On Ricci Soliton metrics conformally equivalent to left invariant metrics

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by  Hamid Reza Salimi Moghaddam

A GENERALIZATION OF INVARIANT RIEMANNIAN METRICS AND
ITS APPLICATIONS TO RICCI SOLITONS
H.R. SALIMIMOGHADDAM
Abstract. InthispaperweintroduceageneralizationoftheconceptofinvariantRiemannian
4 metric. This new definition can help usto find Riemannian manifolds with special geometric
1
properties. As an application, by this method, we give many explicit examples of shrinking,
0
steady and expandingRicci solitons.
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n
a
J
3 1. Introduction

] The geometry of Lie groups and homogeneous spaces equipped with invariant Riemannian
G
metrics is a basic part of Riemannian geometry which have been considered by many great
D
mathematicians (see [1], [2], [3], [5], [7], [9]). This importance comes from the simplicity of the
.
h
structures of these spaces (versus the general case) and their applications in other fields such
t
a as theoretical physics. Although, mathematicians have found many interesting and important
m
applications of these spaces but it seems that such spaces are too special, because the study
[
of their geometry is limited to the study of an inner product on a vector space (Lie algebra).

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For example anyone knows that every commutative Lie group equipped with a left invariant
v
4 Riemannian metric is flat, but one may want to work on a commutative Lie group such that
4
its geometric structures have an interaction with its group structure and is not flat. In the
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0 present article we give a very simple generalization of invariant Riemannian metrics on Lie
.
1 groups such that the inner product on any tangent space is controlled by a smooth function
0
which varies on the base manifold. At the first look it seems that it is a slight generalization,
4
1 but we can see by this method we can construct many examples of Riemannian manifolds
:
v with special curvature properties or other applications. For example by this generalization we
i
X construct many different examples of shrinking, steady and expanding Ricci solitons, which
r show the significance of our definition.
a
A Ricci soliton with expansion constant λ is a Riemannian manifold (M,g) together with a
smooth vector field X on M such the following equation holds;
(1.1) L g = 2(λg−Ric(g)),
X
where L g denotes the Lie derivative of g with respect to X. For a Ricci soliton the Ricci flow
X
equation
d
(1.2) g = −2Ric(g )
t t
dt
with initial condition g0 = g has the solution
(1.3) g = (1−2λt)φ g,
t ∗t
Key words and phrases. f−invariant Riemannian metric, Lie group, Ricci soliton.
AMS 2010 Mathematics Subject Classification: 22E60, 53C44, 53C21.
1
2 H.R.SALIMIMOGHADDAM
in the t-interval on which 1−2λt > 0, where φ :M −→ M is the time t flow of X.
t
The Ricci soliton is called shrinking if λ > 0, steady if λ = 0 and expanding if λ < 0. If
X = gradΦ for some smooth real-valued function Φ on M, then the Ricci soliton is called
gradient and the function Φ is called a potential function of the Ricci soliton.
In the recent years the geometry of homogeneous Ricci solitons is studied by some mathe-
maticians (for example see [4] and [6]). It is shown that any shrinking or steady homogeneous
Ricci soliton must be either a compact Einstein manifold or the productof a compact Einstein
manifold with Euclidean space (see [4], [8] and [11]). On the other hand all known examples
of expanding homogeneous Ricci solitons are isometric to algebraic Ricci soliton metrics on
solvable Lie groups (for more details see [4] and [6]).
Inthispaperbyusingtheconcept f−invariantRiemannianmetricon Liegroupswegive many
examples of shrinking,steady andexpandingRicci solitons whichareconstructed on (solvable)
Liegroups. Theotherbenefitofourmethodisthatanyone, byusingasoftware(suchasMaple)
and our method, can find new examples of Ricci solitons with different properties.
2. f−invariant Riemannian metrics
In this paper, the statement Riemannian metric can be replaced with the statement semi-
Riemannian.
Definition 2.1. Let f bea smooth real positive function on a Liegroup G such that f(e)= 1,
where e is the unit element of G. Suppose that < ·,· > is a Riemannian metric on G. The
metric < ·,· > is said to be f−left invariant if
f(ba)
(2.1) L <X ,Y > :=< L X ,L Y > =< X ,Y > . ,
∗b a a a b∗ a b∗ a ba a a a f(a)
for all a,b ∈ G. Similarly, the Riemannian metric < .,. > is said to be f−right invariant if
f(ab)
(2.2) R < X ,Y > :=< R X ,R Y > =< X ,Y > . ,
b∗ a a a b∗ a b∗ a ba a a a f(a)
for all a,b ∈ G.
Definition 2.2. A Riemannian metric which is both f−left invariant and f−right invariant
is called f−bi-invariant. In this case we can see f must be symmetric which means that
f(ab)= f(ba), for all a,b ∈ G.
Theorem 2.3. Let f be a smooth positive real function on a Lie group G, such that f(e) = 1.
Then G admits a f−leftinvariant Riemannian metric (f−rightinvariant Riemannian metric).
Proof. Suppose that < .,. > is any inner product on T G. For any a ∈ G let
e e
(2.3) < Xa,Ya >a:= f(a)< La−1 Xa,La−1 Ya >e .
∗ ∗
Notice that < .,. > is smooth on TG because the multiplication of G, and f are smooth. Also
< .,. > is f−left invariant because
L∗b < Xa,Ya >a = <Lb La (La−1 Xa),Lb La (La−1 Ya)>ba
∗ ∗ ∗ ∗ ∗ ∗
(2.4) = <Lba (La−1 Xa),Lba (La−1 Ya) >ba
∗ ∗ ∗ ∗
f(ba)
= f(ba)< La−1∗Xa,La−1∗Ya >e= f(a) < Xa,Ya >a .
A GENERALIZATION OF INVARIANT RIEMANNIAN METRICS AND ITS APPLICATIONS TO RICCI SOLITONS3
The proof of the f−right invariant case is exactly similar. (cid:3)
Theorem 2.4. Let f : G −→ (R+,.) be a homomorphism of Lie groups. The Lie group G
has a f−bi-invariant Riemannian metric if and only if g = T G has an inner product which is
e
Ad−invariant.
Proof. Let < .,. > be a f−bi-invariant Riemannian metric on the Lie group G. Suppose that
< .,. > is the inner product induced by < .,. > on T G.
e e
(2.5) < Ad X ,Ad Y > = R < L X ,L Y >
a e a e e a∗−1 a e a e a
∗ ∗
= f(a) < Ra−1 Xe,Ra−1 Ye >a−1
∗ ∗
1
= f(a)f(a ) < X ,Y > =< X ,Y > .
− e e e e e e
Therefore < .,. > is Ad−invariant.
e
Conversely, let < .,. > be an Ad−invariant inner producton T G. Now we define a Riemann-
e e
ian metric on G as follows
(2.6) < Xa,Ya >a:= f(a) < La−1 Xa,La−1 Ya >e,
∗ ∗
obviously < .,. > is a f−left invariant Riemannian metric on G.
Also it is f−right invariant because
Rb∗ < Xa,Ya >a = Rb∗ < La (La−1 Xa),La (La−1 Ya) >a
∗ ∗ ∗ ∗
= Rb∗L∗a < La−1 Xa,La−1 Ya >e
∗ ∗
(2.7) = (LaRb)∗(LbRb−1)∗ < La−1 Xa,La−1 Ya >e
∗ ∗
= (LbRb−1RbLa)∗ < La−1 Xa,La−1 Ya >e
∗ ∗
= L∗aL∗b < La−1 Xa,La−1 Ya >e
∗ ∗
= f(b)f(a)< La−1 Xa,La−1 Ya >e
∗ ∗
f(ab)
= f(b)< X ,Y > = <X ,Y > .
a a a a a a
f(a)
(cid:3)
Remark 2.5. There is not any nontrivial f−bi-invariant Riemannian metric on a compact Lie
group G such that f : G−→ (R+,.) is a homomorphism of Lie groups.
Theorem 2.6. Let G be a connected Lie group equipped with a f−left invariant Riemannian
metric < .,. >. Then the following are equivalent:
(1) < .,. > is f−right invariant, hence f−bi-invariant.
(2) < .,. > is Ad(G)−invariant.
(3) f(a) < ζ Xa,ζ Ya >a−1= f(a−1) < Xa,Ya >a, for all a ∈ G, where ζ is the inversion
∗ ∗
map.
(4) < X,[Y,Z] >=<[X,Y],Z >, for all X,Y,Z ∈ g.
4 H.R.SALIMIMOGHADDAM
Proof. (1) −→ (2)
< AdaXe,AdaYe >e = < La Ra−1 Xe,La Ra−1 Ye >e
∗ ∗ ∗ ∗
f(aa 1)
−
(2.8) = < Ra−1∗Xe,Ra−1∗Ye >a−1 f(a−1)
f(ea 1) f(e)
−
= < X ,Y > =<X ,Y > .
e e e f(e) f(a 1) e e e
−
(2) −→ (1)
f(a 1)
−
< Ra−1∗Xe,Ra−1∗Ye >a−1 = < La∗Ra−1∗Xe,La∗Ra−1∗Ye >e f(aa−1)
1
(2.9) = < Ad X ,Ad Y > f(a )
a e a e e −
1
= < X ,Y > f(a ).
e e e −
(1) −→ (3)
< ζ Xa,ζ Ya >a−1 = <Ra−1 ζ La−1 Xa,Ra−1 ζ La−1 Ya >a−1
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
1
= <ζ La−1 Xa,ζ La−1 Ya >e f(a− )
∗ ∗ ∗ ∗
1
(2.10) = <−La−1 Xa,−La−1 Ya >e f(a− )
∗ ∗
f(a 1)
−
= <X ,Y >
a a a
f(a)
(3) −→ (1)
< Ra Xe,Ra Ye > = < ζ La−1 ζ Xe,ζ La−1 ζ Ye >a
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
f(a)
(2.11) = < La−1∗ζ∗Xe,La−1∗ζ∗Ye >a−1 f(a−1)
f(a 1e) f(a)
−
= < ζ X ,ζ Y >
∗ e ∗ e f(e) f(a−1)
= < X ,Y > f(a).
e e e
(2) ←→ (4)
The proof is similar to invariant Riemannian metric case (see Lemma 3 page 302 of [10].). (cid:3)
Corollary 2.7. Let f be a smooth positive real function on a compact Lie group G, such that
f(e) = 1. Then G admits a f−bi-invariant Riemannian metric.
3. Curvature of f−invariant Riemannian metrics
In this article we use the notation {E1,··· ,En} for a set of left invariant vector fields on a
Lie group G which is an orthogonal basis at any point of G and is an orthonormal basis at the
unit element e, with respect to a f−left invariant Riemannian metric < .,. >.
Theorem 3.1. Let G be a Lie group equipped with a f−left invariant Riemannian metric
< .,. >. Suppose that α are structure constants defined by [E ,E ] = n α E . Then
ijk i j k=1 ijk k
P
A GENERALIZATION OF INVARIANT RIEMANNIAN METRICS AND ITS APPLICATIONS TO RICCI SOLITONS5
the sectional curvature K(E ,E ) is given by the following formula:
p q
1
2 2
K(E ,E ) = 2fδ (f +f )−2f(f +f )−4δ f f +2f +2f
p q 4f3 pq qp pq pp qq pq p q p q
(cid:16)
n
(3.1) + (2δ f −f +2fα )(f +2fδ )
rq q r rqq r prp
r=1
X
2 2
−(δ f +fα ) + δ f −δ f +f(α −α )
rp q pqr qr p pq r rpq qrp
−2fα δ f −δ f(cid:0) +f(α +α −α ) , (cid:1)
pqr pr q rq p rqp prq qpr
(cid:17)
where f := E f and f := E E f. (cid:0) (cid:1)
i i ij j i
Proof. The relation [E ,E ]= n α E shows that
i j k=1 ijk k
P 1
(3.2) α = < [E ,E ],E > .
ijk i j k a
f(a)
Therefore we have
2< ∇ E ,E > = E < E ,E > +E < E ,E > −E < E ,E >
Ei j k i j k j i k k i j
(3.3) − < E ,[E ,E ]> + < E ,[E ,E ]> + < E ,[E ,E ]>
i j k j k i k i j
= δ f +δ f −δ f +f(−α +α +α ),
jk i ik j ij k jki kij ijk
and so,
n
1
(3.4) ∇ E = δ f +δ f −δ f +f(α +α −α ) E .
Ei j 2 jk i ik j ij k ijk kij jki k
Xk=1(cid:16) (cid:17)
Now for the curvature tensor we have:
n
1
R(E ,E )E = 2f(δ f −δ f −δ f +δ f )
i j k 4f2 lj ki jk li li kj ik lj
Xl=1n
−2f (δ f −δ f )+2f (δ f −δ f )
i lj k jk l j li k ik l
n
(3.5) + δ f +δ f −δ f +f(α +α −α )
kr j rj k jk r jkr rjk krj
r=1
X(cid:0) (cid:1)
× δ f +δ f −δ f +f(α +α −α )
rl i li r ir l irl lir rli
−(cid:0)δkrfi+δrifk −δikfr +f(αikr +αrik −αk(cid:1)ri)
×(cid:0)δrlfj +δljfr −δjrfl+f(αjrl+αljr −αrlj) (cid:1)
−2(cid:0)fα δ f −δ f +f(α +α −α )(cid:1) E .
ijr lr k rk l rkl lrk klr l
o
On the other hand, (cid:0) (cid:1)
2 2
(3.6) < E ,E ><E ,E > − < E ,E > = f .
p p q q p q
Now the formula of sectional curvature completes the proof. (cid:3)
Remark 3.2. If we consider the left invariant Riemannian metrics then the formula given in
Theorem(3.1)forthesectionalcurvaturereducestoMilnor’sformulagivenin[7]. Itissufficient
to consider f is the constant function f ≡ 1.
6 H.R.SALIMIMOGHADDAM
Remark 3.3. If the Riemannian metric <.,. > is f−bi-invariant then the array α is skew in
ijk
the last two indices for any i. Therefore in this case we have a simpler formula for sectional
curvature.
Theorem3.4. LetGbeacommutative Liegroup equipped with af−leftinvariant Riemannian
metric < .,. >. Then for the sectional curvature we have
n
1 3 1
2 2 2
(3.7) K(E ,E ) = −f(f +f )+ (f +f )− f .
p q 2f3 pp qq 2 p q 2 l
l=1
(cid:0) X (cid:1)
Proof. In formula (3.1) it is sufficient to consider all structure constants are zero. (cid:3)
Theorem 3.5. Consider the assumptions of theorem (3.1). Then the Ricci curvature tensor
Ric is given by the following formula:
n
1
Ric(E ,E ) = 2f(δ f −δ f −f +δ f )−2f (δ f −δ f )+2f (f −δ f )
p q 4f2 jp qj pq jj qp jq jp j jp q pq j p q jq j
Xj=1(cid:16)
n
(3.8) + (δ f +δ f −δ f +f(α +α −α ))(f +2fα )
qr p rp q pq r pqr rpq qrp r jrj
r=1
X
−(δ f +δ f −δ f +f(α +α −α ))
qr j rj q jq r jqr rjq qrj
×(δ f +δ f −δ f +f(α +α −α ))
rj p jp r pr j prj jpr rjp
−2fα δ f −δ f +f(α +α −α ) .
jpr jr q rq j rqj jrq qjr
(cid:17)
(cid:0) (cid:1)
Proof. Assume that e := Ei, for i = 1,··· ,n. Note that, in general case, the vector fields e
i √f i
are not left invariant. We can see the set {e1,··· ,en} is an orthonormal basis at every point
of G, with respect to the f−left invariant metric. Now the equations
n n
1
(3.9) Ric(E ,E ) = fRic(e ,e )= f < R(e ,e )e ,e >= < R(E ,E )E ,E >,
p q p q j p q j j p q j
f
j=1 j=1
X X
together with the formula (3.5) complete the proof. (cid:3)
Remark 3.6. In general case, if f is a positive function on an open set M containing e ∈ G
such that f(e) = 1 then all previous results are true for the Riemannian manifold M.
4. Applications to constructing Ricci Solitons
In this section, by using the concept f−left invariant Riemannian metric, we give some
examples of Ricci solitons constructed on some solvable Lie groups in dimensions two to four.
In each case we show that the equation (1.1) reduces to a system of equations which is more
simpler than (1.1).
4.1. Lie group G = R2. SupposethattheLiegroup G= R2 equippedwith af−leftinvariant
Riemannian metric g such that the set {E1 := ∂ ,E2 := ∂ } is an orthogonal basis at any
∂x ∂y
point and is orthonormal at e = (0,0). Suppose that X = θ ∂ +η ∂ is an arbitrary vector
∂x ∂y
A GENERALIZATION OF INVARIANT RIEMANNIAN METRICS AND ITS APPLICATIONS TO RICCI SOLITONS7
field on G, where θ and η are smooth real functions on G. Then easily we can see the equation
(1.1) reduces to the following system of three equations,
θf +ηf +2fθ = 2(λ−κ)f
x y x
(4.1) θf +ηf +2fη = 2(λ−κ)f
 x y y
 ηx = −θy,
whereκis theGaussian curvatureof G. In fact theRiemannian manifold (G,< ·,· >)together

with the vector field X and expansion constant λ is a Ricci soliton if and only if the system
(4.1) holds. Also we can see X = gradΦ if and only if
Φ = fθ
x
(4.2)
Φ = fη.
( y
Example 4.1. If we let f(x,y) = 1 , λ = 0, θ = −2x and η = −2y then we have the
1+x2+y2
Hamilton’s cigar which is a steady gradient Ricci soliton with Gaussian curvature κ = 2
1+x2+y2
and potential function Φ(x,y) = −ln(1+x2 +y2) (one may want to use the equations (4.1),
(4.2) and (3.7).).
Example 4.2. Suppose that f(x,y) = exp(x+y) and X = ∂ + ∂ . Then equations (4.1),
∂x ∂y
(4.2)and(3.7)showthat wehaveaflatshrinkinggradientRicci soliton withpotential function
Φ = f and expansion constant λ = 1.
Example 4.3. In the previous example if we consider X = ∂ − ∂ then we have a flat steady
∂x ∂y
Ricci soliton which is not gradient.
Remark 4.4. In two previous examples, one may want to work with g 1 or equivalently
−
f(x,y)= exp(−x−y).
4.2. Lie group G= R⋊R+. Now consider the Lie group G = R⋊R+ equippedwith a f−left
invariant Riemannian metric g such that the set {E1 := y∂∂y,E2 := y∂∂x} is an orthogonal
basis at any point and is orthonormal at e = (0,1), where we have considered the natural
coordinates (x,y) for G = R⋊R+, y > 0. In this case we have α122 = −α212 = 1 and the other
structural constants are zero. Suppose that X = θE1+ηE2 is an arbitrary vector field on G.
Then the equation (1.1) shows that the Riemannian manifold (G,< ·,· >) together with the
vector field X and expansion constant λ is a Ricci soliton if and only if the following system
holds,
yηf +yθf +2yθ f = 2(λ−κ)f
x y y
(4.3) yηf +yθf +2f(yη −θ)= 2(λ−κ)f
 x y x
 η = −y(ηy +θx).
Also we can see X = gradΦ if and only if

Φ = fη
(4.4) x y
Φ = fθ.
( y y
Also by using equation (3.1) for Gaussian curvature of this manifold we have
1
2 2 2
(4.5) κ = 2f3(−f(f11+f22)+f1 +f2 +ff1−2f )
1
2 2 2 2 2
= (−f(yf +y f )+y (f +f )−2f ).
2f3 yy xx x y
8 H.R.SALIMIMOGHADDAM
Example 4.5. WestartwithleftinvariantRiemannianmetric, infactweconsiderf(x,y)= 1,
forany(x,y) ∈ G. Inthiscase, theequation (4.5)showsthatthisspaceisofconstantGaussian
curvature κ = −1. Now if we consider θ = 0, η = 1 and λ = −1 then the system of equations
y
(4.3) shows that G with the vector field X = ∂ is an expanding Ricci soliton. The equations
∂x
(4.4) show that this Ricci soliton is not gradient.
Example 4.6. Now consider the case that f(x,y)= y. In this case the equation (3.1) or (4.5)
shows that the Gaussian curvature of G is κ = − 1 . Suppose that X = ∂ − ∂ (in fact let
2y ∂x ∂y
η = −θ = 1). Then G with the vector field X is a steady Ricci soliton which with attention
y
to equations (4.4) is not gradient.
Example 4.7. In the previous example if we consider X = − ∂ (or equivalently if θ = −1
∂y y
and η = 0) then by the systems (4.3) and (4.4) we have a gradient steady Ricci soliton with
1
potential function Φ = ln .
y
Example 4.8. Assumethatf(x,y) = y2 thenwehave aflattwodimensionalnoncommutative
Lie group. If X = ∂ + ∂ (equivalently η = θ = 1) then we have a steady gradient Ricci
∂x ∂y y
soliton with potential function Φ = x+y.
Example 4.9. For f(x,y) = y2 if X = x ∂ +y ∂ (equivalently η = x and θ = 1) then we
∂x ∂y y
have a flat two dimensional shrinking gradient Ricci soliton with λ = 1 and potential function
Φ = 1(x2+y2).
2
Example 4.10. In two previous examples if X = ∂ (equivalently η = 0 and θ = 1) then we
∂y y
have a flat two dimensional steady gradient Ricci soliton with potential function Φ = y +c,
where c is an arbitrary constant real number.
4.3. Lie group G = R2 ⋊ R+. In this subsection we consider the Lie group G = R ⋊ R+
with natural coordinates (x,y,z) such that z > 0. Similar to above we consider a f−left
invariant Riemannian metric < ·,· > such that the set {E1 := z∂∂z,E2 := z∂∂x,E3 := z∂∂y}
is an orthogonal basis at any point and is orthonormal at e = (0,0,1). Easily we can see
α122 = α133 = 1, α212 = α313 = −1 and the other structural constants are zero. Suppose that
X = θE1+ηE2+µE3 is an arbitrary vector field on G. Thenthe equations (1.1) together with
(3.8) show that the Riemannian manifold (G,< ·,· >) with the vector field X and expansion
constant λ is a Ricci soliton if and only if the following system holds,
(4.6)
X(f)+2θ1f = 2 λf − 4f12 −2f(2f11+f22+f33)+4f12+f22+f32+4f(f1−2f)
 f(η+η1+θ2) =(cid:16)−2f12(3f1f(cid:0)2−2ff21) (cid:1)(cid:17)
 f(µ+µ1+θ3) = −2f12(3f1f3−2ff31)

 X(f)+2f(η2−θ)= 2 λf − 4f12 −2f(f11+2f22+f33)+f12+4f22+f32+2f(3f1−4f)

f(µ2+η3) = −2f12(3f2(cid:16)f3−2ff32(cid:0)) (cid:1)(cid:17)

 X(f)+2f(µ3−θ)= 2 λf − 4f12 −2f(f11+f22+2f33)+f12+f22+4f32+2f(3f1−4f) .


A simple computation s(cid:16)hows that(cid:0)X =gradΦ if and only if (cid:1)(cid:17)

Φ = fη
x z
(4.7) Φ = fµ
 y z
 Φz = fzθ.


A GENERALIZATION OF INVARIANT RIEMANNIAN METRICS AND ITS APPLICATIONS TO RICCI SOLITONS9
Example4.11. InaspecialcaseforleftinvariantRiemannianmetricinducedbyf(x,y,z) = 1,
if X = ∂ + ∂ (equivalently η = µ = 1 and θ = 0) then by using the systems (4.6) and (4.7)
∂x ∂y z
and the equation (3.1) we have a three dimensional expanding Ricci soliton with λ = −2
which is not gradient. Also by using (3.1) for its sectional curvature we have K(E1,E2) =
K(E1,E3)= K(E2,E3)= −1.
Example 4.12. In the previous example if X = ∂ (equivalently η = 1 and θ = µ = 0) then
∂x z
we have a three dimensional expanding Ricci soliton with λ = −2 which is not gradient.
Example 4.13. In two previous examples if X = ∂ (equivalently µ = 1 and θ = η = 0)
∂y z
again we have a three dimensional expanding Ricci soliton with λ =−2 which is not gradient.
Example 4.14. Consider the f−left invariant Riemannian metric induced by f(x,y,z) = z2,
ifX = ∂ + ∂ + ∂ (equivalently θ = η = µ = 1)thenthesystems(4.6)and(4.7)togetherwith
∂x ∂y ∂z z
(3.1) show that the Riamnnian manifold G with the vector field X is a flat three dimensional
steady gradient Ricci soliton with potential function Φ = x+y+z.
4.4. Lie group G = R ⋊ R+ × R. Now we consider another noncommutative Lie group
G = R⋊R+ ×R with natural coordinates (x,y,z) such that y > 0. Then consider a f−left
invariant Riemannian metric < ·,· > such that the set {E1 := y ∂ ,E2 := y ∂ ,E3 := ∂ } is an
∂y ∂x ∂z
orthogonal basis at any point and is orthonormal at e = (0,1,0). In this case we have α122 =
−α212 = 1 and the other structural constants are zero. Suppose that X = θE1+ηE2+µE3 is
an arbitrary vector field on G. Then the equations (1.1) and (3.8) show that the Riemannian
manifold (G,< ·,· >) with the vector field X and expansion constant λ is a Ricci soliton if
and only if the following system holds,
(4.8)
X(f)+2θ1f = 2 λf − 4f12 −2f(2f11+f22+f33)+4f12+f22+f32+2f(f1−2f)
 f(η+η1+θ2) =(cid:16)−2f12(3f1f(cid:0)2−2ff21) (cid:1)(cid:17)
 f(µ1+θ3)= −2f12(3f1f3−2ff31)

 X(f)+2f(η2−θ)= 2 λf − 4f12 −2f(f11+2f22+f33)+f12+4f22+f32+4ff1−4f2

f(µ2+η3) = −2f12(3f2(cid:16)f3−2ff32(cid:0)) (cid:1)(cid:17)

 X(f)+2µ3f = 2 λf − 4f12 −2f(f11+f22+2f33)+f12+f22+4f32+2ff1 .


Also we can see X(cid:16)= gradΦ(cid:0)if and only if (cid:1)(cid:17)

Φ = fη
x y
(4.9) Φ = fθ
 y y
 Φz = µf.
Example 4.15. If f(x,y,z) = 1 then we have K(E1,E2) = −1, K(E1,E3) = K(E2,E3) = 0
and the only non zero Ric(Ei,Ej)’s are Ric(E1,E1) = Ric(E2,E2) = −1. Suppose that X =
−z ∂ (equivalently θ = η = 0 and µ = −z) then by using the systems (4.8) and (4.9) we can
∂z
see (G,< ·,· >) with X is a three dimensional expanding gradient Ricci soliton with λ = −1
and potential function Φ = −1z2.
2
Example 4.16. In the previous example if we consider X = ∂ −z ∂ (equivalently θ = 0,
∂x ∂z
η = 1 and µ = −z) then we have a three dimensional expanding Ricci soliton with λ = −1
y
which is not gradient.
10 H.R.SALIMIMOGHADDAM
Example 4.17. Assume that f(x,y,z) = y2 then the equation (3.1) shows that (G,< ·,· >)
is flat. Suppose that X = x ∂ +y ∂ (equivalently θ = 1, η = x and µ = 0) then the systems
∂x ∂y y
(4.8) and (4.9) show that (G,< ·,· >) with X is a three dimensional shrinking gradient Ricci
soliton with λ = 1 and potential function Φ = 1(x2+y2).
2
4.5. Lie group G = R⋊R+ ×R2. In this subsection we consider the noncommutative four
dimensional Lie group G = R⋊R+×R2 with natural coordinates (x,y,z,w) such that y > 0.
Now suppose that < ·,· > is a f−left invariant Riemannian metric on G such that the set
{E1 := y ∂ ,E2 := y ∂ ,E3 := ∂ ,E4 := ∂ } is an orthogonal basis at any point and is
∂y ∂x ∂z ∂w
orthonormal at e = (0,1,0,0). For this Lie group we have α122 = −α212 = 1 and the other
structural constants are zero. Assume that X = θE1+ηE2+µE3+νE4 is an arbitrary vector
fieldonG. Thentheequations (1.1)and(3.8)showthattheRiemannianmanifold (G,< ·,· >)
with the vector field X and expansion constant λ is a Ricci soliton if and only if the following
system holds,
(4.10)
X(f)+2θ1f = 2 λf − 2f12 −f(3f11+f22+f33+f44)+3f12+f(f1−2f)
 f(η+η1+θ2)=(cid:16)−f12(3f1f2(cid:0)−2ff21) (cid:1)(cid:17)
 f(µ1+θ3)= −f12(3f1f3−2ff31)
 f(ν1+θ4) = −f12(3f1f4−2ff41)

 X(f)+2f(η2−θ)= 2 λf − 2f12 −f(f11+3f22+f33+f44)+3f22+f(3f1−2f)

 f(µ2+η3) = −f12(3f2f(cid:16)3−2ff32)(cid:0) (cid:1)(cid:17)

f(ν2+η4) = −f12(3f2f4−2ff42)
 X(f)+2µ3f = 2 λf − 2f12 −f(f11+f22+3f33+f44)+3f32+ff1
 f(ν3+µ4) = −f12(cid:16)(3f3f4−2(cid:0)ff43) (cid:1)(cid:17)

 X(f)+2ν4f = 2 λf − 2f12 −f(f11+f22+f33+3f44)+3f42+ff1 .

Also we can see X =(cid:16)gradΦ if(cid:0)and only if (cid:1)(cid:17)

Φ = fη
x y
Φ = fθ
(4.11)  y y
 Φz = µf

Φ = νf.
w


Example 4.18. Suppose that the leftinvariant Riemannian metric < ·,· > generated by
f(x,y,z,w) = 1 on G. Then equation (3.1) shows that K(E1,E2) = −1 and K(Ei,Ej) = 0
in other cases. Now consider X = −z ∂ −w ∂ (in fact we assumed that θ = η = 0, µ = −z
∂z ∂w
and ν = −w). The systems (4.10) and (4.11) show that (G,< ·,· >) with X is an expanding
gradient Ricci soliton with λ = −1 and potential function Φ = −1(z2+w2).
2
Example 4.19. In thepreviousexample if wesupposethatX = ∂ −z ∂ −w ∂ (equivalently
∂x ∂z ∂w
θ = 0, η = 1, µ = −z and ν = −w) then we have an expanding Ricci soliton with λ = −1
y
which is not gradient.
4.6. Lie group G = R⋊R+ ×R⋊R+. Now we give another examples of four dimensional
Ricci solitons. Suppose that G = R ⋊ R+ × R ⋊ R+ and consider the natural coordinates
(x,y,z,w), such that y > 0 and w > 0, for it. Consider the f−left invariant Riemannian