In the framework of Riemannian geometry, Blair et al. [2] introduced a (κ,μ)-space in contact geometry as contact manifold M whose curvature tensor R satisfies

for any U and W∈TM and for h a symmetric operator given by h=12Lξψ, where ψ is a (1,1) tensor field and κ,μ are constants. If κ = 1 and h = 0, then (κ,μ)-spaces reduces to the Sasakian manifolds. Non-Sasakian manifolds have proven to be more interesting in this context. The unit tangent sphere bundle of a flat Riemannian manifold with the usual contact metric structure is an example of non-Sasakian spaces of this type. Moreover, this type of spaces is invariant under D-homothetic transformations. These factors drive the study of this type of manifold. Boeckx proved that a non-Sasakian contact metric manifold satisfying (1.1) is completely determined locally by its dimension for the constant values of κ,μ [3].

If κ,μ are functions, then a contact metric manifold satisfying (1.1) is called a generalized (κ,μ)-space [16]. Koufogiorgos et al. introduced a (κ,μ,ν)-contact metric manifold which satisfies [17]

for smooth functions κ,μ,ν on M2n+1 and they proved that it reduces to a (κ,μ)-manifold in dimension 2n+1≥5. Later, the generalized (κ,μ)-space with divided R₅ was introduced in [5]. This further generalizes generalized (κ,μ)-spaces. Here R5=R5,1−R5,2 is divided into R5,1 and R5,2 such that

for any U, W, X ∈ TM. Sharma and Vrancken [19] studied (κ,μ)-contact manifolds with non-Killing conformal vector fields. Chen [7] examined a closed Einstein-Weyl structure and two Einstein-Weyl structures on an acs (κ<0,μ)-manifold. Ghosh and Sharma [12] investigated a (κ,μ)-contact manifold with a divergence free Cotton tensor. De and Sardar [10], studied Bach-flat (κ,μ)-almost co-Kähler manifolds.

In the last few decades there has been extensive study about Ricci solitons and *-Ricci solitons on manifolds. The notion of Ricci solitons was introduced by Hamilton as a natural generalization of Einstein metrics.

A Ricci soliton on a Riemannian manifold satisfies the following equation [8]

where LY is the Lie-derivative along a smooth potential field Y, g is the Riemannian metric, ρ is a real scalar and Ric is the Ricci tensor. Ricci solitons serve as solutions to the Ricci flow of Hamilton \cite{18, which evolve along the symmetries of the flow. The soliton is steady, expanding or shrinking if ρ = 0, <0 or >0, respectively.

In 1959, Tachibana [21] introduced the notion of *-Ricci tensors on almost Hermitian manifolds. Later, Hamada [13] defined *-Ricci tensors of real hypersurfaces in non-flat complex space forms, and then Kaimakamis et al. [15] introduced the notion of *-Ricci solitons in non-flat complex space forms.

The *-Ricci tensor on an almost contact metric (a.c.m) manifold M ([13]) is defined by

where ψ is a (1, 1)-tensor field and R is a Riemann curvature tensor.

A *-Ricci soliton on a Riemannian manifold (M, g) is a generalisation of the *-Einstein manifold and defined as [15]:

Many authors have studied solitons on a.c.m manifolds: Sharma initiated the study of Ricci solitons in contact geometry as a K-contact and (κ,μ)-contact metric [18]. Suh et al. studied Ricci solitons on almost co-Kähler manifolds [20]. Dai [9] investigated *-Ricci soliton on a (κ<0,μ)-acs manifold and proved that there do not exist *-Ricci soliton on a (κ<0,μ)-acs manifold.

In view of the above, we study the existence of *-Ricci solitons on a non-cosymplectic (κ,μ)-acs manifold. Contrary to the non-existence of *-Ricci soliton in [9] we show that there exists a steady *-Ricci soliton. Further, we study the existence/non-existence of some particular type of potential vector field Y on a non-cosymplectic (κ,μ)-acs manifold admitting Ricci solitons and *-Ricci solitons.

This paper is organised as follows: in Section 2, we give some background which is necessary to understand the subsequent sections. In Section 3, we study *-Ricci solitons on non-cosymplectic (κ,μ)-almost cosymplectic manifolds. Section 4 deals with the study of a potential vector field Y as concurrent, concircular, torse forming and torqued vector field. In Section 5, we give some examples.

A smooth Riemannian manifold M2n+1 is called an a.c.m manifold if there exists structure tensors (ψ,ξ,η,g) satisfying [1]

for any U,W∈TM, where ψ is a tensor field of type (1,1), ξ a global vector field and η a 1-form. We denote by 𝚽 the fundamental 2-form which is defined as Φ(U,W)=g(U,ψW). An a.c.m manifold M2n+1 with dη=Φ is called contact manifold. An a.c.m manifold with η and 𝚽 closed is called an almost cosymplectic manifold. A normal almost cosymplectic manifold is called cosymplectic manifold.

On an acs manifold, we have [4]

where l=R(.,ξ)ξ and both h, h' are symmetric operators with respect to metric g.

Also, on an acs manifold, we have [11]

for any U,W∈TM.

The (κ,μ)-nullity distribution of an acs manifold M2n+1 for (κ,μ)∈R2 is a distribution [11]

Endo [11] introduced (κ,μ)-acs manifolds with the Reeb vector ξ in the (κ,μ)-nullity distribution, which satisfy (1.1). If ξ is in the (κ≠0,μ)-nullity distribution, then such manifolds are called non-cosymplectic (κ,μ)-acs manifolds. For more work about non-cosymplectic (κ,μ)-acs manifolds, please see [11].

On (κ<0,μ)-acs manifold using (1.1) and (2.7), we have

Also, on a Riemannian manifold M we have following [23]:

∀U,W,X∈TM.

Let M2n+1 be a (κ,μ)-acs manifold. Then, from (1.1), we obtain

Using (3.1) in (2.5), we get

Let U be an eigenvector of h for eigenvalue θ with U⊥ξ, then using (2.3), (2.4) and (3.2), we get

From (3.3), we find that κ≤0. However, κ = 0 if and only if h=0. Here, we study non-cosymplectic (κ<0,μ)-acs manifolds.

On an acs manifold M2n+1(κ<0,μ), we have([4, 9])

where Q denotes Ricci operator and U,W∈TM.

Lemma 3.1. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5), then

for any U,W,X∈TM.

Proof. Using (3.5) in (1.5), we obtain

Differentiating (3.7) with respect to U on M and using (2.6), we find

Using (3.8) in (2.9), we obtain

Similarly, we get

Adding (3.9) and (3.10) and subtracting (3.11), we get

which gives

Differentiating (3.12) along U on M, we find

Using (3.13) in (3.10), we obtain

Using (2.8) in (3.14), we get (3.6). Hence, the proof of Lemma is complete.

Theorem 3.2. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5). Then *-soliton is steady.

Proof. Let {ei}i=02n be a local orthonormal basis for TM. From (3.6), we obtain

for U,W,X∈TM.

Contracting (3.15) over U, we get

which gives

Using (3.7) in (3.16), we obtain

Using (3.2) and (3.4) in (3.17), we find

From (3.4), we get

Lie-derivative of (3.19) along Y gives

Comparing (3.18) and (3.20), we obtain

Putting W=ξ in (3.21), we get

Taking inner product of (3.22) with ξ, we get

which implies ρ=0 as κ<0. Hence the proof of the Theorem.

Using (1.5), (2.4), (3.4), (3.5), and Theorem 3.2, we get r=2κn, and divY=2κn, where r and div denote the scalar curvature and divergence, respectively. Therefore, we have

Corollary 3.3. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5), then divY=r<0.

Corollary 3.4. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5), then Y cannot be ξ.

Proof. Suppose Y=ξ. Then, putting potential vector field Y=ξ in (1.5) and using (2.6), (3.5) and Theorem 3.2, we get

Taking trace of (3.23) and using (2.4), we obtain 2κn=0, which is not possible as κ<0. Hence the result.

Remark 3.5. In [9] the author proved that there do not exist *-Ricci soliton on non-cosymplectic (κ,μ)-acs manifolds. Unfortunately, there is a crucial error in their proofs. In page 4 of [9], the equation (3.5) should be corrected as

In page 5 of [9], the equation (3.12) should be corrected as

Thereafter, the argument given in [9] is not useful to obtain correct result.

In 1944, Yano [22] introduced a torse-forming vector field as a generalization of concircular, concurrent and parallel vector fields.

Definition 4.1. A vector field V is called torse forming if

where f∈C∞(M), U∈TM and ωis a 1-form.

The vector field V is called concircular if ω in (4.1) vanishes identically. Concircular vector field is also known as geodesic vector field as its integral curve forms geodesics. It has interesting applications in general relativity and in the theory of conformal and projective transformation.

If V satisfies

then V is called concurrent. If V satisfies (4.1) with f=0, then V is called recurrent. Also, if f=ω=0 in (4.1), then V is called parallel.

Recently, in 2017, Chen [6] introduced a torqued vector field. If a non-vanishing V satisfies (4.1) with ω(V)=0, then V is called torqued, f the torqued function and ω torqued form of V. For more details about these vector fields (please see [6, 22]) and references therein.

Now, we have

Theorem 4.2. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.3). Then the potential vector field Y cannot be concurrent.

Proof. Suppose Y be a concurrent field, then using (3.4) and (4.2) in (1.3), we have

Contracting (4.3) over U and W, we obtain

From (4.4), we find that

Using ρ=2nκ (cf. [20], Theorem 5.1) in (4.5), we get κ=2n+14n2>0, a contradiction as κ<0. Thus proof is complete.

Theorem 4.3. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5), then the potential vector field Y cannot be concurrent.

Proof. Suppose Y be a concurrent field. Using (3.5), (4.2) and Theorem 3.2 in (1.5), we get

Contracting (4.6) over U and W, we obtain κ=2n+12n>0, which is not possible as κ<0. Hence the result.

Theorem 4.4. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5). If Y be a torse forming vector field, then

where f∈C∞(M) satisfying (4.1).

Proof. Let Y be a torse forming field, then using (3.5), (4.1) and Theorem 3.2 in (1.5), we get

Putting U=ei and W=ei and tracing i=1 to 2n+1, we obtain (4.7).

Corollary 4.5. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5). If Y be a torqued vector field, then torqued function f is a constant and ξ cannot be torqued.

Proof. Suppose Y be a torqued field then ω(Y)=0. Using this condition in (4.7) we obtain f=2nκ2n+1. Whereby, we get f is a constant. As ω(ξ)≠0 so ξ cannot be torqued.

Theorem 4.6. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5). If Y be a concircular vector field, then f is a constant given by

Proof. Let Y be a concircular field, then

Using (3.5), (4.10) and Theorem 3.2 in (1.4), we get

Contracting (4.11) over U and W, we obtain (4.9).

Theorem 4.7. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.3). If Y be a torse forming vector field, then

Proof. Let Y be a torse forming field. Using (4.1) and (3.4) in (1.3), we get

Taking trace of (4.13) over U and W, we obtain (4.12).

Theorem 4.8. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.3). If Y be a concircular vector field, then f is a constant given by

Proof. Let Y be a concircular field. Using (3.4) and (4.10) in (1.3), we get

Contracting (4.15) over U and W, we obtain (4.14).

Corollary 4.9. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.3). If Y be a torqued vector field, then torqued function f is a constant.

Proof. Suppose Y be a torqued field then ω(Y)=0. Using this condition in (4.12) we obtain f=ρ−2nκ2n+1. Hence f is a constant.

In the following examples, the (1,1)-tensor ψ is defined as

Example 5.1. Consider M={(x,y,z)∈R3:x≠0} with structure tensors

From (5.1), we have

The Koszul's formula with Riemannian connection ∇ is given by

∀U,W,X∈TM.

From (5.2) and (5.3), we find

Computing Riemann curvature tensors using (5.4), we get

Further, from (1.1), we obtain

From (2.4) and (3.3), we find that h on M satisfies

Using (5.1), (5.5), (5.6) and (5.7), we find that M is a (κ<0,μ)-acs manifold with κ=−14,μ=1.

Further, from (3.5), we obtain

Also, we have

Now, we can see that

for ρ=0 and p,q=1,2,3.

Hence M is a non-cosymplectic (−14,1)-acs manifold admitting steady *-Ricci soliton. Also, divY=−12.

Example 5.2. Consider M={(x,y,z)∈R3:x,y≠0} with

where Y is potential field and c₁ is an arbitrary constant. Then similar to Example 5.1, computation can be done to show that M is a non-cosymplectic (−916,12)-acs manifold admitting steady *-Ricci soliton.

Example 5.3. Consider M={(x,y,z)∈R3:x,y≠0} with structure tensors

Then, M is a non-cosymplectic (-1, 0)-acs manifold. Further, M admits steady *-Ricci soliton with potential vector field

Moreover, M admits expanding Ricci soliton with potential vector field

The authors are thankful to the referees for their valuable suggestions for improvement of the article. The first author is thankful to GGSIP University for research fellowship F.No. GGSIPU/DRC/2021/685.