The notion of a κ-nullity distribution (in brief, KND) on a Riemannian manifold (RM, in short) was coined by Tanno [12]. A KND in a RM M is described by

for vector fields V1,V2∈TqM, κ being a real number, and TqM being the tangent space of M at q. A (2m+1)-dimensional contact metric manifold (CMM, in short) is called N(κ)-contact metric manifold (NCMM, in short) if the Reeb vector field 𝜃 satisfies KND. So, for a NCMM, we have

For κ=1, the manifold is a Sasakian manifold and when κ=0 it is locally isometric to the product of an m-dimensional manifold of scalar curvature 4 with a flat (m+1)-dimensional manifold, provided m>1. If m=1 and κ=0, the manifold is flat [1]. NCMMs have been studied by many authors [1, 2, 3, 5].

Hamilton [8] introduced the famous geometric flow, named as Ricci flow, which is a kind of pseudo parabolic heat equation defined on a RM as

where g and Ric indicate, respectively, the Riemannian metric and the (0,2) Ricci tensor.

A Ricci soliton (RS, in short) is a fixed point of Ricci flow (RF, in short) equation (1.2). At the same time it is also a generalization of the Einstein metric. A RS on a (2m+1)-dimensional RM is given by

L being the Lie-derivative operator and λ is a constant. The nature of a RS is described by the value of λ, that means, a RS is shrinking if λ>0, it is steady if λ=0, for λ<0 it is expanding. For more about RSs, one can see the papers [13].

In [10], the authors extended the idea of an RS to a generalized Ricci soliton (GRS, in short). On a RM it is given by

where a, b, λ∈ℝ and E♯ is the canonical 1-form related with E i.e., E♯(V1)=g(V1,E). Similar to a RS, a GRS is shrinking or steady or expanding according as λ takes positive, zero or negative value. Here the potential vector field (PVF, in short) is termed as

• Homothetic vector field if a=b=0

• Killing vector field if a=b=λ=0

and the equation (1.3) is called

• Ricci soliton when a=1 and b=0

• Einstein-Weyl equation when a=1n−1 and b=1.

GRS on different types of RS have been discussed by many authors like Ghosh and De [6, 7], Kumara, Naik and Venkatesha [9].

If the PVF be taken as the gradient of a smooth function, the GRS reduces into generalized gradient Ricci soliton (GGRS, in short). Thus the GGRS on a RM M is given by

where ∇2 being the Hessian operator and ψ is a smooth function on M.

In a RM a vector field E is called concircular vector field [4] if

for any vector field V₁ on the manifold, where ∇ is the Levi-Civita connection and f is a smooth function.

The paper is embodied as follows: after a brief review of literature, we give some basic definition and curvature properties of NCMMs in the Section 2. In Section 3, we deduce certain characterizations of GRSs on NCMMs. The next section deals with generalized gradient Ricci solitons. In the last section, we give an example to support our results.

A (2m+1)-dimensional differentiable manifold ℕ endowed with a tensor field ϕ of type (1,1), a vector field 𝜃, a 1-form τ satisfying [5]

for any vector field V1∈χ(N), the set of all vector fields on N, is known as an almost contact manifold. An almost contact manifold is called almost contact metric manifold if it admits a Riemannian metric g such that

As a consequence of (2.2) and (2.3), we get the following:

for any vector fields V1,V2∈χ(N).

An almost contact metric manifold is called CMM whenever the almost contact metric structure (ϕ,θ,τ,g) satisfies the following condition [5]

for every vector fields V1,V2∈χ(N). For a CMM N, we determine a symmetric (1,1)-tensor field h by h=12Lθϕ, Lθϕ indicates the Lie differentiation of ϕ in the direction 𝜃 and satisfying the following conditions

For a NCMM N of dimension 2m+1, m ≥ 1, we have [5]

for every vector fields V1,V2,∈χ(N); R, Ric and r are the Riemannian curvature, Ricci tensor and scalar curvature, respectively.

Lemma 2.1. In a (2m+1)-dimensional NCMM N, the following holds

for any vector fields V₁, V₂ on the manifold.

Proof. By a straightforward calculation, we obtain

Using (2.4) and (2.9) in the previous relation, the desired result is obtained.

Let N be a NCMM of dimension (2m+1) admitting generalized Ricci soliton. Applying covariant derivative on (1.3) in the direction V₃, we obtain

According to Yano [14], we infer

As ∇g=0 and from the above equation, we have

Due to symmetry property of LE∇, the above equation reduces to

Using (3.1) in (3.2), we get

Assuming that the PVF E as concircular vector field. Then using (1.5) in (3.3), we get

Differentiating (2.6) covariantly with respect to V₃ and utilizing (2.3), one obtains

Applying (3.5) in (3.4), we obtain

which implies

Differentiating above equation covariantly with respect to V₂ and applying (1.5), we infer

Due to Yano ([14], p-23), we have

Applying (3.8) in the above equation, we have

Using (2.11) in (3.10), we get

Setting V2=V3=θ in the above equation, we obtain

Taking Lie derivative of R(V1,θ)θ=κ(V1−τ(V1)θ) along the vector field E, we get

Putting V2=θ in (1.3) and using (2.7), we infer

Applying (3.14) in (3.13), we have

Comparing (3.12) and (3.15), we obtain

Contracting the above equation, we get

Using (2.7) in the above equation, we get

Again, putting V1=V2=θ in (1.3), we obtain

Comparing (3.18) and (3.19) and putting E=θ, we infer

Thus we may assert the following theorem.

Theorem 3.1. If a (2m+1)-dimensional NCMM admits GRS where the PVF being the concircular vector field, then f2=(2m−1)κ2mb(2maκ−λ+b).

Let us consider the PVF be pointwise collinear with the Reeb vector field 𝜃, i.e., E=ψθ, ψ being a smooth function on the manifold. Then, from (1.2), we have

Using (2.3) in (3.21), we infer

Setting V2=θ in the foregoing equation, we obtain

Again, putting V1=θ in the above equation, we have

Applying (3.24) in (3.23), we get

which implies

Taking exterior derivative if (3.26) and then taking wedge product with τ, we get

Since τ∧dτ is the volume element, τ∧dτ≠0. Thus, from (3.27), we infer

which indicates that ψ is a constant. Thus we assert the following

Theorem 3.2. If a (2m+1)-dimensional NCMM admits GRS and the PVF is pointwise collinear with the Reeb vector field, then the potential vector field is a constant multiple of the Reeb vector field.

Let us suppose that the PVF be the Reeb vector field θ, then from (1.2), we infer

Assume an orthonormal frame field {ei}, i=1,2,…,(2m+1) at any point on the manifold and contracting V₁ and V₂, yields

Since tr(ϕh)=0, we obtain from above equation

where we used (2.10). Again putting V1=V2=θ in (3.29) and using (2.7), we have

Comparing (3.31) and (3.32), we get

Thus we may assert the following

Theorem 3.3. If a (2m+1)-dimensional N(κ)-contact metric manifold admits generalized Ricci soliton and the potential vector is the Reeb vector field θ, then κ=−b2ma−m−1m.

In the current section we investigate the behaviour of generalized gradient Ricci solitons on N(κ)-contact metric manifolds.

Let us consider that a (2m+1)-dimensional NCMM admitting generalized gradient Ricci solitons. Then equation (1.4) can be written as

where D indicates the gradient operator. With the help of (2.6), the above equation reduces to

Differentiating covariantly of the equation (4.2), we infer

Altering V₁ and V₂ in the foregoing equation, we have

Also, from (4.2), we get

Using equations (4.3)-(4.5), we have

Remembering (2.3), (2.8), (2.9) and (4.2), the above equation reduces to

Taking inner product with θ, we have

Again, taking inner product of (2.5) with Dψ, we get

Equations (4.8) and (4.9) together give

Setting V2=θ in (4.10), we infer

which indicates that either λ=2amκ−κb or Dψ=(θψ)θ. Thus we conclude that

Theorem 4.1. If a (2m+1)-dimensional NCMM admits GGRS, then either λ=2amκ−κb or the potential vector field is pointwise collinear with the characteristic vector field θ.

In [5], De at el. initiated an example of a N(κ)-contact metric manifold. Following that example we construct the following.

Let us consider the manifold M={x1,x2,x3∈ℝ3:x3≠0} of dimension 3, where (x₁,x₂,x₃) are standard co-ordinates in ℝ3. We choose the vector fields W₁, W₂ and W₃ which satisfy

The Riemannian metric tensor g is considered as

and

The 1-form τ is defined by

for every vector field V on M. The (1,1) type tensor field ϕ is given by

Then we find that

for every vector fields V₁, V₂ on M. Thus (ϕ,W1,τ,g) defines a contact structure.

Due to Koszul's famous formula, we obtain the following

From the above expressions of ∇, we obtain

We also have

Thus the manifold is an N(κ)-contact metric manifold with κ=−3.

On contraction of curvature tensor, we infer

The scalar curvature r of the manifold is given by

Let the potential vector field E=W₁, then

The above data indicates that the given manifold admits generalized Ricci soliton with λ=0 and hence the soliton is steady type. Also, from (1.3), we see that 6a=b and these data satisfy equation (3.33). Hence Theorem 3.3 is verified.

The authors are thankful to the referee for his/her valuable suggestions towards the improvement of the paper.