

Using (2.3) and (2.6), we acquire

(2.9) ( ∇ X Q ) ξ = ∇ X Q ξ − Q ∇ X ξ = ∇ X ( − 2 n ξ ) − Q ( X − η ( X ) ξ ) = − 2 n ( X − η ( X ) ξ ) − Q X + η ( X ) Q ξ = − 2 n X + η ( X ) 2 n ξ − Q X − 2 n ξ η ( X ) = − Q X − 2 n X ,

which gives proof of the first part of the lemma. Now differentiating (2.4) along Z , we obtain

(2.10) ( ∇ Z R ) ( X , Y ) ξ = − R ( X , Y ) Z + g ( X , Z ) Y − g ( Y , Z ) X .

Now, let { e i } i = 1 2 n + 1 be a orthonormal basis on M . Take X = Z = e i in (2.10) and summing up over i , we achieve

(2.11) ∑ i = 1 2 n + 1 g ( ( ∇ e i R ) ( e i , Y ) ξ , W ) = S ( Y , W ) + 2 n g ( Y , W ) .

From second Bianchi identity, we infer

(2.12) ∑ i = 1 2 n + 1 g ( ( ∇ e i R ) ( W , ξ ) Y , e i ) = g ( ( ∇ W Q ) ξ , Y ) − g ( ( ∇ ξ Q ) W , Y ) .

Putting equation (2.12) in (2.11), we obtain

(2.13) g ( ( ∇ W Q ) ξ , Y ) − g ( ( ∇ ξ Q ) W , Y ) = S ( Y , W ) + 2 n g ( Y , W ) , g ( − Q W − 2 n W , Y ) − g ( ( ∇ ξ Q ) W , Y ) = S ( Y , W ) + 2 n g ( Y , W ) ( using ( 2.7 ) ) , − S ( Y , W ) − S ( W , Y ) − 2 n g ( Y , W ) − 2 n g ( Y , W ) = g ( ( ∇ ξ Q ) W , Y ) , ( ∇ ξ Q ) W = − 2 Q W − 4 n W ,

which gives our complete proof.□

3 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 2 n + 1 is said to be η -Einstein, if the Ricci tensor Ric can be written as

(3.1) Ric ( X , Y ) = α g ( X , Y ) + β η ( X ) η ( Y ) ,

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.

Lemma 3.1

Let M 2 n + 1 ( φ , ξ , η , g ) be a para-Kenmotsu manifold. If g represents a conformal η -Ricci soliton with potential vector field V, then we have for any X ∈ χ ( M ) ,

(3.2) ( £ V R ) ( X , ξ ) ξ = 0 .

Proof

Taking the covariant derivative of (1.3) along an arbitrary vector field Z on M and using (2.3) we obtain

(3.3) ( ∇ Z £ V g ) ( X , Y ) = − 2 ( ∇ Z Ric ) ( X , Y ) − 2 μ { g ( X , Z ) η ( Y ) + g ( Y , Z ) η ( X ) − 2 η ( X ) η ( Y ) η ( Z ) } ,

for any X , Y ∈ χ ( M ) . Next, recalling the following commutation formula (see Yano [49], p. 23)

(3.4) ( £ V ∇ Z g − ∇ Z £ V g − ∇ [ V , Z ] g ) ( X , Y ) = − g ( ( £ V ∇ ) ( Z , X ) , Y ) − g ( ( £ V ∇ ) ( Z , Y ) , X ) ,

for all X , Y , Z ∈ χ ( M ) . In view of the parallel Riemannian metric g , it follows from (2.6) that

(3.5) ( ∇ Z £ V g ) ( X , Y ) = g ( ( £ V ∇ ) ( Z , X ) , Y ) + g ( ( £ V ∇ ) ( Z , Y ) , X ) ,

for all X , Y , Z ∈ χ ( M ) . Plugging it into (3.3) we achieve

(3.6) g ( ( £ V ∇ ) ( Z , X ) , Y ) + g ( ( £ V ∇ ) ( Z , Y ) , X ) = − 2 ( ∇ Z Ric ) ( X , Y ) − 2 μ { g ( X , Z ) η ( Y ) + g ( Y , Z ) η ( X ) − 2 η ( X ) η ( Y ) η ( Z ) } ,

for any X , Y , Z ∈ χ ( M ) . Interchanging cyclicly the roles of X , Y , Z in (3.6) we can compute

(3.7) g ( ( £ V ∇ ) ( X , Y ) , Z ) = ( ∇ Z Ric ) ( X , Y ) − ( ∇ X Ric ) ( Y , Z ) − ( ∇ Y Ric ) ( Z , X ) − 2 μ { g ( X , Y ) η ( Z ) − η ( X ) η ( Y ) η ( Z ) } ,

for all Y , Z ∈ χ ( M ) . Now, substituting ξ for Y in (3.7) and using (2.8), (2.7) yields

(3.8) ( £ V ∇ ) ( X , ξ ) = 2 Q X + 4 n X ,

for all X ∈ χ ( M ) . Next, using (2.3), (3.8) in the covariant derivative of (3.8) along Y provides

( ∇ Y £ V ∇ ) ( X , ξ ) = ( £ V ∇ ) ( X , Y ) − 2 ( ∇ Y Q ) X + 2 η ( Y ) ( Q X + 2 n X )

for any X ∈ χ ( M ) . Using this in the following commutation formula (see Yano [49], p. 23)

(3.9) ( £ V R ) ( X , Y ) Z = ( ∇ X £ V ∇ ) ( Y , Z ) − ( ∇ Y £ V ∇ ) ( X , Z ) ,

we can derive

(3.10) ( £ V R ) ( X , Y ) ξ = 2 { ( ∇ Y Q ) X − ( ∇ X Q ) Y } + 2 { η ( X ) Q Y − η ( Y ) Q X } + 4 n { η ( X ) Y − η ( Y ) X }

for all vector fields X , Y ∈ χ ( M ) . Substituting Y by ξ in (3.10) and using (2.6), (2.8), and (2.7) we have the required result.□

Theorem 3.1

Let M 2 n + 1 ( φ , ξ , η , g ) , n > 1 , be a η -Einstein para-Kenmotsu manifold. If g represents a conformal η -Ricci soliton with potential vector field V , then g is Einstein with constant scalar curvature r = − 2 n ( 2 n + 1 ) .

Proof

First, tracing (3.1) gives r = ( 2 n + 1 ) α + β and putting X = Y = ξ in (3.1) and using (2.6) we obtain α + β = − 2 n . Therefore, by computation, (3.1) transforms into

(3.11) Ric ( X , Y ) = 1 + r 2 n g ( X , Y ) − ( 2 n + 1 ) + r 2 n η ( X ) η ( Y ) ,

for all X , Y on M . This gives

(3.12) ( ∇ Y Q ) X = ( Y r ) 2 n { X − η ( X ) ξ } + ( 2 n + 1 ) + r 2 n { g ( X , Y ) ξ + η ( X ) ( Y − 2 η ( Y ) ξ ) } ,

for all X , Y ∈ χ ( M ) . By virtue of this, equation (3.10) provides

(3.13) ( £ V R ) ( X , Y ) ξ = 1 n { ( X r ) ( Y − η ( Y ) ξ ) − ( Y r ) ( X − η ( X ) ξ ) } ,

for all X , Y ∈ χ ( M ) . Setting Y = ξ in (3.13) and using Lemma 3.1 we obtain ( ξ r ) φ 2 X = 0 for any X ∈ χ ( M ) . Using this in the trace of (2.8) we obtain r = − 2 n ( 2 n + 1 ) . It follows from (3.11) that M is Einstein and hence the proof.□

Now, taking the Lie derivative of g ( ξ , ξ ) = 1 along the potential vector field V and applying (1.3) one can obtain

(3.14) η ( £ V ξ ) = λ + μ − 2 n − 1 2 p + 2 2 n + 1 .

Furthermore, from (2.3) we obtain R ( X , ξ ) ξ = − X + η ( X ) ξ and the Lie derivative of this along V yields

(3.15) ( £ V R ) ( X , ξ ) ξ + R ( X , £ V ξ ) ξ + R ( X , ξ ) £ V ξ = { ( £ V η ) X } ξ + η ( X ) £ V ξ

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., ( £ V R ) ( X , ξ ) ξ = 0 . Plugging it into (3.15) and using (2.4) provides

(3.16) ( £ V g ) ( X , ξ ) + 2 η ( £ V ξ ) X = 0 ,

for any X ∈ χ ( M ) . Again, applying (1.3) and (3.14) in (3.16) yields 2 n − λ + 1 2 p + 2 2 n + 1 − μ φ 2 X = 0 for any X ∈ χ ( M ) . Next using (2.1) and then tracing yields 2 n 2 n − λ + 1 2 p + 2 2 n + 1 − μ = 0 . This implies that

(3.17) λ + μ = 2 n + 1 2 p + 2 2 n + 1 .

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.

Theorem 3.2

Let M 2 n + 1 ( φ , ξ , η , g ) be a para-Kenmotsu manifold. If g represents a conformal η -Ricci soliton with nonzero potential vector field V is collinear with ξ , then g is Einstein with constant scalar curvature r = − 2 n ( 2 n + 1 ) .

Proof

Since the potential vector field V is collinear with ξ , i.e., V = ν ξ for some smooth function ν on M . Using (2.3) in the covariant derivative of V = ν ξ along X yields

∇ X V = ( X ν ) ξ + ν { ( X − η ( X ) ξ ) } ,

for any X ∈ χ ( M ) . By virtue of this, the soliton equation (1.3) reduces to

(3.18) 2 Ric ( X , Y ) + ( X ν ) η ( Y ) + ( Y ν ) η ( X ) + 2 ( λ − 1 2 p + 2 2 n + 1 + ν ) g ( X , Y ) − 2 ( ν − μ ) η ( X ) η ( Y ) = 0 ,

for all X , Y ∈ χ ( M ) . Setting X = Y = ξ in (3.18) and using (2.6), (3.17) we obtain ξ ν = 0 . It follows from (3.18) that X ν = 0 . Putting it into (3.18) provides

(3.19) Ric ( X , Y ) = − ν + λ − 1 2 p + 2 2 n + 1 g ( X , Y ) + ( ν − μ ) η ( X ) η ( Y ) ,

for all X , Y ∈ χ ( M ) . This shows that M is η -Einstein and therefore from Theorem 3.1 we conclude that M is Einstein. Thus, from (3.18) we have ν = μ and therefore ν + λ = 2 n + 1 2 p + 2 2 n + 1 (follows from (3.17)). Hence, we have from (3.19) that Ric = − 2 n g and therefore r = − 2 n ( 2 n + 1 ) , as required. Hence the proof.□

Remark 3.1

In Theorem 3.2, we see that X ν = 0 for any X ∈ χ ( M ) and therefore the smooth function ν reduces to a constant and it equals to μ and hence V = μ ξ .

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 = − 2 n ( 2 n + 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

(3.20) £ X η ( Y ) = ρ η ( Y ) ,

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.

Theorem 3.3

Let M 2 n + 1 ( φ , ξ , η , g ) , 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 = − 2 n ( 2 n + 1 ) .

Proof

First, recalling the well-known formula (see page no. 23 of [49]):

(3.21) £ V ∇ X Y − ∇ X £ V Y − ∇ [ V , X ] Y = ( £ V ∇ ) ( X , Y ) ,

for all X , Y ∈ χ ( M ) .

Using (3.15), (3.20), Lemma 3.1 in the Lie derivative of η ( X ) = g ( X , ξ ) along V , we acquire £ V ξ = ρ ξ . Thus, equations (3.14) and (3.17) entail that ρ = 0 and therefore £ V ξ = 0 and V is strict. Furthermore, equation (3.20) gives £ V η = 0 . So, we obtain that ( £ V ∇ ) ( X , ξ ) = 0 for any X ∈ χ ( M ) . Thus, from (3.8) we conclude the rest part of this theorem. Hence the proof.□

4 On conformal η -Ricci almost soliton

In this section, we consider conformal η -Ricci almost soliton on para-Kenmotsu manifold. It follows from (1.4) that conformal η -Ricci almost soliton is the generalization of conformal Ricci almost soliton because it involves two smooth functions λ and μ . First, we consider gradient conformal η -Ricci almost soliton on para-Kenmotsu manifold in order to extend the result of gradient Ricci almost soliton on para-Kenmotsu manifold [16] taking into account of equations (1.3) and (1.4) hold for smooth functions λ , μ . First we prove the following.

Theorem 4.1

Let M 2 n + 1 ( φ , ξ , η , g ) be a para-Kenmotsu manifold. If M admits a gradient conformal η -Ricci almost soliton and the Reeb vector field ξ leaves the scalar curvature r invariant, then it is Einstein with constant scalar curvature − 2 n ( 2 n + 1 ) .

Proof

Equation (1.4) can be exhibited as

(4.1) ∇ X D f + Q X + λ − 1 2 p + 2 2 n + 1 X + μ η ( X ) ξ = 0 ,

for any X ∈ χ ( M ) . Using this in R ( X , Y ) = [ ∇ X , ∇ Y ] − ∇ [ X , Y ] , we can easily obtain the curvature tensor expression in the following form:

(4.2) R ( X , Y ) D f = ( ∇ Y Q ) X − ( ∇ X Q ) Y + ( Y λ ) X − ( X λ ) Y + ( Y μ ) η ( X ) ξ − ( X μ ) η ( Y ) ξ + μ { η ( Y ) X − η ( X ) Y }

for all X , Y ∈ χ ( M ) . Taking contraction of (4.2) over X with respect to an orthonormal basis { e i : i = 1 , 2 , … , 2 n + 1 } , we compute

Ric ( Y , D f ) = − ∑ i = 1 2 n + 1 g ( ( ∇ e i Q ) Y , e i ) + ( Y r ) + 2 n ( Y λ ) + ( Y μ ) + { ( − ξ μ ) + 2 n μ } η ( Y )

for any Y ∈ χ ( M ) . Now, contracting Bianchi’s second identity we have ∑ i = 1 2 n + 1 g ( ( ∇ e i Q ) Y , e i ) = 1 2 ( Y r ) . Plugging it into the previous equation gives

(4.3) Ric ( Y , D f ) = 1 2 ( Y r ) + 2 n ( Y λ ) + ( Y μ ) + { ( − ξ μ ) + 2 n μ } η ( Y )

for any Y ∈ χ ( M ) . Now, substituting ξ for Y in (4.2) and using Lemma 2.1 provides

(4.4) R ( X , ξ ) D f = − Q X − 2 n X + ( ξ λ ) X − ( X λ ) ξ − ( X μ ) ξ + ( ξ μ ) η ( X ) ξ + μ φ 2 X

for any X ∈ χ ( M ) . Next, taking inner product of (4.4) with ξ and using (2.6), we obtain g ( R ( X , ξ ) D f , ξ ) = ξ λ − 1 2 p + 2 2 n + 1 + μ η ( X ) − X λ − 1 2 p + 2 2 n + 1 + μ for any X ∈ χ ( M ) . By virtue of (2.5), the preceding equation reduces to

(4.5) X ( f + λ + μ ) = ξ ( f + λ + μ ) η ( X )

for any X ∈ χ ( M ) . Furthermore, using (2.5) in (4.4), we infer

(4.6) X ( f + λ + μ ) ξ = − Q X + { ξ ( f + λ ) + μ − 2 n } X + { − μ + ( ξ μ ) } η ( X ) ξ ,

for any X ∈ χ ( M ) . By virtue of (4.5), equation (4.6) reduces to

(4.7) Q X = { − μ + ξ ( λ + f ) − 2 n } X − { μ + ξ ( λ + f ) } η ( X ) ξ ,

for any X ∈ χ ( M ) . Thus, M is η -Einstein. Substituting Y by D f in (2.5) and contracting we obtain Ric ( ξ , D f ) = − 2 n ( ξ f ) . Plugging it into (4.3) yields ( ξ r ) + 4 n ξ ( f + λ ) = − 4 n μ . Using it in the trace of (2.8) we have ξ ( λ + f ) = − μ + 2 n + 1 + r 2 n . By virtue of this, equation (4.7) transforms into

(4.8) Q X = 1 + r 2 n X − ( 2 n + 1 ) + r 2 n η ( X ) ξ ,

for any X on ∈ χ ( M ) . By hypothesis: ξ r = 0 and therefore, the trace of (2.8) gives r = − 2 n ( 2 n + 1 ) . It follows from (4.8) that Q X = − 2 n X , 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.

Corollary 4.1

Let M 2 n + 1 ( φ , ξ , η , g ) be a para-Kenmotsu manifold. If M admits g represents a gradient conformal η -Ricci soliton and the Reeb vector field ξ leaves the scalar curvature r invariant, then M is Einstein with constant scalar curvature − 2 n ( 2 n + 1 ) .

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.

Theorem 4.2

Let M 2 n + 1 ( φ , ξ , η , g ) be a para-Kenmotsu manifold. If M admits a conformal η -Ricci almost soliton with nonzero potential vector field V collinear with ξ , then g is η -Einstein. Moreover, if the Reeb vector field ξ leaves the scalar curvature r invariant, then g is Einstein with constant scalar curvature − 2 n ( 2 n + 1 ) .

Proof

By hypothesis: V = σ ξ for some smooth function σ on M . It follows that

( £ V g ) ( X , Y ) = ( X σ ) η ( Y ) + ( Y σ ) η ( X ) + 2 σ { g ( X , Y ) − η ( X ) η ( Y ) } ,

for all X , Y ∈ χ ( M ) . By virtue of this, the soliton equation (1.3) transforms into

(4.9) 2 Ric ( X , Y ) + ( X σ ) η ( Y ) + ( Y σ ) η ( X ) + 2 σ + λ − 1 2 p + 2 2 n + 1 g ( X , Y ) = 2 ( σ − μ ) η ( X ) η ( Y ) ,

for all X , Y ∈ χ ( M ) . Now, putting X = Y = ξ in (4.9) and using (2.6) yields ξ σ = 2 n − λ + 1 2 p + 2 2 n + 1 − μ . Thus, equation (4.9) gives X σ = 2 n − λ + 1 2 p + 2 2 n + 1 − σ η ( X ) . Using this in (4.9) entails that

(4.10) Ric ( X , Y ) = − σ + λ − 1 2 p + 2 2 n + 1 g ( X , Y ) − ( 2 n − λ + 1 2 p + 2 2 n + 1 − σ ) η ( X ) η ( Y ) ,

for all X , Y ∈ χ ( M ) . Hence, M is η -Einstein. Moreover, if the Reeb vector field ξ leaves the scalar curvature r invariant, i.e., ξ r = 0 . Now, tracing (2.7) yields ( ξ r ) = − 2 { r + 2 n ( 2 n + 1 ) } and therefore, r = − 2 n ( 2 n + 1 ) . Using this in the trace of (4.10) gives λ − σ = 2 n + 1 2 p + 2 2 n + 1 . Thus, from (4.10) we have Q X = − 2 n X and therefore M is Einstein. This completes the proof.□

If we consider V = σ ξ for some constant σ instead of a function, then (4.9) holds good and therefore inserting X = Y = ξ in (4.9) and using (2.6) gives ξ σ = 2 n − λ + 1 2 p + 2 2 n + 1 − μ . Using this in (4.9) yields λ − σ = 2 n + 1 2 p + 2 2 n + 1 , where we have used σ is a constant. Thus, from (4.10) we can conclude the following corollary.

Corollary 4.2

Let M 2 n + 1 ( φ , ξ , η , g ) be a para-Kenmotsu manifold. If M admits a nontrivial conformal η -Ricci almost soliton with V = σ ξ for some constant σ , then it is Einstein with constant scalar curvature r = − 2 n ( 2 n + 1 ) .

Now, we present an example of para-Kenmotsu manifold that admits a gradient conformal η -Ricci soliton.

Example 4.1

Let ( x , y , z ) be the standard coordinates in R 3 and M 3 = { ( x , y , z ) ∈ R 3 } be a three-dimensional manifold. Now, consider a orthonormal basis { e 1 , e 2 , e 3 } of vector fields on M 3 , where e 1 = ∂ ∂ x , e 2 = ∂ ∂ y , e 3 = x ∂ ∂ x + y ∂ ∂ y + ∂ ∂ z . Define ( 1 , 1 ) tensor field φ as follows:

φ ( e 2 ) = e 1 , φ ( e 1 ) = e 2 , φ ( e 3 ) = 0 .

The pseudo-Riemannian metric is given by

( g i j ) = 1 0 0 0 − 1 0 0 0 1

and η ( X ) = g ( X , e 5 ) for any X ∈ χ ( M ) . Then η ( e 3 ) = 1 , φ 2 = X − η ( X ) e 3 , and g ( φ X , φ Y ) = − g ( X , Y ) + η ( X ) η ( Y ) for all X , Y ∈ χ ( M ) . Thus, ( φ , ξ , η , g ) is an almost paracontact structure. The nonzero components of the Levi-Civita connection ∇ (using Koszul’s formula) are

(4.11) ∇ e 1 e 3 = e 1 ∇ e 1 e 1 = − ∇ e 2 e 2 = − e 3 ∇ e 2 e 3 = e 2 .

By virtue of this we can verify (2.2) and therefore M 3 ( φ , ξ , η , g ) is a para-Kenmotsu manifold. Using the well-known expression of curvature tensor R ( X , Y ) = [ ∇ X , ∇ Y ] − ∇ [ X , Y ] , we now compute the following nonzero components

R ( e 1 , e 2 ) e 1 = e 2 , R ( e 1 , e 2 ) e 2 = e 1 , R ( e 1 , e 3 ) e 1 = e 3 , R ( e 1 , e 3 ) e 3 = − e 1 , R ( e 2 , e 3 ) e 2 = − e 3 , R ( e 2 , e 3 ) e 3 = − e 2 .

Using these, we compute the components of the Ricci tensor

Ric ( e i , e i ) = − 2 for i = 1 , 3 Ric ( e 2 , e 2 ) = 2 .

Therefore, the Ricci tensor is given by Ric = − 2 g and the scalar curvature r = − 6 . Hence, ( M 3 , g ) is Einstein with constant scalar curvature r = − 2 n ( 2 n + 1 ) for n = 1 . Let us consider a potential vector field V = ( x − 1 ) ∂ ∂ x + ( y − 1 ) ∂ ∂ y + ∂ ∂ z on M 3 . Then using (4.11) we obtain

(4.12) 1 2 ( £ V g ) ( X , Y ) = { g ( X , Y ) − η ( X ) η ( Y ) } ,

for all X , Y ∈ χ ( M 3 ) . If we choose the potential function f ( x , y , z ) = ( x − 1 ) 2 2 + ( y − 1 ) 2 2 + z , then from Ric = − 2 g and (4.12) we can conclude that the metric g is a gradient conformal η -Ricci soliton with constants λ = p 2 + 4 3 and μ = 1 .

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 ) by Φ ( X , Y ) = g ( X , ϕ Y ) 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., d η = 0 and d Φ = 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

(5.1) R ( X , Y ) ξ = 0 ,

(5.2) ( ∇ X ϕ ) = 0 ,

(5.3) ∇ X ξ = 0 ,

(5.4) S ( X , ξ ) = 0 ,

(5.5) Q ξ = 0 ,

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

(5.6) R ( X , Y ) Z = g ( Y , Z ) Q X − g ( X , Z ) Q Y + g ( Q Y , Z ) X − g ( Q X , Z ) Y − r 2 { g ( Y , Z ) X − g ( X , Z ) Y }

for arbitrary vector fields X , Y , Z . Using this result we deduce that three-dimensional para-cosymplectic manifold satisfies

(5.7) S ( X , Y ) = r 2 { g ( X , Y ) − η ( X ) η ( Y ) } ,

(5.8) Q X = r 2 { X − η ( X ) ξ } ,

for any X , Y ∈ χ ( M ) .

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

(5.9) £ V g = 2 ρ g .

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

(5.10) ( £ V S ) ( X , Y ) = − ( n − 2 ) g ( ∇ X D ρ , Y ) + ( Δ ρ ) g ( X , Y ) ,

(5.11) £ V r = 2 ( n − 1 ) Δ ρ − 2 ρ r ,

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

(5.12) η ( £ V ξ ) = λ − p 2 − 1 3 + μ ,

(5.13) ( £ V η ) ξ = − λ + p 2 + 1 3 − μ .

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 ( £ V g ) ( ξ , ξ ) + 2 η ( £ V ξ ) = 0 . Using (1.3), (2.1), and (5.7), we achieve

η ( £ V ξ ) = λ − p 2 − 1 3 .

Taking Lie derivative of (2.1) along the direction of V and using (5.12), we arrive at

( £ V η ) ξ = − λ + p 2 + 1 3 . □

Lemma 5.3

For a three-dimensional para-cosymplectic manifold, we have

(5.14) ξ ( r ) = 0 .

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

(5.15) £ V g = 2 ρ g ,

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

(5.16) ( £ V S ) ( X , Y ) = − ( n − 2 ) g ( ∇ X D ρ , Y ) + ( Δ ρ ) g ( X , Y ) ,

(5.17) £ V r = − 2 ρ r + 2 ( n − 1 ) ( Δ ρ ) .

Now, we prove the following lemma.

Lemma 5.5

For a three dimensional para-cosymplectic manifold which admits a conformal η -Ricci soliton and satisfies p = 2 λ + 2 μ − 2 3 has always harmonic scalar curvature, i.e., Δ r = 0 .

Proof

Equations (1.3) and (5.15) enable us to obtain

(5.18) 2 ρ g + 2 Ric + 2 λ − p + 2 3 g + 2 μ η ⊗ η = 0 .

Now, we contract the previous equation to yield

(5.19) 2 r + 6 ρ + 6 λ − p + 2 3 3 + 2 μ = 0 .

Using (5.19), we achieve

ρ = − r + 3 λ − 3 2 p + 2 3 + μ 3 .

We plug n = 3 into identity (5.16) to infer

(5.20) ( £ V S ) ( X , Y ) = − g ( ∇ X D ρ , Y ) + ( Δ ρ ) g ( X , Y ) ,

D ρ = − D r 3 .

Also,

Δ ρ = − Δ r 3 .

So,

(5.21) ( £ V S ) ( X , Y ) = 1 3 g ( ∇ X D r , Y ) − Δ r 3 g ( X , Y ) .

Again,

(5.22) £ V r = − 2 ρ r + 4 Δ ρ = 2 r 3 r + 3 λ − 3 2 p + 2 3 + μ − 4 3 Δ r .

Now, from the identity (5.7), we arrive at

(5.23) ( £ V S ) ( X , Y ) = £ V r 2 g ( X , Y ) − £ V r 2 η ( X ) η ( Y ) + r 2 [ ( £ V g ) ( X , Y ) − ( £ V η ) ( X ) η ( Y ) − ( £ V η ) ( X ) η ( Y ) ] = 1 2 2 3 r r + 3 λ − 3 2 p + 2 3 + μ − 4 3 Δ r g ( X , Y ) + r 2 − 2 S ( X Y ) − 2 λ − p + 2 3 g ( X , Y ) − 2 μ η ( X ) η ( Y ) − 1 2 2 r 3 r + 3 λ − 3 2 p + 2 3 + μ − 4 3 Δ r η ( X ) η ( Y ) − r 2 [ ( £ V η ) ( X ) η ( Y ) − ( £ V η ) ( X ) η ( Y ) ] .

Now, we put the value of (5.21) in (5.23) to yield

(5.24) 1 3 g ( ∇ X D r , Y ) − Δ r 3 g ( X , Y ) = £ V r 2 g ( X , Y ) − £ V r 2 η ( X ) η ( Y ) + r 2 [ ( £ V g ) ( X , Y ) − ( £ V η ) ( X ) η ( Y ) − ( £ V η ) ( X ) η ( Y ) ] = 1 2 2 3 r r + 3 λ − 3 2 p + 2 3 + μ − 4 3 Δ r g ( X , Y ) + r 2 − 2 S ( X Y ) − 2 λ − p + 2 3 g ( X , Y ) − 2 μ η ( X ) η ( Y ) − 1 2 2 r 3 r + 3 λ − 3 2 p + 2 3 + μ − 4 3 Δ r η ( X ) η ( Y ) − r 2 [ ( £ V η ) ( X ) η ( Y ) − ( £ V η ) ( X ) η ( Y ) ] .

Now, we plug X = Y = ξ into the identity (5.24) and using the condition p = 2 λ + 2 μ − 2 3 to arrive Δ r = 0 .□

Theorem 5.1

If the semi-Riemannian metric of a three-dimensional para-cosymplectic manifold is a conformal η -Ricci soliton satisfying p = 2 λ + 2 μ − 2 3 , then it has a constant scalar curvature.

  1. If r ≠ 0 , then the manifold is an Einstein manifold.

  2. If r = 0 , then the manifold is Ricci-flat.

Proof

We have

(5.25) g ( ∇ X D r , ξ ) = 0 .

From equations (5.21) and (5.23), we obtain

(5.26) 1 3 g ( ∇ X D r , Y ) − Δ r 3 g ( X , Y ) = r 2 − 2 λ − p + 2 3 − 2 μ − r 2 [ ( £ V η ) ( X ) + η ( X ) ( £ V η ) ( ξ ) ] .

Now, we utilize Lemma 5.3, Lemma 5.4, (5.25) and using the given condition to find

(5.27) r 2 [ ( £ V η ) ( X ) ] = 0 .

As r ≠ 0 , we arrive ( £ V η ) ( X ) = 0 . Again

(5.28) 1 3 ∇ X D r = 1 2 2 3 r r + 3 λ − 3 2 p + 2 3 + μ X + r 2 − 2 Q X − 2 λ − p + 2 3 X − 2 μ η ( X ) ξ − 1 2 2 r 3 r + 3 λ − 3 2 p + 2 3 + μ η ( X ) ξ .

Now using (5.8), we infer

(5.29) ∇ X D r = r r + 3 λ − 3 2 p + 2 3 + μ [ X − η ( X ) ξ ] + 3 2 r − r ( X − η ( X ) ξ ) − 2 λ − p + 2 3 X − 2 μ η ( X ) ξ = r 3 ( 3 r + 5 λ − μ ) + r 2 ( − 3 r + 6 μ ) [ X − η ( X ) ξ ] .

Now, we take covariant derivative in the previous equation with respect to Y and achieve

(5.30) ∇ Y ∇ X D r = Y r 3 ( 3 r + 5 λ − μ ) + r 3 ( Y r ) + Y r 2 ( − 3 r + 6 μ ) + r 2 ( − 3 Y r ) [ X − η ( X ) ξ ] + r 3 ( 3 r + 5 λ − μ ) + r 2 ( − 3 r + 6 μ ) [ ∇ Y X − η ( ∇ Y X ) ξ ] .

Similarly, we have

(5.31) ∇ X ∇ Y D r = X r 3 ( 3 r + 5 λ − μ ) + r 3 ( X r ) + X r 2 ( − 3 r + 6 μ ) + r 2 ( − 3 Y r ) [ Y − η ( Y ) ξ ] + r 3 ( 3 r + 5 λ − μ ) + r 2 ( − 3 r + 6 μ ) [ ∇ X Y − η ( ∇ X Y ) ξ ] .

Also, we obtain

(5.32) ∇ [ X , Y ] D r = r 3 ( 3 r + 5 λ − μ ) + r 2 ( − 3 r + 6 μ ) [ ∇ X Y − ∇ Y X − g ( ∇ X Y , ξ ) ξ + g ( ∇ Y X , ξ ) ξ ] .

Now

(5.33) g ( R ( X , Y ) D r , W ) = X r 3 ( 3 r + 5 λ − μ ) + r 3 ( X r ) + X r 2 ( 6 μ − 3 r ) − 3 ( X r ) r 2 [ g ( Y , W ) − η ( Y ) η ( W ) ] − Y r 3 ( 3 r + 5 λ − μ ) + r 3 ( Y r ) + Y r 2 ( − 3 r + 6 μ ) + r 2 ( − 3 Y r ) [ g ( X , W ) − η ( X ) η ( W ) ] .

Now, we set X = W = e i in the previous equation to yield

(5.34) S ( Y , D r ) = − Y r 3 ( 3 r + 5 λ − μ ) + r 3 ( Y r ) + Y r 2 ( − 3 r + 6 μ ) − 3 ( Y r ) ( r 2 + 1 ) .

So, it can be written as

S ( Y , D r ) = k g ( Y , D r ) ,

where k is a scalar function.

Hence, if r ≠ 0 , then the manifold is an Einstein manifold, and if r = 0 , then the manifold is Ricci-flat.□

Acknowledgements

Dr. Akram Ali extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the research group program under grant number R.G.P2/130/43. The authors are thankful to the referee for his/her valuable suggestions toward the improvement of the paper. This research was funded by National Natural Science Foundation of China (Grant No. 12101168) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ22A010014).

  1. Conflict of interest: The authors declare that they have no conflicts of interest.

References

[1] R. S. Hamilton, The Ricci flow on surfaces, Amer. Math. Soc. 71 (1988), 237–261.10.1090/conm/071/954419Search in Google Scholar

[2] S. Pigola, M. Rigoli, M. Rimoldi, and A. Setti, Ricci almost solitons, Ann. Scuola Norm. Sup. Pisa. CL Sc. X (2011), no. 5, 757–799. 10.2422/2036-2145.2011.4.01Search in Google Scholar

[3] A. Barros and E. Ribeiro Jr., Some characterizations for compact almost Ricci solitons, Proc. Amer. Math. Soc. 140 (2012), no. 3, 1033–1040. 10.1090/S0002-9939-2011-11029-3Search in Google Scholar

[4] A. Barros, R. Batista, and E. Ribeiro Jr., Compact almost Ricci solitons with constant scalar curvature are gradient, Monatsh Math. 174 (2014), no. 1, 29–39, https://doi.org/10.1007/s00605-013-0581-3. Search in Google Scholar

[5] J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J. (2) 61 (2009), no. 2, 205–212. 10.2748/tmj/1245849443Search in Google Scholar

[6] C. Calin and M. Crasmareanu, η-Ricci solitons on Hopf hypersurfaces in complex space forms, Rev. Roumaine Math. Pures Appl. 57 (2012), no. 1, 55–63. Search in Google Scholar

[7] A. E. Fischer, An introduction to conformal Ricci flow, Class. Quantum Grav. 21 (2004), S171–S218. 10.1088/0264-9381/21/3/011Search in Google Scholar

[8] N. Basu and A. Bhattacharyya, Conformal Ricci soliton in Kenmotsu manifold, Glob. J. Adv. Res. Class. Mod. Geom. 4 (2015), 15–21. Search in Google Scholar

[9] M. D. Siddiqi, Conformal η-Ricci solitons in δ-Lorentzian trans Sasakian manifolds, Int. J. Maps Math. 1 (2018), no. 1, 15–34. Search in Google Scholar

[10] R. Sharma, Certain results on K-contact and (κ,μ)-contact manifolds, J. Geom. 89 (2008), no. 1–2, 138–147. 10.1007/s00022-008-2004-5Search in Google Scholar

[11] J. T. Cho and R. Sharma, Contact geometry and Ricci solitons, Int. J. Geom. Methods Math. Phys. 7 (2010), no. 6, 951–960, https://doi.org/10.1142/S0219887810004646. Search in Google Scholar

[12] Y. Wang and X. Liu, Ricci solitons on three-dimensional η-Einstein almost Kenmotsu manifolds, Taiwanese J. Math. 19 (2015), no. 1, 91–100, https://doi.org/10.11650/tjm.19.2015.4094. Search in Google Scholar

[13] A. Ghosh, Kenmotsu 3-metric as a Ricci soliton, Chaos Solitons Fractals 44 (2011), no. 8, 647–650, DOI: https://doi.org/10.1016/j.chaos.2011.05.015. 10.1016/j.chaos.2011.05.015Search in Google Scholar

[14] A. Ghosh, An η-Einstein Kenmotsu metric as a Ricci soliton, Publ. Math. Debrecen 82 (2013), no. (3–4), 591–598. 10.5486/PMD.2013.5344Search in Google Scholar

[15] G. Calvaruso and A. Perrone, Ricci solitons in three-dimensional paracontact geometry, J. Geom. Phys. 98 (2015), 1–12, https://doi.org/10.1016/j.geomphys.2015.07.021. Search in Google Scholar

[16] D. S. Patra, Ricci solitons and Ricci almost solitons on para-Kenmotsu manifold, Bull. Korean Math. Soc. 56 (2019), no. 5, 1315–1325, https://doi.org/10.4134/BKMS.b181175. Search in Google Scholar

[17] A. Ali, F. Mofarreh, and D. S. Patra, Geometry of almost Ricci solitons on paracontact metric manifolds, Quaest. Math. (2021), 1–14, https://doi.org/10.2989/16073606.2021.1929539. Search in Google Scholar

[18] D. M. Naik and V. Venkatesha, η-Ricci soliton and almost η-Ricci soliton on para-Sasakian manifolds, Int. J. Geom. Methods Mod. Phys. 16 (2019), no. 9, 1950134, https://doi.org/10.1142/S0219887819501342. Search in Google Scholar

[19] J. Wełyczko, Slant curves in three-dimensional normal almost paracontact metric manifolds, Mediterr. J. Math. 11 (2014), no. 3, 965–978, https://doi.org/10.1007/s00009-013-0361-2. Search in Google Scholar

[20] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tôhoku Math. J. 24 (1972), 93–103. 10.2748/tmj/1178241594Search in Google Scholar

[21] S. Zamkovoy, On para-Kenmotsu manifolds, Filomat 32 (2018), no. 14, 4971–4980, DOI: https://doi.org/10.2298/FIL1814971Z. 10.2298/FIL1814971ZSearch in Google Scholar

[22] A. M. Blaga, Almost η-Ricci solitons in (LCS)n-manifolds, Bull. Belgian Math. Soc. Simon Stevin 25 (2018), no. 5, 641–653, https://doi.org/10.36045/bbms/1547780426. Search in Google Scholar

[23] A. M. Blaga, η-Ricci solitons on para-Kenmotsu manifolds, Balkan J. Geom. Appl. 20 (2015), no. 1, 1–13. Search in Google Scholar

[24] A. M. Blaga and C. Ozgurr, Almost η-Ricci and almost η-Yamabe solitons with torse forming potential vector field, Quaest. Math. 45 (2022), no. 1, 143–163, https://doi.org/10.2989/16073606.2020.1850538. Search in Google Scholar

[25] Y. L. Li, D. Ganguly, S. Dey, and A. Bhattacharyya, Conformal η-Ricci solitons within the framework of indefinite Kenmotsu manifolds, AIMS Math. 7 (2022), no. 4, 5408–5430, https://doi.org/10.3934/math.2022300. Search in Google Scholar

[26] S. Roy, S. Dey, and A. Bhattacharyya, A Kenmotsu metric as a conformal η-Einstein soliton, Carpathian Math. Publ. 13 (2021), no. 1, 110–118, https://doi.org/10.15330/cmp.13.1.110-118. Search in Google Scholar

[27] S. Dey and S. Uddin, Conformal η-Ricci almost solitons on Kenmotsu manifolds, Int. J. Geom. Methods Mod. Phys. 19 (2022), no. 8, 2250121, https://doi.org/10.1142/S0219887822501213. Search in Google Scholar

[28] S. Dey and S. Roy, ∗-η-Ricci Soliton within the framework of Sasakian manifold, J. Dyn. Syst. Geom. Theor. 18 (2020), no. 2, 163–181, https://doi.org/10.1080/1726037X.2020.1856339. Search in Google Scholar

[29] S. Roy, S. Dey, A. Bhattacharyya, and S. K. Hui, ∗-Conformal η-Ricci soliton on Sasakian manifold, Asian-Eur. J. Math. 15 (2022), no. 2, 2250035, https://doi.org/10.1142/S1793557122500358. Search in Google Scholar

[30] D. Ganguly, S. Dey, A. Ali, and A. Bhattacharyya, Conformal Ricci soliton and Quasi-Yamabe soliton on generalized Sasakian space form, J. Geom. Phys. 169 (2021), 104339, https://doi.org/10.1016/j.geomphys.2021.104339. Search in Google Scholar

[31] S. K. Hui, S. K. Yadav, and A. Patra, Almost conformal Ricci solitons on f-Kenmotsu manifolds, Khayyam J. Math. 5 (2019), no. 1, 89–104. Search in Google Scholar

[32] H. G. Nagaraja and K. Venu, f-Kenmotsu metric as conformal Ricci soliton, An. Univ. Vest. Timis. Ser. Mat.-Inform. 55 (2017), 119–127. 10.1515/awutm-2017-0009Search in Google Scholar

[33] S. Roy, S. Dey, and A. Bhattacharyya, Conformal Einstein soliton within the framework of para-Kähler manifold, Differ. Geom. Dyn. Syst. 23 (2021), 235–243. Search in Google Scholar

[34] M. D. Siddiqi, Almost conformal Ricci solitons in (κ,μ)-paracontact metric manifolds, Palest. J. Math. 9 (2020), no. 2, 832–840. Search in Google Scholar

[35] V. Venkatesha, H. A. Kumara, and D. M. Naik, Almost ∗-Ricci soliton on para-Kenmotsu manifolds, Arab J. Math. 9 (2020), 715–726, https://doi.org/10.1007/s40065-019-00269-7. Search in Google Scholar

[36] I. K. Erken, Yamabe solitons on three-dimensional normal almost paracontact metric manifolds, Period. Math. Hungar. 80 (2020), no. 2, 172–184, https://doi.org/10.1007/s10998-019-00303-3. Search in Google Scholar

[37] Y. L. Li, Y. S. Zhu, and Q. Y. Sun, Singularities and dualities of pedal curves in pseudo-hyperbolic and de Sitter space, Int. J. Geom. Methods Mod. Phys. 18 (2021), no. 1, 2150008, https://doi.org/10.1142/S0219887821500080. Search in Google Scholar

[38] Y. L. Li, A. H. Alkhaldi, A. Ali, and P. Laurian-Ioan, On the topology of warped product pointwise semi-slant submanifolds with positive curvature, Mathematics 9 (2021), 3156, https://doi.org/10.3390/math9243156. Search in Google Scholar

[39] Y. L. Li, M. A. Lone, and U. A. Wani, Biharmonic submanifolds of Kähler product manifolds, AIMS Math. 6 (2021), 9309–9321, https://doi.org/10.3934/math.2021541. Search in Google Scholar

[40] Y. L. Li, A. Ali, and R. Ali, A general inequality for CR-warped products in generalized Sasakian space form and its applications, Adv. Math. Phys. 2021 (2021), 5777554, https://doi.org/10.1155/2021/5777554. Search in Google Scholar

[41] Y. L. Li, A. Ali, F. Mofarreh, A. Abolarinwa, and R. Ali, Some eigenvalues estimate for the ϕ-Laplace operator on slant submanifolds of Sasakian space forms, J. Funct. Space 2021 (2021), 6195939. Search in Google Scholar

[42] Y. L. Li, A. Ali, F. Mofarreh, and N. Alluhaibi, Homology groups in warped product submanifolds in hyperbolic spaces, J. Math. 2021 (2021), 8554738. 10.1155/2021/8554738Search in Google Scholar

[43] Z. C. Yang, Y. L. Li, M. Erdogdu, and Y. S. Zhu, Evolving evolutoids and pedaloids from viewpoints of envelope and singularity theory in Minkowski plane, 176 (2022), 104513, https://doi.org/10.1016/j.geomphys.2022.104513. Search in Google Scholar

[44] G. Calvaruso and D. Perrone, Geometry of H-paracontact metric manifolds, Publ. Math. Debrecen 86 (2015), 325–346. 10.5486/PMD.2015.6078Search in Google Scholar

[45] S. Zamkovoy, Canonical connections on paracontact manifolds, Ann. Glob. Anal. Geom. 36 (2009), no. 1, 37–60, https://doi.org/10.1007/s10455-008-9147-3. Search in Google Scholar

[46] A. Perrone, Some results on almost paracontact metric manifolds, Mediterr. J. Math. 13 (2016), no. 5, 3311–3326, https://doi.org/10.1007/s00009-016-0687-7. Search in Google Scholar

[47] S. Sarkar, S. Dey and X. Chen, Certain results of conformal and *-conformal Ricci soliton on para-cosymplectic and para-Kenmotsu manifolds, Filomat 35 (2021), no. 15, 5001–5015, https://doi.org/10.2298/FIL2115001S. Search in Google Scholar

[48] K. Yano and M. Kon, Structures on manifolds, Series in Pure Mathematics 3, World Scientific Pub. Co., Singapore, 1984. 10.1142/0067Search in Google Scholar

[49] K. Yano, Integral Formulas in Riemannian Geometry, Marcel Dekker, New York, 1970. Search in Google Scholar

[50] P. Dacko, On almost para-cosymplectic manifolds, Tsukuba J. Math. 28 (2004), no.1, 193–213. 10.21099/tkbjm/1496164721Search in Google Scholar

[51] S. Erdem, On almost (para)contact (hyperbolic) metric manifolds and harmonicity of (φ,φ′) holomorphic maps between them, Houston J. Math. 28 (2002), 21–45. Search in Google Scholar

[52] A. Ghosh, Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold, Carpathian Math. Publ. 11 (2019), no. 1, 59–69, https://doi.org/10.15330/cmp.11.1.59-69. Search in Google Scholar

[53] S. Kaneyuki and F. L. Willams, Almost paracontact and parahodge structure on manifolds, Nagoya Math. J. 99 (1985), 173–187. 10.1017/S0027763000021565Search in Google Scholar

[54] D. S. Patra, Ricci soliton and paracontact geometry, Mediterr. J. Math. 16 (2019), no. 6, 1–3, DOI: https://doi.org/10.1007/s00009-019-1419-6. 10.1007/s00009-019-1419-6Search in Google Scholar

[55] S. Y. Prktas and S. Keles, Ricci solitons in three-dimensional normal almost paracontact metric manifolds, Int. Electron. J. Geom. 8 (2015), no. 2, 34–45. 10.36890/iejg.592276Search in Google Scholar

[56] J. Welyczko, Legendre curves in three-dimensional normal almost paracontact metric manifolds, Result. Math. 54 (2009), 377–387, https://doi.org/10.1007/s00025-009-0364-2. Search in Google Scholar
Received: 2021-09-04
Accepted: 2022-03-23
Published Online: 2022-08-09

© 2022 Yanlin Li et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
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  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
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  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
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  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
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  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
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  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
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  127. Review Article
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  129. Rapid Communication
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  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
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  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0048
Keywords for this article
conformal η-Ricci soliton; Kenmotsu manifold; Einstein manifold; infinitesimal contact transformation; para-cosymplectic manifold
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Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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