Paracontact geometry methods plays an important role in modern mathematics. In the sam way that almost contact manifolds extend almost Hermitian manifolds, the geometry of almost paracontact manifolds is a natural extension of almost paraHermitian geometry. Over the last few years, the study of paracontact geometry has evolved from the mathematical formalism of classical mechanics (see [13, 21]). The concept of Ricci flow, is an evolution equation for metrics defined on connected almost contact metric manifolds whose automorphism groups have maximal dimensions.

Very recently, in [14], Güler and Crasmareanu studied Ricci-Yamabe flow of the type (α, β). A soliton to the Ricci-Yamabe flow is called Ricci-Yamabe soliton (abbreviated to RYS) if it moves only by one parameter group of diffeomorphism and scaling. The metric of the Riemannain manifold (Mn,g), n>2 is said to admit a (α,β)-Ricci-Yamabe soliton or simply a Ricci-Yamabe soliton (g,V,λ,α,β) if it satisfies the equation

where LVg denotes the Lie derivative of the metric g along the vector field V, Ric_g is the Ricci tensor, r is the scalar curvature and λ, α, β are real scalars.

In [8], Dey et al. defined a δ-Ricci-Yamabe soliton (in short δ-RYS). A complete Riemannian manifold (Mⁿ,g) is said to be a δ-Ricci-Yamabe almost soliton, denoted by (Mn,g,V,δ,λ), if there exists smooth vector field V on Mⁿ, a soliton function λ∈C∞(Mn) and a non-zero real valued function δ on Mⁿ such that

This soliton is called shrinking, steady or expanding according as λ is negative, zero or positive respectively. If the potential vector field V can be written as a gradient of a smooth function u on Mⁿ, then the δ-Ricci-Yamabe almost soliton is called a gradient δ-Ricci-Yamabe almost soliton. In this case, (1.1) can be expressed as

where ∇2u be the Hessian of u. We denote this as (Mn,g,Du,λ). Now, the identity (1.2) can be written as

There are many papers that prove the existence of Ricci solitons and gradient Ricci solitons on paracontact manifolds. In particular, Calvaruso et al. [3] exhibited Ricci solitons on 3-dimensional almost paracontact manifolds. Ricci solitons and their generalizations have been well studied within the framework of contact and paracontact metric manifolds. See [2] for fundamental background, and say, [9] which list many recent related papers in it extensive references. Recently, Erken [11] demonstrated Yamabe solitons on 3-dimensional para-cosymplectic manifold and proved, for example, that the manifold is either η-Einstein or Ricci flat.

In [17, 19], Patra gave answers to the following important questions associated to almost Ricci, and almost Ricci–Bourguignon, solitons: Under which conditions is a (gradient) Ricci almost soliton Einstein? ...trivial? and Under which conditions is a (gradient) Ricci–Bourguignon almost soliton Einstein (trivial) on a paracontact metric manifold?. It is natural to ask the same questions about more general solitons.

Question. Under which conditions is a (gradient) δ-Ricci-Yamabe almost soliton on a paracontact metric manifold Einstein (trivial)?

We find sufficient conditions under which a paracontact metric manifold admitting a δ-Ricci-Yamabe almost soliton or a gradient δ-Ricci-Yamabe almost soliton is Einstein (trivial). We prove the following.

Lemma 1. If a K-paracontact metric g is a δ-Ricci-Yamabe almost soliton, then

Patra [17] proved that "if a paracontact metric manifold endows a Ricci soliton with nonzero potential vector field V parallel to the Reeb vector field ξ and the Ricci operator commutes with paracontact structure ϕ, then the manifold is Einstein with Einstein constant -2n". Here, we generalize this result for δ-Ricci-Yamabe almost soliton and, removing the commutativity condition, prove that the potential vector field V being parallel to ξ is a sufficient condition under which a K-paracontact manifold admitting a δ-Ricci-Yamabe almost soliton is Einstein (trivial). So, we have the following.

Theorem 1. If K-paracontact metric g endows a δ-Ricci-Yamabe almost soliton with the non-zero potential vector field V is parallel to ξ, then g is Einstein with Einstein constant -2n. Moreover, V is a constant multiple of ξ.

After Theorem 1, we prove the following result.

Proposition 1. Let M2n+1(ϕ,ξ,η,g) be a para-Sasakian manifold. If the metric g represents a δ-Ricci-Yamabe almost soliton with the potential vector field V, then the following relation holds:

Next, we get results on K-paracontact manifold and para-Sasakian manifold whose metric endows a gradient δ-Ricci-Yamabe almost soliton. We state this as follows.

Theorem 2. Let M2n+1(ϕ,ξ,η,g) be a K-paracontact manifold. If the metric g represents a gradient δ-Ricci-Yamabe almost soliton, then M²ⁿ⁺¹ satisfies either

or α=0, that is, it becomes a gradient δ-Yamabe almost soliton, provided β=2.

In [12], Ghosh proved that if a K-contact manifold endows a gradient Ricci almost soliton, then it is of constant scalar curvature. Recently, Patra [18] generalized this result and proved that if a K-contact manifold admits a non-trivial gradient Ricci almost soliton, then the manifold becomes an Einstein metric with constant scalar curvature 2n(2n+1). Here, we prove the nonexistence of a para-Sasakian metric g admitting a gradient Ricci–Yamabe almost soliton with a Ricci operator Q which commutes with a paracontact metric structure ϕ.

Theorem 3. There does not exist a para-Sasakian manifold M2n+1(ϕ,ξ,η,g) with gradient δ-Ricci-Yamabe almost soliton.

As every para-Sasakian manifold is always K-paracontacto, this theorem also holds for K-paracontact manifolds.

Now, we turn our attention to a gradient δ-Ricci-Yamabe almost soliton on a (κ,μ)-paracontact manifold, and state the following results.

Lemma 5. If a (κ,μ)-paracontact manifold (dimension (2n+1)) with κ>−1 endows a gradient δ-Ricci-Yamabe almost soliton, then we have

By virtue of Lemma 5 and Theorem 3, we can assert the following:

Theorem 4. If a (κ,μ)-paracontact manifold (dimension (2n+1)) with κ>−1 admits a gradient δ-Ricci-Yamabe almost soliton, then the manifold is locally isometric to the product of a flat (n+1)-dimensional manifold and an n-dimensional manifold of negative constant curvature -4.

The structure of this paper is the following. In Section 2, after a brief introduction, we discuss some preliminaries of paracontact metric manifolds. In Section 3, we examine δ-Ricci-Yamabe almost solitons on K-paracontact and para-Sasakian manifolds. Also, we show that if K-paracontact metric g represents δ-Ricci-Yamabe almost soliton with the non-zero potential vector field V is parallel to ξ, then g is Einstein with Einstein constant -2n. Section 4 deals with a gradient δ-Ricci-Yamabe almost solitons on K-paracontact and para-Sasakian manifolds, and proves that there does not exist such a manifold. In the last section, we study δ-Ricci-Yamabe almost solitons within the framework of (κ,μ)-paracontact manifold. Here, we prove that if a (κ,μ)-paracontact manifold with κ>−1 admits a gradient δ-Ricci-Yamabe almost soliton, then the manifold is locally isometric to the product of a flat (n+1)-dimensional manifold and a n-dimensional manifold of negative constant curvature -4.

In this section, we discuss some definitions and identities of paracontact metric manifolds (for more details see [4, 5, 15, 23]). A dimensional smooth manifold M is said to be an almost paracontact structure (ϕ,ξ,η) if it endows a (1,1)-tensor field ϕ, a vector field ξ and a 1-form η such that

and there is a paracontact distribution D:q∈M→Dq⊂TqM:Dq=Ker(η)={x∈TqM:η(x)=0} generated by η. If an almost paracontact manifold admits a pseudo-Riemannian metric g such that

for all V₁,V₂ on M, then M has an almost paracontact metric structure (ϕ,ξ,η,g) and g is called a compatible metric. Notice that, since Eq. (2.2) holds any compatible metric g has signature (n+1,n). The fundamental 2-form 𝚽 of an almost paracontact metric structure (ϕ,ξ,η,g) is defined by Φ(V1,V2)=g(V1,ϕV2) for all vector fields V₁, V₂ on M. The manifold M2n+1(ϕ,ξ,η,g) is called paracontact metric manifold, if 𝚽=dη. Here, η is a contact form, i.e., η∧(dη)n≠0, ξ is its Reeb vector field and M is a contact manifold (see [5]). We define self-adjoint operators h=12Lξϕ and l=R(.,ξ)ξ, where Lξ is the Lie-derivative along ξ and R is the Riemannian curvature tensor of g on a paracontact metric manifold. The operators h and l satisfy [23]:

The following results hold on a paracontact metric manifold [23]:

for all V₁, V₂ on M²ⁿ⁺¹, where ∇ is the operator of covariant differentiation of g and Q denotes the Ricci operator given by Ricg(V1,V2)=g(ϕV1,V2)∀V1,V2 on M²ⁿ⁺¹. M is said to be a K-paracontact manifold if the vector field ξ is a killing (equivalently h=0). On a K-paracontact manifold the following formula holds [23]:

for any vector fields V1,V2 on M²ⁿ⁺¹. Moreover, from [23] we have (Lξg)(V1,V2)=2g(V1,ϕhV2) and therefore, M is K-paracontact if and only if ϕ h=0.

A paracontact metric structure on M is said to be normal if the almost paracomplex structure on M×R defined by

where f is a real function on M×R, is integrable. A normal paracontact metric manifold is said to be para-Sasakian. A para-Sasakian manifold is always a K-paracontact manifold. A 3-dimensional K-paracontact manifold is a para-Sasakian manifold [3], which may not be true in higher dimensions [16]. Equivalently, a paracontact metric manifold is said to para-Sasakian if [23]:

for any vector fields V1,V2 on M²ⁿ⁺¹. Further, on any para-Sasakian manifold [23]:

for any vector fields V1,V2 on M²ⁿ⁺¹.

We recall the following commutation formula from [22]

for all vector fields V1,V2 on M²ⁿ⁺¹. By virtue of parallelism of the pseudo-Riemannian metric g, this formula yields

for all vector fields V1,V2 on M²ⁿ⁺¹. We also recall the following from [10, p. 39]

for any vector fields V₁, V₂, V on M²ⁿ⁺¹.

Let R be the Riemannian curvature tensor of the Levi-Civita connection ∇ of g, given by

where χ(M) is the set of all vectors fields on M. On a paracontact metric manifold the following formula holds

for any V₁, V2∈χ(M),

for any vector fields V₁, V2∈χ(M).

The reading of nullity conditions on paracontact geometry is an attractive topic in paracontact geometry. In [6], Cappelletti-Montano et al. initiated the notion of (κ,μ)-paracontact structure. They defined a (κ,μ)-paracontact manifold as a paracontact metric manifold M2n+1(ϕ,ξ,η,g) whose curvature tensor satisfies

for some real numbers (κ,μ). Many geometers have studied (κ,μ)-paracontact manifolds and attained several significant properties of these manifold (see [7, 20]). On a (κ,μ)-paracontact manifold one has [6]

for V1∈χ(M). And also we have the following

where V₁ and V₂ are any vector fields on M.

In this section, we prove the results we stated about δ-Ricci-Yamabe almost solitons on K-paracontact and para-Sasakian manifolds. We begin with the following.

Proof of the Lemma 1. In light of identity (2.10), the soliton equation (1.1) gives

Taking the Lie differentiation of η(V1)=g(V1,ξ) by the vector field V, we achieve (LVη)(V1)−g(LVξ,V1)=(LVg)(V1,ξ). By using (3.1), we acquire

The result then follows using (3.2) with g(ξ,ξ)=1.

Lemma 2. [17] On a K-paracontact manifold M2n+1(ϕ,ξ,η,g), we have

for all vector fields V₁ on M2n+1(ϕ,ξ,η,g).

Proof of the Theorem 1. Since the potential vector field V is parallel to ξ, i.e., V=σξ for a non-zero smooth function σ on M, we acquire ∇V1V=V1(σ)ξ−σ(ϕV1) by the derivative of V=σξ covariantly by V1∈χ(M) and using the identity (2.8). Thus, the equation (1.1) reduces to

for all V1,V2∈χ(M). Now, we insert V1=V2=ξ into (3.3) and use fact (2.10) to infer ξ(σ)=12δ{4nα−(2λ−βr)}. Setting V₂=ξ in (3.3) and recalling (2.10), we get

and therefore, by (2.8), get

Since Hessσ is symmetric and ϕ is skew-symmetric, by (2.1) and (3.4), we get

as dη(V1,V2)=g(V1,ϕV2). This exposes that ξ(σ)=0, as dη is a non-zero

on M, hence, ∇σ=0. Hence, σ is constant on M. This simplifies the equation (3.4) to

using Qξ=−2nξ and hence (M,g) is an Einstein with Einstein constant -2n. This finishes the proof.

Proof of the Proposition 1.

Now, we use identities (1.1) and (2.14) to acquire

for all vector fields V1,V2,V3 on M2n+1. Interchanging cyclicly the roles of V₁, V₂ and V₃ in the upstairs equalization and with the straight enumeration we gain

∀ V1,V2,V3 on M2n+1. Recall the following from [23, Lemma 3.15]:

Using (∇V3Ricg)(V1,V2)=g((∇V3Q)V1,V2) and the identity (2.1) of Lemma 2, we find ∇ξQ=Qϕ−ϕQ=2nη⊗ξ after putting V₃=ξ into (3.5). With this, Lemma 2 and substituting V₂ by ξ in (3.5) we can get

for all V₁ on M. Now, we taking the covariant differentiation of (3.6) by a vector field V₂ on M2n+1 and applying (2.8), (2.10) and (2.11) we obtain

Now, we plug V1=ξ, V2=ξ in the equation (3.7) and using (2.1), (2.8) and (2.11) to achieve

This completes the proof.

In this section, we take into account the gradient almost δ-Ricci-Yamabe solitons on paracontact metric manifolds.

Let a paracontact metric manifold M2n+1(ϕ,ξ,η,g) admits a gradient almost δ-Ricci-Yamabe soliton. Then the soliton equation (1.2) can be demonstrated as

for all V1∈χ(M) and hence the curvature tensor gained from (4.1) and (2.16) satisfies

Proof of the Theorem 2. First, we take the covariant differentiation of 2.10 through the vector field V1∈χ(M), and apply (2.8) to yield

Since ξ is killing, we have

It follows from (2.8) that ∇ξQ=Qϕ−ϕQ. Now, we replace V₁ by ξ into identity (4.2) and then replace the scalar product with V1∈χ(M) to yield

Now, by identity (4.3) and equation (2.8) we get

Using the Bianchis's first identity, we achieve

We insert the above identity into (4.4) and ξ(r)=0 (which holds since ∇ξQ=Qϕ−ϕQ) to get

Now setting V1=ϕV1, and V2=ϕV2 in (4.5) and eliminating (4.5) from the results expression, we get

Herre we also used (2.10). The following formula for paracontact metric manifolds is from [23, Lemma 2.7]:

Using (4.6) and (4.7), we can infer that

since h=0 for K-paracontact manifold. At this point, placing V₂ by ξ in (4.2), we get

replacing the scalar product in the above result with ξ and using (2.9) and (4.3) we get

by ∇ξQ=Qϕ−ϕQ. Let σ=λ−uδ−βr2. Equation (4.9) becomes V1(σ)=ξ(σ)η(V1), for V1∈χ(M) as ξ(r)=0. In this manner, by the argument in Section 3, we get that σ=λ−uδ−βr2 is constant on M. Using ∇ξQ=Qϕ−ϕQ which follows from (4.8), we get

This implies that α{g((∇ξQ)V2+2ϕQV2+4nϕV2,V1)}=0. So either α=0 or g((∇ξQ)V2+2ϕQV2+4nϕV2,V1)=0, which completes the proof.

Proof of the Theorem 3. On a para-Saskian manifold, a Ricci operator satisfies the following (see [23, Lemma 3.15])

With this and (2.1) we see that the Ricci operator Q deflects the paracontact structure ϕ.

On the other hand, para-Sasakian manifolds are K-paracontact. So one has ∇ξQ=Qϕ−ϕQ=2nη⊗ξ, which it implies the contradiction η⊗ξ=0. This finishes the proof.

By virtue of this formula ∇ξQ=Qϕ−ϕQ=2nη⊗ξ, Theorem 3 gives a non-existence theorem.

In this last section, we discuss the nullity agreements on paracontact geometry. In [5], Cappelletti-Montano et al. introduced the notion of (κ,μ)-paracontact structures. According to them a (κ,μ)-paracontact manifold is a paracontact metric manifold M2n+1(ϕ,ξ,η,g) whose curvature tensor satisfies

for all V1,V2∈χ(M) and for some real numbers (κ,μ). Equivalently, this equation can be written as

for all V1,V2∈χ(M). On a (κ,μ)-paracontact manifold one has [5]:

Lemma 3. (see [5]). In any (κ,μ)-paracontact manifold M2n+1(ϕ,ξ,η,g), the Ricci operator Q of M can be written as

for any vector field V₁ on M²ⁿ⁺¹. Moreover, the scalar curvature of M is 2n(2(1−n)+κ+nμ).

Lemma 4. (see [5]). On a (κ,μ)-paracontact manifold M2n+1(ϕ,ξ,η,g), we have

for any vector field V₁ in M2n+1.

Proof of the Lemma 5. First, taking the covariant derivative of (5.4) through an arbitrary vector field V₁ on M and applying (2.4) we get

With (1.2) this can be written as

for all vector fields V₂ on M²ⁿ⁺¹. Using the foregoing equation in the famous manifestation of the curvature tensor R(V1,V2)=[∇V1,∇V2]−∇[V1,V2], we can easily derive

for all V₁ and V₂ on M²ⁿ⁺¹. In this manner, taking the scalar product of (5.9) along ξ and making use of (5.3) and (5.7) leaves that

Replacing V₁ by ϕ V₁ and V₂ by ϕ V₂ in (5.10) and noting that R(ϕV1,ϕV2)ξ=0 (from (5.1)) and (2.1), we get

Now, we put V1=ϕV1 into (5.5) and use ϕξ=0 to obtain

Dy acting h on the last equation and making use of (2.1), (5.3) and hξ=0 leaves

In addition, operating ϕ on (5.5) and using ϕξ=0, we get

Now, we replace V₁ by hV₁ in the foregoing equation and use (5.3) to yield

Applying the last four equations in (5.9) and also using ϕh=−hϕ we obtain 5.7. This completes the proof.

Proof of the Theorem 4. First, we substitute ξ for V₁ in (5.10) and use the identity (5.4) and hξ=ϕξ=0 to acquire δg(R(ξ,V2)ξ,Du)=0. With this and (5.3) we get

where we have used g(V1,Du)=V1u. Now, we take a covariant differentiation of the equation (5.12) by ξ and use the relation (5.6), ∇ξξ=0 to achieve

By equations (5.5) and (5.8), we have

and also