Ricci Soliton

NEW METHODS OF CONSTRUCTION FOR RICCI SOLITONS
DELLOUM ADEL1 , OUBBICHE NOUR2 AND BELDJILALI GHERICI3
Abstract. The aim of this paper is to construct Ricci solitons on Riemennian
manifolds using a new deformation of the metric. Concrete examples are given.
1. Introduction
In general, the construction in geometry is a very wide field of research, the
construction of structures on Riemanian manifolds will be specifically mention here,
particularly the structure of Ricci Soliton. The structure of Ricci Soliton is recently
developed in several works, for example as in [2, 5, 7, 10, 9, 20]. The majority of this
research are use in the construction of the important geometric tools such as special
vector fields, the deformation of metrics, the product of manifolds and others.
In 1982, Hamilton [12] introduced the notion of Ricci flow to find a canonical
metric on a smooth manifold. Then Ricci flow has become a powerful tool to study
Riemannian manifolds, especially for manifolds with positive curvatures.
The vector field V generates the Ricci soliton that is viewed as a special solution
of the Ricci flow, and is called the generating vector field. A Ricci soliton is said
to be a gradient Ricci soliton if the generating vector field V is the gradient of a
potential function.
This paper focused on the geometry of transformations that is related to the
(n − 1)−dimensional distribution D of n-dimensional Riemannian manifold M .
Precisely, the goal of this work is the construction of a Ricci soliton from a unit
global vector field which will be the main tool for the deformation of the metric.
The paper is organized as follows:
Section 2 introduces the background of the formulas which will be used in the
sequel. In Section 3, an interesting deformation of metrics is established and some
basic properties are proved. In Section 4, the necessary and sufficient techniques
are given to construct Ricci solitons in addition to a class of examples. In the last
Section , we focus on the case of three-dimensional with concrete examples.
2. Review of needed notions
By R, Q, S and r we denote respectively the Riemannian curvature tensor, the
Ricci operator, the Ricci tensor and the scalar curvature of a Riemannian manifold
(M n , g). Then R, Q S and r are defined by:
(2.1)
(2.2)
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,
QX =
n
X
R(X, ei )ei ,
i=1
2000 Mathematics Subject Classification. 53C15, 53C25, 53D10.
Key words and phrases. Riemannian manifolds, Ricci solitons, Deformation of metrics.
1
DELLOUM ADEL1 , OUBBICHE NOUR2 AND BELDJILALI GHERICI3
2
(2.3)
S(X, Y ) = g(QX, Y ) =
n
X
g R(X, ei )ei , Y ,
i=1
(2.4)
r=
n
X
S(ei , ei ),
i=1
where ∇ is the Levi-Civita connection with respect to g, {ei } is an orthonormal
frame, and X, Y, Z ∈ Γ(T M ). For all vector field X on M , we have
n
X
(2.5)
X=
g(X, ei )ei ,
i=1
the divergence of a vector field V is defined by:
n
X
(2.6)
divV =
g(∇ei V, ei ),
i=1
and for a unit closed 1-form η (i.e. dη = 0) one can get
(2.7)
g(∇X ξ, Y ) = g(∇Y ξ, X),
g(∇X ξ, ξ) = 0
and
∇ξ ξ = 0.
where η be the g-dual of ξ which means η(X) = g(X, ξ) for all vector field X on
M . We recall that (M n , g), n ≥ 3, is said to be Einstein if at every x ∈ M its Ricci
tensor S has the form
(2.8)
S = ρg,
ρ ∈ R.
(For more details, see for example [19]).
A smooth vector field V on a Riemannian manifold (M, g) is said to define a
Ricci soliton if it satisfies the following Ricci soliton equation:
(2.9)
(LV g)(X, Y ) + 2S(X, Y ) = 2λg(X, Y ),
where LV g denotes the Lie derivative of g along a vector field V and λ is a constant.
We shall denote a Ricci soliton by (M, g, V, λ). If λ is a smooth function on M ,
we say that (M, V, λ, g) is an almost Ricci soliton. We call the vector field V the
potential field. A Ricci soliton (M, g, V, λ) is called shrinking, steady or expanding
according to λ > 0, λ = 0, or λ < 0, respectively. A trivial Ricci soliton is one for
which V is zero or Killing, in which case the metric is Einsteinian (see, for instance,
[7, 11, 17]).
3. Deformations of Metrics
n
Let M be a Riemannian manifold equipped with Riemannian metric g, and η
a unit global closed 1-forme on M . We denote by ξ the vector field corresponding
to η, i.e for all X vector field on M ,
η(X) = g(ξ, X)
and
η(ξ) = 1.
We define on M a Riemannian metric, denoted g̃, by
(3.1)
g̃(X, Y ) = g(X, Y ) + η(X)η(Y ).
The equation η = 0 defines a (n − 1)-dimensional distribution D on M . Then, we
have
g̃(ξ, ξ) = 2
g̃(X, X) = g(X, X)
∀X ∈ D
That is why, we refer to this construction as D-isometric deformation.
NEW METHODS OF CONSTRUCTION FOR RICCI SOLITONS
3
Note that the simplest case for this deformation is for η = df where f ∈ C ∞ (M ).
This case has been studied in [3] and [4].
In [13], Innami proved that M admits a non constant affine function if and only
if M splits as a Riemannian product M = N × R. In our situation, since dη = 0
then ξ is locally of type gradient, what means that at each point p on M there
exists a function f such that ξ = ∇f on a neighborhood at p, where ∇f denotes
the gradient vector field of f .
˜ denote the Riemannian connections of g and g̃
Proposition 3.1. Let ∇ and ∇
respectively. Then, for all X, Y vector fields on M , we have the relation:
˜ X Y = ∇X Y + 1 g(∇X ξ, Y )ξ.
(3.2)
∇
2
Proof. Using Koszul’s formula for the metric g̃,
˜ X Y, Z) = X g̃(Y, Z) + Y g̃(Z, X) − Z g̃(X, Y )
2g̃(∇
− g̃(X, [Y, Z]) + g̃(Y, [Z, X]) + g̃(Z, [X, Y ],
one can obtain
˜ X Y, Z)
g̃(∇
= g̃(∇X Y, Z)
1
(∇X η)Y + (∇Y η)X η(Z)
+
2
Knowing that dη = 0, we get
˜ X Y, Z) = g̃(∇X Y, Z) + (∇X η)(Y )η(Z),
g̃(∇
with
(∇X η)(Y ) = g(∇X ξ, Y ).
On the other hand, we have
η(Z)
=
g(ξ, Z)
=
g̃(ξ, Z) − η(Z),
which gives
η(Z) =
Therefore
1
g̃(ξ, Z).
2
˜ X Y = ∇X Y + 1 g(∇X ξ, Y )ξ.
∇
2
□
Theorem 3.1. Let R̃, Q̃ and S̃ be the Riemannian curvature tensor, the Ricci
operator and the Ricci tensor of the Riemannian manifold (M n , g̃) respectively.
For all X, Y and Z vector fields on M , we have
2R̃(X, Y )Z = 2R(X, Y )Z − g R(X, Y )Z, ξ ξ
(3.3)
+g(∇Y ξ, Z)∇X ξ − g(∇X ξ, Z)∇Y ξ,
2Q̃X = 2QX − R(X, ξ)ξ − S(X, ξ)ξ + (divξ)∇X ξ − ∇∇X ξ ξ,
(3.5) 2S̃(X, Y ) = 2S(X, Y ) − g R(X, ξ)ξ, Y + divξg(∇X ξ, Y ) − g(∇X ξ, ∇Y ξ).
(3.4)
4
DELLOUM ADEL1 , OUBBICHE NOUR2 AND BELDJILALI GHERICI3
Proof. By the definition of the curvature tensor R̃
˜ X∇
˜Y Z − ∇
˜Y ∇
˜ XZ − ∇
˜ [X,Y ] Z,
(3.6)
R̃(X, Y )Z = ∇
using formula 3.2, the first term of (3.6) is given by
˜ X∇
˜Y Z = ∇
˜ X ∇Y Z + 1 g(∇Y ξ, Z)ξ
∇
2
1
˜ X ∇Y Z + g(∇X ∇Y ξ, Z)ξ
= ∇
2
1
1
˜ Xξ
+ g(∇Y ξ, ∇X Z)ξ + g(∇Y ξ, Z)∇
2
2
1
= ∇X ∇Y Z + g(∇X ξ, ∇Y Z)ξ
2
1
1
+ g(∇X ∇Y ξ, Z)ξ + g(∇Y ξ, ∇X Z)ξ
2
2
1
(3.7)
+ g(∇Y ξ, Z)∇X ξ.
2
With the similar method, the second term of (3.6) is given by
˜Y ∇
˜ X Z = ∇Y ∇X Z + 1 g(∇Y ξ, ∇X Z)ξ
∇
2
1
1
+ g(∇Y ∇X ξ, Z)ξ + g(∇X ξ, ∇Y Z)ξ
2
2
1
+ g(∇X ξ, Z)∇Y ξ.
(3.8)
2
Knowing that dη = 0, the last term of (3.6) becomes
˜ [X,Y ] Z = ∇[X,Y ] + 1 g(∇[X,Y ] ξ, Z)ξ
∇
2
1
1
= ∇[X,Y ] + g(∇∇X Y ξ, Z)ξ − g(∇∇Y X ξ, Z)ξ
2
2
1
1
= ∇[X,Y ] + g(∇Z ξ, ∇X Y )ξ − g(∇Z ξ, ∇Y X)ξ.
(3.9)
2
2
Formula (3.3) follows from equations (3.6)-(3.9).
To show the second formula (3.4), consider {ξ, ei }2≤i≤n the orthonormal basis on
M with respect to the metric g, it is easy to prove that { √12 ξ, ei }2≤i≤n is an
orthonormal frame on M with respect to the metric g̃. Using formula (2.2), we
have
n
X
1
Q̃X =
R̃(X, ξ)ξ +
R̃(X, ei )ei
2
i=2
=
n
X
1
− R̃(X, ξ)ξ +
R̃(X, ei )ei
2
i=1
n X
1
1
= − R(X, ξ)ξ +
R(X, ei )ei − g R(X, ei )ei , ξ ξ
2
2
i=1
1
1
+ g(∇ei ξ, ei )∇X ξ − g(∇X ξ, ei )∇ei ξ .
(3.10)
2
2
Using equations (2.2), (2.3), (2.5) and (2.6), we get (3.4).
For the third formula (3.5), just use the two formulas (2.3) and (3.4).
□
NEW METHODS OF CONSTRUCTION FOR RICCI SOLITONS
5
4. Construction of Ricci solitons
In this section we will present three methods for the construction of a Ricci
soliton on a Riemannian manifold with the D-isometric deformation.
For our first construction of Ricci soliton we consider the case where the potential
field V be pointwise collinear with the vector field ξ i.e. V = f ξ, where f is a
function on M . We compute
˜ X V, Y ) + g̃ ∇
˜ Y V, X
(LV g̃)(X, Y ) = g̃(∇
˜ X (f ξ), Y + g̃ ∇
˜ Y (f ξ), X
= g̃ ∇
(4.1)
=
2f g(∇X ξ, Y ) + 2X(f )η(Y ) + 2Y (f )η(X).
Replacing (3.1), (3.5) and (4.1) in (2.9), we obtain
(LV g̃)(X, Y ) + 2S̃(X, Y ) − 2λg̃(X, Y )
=
(2f + divξ)g(∇X ξ, Y ) + 2S(X, Y )
−g R(X, ξ)ξ, Y − g(∇X ξ, ∇Y ξ)
−2λg(X, Y ) − 2λη(X)η(Y )
+2X(f )η(Y ) + 2Y (f )η(X)
= g (2f + divξ)∇X ξ + 2QX − R(X, ξ)ξ
−∇∇X ξ ξ − 2λX − 2λη(X)ξ
+2X(f )ξ + 2η(X)gradf, Y .
(4.2)
Thus (M, g̃, f ξ, λ) is a Ricci soliton if and only if
(2f + divξ)∇X ξ + 2QX − R(X, ξ)ξ
(4.3)
−
∇∇X ξ ξ − 2λX − 2λη(X)ξ
+
2X(f )ξ + 2η(X)gradf = 0.
By taking X = ξ in (4.3), we obtain
Qξ − 2λξ + ξ(f )ξ + gradf = 0,
which gives
1
S(ξ, ξ) + ξ(f ).
2
Taking X = ei ⊥ ξ with the equation (4.4), the equation (4.3) becomes
(4.4)
λ=
(4.5) (2f +divξ)∇ei ξ+2Qei −R(ei , ξ)ξ−∇∇ei ξ ξ−S(ξ, ξ)ei −2ξ(f )ei +2ei (f )ξ = 0,
applying η we have
(4.6)
ei (f ) = −S(ei , ξ).
Therefore, summing up the arguments above, we have the following main theorem:
Theorem 4.1. Let (M n , g) be a Riemannian manifold and {ei , ξ}1≤i≤n−1 be an
orthonormal frame on (M, g). Then (M, g̃, f ξ, λ) is a Ricci soliton with f is a
function on M if and only if
(4.7) (2f +divξ)∇ei ξ+2Qei −R(ei , ξ)ξ−∇∇ei ξ ξ−S(ξ, ξ)ei +2ei (f )ξ−2ξ(f )ei = 0.
where ei (f ) = −S(ei , ξ) and λ = 12 S(ξ, ξ) + ξ(f ).
DELLOUM ADEL1 , OUBBICHE NOUR2 AND BELDJILALI GHERICI3
6
5. A class of examples
Let M = Hn × R = {(xi , z) ∈ Rn+1 /xn > 0}, where (xi , z)1≤i≤n are standard
co-ordinate in Rn+1 . Let {ei , ξ}1≤i≤n be linearly independent vector fields given
by
∂
∂
ei = xn
,
ξ=
.
∂xi
∂z
We define a Riemannian metric g by
g=
n
1 X 2
dx + dz 2 .
x2n i=1 i
Let ∇ be the Riemannian connection of g, then we have
[ei , en ] = −ei ,
[ei , ξ] = [ei , ej ] = 0,
∀i ̸= j ∈ {1, .., n − 1}.
By using the Koszul formula for the Riemannian metric g,
2g(∇ei ej , ek ) = −g(ei , [ej , ek ]) + g(ej , [ek , ei ]) + g(ek , [ei , ej ]),
the non zero components of the Levi-Civita connection corresponding to g are given
by:
∇ ei e i = e n ,
∇ei en = −ei ,
∀i ∈ {1, .., n − 1}.
The non-vanishing curvature tensor R components are computed as
R(ei , ej )ej = −ei
R(ei , en )ei = en ,
R(ei , en )en = −ei ,
∀i ̸= j ∈ {1, .., n−1}.
The Ricci operator Q and the Ricci curvature S components are computed as
Qei = (1 − n)ei ,
S(ei , ej ) = (1 − n)δij ,
S(ξ, ξ) = 0,
∀i ∈ {1, .., n}.
Since divξ = 0, using formula (4.7), we obtain the following equation
n − 1 + ξ(f ) ei − ei (f )ξ = 0.
Knowing that {ei }1≤i≤n and ξ are linearly independent vector fields then we get
ξ(f ) = 1 − n
fz = 1 − n
⇔
ei (f ) = 0
fi = 0
where fi =
∂f
∂xi
and fz =
∂f
∂z
which gives
f = (1 − n)z + a,
a ∈ R.
Based on Theorem 4.1, we declare that (M, g̃, f ξ, λ) is a Ricci soliton shuch that
g̃ =
n
1 X 2
dx + 2dz 2 ,
x2n i=1 i
f = (1 − n)z + a,
λ=
1
S(ξ, ξ) + ξ(f ) = 1 − n.
2
Notice that for n = 1 the Ricci soliton is steady and for n > 1 it is expanding.
Proposition 5.1. Let (M n , g) be a Riemannian manifold of constant curvature
c and {ei , ξ}1≤i≤n−1 be an orthonormal frame on (M, g). Then (M, g̃, f ξ, λ) is a
Ricci soliton if and only if
(5.1)
(2f + divξ)∇ei ξ + c(n − 2)ei − ∇∇ei ξ ξ − 2ξ(f )ei = 0.
where λ = 2c (n − 2) + ξ(f ), ei (f ) = 0 and ξ(f ) = constant.
NEW METHODS OF CONSTRUCTION FOR RICCI SOLITONS
7
Proof. Let (M n , g) be a Riemannian manifold of constant curvature c is known as
real space form and its curvature tensor is given by
(5.2)
R(X, Y )Z = c g(Y, Z)X − g(X, Z)Y ,
for all vector field X, Y and Z of M . One can easily check that
R(ei , ξ)ξ = cei ,
Qei = c(n − 1)ei ,
S(ξ, ξ) = c(n − 2),
S(ei , ξ) = 0.
Substituting into formula (4.7), we get the required.
□
A second technique to build a Ricci soliton from another Ricci soliton. Suppose
that (M, g, f ξ, λ) is a Ricci soliton then, the equation (4.2) becomes
(Lf ξ g̃) + 2S̃ − 2λg̃ (X, Y ) = (Lf ξ g) + 2S − 2λg (X, Y )
+g divξ∇X ξ − R(X, ξ)ξ − ∇∇X ξ ξ
(5.3)
−2λη(X)ξ + X(f )ξ + η(X)gradf, Y
Suppose that (M, g̃, f ξ, λ) is a Ricci soliton, we get
(5.4)
divξ∇X ξ − R(X, ξ)ξ − ∇∇X ξ ξ − 2λη(X)ξ + X(f )ξ + η(X)gradf = 0.
Setting X = ξ and applying η we have
(5.5)
λ = ξ(f ).
Again, taking X = ei we obtain
(5.6)
divξ∇ei ξ − R(ei , ξ)ξ − ∇∇ei ξ ξ + ei (f )ξ = 0,
and applying η we find
(5.7)
ei (f ) = 0,
Theorem 5.1. Let (M, g, f ξ, λ) be a Ricci soliton and {ei , ξ}1≤i≤n−1 be an orthonormal frame on (M, g). If
(5.8)
divξ∇ei ξ − R(ei , ξ)ξ − ∇∇ei ξ ξ = 0,
then (M, g̃, f ξ, λ) is a Ricci soliton on the deformed manifold with ei (f ) = 0 and
λ = ξ(f ).
By considering the above two Theorems 4.1 and 5.1, it is proved that there exists
an infinite number of Ricci solitons (M, gm , f ξ, λ) where gm = g + mη ⊗ η.
We start with the construction of the Ricci soliton (M, g1 = g + η ⊗ η, f ξ, λ)
from a Riemanian manifold (M, g) using the techniques of Theorem 4.1, then using
Theorem 5.1 we obtain a second Ricci soliton (M, g2 = g1 +η ⊗η, f ξ, λ). Continuing
to use the theorem 5.1 with each new Ricci soliton we get another Ricci soliton,
after m times we get the Ricci soliton (M, gm = g + mη ⊗ η, f ξ, λ).
For the third construction of Ricci soliton we consider the case where V is orthogonal to ξ i.e. η(V ) = 0. We compute
˜ X V, Y ) = g̃ ∇X V + 1 g(∇X ξ, V )ξ, Y
g̃(∇
2
1
= g̃(∇X V, Y ) + g(∇X ξ, V )g̃(ξ, Y ),
2
knowing that g̃ = g + η ⊗ η and g(∇X ξ, V ) = −η ∇X V ), we get
˜ X V, Y ) = g(∇X V, Y ),
g̃(∇
8
DELLOUM ADEL1 , OUBBICHE NOUR2 AND BELDJILALI GHERICI3
then
(LV g̃)(X, Y )
(5.9)
=
g(∇X V, Y ) + g(∇Y V, X)
=
(LV g)(X, Y ).
Replacing (3.1), (3.5) and (5.9) in (2.9), we obtain
LV g̃ + 2S̃ − 2λg̃ (X, Y ) = LV g + 2S − 2λg (X, Y )
(5.10)
− g R(X, ξ)ξ + ∇∇X ξ ξ − divξ∇X ξ + 2λη(X)ξ, Y .
If (M, g, V, λ) is a Ricci soliton, the above equation takes the form
LV g̃ + 2S̃ − 2λg̃ (X, Y ) = −g R(X, ξ)ξ + ∇∇X ξ ξ − divξ∇X ξ + 2λη(X)ξ, Y .
Thus (M, g̃, V, λ) is a Ricci soliton if and only if
(5.11)
R(X, ξ)ξ + ∇∇X ξ ξ − divξ∇X ξ + 2λη(X)ξ = 0.
By taking X = ξ in (5.11), we obtain
(5.12)
λ = 0,
Substituting equation (5.12) in (5.11) we get
(5.13)
R(X, ξ)ξ + ∇∇X ξ ξ − divξ∇X ξ = 0.
Substituting equations (5.12) and (5.13) in (3.5), we obtain
(5.14)
S̃ = S.
Then we get
(5.15)
LV g̃ + 2S̃ − 2λg̃ = LV g + 2S − 2λg − 2λ η ⊗ η.
Thus we have the following:
Theorem 5.2. Let (M, g, V, λ) be a Ricci soliton with the potential vector field V
orthogonal to ξ. If
(5.16)
R(ei , ξ)ξ + ∇∇ei ξ ξ − divξ∇ei ξ = 0,
for all X vector field on M then (M, g̃, V, λ) is a steady Ricci Soliton.
Example 5.1. (3D cigar soliton)
Let M = S1 × R × R = {(x, r, t) ∈ R3 /x > 0}. and let {e1 , e2 , e3 } be linearly
independent vector fields given by
∂
∂
1 ∂
e1 = (1 + x2 ) ,
e2 =
,
e3 =
.
∂x
∂r
x ∂t
and {θ1 , θ2 , θ3 } be the dual frame of differential 1-forms such that
1
θ1 =
dx,
θ2 = dr,
θ3 = xdt.
1 + x2
P3
We define a Riemannian metric g by g = i=1 θi ⊗ θi , That is the form


1
0 0
(1+x2 )2
g=
0
1 0 .
0
0 x2
The potential vector field is given by V = gradf = 2x(1 + x2 )e1 where the potential
function is f = ln(1 + x2 ).
NEW METHODS OF CONSTRUCTION FOR RICCI SOLITONS
With simple computations one
we find

1
S = −2(1 + x2 )  0
0
9
can prove that the condition (5.16) is satisfied and
0
0
0


0
1
0  , LV g = 4(1 + x2 )  0
1
0
0
0
0

0
0 .
1
We can easily notice that LV g + 2S = 0 which implies that (M, g, V ) is a steady
Ricci soliton.
We take ξ = e2 to ensure the conditions V ⊥ ξ and dθ2 = 0 then we deform the
metric as follows


1
0 0
(1+x2 )2
g̃ = g + θ2 ⊗ θ2 = 
0
2 0 .
0
0 x2
Using formulas (5.9) and (5.14) we conclude that (M, g̃, V ) is a steady Ricci soliton
too.
6. Ricci solitons on 3-dimensional Riemannian manifolds
In this section, we’ll focus on the 3-dimensional Riemannian manifolds (M, g).
So, in the following, we consider {e1 , e2 , ξ} an orthonormal frame on (M, g) and
g̃ = g + η ⊗ η where η is a closed 1-form on M such that η(ξ) = 1.
Recall that, the conformal curvature tensor is vanishngs in the 3-dimensional
Riemannian manifold, results in [1]
g(Y, Z)QX − g(X, Z)QY + S(Y, Z)X − S(X, Z)Y
r
(6.1)
g(Y, Z)X − g(X, Z)Y .
−
2
where r indicates to the scalar curvature. Putting X = ei and Y = Z = ξ in (6.1)
we get
r
(6.2)
R(ei , ξ)ξ = Qei + S(ξ, ξ)ei − S(ei , ξ)ξ − ei .
2
Suppose that (M, g̃, f ξ, λ) is a Ricci soliton. Then, from (4.7) and (6.2), we have
(6.3)
r
∇∇ei ξ ξ = (2f + divξ)∇ei ξ + Qei − 2S(ξ, ξ)ei + S(ei , ξ)ξ − 2ξ(f )ei + 2ei (f )ξ + ei .
2
Substituting this in (4.7), we get
r
(6.4)
R(ei , ξ)ξ − Qei + S(ei , ξ)ξ − S(ξ, ξ)ei + ei = 0.
2
This leads to the following:
R(X, Y )Z
=
Theorem 6.1. Let (M 3 , g) be a Riemannian manifold and let {ei , ξ}1≤i≤2 be an
orthonormal frame on (M 3 , g). Then, (M, g̃, f ξ, λ) is a Ricci soliton if and only if
r
(6.5)
R(ei , ξ)ξ − Qei + S(ei , ξ)ξ − S(ξ, ξ)ei + ei = 0,
2
where ei (f ) = −S(ei , ξ) and λ = 12 S(ξ, ξ) + ξ(f ).
Example 6.1. Let M = S2 × R = {(x, y, t) ∈ R3 }. Let {e1 , e2 , ξ} be the frame of
vector fields on M given by
∂
∂
∂
e1 = (1 + x2 + y 2 ) ,
e2 = (1 + x2 + y 2 ) ,
ξ= .
∂x
∂y
∂t
10
DELLOUM ADEL1 , OUBBICHE NOUR2 AND BELDJILALI GHERICI3
We define a Riemannian metric g by
1
g=
(dx2 + dy 2 ) + dt2 .
2
(1 + x + y 2 )2
Let ∇ be the Riemannian connection of g, then we have
[e1 , e2 ] = 2xe2 − 2ye1 ,
[e1 , ξ] = [e2 , ξ] = 0.
By using the Koszul formula for the Riemannian metric g, the non zero components
of the Levi-Civita connection corresponding to g are given by:
∇e1 e1 = 2ye2 ,
∇e1 e2 = −2ye1 ,
∇e2 e1 = −2xe2 ,
∇e2 e2 = 2xe1 .
The non-vanishing curvature tensor R components are computed as
R(e1 , e2 )e1 = −4e2
R(e1 , e2 )e2 = 4e1 .
The Ricci operator Q, the Ricci curvature S components and the scalar curvature
r are computed as
Qei = 4ei ,
S(ei , ei ) = 4,
S(ξ, ξ) = 0
r = 8.
Now, we can easily see that condition (6.5) is satisfied which allows us to conclude
that (M, g̃, f ξ, λ) is a Ricci soliton with λ = 4.
Thus, we get f = 4t + c with c ∈ R. To check we give:




0 0 0
4 0 0
S̃ =  0 4 0  and LV g̃ =  0 0 0  .
0 0 8
0 0 0
Corollary 6.1. Let (M 3 , g) be an Einstein manifold i.e S = ρg with ρ is a certain
constant. Then (M, g̃, f ξ, λ) is a Ricci soliton with λ = ρ2 + ξ(f ) if and only if
ρ
(6.6)
R(ei , ξ)ξ = ei ,
2
where f is a function on M such that ei (f ) = 0 and ξ(f ) = constant.
Proof. Suppose that (M, g) is an Einstein manifold. Using (2.8) with simple calculations we find
S(ei , ξ) = 0,
S(ξ, ξ) = ρ,
Qei = ρei ,
r = 3ρ.
Substituting these arguments in (6.5), we obtain
ρ
R(ei , ξ)ξ = ei ,
2
□
It’s easy to notice that the condition (6.5) is fulfilled for any 3-dimentional flat
Riemannian manifold with λ = ξ(f ). This remark can be generalized as follows:
Theorem 6.2. Let (M, g) be a 3-dimentional Riemannian manifold of constant
curvature c. Then, (M, g̃, f ξ, λ) is a Ricci soliton with λ = c + ξ(f ) and f is a
function on M such that ξ(f ) = constant and ei (f ) = 0.
Proof. Let (M 3 , g) be a Riemannian manifold of constant curvature c. The Riemannian curvature tensor of M is given by the following formula (5.2). One can
easily check that
R(ei , ξ)ξ = cei ,
Qei = 2cei ,
S(ξ, ξ) = 2c,
Substituting into formula (6.5), we get the required.
S(ei , ξ) = 0,
r = 6c.
□
NEW METHODS OF CONSTRUCTION FOR RICCI SOLITONS
11
We point out here that ξ(f ) is not always a constant but it can be a smooth
function and then we say that (M, g̃, f ξ, λ) is an almost Ricci soliton.
Example 6.2. Let M = R2 × R = {(x, y, t) ∈ R3 }. Let {e1 , e2 , ξ} be the frame of
vector fields on M given by
e1 = e−ct
∂
,
∂x
e2 = e−ct
∂
,
∂y
∂
,
∂t
ξ=
η = ξ ♭ = dt.
where c ∈ R. We define a Riemannian metric g by
g = e2ct (dx2 + dy 2 ) + dz 2 .
Let ∇ be the Riemannian connection of g, then we have
[e1 , e2 ] = 0,
[e1 , ξ] = ce1 ,
[e2 , ξ] = ce2 .
By using the Koszul formula for the Riemannian metric g, the non zero components
of the Levi-Civita connection corresponding to g are given by:
∇e1 e1 = ∇e2 e2 = −cξ,
∇e1 ξ = ce1 ,
∇e2 ξ = ce2 .
The non-vanishing curvature tensor R components are computed as
R(e1 , e2 )e1 = −R(e2 , ξ)ξ = c2 e2 ,
R(e1 , e2 )e2 = R(e1 , ξ)x i = −c2 e1 ,
R(e1 , ξ)e1 = R(e2 , ξ)e2 = c2 ξ.
One can check that (M, g) is of constant curvature −c2 .
With simple computations one can get



−c2
0
0
2cf
−c2
0  and LV g̃ =  0
S̃ =  0
0
0
−c2
0
where f3 =
∂f
∂t .
0
2cf
0

0
0 ,
2f3
So, (M, g̃, f ξ, λ) is an almost Ricci soliton if and only if
2cf − 2c2 = λ
2f3 − 2c2 = λ,
implies f3 = cf then f = aect with a ∈ R. Notice that λ = −c2 + ξ(f ) = c(aect − c)
therefore the structure is an almost Ricci soliton. For c = 0 we get a steady Ricci
soliton and for a = 0 we get λ = −c2 i.e. an expanding Ricci soliton.
Three open questions:
1) What would happen if the vector field ξ was not unitary?.
2) The existence of a global unit vector field ξ is assured in the almost contact metric
manifolds, do these results appear in such manifolds in particular the Kenmotsu
manifold?
3) Do the techniques introduced in this work allow the construction of Yamabe
solitons?.
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DELLOUM ADEL1 , OUBBICHE NOUR2 AND BELDJILALI GHERICI3
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1,2 Department of Mathematics, University Mustapha Stambouli, Mascara, 29000, Algeria.
Email address: [email protected]
Email address: [email protected]
3 Laboratory
of Quantum Physics and Mathematical Modeling (LPQ3M), University
of Mascara, Algeria.
Email address: [email protected]