Nowadays, the Ricci solitons and their generalizations are enjoying rapid growth by providing new techniques in understanding the geometry and topology of arbitrary Riemannian manifolds. Ricci soliton is a natural generalization of Einstein metric, and is also a self-similar solution to Hamilton's Ricci flow [20, 21]. It plays a specific role in the study of singularities of the Ricci flow. A solution g(t) of the non-linear evolution PDE: ∂∂tg(t)=−2S(g(t)) is called the Ricci flow, where S is the Ricci tensor associated to the metric g. In differential geometry, the Ricci flow is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric. A Riemannian manifold (M, g) is called a Ricci soliton if there are a smooth vector field V and a scalar λ∈ℝ such that

on M, where S is the Ricci tensor and £Vg is the Lie derivative of the metric g along V. If the potential vector field V vanishes identically, then the Ricci soliton becomes trivial, and in this case manifold is an Einstein one. A Ricci soliton is said to be a gradient Ricci soliton if the potential vector field V can be expressed as a gradient of a smooth function u on M, i.e., V=Du, where D is the gradient operator of g on M. An important application of the Ricci flow is the proof for Thurston's Conjecture given recently by Perelman [32]: A Ricci soliton on any compact Riemannian manifold is always a gradient Ricci soliton. We recommend the papers [8, 11, 12, 13, 17, 18, 28, 40] and the references therein for more details about the study of Ricci solitons, gradient Ricci solitons and their generalizations in the context of contact Riemannian geometry.

The generalized version of Ricci soliton, so called almost Ricci soliton was introduced in the paper [33] by treating the soliton constant λ as a smooth function. It is noted from [5] that a compact almost Ricci soliton with constant scalar curvature is isometric to Euclidean sphere. Later in [38], Sharma studied Ricci almost solitons in K-contact geometry and Ghosh [16] studied Ricci almost solitons and gradient Ricci almost solitons in (κ,μ)-contact geometry. Recently, Wang-Gomes-Xia [39] extended the notion of almost Ricci soliton to h-almost Ricci soliton. According to [39], a complete connected Riemannian manifold (M2n+1,g) is said to be h-almost Ricci soliton if there exists a smooth vector field V on M2n+1 such

that

where λ and h are smooth functions on M2n+1. Here, λ is called soliton function and V is called the potential vector field of h-almost Ricci soliton. This notion is denoted by (M2n+1,g,V,h,λ). An h-almost Ricci soliton is called: (i) shrinking, when the soliton constant λ is positive; (ii) steady, when λ is zero and (iii) expanding, when λ is negative. If the potential vector field V can be expressed as a gradient of a smooth function u on M, i.e., V = Du, where D is the gradient operator of g on M, then the h-almost Ricci soliton equation becomes

(where, Hess u=∇2u denotes the Hessian of the smooth function u) and characterizes what is called h-almost gradient Ricci soliton. The problem of studying h-almost Ricci solitons and h-almost gradient Ricci solitons in the context of contact metric geometry was initiated by Ghosh-Patra [19]. In particular, they studied h-almost Ricci solitons and h-almost gradient Ricci solitons on a K-contact manifold and proved that if a compact K-contact metric is h-almost gradient Ricci soliton then it is isometric to a unit sphere S²ⁿ⁺¹. More recently, Kar-Majhi [26] studied (κ,μ)-almost co-Kähler manifold which admits h-almost Ricci soliton and h-almost gradient Ricci soliton. Also, h-almost Ricci solitons on Sasakian 3-manifolds was studied in the paper [29]. Motivated by the above studies, in this paper we undertake the study of h-almost Ricci solitons on almost contact metric manifolds, particularly, on generalized Saskian-space-forms. Before to proceed further, we recall that, Ricci solitons and η-Ricci solitons on generalized Sasakian-space-forms were studied in the paper [31]. Further, the study of invariant submanifolds of generalized Sasakian-space-forms was recorded in [25].

The paper is organized as follows: Section 2 is concerned with the preliminaries on generalized Sasakian-space-forms. In section 3, h-almost Ricci soliton on a (2n+1)-dimensional (n>1) generalized Sasakian-space-form M2n+1(f1,f2,f3) with the potential vector field as a contact vector field is being considered and prove that in such a case the manifold has a constant scalar curvature and the flow vector field is Killing. Next, we also show that the manifold M2n+1(f1,f2,f3) is locally symmetric and has a constant ϕ-sectional curvature provided the characteristic vector field ξ is Killing. In the last section, we study three-dimensional quasi-Sasakian generalized Sasakian-space-form M3(f1,f2,f3) with f1≠f3 admitting h-almost gradient Ricci soliton and prove that in this situation the manifold M3(f1,f2,f3) is of constant curvature f1−f3.

A (2n+1)-dimensional differentiable manifold M2n+1 is called an almost contact manifold (see, Blair [7]) equipped with the structure (ϕ,ξ,η) where ϕ is a tensor field of type (1,1), ξ a characteristic or Reeb vector field and η is a 1-form satisfying

for all vector field X on M2n+1. In general, a differentiable manifold M2n+1 together with the almost contact structure (ϕ,ξ,η) is said to be an almost contact manifold and it is denoted by (M2n+1,ϕ,ξ,η). If an almost contact manifold (M2n+1,ϕ,ξ,η) admits a Riemannian metric g satisfying

for any vector fields X, Y on M2n+1, then the manifold is called an almost contact metric manifold and is denoted by (M2n+1,ϕ,ξ,η,g). Then from (2.2), it can be easily deduced that g(ϕX,Y)=−g(X,ϕY). The fundamental 2-form dη associate with the almost contact metric structure is defined by

for any vector fields X and Y.

An almost contact metric manifold (M2n+1,ϕ,ξ,η) is said to be a generalized Sasakian-space-form if the curvature tensor of the manifold satisfies

for some smooth functions f₁, f₂ and f₃ on M2n+1, where R₁, R₂ and R₃ are curvature-like tensors given by

for any vector fields X, Y, Z on M2n+1. In such case we will write the manifold as M2n+1(f1,f2,f3). Moreover, Sasakian, cosymplectic or/and Kenmotsu space forms are the typical examples of generalized Sasakian-space-forms. This almost contact counterpart was introduced and studied by Alegre-Blair-Carriazo [1] in 2004. Since then, several papers have appeared concerning different aspects of this topic. At this point, we recommend the papers [2, 3, 4, 10, 14, 22, 23, 24, 27, 34, 35, 36, 37] and the references therein to reader for a wide and detailed overview of the results on generalized Sasakian-space-forms.

In addition to the relation (2.4), for a (2n+1)-dimensional (n>1) generalized Sasakian-space-form M2n+1(f1,f2,f3) the following relations also hold [1]:

for any vector fields X, Y on M2n+1(f1,f2,f3), where R, S and r are the curvature tensor, Ricci tensor and scalar curvature of the space-form, respectively.

Also, for a generalized Sasakian-space-form M3(f1,f2,f3) of dimension three the Ricci operator Q and the curvature tensor R are given by [28]:

and

respectively, for any vector fields X, Y, Z on M3(f1,f2,f3).

Definition 2.1. ([7]) A vector field V on a contact manifold M2n+1 is said to be a contact vector field if it preserves the contact form η, that is,

for some smooth function ρ on M2n+1. When ρ = 0 on M2n+1, the vector field V is called a strict contact vector field.

Definition 2.2. ([15]) An infinitesimal automorphism V is a smooth vector field such that Lie derivatives of all structure tensor along V vanishes, that is,

Let g be an h-almost Ricci soliton on a (2n+1)-dimensional (n>1) generalized Sasakian-space-form M2n+1(f1,f2,f3). Then we have from (1.2) that

Replacing ξ instead of X and Y in (3.1) we get

Plugging Y by ξ in (3.1) and then using (2.8) and (2.13) gives

Observing (3.2) in (3.3) we have

Making use of (3.4) in (3.3) we get

On the other hand, from (2.3) we deduce that

Multiplying both sides of (3.6) by h and then using (3.1) we infer

Feeding (2.7) in (3.7) we get

Let us suppose that the potential vector field V of M2n+1 be a contact vector field. Then, with the aid of (2.13) we have

This together with (3.8) provides

Inserting ξ in place of Y we get

With the help of ϕξ=0 and (3.5) we obtain

Applying (3.12) in (3.11) we have

Taking the inner product of (3.13) with X gives

or equivalently,

Taking exterior derivative of (3.15) we get

which implies

Taking wedge product of (3.16) with η we gave

from which it follows that ξρ=0. Since η∧(dη)n≠0, and by (3.15) one can obtain dρ=0 and hence ρ is constant.

Further, with the help of (2.13) and noting that £V and d commutes, we have

As a volume form, ω is closed and by thus the Cartan's formula provides

Next, taking the Lie differentiation to volume form ω=η∧(dη)n and then using (3.18) and (3.19) we obtain

Integrating (3.20) over M2n+1 and then applying Divergence theorem, we infer

(and so div V=0). Thus, we obtain from (3.4) that

Theorem 3.1. Let M2n+1(f1,f2,f3) be a (2n+1)-dimensional generalized Sasakian-space-form with the potential vector field V as a contact vector field. If g is h-almost Ricci soliton on M2n+1(f1,f2,f3), then the soliton is shrinking, steady or expanding accordingly as f1−f3 is positive, zero or negative.

The trace of (3.1) and with the fact that ∑ i=13(£Vg)(ei,ei)=2divV and (3.20) we deduce

Now it is easy to check from (2.10), (3.22) and (3.23) that

By virtue of (3.24) and (2.7) we have

That is, M2n+1(f1,f2,f3) is Einstein with Einstein constant 2n(f1−f3). Contracting (3.25) we obtain

Therefore, the scalar curvature r is constant.

Next, with the help of (3.22) and (3.25), from (3.1) we get £Vg=0, which implies that V is Killing.

Since ρ is constant, it follows from (3.10) that

We employ (3.22) and (3.24) in the above equation to achieve £Vϕ=0 as h is positive. Further, by taking account of (3.22) in (3.5), we have £Vξ=0. Finally, we substitute (3.21) in (2.13) to deduce £Vη=0. Thus, Lie derivatives of all structure tensor along V vanishes and from (2.14), the flow vector field V is an infinitesimal automorphism of the almost contact metric structure of M2n+1(f1,f2,f3). Hence, we summarize the above in the form of a theorem which is as follows:

Theorem 3.2. Let M2n+1(f1,f2,f3) be a (2n+1)-dimensional (n>1) generalized Sasakian-space-form with the potential vector field V as a contact vector field. If g is h-almost Ricci soliton on M2n+1(f1,f2,f3), then the scalar curvature of M2n+1(f1,f2,f3) is constant and the flow vector field V is Killing. Moreover, V is an infinitesimal automorphism of the almost contact metric structure of M2n+1(f1,f2,f3).

Remark 3.3. In [14], De and Sarkar studied projective curvature tensor on a generalized Sasakian-space-forms and proved that a (2n+1)-dimensional (n>1) generalized Sasakian-space-form is projectively flat if and only if 3f2+(2n−1)f3=0. So by virtue of (3.24) it is evident that a (2n+1)-dimensional (n>1) generalized Sasakian-space-form M2n+1(f1,f2,f3) admits an h-almost Ricci soliton is projectively flat.

Now, at this junction, we recall the following theorem due to Kim [27]:

Theorem 3.4. Let M2n+1 be a (2n+1)-dimensional generalized Sasakian-space-form. Then we have following:

(i) If n>1, then M2n+1 is conformally flat if and only if f₂ =0.

(ii) If M2n+1 is conformally flat and ξ is a Killing vector field, then M2n+1 is locally symmetric and has constant ϕ-sectional curvature.

Also, it is known that projectively flat and conformally flat conditions for a generalized Sasakian-space-form of dimension greater than three are equivalent. By taking account of this fact along with previous discussion, we are able to conclude the following:

Theorem 3.5. Let M2n+1(f1,f2,f3) be a generalized Sasakian-space-form of dimension greater than 3. If (g, V) is an h-almost Ricci soliton on M2n+1(f1,f2,f3) with the potential vector field V as a contact vector field, then M2n+1(f1,f2,f3) is conformally flat. In addition, if the characteristic vector field ξ of M2n+1 is a Killing vector field, then M2n+1(f1,f2,f3) is locally symmetric and has constant ϕ-sectional curvature.

In [28], authors have studied the notion of quasi-Sasakian generalized Sasakian-space-forms. This notion is an analogous version of the trans-Sasakian generalized Sasakian-space-forms studied in [2]. An almost contact metric manifold M³ is a three-dimensional quasi-Sasakian

manifold if and only if [30]

for any vector field X on M³ and for a certain function β, such that ξβ=0. Here, ∇ denotes the operator of the covariant differentiation with respect to the Levi-Civita connection of M³. If β = constant, then the manifold reduces to a β-Sasakian manifold and if in particular β = 1, the manifold becomes a Sasakian manifold. As a consequence of (4.1), we have

From (4.2) it follows that

for any vector field X on M³, orthogonal to ξ. Also, from (2.4) we obtain

Therefore (4.3) and (4.4) give us β2=f1−f3 and β is constant. Also, in a three-dimensional quasi-Sasakian generalized Sasakian-space-form β is non-zero, provided f1≠f3.

In this section, before entering into the main part we prove the following:

Lemma 4.1. On a three-dimensional quasi-Sasakian generalized Sasakian-space-form M3(f1,f2,f3), we have

for any vector field X on M3(f1,f2,f3).

Proof. For a three-dimensional quasi-generalized generalized Sasakian-space-form M3(f1,f2,f3) we have

Taking covariant differentiation of (4.1) along an arbitrary vector field X on M3(f1,f2,f3) and using (4.1), we get

Since ξ is Killing on a three-dimensional quasi-Sasakian generalized Sasakian-space-form M3(f1,f2,f3), we have (£ξQ)=0 on M3(f1,f2,f3). This follows that £ξ(QX)=Q(£ξX). Now, taking into account (4.1) it follows that

for any vector field X on M3(f1,f2,f3). Subtraction of (4.8) from (4.7) gives (4.5). This completes the proof.

Next, suppose that in a three-dimensional quasi-Sasakian generalized Sasakian-space-form M3(f1,f2,f3), the metric g admits h-almost gradient Ricci soliton. Then the soliton equation defined by (1.3) with the potential function u can be exhibited as

for any vector field X on M³; where D is the gradient operator of g on M3(f1,f2,f3). By straightforward computations, using the well known expression of the curvature tensor:

and the repeated use of equation (4.9) gives

Replacing ξ instead of X in (4.10) and making use of (2.12) and (4.5) we get

for any vector field Y on M3(f1,f2,f3). Scalar product of the last equation with an arbitrary vector field X and using (2.6), we obtain

for any vector fields X and Y on M3(f1,f2,f3). Next, substituting X by ϕ X and Y by ϕ Y in (4.11) and then using (2.1) provides

for all vector fields X, Y on M3(f1,f2,f3). Adding the preceding equation with (4.11) yields

Anti-symmetrizing the foregoing equation provides

for all vector fields X, Y on M3(f1,f2,f3). Moreover, substituting X by ϕ X and Y by ϕ Y in the last equation and using (2.9) and (2.1) gives

for all vector fields X, Y on M3(f1,f2,f3). It follows from last equation that either β = 0 or

for any vector field X on M3(f1,f2,f3). Let us assume that f1≠f3. Then we know that β is non zero. Hence, the equation (4.13) stands. Let {e,ϕe,ξ} be an orthonormal ϕ-basis of M3(f1,f2,f3) such that Qe=σe. Thus, we have ϕQe=σϕe. Substituting e for X in (4.13) and using the foregoing equation, we obtain Qϕe=(4(f−f3)−σ)ϕe. Using ϕ-basis and (2.9) the scalar curvature is given by

For the scalar curvature r=6(f1−f3), (2.12) gives us

for any vector fields X, Y, Z on M3(f1,f2,f3). Hence from (4.16) it follows that M3(f1,f2,f3) is of constant curvature f1−f3. Thus, we state the following:

Theorem 4.2. Let M3(f1,f2,f3) be a three-dimensional quasi-Sasakian generalized Sasakian-space-form with f1≠f3. If g is an h-almost gradient Ricci soliton, then M3(f1,f2,f3) is of constant curvature f₁ - f₃.

It is noted that the Weyl tensor vanishes on any three dimensional Riemannian manifold. Therefore we may consider Cotton tensor which is another conformal invariant of a three-dimensional Riemannian manifold. The Cotton tensor ℭ(X,Y) of type (1,1) is defined by: (see [6, 9, 41])

for any vector fields X and Y on M³. A three-dimensional Riemannian manifold is said to be conformally flat if the Cotton tensor ℭ vanishes.

Since the manifold under consideration is of constant curvature, that is, the scalar curvature r is constant, therefore the Cotton tensor vanishes. From the above discussions, we conclude the following:

Corollory 4.3. Let M3(f1,f2,f3) be a 3-dimensional quasi-Sasakian generalized Sasakian-space-form with f1≠f3. If g is an h-almost gradient Ricci soliton, then the Cotton tensor ℭ vanishes on M3(f1,f2,f3).

This work was supported by Department of Science and Technology (DST), Ministry of Science and Technology, Government of India, to the second author Amruthalakshmi M. R. (AMR) by providing financial assistance in the form of DST-INSPIRE Ferllowship(N0:DST/INSPIRE Fellowship/[IF 190869]) and also the fifth author Young Jin Suh(YJS) was supported by NRF-2018-R1D1A1B from National Research Foundation of Korea.