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Geometric classifications of k-almost Ricci solitons admitting paracontact metrices

  • Yanlin Li , Dhriti Sundar Patra , Nadia Alluhaibi , Fatemah Mofarreh and Akram Ali EMAIL logo
Published/Copyright: September 18, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

The prime objective of the approach is to give geometric classifications of k -almost Ricci solitons associated with paracontact manifolds. Let M 2 n + 1 ( φ , ξ , η , g ) be a paracontact metric manifold, and if a K -paracontact metric g represents a k -almost Ricci soliton ( g , V , k , λ ) and the potential vector field V is Jacobi field along the Reeb vector field ξ , then either k = λ − 2 n , or g is a k -Ricci soliton. Next, we consider K -paracontact manifold as a k -almost Ricci soliton with the potential vector field V is infinitesimal paracontact transformation or collinear with ξ . We have proved that if a paracontact metric as a k -almost Ricci soliton associated with the non-zero potential vector field V is collinear with ξ and the Ricci operator Q commutes with paracontact structure φ , then it is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) . Finally, we have deduced that a para-Sasakian manifold admitting a gradient k -almost Ricci soliton is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) .

Keywords: k-almost Ricci solitons; Ricci soliton; Einstein manifold; paracontact metric manifold; infinitesimal paracontact transformation
MSC 2010: 53C25; 53C21; 53D15

1 Introduction and main results

Through the last two decades, the Ricci solitons as self-similar solutions of the Ricci flow had been increasingly present in theory and in application to theoretical physics and mathematics. Nowadays, the Ricci solitons and their generalizations on pseudo-Riemannian settings are important research topics on differential geometry. On pseudo-Riemannian geometry, paracontact metric structures were originated by Kaneyuki and Williams [1], as a natural odd-dimensional analogous to para-Hermitian structures. The significance of paracontact geometry is emitted from the para-Kähler manifolds theory, while the theory of Legendre foliations is linked to the geometry of paracontact metric manifolds. Several authors investigated paracontact geometry using various meaningful geometric conditions [2–4]. For recent progress of Ricci solitons on paracontact geometry and related studies, the reader can use [5–35] as references.

In [36], Pigola et al. generalized the notion of Ricci soliton to almost Ricci soliton by imposing the soliton constant λ to be a deferential function and defined as follows: a pseudo-Riemannian manifold ( M n , g ) of dimension n is referred to as an almost Ricci soliton if we find a vector field V on M n and a deferential function λ : M n ⟶ R in which L V g + 2 Ric = 2 λ g , where L V stands for the Lie-derivative in the direction of V and Ric , r are Ricci tensor, scalar curvature of g . If the potential vector field is Killing, then the Ricci soliton is trivial and soliton equation reduces to an Einstein metric (i.e., in case that dim M > 2 , the Ricci tensor is just a constant multiple of the Riemannian metric). Thus, we can say almost Ricci’s soliton is a direct generality of Einstein’s metric. The k -almost Ricci soliton is a generalization of almost Ricci solitons, Ricci solitons, Yamabe solitons, and Einstein metrics. It was first introduced by Gomes et al. [37] who defines a pseudo-Riemannian manifold ( M n , g ) is called a k -almost Ricci soliton if we find a vector field V on M n and two differential functions λ , k : M n ⟶ R in which

(1.1) k 2 L V g + Ric = λ g ,

and it is denoted by ( M n , g , V , k , λ ) . It is clear that k -almost Ricci soliton is a natural extension of the almost Ricci soliton by Pigola et al. [36]. A k -almost Ricci soliton is referred to as steady, shrinking, or expanding according as λ = 0 , λ > 0 , or λ < 0 , respectively. It is trivial (Einstein) if the potential vector field V is homothetic, that is, for some constant c , L V g = c g . Otherwise, it is non-trivial. In particular, for constant λ , it is referred to as k -Ricci soliton, and an almost Ricci soliton is just the 1-almost Ricci soliton. A special case of soliton occurs when the potential vector field V is the gradient of a differential function u on M n , i.e., V = ∇ u , and then the k -almost Ricci soliton is called a gradient k-almost Ricci soliton. Then equation (1.1) is exhibited as follows:

(1.2) k ∇ 2 u + Ric = λ g ,

where ∇ 2 u is the Hessian of u . In this case, we denote ( M n , g , ∇ u , k , λ ) as a gradient k -almost Ricci soliton associated with the potential function u . It was proved [37] that a compact nontrivial k -almost Ricci soliton of dimension ≮ 3 in which k is of defined signal and constant scalar curvature and is isometric to a standard sphere of well-determined potential function. It has been shown that a non-trivial non-compact k -almost Ricci soliton ( M n , g , V , k , λ ) with k defined signal, such that L V r ≤ 0 and ∣ Ric ( V ) ∣ lies in L 1 ( M n ) , is an Einstein manifold of non positive scalar curvature r , that is, isometric to R n if r = 0 . Recently, in [38], Hajar has shown that if a potential vector field V is a non-zero conformal vector field, then a compact k -almost Ricci soliton with n ≥ is isometric to a Euclidean sphere S n . In addition, he has proved that every two-dimensional non-trivial compact gradients k -almost Ricci soliton ( M n , g , ∇ u , k , λ ) is isometric to a Euclidean sphere S n ( r ) if M n is of constant scalar curvature. Faraji et al. [39] studied the k -almost Ricci soliton admitting concurrent potential fields and proved several classifications results on k -almost Ricci soliton immersed as a hypersurface in Euclidean space R n + 1 . On the other hand, many important Ricci solitons were investigated on contact metric structures as a natural odd-dimensional counterpart of complex structures. For example, k -almost Ricci solitons in contact geometry are considered in [40]. It has been proved that there is an isometric mapping between gradient k -almost Ricci soliton and the sphere S 2 n + 1 if K -contact metric endowed with gradient k -almost Ricci soliton. Several results have been derived in contact geometry (see [40] and the references therein).

Recently, the interest in Ricci soliton’s study has come to light. The generality of the study on paracontact metric manifolds is a natural odd-dimensional counterpart to para-Hermitian structures. In this direction, Patra [12,13] showed that if a K -paracontact metric g as a Ricci soliton or an almost Ricci soliton with the non-trivial potential vector field V point-wise collinear with ξ and the paracontact structure φ commutes with Ricci operator Q , then V , a constant multiple of ξ and g , is Einstein with constant scalar curvature. On the other hand, Patra [12] showed that the metric g of K -paracontact manifold is Einstein with constant scalar curvature if it is a gradient Ricci soliton. Such a finding has been generalized by Patra et al. [13] for almost Ricci soliton. Motivated by previous deployment on solitons geometry, we provide proof of the following results.

Theorem 1.1

Assume that M 2 n + 1 ( φ , ξ , η , g ) is a K-paracontact manifold admitting a k-almost Ricci soliton with the potential vector field V is being Jacobi field in direction of ξ . Then, either k = λ − 2 n or g is a k -Ricci soliton.

Theorem 1.2

Assume that a K-paracontact metric g represents a k-almost Ricci soliton and the potential vector field is an infinitesimal paracontact transformation and the paracontact structure φ commutes with Ricci operator Q, then g is η -Einstein and V leaves φ invariant.

Theorem 1.3

Suppose that M 2 n + 1 ( φ , ξ , η , g ) is a K-paracontact manifold admitting a k-almost Ricci soliton with the non-zero potential vector field V is collinear with ξ , then g is Einstein with constant scalar curvature is equal to − 2 n ( 2 n + 1 ) .

To generalize the aforementioned theorem on paracontact metric manifold, we provide proof of the following result.

Theorem 1.4

Suppose that M 2 n + 1 ( φ , ξ , η , g ) is a paracontact manifold admitting a k -almost Ricci soliton associated to the non-trivial potential vector field V, that is collinear with ξ , and the Ricci operator Q commuting with paracontact structure φ , then g is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) .

Next, we consider the gradient k -almost Ricci soliton on K -paracontact manifold and derive the following:

Theorem 1.5

Assume that a K-paracontact metric g represents a gradient k-almost Ricci soliton and the paracontact structure φ commutes with Ricci operator Q, then g is Einstein of constant scalar curvature − 2 n ( 2 n + 1 ) .

For the para-Sasakian manifold, we give the following interesting result.

Theorem 1.6

Assume that M 2 n + 1 ( φ , ξ , η , g ) is a para-Sasakian manifold and g represents a gradient k-almost Ricci soliton. Then, g is Einstein with constant scalar curvature equal to − 2 n ( 2 n + 1 ) .

Remark 1.1

It is noticed that the results of k -almost Ricci solitons in paracontact geometry are different and interesting as compared to contact geometry [40]. Therefore, the paracontact metric is motivational in the context of soliton geometry.

Thus, while gradient Ricci soliton and gradient almost Ricci soliton are covered, k -almost Ricci soliton generalizes the generalized m -quasi Einstein metric. To obtain more information, see [8,37]. Yun et al. recently investigated Bach-flat k -almost gradient Ricci solitons [19]. Geometric flows are a family of partial differential equations that describe the evolution of geometric objects over time. Ricci flow is one such flow that is used to study the behavior of metrics on manifolds. The k -almost Ricci solitons arise naturally as fixed points of certain geometric flows, and their study can shed light on the long-time behavior of these flows. On the other side, k -almost Ricci solitons also arise in various areas of mathematical physics, including string theory and quantum gravity. They provide a mathematical framework for studying the behavior of physical systems at the smallest scales, where the effects of quantum mechanics become important.

2 Preliminaries and notations

Here, we recall some basic conceptions and equations of paracontact metric manifolds [2,5–7]. A ( 2 n + 1 ) -dimensional smooth manifold M (or M 2 n + 1 ) is of an almost paracontact structure ( φ , ξ , η ) if it has a vector field ξ (called Reeb vector field), a ( 1 , 1 ) -tensor field φ and a 1-form η verifying the relations:

  1. (2.1) φ ( ξ ) = 0 , η ( ξ ) = 1 , η ∘ φ = 0 , φ 2 = I − η ∘ ξ ,

  2. 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 } is referred to as paracontact distribution generated by η .

Let X M is the module over C ∞ ( M ) of all vector fields on M . If an almost paracontact manifold has a pseudo-Riemannian metric g in which

(2.2) g ( φ X 1 , φ X 2 ) = − g ( X 1 , X 2 ) + η ( X 1 ) η ( X 2 ) , X 1 , X 2 ∈ X M ,

then M admits an almost paracontact metric structure ( φ , ξ , η , g ) and g is referred to as a compatible metric. Moreover, compatible metrics necessarily have signature ( n + 1 , n ) . The fundamental 2-form Φ of an almost paracontact metric structure ( φ , ξ , η , g ) is given by Φ ( X 1 , X 2 ) = g ( X 1 , φ X 2 ) for any vector fields X 1 and X 2 on M . In case that Φ = d η , then M 2 n + 1 ( φ , ξ , η , g ) is referred to as a paracontact metric manifold and we can define two self-adjoint operators on M 2 n + 1 ( φ , ξ , η , g ) as l = R ( . , ξ ) ξ and h = 1 2 L ξ φ , where L ξ is the Lie-derivative in the direction of ξ and R is the Riemann curvature tensor of g defined as follows:

(2.3) R ( X 1 , X 2 ) = [ ∇ X 1 , ∇ X 2 ] − ∇ [ X 1 , X 2 ] , X 1 , X 2 ∈ X M ,

where ∇ is the operator of covariant differentiation of g . Then the two operators h and l satisfy the following conditions [2]:

Tr g h = 0 = Tr g ( h φ ) , h ξ = 0 = l ξ , h φ + φ h = 0 .

On a paracontact metric manifold, the following formulae are hold [2]:

(2.4) ∇ X 1 ξ = − φ X 1 + φ h X 1 , X 1 ∈ X M ,

(2.5) Ric ( ξ , ξ ) = g ( Q ξ , ξ ) = Tr g l = Tr g ( h 2 ) − 2 n ,

where Q stands for the Ricci operator associated to the Ricci tensor defined as Ric ( X 1 , X 2 ) = g ( Q X 1 , X 2 ) for X 1 , X 2 ∈ X M . If h = 0 , i.e., the vector field ξ is a Killing, then M is referred to as a K-paracontact manifold. On K -paracontact manifold, the following formulae hold [2]:

(2.6) ∇ X 1 ξ = − φ X 1 ,

(2.7) R ( X 1 , ξ ) ξ = − X 1 + η ( X 1 ) ξ ,

(2.8) Q ξ = − 2 n ξ

∀ X 1 ∈ X M . A paracontact metric structure on M is referred to as normal if the almost para-complex structure J on M × R given as follows:

J ( X 1 , f d ⁄ d t ) = ( φ X 1 + f ξ , η ( X 1 ) d ⁄ d t ) ,

where t is f is a real function on M × R and the coordinate of R , is integrable. A normal paracontact metric manifold is referred to as para-Sasakian. Equivalently, a paracontact metric manifold is referred to as para-Sasakian if [2]

(2.9) ( ∇ X 1 φ ) X 2 = − g ( X 1 , X 2 ) ξ + η ( X 2 ) X 1 , X 1 , X 2 ∈ X M .

A para-Sasakian manifold is K -paracontact, but the converse is true only if it is of dimension 3 [5,6]. A para-Sasakian manifold M is referred to as η -Einstein if its Ricci tensor Ric can be written as follows:

Ric = a g + b η ⊗ η ,

in which a , b are differentiable functions on M . A vector field X 1 on a paracontact manifold M is referred to as an infinitesimal paracontact transformation (or a para-contact vector field) if it preserves the paracontact form η , i.e., ∃ a differentiable function f : M ⟶ R , satisfies

(2.10) £ X 1 η = f η .

When f = 0 on M , the vector field X 1 is referred to as a strict.

Lemma 2.1

(Lemma 3.1 of [12]) On a K-paracontact manifold ( M , g ) , we have

(2.11) ( ∇ ξ Q ) X 1 = Q φ X 1 − φ Q X 1 , X 1 ∈ X M ,

(2.12) ( ∇ X 1 Q ) ξ = Q φ X 1 + 2 n φ X 1 , X 1 ∈ X M .

3 Proofs main outcomes

3.1 Proof of Theorem 1.1

Taking covariant derivative of (1.1) along X 3 ∈ X M and using (2.4), we have

(3.1) ( X 3 k ) ( L V g ) ( X 1 , X 2 ) + k ( ∇ X 3 L V g ) ( X 1 , X 2 ) = − 2 ( ∇ X 3 Ric ) ( X 1 , X 2 ) + 2 ( X 3 λ ) g ( X 1 , X 2 ) ,

for all X 1 , X 2 ∈ X M . Next, we recall the following commutation relation (see Yano [41, p.23]):

(3.2) ( L V ∇ X 3 g − ∇ X 3 L V g − ∇ [ V , X 3 ] g ) ( X 1 , X 2 ) = − g ( ( L V ∇ ) ( X 3 , X 1 ) , X 2 ) − g ( ( L V ∇ ) ( X 3 , X 2 ) , X 1 ) .

On pseudo-Riemannian manifold ( M , g ) , we have

(3.3) ( ∇ X 3 L V g ) ( X 1 , X 2 ) = g ( ( L V ∇ ) ( X 3 , X 1 ) , X 2 ) + g ( ( L V ∇ ) ( X 3 , X 2 ) , X 1 ) ,

since the metric g is parallel. By utilizing the symmetry of the ( 1 , 2 ) -type tensor filed L V ∇ , i.e., L V ∇ ( X 1 , X 2 ) = L V ∇ ( X 2 , X 1 ) , and interchanging the roles of X 1 , X 2 , and X 3 in the preceding equation, we can compute

(3.4) g ( ( L V ∇ ) ( X 1 , X 2 ) , X 3 ) = 1 2 ( ∇ X 1 L V g ) ( X 2 , X 3 ) + 1 2 ( ∇ X 2 L V g ) ( X 3 , X 1 ) + 1 2 ( ∇ X 3 L V g ) ( X 1 , X 2 ) .

Next, by combining (3.1) and (3.4), we deduce

(3.5) k g ( ( L V ∇ ) ( X 1 , X 2 ) , X 3 ) = ( ∇ X 3 Ric ) ( X 1 , X 2 ) − ( ∇ X 1 Ric ) ( X 2 , X 3 ) − ( ∇ X 2 Ric ) ( X 3 , X 1 ) + ( X 1 λ ) g ( X 2 , X 3 ) + ( X 2 λ ) g ( X 3 , X 1 ) − ( X 3 λ ) g ( X 1 , X 2 ) + 1 k [ ( X 1 λ ) { Ric ( X 2 , X 3 ) − λ g ( X 2 , X 3 ) } + ( X 2 λ ) { Ric ( X 3 , X 1 ) − λ g ( X 3 , X 1 ) } − ( X 3 λ ) { Ric ( X 1 , X 2 ) − λ g ( X 1 , X 2 ) } ] .

Now by substituting ξ for X 2 in (3.5) and invoking the Lemma 2.1, we obtain

(3.6) k ( L V ∇ ) ( X 1 , ξ ) = − 2 Q φ X 1 − 4 n φ X 1 + ( ξ λ ) X 1 + ( X 1 λ ) ξ − η ( X 1 ) ∇ λ + 1 k [ ( ξ λ ) ( Q X 1 − λ X 1 ) − ( λ + 2 n ) { ( X 1 λ ) ξ − η ( X 1 ) ∇ λ } ] .

Setting X 2 = ξ in the well-known formula [41, p.23]:

(3.7) ∇ X 1 ∇ X 2 V − ∇ ∇ X 1 X 2 V − R ( V , X 1 ) X 2 = ( L V ∇ ) ( X 1 , X 2 ) ,

and by plugging the value of ( L V ∇ ) ( X 1 , ξ ) in (3.6), we obtain

(3.8) k { ∇ X 1 ∇ ξ V − ∇ ∇ X 1 ξ V − R ( V , X 1 ) ξ } = − 2 Q φ X 1 − 4 n φ X 1 + ( ξ λ ) X 1 + ( X 1 λ ) ξ − η ( X 1 ) ∇ λ + 1 k [ ( ξ λ ) ( Q X 1 − λ X 1 ) − ( λ + 2 n ) { ( X 1 λ ) ξ − η ( X 1 ) ∇ λ } ] .

Next taking ξ instead of X 1 and using (2.8), then φ ξ = 0 gives

(3.9) k 2 { ∇ ξ ∇ ξ V − R ( V , ξ ) ξ } = ( k − λ − 2 n ) { 2 ( ξ λ ) ξ − ∇ λ } .

Assume the potential vector V is a Jacobi field in direction of ξ , That is, ∇ ξ ∇ ξ V − R ( V , ξ ) ξ = 0 . It follows from (3.9) that

(3.10) ( k − λ − 2 n ) { ∇ λ − 2 ( ξ λ ) ξ } = 0 .

Suppose k ≠ λ + 2 n on M . It follows from (3.10) that ∇ λ = 2 ( ξ λ ) ξ . Co-variant derivative of this along X 1 ∈ X M and then inner product with X 2 ∈ X M , we attain

(3.11) g ( ∇ X 1 ∇ λ , X 2 ) = 2 X 1 ( ξ λ ) η ( X 2 ) − 2 ( ξ λ ) g ( φ X 1 , X 2 ) .

By making use of the property g ( ∇ X 1 ∇ λ , X 2 ) = g ( ∇ X 2 ∇ λ , X 1 ) and the formula d η ( X 1 , X 2 ) = g ( X 1 , φ X 2 ) , we can conclude from (3.11) that ( ξ λ ) d η ( X 1 , X 2 ) = 0 all X 1 , X 2 orthogonal to ξ . This gives that ξ λ = 0 on M , as d η is non-zero everywhere on M . It follows from (3.10) that ∇ λ = 0 on M , and therefore, λ is constant on M . Hence, g is a k -Ricci soliton. The rest of the proof easily follows from (3.9).

3.2 Proof of Theorem 1.2

By utilizing the property of exterior product and the Lie-derivative of (2.10) in direction of V , we have

(3.12) ( L V d η ) ( X 1 , X 2 ) = d ( L V η ) ( X 1 , X 2 ) = 1 2 { g ( X 1 , ∇ f ) η ( X 2 ) − g ( X 2 , ∇ f ) η ( X 1 ) } + f d η ( X 1 , X 2 ) , X 1 , X 2 ∈ X M .

Now, recalling a formula on paracontact metric manifold: d η ( X 1 , X 2 ) = g ( X 1 , φ X 2 ) , by taking its Lie-derivative along V and using the soliton equation (1.1), we obtain

(3.13) k ( L V d η ) ( X 1 , X 2 ) + 2 g ( Q X 1 , φ X 2 ) − 2 λ g ( X 1 , φ X 2 ) = k g ( X 1 , ( L V φ ) X 2 ) .

Plugging (3.12) and d η ( X 1 , X 2 ) = g ( X 1 , φ X 2 ) in (3.13) gives us

(3.14) k ( L V φ ) ( X 1 ) + 2 Q φ X 1 + ( 2 λ − k f ) φ X 1 = k 2 { η ( X 1 ) ∇ f − g ( X 1 , ∇ f ) ξ } .

By utilizing (2.8), the soliton equation (1.1) reduces to

(3.15) k ( L V g ) ( X 1 , ξ ) = 2 ( λ + 2 n ) η ( X 1 ) , X 1 ∈ X M .

Then, by taking Lie-derivative of η ( X 1 ) = g ( X 1 , ξ ) and utilizing (3.15), we obtain

(3.16) k g ( L V ξ , X 1 ) = ( k f − 2 λ − 4 n ) η ( X 1 ) .

As a result φ ξ = 0 , we obtain ( L V φ ) ξ + φ ( L V ξ ) = 0 , which yields from (3.16) that ( L V φ ) ξ = 0 . Furthermore, if we replace X 2 by ξ in (3.14), we have

2 ( L V φ ) ( ξ ) = ∇ f − ( ξ f ) ξ ,

as k is a non-zero differentiable function on M . Therefore, we must have ∇ f = ( ξ f ) ξ . Covariantly differentiating it along X 1 ∈ X M and using (2.8), we acquire

∇ X 1 ∇ f = X 1 ( ξ f ) ξ − ( ξ f ) φ X 1 .

By utilizing this in the formula g ( ∇ X 1 ∇ f , X 2 ) = g ( ∇ X 2 ∇ f , X 1 ) , we deduce

(3.17) X 1 ( ξ f ) η ( X 2 ) − X 2 ( ξ f ) η ( X 1 ) = 2 ( ξ f ) g ( φ X 1 , X 2 ) .

Now replacing X 1 by φ X 1 and X 2 by φ X 2 and recalling (2.1) and d η ( X 1 , X 2 ) = g ( X 1 , φ X 2 ) , we achieve ( ξ f ) d η ( X 1 , X 2 ) = 0 for any vector fields X 1 , X 2 on M . It follows that ξ f = 0 , as d η is non-vanishing on M . As a result, ∇ f = 0 on M . Thus, we can conclude that f is a constant on M . Further, taking Lie-derivative of g ( ξ , ξ ) = 1 and using (3.15) and (3.16), we derive k f = λ + 2 n . By again taking the Lie-derivative of φ 2 X 1 = X 1 − η ( X 1 ) ξ along V , we obtain

(3.18) ( L V φ ) φ X 1 + φ ( L V φ ) X 1 + ( L V η ) ( X 1 ) ξ + η ( X 1 ) L V ξ = 0 .

By using (2.10) and (3.16) in (3.18), we arrive at

(3.19) ( L V φ ) φ X 1 + φ ( L V φ ) X 1 + 2 ( k f − λ − 2 n ) η ( X 1 ) ξ = 0 .

As f is constant, equation (3.14) reduces to

(3.20) k ( L V φ ) X 1 + 2 Q φ X 1 + ( 2 λ − k f ) φ X 1 = 0 .

By making use of (3.20) and k f = λ + 2 n in (3.19), we compute

(3.21) Q φ 2 X 1 + φ Q φ X 1 + ( λ − 2 n ) φ 2 X 1 = 0 .

By assumption, φ commutes with the Ricci operator Q , i.e., Q φ = φ Q , plugging it into (3.21) and using (2.1) yields that

(3.22) Ric = n − λ 2 g + 3 n − λ 2 η ⊗ η .

Thus, ( M , g ) is η -Einstein. Replacing X 1 by φ X 1 in (3.22), we have 2 Q φ X 1 = ( 2 n − λ ) φ X 1 . By putting it into (3.20) and using k f = λ + 2 n , we obtain L V φ = 0 , and hence, V leaves φ invariant. The proof is completed.

3.3 Proof of Theorem 1.3

Since the potential vector field V is collinear with ξ , i.e., V = σ ξ for a non-zero differentiable function σ on M . In view of its covariant derivative and equation (2.6), we can compute

( £ V g ) ( X 1 , X 2 ) = g ( ∇ X 1 V , X 2 ) + g ( ∇ V X 2 , X 1 ) = X 1 ( σ ) η ( X 2 ) + X 2 ( σ ) η ( X 1 ) ,

where we have utilized the anti-symmetric property of φ . In view of this, equation (1.1) becomes

(3.23) k ( X 1 σ ) η ( X 2 ) + k ( X 2 σ ) η ( X 1 ) + 2 Ric ( X 1 , X 2 ) = 2 λ g ( X 1 , X 2 )

for all X 1 , X 2 ∈ X M . At this point, replacing X 2 by ξ (3.23) and recalling (2.8) yields that

(3.24) k X 1 ( σ ) = ( 2 λ + 4 n − k ξ ( σ ) ) η ( X 1 ) , X 1 ∈ X M .

Again, by substituting ξ for both X 1 and X 2 and noting that (2.8), we acquire k ξ ( σ ) = λ + 2 n , and hence, equation (3.24) follows that X 1 ( σ ) = ξ ( σ ) η ( X 1 ) , as k is a non-zero differentiable function on M . By the idea of the previous theorem, we deduce that σ is a constant on M , and hence, equation (3.23) reduces to Ric = λ g . With the addition of (2.8), it follows that λ = − 2 n , consequently, Ric = − 2 n g , which settle our claim.

3.4 Proof of Theorem 1.4

Since V is collinear with ξ , i.e., V = σ ξ for a non-zero differentiable function σ on M . By using its covariant derivative and equation (2.4), we can compute

( L V g ) ( X 1 , X 2 ) = g ( ∇ X 1 V , X 2 ) + g ( ∇ V X 2 , X 1 ) = ( X 1 σ ) η ( X 2 ) + ( X 2 σ ) η ( X 1 ) + 2 σ g ( φ h X 1 , X 2 ) ,

by using the anti-symmetric property of φ and φ h = − h φ . From virtue of this, the soliton equation (1.1) reduces to the following:

(3.25) k ( X 1 σ ) η ( X 2 ) + k ( X 2 σ ) η ( X 1 ) + 2 Ric ( X 1 , X 2 ) + 2 k σ g ( φ h X 1 , X 2 ) = 2 λ g ( X 1 , X 2 )

for all X 1 , X 2 ∈ X M . Now, by substituting ξ for X 2 in (3.25) and using φ ( ξ ) = 0 , we obtain

(3.26) k { D σ + ( ξ σ ) ξ } + 2 Q ξ = 2 λ ξ .

Again, setting X 1 = X 2 = ξ in (3.25) and applying (2.5), φ ( ξ ) = 0 , we derive

(3.27) k ( ξ σ ) + Tr g l = λ .

By the hypothesis of the theorem: Q φ = φ Q . Combining this with (2.5) yields that

(3.28) Q ξ = ( Tr g l ) ξ .

From the aforementioned equation together with (3.27) inserting in (3.26), we achieve

k ∇ σ = k ( ξ σ ) ξ .

This implies that ∇ σ = ( ξ σ ) ξ . Covariantly by differentiating it along X 1 ∈ X M and utilizing (2.6), we attain

∇ X 1 ∇ σ = X 1 ( ξ σ ) ξ − ( ξ σ ) ( φ X 1 + φ h X 1 ) .

By using this in the formula g ( ∇ X 1 ∇ σ , Y 1 ) = g ( ∇ Y 1 ∇ σ , X 1 ) and noting that h φ + φ h = 0 , we deduce

(3.29) X 1 ( ξ σ ) η ( Y 1 ) − Y 1 ( ξ σ ) η ( X 1 ) = 2 ( ξ σ ) g ( φ X 1 , Y 1 ) .

Now replacing X 1 by φ X 1 and Y 1 by φ Y 1 in (3.29) and recalling (2.1) and d η ( X 1 , Y 1 ) = g ( X 1 , φ Y 1 ) , we achieve ( ξ σ ) d η ( X 1 , Y 1 ) = 0 , ∀ X 1 , Y 1 on M . It follows that ξ σ = 0 , as d η is non-zero on M . Consequently, ∇ σ = 0 on M . Thus, we deduce that σ is a constant on M . This transforms equation (3.25) into the following:

(3.30) k σ g ( φ h X 1 , X 2 ) + Ric ( X 1 , X 2 ) = λ g ( X 1 , X 2 ) .

Now, by replacing X 1 by φ X 1 and X 2 by φ X 2 in (3.30) and using (2.1), φ h = − h φ , we achieve

(3.31) k σ g ( φ h X 1 , X 2 ) + Ric ( φ X 1 , φ X 2 ) = − λ { g ( X 1 , X 2 ) − η ( X 1 ) η ( X 2 ) } .

By combining (3.30) and (3.31), we obtain

(3.32) Ric ( X 1 , X 2 ) − Ric ( φ X 1 , φ X 2 ) = λ { 2 g ( X 1 , X 2 ) − η ( X 1 ) η ( X 2 ) } .

Now, by inserting X 1 = φ X 1 in (3.32) and recalling (2.1), we acquire

(3.33) Q φ X 1 + φ Q ( X 1 − η ( X 1 ) ξ ) = 2 λ φ X 1 .

Since, the paracontact structure φ commutes with Ricci operator Q , equation (3.33) entails that Q φ X 1 = λ φ X 1 . Operating φ and using Q φ = φ Q and Q ξ = ( Tr g l ) ξ = λ ξ (follows from (3.27) and σ constant), we obtain Q X 1 = λ X 1 , and hence, ( M , g ) is Einstein. This reduces equation (3.30) to φ h X 1 = 0 , as σ is a non-zero constant. Operating this by φ and using (2.1) and h ξ = 0 , we obtain h = 0 . Therefore, M is K -paracontact manifold and equation (2.5) follows that Tr g l = − 2 n . Since σ is a non-zero constant, equation (3.27) yields Tr g l = λ ; consequently, λ = − 2 n , and therefore, we obtain Q X 1 = − 2 n X 1 and r = − 2 n ( 2 n + 1 ) , as required. The proof is finished.

3.5 Proof of Theorem 1.5

Gradient k -almost Ricci soliton equation (1.2) can be exhibited as k ∇ X 1 ∇ u + Q X 1 = λ X 1 for all X 1 ∈ X M . By applying this in the curvature tensor expression (2.3), we obtain

(3.34) k R ( X 2 , X 1 ) ∇ u = ( ∇ X 1 Q ) X 2 − ( ∇ X 2 Q ) X 1 + ( X 2 λ ) X 1 − ( X 1 λ ) X 2 + 1 k { ( X 2 k ) ( Q X 1 − λ X 1 ) − ( X 1 k ) ( Q X 2 − λ X 2 ) } .

Now, setting X 2 = ξ in (3.34) and then taking scalar product with X 2 ∈ X M and using (2.11), (2.12), we deduce

(3.35) g ( R ( ξ , X 1 ) ∇ u , X 2 ) = g ( φ Q X 1 , X 2 ) + 2 n g ( φ X 1 , X 2 ) + ( ξ k ) k g ( Q X 1 , X 2 ) + ξ λ − ( ξ k ) k λ g ( X 1 , X 2 ) + λ + 2 n k ( X 1 k ) − ( X 1 λ ) η ( X 2 ) .

With the aid of (2.8), by substituting X 1 = φ X 1 and X 2 = φ X 2 in (3.35), we acquire

(3.36) g ( R ( ξ , φ X 1 ) ∇ u , φ X 2 ) = − g ( Q φ X 1 , X 2 ) − 2 n g ( φ X 1 , X 2 ) + ( ξ k ) k g ( Q φ X 1 , φ X 2 ) + ξ λ − ( ξ k ) k λ g ( φ X 1 , φ X 2 ) .

At this point, by applying (2.6) and (2.3), one can easily derive that

g ( R ( X 1 , X 2 ) ξ , X 3 ) = g ( ( ∇ X 2 φ ) X 1 , X 3 ) − g ( ( ∇ X 1 φ ) X 2 , X 3 ) .

By using this in the Bianchis’s first identity, we obtain g ( R ( ξ , X 3 ) X 1 , X 2 ) = g ( ( ∇ X 3 φ ) X 1 , X 2 ) for all X 1 , X 2 , X 3 ∈ X M . By using this in a formula on K -paracontact manifold (see Zamkovoy [2, Lemma 2.7]):

( ∇ φ X 1 φ ) φ X 2 − ( ∇ X 1 φ ) X 2 = 2 g ( X 1 , X 2 ) ξ − η ( X 2 ) ( X 1 + η ( X 1 ) ξ ) .

The aforementioned equation reduces to the following:

g ( R ( ξ , X 1 ) ∇ u , X 2 ) − g ( R ( ξ , φ X 1 ) ∇ u , φ X 2 ) = 2 ( ξ u ) g ( X 1 , X 2 ) − ( X 1 u ) η ( X 2 ) − ( ξ u ) η ( X 1 ) η ( X 2 ) .

By plugging the value of g ( R ( ξ , X 1 ) ∇ u , X 2 ) , g ( R ( ξ , φ X 1 ) ∇ u , φ X 2 ) together with (3.35) and (3.36) in the proceeding equation, we obtain

k ( X 1 u ) + ( X 1 λ ) + λ + 2 n k ( X 1 k ) η ( X 2 ) + k ( ξ λ ) η ( X 1 ) η ( X 2 ) = ξ λ − ( ξ k ) k λ { g ( φ X 1 , φ X 2 ) − g ( X 1 , X 2 ) } + ( ξ k ) k { g ( Q φ X 1 , φ X 2 ) − g ( Q X 1 , X 2 ) } + 2 k ( ξ k ) g ( X 1 , X 2 ) − g ( Q φ X 1 + φ Q X 1 , X 2 ) − 4 n g ( φ X 1 , X 2 ) .

At this point, by anti-symmetrizing of the last equation and using an anti-symmetric property of φ , we obtain

(3.37) 2 g ( Q φ X 1 + φ Q X 1 , X 2 ) + 8 n g ( φ X 1 , X 2 ) = k ( X 2 u ) + ( X 2 λ ) + λ + 2 n k ( X 2 k ) η ( X 1 ) − k ( X 1 u ) + ( X 1 λ ) + λ + 2 n k ( X 1 k ) η ( X 2 ) ,

where we have utilized the symmetric aspect of the Ricci operator Q . Next by substituting φ X 1 for X 1 and φ X 2 for X 2 in (3.37) and again using the symmetric property of Q and (2.1), we obtain

(3.38) Q φ X 1 + φ Q X 1 = − 4 n φ X 1 , X 1 ∈ X M .

By hypothesis: Q φ = φ Q , thus, equation (3.38) gives φ Q X 1 = − 2 n φ X 1 . Operating this by φ and using (2.1) and (2.8), we obtain Q X 1 = − 2 n X 1 , and therefore, ( M , g ) is Einstein with Einstein constant − 2 n . The proof gets completed.

3.6 Proof of Theorem 1.6

On the para-Sasakian manifold, the Ricci tensor satisfies (see Lemma 3.15 of [2]):

(3.39) Ric ( φ X 1 , φ X 2 ) = − Ric ( X 1 , X 2 ) − 2 n η ( X 1 ) η ( X 2 ) , X 1 , X 2 ∈ X M .

Since the Ricci operator Q is symmetric, replacing X 2 by φ X 2 in (3.39) and using (2.1), (2.8), it follows that Q φ = φ Q . The rest of the proof imitates the one of Theorem 1.5.

4 Conclusions and discussions

For K -paracontact manifold admitting a gradient Ricci soliton is Einstein [12], and this holds for gradient k -almost Ricci soliton (in particular, gradient almost Ricci soliton, see [13]) with the additional condition: φ commutes with Ricci operator Q . In [40], Ghosh and Patra proved that if a complete K -contact (Sasakian) metric g represents a gradient k -almost Ricci soliton, then it is isometric to a unit sphere S 2 n + 1 . Here, we have proven if a K -paracontact manifold has a gradient k -almost Ricci soliton satisfying Q φ = φ Q is an Einstein manifold with Einstein constant − 2 n .

For more consequences, a k -almost Ricci soliton is a generalization of a Ricci soliton, which is a solution to a certain geometric flow equation in differential geometry. In a k -almost Ricci soliton, the soliton equation is modified by adding a vector field that satisfies certain properties. k -almost Ricci solitons can be applied in various areas of geometry and physics. Here are a few examples: An Einstein manifold is a Riemannian manifold whose Ricci tensor is proportional to the metric tensor. It is a special case of a Ricci soliton, where the soliton vector field is zero. k -almost Ricci solitons provide a natural generalization of Einstein manifolds, where the soliton vector field is nonzero. It is also noticed that general relativity is a theory of gravitation that describes the behavior of spacetime in the presence of matter and energy. Some k -almost Ricci solitons can be used to study the geometry of spacetime in the presence of matter and energy and provide a framework for understanding the dynamics of gravitational waves and black holes.

Acknowledgments

The last author would like to express his gratitude to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for providing a funding research group under the research grant number R.G.P2/91/44.

  1. Funding information: This research was funded by the National Natural Science Foundation of China (Grant No. 12101168) and the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ22A010014). The authors also express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

  2. Author contributions: DSP and AA conceived the study and the manuscript design as well as classified the contact setting of the k -Almost Ricci soliton equation. YL and AA constructed the results and computed the scalar curvature equations. FM constructed Ricci operators and obtained commutative law. NA and FM introduced the applications and theories of the Ricci operators equations in various fields.

  3. Conflict of interest: The authors state no conflicts of interest.

  4. Data availability statement: There is no data used for this manuscript.

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Received: 2022-10-26
Revised: 2023-03-20
Accepted: 2023-06-27
Published Online: 2023-09-18

© 2023 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/math-2022-0610
Keywords for this article
k-almost Ricci solitons; Ricci soliton; Einstein manifold; paracontact metric manifold; infinitesimal paracontact transformation
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  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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