CUBO A Mathematical Journal
Vol.20, No¯ 3, (37–47). October 2018

http: // dx. doi. org/ 10. 4067/ S0719-06462018000300037

Yamabe Solitons with potential vector field as torse forming

Yadab Chandra Mandal and Shyamal Kumar Hui

Department of Mathematics,

The University of Burdwan,

Burdwan, 713104, West Bengal, India

myadab436@gmail.com, skhui@math.buruniv.ac.in

ABSTRACT

The Riemannian manifolds whose metric is Yamabe soliton with potential vector field

as torse forming admitting Riemannian connection, semisymmetric metric connection

and projective semisymmetric connection have been studied. An example is constructed

to verify the theorem concerning Riemannian connection.

RESUMEN

Se estudian las variedades Riemannianas cuya métrica es un solitón de Yamabe con

vector de potencial que forma un virol (superficie desarrollable) con respecto a conex-

iones Riemanniana, semisimétrica métrica y proyectiva semisimétrica. Se construye

un ejemplo expĺıcito para verificar las hipótesis del teorema en el caso de la conexión

Riemanniana.

Keywords and Phrases: Yamabe soliton, torse forming vector field, torqued vector field,

semisymmetric metric connection, projective semisymmetric connection.

2010 AMS Mathematics Subject Classification: 53C21, 53C25.

http://dx.doi.org/10.4067/S0719-06462018000300037


38 Yadab Chandra Mandal and Shyamal Kumar Hui CUBO
20, 2 (2018)

1 Introduction

The curvature tensor, Ricci tensor and scalar curvature of a Riemannian manifold M of dimension

n equipped with Riemannian metric g with respect to Levi-Civita connection ∇ are denoted by

R, S and r respectively. Hamilton ([5], [6]) introduced the notion of Yamabe flow, which is an

evolution equation for metrics on M as follows:

∂

∂t
g = −rg.

When n = 2, the Yamabe flow is equivalent to the Ricci flow. However, for n > 2, they do not

agree.

A Yamabe soliton on M is, a special solution of the Yamabe flow, a triplet (g,V,σ) such that

1

2
£Vg = (r − σ)g, (1.1)

where £V is the Lie derivative in the direction of V ∈ χ(M) and σ is a constant. The nature of

such soliton depends on the behaviour of σ. The Yamabe soliton is said to be shrinking, steady

and expanding according as σ < 0, = 0 and > 0 respectively. If σ ∈ C∞(M) then the metric

satisfying (1.1) is called almost Yamabe soliton [1]. For n = 2 such soliton is equivalent with Ricci

soliton, but for n > 2, they do not. Yamabe solitons have been studied by several authors such as

[5], [6], [9], [10] and references there in.

As a generalization of concircular, concurrent and parallel vector field, Yano [14] introduced

the torse-forming vector field. A nowhere vanishing vector field τ is said to be a torse-forming on

M if

∇Xτ = fX + γ(X)τ, (1.2)

where f ∈ C∞(M) and γ is an 1-form.

If the 1-form γ in (1.2) vanishes identically, then τ is concircular [13]. Concircular vector

fields also known as geodesis vector fields since integral curves of such vector fields are geodesis.

Recently, Chen [2] studied Ricci solitons with concircular vector field. If f = 1 and γ = 0 then τ

is concurrent [16]. The vector field τ is recurrent if it satisfies (1.2) with f = 0. Also if f = γ = 0,

the vector field τ in (1.2) is parallel vector field.

As a consequence of torse forming vector field, recently Chen [3] introduced a new vector field,

called torqued vector field. If the vector field τ satisfies (1.2) with γ(τ) = 0 then τ is called torqued

vector field. Here, f is known as the torqued function and the 1-form is the torqued form of τ.

In this paper we have studied Yamabe solitons, whose potential vector field is torse forming,

on Riemannian manifolds with respect to Riemannian connection (RC), semisymmetric metric

connection (SSMC) and projective semisymmetric connection (PSSC) and prove the following:



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Yamabe Solitons with potential vector field as torse forming 39

Theorem 1.1. Let (g,τ,σ) be a Yamabe soliton on M with respect to RC ∇. Then the following

holds:

τ condition of existence conditions of shrinking,

steady and expanding

torse-forming r − f − 1
n
γ(τ)=constant r − f − 1

n
γ(τ) S 0

concircular r − f= constant r − f S 0
concurrent r=constant r S 1
recurrent r − 1

n
γ(τ)= constant r − 1

n
γ(τ) S 0

parallel r = constant r S 0
torqued r − f =constant r − f S 0

Theorem 1.2. Let (g,τ,σ) be a Yamabe soliton on M with respect to SSMC ∇̄. Then the following

holds:

τ condition of existence conditions of shrinking,

steady and expanding

torse-forming r − f − 2(n − 1)a r − f − 2(n − 1)a

− 1
n
{(n − 1)π(τ) + γ(τ)}=constant − 1

n
{(n − 1)π(τ) + γ(τ)} S 0

concircular r − f − (n − 1){2a + 1
n
π(τ)}=constant r − f − (n − 1){2a + 1

n
π(τ)} S 0

concurrent r − 1 − (n − 1){2a + 1
n
π(τ)} = constant r − 1 − (n − 1){2a + 1

n
π(τ)}S 0

recurrent r − 2(n − 1)a r − 2(n − 1)a

− 1
n
{(n − 1)π(τ) + γ(τ)}= constant − 1

n
{(n − 1)π(τ) + γ(τ)} S 0

parallel r − (n − 1){2a + 1
n
π(τ)} = constant r − (n − 1){2a + 1

n
π(τ)} S 0

torqued r − f − (n − 1){2a + 1
n
π(τ)} =constant r − f − (n − 1){2a + 1

n
π(τ)}S 0



40 Yadab Chandra Mandal and Shyamal Kumar Hui CUBO
20, 2 (2018)

Theorem 1.3. Let (g,τ,σ) be a Yamabe soliton on M with respect to PSSC ∇̃. Then the following

holds:

τ condition of existence conditions of shrinking,

steady and expanding

torse-forming r − f + Tr · β − (n − 1)Tr · α r − f + Tr · β − (n − 1)Tr · α

− 1
n
{(n − 1)π(τ) + γ(τ)}=constant − 1

n
{(n − 1)π(τ) + γ(τ)} S 0

concircular r − f + Tr · β − (n − 1){Tr · α r − f + Tr · β − (n − 1){Tr · α

+ 1
n
π(τ)} =constant + 1

n
π(τ)} S 0

concurrent r + Tr · β − (n − 1){Tr · α r − 1 + Tr · β

+ 1
n
π(τ)} =constant −(n − 1){Tr · α + 1

n
π(τ)} S 0

recurrent r + Tr · β − (n − 1)Tr · α r + Tr · β − (n − 1)Tr · α

− 1
n
{(n − 1)π(τ) + γ(τ)}= constant − 1

n
{(n − 1)π(τ) + γ(τ)} S 0

parallel r + Tr · β − (n − 1){Tr · α + 1
n
π(τ)} r + Tr · β − (n − 1){Tr · α+

=constant 1
n
π(τ)} S 0

torqued r − f + Tr · β − (n − 1){Tr · α+ r − f + Tr · β − (n − 1){Tr · α+
1
n
π(τ)}=constant 1

n
π(τ)} S 0

Section 2 consists with preliminaries. The proof of our theorems are given in section 3. In

section 4, we have constructed an example to verify Theorem 1.1.

Remark. The conditions of existence of Theorem 1.1, Theorem 1.2 and Theorem 1.3 are only

necessary. Finding sufficient conditions for the existence of solitons is a much deeper problem and

this is not addressed in the present manuscript.

2 Preliminaries

The relation between the semisymmetric metric connection (SSMC) ∇̄ and ∇ of M is given by

([4], [7], [15])

∇̄XY = ∇XY + π(Y)X − g(X,Y)ρ, (2.1)

where π(X) = g(X,ρ) for all X ∈ χ(M). If R̄ (resp. S̄ and r̄) are the curvature tensor (respectively

Ricci tensor and scalar curvature) of M with respect to SSMC, then [4]

R̄(X,Y)Z = R(X,Y)Z − P(Y,Z)X + P(X,Z)Y − g(Y,Z)LX + g(X,Z)LY, (2.2)

S̄(Y,Z) = S(Y,Z) − (n − 2)P(Y,Z) − ag(Y,Z), (2.3)

r̄ = r − 2(n − 1)a, (2.4)



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Yamabe Solitons with potential vector field as torse forming 41

where P is a tensor field of type (0,2) given by

P(X,Y) = g(LX,Y) = (∇Xπ)(Y) − π(X)π(Y) +
1

2
π(ρ)g(X,Y)

and a = Tr.P for any X,Y ∈ χ(M). The relation between projective semisymmetric connection ∇̃

and ∇ is [17]

∇̃XY = ∇XY + ψ(Y)X + ψ(X)Y + φ(Y)X − φ(X)Y, (2.5)

where the 1-forms φ and ψ are given by φ(X) = 1
2
π(X) and ψ(X) = n−1

2(n+1)
π(X). If R̃ (resp. r̃ and

S̃) are the curvature tensor, Ricci tensor and scalar curvature of M with respect to ∇̃, then ([12],

[17])

R̃(X,Y)Z = R(X,Y)Z + β(X,Y)Z + α(X,Z)Y − α(Y,Z)X, (2.6)

S̃(Y,Z) = S(Y,Z) + β(Y,Z) − (n − 1)α(Y,Z), (2.7)

r̃ = r + Tr.β − (n − 1)Tr.α, (2.8)

for all X,Y,Z ∈ χ(M), where

β(X,Y) =
1

2
[(∇Yπ)(X) − (∇Xπ)(Y)],

α(X,Y) =
n − 1

2(n + 1)
(∇Xπ)(Y) +

1

2
(∇Yπ)(X) −

n2

(n + 1)2
π(X)π(Y).

3 Proof of the Theorems

Proof of the Theorem 1.1. Let (g,τ,σ) be a Yamabe soliton on M. Then from (1.1) we get

1

2
(£τg)(X,Y) = (r − σ)g(X,Y). (3.1)

Now from (1.2) we have

(£τg)(X,Y) = g(∇Xτ,Y) + g(X,∇Yτ) (3.2)

= 2fg(X,Y) + γ(X)g(τ,Y) + γ(Y)g(τ,X)

for all X,Y ∈ χ(M). In view of (3.2), (3.1) yields

(r − σ − f)g(X,Y) =
1

2
{γ(X)g(τ,Y) + γ(Y)g(τ,X)}. (3.3)

Taking contraction of (3.3) over X and Y we get

n(r − σ − f) = γ(τ). (3.4)

This leads to the following:



42 Yadab Chandra Mandal and Shyamal Kumar Hui CUBO
20, 2 (2018)

Proposition 3.1. Let (g,τ,σ) be a Yamabe soliton on M with respect to RC ∇. If τ is torse-

forming then this soliton is shrinking, steady and expanding according as

r − f −
1

n
γ(τ) S 0,

provided as r − f − 1
n
γ(τ) is constant.

From Proposition 3.1, we obtain Theorem 1.1.

Proof of the Theorem 1.2. We now consider (g,τ,σ) is a Yamabe soliton on M with respect

to semisymmetric metric connection. Then we have

1

2
(£̄τg)(X,Y) = (r̄ − σ)g(X,Y), (3.5)

where £̄τ is the Lie derivative along τ of ∇̄. From (2.1) we get

(£̄τg)(X,Y) = g(∇̄Xτ,Y) + g(X,∇̄Yτ) (3.6)

= g(∇Xτ + π(τ)X − g(X,τ)ρ,Y)

+ g(X,∇Yτ + π(τ)Y − g(Y,τ)ρ)

= (£τg)(X,Y) + 2π(τ)g(X,Y)

− [g(X,τ)π(Y) + g(Y,τ)π(X)].

Using (2.4) and (3.6) in (3.5), we get

1

2
(£τg)(X,Y) = (r − σ)g(X,Y) − {2(n − 1)a + π(τ)}g(X,Y) (3.7)

+
1

2
[g(X,τ)π(Y) + g(Y,τ)π(X)].

In view of (3.2), (3.7) yields

{r − σ − f − 2(n − 1)a − π(τ)}g(X,Y) (3.8)

+
1

2
[{π(Y) − γ(Y)}g(τ,X) + {π(X) − γ(X)}g(τ,Y)] = 0.

Contracting (3.8) over X and Y, we get

n{r − σ − f − 2(n − 1)a} − (n − 1)π(τ) − γ(τ) = 0. (3.9)

This leads to the following:

Proposition 3.2. Let (g,τ,σ) be a Yamabe soliton on M with respect to SSMC ∇̄. If τ is torse-

forming then this soliton is shrinking, steady and expanding according as

r − f − 2(n − 1)a −
1

n
{(n − 1)π(τ) + γ(τ)} S 0,



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Yamabe Solitons with potential vector field as torse forming 43

provided r − f − 2(n − 1)a − 1
n
{(n − 1)π(τ) + γ(τ)} is constant.

From Proposition 3.2, we obtain Theorem 1.2.

Proof of the Theorem 1.3. We now consider (g,τ,σ) is a Yamabe soliton on M with respect

to ∇̃. Then we have
1

2
(£̃τg)(X,Y) = (̃r − σ)g(X,Y), (3.10)

where £̃τ is the Lie derivative along τ of ∇̃. From (2.5) we get

(£̃τg)(X,Y) = g(∇̃Xτ,Y) + g(X,∇̃Yτ) (3.11)

= (£τ)(X,Y) +
1

n + 1
{2nπ(τ)g(X,Y)

− π(X)g(τ,Y) − π(Y)g(X,τ)}.

Using (2.8) and (3.11) in (3.10), we get

1

2
(£τg)(X,Y) = (r − σ)g(X,Y) (3.12)

+ [Tr · β − (n − 1)Tr · α]g(X,Y)

−
1

2(n + 1)
{2nπ(τ)g(X,Y)

− π(X)g(Y,τ) − π(Y)g(X,τ)}.

In view of (3.2), (3.12) yields

{r − σ − f + Tr · β − (n − 1)Tr · α −
n

n + 1
π(τ)}g(X,Y) (3.13)

+
1

2
[{
π(Y)

n + 1
− γ(Y)}g(τ,X) + {

π(X)

n + 1
− γ(X)}g(τ,Y)] = 0.

Contracting (3.13) over X and Y, we get

n{r − σ − f + Tr · β − (n − 1)Tr · α} − (n − 1)π(τ) − γ(τ) = 0. (3.14)

This leads to the following:

Proposition 3.3. Let (g,τ,σ) be a Yamabe soliton on M with respect to ∇̃. If τ is torse-forming

then this soliton is shrinking, steady and expanding according as

r − f + Tr · β − (n − 1)Tr · α −
1

n
{(n − 1)π(τ) + γ(τ)} S 0,

provided r − f + Tr · β − (n − 1)Tr · α − 1
n
{(n − 1)π(τ) + γ(τ)} is constant.

From Proposition 3.3, we obtain Theorem 1.3.



44 Yadab Chandra Mandal and Shyamal Kumar Hui CUBO
20, 2 (2018)

4 Example

Here we construct an example to verify Theorem 1.1.

Example: Let us consider a 3-dimensional manifold M = {(x,y,z) ∈ R3 : z 6= 0}. Let {e1,e2,e3}

be a linearly independent global frame on M given by

e1 = z
2 ∂

∂x
, e2 = z

2 ∂

∂y
, e3 =

∂

∂z
.

Let g be the Riemannian metric defined by g(ei,ej) =

{

1, i=j

0, i 6= j
.

These vector field and such metric is used in ([8], [11]). Using Koszul formula, we have [11]

∇e1e1 =
2

z
e3, ∇e1e2 = 0, ∇e1e3 = −

2

z
e1,

∇e2e1 = 0, ∇e2e2 =
2

z
e3, ∇e2e3 = −

2

z
e2,

∇e3e1 = 0, ∇e3e2 = 0, ∇e3e3 = 0.

The scalar curvature of this manifold is also computed in [11] and it is r = −32
z2
.

Since {e1,e2,e3} forms a basis, any vector field X,Y,U ∈ χ(M) can be written as

X = a1e1 + b1e2 + c1e3, Y = a2e1 + b2e2 + c2e3, U = a3e1 + b3e2 + c3e3,

where ai,bi,ci ∈ R
+ for i = 1,2,3 such that

a1a2 + b1b2

c1
+ c1(

b2

b1
−
a2

a1
− 1) 6= 0.

If we choose the 1-form γ by γ(W) = g(W, 2
z
e3) for any W ∈ χ(M) and considering f ∈ C

∞(M) as

f =
2

z

{

a1a2 + b1b2

c1
+ c1

(
b2

b1
−
a2

a1
− 1

)}
.

Then the relation

∇XY = fX + γ(X)Y (4.1)

holds. Consequently Y is a torse-forming vector field. Now from (4.1) we get

(£Yg)(X,U) = g(∇XY,U) + g(X,∇UY) (4.2)

= 2fg(X,U) + γ(X)g(Y,U) + γ(U)g(Y,X).

Also we can calculate










g(X,U) = a1a3 + b1b3 + c1c3,

g(Y,U) = a2a3 + b2b3 + c2c3,

g(Y,X) = a1a2 + b1b2 + c1c2,

(4.3)

γ(X) =
2c1

z
,γ(Y) =

2c2

z
,γ(U) =

2c3

z
. (4.4)



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Yamabe Solitons with potential vector field as torse forming 45

In view of (4.3) and (4.4), (4.2) yields

1

2
(£Yg)(X,U) =

1

z

[{
2(a1a2 + b1b2)

c1
+ 2c1

(
b2

b1
−
a2

a1
− 1

)}
(a1a3 + b1b3 + c1c3)

+ c1(a2a3 + b2b3 + c2c3) + c3(a1a2 + b1b2 + c1c2)
]
. (4.5)

Also

(r − σ)g(X,U) = (−
32

z2
− σ)(a1a3 + b1b3 + c1c3). (4.6)

Assuming that a1a3 + b1b3 + c1c3 6= 0 and

3c1(a2a3 + b2b3 + c2c3) + 3c3(a1a2 + b1b2 + c1c2) − 2c2(a1a3 + b1b3 + c1c3) = 0,

we get (g,Y,σ) is an Yamabe soliton, i.e 1
2
(£Yg)(X,U) = (r − σ)g(X,U) holds, provided

σ = −
32

z2
−
2

z

{

(a1a2 + b1b2)

c1
+ c1

(
b2

b1
−
a2

a1
− 1

)}

−
c1(a2a3 + b2b3 + c2c3) + c3(a1a2 + b1b2 + c1c2)

(a1a3 + b1b3 + c1c3)z

= r − f −
1

3
γ(Y)

= constant.

Thus the condition of existence of Yamabe soliton (g,Y,σ) on a 3-dimensional Riemannian mani-

fold with potential vector field Y as torse forming in Theorem 1.1 is verified.

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

towards to the improvement of the paper. The first author (Y. C. Mandal) gratefully acknowl-

edges The University Grants Commission, Government of India, for the award of Junior Research

Fellowship.



46 Yadab Chandra Mandal and Shyamal Kumar Hui CUBO
20, 2 (2018)

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Yamabe Solitons with potential vector field as torse forming 47

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	Introduction
	Preliminaries
	Proof of the Theorems
	Example