Non-trivial $m$-quasi-Einstein metrics on quadratic Lie groups

0.14 MB·

by  Zhiqi Chen

NON-TRIVIAL m-QUASI-EINSTEIN METRICS ON QUADRATIC
LIE GROUPS
ZHIQICHEN,KELIANG,ANDFAHUAIYI
Abstract. Wecallametricm-quasi-EinsteinifRicmX (amodification ofthe
5 m-Bakry-EmeryRiccitensorintermsofasuitablevectorfieldX)isaconstant
1 multipleof themetrictensor. Itisageneralization ofEinsteinmetricswhich
0 contains Ricci solitons. In this paper, we focus on left-invariant vector fields
2 andleft-invariantRiemannianmetricsonquadraticLiegroups. Firstweprove
that anyleft-invariantvector fieldX suchthat the left-invariantRiemannian
n
metric on a quadratic Lie group is m-quasi-Einstein is a Killing field. Then
u
weconstruct infinitely manynon-trivial m-quasi-Einsteinmetrics onsolvable
J
quadraticLiegroupsG(n)formfinite.
0
3

]
G 1. Introduction
D
A natural extension of the Ricci tensor is the m-Bakry-EmeryRicci tensor
.
h 1
at (1.1) Ricmf =Ric+∇2f − mdf ⊗df
m where 0<m6 , f is a smooth function on Mn, and 2f stands for the Hessian
∞ ∇
[ form. Instead of a gradient of a smooth function f by a vector field X, m-Bakry-

Emery Ricci tensor was extended by Barros and Ribeiro Jr in [3] and Limoncu in
2
v [15] for an arbitrary vector field X on Mn as follows:
2
1 1
2 (1.2) Ricm =Ric+ L g X∗ X∗
9 X 2 X − m ⊗
1 where L g denotes the Lie derivative onMn andX∗ denotes the canonical1-form
X
. associated to X. With this setting (Mn,g) is called an m-quasi-Einstein metric, if
1
0 there exist a vector field X X(Mn) and constants m and λ such that
∈
4
(1.3) Ricm =λg.
1 X
v: Anm-quasi-EinsteinmetriciscalledtrivialwhenX 0. Thetrivialitydefinition
≡
i is equivalent to say that Mn is an Einstein manifold. When m= , the equation
X ∞
(1.3) reduces to a Ricci soliton, for more details see [5] and the references therein.
r IfmisapositiveintegerandX isagradientvectorfield,theconditioncorresponds
a
toawarpedproductEinsteinmetric,formoredetailssee[11]. Classicallythestudy
on m-quasi-Einstein are considered when X is a gradient of a smooth function f
on Mn, see [1, 2, 6, 8, 10, 14].
In this paper, we focus on left-invariant Riemannian metrics on quadratic Lie
groups which include compact Lie groups and semisimple Lie groups as special
classes. First, we have the following theorem.
Theorem 1.1. Let G be a quadratic Lie group with a left-invariant Riemannian
metric , and the Lie algebra g of left-invariant vector fields. If X g and
Ricm =hλ· ·i, , then X is a Killing field. ∈
X h· ·i
2010 Mathematics Subject Classification. Primary53C25, 53C20,53C21;Secondary17B30.
Key words and phrases. m-quasi-Einstein metric, left-invariant metric, quadratic Lie group,
quadraticLiealgebra,Killingfield.
1
2 ZHIQICHEN,KELIANG,ANDFAHUAIYI
Theorem 1.1 for compact Lie groups is proved in [7], and there are non-trivial
m-quasi-Einstein metrics on homogeneous manifolds [4, 7]. Moreover, compact
homogeneous Ricci solitons are Einstein, which is equivalent with the conclusion
that the vector field is a Killing field, by the work of Petersen-Wylie [18] and
Perelman[17]. In [13],Jablonskigivesa new proof. Essentially,Theorem1.1for m
infinite holds for semisimple Lie groups by Jablonski’s proof.
NextwepayattentiontosolvablequadraticLiegroups. Herewestudyaclassof
simplyconnectedsolvablequadraticLiegroupsG(n)forn 1,thederivedalgebras
≥
of whose Lie algebras are Heisenberg Lie algebras of dimension 2n+1, and prove
Theorem 1.2. Every solvable quadratic Lie groups G(n) admits infinitely many
non-equivalent non-trivial m-quasi-Einstein metrics for m finite.
In order to prove Theorem 1.2, we first obtain a formula of the Ricci curvature
with respect to a left-invariant Riemannian metric which holds for any quadratic
Lie group. It is a natural extension of the formula on compact semisimple Lie
groups,which is given by Sagle [19] and simpler provedby D’Atri and Ziller in [9].
Based on it, we get a computable formula of the Ricci curvature on quadratic Lie
groups, i.e. Theorem 3.2, which is the fundament of the proof for Theorem 1.2.
2. The proof of Theorem 1.1
A quadratic Lie algebra is a Lie algebra g together with a pseudo-Riemannian
metric (, ): g g R that is invariant under the adjoint action, i.e.
· · ⊗ →
([X,Y],Z)+(Y,[X,Z])=0 for any X,Y,Z g.
∈
ALiegroupGiscalledaquadraticLiegroupifandonlyiftheLiealgebragofleft-
invariant vector fields is a quadratic Lie algebra. It is easy to see that tradX = 0
for any X g, i.e. a quadratic Lie group is unimodular.
∈
Let M denote the set of left-invariant Riemannian metrics on a unimodular Lie
groupG. Foranyleft-invariantRiemannianmetricQonG,thetangentspaceT M
Q
atQisthesetofleft-invariantsymmetric,bilinearformsong. DefineaRiemannian
metric on M by
(v,w) =tr vw = v(e ,e )w(e ,e ),
Q X i i i i
i
where v,w T M and e is a Q-orthonormalbasis of g.
Q i
∈ { }
Let G be a unimodular Lie group with a left-invariant Riemannian metric , .
h· ·i
Given Q M, denote by ric and sc the Ricci and scalar curvatures of (G,Q),
Q Q
respective∈ly. The gradient of the function sc:M R is
→
(2.1) (grad sc) = ric
Q Q
−
relative to the above Riemannian metric on M, see [12] or [16].
Assume that , is m-quasi-Einstein. That is, there exists a vector field X on
G such that Rich+· ·i1L , 1X∗ X∗ = λ , . In addition, assume that X is
2 Xh i− m ⊗ h· ·i
a left-invariant vector field, i.e. X g. Since , is a left-invariant Riemannian
∈ h· ·i
metric, for an orthonormalbasis relative to , , we have
h· ·i
1 X 2
(2.2) Ric=λId [(adX)+(adX)t]+ | | Pr .
X
− 2 m |
Here Pr is the orthogonalprojection onto RX.
X
|
Lemma 2.1 ([7]). Let Gbe a unimodular Lie group with a left-invariant Riemann-
ian metric , and the Lie algebra g. If X g and Ricm = λ , , then X is a
h· ·i ∈ X h· ·i
Killing field.
Since a quadratic Lie group is unimodular, we know that Theorem 1.1 follows
from Lemma 2.1.
m-QUASI-EINSTEIN METRICS ON QUADRATIC LIE GROUPS 3
3. Ricci curvature on quadratic Lie groups
Let , be a left-invariant Riemannian metric on G, i.e.
h· ·i
Y,Z + Y, Z =0 for any X,Y,Z g.
X X
h∇ i h ∇ i ∈
In fact, the Levi-Civita connection corresponding to the left-invariant Riemannian
metric , is determined by the following equation
h· ·i
1
(3.1) Y,Z = [X,Y],Z [Y,Z],X + [Z,X],Y .
X
h∇ i 2{h i−h i h i}
There is a linear map D of g satisfying
(X,Y)= X,θ(Y) for any X,Y g.
h i ∈
It follows that θ is invertible, and symmetric with respect to , . Since , is
h· ·i h· ·i
positive definite and(, ) is symmetric, by a resultin linear algebra,we canchoose
· ·
an orthonormal basis X , ,X of g with respect to , such that
1 n
{ ··· } h· ·i
(X ,X )=λ δ ,
i j i ij
for some λ R. Then θ(X )=λ (X ).
i i i i
∈
By the equation (3.1) and the ad-invariance of (, ), we have
· ·
1
Y,Z = [X,Y],Z [Y,Z],X + [Z,X],Y
X
h∇ i 2{h i−h i h i}
1
= ([X,Y],θ−1Z) ([Y,Z],θ−1X)+([Z,X],θ−1Y)
2{ − }
1
= (θ−1[X,Y],Z) ([θ−1X,Y],Z)+([X,θ−1Y],Z)
2{ − }
1
= θ−1[X,Y] [θ−1X,Y]+[X,θ−1Y],θZ .
2h − i
It follows that
1
(3.2) Y = [X,Y] θ[θ−1X,Y]+θ[X,θ−1Y] ,
X
∇ 2{ − }
that is,
1
= adX θad(θ−1X)+θadXθ−1 .
X
∇ 2{ − }
One has a simple formula of the Ricci curvature on a compact semisimple Lie
groupwithrespecttoaleft-invariantmetricwhichwasderivedin[19],andasimpler
proof is given in [9]. The proof given in [9] is easily extended to a quadratic Lie
group. That is,
Lemma 3.1. For any X,Y g, Ric(X,Y)= tr( adX)( adY).
X Y
∈ − ∇ − ∇ −
Proof. For any X g, we have
i
∈
Ric(X,Y) = tr(X Y Y Y)
i 7→∇Xi∇X −∇X∇Xi −∇[Xi,X]
= tr(∇∇XY −ad(∇XY)−∇X∇Y +∇XadY +∇YadX −adYadX)
= tr(∇∇XY)−tr(ad(∇XY))−tr(∇X −adX)(∇Y −adY).
By the left-invariance of , , we know that is skew-symmetric with respect
x
h· ·i ∇
to h·,·i. It follows that tr∇Z =0 for any Z ∈g. In particular, tr∇∇XY =0. Since
Gisquadratic,wehavetradX =0foranyX g. Inparticular,tr(ad( Y))=0.
X
∈ ∇
Then the theorem follows. (cid:3)
Assume that X ,...,X is the above orthonormal basis of g with respect to
1 n
{ }
, , and Ck are the structure constants with respect to the basis. That is,
h· ·i ij
n
[X ,X ]= Cl X .
i j X ij l
l=1
4 ZHIQICHEN,KELIANG,ANDFAHUAIYI
By the ad-invariance of (, ), we have
· ·
(3.3) Cl λ =Ci λ =Cjλ .
ij l jl i li j
Let µ = 1 . Then we have
i λi
1
X = ([X ,X ] θ[θ−1X ,X ]+θ[X ,θ−1X ])
∇Xi j 2 i j − i j i j
1
= (Id µ θ+µ θ)([X ,X ])
i j i j
2 −
n
1 µ µ +µ
= l− i jCl X .
2X µ ij l
l
l=1
Then by Lemma 3.1 and the equation (3.3), we have
Ric(X ,X ) = tr( adX )( adX )
j k − ∇Xj − j ∇Xk − k
n
= ( adX )( adX )X ,X
−Xh ∇Xj − j ∇Xk − k i ii
i=1
n n
1 µ µ +µ
= ( l− k i 1)Cl ( adX )X ,X
−Xh{ 2X µ − ki} ∇Xj − j l ii
l
i=1 l=1
n n
1 µ µ +µ µ µ +µ
= − l− k i− i− j lCl Ci
−4XX µ µ ki jl
l i
i=1 l=1
n n
1 µ µ +µ µ µ µ
= − l− k i l− j − iCl Cl
4XX µ µ ki ji
l l
i=1 l=1
Furthermore, by the equation (3.3), we know
1 µ µ +µ µ µ µ
− l− k i l− j − iCl Cl
4X µ µ ki ji
l l
i>l
1 Ci Ci
= ( µ µ +µ )(µ µ µ ) kl jl
4X − i− k l i− j − l µ µ
i i
i<l
1 Cl Cl
= ( µ µ +µ )(µ µ µ ) ki ji.
4X − i− k l i− j − l µ µ
l l
i<l
Thus we have the following theorem.
Theorem 3.2. Let X ,...,X be as above. Then
1 n
{ }
1 Cl Cl
Ric(X ,X )= ((µ µ )2 µ µ ) ki ji.
j k −2X l− i − k j µ µ
l l
i<l
4. m-quasi-Einstein metrics on G(n) for m finite
LetGbeasimplyconnectedLiegroupwiththeLiealgebrag,where D,X,Y,Z
{ }
is a basis of g such that the non-zero brackets are given by
(4.1) [D,X]=X,[D,Y]= Y,[X,Y]=Z.
−
It is easy to check that the symmetric bilinear form on g satisfying
(4.2) (D,Z)=(X,Y)
is invariant. Thus G is a quadratic Lie group if we take
1
(4.3) (D,Z)=(X,Y)= and other brackets equal to zero.
2
m-QUASI-EINSTEIN METRICS ON QUADRATIC LIE GROUPS 5
Lete =D+Z,e =D Z,e =X+Y ande =X Y. Considertheleft-invariant
1 2 3 4
− −
Riemannian metric , on G defined by
h· ·i
(4.4) e ,e =δ λ2,λ =0, for any i,j =1,2,3,4.
h i ji ij i i 6
Let fi = λeii for any 1≤i≤4. Then we have
1 1 1 1
(4.5) f ,f =δ ,(f ,f )= ,(f ,f )= ,(f ,f )= ,(f ,f )= ,
h i ji ij 1 1 λ2 2 2 −λ2 3 3 λ2 4 4 −λ2
1 2 3 4
and the non-zero structure constants corresponding to the basis f are
i i=1,2,3,4
{ }
(4.6)
λ λ λ λ λ λ
C4 = 4 ,C4 = 4 ,C3 = 3 ,C3 = 3 ,C2 = 2 ,C1 = 1 .
13 λ λ 23 λ λ 14 λ λ 24 λ λ 34 λ λ 34 −λ λ
1 3 2 3 1 4 2 4 3 4 3 4
Any left-invariant Killing vector field with respect to , is of the form
h· ·i
(4.7) a(λ f λ f ).
1 1 2 2
−
By Theorem 3.2, the Ricci curvatures are given by
Ric(f ,f )=Ric(f ,f )=Ric(f ,f )=Ric(f ,f )=Ric(f ,f )=0,
1 3 1 4 2 3 2 4 3 4

1 1
RRRiiiccc(((fff11,,,fff12)))===−−1122((((((λλλ23232+++λλλ24242)))222+−λλλ21414λ))λ22)21λλ11231λλ242λ,,23λ24,
2 2 −2 3 4 − 2 λ2λ2λ2
2 3 4
RRiicc((ff43,,ff43))==−−1122((((λλ2121+−λλ2423))22−−λλ4344))λλ2121λλ112323λλ2424 −− 2211((((λλ2222−+λλ2423))22−−λλ4344))λλ2222λλ112323λλ2424,.
Assume that , is m-quasi-Einsteinfor some X g. Then by Theorem 1.1, X is
h· ·i ∈
a left-invariant Killing vector field with respect to , . That is,
h· ·i
(4.8) X =a(λ f λ f ) for some a.
1 1 2 2
−
Thus , is m-quasi-Einstein with the constant λ and a = 1 if and only if the
h· ·i
following equations holds, i.e.
1 1 λ λ
 − 2((λ23+λ24)2+λ21λ22)λ λ λ2λ2 + 1m2 =0,
 −− 2211((((λλ2323++λλ2424))22−−λλ4142))λλ212λλ111232λλ2242 −−3 λλmm42122 ==λλ,,
2 3 4
 −− 2211((((λλ2121+−λλ2423))22−−λλ4344))λλ2121λλ112323λλ2424 −− 2211((((λλ2222−+λλ2423))22−−λλ4344))λλ2222λλ112323λλ2424 ==λλ,.
By the first equation, we can represent m by λ . Put the representation
i i=1,2,3,4
{ }
of m into the second and third equations, and then eliminate λ. Finally we have
(4.9) λ2 =λ2 and λ2λ2 =4λ4.
3 4 1 2 4
It is easy to check that λ satisfying the equations (4.9) are the solutions
i i=1,2,3,4
{ }
of the above equations. On the other hand, non-zero λ satisfying the
i i=1,2,3,4
{ }
equations (4.9) give an m-quasi-Einstein metric on G.
6 ZHIQICHEN,KELIANG,ANDFAHUAIYI
Fortheabovecase,[g,g]istheHeisenbergLiealgebraofdimension3. Motivated
by the above example, we try to give examples when [g,g] is the Heisenberg Lie
algebra of higher dimension.
Let G(n) be a simply connected Lie group with the Lie algebra g(n), where
D,X ,Y ,Z is a basis of g(n) such that non-zero brackets are given by
s s s=1,...,n
{ }
(4.10) [D,X ]=a X ,[D,Y ]= a Y ,[X ,Y ]=Z.
s s s s s s s s
−
We can assume that 0<a a ... a by adjusting the order and the sign of
1 2 n
≤ ≤ ≤
the basis if necessary. It is easy to check that the symmetric bilinear form on g(n)
satisfying
(4.11) (D,Z)=a (X ,Y ) for any s=1,2,...,n
s s s
is ad-invariant. Thus G is a quadratic Lie group if we take
1
(4.12) (D,Z)=a (X ,Y )= for any s and other brackets equal to zero.
s s s
2
Let e1 =D+Z, e2 =D Z, e2s+1 =√as(Xs+Ys) and e2s+2 =√as(Xs Ys) for
− −
any1 s n. Considertheleft-invariantRiemannianmetric , onG(n)defined
≤ ≤ h· ·i
by
(4.13) e ,e =δ λ2,λ =0, for any i,j =1,2,...,2n+2.
h i ji ij i i 6
Let fi = λeii for any 1≤i≤2n+2. Then we have
1 1
(4.14) f ,f =δ ,(f ,f )= ,(f ,f )= , 0 s n
h i ji ij 2s+1 2s+1 λ2 2s+2 2s+2 −λ2 ∀ ≤ ≤
2s+1 2s+2
and the non-zerostructure constants correspondingto the basis f are
i i=1,...,2n+2
{ }
a λ a λ
(4.15) C2s+2 = s 2s+2,C2s+2 = s 2s+2
1(2s+1) λ λ 2(2s+1) λ λ
1 2s+1 2 2s+1
a λ a λ
(4.16) C2s+1 = s 2s+1 ,C2s+1 = s 2s+1
1(2s+2) λ λ 2(2s+2) λ λ
1 (2s+2) 2 (2s+2)
a λ a λ
(4.17) C2 = s 2 ,C1 = s 1 .
(2s+1)(2s+2) λ λ (2s+1)(2s+2) −λ λ
2s+1 2s+2 2s+1 2s+2
Any left-invariant Killing vector field with respect to , is of the form
h· ·i
(4.18) a(λ f λ f ).
1 1 2 2
−
By Theorem 3.2, Ric(f ,f )=0 except i=j or (i,j)=(1,2). Furthermore,
i j
n 1 a2
Ric(f ,f )= ((λ2 +λ2 )2+λ2λ2) s ,
Ric(f11,f12)=XXs=n1−−221((λ222ss++11+λ222ss++22)2−λ141)λ22λλ21λa22sλλ222s+1λ,22s+2
s=1 1 2s+1 2s+2
Ric(f2,f2)=Xs=n1−21((λ22s+1+λ22s+2)2−λ42)λ22λ22s+a2s1λ22s+2,
m-QUASI-EINSTEIN METRICS ON QUADRATIC LIE GROUPS 7
and for any 1 s n,
≤ ≤
1 a2
Ric(f ,f )= ((λ2+λ2 )2 λ4 ) s
 2s+1 2s+1 −−1221((λ221−λ222ss++22)2−−λ422ss++11)λλ2221λλ2222ss+a+a2s121λλ2222ss++22,
Ric(f ,f )= ((λ2 λ2 )2 λ4 ) s
 2s+2 2s+2 −−122((λ221+−λ222ss++11)2−−λ422ss++22)λλ2221λλ2222ss+a+2s11λλ2222ss++22.
Assume that , is m-quasi-Einstein for some X g(n). By Theorem 1.1, X is a
h· ·i ∈
left-invariant Killing vector field with respect to , . That is,
h· ·i
(4.19) X =a(λ f λ f ) for some a.
1 1 2 2
−
Assume that , is m-quasi-Einsteinwiththe constantλ anda=1. Thenfor any
h· ·i
1 s n, Ric(f ,f )=Ric(f ,f ). It follows that
2s+1 2s+1 2s+2 2s+2
≤ ≤
(4.20) λ2 =λ2 .
2s+1 2s+2
For any 1 i=j n, since Ric(f ,f )=Ric(f ,f ), we have
2i+1 2i+1 2j+1 2j+1
≤ 6 ≤
λ2 λ2
(4.21) 2i+1 = 2j+1.
a a
i j
By the first equation, we can represent m by λ , and then put the
i i=1,...,2n+2
representation of m into Ricm(f ,f ) and Ricm({f ,}f ). It is easy to check that
X 1 1 X 2 2
Ricm(f ,f )=Ricm(f ,f ).
X 1 1 X 2 2
Since Ricm(f ,f )=Ricm(f ,f ), we have
X 1 1 X 3 3
4 n a2
(4.22) λ2λ2 = Ps=1 sλ4.
1 2 a2 3
1
It is easy to check that λ satisfying the equations (4.20), (4.21) and
i i=1,...,2n+2
{ }
(4.22) are the solutions. Onthe other hand, non-zero λ satisfying the
i i=1,...,2n+2
{ }
equations (4.20), (4.21) and (4.22) give an m-quasi-Einstein metric on G(n). It is
easy to check that
2 a2
(4.23) S = Ric(f ,f )= n(λ2+λ2)( + 1 ).
X i i − 1 2 λ2λ2 2λ4
i 1 2 3
Withrespecttotheorthonormalbasis f ,thedeterminantofthemetric
i i=1,...,2n+2
{ }
matrix with respect to , is 1. Take λ = c = 0. Then by the equation (4.22),
3
h· ·i 6
λ2λ2 is a constant. It follows that S is based on the choice of λ2 and λ2. That
1 2 1 2
is,G(n)admitsinfinitelymanynon-equivalentnon-trivialm-quasi-Einsteinmetrics
for m finite, i.e. Theorem 1.2 holds.
5. Acknowledgments
This workis supportedby NSFC (No. 11001133,No. 11271143,No. 11371155),
andUniversitySpecialResearchFundforPh.D.ProgramofChina(20124407110001
and 20114407120008). We would like to thank F.H. Zhu for the helpful comments,
conversationand suggestions.
8 ZHIQICHEN,KELIANG,ANDFAHUAIYI
References
[1] M.Anderson,Scalarcurvature,metricdegenerations andthestaticvacuum Einsteinequa-
tionson3-manifolds,Geom. Funct. Anal.9(1999), 855–967.
[2] M. Anderson and M. Khuri, The static extension problem in General relativity, arXiv:
0909.4550v1 [math.DG],2009.
[3] A. Barros and E. Ribeiro Jr, Integral formulae on quasi-Einstein manifolds and applica-
tions,Glasgow Math. J.54(2012), 213–223.
[4] A. Barros, E. Ribeiro Jr and S. Joa˜o, Uniqueness of quasi-Einstein metrics on 3-
dimensionalhomogenous manifolds,DifferentialGeom. Appl.35(2014), 60–73.
[5] H.D. Cao,Recent progressonRiccisoliton,Adv. Lect. Math. (ALM),11(2009), 1–38.
[6] J.Case,Y.ShuandG.Wei,Rigityofquasi-Einsteinmetrics,DifferentialGeom. Appl.,29
(2011), 93–100.
[7] Z. Chen,K. LiangandF. Zhu,Non-trivialm-quasi-EinsteinmetricsonsimpleLiegroups,
Ann. Mat. Pura. Appl.,2015,doi:10.1007/s10231-015-0506-y.
[8] J. Corvino, Scalar curvature deformations and a gluing construction for the Einstein con-
straintequations,Comm. Math. Phys.214(2000), 137–189.
[9] J.E. D’Atri and W. Ziller, Naturally reductive metrics and Einstein metrics on compact
Liegroups,Memoirs of Amer. Math. Soc.215(1979).
[10] M.Eminenti,G.La NaveandC.Mantegazza,Riccisolitons: theequationpointofview,
Manuscripta Math.127(2008), 345–367.
[11] C.He,P.PetersenandW.Wylie,OntheclassificationofwarpedproductEinsteinmetrics,
Commun. Anal. Geom.20(2012), 271–312.
[12] J.Heber,NoncompacthomogeneousEinsteinspaces,Invent.Math.133(1998),no.2,279–
352.
[13] M.Jablonski,Homogeneous Riccisolitons,J. Reine Angew. Math.699(2015), 159–182.
[14] D.S. Kim and Y.H. Kim, Compact Einstein warped product spaces with nonpositive scalar
curvature,Proc. Amer. Math. Soc.131(2003), 2573–2576.
[15] M. Limoncu, Modifications of the Ricci tensor and applications, Arch. Math. 95 (2010),
191–199.
[16] Yu.G. Nikonorov, The scalar curvature functional and homogeneous Einstein metrics on
Liegroups,Sibirsk. Mat. Zh.39(1998), no.3,583–589.
[17] G.Perelman,TheentropyformulafortheRicciflowanditsgeometricapplications,Prerpint,
arXivmath/0211159,2002.
[18] P.PetersenandW.Wylie,OngradientRiccisolitonswithsymmetry,Proc. Amer.Math.
Soc.137(2009), no.6,2085–2092.
[19] A.Sagle,Somehomogenous Einsteinmanifolds,Nogaya Math. J.39(1970), 81–106.
SchoolofMathematicalSciencesandLPMC,NankaiUniversity,Tianjin300071,P.R.
China
E-mail address: [email protected]
SchoolofMathematicalSciencesandLPMC,NankaiUniversity,Tianjin300071,P.R.
China
E-mail address: [email protected]
SchoolofMathematicalSciences,SouthChinaNormalUniversity,Guangzhou510631,
P.R. China
E-mail address: [email protected]