Ricci Soliton

NEW METHODS OF CONSTRUCTION FOR RICCI SOLITONS
DELLOUM ADEL1, OUBBICHE NOUR2AND 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 Vgenerates 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 Vis the gradient of a
potential function.
This paper focused on the geometry of transformations that is related to the
(n−1)−dimensional distribution Dof 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,Sand rwe denote respectively the Riemannian curvature tensor, the
Ricci operator, the Ricci tensor and the scalar curvature of a Riemannian manifold
(Mn, g). Then R,Q S and rare defined by:
(2.1) R(X, Y )Z=∇X∇YZ− ∇Y∇XZ− ∇[X,Y ]Z,
(2.2) QX =
n
X
i=1
R(X, ei)ei,
2000 Mathematics Subject Classification. 53C15, 53C25, 53D10.
Key words and phrases. Riemannian manifolds, Ricci solitons, Deformation of metrics.
1

2 DELLOUM ADEL1, OUBBICHE NOUR2AND BELDJILALI GHERICI3
(2.3) S(X, Y ) = g(QX, Y ) =
n
X
i=1
gR(X, ei)ei, Y ,
(2.4) r=
n
X
i=1
S(ei, ei),
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 Xon M, we have
(2.5) X=
n
X
i=1
g(X, ei)ei,
the divergence of a vector field Vis defined by:
(2.6) divV=
n
X
i=1
g(∇eiV, ei),
and for a unit closed 1-form η(i.e. dη= 0) one can get
(2.7) g(∇Xξ, Y ) = g(∇Yξ, X), g(∇Xξ, ξ) = 0 and ∇ξξ= 0.
where ηbe the g-dual of ξwhich means η(X) = g(X, ξ) for all vector field Xon
M. We recall that (Mn, g), n≥3, is said to be Einstein if at every x∈Mits Ricci
tensor Shas the form
(2.8) S=ρg, ρ ∈R.
(For more details, see for example [19]).
A smooth vector field Von a Riemannian manifold (M, g) is said to define a
Ricci soliton if it satisfies the following Ricci soliton equation:
(2.9) (LVg)(X, Y )+2S(X, Y ) = 2λg(X, Y ),
where LVgdenotes the Lie derivative of galong a vector field Vand λis a constant.
We shall denote a Ricci soliton by (M, g, V, λ). If λis a smooth function on M,
we say that (M, V, λ, g) is an almost Ricci soliton. We call the vector field Vthe
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 Vis zero or Killing, in which case the metric is Einsteinian (see, for instance,
[7, 11, 17]).
3. Deformations of Metrics
Let Mnbe a Riemannian manifold equipped with Riemannian metric g, and η
a unit global closed 1-forme on M. We denote by ξthe vector field corresponding
to η, i.e for all Xvector field on M,
η(X) = g(ξ, X)and η(ξ) = 1.
We define on Ma Riemannian metric, denoted ˜g, by
(3.1) ˜g(X, Y ) = g(X, Y ) + η(X)η(Y).
The equation η= 0 defines a (n−1)-dimensional distribution Don M. Then, we
have ˜g(ξ, ξ)=2
˜g(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= dfwhere f∈ C∞(M).
This case has been studied in [3] and [4].
In [13], Innami proved that Madmits a non constant affine function if and only
if Msplits as a Riemannian product M=N×R. In our situation, since dη= 0
then ξis locally of type gradient, what means that at each point pon Mthere
exists a function fsuch that ξ=∇fon a neighborhood at p, where ∇fdenotes
the gradient vector field of f.
Proposition 3.1. Let ∇and ˜ ∇denote the Riemannian connections of gand ˜g
respectively. Then, for all X, Y vector fields on M, we have the relation:
(3.2) ˜ ∇XY=∇XY+1
2g(∇Xξ, Y )ξ.
Proof. Using Koszul’s formula for the metric ˜g,
2˜g(˜ ∇XY, Z) = X˜g(Y, Z) + Y˜g(Z, X)−Z˜g(X, Y )
−˜g(X, [Y, Z]) + ˜g(Y, [Z, X]) + ˜g(Z, [X, Y ],
one can obtain
˜g(˜ ∇XY, Z) = ˜g(∇XY, Z)
+1
2(∇Xη)Y+ (∇Yη)Xη(Z)
Knowing that dη= 0, we get
˜g(˜ ∇XY, Z) = ˜g(∇XY, Z)+(∇Xη)(Y)η(Z),
with
(∇Xη)(Y) = g(∇Xξ, Y ).
On the other hand, we have
η(Z) = g(ξ, Z)
= ˜g(ξ, Z)−η(Z),
which gives
η(Z) = 1
2˜g(ξ, Z).
Therefore
˜
∇XY=∇XY+1
2g(∇Xξ, Y )ξ.
□
Theorem 3.1. Let ˜ R,˜ Qand ˜ Sbe the Riemannian curvature tensor, the Ricci
operator and the Ricci tensor of the Riemannian manifold (Mn,˜g)respectively.
For all X, Y and Zvector fields on M, we have
2˜ R(X, Y )Z= 2R(X, Y )Z−gR(X, Y )Z, ξξ
+g(∇Yξ, Z)∇Xξ−g(∇Xξ, Z)∇Yξ,(3.3)
(3.4) 2 ˜ QX = 2QX −R(X, ξ)ξ−S(X, ξ)ξ+ (divξ)∇Xξ− ∇∇Xξξ,
(3.5) 2 ˜ S(X, Y )=2S(X, Y )−gR(X, ξ)ξ, Y + divξg(∇Xξ, Y )−g(∇Xξ, ∇Yξ).

4 DELLOUM ADEL1, OUBBICHE NOUR2AND BELDJILALI GHERICI3
Proof. By the definition of the curvature tensor ˜ R
(3.6) ˜ R(X, Y )Z=˜ ∇X˜ ∇YZ−˜ ∇Y˜ ∇XZ−˜ ∇[X,Y ]Z,
using formula 3.2, the first term of (3.6) is given by
˜
∇X˜ ∇YZ=˜ ∇X∇YZ+1
2g(∇Yξ, Z)ξ
=˜ ∇X∇YZ+1
2g(∇X∇Yξ, Z)ξ
+1
2g(∇Yξ, ∇XZ)ξ+1 2g(∇Yξ, Z)˜
∇Xξ
=∇X∇YZ+1
2g(∇Xξ, ∇YZ)ξ
+1
2g(∇X∇Yξ, Z)ξ+1 2g(∇Yξ, ∇XZ)ξ
+1
2g(∇Yξ, Z)∇Xξ.(3.7)
With the similar method, the second term of (3.6) is given by
˜
∇Y˜ ∇XZ=∇Y∇XZ+1
2g(∇Yξ, ∇XZ)ξ
+1
2g(∇Y∇Xξ, Z)ξ+1 2g(∇Xξ, ∇YZ)ξ
+1
2g(∇Xξ, Z)∇Yξ.(3.8)
Knowing that dη= 0, the last term of (3.6) becomes
˜
∇[X,Y ]Z=∇[X,Y ]+1
2g(∇[X,Y ]ξ, Z)ξ
=∇[X,Y ]+1
2g(∇∇XYξ, Z)ξ−1 2g(∇∇YXξ, Z)ξ
=∇[X,Y ]+1
2g(∇Zξ, ∇XY)ξ−1 2g(∇Zξ, ∇YX)ξ.(3.9)
Formula (3.3) follows from equations (3.6)-(3.9).
To show the second formula (3.4), consider {ξ, ei}2≤i≤nthe orthonormal basis on
Mwith respect to the metric g, it is easy to prove that {1
√2ξ, ei}2≤i≤nis an
orthonormal frame on M with respect to the metric ˜g. Using formula (2.2), we
have
˜
QX =1
2˜
R(X, ξ)ξ+
n
X
i=2
˜
R(X, ei)ei
=−1
2˜
R(X, ξ)ξ+
n
X
i=1
˜
R(X, ei)ei
=−1
2R(X, ξ)ξ+
n
X
i=1 R(X, ei)ei−1
2gR(X, ei)ei, ξξ
+1
2g(∇eiξ, ei)∇Xξ−1 2g(∇Xξ, ei)∇eiξ.(3.10)
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 Vbe pointwise collinear with the vector field ξi.e. V=fξ, where fis a
function on M. We compute
(LV˜g)(X, Y ) = ˜g(˜ ∇XV, Y ) + ˜g˜ ∇YV, X
= ˜g˜ ∇X(fξ), Y + ˜g˜ ∇Y(fξ), X
= 2fg(∇Xξ, Y )+2X(f)η(Y)+2Y(f)η(X).(4.1)
Replacing (3.1), (3.5) and (4.1) in (2.9), we obtain
(LV˜g)(X, Y )+2˜ S(X, Y )−2λ˜g(X, Y ) = (2f+ divξ)g(∇Xξ, Y )+2S(X, Y )
−gR(X, ξ)ξ, Y −g(∇Xξ, ∇Yξ)
−2λg(X, Y )−2λη(X)η(Y)
+2X(f)η(Y)+2Y(f)η(X)
=g(2f+ divξ)∇Xξ+ 2QX −R(X, ξ)ξ
−∇∇Xξξ−2λX −2λη(X)ξ
+2X(f)ξ+ 2η(X)gradf, Y .(4.2)
Thus (M, ˜g, fξ, λ) is a Ricci soliton if and only if
(2f+ divξ)∇Xξ+ 2QX −R(X, ξ)ξ− ∇∇Xξξ−2λX −2λη(X)ξ
+ 2X(f)ξ+ 2η(X)gradf= 0.(4.3)
By taking X=ξin (4.3), we obtain
Qξ −2λξ +ξ(f)ξ+ gradf= 0,
which gives
(4.4) λ=1
2S(ξ, ξ) + ξ(f).
Taking X=ei⊥ξwith the equation (4.4), the equation (4.3) becomes
(4.5) (2f+divξ)∇eiξ+2Qei−R(ei, ξ)ξ−∇∇eiξξ−S(ξ, ξ)ei−2ξ(f)ei+2ei(f)ξ= 0,
applying ηwe have
(4.6) ei(f) = −S(ei, ξ).
Therefore, summing up the arguments above, we have the following main theo-
rem:
Theorem 4.1. Let (Mn, g)be a Riemannian manifold and {ei, ξ}1≤i≤n−1be an
orthonormal frame on (M, g). Then (M, ˜g, fξ, λ)is a Ricci soliton with fis a
function on Mif and only if
(4.7) (2f+divξ)∇eiξ+2Qei−R(ei, ξ)ξ−∇∇eiξξ−S(ξ, ξ)ei+2ei(f)ξ−2ξ(f)ei= 0.
where ei(f) = −S(ei, ξ)and λ=1
2S(ξ, ξ) + ξ(f).