CUBO A Mathematical Journal
Vol.21, No¯ 03, (63–74). December 2019

http: // dx. doi. org/ 10. 4067/ S0719-06462019000300063

Beta-almost Ricci solitons on Sasakian 3-manifolds

Pradip Majhi and Debabrata Kar

Department of Pure Mathematics,

University of Calcutta,

35, Ballygaunge Circular Road,

Kolkata 700019, West Bengal, India

mpradipmajhi@gmail.com, debabratakar6@gmail.com

ABSTRACT

In this paper we characterize the Sasakian 3-manifolds admitting β-almost Ricci solitons

whose potential vector field is a contact vector field. Among others we prove that a β-

almost Ricci soliton whose potential vector field is a contact vector field on a Sasakian

3-manifold is shrinking, Einstein and non-trivial. Moreover, we prove that this type of

manifolds are isometric to a sphere of radius
√
7.

RESUMEN

En este art́ıculo caracterizamos las 3-variedades Sasakianas que admiten solitones β-

casi Ricci cuyo campo de vectores potencial es un campo de vectores de contacto. Entre

otros, probamos que un solitón β-casi Ricci cuyo campo de vectores potencial es un

campo de vectores de contacto en una 3-variedad Sasakiana se contrae, es Einstein y

no trivial. Más aún, probamos que este tipo de variedades son isométricas a una esfera

de radio
√
7.

Keywords and Phrases: Ricci soliton, β-almost Ricci soliton, Sasakian 3-manifolds, Einstein.

2010 AMS Mathematics Subject Classification: 53C15, 53C25.

http://dx.doi.org/10.4067/S0719-06462019000300063


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1 Introduction

In 1982, R. S. Hamilton [17] introduced the notion of Ricci flow to find a canonical metric on

a smooth manifold. The Ricci flow is an evolution equation for metrics on a Riemannian manifold

defined as follows:

∂

∂t
g = −2S, (1)

where S denotes the Ricci tensor of g. Ricci solitons are special solutions of the Ricci flow

equation (1) of the form g = σ(t)ψ∗tg with the initial condition g(0) = g, where ψt are diffeomor-

phisms of M and σ(t) is the scaling function. A Ricci soliton is a generalization of an Einstein

metric. We recall the notion of Ricci soliton according to [5]. On the manifold M, a Ricci soliton

is a triple (g,V,λ) with g, a Riemannian metric, V a vector field, called the potential vector field

and λ a real scalar such that

£Vg + 2S + 2λg = 0, (2)

where £ is the Lie derivative. Metrics satisfying (2) are interesting and useful in physics and are

often referred as quasi-Einstein ([6],[7]). Compact Ricci solitons are the fixed points of the Ricci

flow ∂
∂t
g = −2S projected from the space of metrics onto its quotient modulo diffeomorphisms

and scalings, and often arise blow-up limits for the Ricci flow on compact manifolds. Theoretical

physicists have also been looking into the equation of Ricci soliton in relation with string theory.

The initial contribution in this direction is due to Friedan [14] who discusses some aspects of it.

Recently, the notion of almost Ricci soliton has been introduced in [24] by Piagoli, Riogoli, Rimoldi

and Setti.

The Ricci soliton is said to be shrinking, steady or expanding according as λ is negative, zero or

positive respectively. Ricci solitons have been studied by several authors ([8], [9], [18], [19], [20],

[27], [28], and many others).

Recently, Gomes, Wang and Xia [26] generalized almost Ricci soliton to h -almost Ricci soliton as

follows:

Definition 1.1. A complete connected Riemannian manifold (M2n+1,g) is said to be a β-almost

Ricci soliton, denoted by (M2n+1,g,V,β,λ), if there exist a smooth vector field V on M2n+1 such

that

S +
β

2
£Vg + λg = 0, (3)

where λ and β are smooth functions on M2n+1. λ is called soliton function and V is called the

potential vector field.



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Beta-almost Ricci solitons on Sasakian 3-manifolds 65

A β-almost Ricci soliton is said to be shrinking, steady or expanding according as λ is negative,

zero or positive respectively. A β-almost Ricci soliton is called β-Ricci soliton if λ is constant. A

β-almost Ricci soliton is said to be trivial, that is, Einstein if the flow vector field V is homothetic,

that is, £Vg = cg, for some constant c. Otherwise, it is called non-trivial. A β-almost Ricci

soliton is said to be β-almost gradient Ricci soliton if the potential vector field V is the gradient of

a smooth function f on M2n+1, that is, V = Df, where D is the gradient operator of g on M2n+1.

For convenience, we denote (M2n+1,g,Df,β,λ) as a β-almost gradient Ricci soliton with potential

function f.

In particular, a Ricci soliton is a 1-almost Ricci soliton with constant soliton λ and an almost

Ricci soliton is nothing but a 1-almost Ricci soliton. Recently, Ghosh and Patra studied [16] the

k-almost Ricci solitons on contact geometry. In [1], Barros and Ribeiro proved that a compact

almost Ricci soliton with constant scalar curvature is isometric to an Euclidean sphere. In this

connection, a theorem has also been proved by Gomes, Wang and Xia in [26] for β-almost Ricci

soliton which is given as follows:

Theorem 1.1. [26] Let (Mn,g,V,β,λ),n ≥ 3 be a non-trivial β-almost Ricci soliton with constant
scalar curvature r. If Mn is compact, then it is isometric to a standard sphere Sn(c) of radius

c =

√

2n(2n+1)

r
.

The above Theorem will be used in later to prove our results.

The paper is organized as follows:

After preliminaries in Section 2, we study β-almost Ricci solitons on a Sasakian 3-manifold.

Among others we prove that β-almost Ricci solitons whose potential vector field is a contact vector

field on Sasakian 3-manifolds are shrinking and Einstein. Beside these, we prove that this type

of manifolds are isometric to a sphere of radius
√
7. Also we prove that a β-almost Ricci soliton

whose potential vector field is a contact vector field on a Sasakian 3-manifold is non-trivial.

2 Preliminaries

An odd dimensional smooth manifold M2n+1 (n ≥ 1) is said to admit an almost contact structure,
sometimes called a (φ,ξ,η)-structure, if it admits a tensor field φ of type (1,1), a vector field ξ

and a 1-form η satisfying ([2], [3])

φ2 = −I + η ⊗ ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0. (4)



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The first and one of the remaining three relations in (4) imply the other two relations in (4). An

almost contact structure is said to be normal if the induced almost complex structure J on Mn ×R
defined by

J(X,f
d

dt
) = (φX − fξ,η(X)

d

dt
) (5)

is integrable, where X is tangent to M, t is the coordinate of R and f is a smooth function on

Mn × R. Let g be a compatible Riemannian metric with the (φ,ξ,η)-structure, that is,

g(φX,φY) = g(X,Y) − η(X)η(Y) (6)

or equivalently,

g(X,φY) = −g(φX,Y) (7)

and

g(X,ξ) = η(X), (8)

for all vector fields X, Y tangent to M. Then M becomes an almost contact metric manifold

equipped with an almost contact metric structure (φ,ξ,η,g).

An almost contact metric structure becomes a contact metric structure if

g(X,φY) = dη(X,Y), (9)

for all X, Y tangent to M. The 1-form η is then a contact form and ξ is its characteristic vector

field.

Given the contact metric manifold (M,η,ξ,φ,g), we define a symmetric (1,1)-tensor field h as

h = 1
2
Lξφ, where Lξφ denotes Lie differentiation in the direction of ξ. We have the following

identities ([2], [3]):

hξ = 0, hφ + φh = 0, (10)

∇Xξ = −φX − φhX, (11)

∇ξφ = 0, (12)

R(ξ,X)ξ − φR(ξ,φX)ξ = 2(h2 + φ2)X, (13)

(∇ξh)X = φX − h2φX + φR(ξ,X)ξ, (14)

S(ξ,ξ) = 2n − trh2, (15)

R(X,Y)ξ = −(∇Xφ)Y + (∇Yφ)X − (∇Xφh)Y + (∇Yφh)X. (16)

Here, ∇ is the Levi-Civita connection and R is the Riemannian curvature tensor of (M,g) with
the sign convention defined by

R(X,Y)Z = ∇X∇YZ − ∇Y∇XZ − ∇[X,Y]Z (17)



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Beta-almost Ricci solitons on Sasakian 3-manifolds 67

for vector fields X, Y, Z on M. The tensor l = R(.,ξ)ξ is the Jacobi operator with respect to the

characteristic field ξ.

If the characteristic vector field ξ is a Killing vector field, the contact metric manifold (M,η,ξ,φ,g)

is called K-contact manifold. This is the case if and only if h = 0. The contact structure on M

is said to be normal if the almost complex structure on M × R defined by (5), is integrable. A
normal contact metric manifold is called a Sasakian manifold. Sasakian metrices are K-contact

and K-contact metrics on 3-manifolds are Sasakian. For a Sasakian manifold, the following hold

([2], [3]):

∇Xξ = −φX, (18)

(∇Xφ)Y = g(X,Y)ξ − η(Y)X, (19)

R(X,Y)ξ = η(Y)X − η(X)Y, (20)

Qξ = 2nξ, (21)

where Q denotes the (1,1)-tensor metrically equivalent to the Ricci tensor of g. The curvature

tensor of a 3-dimensional Riemannian manifold is given by

R(X,Y)Z = [S(Y,Z)X − S(X,Z)Y + g(Y,Z)QX − g(X,Z)QY]

−
r

2
[g(Y,Z)X − g(X,Z)Y], (22)

where S and r are the Ricci tensor and scalar curvature respectively and Q is the Ricci operator

defined by g(QX,Y) = S(X,Y).

It is known that the Ricci tensor of a Sasakian 3-manifold is given by [4]

S(X,Y) =
1

2
{(r − 2)g(X,Y) + (6 − r)η(X)η(Y)} (23)

where r is the scalar curvature which need not be constant, in general. So, g is Einstein (hence

has constant curvature 1) if and only if r = 6.

As a consequence of (23), we have

S(X,ξ) = 2η(X). (24)

Contact metric manifolds have also been studied by several authors ([4], [10], [11], [12], [13],

[21], [22], [23], [25], and many others).



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Definition 2.1. ([16]) A vector field V on a contact manifold is said to be a contact vector field

if it preserves the contact form η, that is

£Vη = ψη, (25)

for some smooth function ψ on M. When ψ = 0 on M, the vector field V is called a strict contact

vector field.

Lemma 2.1. ([15]) If a vector field X leaves the structure tensor φ of the contact metric manifold

M invariant, then there exists a constant c such that £Xg = c(g + η ⊗ η).

3 β-almost Ricci solitons on Sasakian 3-manifolds

In this section we characterize Sasakian 3-manifolds M3 admitting β-almost Ricci solitons whose

potential vector field V is a contact vector field. Then the equations (3) and (25) hold good. The

equation (3) can be exhibited as

S(X,Y) +
β

2
{g(∇XV,Y) + g(X,∇YV)} + λg(X,Y) = 0. (26)

Using (23) in the above equation we get

β{g(∇XV,Y) + g(X,∇YV)} = −(r + 2λ − 2)g(X,Y)

+(r − 6)η(X)η(Y). (27)

Tracing the equation (27) we obtain

βdivV = −(r + 3λ). (28)

With the help of (25) we have

£Vdη = d£Vη = (dψ) ∧ η + ψ(dη). (29)

Let us consider ω as the volume form of the manifold M3, that is,

ω = η ∧ dη 6= 0. (30)

Taking Lie derivative of the preceding equation along the potential vector field V and using (25)

and (29) we have £Vω = 2ψω, and hence

divV = 2ψ. (31)



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Beta-almost Ricci solitons on Sasakian 3-manifolds 69

Using the foregoing equation in (28) we infer

r = −2ψβ − 3λ. (32)

The soliton equation (3) also can be represented as

S(X,Y) +
β

2
(£Vg)(X,Y) + λg(X,Y) = 0. (33)

Substituting X = Y = ξ in (33) we get

βg(£Vξ,ξ) = λ + 2. (34)

Putting Y = ξ in (33) and using (24),

β

2
(£Vη)(X) −

β

2
g(X,£Vξ) + (λ + 2)η(X) = 0. (35)

Making use of (25) we obtain

β£Vξ = (ψβ + 2λ + 4)ξ. (36)

By the virtue of (34) and (36) we have

ψβ = −λ − 2. (37)

Using (37), (36) entails

β£Vξ = (λ + 2)ξ. (38)

From (9) we deduce that

(£Vdη)(X,Y) = (£Vg)(X,φY) + g(X,(£Vφ)Y). (39)

Multiplying both sides of (39) by β and then using (33) we infer

β(£Vdη)(X,Y) = −2S(X,φY) − 2λg(X,φY) + βg(X,(£Vφ)Y). (40)

In view of (23) and (40) we get

β(£Vdη)(X,Y) = −(r + 2λ − 2)g(X,φY) + βg(X,(£Vφ)Y). (41)

From (29) we derive

(£Vdη)(X,Y) =
1

2
{dψ(X)η(Y) − dψ(Y)η(X)} + ψg(X,φY). (42)

Comparing (41) and (42), after simplification we obtain

2β(£Vφ)Y = 2(r + 2λ − 2)φY + βη(Y)Dψ − β(Yψ)ξ + 2ψβφY. (43)



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Replacing Y by ξ we get

2β(£Vφ)ξ = βDψ − β(ξψ)ξ. (44)

With the help of (4) and (36) we find that

β(£Vφ)ξ = 0. (45)

Applying (45) on (44) we have

Dψ = (ξψ)ξ. (46)

Taking inner product of (46) with X gives

dψ(X) = (ξψ)η(X), (47)

or equivalently,

dψ = (ξψ)η. (48)

Taking exterior derivative we get

d(ξψ) ∧ η + (ξψ)dη = 0. (49)

Taking wedge product of (49) with η we have

(ξψ)η ∧ dη = 0, (50)

from which it follows that

ξψ = 0, (51)

Since η ∧ dη 6= 0, and by (48),

dψ = 0 (52)

and hence ψ is constant. Integrating (31) and then using Divergence Theorem we infer

ψ = 0. (53)

Thus the potential vector field V becomes a strict contact vector field and hence we have the

following:

Theorem 3.1. Let (M3,g,V,β,λ) be a non-trivial β-almost Ricci soliton whose potential vector

field is a contact vector field on a Sasakian 3-manifold. Then the potential vector field is a strict



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Beta-almost Ricci solitons on Sasakian 3-manifolds 71

contact vector field.

By the virtue of (37) and (53) we find

λ = −2. (54)

Therefore, the β-almost Ricci soliton is shrinking. Thus we are in a position to state that

Theorem 3.2. A non-trivial β-almost Ricci soliton (M3,g,V,β,λ) whose potential vector field is

a contact vector field on a Sasakian 3-manifold is shrinking.

Making use of (53) and (54), from (32) we get

r = 6. (55)

Then we can conclude that

Theorem 3.3. The scalar curvature of a non-trivial β-almost Ricci soliton

(M3,g,V,β,λ) whose potential vector field is a contact vector field on a Sasakian 3-manifold is 6.

With the help of (55) from (23) we have

S(X,Y) = 2g(X,Y). (56)

Hence we can state the following:

Theorem 3.4. A non-trivial β-almost Ricci soliton (M3,g,V,β,λ) whose potential vector field is

a contact vector field on Sasakian 3-manifold is Einstein.

From (55) we can say that r is constant. Then in view of Theorem 1.1 we can conclude the

following:

Theorem 3.5. Let (M3,g,V,β,λ) be a non-trivial β-almost Ricci soliton whose potential vector

field is a contact vector field on a Sasakian 3-manifolds. Then (M3,g,V,β,λ) is isometric to a

sphere S3(c) of radius c =
√
7.

Using (53), (54) and (55) in (43) we infer

(£Vφ)Y = 0, (57)

as we have considered β as positive, that is, V leaves the structure tensor φ of the Sasakian

3-manifold invariant. Then, by Lemma 2.2, exists a constant a such that

£Vg = a(g + η ⊗ η), (58)



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which shows that the β-almost Ricci solitons are non-trivial. Thus our next theorem can be stated

as follows:

Theorem 3.6. Let (M3,g,V,β,λ) be a non-trivial β-almost Ricci soliton whose potential vector

field is a contact vector field on a Sasakian 3-manifolds. Then the β-almost Ricci solitons are

non-trivial.

Acknowledgement: The authors are thankful to the referee for his/her valuable suggestions and

comments towards the improvement of the paper. The author Debabrata Kar is supported by the

Council of Scientific and Industrial Research, India (File no : 09/028(1007)/2017-EMR-1).



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Beta-almost Ricci solitons on Sasakian 3-manifolds 73

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	Introduction
	Preliminaries
	-almost Ricci solitons on Sasakian 3-manifolds