arXiv:2002.12175v1 [math.DG] 26 Feb 2020arXiv:2002.12175v1 [math.DG] 26 Feb 2020 RICCI FLOW ON...

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RICCI FLOW ON CERTAIN HOMOGENEOUS SPACES

MARINA STATHA

Abstract. We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous

metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands

and second Betti number equal to one, and the Stiefel manifolds V2Rn and V1+k2

Rn, with n = 1+ k2 + k3.

We use techniques from the theory of differential equations, in particular the Poincare compactification.

2020 Mathematics Subject Classification. Primary 53C25; Secondary 53C30, 53E20, 34A26.

Keywords: Ricci flow, Einstein metric, Poincare compactification, generalized Wallach space, Stiefel mani-

fold, Generalized flag manifold, Grobner basis.

1. Introduction

The Ricci flow equation was introduced by Hamilton in 1982 ([Ha]), and is defined by

∂g

∂t= −2Ricg, (1)

where g = g(t) is a curve on the space of Riemannian metrics M on a smooth manifold Mn and Ricg is

the Ricci tensor of the Riemannian metric g. The solution of this equation, the so called Ricci flow, is a

1-parameter family of metrics g(t) in Mn. Intuitively, this is the heat equation for the metric g.

The Ricci flow (1) in general does not preserve the volume. In the case of a compact manifold Mn we

consider the normalized Ricci flow∂g

∂t= −2Ricg +

2r

ng, (2)

where r = r(g(t)) =∫MSgdug/

∫Mdug, dug is the volume element of g, and Sg denotes the scalar curvature

function of g. Under this normalized flow, the volume of the solution metric is constant in time. Equations

(1) and (2) can be shown to be equivalent by reparametrizing time t and by scaling the metric in space by a

function of t. The Einstein metrics, that is the Riemannian metrics of constant Ricci curvature which satisfy

Ricg = λg (from now on call it Einstein equation) for some constant λ ∈ R, are in this case the fixed points of

the normalized Ricci flow (2). In general, the Einstein equation reduces to a second order PDE and general

existence results are difficult to be obtained. Some methods are described in [Bo], [BoWaZi] and [WaZi].

Besides the detailed exposition on Einstein manifolds in [Be], we refer to [Wa1], [Wa2] and [Arv3] for more

recent results. For the case of homogeneous spaces G/H the problem of finding all invariant Einstein metrics

becomes slightly more accesible, due to the possibility of making symmetry assumptions, but still it is not

easy. A special class of homogeneous spaces for which Einstein metrics have been completely classified are the

generalized flag manifolds G/K (of a compact simple Lie group G) with two ([ArCh1]) three ([Arv2], [Ki])

four and five isotropy summands ([ArCh2], [ArChSa1]). There is also some classification of Einstein metrics

on flag manifolds with six isotropy summands (see in [Arv3] for more details). Also, the classification of

invariant Einstein metrics for another class of homogeneous space, the generalized Wallach spaces, was only

recently achieved ([ChNi]). The problem becomes more difficult in case where the isotropy representation

χ : H → Aut(ToG/H), (o = eH) of a homogeneous space G/H contains equivalent summands. This

happens for example for Stiefel manifolds VkFn, where F ∈ {R,C,H}. In the papers [ArSaSt1], [ArSaSt2]

and [ArSaSt3], A. Arvanitoyergos, Y. Sakane and the author found (by using a technique which is described

in detail in [St]), Einstein metrics for several classes of Stiefel manifolds.

An important property of the Ricci flow is that it preserves symmetries of the initial metric g. This is

due to the fact that the Ricci tensor is invariant under diffeomorphisms of the manifold M . In general,

the normalized Ricci flow (2) for an arbitrary manifold is a non-linear system of PDEs. When restricted1

http://arxiv.org/abs/2002.12175v2

2 Marina Statha

to the set of invariant metrics, such system reduces to an non-linear system of ODEs. For this reason it is

natural to study the Ricci flow on homogeneous spaces, that is a Riemmanian manifold (M, g) with a closed

subgroup G of the isometries Iso(M, g), such that for any p and q in M , there exists a g ∈ G with g(p) = q.

In this case M = G/H , where H = {g ∈ G : g · p = p} is the isotropy subgroup at the point p ∈M . On such

spaces we work with G-invariant metrics, i.e. metrics for which the map τα : G/H → G/H , gH 7→ αgH is

an isometry. It is natural to proceed the study of the Ricci flow using tools from the theory of dynamical

systems. As mentioned above, to show existence of Einstein metrics is a not an easy task. The use of the

normalized Ricci flow on homogeneous spaces towards a qualitative study of homogeneous invariant Einstein

metrics, has been used by various works, such as [Ab], [AANS], [AbNi], [AnCh], [BW], [Bu], [GlPa], [GrMa1]

and [GrMa2].

In this paper we study the normalized Ricci flow of invariant metrics on certain homogeneous spaces with

three and four isotropy summands, such as generalized Wallach spaces (that is a homogeneous space G/H

for which the tangent space To(G/H) ∼= m written as direct sum of three summands mi, i = 1, 2, 3 with

the property [mi,mi] ⊂ h), certain Stiefel manifolds VkRn (the set of all orthonormal k-frames on Rn) and

generalized flag manifolds (this is homogeneous spaces M = G/K where G is a compact semisimple Lie

group and K is the cetralizer of a torus in G. Equivalently, these are diffeomorphic to the adjoint orbit

Ad(G)w, for some w ∈ g, the Lie algebra of G). On such spaces the normalized Ricci flow (2) is equivalent

to a homogeneous system of differential equations in R3 and R4. So in order to study the behavior of such

systems at infinity, we will use a method introduced by Poincare, the so called Poincare compactification.

This method allows us to study global phase portraits for polynomial systems.

The main contribution of the present work is that by using the Poincare compactification we not only

confirm previously obtained Einstein metrics as fixed points of dynamical systems deduced from normalized

Ricci flow, but we also detect new homogeneous Einstein metrics on certain spaces. The main theorems are

the following:

Theorem A. Let G/H be a generalized Wallach space. The normalized Ricci flow of a G-invariant Rie-

mannian metric on G/H has a finite number of singularities at infinity (in fact one to four). These fixed

points determine the invariant Einstein metrics of G/H .

We also give some further analysis for the generalized Wallach space SO(k+ l+m)/ SO(k)×SO(l)×SO(m),

for l = m.

Theorem B. (1) Let V2Rn = SO(n)/ SO(n − 2) be a Stiefel manifold. The normalized Ricci flow of the

SO(n)-invariant Riemannian metrics on V2Rn has exactly one singularity at infinity. This corresponds (up

to scale) to the unique invariant Einstein metric.

(2) Let the Stiefel manifold V1+k2Rn ∼= SO(n)/ SO(k3), with n = 1 + k2 + k3. The normalized Ricci flow

on the space of invariant Riemannian metrics on V1+k2R1+2k2 , for k2 ≥ 6 has at least four singularities at

infinity. These fixed points correspond (up to scale) to new G-invariant Einstein metrics on V1+k2R1+2k2 .

(3) Let G/H be the Stiefel manifold V5R7. The normalized Ricci flow on the space of invariant Riemannian

metrics on G/H possesses exactly four singularities at infinity. These fixed points correspond (up to scale)

to the G-invariant Einstein metrics on G/H .

We are also motivated to state the following:

Conjecture 1. Let G/H be the Stiefel manifold V1+k2Rn, with n = 1 + k2 + k3. Then, for k2 ≥ 4 and

k3 > 1, G/H has four singularities at infinity, and for k2 ≥ 6 and k3 = 2 it has six singularities at infinity.

These fixed points correspond to the G-invariant Einstein metrics on G/H .

Theorem C. Let G/K be a generalized flag manifold with four isotropy summands and b2(G/K) = 1.

The normalized Ricci flow of G-invariant Riemannian metrics on G/K has, for the case of exceptional flag

manifold F4, E7 and E8(α6), exactly three singularities at infinity. For the case of flag manifold corresponding

to E8(α3) it has exactly five singularities at infinity. These fixed points determine explicitly the three and

five (up to scale) invariant Einstein metrics of G/K, respectively.

Ricci flow on certain homogeneous spaces 3

2. The normalized Ricci flow

2.1. Ricci tensor and scalar curvature. Let G be a compact semisimple Lie group, H a connected closed

subgroup of G and let g and h be the corresponding Lie algebras. The Killing form B of g is negative definite,

so we can define an Ad(G)-invariant inner product −B on g. Let g = h ⊕ m be a reductive decomposition

of g with respect to −B so that [ h, m ] ⊂ m and m ∼= To(G/H) where o is the identity coset of G/H . Any

G-invariant metric g on G/H corresponds to an Ad(H)-invariant inner product 〈·, ·〉 on m and vice versa.

Let {Xj} be a 〈·, ·〉-orthonormal basis of m. The Ricci tensor Ricg of the metric g is given as follows ([Be,

p. 185]):

Ricg(X,Y ) = −1

2

∑

i

〈[X,Xi], [Y,Xi]〉+1

2B(X,Y ) +

1

4

∑

i,j

〈[Xi, Xj], X〉〈[Xi, Xj], Y 〉. (3)

The scalar curvature Sg = trRicg of g is given by ([Be, p. 186]):

Sg =1

4

∑

i,j

|[Xi, Xj ]m|2 −1

2

∑

i

B(Xi, Xi). (4)

If the isotropy representation of G/H is decomposed into a sum of mutually non equivalent irreducible

summands, then we will also use the following alternative expression for the Ricci tensor. Let m = m1 ⊕· · ·⊕mq be a decomposition into mutually non equivalent irreducible Ad(H)-modules. Then any G-invariant

metric on G/H can be expressed as follows:

〈·, ·〉 = x1(−B)|m1+ x2(−B)|m2

+ · · ·+ xq(−B)|mq, (5)

for positive real numbers (x1, . . . , xq) ∈ Rq+. Note that G-invariant symmetric covariant 2-tensors on G/H

are of the same form as the Riemannian metrics (although they are not necessarilly positive definite). In

particular, the Ricci tensor Ricg of a G-invariant Riemannian metric on G/H is of the same form as (5),

that is Ricg = y1(−B)|m1+ y2(−B)|m2

+ · · ·+ yq(−B)|mqfor some yi ∈ R, i = 1, 2, . . . , q.

Let {e(k)α }dk

α=1, where dk = dimmk, be a (−B)-orthonormal basis of mk. Then the set {X(k)α = e

(k)α /

√xk} is

a 〈·, ·〉-orthonormal basis of mk. If we denote by rk = Ricg(X(k)α , X

(k)α ), then we obtain rk = (1/xk) Ricg(e

(k)α ,

e(k)α ) that is Ricg(e

(k)α , e

(k)α ) = xkrk. Thus the Ricci tensor is written as Ricg = x1r1(−B)|m1

+x2r2(−B)|m2+

· · ·+xqrq(−B)|mq, where the ri’s are the components of the Ricci tensor on each mi for i = 1, 2, . . . , k. Now

let Aγαβ = −B([e

(i)α , e

(j)β ], e

(k)γ ) so that [e

(i)α , e

(j)β ] =

∑

γ

Aγαβe

(k)γ and set Aijk :=

[k

ij

]=

∑(Aγ

αβ)2, where the

sum is taken over all indices α, β, γ with e(i)α ∈ mi, e

(j)β ∈ mj , e

(k)γ ∈ mk (cf. [WaZi]). Then the positive

numbers Aijk are independent of the B-orthonormal bases chosen for mi,mj,mk, and Aijk = Ajik = Akij .

We have the following:

Lemma 2.1. ([PaSa]) Let G/H be a homogeneous space where G is compact and semisimple Lie group. Let

g the G-nvariant metric on G/H given by the Ad(H)-invariant inner products (5). Then:

(1) The components r1, . . . , rs of the Ricci tensor Ricg are given by

rk =1

2xk+

1

4dk

∑

j,i

xkxjxi

Ajik − 1

2dk

∑

j,i

xjxkxi

Akij (k = 1, . . . , q) (6)

where the sum is taken over i, j = 1, . . . , q.

(2) The scalar curvature Sg = trRicg =∑s

i=1 diri is given by

Sg =1

2

s∑

i=1

dixi

− 1

4

∑

i,j,k

xkxixj

Aijk (7)

Remark 2.2. Since by assumption the tangent space m of G/H decomposes into Ad(H)-modules mi,mj

which are mutually non equivalent for any i 6= j, it is Ricg(mi,mj) = 0 whenever i 6= j. Thus, by Lemma 2.1 it

follows that G-invariant Einstein metrics onG/H are exactly the positive real solutions g = (x1, . . . , xq) ∈ Rq+

of the polynomial system {r1 = λ, r2 = λ, . . . , rq = λ}, where λ ∈ R+ is the Einstein constant.

4 Marina Statha

2.2. The normalized Ricci flow. Let (M = G/H, g) a Riemannian homogeneous space and let MG1 be

the set of G-invariant metrics with total volume 1, that is∫Mdvg = 1. Then for every g in MG

1 its scalar

curvature Sg is a constant function on M . Therefore we have r =∫MSgdvg/

∫Mdvg = Sg. Thus, for a

G-invariant metric on G/H the normalized Ricci flow (2) is equivalent to

∂g

∂t= −2Ricg +

2Sg

ng,

where n is the dimension of G/H . Actually for the G-invariant metric (5) the normalized Ricci flow reduces

to the following system of ODEs:{x1 = 2x1r1 +

2Sg

nx1, x2 = 2x2r2 +

2Sg

nx2, . . . , xq = 2xqrq +

2Sg

nxq

}. (8)

3. Generalized Wallach spaces

Let G/H be a reductive homogeneous space with G a compact and semisimple Lie group and H a

compact subgroup of G. Let g = h ⊕ m the reductive decomposition of G/H , that is Ad(H)m ⊂ m, with

m ∼= To(G/H). Then G/H is called a generalized Wallach space if the module m decomposes into a direct

sum of three Ad(H)-invariant irreducible modules pairwise orthogonal with respect to −B (the Killing form

of G), i.e. m = m1 ⊕m2 ⊕m3, such that [mi,mi] ⊂ h, for i = 1, 2, 3. Every generalized Wallach space admits

a three parameter family of G-invariant Riemannian metrics determined by Ad(H)-invariant inner products:

〈·, ·〉 = x1(−B)|m1+ x2(−B)|m2

+ x3(−B)|m3, xi ∈ R+, i = 1, 2, 3, (9)

where x1, x2, x3 are positive real numbers. We will denote such metrics with g = (x1, x2, x3).

The classification of generalized Wallach spaces G/H was obtained in [Ni1] and [ChKaLi]:

Theorem 3.1. Let G/H be a connected and simply connected compact homogeneous space. Then G/H is a

generalized Wallach space if and only if it is one of the following types:

(1) G/H is a direct product of three irreducible symmetric spaces of compact type.

(2) The group G is simple and the pair (g, k) is one of the pairs in Table 1.

(3) G = F × F × F × F and H = diag(F ) ⊂ G for some connected, compact, simple Lie group F , with the

following description on the Lie algebra level: (g, h) = (f⊕f⊕f⊕f, diag(f) = {(X,X,X,X)|X ∈ f}), where f isthe Lie algebra of F , and (up to permutation) m1 = {(X,X,−X,−X)|X ∈ f}, m2 = {(X,−X,X,−X)|X ∈f}, m3 = {(X,−X,−X,X)|X ∈ f}.

Table 1. The pairs (g, h) corresponding to generalized Wallach spaces G/H with G simple

GWS. g h d1 d2 d3

1 so(k + l +m) so(k)⊕ so(l)⊕ so(m) kl km lm

2 su(k + l +m) su(k)⊕ su(l)⊕ su(m) 2kl 2km 2lm

3 sp(k + l +m) sp(k)⊕ sp(l)⊕ sp(m) 4kl 4km 4lm

4 su(2l), l ≥ 2 u(l) l(l − 1) l(l + 1) l2 − 1

5 so(2l), l ≥ 4 u(1)⊕ u(l − 1) 2(l − 1) 2(l + 1) (l − 1)(l − 2)

6 e6 su(4)⊕ 2sp(1)⊕ R 16 16 24

7 e6 so(8)⊕ R2 16 16 16

8 e6 sp(3)⊕ sp(1) 14 28 12

9 e7 so(8)⊕ 3sp(1) 32 32 32

10 e7 su(6)⊕ sp(1)⊕ R 30 40 24

11 e7 so(8) 35 35 35

12 e8 so(12) ⊕ 2sp(1) 64 64 48

13 e8 so(8)⊕ so(8) 64 64 64

14 f4 so(5)⊕ 2sp(1) 8 8 20

15 f4 so(8) 8 8 8


3.1. Ricci tensor for generalized Wallach spaces. Let di = dimmi, i = 1, 2, 3. From the definition of

Aijk it is easy to see that for the generalized Wallach spaces, the Aijk = 0 if two of the indicies are equal.

Therefore, we only need to compute the number A123. It is di ≥ 2A123 for any i = 1, 2, 3 (see [Ni2]). Hence

the constants ai = A123/di, i ∈ {1, 2, 3} are such that (a1, a2, a3) ∈ (0, 1/2]3. In Table 2 we give these

numbers for the generalized Wallach spaces listed in Table 1 (these numbers were computed in [Ni1] and

[ChKaLi])

Table 2. The numbers ai, i = 1, 2, 3 for the generalized Wallach spaces G/H with G simple

GWS. a1 a2 a3 GWS. a1 a2 a3

1 m/2(k + l +m− 2) l/2(k + l +m− 2) k/2(k + l +m− 2) 9 2/9 2/9 2/9

2 m/2(k + l +m) l/2(k + l +m) k/2(k + l +m) 10 2/9 1/6 5/18

3 m/2(k + l +m+ 1) l/2(k + l +m+ 1) k/2(k + l +m+ 1) 11 5/18 5/18 5/18

4 (l + 1)/4l (l − 1)/4l 1/4 12 1/5 1/5 4/15

5 (l − 2)/4(l − 1) (l − 2)/4(l − 1) 1/2(l − 1) 13 4/15 4/15 4/15

6 1/4 1/4 1/6 14 5/18 5/18 1/9

7 1/6 1/6 1/6 15 1/9 1/9 1/9

8 1/4 1/8 7/24

From Lemma 2.1 the components of the Ricci tensor for the metric which corresponds to the Ad(H)-

invariant inner products (9), are given as follows

ri =1

2xi+ai2

(xixjxk

− xkxixj

− xjxixk

),

where i, j, k ∈ {1, 2, 3} with i 6= j 6= k 6= i. The scalar curvature is given by the following

Sg =1

2

(d1x1

+d2x2

+d3x3

)−A123

(x1x2x3

+x2x1x3

+x3x1x2

).

The G-invariant metric on a generalized Wallach space G/H corresponding to the inner product (9) is

Einstein if and only if r1 = r2 = r3. This is equivalent to the polynomial system

(a2 + a3)(a1x22 + a1x

23 − x2x3) + (a2x2 + a3x3)− (a1a2 + a1a3 + 2a2a3)x

21 = 0

(a1 + a3)(a2x21 + a2x

23 − x1x3) + (a1x1 + a3x3)− (a1a2 + 2a1a3 + a2a3)x

22 = 0. (10)

The first general result about the number of invariant Einstein metrics on generalized Wallach spaces is

the following:

Theorem 3.2. ([LoNiFi]) Let G/H be a generalized Wallach space with pairwise non equivalent submodules

mi. Then G/H admits from one to four invariant Einstein metrics (up to scale and homothety).

We take the G-invariant metric of the form g = (1, y1, y2), where yi = xi/x1, i = 1, 2 on generalized

Wallach spaces. Then from [LoNiFi] and [ChKaLi] we have the following Einstein metrics for GWS corre-

sponding to exceptional groups.

Table 3. The Einstein metrics for the generalized Wallach spaces G/H

GWS. g1 = (1, y2, y3) g2 = (1, y2, y3) g3 = (1, y2, y3) g4 = (1, y2, y3)

6 (1, 0.6, 0.8) (1, 1.66667, 1.33333) − −

7 (1, 1, 1) (1, 0.5, 0.5) (1, 2, 1) (1, 1, 2)

8 (1, 1.4618, 1.88845) (1, 0.8640, 0.4838) − −

9 (1, 1, 1) (1, 1.25, 1) (1, 0.8, 0.8) (1, 1, 1.25)

11 (1, 1, 1) (1, 1, 0.8) (1, 1.25, 1.25) (1, 0.8, 1)

12 (1, 1, 1.45608) (1, 1, 0.68677) − −

13 (1, 1, 1) (1, 1, 0.875) (1, 1.14285, 1.14285) (1, 0.875, 1)

14 (1, 0.4852, 0.8251) (1, 2.0606, 1.700349) − −

15 (1, 1, 1) (1, 3.5, 1) (1, 1, 3.5) (1, 0.28571, 0.28571)

6 Marina Statha

The above list does not include case GWS.10, that has been studied in [AnCh]. Note that the spaces

GWS.2 and GWS.5 are also generalized flag manifolds with three isotropy summands (this is also true for

spaces GWS.7 and GWS.10). The classification of Einstein metrics on these flag manifolds was given in

[Ki] and [Arv2], which agrees with the following theorem:

Theorem 3.3. ([LoNiFi]) If a generalized Wallach space G/H with pairwise non-isomorphic modules mi

satisfy the equality a1 + a2 + a3 = 1/2, then G/H admits four families of proportional invariant Einstein

metrics. These metrics have the form

(1) ((1− 2a1)q, (1− 2a2)q, 2(a1 + a2)q), (2) ((1 − 2a1)q, (1 − 2a2)q, 2(1− a1 − a2)q),

(3) ((1− 2a1)q, (1 + 2a2)q, 2(a1 + a2)q), (4) ((1 + 2a1)q, (1 − 2a2)q, 2(a1 + a2)q), (11)

where q ∈ R.

The following theorem gives a good insight about the number of invariant Einstein metrics on the gener-

alized Wallach space SO(k + l +m)/ SO(k)× SO(l)× SO(m).

Theorem 3.4. ([ChNi]) Assume that 1 ≤ k ≤ l ≤ m and l ≥ 2 and let G/H = SO(k + l +m)/ SO(k) ×SO(l) × SO(m). Then, for k >

√2m+ 2l− 4 the number of invariant Einstein metrics on G/H is 4, and

for k <√m+ l the number of invariant Einstein metrics on G/H is 2.

Finally, for the generalized Wallach spacesGWS.3 and GWS.4, the Einstein metrics are given as solutions

of equation (9) in the paper [LoNiFi, p. 51]. We give for GWS.3 some examples of Einstein metrics (up to

scale):

GWS.3a For Sp(6)/ Sp(1) × Sp(2) × Sp(3) we have the following metrics gi = (1, x1, x2), i = 1, 2, 3, 4

(1, 0.38050, 0.46780), (1, 1.23251, 1.39606), (1, 3.26361, 1.60389), (1, 1.30670, 3.18223).

GWS.3b For Sp(14)/ Sp(2) × Sp(5) × Sp(7) we have the following metrics gi = (1, x1, x2), i = 1, 2, 3, 4

(1, 0.40168, 0.52944), (1, 1.24716, 1.53155), (1, 2.94748, 1.67504), (1, 1.27217, 2.71689).

4. Stiefel manifolds

We embed the group SO(n− k) in SO(n) as

(1k 0

0 C

)where C ∈ SO(n− k). The Killing form of so(n) is

B(X,Y ) = (n−2) trXY . Then with respect to −B the subspace m = so(n−k)⊥ in so(n), may be identified

with the set of matrices of the form

{(Dk A

−At 0n−k

): Dk ∈ so(k), A ∈Mk×(n−k)(R)

}. Let Eab denote the

n× n matrix with 1 at the (ab)-entry and 0 elsewhere. Then the set B = {eab = Eab −Eba : 1 ≤ a ≤ k, 1 ≤a < b ≤ n} constitutes a −B-orthogonal basis of m. Note that eba = −eab, thus we have:

Lemma 4.1. If all four indices are distinct, then the Lie brackets in B are zero. Otherwise, [eab, ebc] = eac,

where a, b, c are distinct.

Next, we study the isotropy representation of VkRn = G/H = SO(n)/ SO(n − k). Let λn denote the

standard representation of SO(n) (given by the natural action of SO(n) on Rn). If ∧2λn denotes the

second exterior power of λn, then AdSO(n) = ∧2λn. The isotropy representation χ : SO(n) → Aut(m)

(m ∼= To(G/H)) of G/H is characterized by the property AdSO(n)∣∣∣SO(n−k)

= AdSO(n−k) ⊕χ. We compute

AdSO(n)∣∣∣SO(n−k)

= ∧2λn∣∣SO(n−k)

= ∧2(λn−k ⊕ k) = ∧2λn−k ⊕ ∧2k ⊕ (λn−k ⊗ k), (12)

where k denotes the trivial k-dimensional representation. Therefore, the isotropy representation is given by

χ = 1 ⊕ · · · ⊕ 1 ⊕ λn−k ⊕ · · · ⊕ λn−k. This decomposition induces an Ad(H)-invariant decomposition of

m given by m = m1 ⊕ · · · ⊕ ms, where the first(k2

)Ad(H)-modules are 1-dimensional and the rest k are

(n − k)-dimensional. It is clear that the isotropy representation of VkRn contains equivalent summands,

so a complete description of all G-invariant metrics is rather hard. In [ADN1] the authors introduced a

method for proving existence of homogeneous Einstein metrics by assuming additional symmetries. In [St]

is presented a systematic and organized description of such metrics.


4.1. The Stiefel manifold V2Rn ∼= SO(n)/ SO(n − 2). The isotropy representation of V2R

n, is expressed

as a direct sum χ = 1 ⊕ χ1 ⊕ χ2, where χ1 ≈ χ2 = λn−2 is the standard representation of SO(n− 2). This

decomposition induces an Ad(SO(n − 2))-invariant decomposition of m given by m = m0 ⊕ m1 ⊕ m2. Even

though an SO(n)-invariant metric on V2Rn depends on four parameters, it can be shown (cf. [Ke]) that it

can be descrided by an Ad(SO(n− 2))-invariant inner product of m of the form:

〈·, ·〉 = x0(−B)|m0+ x1(−B)|m1

+ x2(−B)|m2, xi ∈ R+, i = 1, 2, 3. (13)

Therefore, for the Ricci tensor of metrics corresponding to inner products (13) we can use the Lemma 2.1.

By using Lemma 4.1 the only non-zero number is A012 and equals to 1/2. Hence we have,

Proposition 4.2. (1) The components of the Ricci tensor for the metric (13) are given as follows

r0 =1

2x0

−1

4

(

x1

x0x2

+x2

x0x1

−x0

x1x2

)

, r1 =1

2x1

−1

4(n− 2)

(

x0

x1x2

+x2

x0x1

−x1

x0x2

)

r2 =1

2x2

−1

4(n− 2)

(

x0

x1x2

+x1

x0x2

−x2

x0x1

)

(14)

(2) The scalar curvature Sg is given by

Sg =1

2x0+

1

4

(− x1x0x2

− x2x0x1

− x0x1x2

)+n− 2

2

(1

x2+

1

x1

)(15)

Theorem 4.3. ([Arv1], [Ke]) The Stiefel manifold V2Rn = SO(n)/ SO(n − 2) admits (up to scale) exactly

one SO(n)-invariant Einstein metric which is given explicitly as (1, (n− 1)/2(n− 2), (n− 1)/2(n− 2)).

4.2. The Stiefel manifolds V1+k2Rn. Let G/H be the Stiefel manifold V1+k2

Rn = SO(n)/ SO(k3), with

n = 1+ k2 + k3. The isotropy representation for this case according to (12) contains equivalent summands.

Next, we describe a special class of invariant metrics on this space (for more details see for example [St],

[ArSaSt1] and [ArSaSt2]). The basic approach is to use an appropriate subgroupK ofG, such that the special

class of Ad(K)-invariant inner products, which are a subset of Ad(H)-invariant inner products, are diagonal.

In order to have this, it is sufficient for the subgroup K to satisfy the condition H ⊂ K ⊂ NG(H) ⊂ G.

We take the subgroup K = SO(k2) × SO(k3) of SO(n). Then, for the tangent space m ∼= To(G/H), we

consider the irreducible, Ad(K)-invariant and non-equivalent decomposition: m = so(k2)⊕m12⊕m13⊕m231.

Then the G-invariant metrics on G/H determined by the Ad(SO(k2)× SO(k3))-invariant scalar products

on m are given by

〈·, ·〉 = x2(−B)|so(k2) + x12(−B)|m12+ x13(−B)|m13

+ x23(−B)|m23(16)

Then by using Lemma 4.1 it follows that the only non zero triplets (up to permutation of indices) are

A222, A2(12)(12), A2(23)(23), A(13)(12)(23), where Aiii is non zero only for k2 ≥ 3.

Lemma 4.4. ([ArSaSt1]) For a, b, c = 1, 2, 3 and (a− b)(b− c)(c− a) 6= 0 the following relations hold:

Aaaa =ka(ka − 1)(ka − 2)

2(n− 2), Aa(ab)(ab) =

kakb(ka − 1)

2(n− 2), A(ac)(ab)(bc) =

kakbkc2(n− 2)

.

Lemma 4.5. (1) The components of the Ricci tensor Ric for the invariant metric 〈·, ·〉 on G/H defined by

(16), are given as follows:

r2 =k2 − 2

4(n− 2)x2

+1

4(n− 2)

(

x2

x122+ k3

x2

x232

)

,

r12 =1

2x12

+k3

4(n− 2)

(

x12

x13x23

−x13

x12x23

−x23

x12x13

)

−1

4(n− 2)

(

(k2 − 1)x2

x122

)

,

r23 =1

2x23

+1

4(n− 2)

(

x23

x13x12

−x13

x12x23

−x12

x23x13

)

−1

4(n− 2)

(

(k2 − 1)x2

x232

)

,

r13 =1

2x13

+k2

4(n− 2)

(

x13

x12x23

−x12

x13x23

−x23

x12x13

)

(17)

1The direct sum m12 ⊕m13 ⊕ m23 is the tangent space of generalized Wallach space GWS.1

8 Marina Statha

where n = 1 + k2 + k3.

(2) The scalar curvature Sg is given by

Sg =1

2

(d2x2

+d12x12

+d13x13

+d23x23

)− 1

4x2

(A2(12)(12) +A2(13)(13) +A2(23)(23)

)

−A(12)(13)(23)

2

(x12

x13x23+

x13x12x23

+x23

x12x13

)(18)

where d2 = dimm2 and dij = dimmij, i 6= j = 1, 2, 3.

We normalize the metric g = (x2, x12, x13, x23) by setting x23 = 1, then by solving the system {r2− r12 =

0, r12 − r23 = 0, r13 − r23 = 0} for k2 = 4 and k3 = 2, we take the following:

Theorem 4.6. ([ArSaSt1]) The Stiefel manifold V5R7 = SO(7)/ SO(2) admits at least four invariant Ein-

stein metrics, which are determined by the Ad(SO(4) × SO(2))-invariant inner products of the form (16)

given as: g1 = (1.27429, 1.27429, 1, 1), g2 = (0.392375, 0.392375, 1, 1), g3 = (0.245146, 1.01652, 0.253386, 1),

and g4 = (0.291175, 0.669071, 1.16137, 1).

5. Generalized flag manifolds

5.1. Description of flag manifolds in terms of painted Dynkin diagrams. Let g and k be the Lie

algebras of G and K respectively and gC, kC be their complexifications. We choose a maximal torus T in G

and let h be the Lie algebra of T . Then the complexification hC is a Cartan subalgebra of gC. Let R ⊂ (hC)∗

be the root system of gC relative to the Cartan subalgebra hC and consider the root space decomposition

gC = hC ⊕∑

α∈R gCα, where gCα = {X ∈ gC : ad(H)X = α(H)X, for all H ∈ hC} denotes the root space

associated to a root α. Assume that gC is semisimple, so the Killing form B of gC is non degenerate, and

we establish a natural isomorphism between hC and the dual space (hC)∗ as follows: for every α ∈ (hC)∗

we define Hα ∈ hC by the equation B(H,Hα) = α(H), for all H ∈ hC. Choose a basis Π = {α1, . . . , αℓ}(dim hC = ℓ) of simple roots for R, and let R+ be a choise of positive roots.

Since hC ⊂ kC ⊂ gC, there is a closed subsystem RK of R such that kC = hC ⊕∑

α∈RKgCα. In particular,

we can always find a subset ΠK ⊂ Π such that RK = R ∩ 〈ΠK〉 = {β ∈ R : β =∑

αi∈ΠKkiαi, ki ∈ Z},

where 〈ΠK〉 is the space of roots generated by ΠK with integer coefficients. The complex Lie algebra kC is a

maximal reductive subalgebra of gC and thus it admits the decomposition kC = z(kC)⊕ kCss, where z(kC) is the

center of kC and kCss = [kC, kC] is its semisimple part. Note that kCss is given by kCss = h′ ⊕∑α∈RK

gCα, where

h′ =∑

α∈ΠKCHα ⊂ hC is a Cartan subalgebra of kCss. In fact, RK is the root system of the semisimple

part kCss and ΠK is a corresponding basis. Thus we easily conclude that dimC h′ = cardΠK , where cardΠK

denotes the cardinality of the set ΠK . Let K be the connected Lie subgroup of G generated by k = kC ∩ g.

Then the homogeneous manifold M = G/K is a flag manifold, and any flag manifold is defined in this way,

i.e. by the choise of a triple (g,Π,ΠK).

Set ΠM = Π\ΠK , and RM = R\RK , such that Π = ΠK ∪ ΠM , and R = RK ∪ RM , respectively. Roots

in RM are called complementary roots, and they play an important role in the geometry of M = G/K. For

example, let m the orthogonal complement of k in g with respect to B. Then we have [k,m] ⊂ m where

m ∼= To(G/K). We set R+M = R+\R+

K where R+K is the system of positive roots of kC (R+

K ⊂ R+). Then

mC =∑

α∈RMgCα.

We conclude that all information contained in Π = ΠK ∪ΠM can be presented graphically by the painted

Dynkin diagram of M = G/K.

Definition 5.1. Let Γ = Γ(Π) be the Dynkin diagram of the fundamental system Π. By painting in black

the nodes of Γ corresponding to ΠM , we obtain the painted Dynkin diagram of the flag manifold G/K. In

this diagram the subsystem ΠK is determined as the subdiagram of white roots.

Conversely, given a painted Dynkin diagram, in order to obtain the corresponding flag manifoldM = G/K

we are working as follows: We define G as the unique simply connected Lie group corresponding to the

underlying Dynkin diagram Γ = Γ(Π). The connected Lie subgroupK ⊂ G is defined by using the additional

information Π = ΠK ∪ΠM encoded into the painted Dynkin diagram. The semisimple part of K is obtained


from the (not necessarily connected) subdiagram of white roots, and each black root, i.e. each root in ΠM ,

gives rise to one U(1)-summand. Thus the painted Dynkin diagram determines the isotropy subgroup K

and the space M = G/K completely. By using certain rules to determine whether different painted Dynkin

diagrams define isomorphic flag manifolds (see [AlAr]), one can obtain all flag manifolds G/K of a compact

simple Lie group G.

5.2. t-roots and isotropy summands. We study the isotropy representation of a generalized flag manifold

M = G/K of a compact simple Lie group G in terms of t-roots. In order to realise the decomposition of

m into irreducible Ad(K)-modules we use the center t of the real Lie algebra k. For simplicity, we fix a

system of simple roots Π = {α1, . . . , αr, φ1, . . . , φk} of R, such that r + k = ℓ = rk gC and we assume that

ΠK = {φ1, . . . , φk} is a basis of the root system RK of K so ΠM = Π\ΠK = {α1, . . . , αr}. Let Λ1, . . . ,Λr

be the fundamental weights corresponding to the simple roots of ΠM , i.e. the linear forms defined by2(Λi,αj)(αj ,αj)

= δij , (Λj , φi) = 0, where (α, β) denotes the inner product on (hC)∗ given by (α, β) = (Hα, Hβ),

for all α, β ∈ (hC)∗. Then the {Λi : 1 ≤ i ≤ r} is a basis of the dual space t∗ of t, t∗ =∑r

i=1 RΛi and

dim t∗ = dim t = r.

Consider now the linear restriction map κ : h∗ → t∗ defined by κ(α) = α|t, and set Rt = κ(R) = κ(RM ).

Definition 5.2. The elements of Rt are called t-roots.

As we saw the flag manifolds G/K are determined by pairs (g,Π,ΠK). The number of ad(k)-submodules

of m ∼= To(G/K) correspond to the Dynkin mark of the simple root we paint black on the Dynkin diagram.

We recall the following definition

Definition 5.3. The Dynkin mark of a simple root αi ∈ Π (i = 1, . . . , ℓ), is the positive integer mi in the

expression of the highest root α =∑ℓ

k=1mkαk in terms of simple roots. We will denote by Mrk the function

Mrk : Π → Z+ with Mrk(αi) = mi.

A fundamental result about t-root is the following:

Proposition 5.4. ([AlPe]) There exists a one-to-one correspondence between t-roots ξ and irreducible ad(kC)-

submodules mξ2 of the isotropy representation of mC, which is given by Rt ∋ ξ ↔ mξ =

∑α∈RM :κ(α)=ξ CEα.

Thus mC =⊕

ξ∈Rtmξ. Moreover, these submodules are non equivalent as ad(kC)-modules.

5.3. Flag manifolds with four isotropy summands. The generalized flag manifolds with four isotropy

summands can be separated into two types I and II. Type I is defined by the set: Π\ΠK = {αi : Mrk(αi) = 4}(that is b2(G/K) = 1) and Type II is given by Π\ΠK = {αi, αj : Mrk(αi) = 1,Mrk(αi) = 2} (however this

set may be define flag manifolds with four or five isotropy summands). The classification of those spaces was

given by A. Arvanitoyeorgos and I. Chrysikos in [ArCh2]. Below we give the flag manifolds together with

the Dynkin diagram of Type I.

F4 / SU(3)× SU(2)×U(1) E7 / SU(4)× SU(3)× SU(2)×U(1)

❝

α1

2❝

α2

3>sα3

4❝

α4

2❝

α1

1❝

α2

2❝

α3

3s

α4

4❝α72

❝

α5

3❝

α6

2

E8(α3)/ SO(10)× SU(3)×U(1) E8(α6)/ SU(7)× SU(2)×U(1)

❝

α1

2❝

α2

3s

α3

4❝

α4

5❝α83

❝

α5

6❝

α6

4❝

α7

2❝

α1

2❝

α2

3❝

α3

4❝

α4

5❝α83

❝

α5

6s

α6

4❝

α7

2

LetM = G/K be a generalized flag manifold of Type I and let m = m1⊕m2⊕m3⊕m4 be a decomposition

of m = ToM into irreducible non-equivalent Ad(K)-modules, with respect to the negative of the Killing form

2We mean that [kC,mξ] ⊂ mξ for all ξ ∈ Rt.

10 Marina Statha

B of G. Then, a G-invariant metric on M = G/K is given by

g = 〈·, ·〉 = x1(B)|m1+ x2(B)|m2

+ x3(B)|m3+ x4(B)|m4

(19)

where xi ∈ R+, i = 1, 2, 3, 4. Very often we will denote such metrics with g = (x1, x2, x3, x4). The Ricci

tensor for the above metric has (as symmetric, covariant 2-tensor) the same expression that is: Ric〈·,·〉 =∑4i=1 xiri(−B)|mi

, where ri, i = 1, 2, 3, 4 are the components of the Ricci tensor and are given as follows:

Proposition 5.5. ([ArCh2]) (1) The components of the Ricci tensor Ric for the invariant metric 〈·, ·〉 on

G/K defined by (19), are given as follows:

r1 =1

2x1

−A112

2d1

x2

x21

+A123

2d1

( x1

x2x3

−x2

x1x3

−x3

x1x2

)

+A134

2d1

( x1

x3x4

−x3

x1x4

−x4

x1x3

)

r2 =1

2x2

−A224

2d2

x4

x22

+A112

4d2

(x2

x21

−2

x2

)

+A123

2d2

( x2

x1x3

−x1

x2x3

−x3

x1x2

)

r3 =1

2x3

+A123

2d3

( x3

x1x2

−x2

x1x3

−x1

x2x3

)

+A134

2d3

( x3

x1x4

−x1

x3x4

−x4

x1x3

)

r4 =1

2x4

+A224

4d4

(x4

x22

−2

x4

)

+A134

2d4

( x4

x1x3

−x1

x3x4

−x3

x1x4

)

.

(20)

(2) The scalar curvature is given by

Sg =1

2

(d1x1

+d2x2

+d3x3

+d4x4

)+A112

4

(− 2

x2− x2x21

)+A123

2

(− x1x2x3

− x2x1x3

− x3x1x2

)

+A224

4

(− 2

x4− x4x22

)+A134

2

(− x1x3x4

− x3x1x4

− x4x1x3

), (21)

where di = dimmi, i = 1, 2, 3, 4.

In [ArCh2] the authors computed the numbers Aijk using the twistor fibration which admits any flag

manifold M = G/K of a compact (semi)-simple Lie group G, over an irreducible symmetric space G/L of

compact type. In particular we have the following table for Aijk and the dimensions di.

Table 4. The numbers Aijk and the dimensions for the flag manifold of Type I

M = G/K A224 A112 A123 A134 d1 d2 d3 d4

F4 /SU(3)× SU(2)× U(1) 2 2 1 2/3 12 18 4 6

E7 /SU(4)× SU(3)× SU(2)× U(1) 2 8 4 4/3 48 36 16 6

E8(α3)/SO(10)× SU(3)× U(1) 2 16 8 8/5 96 60 32 6

E8(α6)/SU(7) × SU(2)× U(1) 14/3 14 7 14/5 84 70 28 14

We normalize the metric g = (x1, x2, x3, x4) by setting x1 = 1. Then g is Einstein if and only the system:

{r1 − r2 = 0, r2 − r3 = 0, r3 − r4 = 0} has positive solution. After solving the previous system for each flag

manifold separately we obtain the following theorem

Theorem 5.6. ([ArCh2]) (1) The generalized flag manifolds G/K associated to the exceptional Lie groups

F4,E7 and E8(α3) admits (up to scale) three G-invariant Einstein metrics. One is Kahler given by g =

(1, 2, 3, 4) and other two are non-Kahler given approximatelly as follows:

F4 : g1 = (1, 1.2761, 1.9578, 2.3178), g2 = (1, 0.9704, 0.2291, 1.0097)

E7 : g1 = (1, 0.8233, 1.2942, 1.3449), g2 = (1, 0.9912, 0.5783, 1.1312)

E8(α3) : g1 = (1, 0.9133, 1.4136, 1.5196), g2 = (1, 0.9663, 0.4898, 1.0809)

(2) If G/K = E8(α6)/ SO(10) × SU(3) × U(1) then G/K admits (up to scale) five E8-invariant Einstein

metrics. One is Kahler given by g = (1, 2, 3, 4) and other four are non-Kahler given approximatelly as

follows: g1 = (1, 0.6496, 1.1094, 1.0610), g2 = (1, 1.1560, 1.0178, 0.2146), g3 = (1, 1.0970, 0.7703, 1.2969), g4 =

(1, 0.7633, 1.0090, 0.1910).


6. Poincare compactification

The method of Poincare compactification dates back to 1881. The main idea is to pass the study of a

vector field on a non compact manifold, to its study on the sphere (compact manifold). This allows us to

better understand its behavior at infinity. Poincare was studying the behavior of polynomial planar vector

fields at infinity by means of the central projection. For more details of the description of this method for

n-dimensional case can be found in [Ve]. Next, we will describe the method in three dimensions.

Let (x1, x2, x3) be the coordinates of R3 and X = P1(x1, x2, x3)

∂∂x1

+P2(x1, x2, x3)∂

∂x2

+P3(x1, x2, x3)∂

∂x3

be a polynomial vector field of degree d = max{deg(P1), deg(P2), deg(P3)}. We consider the sphere S3 =

{(y1, y2, y3, y4) ∈ R4 : y21 + y22 + y23 + y24 = 1}, which we shall call the Poincare sphere with north hemisphere

S3+ = {y ∈ S3 : y4 > 0}, south hemishpere S3− = {y ∈ S3 : y4 < 0} and equator S2 = {y ∈ S3 : y4 = 0}.The central projection from R3 to the Poincare sphere is defined as follows:

f+ : R3 → S3, (x1, x2, x3) 7→(

x1∆(x)

,x2

∆(x),x3

∆(x),

1

∆(x)

)

f− : R3 → S3, (x1, x2, x3) 7→( −x1∆(x)

,−x2∆(x)

,−x3∆(x)

,−1

∆(x)

), (22)

where ∆(x) =√x21 + x22 + x23 + 1. The maps f+ and f− define the following two vectors fields on each

hemisphere

(df+)xX(x) at y = f+(x), and (df−)xX(x) at y = f−(x).

However, these two vector fields can not be extended to the equator S2. Indeed, in the case where y ∈ U1 =

{y ∈ S3 : y1 > 0} where U1 is the local chart with corresponding map:

ϕ1 : U1 → R3, ϕ1(y) =

(y2y1,y3y1,y4y1

)(23)

we shall denote by z = (z1, z2, z3) the value of ϕ1(y), so that z represent different things according to the

case under consideration.

Remark 6.1. There are eight of these charts: Ui = {y ∈ S3 : yi > 0} and Vi = {y ∈ S3 : yi < 0} with

i = 1, 2, 3, 4 and the local maps for the corresponding charts are given by ϕi : Ui → R3, ψi : Vi → R3 with

ϕi(y) = −ψi(y) =

(ynyi,ymyi,ykyi

), for n < m < k and n,m, k 6= i.

By using (22) and (23) we have

z = ϕ1(y) =

(y2y1,y3y1,y4y1

)=

(x2x1,x3x1,1

x1

). (24)

The differential of f+ in the case where y is in the upper hemisphere, is given by the following matrix

{(df+)x}ij =∂zi∂xj

=

−x2

x2

1

1x1

0

−x3

x2

1

0 1x1

− 1x2

1

0 0

.

Therefore

(df+)xX(x) = (1/x1)2 (−x2P1(x) + x2P2(x),−x3P1(x) + x1P3(x),−P1(x)) . (25)

We can write the above expression in terms of the corresponding point on the sphere and we get for (24)

(df+)xX(x) =

(y4y1

)2 {−y2y4P1

(y1y4,y2y4,y3y4

)+y1y4P2

(y1y4,y2y4,y3y4

),

−y3y4P1

(y1y4,y2y4,y3y4

)+y1y4P3

(y1y4,y2y4,y3y4

),−P1

(y1y4,y2y4,y3y4

)}(26)

It is clear that if we multiply the above vector field by the factor yd−14 (it depends only on the degree d of

the polynomial vector field X) then the vector field (26) extends into S2. We will take the same expression

for the induced vector field as (26) in case where we work on lower hemisphere. Actually, if we denote by

12 Marina Statha

X(y) the vector field on S3 \ S2 = S3+ ∪ S3− then to extend X(y) to the Poincare sphere S3, we define the

Poincare compactification of X = (P1, P2, P3) denoted by p(X) and is given as p(X)(y) = yd−14 X(y).

Theorem 6.2. ([Ve]) The vector field p(X) extends X(y) analytically to the whole sphere S3, in such a way

that the equator S2 is invariant.

The important point here is the fact that if we know the behavior of p(X) around the equator, then we

know the behavior of X in the neighborhood of infinity.

Now we will give explicitly the expressions of p(X) in the local charts. It is convenient to express these

fields in terms of the variable z = (z1, z2, z3) of R3. In the chart (U1, ϕ1) we have from (24) that y1/y4 = 1/z3,

y3/y4 = z2/z3, y2/y4 = z1/z3 and also y24 = z23/∆(z)2 where ∆(z) =√z21 + z22 + z23 + 1. From (24) and

because y1 > 0 the y4 has the same sign as z3. Thus y4 = z3/∆(z). Then after some substitutions it turns

out that p(X) can be expressed as

zd3∆(z)d−1

{−z1P1

(1

z3,z1z3,z2z3

)+ P2

(1

z3,z1z3,z2z3

),

−z2P1

(1

z3,z1z3,z2z3

)+ P3

(1

z3,z1z3,z2z3

),−z3P1

(1

z3,z1z3,z2z3

)}. (27)

The expression in the chart (U2, ϕ2) is

zd3∆(z)d−1

{P1

(z1z3,1

z3,z2z3

)− z1P2

(z1z3,1

z3,z2z3

),

−z2P2

(z1z3,1

z3,z2z3

)+ P3

(z1z3,1

z3,z2z3

),−z3P2

(z1z3,1

z3,z2z3

)}. (28)

For the chart (U3, ϕ3) we have

zd3∆(z)d−1

{P1

(z1z3,z2z3,1

z3

)− z1P3

(z1z3,z2z3,1

z3

),

P2

(z1z3,z2z3,1

z3

)− z2P3

(z1z3,z2z3,1

z3

),−z3P3

(z1z3,z2z3,1

z3

)}. (29)

Finally, for the chart (U4, ϕ4) we have {P1(z1, z2, z3), P2(z1, z2, z3), P3(z1, z2, z3)}. We can avoid the use

of the factor 1/∆(z)d−1 in the expression of p(X). Also, note that for the singularities at infinity have

z3 = 0. For the charts (V1, ψ1), (V2, ψ2) and (V3, ψ3) the expression of p(X) is the same as (27), (28) and

(29) multiplied by the factor (−1)d−1.

By the same method we can describe the necessary formulas for the compactified vector field in case

of four-dimensional. We use (z1, z2, z3, z4) as coordinates. If the original polynomial vector field is X =∑4i=1 Pi(x1, x2, x3, x4)

∂∂xi

, with d = max{deg(P1), . . . , deg(P4)} the degree of X then the equations of the

compactified field p(X) are given as follows:

On the chart (U1, ϕ1) it is

zd4∆(z)d−1

{−z1P1

(1

z4,z1z4,z2z4,z3z4

)+ P2

(1

z4,z1z4,z2z4,z3z4

),

−z2P1

(1

z4,z1z4,z2z4,z3z4

)+ P3

(1

z4,z1z4,z2z4,z3z4

),−z3P1

(1

z4,z1z4,z2z4,z3z4

)+ P4

(1

z4,z1z4,z2z4,z3z4

),

−z4P1

(1

z4,z1z4,z2z4,z3z4

)}(30)

The expression in the chart (U2, ϕ2) is

zd4∆(z)d−1

{P1

(z1z4,1

z4,z2z3,z3z4

)− z1P2

(z1z4,1

z4,z2z3,z3z4

),

−z2P2

(z1z4,1

z4,z2z3,z3z4

)+ P3

(z1z4,1

z4,z2z3,z3z4

),−z3P2

(z1z4,1

z4,z2z3,z3z4

)+ P4

(z1z4,1

z4,z2z3,z3z4

),

−z4P2

(z1z4,1

z4,z2z3,z3z4

)}(31)



zd4∆(z)d−1

{P1

(z1z4,z2z4,1

z4,z3z4

)− z1P3

(z1z4,z2z4,1

z4,z3z4

),

P2

(z1z4,z2z4,1

z4,z3z4

)− z2P3

(z1z4,z2z4,1

z4,z3z4

),−z3P3

(z1z4,z2z4,1

z4,z3z4

)+ P4

(z1z4,z2z4,1

z4,z3z4

),

−z4P3

(z1z4,z2z4,1

z4,z3z4

)}(32)


zd4∆(z)d−1

{P1

(z1z4,z2z4,z3z4,1

z4

)− z1P4

(z1z4,z2z4,z3z4,1

z4

),

P2

(z1z4,z2z4,z3z4,1

z4

)− z2P4

(z1z4,z2z4,z3z4,1

z4

),−z3P4

(z1z4,z2z4,z3z4,1

z4

)+ P3

(z1z4,z2z4,z3z4,1

z4

),

−z4P4

(z1z4,z2z4,z3z4,1

z4

)}(33)

Finally for the chart (U5, ϕ5) we have {P1(z1, z2, z3, z4), P2(z1, z2, z3, z4), P3(z1, z2, z3, z4), P4(z1, z2, z3, z4)}.

7. The behavior of the normalized Ricci flow

We analyse the Ricci flow of invariant metrics on the generalizedWallach space, the Stiefel manifolds V2Rn,

V1+k2R1+k2+k3 and the generalized flag manifolds G/K with four isotropy summands and b2(G/K) = 1.

7.1. Ricci flow for the generalized Wallach spaces. By using the Ricci components and scalar curvature

of subsection 3.1 it is easy to see that system (8) reduces to

x1 =1

x2x3(d1 + d2 + d3)

{a1

((d2 + d3)

(x1

2 − x22 − x3

2)− 2d1

(x2

2 + x32))

+x3(2d1x2 + d2(x1 + x2)) + d3x2(x1 + x3)}

x2 =1

x1x3(d1 + d2 + d3)

{− a1d1

(x1

2 + x22 + x3

2)− a2(d1 + d2 + d3)

(x1

2 − x22 + x3

2)

+x3(x1(d1 + 2d2 + d3) + d1x2) + d3x1x2}

x3 =1

x1x2(d1 + d2 + d3)

{− a1d1

(x1

2 + x22 + x3

2)− a3(d1 + d2 + d3)

(x1

2 + x22 − x3

2)

+x1x2(d1 + d2 + 2d3) + x3(d1x2 + d2x1)}.

The above system is not polynomial, hence in order to apply the Poincare compactification, we multiply it

by the factor (d1 + d2 + d3)x1x2x3. Then we obtain the following system:

x1 = x1(−2a1d1x2

2 − 2a1d1x32 + a1d2x1

2 − a1d2x22 − a1d2x3

2 + a1d3x12 − a1d3x2

2 − a1d3x32

+2d1x2x3 + d2x1x3 + d2x2x3 + d3x1x2 + d3x2x3)

x2 = x2(−a1d1x12 − a1d1x2

2 − a1d1x32 − a2d1x1

2 + a2d1x22 − a2d1x3

2 − a2d2x12 + a2d2x2

2

−a2d2x32 − a2d3x12 + a2d3x2

2 − a2d3x32 + d1x1x3 + d1x2x3 + 2d2x1x3 + d3x1x2 + d3x1x3

)

x3 = x3(−a1d1x12 − a1d1x2

2 − a1d1x32 − a3d1x1

2 − a3d1x22 + a3d1x3

2 − a3d2x12 − a3d2x2

2

+a3d2x32 − a3d3x1

2 − a3d3x22 + a3d3x3

2 + d1x1x2 + d1x2x3 + d2x1x2 + d2x1x3 + 2d3x1x2)(34)

If we denote by Pi(x1, x2, x3) the xi, for i = 1, 2, 3 respectively, then the degree of the vector field X =∑3i=1 Pi(x1, x2, x3)

∂∂xi

is 3. Now we study the system (34) at infinity. We apply the Poincare compactification

14 Marina Statha

to the above system written in the chart (U1, ϕ1) as follows:

z1 = z1(d1 + d2 + d3)((z1

2 − 1)(a1 + a2) + z2

2(a1 − a2)− z1z2 + z2)

z2 = z2(d1 + d2 + d3)(a1

(z1

2 + z22 − 1

)+ a3

(−z12 + z2

2 − 1)+ z1(−z2) + z1

)

z3 = z3(a1

(2d1

(z1

2 + z22)+ d2

(z1

2 + z22 − 1

)+ d3

(z1

2 + z22 − 1

))

−z2(z1(2d1 + d2 + d3) + d2)− d3z1) . (35)

In order to find the singularities at infinity of the above system we set z3 = 0. We will substitute into

(35), the values of the dimensions d1, d2, d3 and a1, a2, a3 from Tables 1 and 2 respectively. Then we have

the following:

For the generalized Wallach space GWS.1 SO(k + l+m)/ SO(k)× SO(l)× SO(m) system (35) comes

z1 = z1(−k(l +m)− lm)(2(z1 − 1)z2(k + l +m− 2) +

(z1

2 − 1)(−l−m) + z2

2(l −m))

z2 = z2(−k(l +m)− lm)(k(2z1z2 + (z1 − 1)2 − z2

2)+ 2(l− 2)z1(z2 − 1)

−m(−2z1z2 + z1(z1 + 2) + z2

2 − 1)). (36)

Next, we will study the case l = m.

Proposition 7.1. For the generalized Wallach space SO(k+2m)/ SO(k)×SO(m)×SO(m), the normalized

Ricci flow has for m > 2 and k2 > 4(m− 1), precisely four singularities at infinity.

Proof. We have m = l so system (35) can be written as

z1 = f1(z1, z2) = 2m(z1 − 1)z1(2k +m)(m(z1 + 1)− z2(k + 2m− 2)) = 0

z2 = f2(z1, z2) = −mz2(2k +m)(k(2z1z2 + (z1 − 1)2 − z22) + 4(m− 1)z1z2

−mz1(z1 + 4)−mz22 +m+ 4z1) = 0. (37)

From the first equation we obtain z1 = 1 and by substituting into the (37) we obtain the solutions

z2 =k + 2m− 2±

√k2 − 4m+ 4

k +m,

hence we obtain two solutions (1, z2). Now, if (m(z1 + 1)− z2(k + 2m− 2)) = 0 then it is possible to obtainexplicit solutions for z1, z2, but these are complicated expressions involving radicals. So we use Grobnerbases to prove existence of at least two solutions. We consider a polynomial ring R = Q[z, z1, z2] and anideal I generated by the polynomials {f1(z1, z2), f2(z1, z2), z z1 z2 (z1 − 1) − 1}. We take a lexicographicordering > with z > z1 > z2 for a monomial ordering on R. Then, with the aid of computer, we see that aGrobner basis for the ideal I contains the polynomial H(z2) of z2, which is given by

H(z2) = 8k2m3 + (−8k3m2− 28k2m3 + 24k2m2

− 28km4 + 44km3− 16km2

− 8m5 + 16m4− 8m3)z2

+(2k4m+ 11k3m2− 8k3m+ 19k2m3

− 28k2m2 + 8k2m+ 13km4− 28km3 + 12km2 + 3m5

−8m4 + 4m3)z22 + 12km4 + 8km3 + 4m5

− 4m4.

We compute H(0) = 8k2m3 + 12km4 − 8km3 + 4m5 − 4m4 > 0, H(1) = 2k4m+ 3k3m2 − 8k3m− k2m3 −4k2m2 + 8k2m − 3km4 + 8km3 − 4km2 −m5 + 4m4 − 4m3 < 0 and H(2) = 8k4m + 28k3m2 − 32k3m +28k2m3−64k2m2+32k2m+8km4−32km3+16km2−4m4 which is positive for m > 2. Thus, there exist atleast two solutions z2 = α and z2 = β of H(z2) = 0 with 0 < α < 1 and 1 < β < 2. Now, for positive valuesz2 = α, β we obtain from (m(z1 + 1)− z2(k + 2m− 2)) = 0 real values z1 = γ, δ as solutions of (37). Next,we consider a lexicographic ordering > with z > z2 > z1 for a monomial ordering on the ring R. Then, aGrobner basis for the ideal I contains the polynomial G(z1) of z1, which can be written as

G(z1) = (6k4 + 75k3 + (19k2 + 128k + 178)(m− 3)3 + 285k2 + (11k3 + 143k2 + 462k + 414)(m− 3)2

+(2k4 + 58k3 + 353k2 + 720k + 459)(m − 3) + (13k + 37)(m− 3)4 + 405k + 3(m− 3)5 + 189)z21 + (−12k4

−126k3 + (−46k2− 352k − 556)(m − 3)3 − 558k2 + (−22k3

− 322k2− 1220k − 1316)(m − 3)2 + (−4k4

− 108k3

−738k2− 1888k − 1578)(m − 3) + (−38k − 118)(m− 3)4 − 1110k − 10(m − 3)5 − 774)z1 + 6k4 + 75k3 + (19k2

+128k + 178)(m − 3)3 + 285k2 + (11k3 + 143k2 + 462k + 414)(m − 3)2 + (2k4 + 58k3 + 353k2 + 720k

+459)(m − 3) + (13k + 37)(m− 3)4 + 405k + 3(m− 3)5 + 189.


Then we see that, for m ≥ 3, the coefficients of the polynomial G(z1) are positive for even degree terms and

negative for odd degree terms. Thus, if the equation G(z1) = 0 has real solutions, then these are all positive.

In particular, z1 = γ, δ are positive. So far we have proved existence of at least four solutions. However, by

Theorem 3.2 the number of invariant Einstein metrics is at most four, hence the result follows. �

Remark 7.2. If k2 = 4(m − 1), then k and m are of the form (k,m) = (2ρ, ρ2 + 1), ρ > 1, and according

to Example 1, p. 56 in [LoNiFi], there exist precisely three singularities at infinity. Also, if m = 2 then

k = 2, this corresponds to the space SO(6)/ SO(2) × SO(2) × SO(2). It easy to see that the system (37)

has a unique singularity at infinity. This singularity corresponds to the unique invariant Einstein metric (cf.

[LoNiFi], [ChNi]).

For the space GWS.2 SU(k + l +m)/ SU(k)× SU(l)× SU(m) system (35) becomes:

z1 = z1(−k(l +m)− lm)(2(z1 − 1)z2(k + l +m) +

(z1

2 − 1)(−l −m) + z2

2(l −m)),

z2 = z2(−k(l +m)− lm)(2z1z2(k + l+m)− z22(k +m) + kz1

2 − 2kz1 + k − 2lz1 −mz12 − 2mz1 +m).

The solutions are: ((k+m)/(k+ l), (l+m)/(k+ l)), ((k+m)/(k+ l), (2k+ l+m)/(k+ l)), ((k+2l+m)/(k+

l), (l +m)/(k + l)) and ((k +m)/(k + l + 2m), (l +m)/(k + l + 2m)).

For the generalized Wallach space GWS.3 Sp(k + l +m)/ Sp(k)× Sp(l)× Sp(m) the system (35) comes:

z1 = z1(−k(l +m)− lm)(2(z1 − 1)z2(k + l +m+ 1) +

(z1

2 − 1)(−l−m) + z2

2(l −m))

z2 = z2(−k(l +m)− lm)(k(2z1z2 + (z1 − 1)2 − z2

2)+ 2(l+ 1)z1(z2 − 1)

−m(−2z1z2 + z1(z1 + 2) + z2

2 − 1)).

By similar method as in Proposition 7.1 we can prove the following:

Proposition 7.3. For the generalized Wallach space Sp(k + 2m)/ Sp(k) × Sp(m) × Sp(m), the normalized

Ricci flow has precisely four singularities at infinity.

It is easy to see that for some values of k, l and m the above system has four solutions. For example,

(1) For (k, l,m) = (1, 2, 3) the solutions are: (3.26361, 1.60389), (1.30670, 3.18223), (1.23251, 1.39606)

and (0.38050, 0.46780).

(2) For (k, l,m) = (2, 5, 7) the solutions are: (2.94748, 1.67504), (1.27217, 2.71689), (1.24716, 1.53155)

and (0.40168, 0.52944).

GWS.4 SU(2l)/U(l)

z1 = 2(3l2 − 1

)z1

(l(z1 − 1)(z1 − 2z2 + 1) + z2

2)

z2 =(3l2 − 1

)z2

(2l(−2z1(z2 − 1) + z2

2 − 1)+ z1

2 + z22 − 1

). (38)

Proposition 7.4. For the generalized Wallach space SU(2l)/U(l), the normalized Ricci flow has for l ≥ 2

at least two singularities at infinity.

Proof. We consider a polynomial ring R = Q[z, z1, z2] and an ideal I generated by the polynomials {z1 =(3l2 − 1

)z1

(l(z1 − 1)(z1 − 2z2 + 1) + z2

2)z2 =

(3l2 − 1

)z2

(2l(−2z1(z2 − 1) + z2

2 − 1)+ z1

2 + z22 − 1

),

z z1 z2 − 1}. We take a lexicographic ordering > with z2 > z1 for a monomial ordering on R. Then, with

the aid of computer we see that a Grobner basis for the ideal I contains the polynomial F (z1) of z1, which

is given by

F (z1) = (3l2 − 1)12l4z14 − 48l4z1

3 + 72l4z12 − 48l4z1 + 12l4 − 20l3z1

4 + 40l3z13 − 40l3z1 + 20l3

+7l2z14 + 8l2z1

3 − 34l2z12 + 8l2z1 + 7l2 + 2lz1

4 − 8lz13 + 8lz1 − 2l− z1

4 + 2z12 − 1.

We compute F (0) = 12l4+20l3+7l2−2l−1 > 0,H(1) = −4l2 < andH(4) = 972l4−2700l3+1799l2+30l−225,

which is positive for l ≥ 2. Thus, there exist at least two solutions z1 = α and z1 = β of F (z1) = 0

with 0 < α1 < 1 and 1 < β1 < 4. The Grobner basis for the ideal I contains also the polynomial

(3l2−1)z2− (3l2−1)a(z1, k) = 0. Thus for the positive values z1 = α1, β1 found above, we obtain real values

z2 = α2, β2 as solutions of the system (38). Now we consider a lexicographic ordering > with z1 > z2 for a

16 Marina Statha

monomial ordering on the ring R. Then, a Grobner basis for the ideal I contains the polynomial polynomial

(3l2 − 1)G(z2), where G(z2) is a polynomial of of z2, which can be written as

G(z2) = (12(l− 2)4 + 76(l− 2)3 + 175(l− 2)2 + 174(l− 2) + 63)z42

(−48(l− 2)4 − 336(l− 2)3 − 868(l− 2)2 − 980(l− 2)− 408)z32

(72(l− 2)4 + 540(l− 2)3 + 1508(l− 2)2 + 1856(l− 2) + 848)z22

(−48(l− 2)4 − 376(l− 2)3 − 1104(l− 2)2 − 1440(l− 2)− 704)z2

12(l− 2)4 + 96(l − 2)3 + 288(l− 2)2 + 384(l− 2) + 192.

Then we see that, forl ≥ 2,the coefficients of the polynomial G(z2) are positive for even degree terms and

negative for odd degree terms. Thus if the equation G(z2) = 0 has real solutions, then these are all positive.

In particular, z2 = α2, β2 are positive. �

GWS.5 SO(2l)/U(1)×U(l − 1)

z1 = 2(l + 2)(z1 − 1)z1((l − 2)(z1 + 1)− 2(l − 1)z2),

z2 = (l + 2)z2(l(−4z1z2 + z1(z1 + 4) + z2

2 − 1)− 4z1(z1 − z2 + 1)

)

The solutions are: (1, 2), (1, (2l− 4)/l), (l/(3l− 4), 2(l− 2)/(3l− 4)) and ((3l − 4)/l, 2(l− 2)/l).

GWS.6 E6 / SU(4)× SU(2)× SU(2)×U(1)

z1 = 28(z1 − 1)z1(z1 − 2z2 + 1), z2 = (14/3)z2(z1

2 − 12z1(z2 − 1) + 5z22 − 5

).

The solutions are: (0.6, 0.8) and (1.66667, 1.33333).

GWS.7 E6 / SO(8)× U(1)×U(1)

z1 = 16(z1 − 1)z1(z1 − 3z2 + 1), z2 = 16z2(−3z1(z2 − 1) + z2

2 − 1).

The solutions are: (1, 1), (2, 1), (1, 2) and (1/2, 1/2).

GWS.8 E6 / Sp(3)× SU(2)

z1 = (27/4)z1(3z1

2 − 8z1z2 + z2(z2 + 8)− 3), z2 = −(9/4)z2

(z1

2 + 24z1(z2 − 1)− 13z22 + 13

).


GWS.9 E7 / SO(8)× SU(2)× SU(2)× SU(2)

z1 = (32/3)(z1 − 1)z1(4z1 − 9z2 + 4), z2 = (32/3)(z2 − 1)z2(−9z1 + 4z2 + 4).

The solutions are: (1, 1), (1.25, 1), (1, 1.25) and (0.8, 0.8).

GWS.11 E7 / SO(8)

z1 = (35/3)(z1 − 1)z1(5z1 − 9z2 + 5), z2 = (35/3)(z2 − 1)z2(−9z1 + 5z2 + 5).

The solutions are: (1, 1), (1, 0.8), (0.8, 1) and (1.25, 1.25).

GWS.12 E8 / SO(12)× SU(2)× SU(2)

z1 = (176/5)(z1 − 1)z1(2z1 − 5z2 + 2), z2 = −(176/5)z2(z1

2 + 15z1(z2 − 1)− 7z22 + 7

).

The solutions are: (1, 1.456083) and (1, 0.686773).

GWS.13 E8 / SO(8)× SO(8)

z1 = (64/5)(z1 − 1)z1(8z1 − 15z2 + 8), z2 = (64/5)(z2 − 1)z2(−15z1 + 8z2 + 8).

The solutions are: (1, 1), (1, 0.875), (0.875, 1) and (1.142857, 1.142857).

GWS.14 F4 / SO(5)× SU(2)× SU(2)

z1 = 4(z1 − 1)z1(5z1 − 9z2 + 5), z2 = 2z2(−18z1z2 + 3z1(z1 + 6) + 7z2

2 − 7)


GWS.15 F4 / SO(8)

z1 = (8/3)(z1 − 1)z1(2z1 − 9z2 + 2), z2 = (8/3)(z2 − 1)z2(−9z1 + 2z2 + 2).

The solutions are: (1, 1), (3.5, 1), (1, 3.5) and (0.285714, 0.285714).


Since we work on the chart (U1, ϕ1) that corresponds to the plane y1 = 1, we consider metrics which are

of the form (1, α, β), where α, β are the solutions of system (35) for z3 = 0. These metrics are invariant

Einstein metrics on M . Thus, we have proved the following:

Theorem 7.5. Let G/H be a generalized Wallach space. The normalized Ricci flow, on the space of invariant

Riemannian metrics on G/H, possesses a finite number of singularities at infinity. These fixed points

correspond to the G-invariant Einstein metrics on G/H (cf. Subsection 3.1).

7.2. Ricci flow for the Stiefel manifold V2Rn. By using the Ricci components (14) and scalar curvature

(15) of the metric (13) it is easy to see that, for the Stiefel manifold V2Rn system (8) reduces to

x0 =(n− 2)x0

2 + (n− 2)x0(x1 + x2)− (n− 1)(x1 − x2)2

(2n− 3)x1x2

x1 =(5− 3n)x0

2 + 2(n− 2)x0((n− 2)x1 + (3n− 5)x2) + (x1 − x2)((n− 1)x1 + (3n− 5)x2)

2(n− 2)(2n− 3)x0x2

x2 =(5− 3n)x0

2 + 2(n− 2)x0((3n− 5)x1 + (n− 2)x2)− (x1 − x2)((3n− 5)x1 + (n− 1)x2)

2(n− 2)(2n− 3)x0x1.

We observe that the above system is not a polynomial system, therefore we cannot apply the Poincare

compactification directly. We multiply it by the factor 2(n− 2)(2n− 3)x0x1x2, with this multiplication will

only change the time of parametrization of the orbits and not the structure of the phase portrait. After that

we take the following system:

x0 = (n− 2)(nx0

2 + nx0x1 + nx0x2 − nx12 + 2nx1x2

−nx22 − 2x02 − 2x0x1 − 2x0x2 + x1

2 − 2x1x2 + x22)x0

x1 = 1/2(2n2x0x1 + 6n2x0x2 − 3nx0

2 − 8nx0x1 − 22nx0x2

+nx12 + 2nx1x2 − 3nx2

2 + 5x02 + 8x0x1 + 20x0x2 − x1

2 − 4x1x2 + 5x22)x1

x2 = 1/2(6n2x0x1 + 2n2x0x2 − 3nx0

2 − 22nx0x1 − 8nx0x2

−3nx12 + 2nx1x2 + nx2

2 + 5x02 + 20x0x1 + 8x0x2 + 5x1

2 − 4x1x2 − x22)x2. (39)

If we denote by Pi(x0, x1, x2) the xi, for i = 0, 1, 2 respectively, then the degree of the vector field

X =∑2

i=0 Pi(x0, x1, x2)∂

∂xiis 3. From the above system it is easy to see the following lemma:

Lemma 7.6. The coordinate planes along with the straight line γ(t) = ((2(n− 2)/(n− 1))t, t, t), remain

invariant under the flow defined by the system of (39).

To study the singularities at infinity of (39), we should write this system in the local charts of the Poincare

compactification. Because we are interesting for positive values of x0, x1 and x2 we study the behavior of

the previous system only on the chart (U1, ϕ1). Therefore from (27) we have:

z1 = 1/2(2n− 3)z1((n− 1)

(z1

2 − 1)− 2(n− 2)(z1 − 1)z2 + (n− 3)z2

2)

z2 = 1/2(2n− 3)z2(n(−2z1z2 + z1(z1 + 2) + z2

2 − 1)− 3z1

2 + 4z1(z2 − 1)− z22 + 1

)

z3 = (n− 2)z3(n(z1

2 − z1(2z2 + 1) + (z2 − 1)z2 − 1)− (z1 − z2)

2 + 2(z1 + z2 + 1)). (40)

In order to find the singularities at the infinity of the above system we set z3 = 0. Then it is easy to see that

the system {z1 = 0, z2 = 0} has only one solution namely (z1, z2) = ((n− 1)/2(n− 2), (n− 1)/2(n− 2)).

Since we work on the chart (U1, ϕ1) that corresponds to the plane y1 = 1, we consider metrics whose are

the form (1, α, β), where α, β are the solution of the system (40) for z3 = 0. These metrics are invariant

Einstein metrics on M . We have thus proved the following.

Theorem 7.7. Let G/H be the Stiefel manifold V2Rn. The normalized Ricci flow on the space of invariant

Riemannian metrics on G/H, possesses exactly one singularity at infinity. This fixed point corresponds to

the unique (up to scale) G-invariant Einstein metric on G/H (cf. Theorem 4.3).

18 Marina Statha

7.3. Ricci flow on the Stiefel manifold V1+k2Rn. We study the behavior of the normalized Ricci flow

for the Stiefel manifold V1+k2Rn. For this case we take the Ricci components (17) and scalar curvature (18)

of the metric (16). Then, system (8) reduces to

x2 = − 1

2(k2 + 1)(k2 − 2n+ 2)(n− 2)x122x13x232

{4(x23

2(−x13x22(k22 − (k2 + 1)n+ k2 + 1)

+k2x122x13(k2(n− 3) + n) + 2(n− 2)x12x2(x12(−k2 + n− 1) + k2x13)) + x12

2x13x22(k2

−n+ 1)(k22 − (k2 + 1)n+ k2 + 1) + k2x12x2x23(k2 − n+ 1)(−2(n− 2)x12x13 + x12

2 + x132)

+k2x12x2x233(k2 − n+ 1))

}.

x12 = − 1

2(k2 + 1)(n− 2)(k2 − 2n+ 2)x12x13x2x232

{x23(x12

3(−k2 + n− 1) + x12(x132(k2 − n+ 1)

+x232(k2 − n+ 1) + 2(n− 2)x13x23)− (k2 − 1)x13x2x23) + 2k2x12x2x23(k2 − n+ 1)(−2(n

−2)x12x13 + x122 + x13

2) + x232(4(n− 2)x12x2(x12(−k2 + n− 1) + k2x13) + (k2 − 2)(k2 − 1)k2

x122x13 − (k2 − 1)k2x13x2

2) + (k2 − 1)k2x122x13x2

2(k2 − n+ 1) + 2k2x12x2x233(k2 − n+ 1)

}.

x13 = − 1

2(k2 + 1)(n− 2)(k2 − 2n+ 2)x122x2x232

{2k2x12x2x23(k2 − n+ 1)(−2(n− 2)x12x13 + x12

2

+x132) + x12x23(k2(x12

2 − x132 + x23

2)− 2(n− 2)x12x23) + x232(4(n− 2)x12x2(x12(−k2 + n

−1) + k2x13) + (k2 − 2)(k2 − 1)k2x122x13 − (k2 − 1)k2x13x2

2) + (k2 − 1)k2x122x13x2

2(k2 − n

+1) + 2k2x12x2x233(k2 − n+ 1)

}.

x23 = − 1

2(k2 + 1)(n− 2)(k2 − 2n+ 2)x122x13x2x23

{x12x2x23

3(k22 − k2 + 2n− 2) + x12x2x23(k2(3k2

−4n+ 5)− 2n+ 2)(−2(n− 2)x12x13 + x122 + x13

2) + x232(4(n− 2)x12x2(x12(−k2 + n− 1)

+k2x13) + (k2 − 2)(k2 − 1)k2x122x13 − (k2 − 1)k2x13x2

2)

+(k2 − 1)x122x13x2

2(2(k2 + 1)2 − (3k2 + 2)n)}.

The above system is not polynomial, hence in order to apply the Poincare compactification, we multiply it

by the factor 2(k2 + 1)(k2 + k3 − 1)(−k2 − 2k3)x122x13x2x

223. Then we obtain the following system:

x2 = 2x2(k2k3x12x2x23(2x12x13(k2 + k3 − 1)− x122 − x13

2) + x232((k2 − 2)x12

2x13(k2(k2 + k3)

+k3) + 2x12x2(k2 + k3 − 1)(k2x13 + k3x12) + x13x22(k2k3 + k2 + k3)) +

k3x122x13x2

2(k2k3 + k2 + k3)− k2k3x12x2x233)

x12 = x12(k23x13x23

2(x122 + 2x12x2 − x2

2) + k22(k3x12x2x23(x12

2 + 4x12x13 − x132)

−k3x122x13x22 − x13x232(3x12 − x2)(x12 − (2k3 + 1)x2)− k3x12x2x23

3)

+k2(2k32x12x2x23(x12

2 + 2x12x13 − (x13 − x23)2) + k3x12x2(x12x13x2 + 2x23

2(2x12 + 3x13)

−x23(x12 + x13)(x12 + 3x13)− 3x233) + 2x13x23

2(x122 − 3x12x2 + x2

2)) + 2k3x2x23(k3

x12(x122 + 2x12x23 − (x13 − x23)

2)− 2x12x23(x12 + x13) + x13x2x23))

x13 = x13(k23x12x23(x12

2(−x2) + x12x23(x13 + 2x2) + x2(x13 − x23)(x13 + x23))− k22(k3x12x2(2x23

(x122 − 2x12x13 − x13

2) + x12x13x2 − 6x12x232 + 2x23

3) + x23(x13x23(x12 − x2)(3x12 − x2)

+x12x2(x12 − x13)(x12 + x13) + x12x2x232)) + k2(4k3

2x122x2x23(x13 + x23) + k3x12x2(x12x13x2

+2x232(3x12 + 2x13)− 4x12x23(x12 + x13)− 4x23

3) + x232(2x12

2x13 − 2x12x2(x12 + 2x13)

+x13x22)) + 8(k3 − 1)k3x12

2x2x232)

x23 = x23(x12x2x23(k22 + 4k2k3 + k2 + 2k3)(2x12x13(k2 + k3 − 1)− x12

2 − x132) + (1 − k2)x12

2x13

x22(k2

2 + 3k2k3 + k2 + 2k3) + x12x2x233(k2

2 + k2 + 2k3) + x232(4x12x2(k2 + k3 − 1)(k2x13

+k3x12) + (k2 − 2)(k2 − 1)k2x122x13 − (k2 − 1)k2x13x2

2)) (41)


If we denote by P2(x1, x2, x3, x4) the x2, and by Pij(x2, x12, x13, x23) the xij for i < j = 1, 2, 3 respectively

then the degree of vector field X = P2(x2, x12, x13, x23)∂

∂x2+∑

i<j=1,2,3 Pij(x2, x12, x13, x23)∂

∂xijis 6. Now,

in order to study the singularities at infinity of (41), we should write this system in the local charts of thePoincare compactification. Because we are interested for positive values of x2, x12, x13 and x23 we study thebehavior of the previous system only in to the chart (U1, ϕ1). Therefore from (30) we have:

z1 = (k2 + 1)z1(k2 + 2k3)((z1 − 1)z22z3(2z1 − k2(z1 − 1)) + k3z1(z1

2z2 − z1z3 − z2(z2 − z3)2))

z2 = (k2 + 1)z2(k2 + 2k3)(−z3(z12(−2z2(k2 + k3 − 1) + (k2 − 2)z2

2 + k2 + k3 − 1) + z22)

−z1z2z32 + z1z2(z2 − z1)(z1 + z2))

z3 = (−k2 − 1)z3(k2 + 2k3)(k2z1z2(z1z2z3 + (z1 − z2)2− z3

2) + z12(k3z3 − 2z2

2(k3 + z3 − 1)) + z22z3)

z4 = 2z4(−z22((k2 − 2)z1

2z3(k2(k2 + k3) + k3) + 2z1(k2 + k3 − 1)(k2z3 + k3z1) + z3(k2k3 + k2 + k3))

+k2k3z1z2(−2z1z3(k2 + k3 − 1) + z12 + z3

2)− k3z12z3(k2k3 + k2 + k3) + k2k3z1z2

3) (42)

Theorem 7.8. Let the Stiefel manifold V1+k2Rn ∼= SO(n)/ SO(k3), with n = 1 + k2 + k3. The normalized

Ricci flow on the space of invariant Riemannian metrics on SO(1 + 2k2)/ SO(k2), for k2 ≥ 6 has at least

four singularities at infinity.

Proof. We have k2 = k3 so the system (42) can be written as follows

z1 = 3(−k2 − 1)k2z1(k2z1(z12(−z2) + z1z3 + z2(z2 − z3)

2) + (z1 − 1)z22z3(k2(z1 − 1)− 2z1)) = 0

z2 = 3(−k2 − 1)k2z2(z12z3((k2 − 2)z2

2 + (2− 4k2)z2 + 2k2 − 1) + z13z2 + z1z2(z3

2 − z22) + z2

2z3) = 0

z3 = 3(−k2 − 1)k2z3(k2z1(z12z2 + z1(z2

2(z3 − 4) + z3) + z23 − z2z3

2) + z22(z3 − 2z1

2(z3 − 1))) = 0.(43)

We consider a polynomial ring R = Q[z, z1, z2, z3] and a ideal I generated by the polynomials {z1, z2, z3, z z1z2 z3 − 1}. We take a lexicographic order > with z > z3 > z2 > z1 for a monomial ordering on R. Then, bythe aid of computer, we see that a Grobner basis for the I contains the polynomial k2(1 + k2)(z1 − 1)F(z1)where F1(z1) given by

F1(z1) = 4k212 + 48k2

11− 176k2

10 + 96k29 + 248k2

8− 320k2

7 + 16k26 + 160k2

5− 92k2

4 + 16k23 + (289k2

12

−2414k211 + 7267k2

10− 7656k2

9− 4925k2

8 + 16858k27− 10115k2

6− 1924k2

5 + 4556k24− 4032k2

3

+2864k22− 832k2 + 64)z1

10 + (−3094k212 + 21740k2

11− 53420k2

10 + 43474k29 + 25150k2

8− 59760k2

7

+32856k26− 11070k2

5 + 1676k24 + 4944k2

3− 3104k2

2 + 672k2 − 64)z19 + (11951k2

12− 62636k2

11 + 87727k210

+63904k29− 306211k2

8 + 405020k27− 357771k2

6 + 244112k25− 115968k2

4 + 37280k23− 9200k2

2 + 1664k2

−128)z18 + (−23556k2

12 + 81980k211 + 1572k2

10− 381432k2

9 + 817168k28− 1006372k2

7 + 869836k26

−522872k25 + 212404k2

4− 61192k2

3 + 12816k22− 1760k2 + 128)z1

7 + (27243k212

− 54638k211

− 133224k210

+635796k29− 1177180k2

8 + 1342266k27− 995716k2

6 + 467944k25− 133163k2

4 + 19352k23 + 376k2

2− 672k2

+80)z16 + (−20366k2

12 + 17416k211 + 186738k2

10− 653354k2

9 + 1066178k28− 1003674k2

7 + 537954k26

−128838k25− 18604k2

4 + 23530k23− 7564k2

2 + 1176k2 − 80)z15 + (10457k2

12 + 3504k211

− 152744k210

+427896k29− 550638k2

8 + 347496k27− 51536k2

6− 73272k2

5 + 54525k24− 17048k2

3 + 2464k22− 64k2

−16)z14 + (−3592k2

12− 7144k2

11 + 75380k210

− 160756k29 + 138300k2

8− 14956k2

7− 63900k2

6 + 50796k25

−15124k24 + 196k2

3 + 1016k22− 232k2 + 16)z1

3 + (768k212 + 3264k2

11− 20788k2

10 + 32264k29− 10448k2

8

−20736k27 + 23488k2

6− 7528k2

5− 1728k2

4 + 1904k23− 508k2

2 + 48k2)z12 + (−88k2

12− 656k2

11 + 3024k210

−3152k29− 1552k2

8 + 4816k27− 2496k2

6− 624k2

5 + 1064k24− 384k2

3 + 48k22)z1.

If z1 = 1, then from z1 = 0 in system (43) we take z2 = z3 and z3 = (z22 − 1)/z2 (this does not give a

positive solution). By substituting z2 = z3 back into (43) we find the real solutions

z1 = 1, z2 = z3 =−√2k2

2 − 2k2 + 1 + 2k2 − 1

k2 − 1and z1 = 1, z2 = z3 =

√2k2

2 − 2k2 + 1 + 2k2 − 1

k2 − 1.

These are known as Jensen’s Einstein metrics on Stiefel manifolds. Now if z1 6= 1 then F1(z1) = 0 and we

claim that the equation F1(z1) = 0 has at least two positive roots. We see that F1(1) = 16k212 + 464k2

11 +

1356k210 − 2920k2

9 − 3910k28 + 10638k2

7 − 17384k26 + 16884k2

5 − 10454k24 + 4566k2

3 − 792k22 > 0 and

for k2 ≥ 6 we have F1(3) < 0 and F1(6) > 0. Thus for k2 ≥ 6 we have at least two solutions z1 = α1, β1 ofF1(z1) = 0 with 1 < α1 < 3 and 3 < α2 < 6. Next, we consider the ideal J generated by the {z1, z2, z3, z z1z2 z3 (z1 − 1) − 1} and take a lexicographic order > with z > z2 > z3 > z1 for a monomial ordering on R.

20 Marina Statha

Then, by the aid of computer, we see that a Grobner basis for the ideal J contains the polynomials F1(z1) and

q3(k2)z3 =∑9

ℓ=0 pℓ(k2)zℓ1. Also, for the same ideal J and the lexicographic order > with z > z3 > z2 > z1

for monomials on R, we see that a Grobner basis for J contains the polynomial q2(k2)z2 =∑9

ℓ=0 pℓ(k2)zℓ1,

where qi(k2) (i = 2, 3) and pℓ(k2) (ℓ = 0, 1, . . . , 9) are polynomials of k2 of degree 58 and 60 respectively withinteger coefficients. It is easy to see that qi(k2) 6= 0 (i = 2, 3) for k2 > 1. Thus, for the positive solutionsz1 = α1, β1 found above we obtain real values z2 = α2, β2 and z3 = α3, β3 as solutions of the system (43).We claim that α2, β2, α3, β3 are positive. Take a lexicographic order > with z > z1 > z3 > z2 for a monomialordering on the ring R. Then, a Grobner basis for the ideal J contains a polynomial F2(z2) of z2, which isgiven by

F2(z2) = (2680947582(k2 − 6)7 + 14591972187(k2 − 6)6 + 59640424960(k2 − 6)5 + 180449511648(k2 − 6)4

+392350767872(k2 − 6)3 + 580027773696(k2 − 6)2 + 522221572096(k2 − 6) + 216162222080)z102

+(−50225598762(k2 − 6)7 − 276982406346(k2 − 6)6 − 1149224425652(k2 − 6)5 − 3535515192656(k2 − 6)4

−7827543360064(k2 − 6)3 − 11798003601664(k2 − 6)2 − 10842357528576(k2 − 6) − 4585758048256)z92

+(390159654498(k2 − 6)7 + 2172262119880(k2 − 6)6 + 9119121333384(k2 − 6)5 + 28438722948740(k2 − 6)4

+63932956074624(k2 − 6)3 + 97997952225152(k2 − 6)2 + 91718708609024(k2 − 6) + 39559071548416)z82

+(−1636143378304(k2 − 6)7 − 9149184702208(k2 − 6)6 − 38667393707288(k2 − 6)5 − 121655123121672(k2 − 6)4

−276434836684176(k2 − 6)3 − 429032167222784(k2 − 6)2 − 407238153983232(k2 − 6) − 178416492097536)z72

+(4077237826688(k2 − 6)7 + 22741035201568(k2 − 6)6 + 96082339120904(k2 − 6)5 + 302818376659728(k2 − 6)4

+690563967560064(k2 − 6)3 + 1077488181695360(k2 − 6)2 + 1029907447940096(k2 − 6) + 455097671436288)z62

+(−6367615594352(k2 − 6)7 − 35174632043952(k2 − 6)6 − 147420550162112(k2 − 6)5 − 461548083948832(k2 − 6)4

−1046969395743360(k2 − 6)3 − 1626943248550784(k2 − 6)2 − 1550563723207680(k2 − 6) − 683916568270848)z52

+(6500710198416(k2 − 6)7 + 35468514250320(k2 − 6)6 + 146902006546624(k2 − 6)5 + 454750667573776(k2 − 6)4

+1020485202419072(k2 − 6)3 + 1569594503117888(k2 − 6)2 + 1481383976485632(k2 − 6) + 647378577709056)z42

+(−4376025268608(k2 − 6)7 − 23628587384704(k2 − 6)6 − 96812750676960(k2 − 6)5 − 296392431072480(k2 − 6)4

−657659598211392(k2 − 6)3 − 1000042532332416(k2 − 6)2 − 933014709911808(k2 − 6)− 403027461872640)z32

+(1873316885024(k2 − 6)7 + 10053816339984(k2 − 6)6 + 40911319513376(k2 − 6)5 + 124311949165760(k2 − 6)4

+273618707342592(k2 − 6)3 + 412536333595392(k2 − 6)2 + 381469386539520(k2 − 6) + 163261397836800)z22

+(−460264120480(k2 − 6)7 − 2465535026848(k2 − 6)6 − 10005481382592(k2 − 6)5 − 30298379604480(k2 − 6)4

−66422377562112(k2 − 6)3 − 99697425799680(k2 − 6)2 − 91739835878400(k2 − 6)− 39057790464000)z2

49150850208(k2 − 6)7 + 263520189632(k2 − 6)6 + 1069436792448(k2 − 6)5 + 3236342259456(k2 − 6)4

+7086348357120(k2 − 6)3 + 10618426598400(k2 − 6)2 + 9750544128000(k2 − 6) + 4141186560000

Then we see that, for k2 ≥ 6, the coefficients of the polynomial F2(z2) are positive for even degree andnegative for odd degree terms. Thus if the equation F2(z2) = 0 has real solutions, then these are all positive.In particular z2 = α2, β2 are positive. Now we take a lexicographic order > with z > z2 > z1 > z3 for amonomial ordering on R. Then, by the aid of computer, we see that a Grobner basis for the ideal J containsthe polynomialF3(z3), where the polynomial F3(z3) can be written as

F3(z3) = (21286672726062828160(k2 − 6)7 + 58254716280111893248(k2 − 6)6 + 127455519890994757632(k2 − 6)5

+217655806674022477824(k2 − 6)4 + 279537041132943605760(k2 − 6)3 + 253958157960834252800(k2 − 6)2

+145499460899831808000(k2 − 6) + 39531747173990400000)z103 + (−292777902042514008608(k2 − 6)7

−814588260579033355648(k2 − 6)6 − 1811976784418141538304(k2 − 6)5 − 3146205600377563549696(k2 − 6)4

−4108991272662490095616(k2 − 6)3 − 3796725395351760764928(k2 − 6)2 − 2212811038797252526080(k2 − 6)

−611729579268322099200)z93 + (1660257266722947751552(k2 − 6)7 + 4695714013629690519424(k2 − 6)6

+10616322476744264890880(k2 − 6)5 + 18734391448720331455488(k2 − 6)4 + 24867099690573745029120(k2 − 6)3

+23354475414681440354304(k2 − 6)2 + 13836689687674552975360(k2 − 6) + 3889096495709532979200)z83

+(−5079593933368097151208(k2 − 6)7 − 14594154103096371015040(k2 − 6)6 − 33503861421156201380608(k2 − 6)5

−60018579090397951259648(k2 − 6)4 − 80858976481531529269248(k2 − 6)3 − 77073028202750859501568(k2 − 6)2

−46344650438235365801984(k2 − 6) − 13221646074705577377792)z73 + (9308466298136566608492(k2 − 6)7

+27115170479328586830320(k2 − 6)6 + 63066218770915166201344(k2 − 6)5 + 114399455055489074824960(k2 − 6)4

+156004137935460456838144(k2 − 6)3 + 150475504061699408097280(k2 − 6)2 + 91547705640188726935552(k2 − 6)

+26423046593985210351616)z63 + (−11095597642774291380236(k2 − 6)7 − 32642462593282533158032(k2 − 6)6


−76608697711598428869856(k2 − 6)5 − 140125082414737357727872(k2 − 6)4 − 192577122303638985157632(k2 − 6)3

−187123466429387154372608(k2 − 6)2 − 114646689401743117639680(k2 − 6)− 33314932853414556467200)z53

+(9347621052101913097360(k2 − 6)7 + 27635109722202090645504(k2 − 6)6 + 65135190023242223199008(k2 − 6)5

+119592275581440467432704(k2 − 6)4 + 164920911373642400207616(k2 − 6)3 + 160749870833420529548288(k2 − 6)2

+98770860152536885063680(k2 − 6) + 28778321592849865523200)z43 + (−5615419680567151075808(k2 − 6)7

−16600290237902267697984(k2 − 6)6 − 39112773412702956110336(k2 − 6)5 − 71771758931254298414464(k2 − 6)4

−98898714775619171369216(k2 − 6)3 − 96307590510023343088640(k2 − 6)2 − 59111874345673483827200(k2 − 6)

−17202705345352952832000)z33 + (2402583297061148772864(k2 − 6)7 + 7056849843649244636416(k2 − 6)6

+16525930989063218544768(k2 − 6)5 + 30148489433708152475392(k2 − 6)4 + 41309556136774635555840(k2 − 6)3

+40006454668872611788800(k2 − 6)2 + 24423031403576407296000(k2 − 6) + 7069849819878827520000)z23

+(−688999017327743752704(k2 − 6)7 − 1994617060599839321088(k2 − 6)6 − 4609779004452472829952(k2 − 6)5

−8307601122406729420800(k2 − 6)4 − 11253654298468482048000(k2 − 6)3 − 10781346008039639040000(k2 − 6)2

−6514118885048832000000(k2 − 6)− 1867029805159833600000)z3 + 109572943236302618624(k2 − 6)7

+309741220607312068608(k2 − 6)6 + 700554453138220400640(k2 − 6)5 + 1237766738525319168000(k2 − 6)4

+1646233648197304320000(k2 − 6)3 + 1550332512446054400000(k2 − 6)2 + 921701573594726400000(k2 − 6)

+260150759534592000000.

Then, we see that, for k2 ≥ 6, the coefficients of the polynomial F3(z3) are positive for even degree terms

and negative for odd degree terms. Thus if the equation F3(z3) = 0 has real solutions, then these are all

positive. In particular, the solutions z3 = α3, β3 are positive, thus we get our claim. �

In order to find the singularities at infinity of the above system we set z4 = 0.

Next, we compute the fixed points of the system z1 = 0, z2 = 0, z3 = 0 for specific values of k2 and k3 > 1.

We have:

• V5R7: (4.1466, 4.07919, 1.03361), (2.29783, 3.43436, 3.98856), (1, 2.54858, 2.54858), (1, 0.78475, 0.78475).

• V5R8: (5.39567, 4.8672, 2.16024), (2.31234, 4.49843, 4.93295), (1, 3.29099, 3.29099), (1, 0.709006, 0.709006).

. . .

• V6R8: (3.19365, 3.15771, 0.674502), (1.86343, 2.64311, 3.07833), (1, 2.20711, 2.20711), (1, 0.792893, 0.792893).

• V6R9: (3.99996, 3.71213, 1.41708), (1.89382, 3.36866, 3.73723), (1, 2.78078, 2.78078) (1, 0.719224, 0.719224).

. . .

• V7R9: (2.71186, 2.68928, 0.499721), (1.64442, 2.25706, 2.6166), (1, 2, 2), (0.95544, 0.798009, 0.734193),

(0.805105, 0.771014, 0.379868), (1, 0.8, 0.8).

• V7R10: (3.30651, 3.12526, 1.05079), (1.67763, 2.81537, 3.13489), (1, 2.47178, 2.47178), (1, 0.72822, 0.72822).

. . .

• V8R10: (2.41937, 2.40377, 0.396819), (1.51286, 2.02874, 2.33539), (1, 1.86038, 1.86038), (1, 0.806287, 0.806287),

(0.98506, 0.805755, 0.78523), (0.791817, 0.770023, 0.312754).

• V8R11: (2.89171, 2.76727, 0.833614), (1.54531, 2.48719, 2.76871), (1, 2.26376, 2.26376), (1, 0.736237, 0.736237).

. . .

From the above results we have:

Theorem 7.9. The normalized Ricci flow on the space of invariant Riemannian metrics on V5R7, possesses

exactly four singularities at infinity. These fixed points correspond (up to scale) to the G-invariant Einstein

metrics on V5R7 (cf. Theorem 4.6).

We can also make the following:

Conjecture 7.10. Let G/H be the Stiefel manifold V1+k2Rn, with n = 1 + k2 + k3. Then, for k2 ≥ 4 and

k3 > 1, G/H has four singularities at infinity, and for k2 ≥ 6 and k3 = 2 it has six singularities at infinity.

These fixed points correspond to the G-invariant Einstein metrics on G/H.

22 Marina Statha

7.4. Ricci flow for the generalized flag manifolds. We study the behavior of the normalized Ricci flow

for the generalized flag manifold G/K with four isotropy summands and b2(G/K) = 1. For this case we

take the Ricci components (20) and scalar curvature (21) of the metric (19). Then, system (8) reduces to

x1 = − 1

2d1x1x22x3x4N

{x2x4(x3(A112x2

2(3d1 + 2(d2 + d3 + d4)) + 2d1x12(A112 − d2)

−2d1x1x2(2d1 + d2 + d3 + d4)) + 2A123x1(2d1(x22 + x3

2) + (−d2 − d3 − d4)(x12 − x2

2 − x32))

−2d1d3x12x2) + 2x1x2

2(A134x32(2d1 + d2 + d3 + d4)−A134x1

2(d2 + d3 + d4)

+d1x1x3(A224 − d4)) + x1x42(2A134x2

2(2d1 + d2 + d3 + d4) +A224d1x1x3)}

x2 =1

2d2x12x2x3x4N

{x2x4(x3(−2x1

2(A112 − d2)(d1 + 2d2 + d3 + d4) +A112x22(d1 + d3 + d4)

+2d1d2x1x2) + 2x1(−A123x12(d1 + 2d2 + d3 + d4) +A123x2

2(d1 + d3 + d4) + d2d3x1x2)

−2A123x1x32(d1 + 2d2 + d3 + d4))− 2A134d2x1x2

2(x12 + x3

2 + x42)

−A224x12x3x4

2(2d1 + 3d2 + 2(d3 + d4)) + 2d2x12x2

2x3(d4 −A224)}

x3 = − 1

2d3x12x22x4N

{x2x4(d3x3(A112(2x1

2 + x22)− 2x1(d1x2 + d2x1)) + 2x1(A123(x1

2 + x22)

−d3x1x2)(d1 + d2 + 2d3 + d4)− 2A123x1x32(d1 + d2 + d4)) + 2x1x2

2(A134x12(d1 + d2

+2d3 + d4)−A134x32(d1 + d2 + d4) + d3x1x3(A224 − d4)) + x1x4

2(2A134x22(d1 + d2

+2d3 + d4) +A224d3x1x3)}

x4 =1

2d4x12x22x3N

{− d4x2x4(x3(2x1

2(A112 − d2) +A112x22 − 2d1x1x2) + 2A123x1(x1

2 + x22

+x32)− 2d3x1

2x2)− 2x1x22(d1 + d2 + d3 + 2d4)(A134(x1

2 + x32) + x1x3(A224 − d4))

+x1x42(d1 + d2 + d3)(2A134x2

2 +A224x1x3)},

where N = d1 + d2 + d3 + d4. The above system is not polynomial, hence in order to apply the Poincare

compactification, we multiply it by the factor x21x22x3x4d1d2d3d4N . After that we take the following system:

x1 = −d2d3d4x1(x2x4(x3(A112x22(3d1 + 2(d2 + d3 + d4)) + 2d1x1

2(A112 − d2)− 2d1x1x2(2d1 + d2

+d3 + d4)) + 2A123x1(2d1(x22 + x3

2) + (−d2 − d3 − d4)(x12 − x2

2 − x32))− 2d1d3x1

2x2)

+2x1x22(A134x3

2(2d1 + d2 + d3 + d4)−A134x12(d2 + d3 + d4) + d1x1x3(A224 − d4))

+x1x42(2A134x2

2(2d1 + d2 + d3 + d4) +A224d1x1x3))

x2 = d1d3d4x2(x2x4(x3(−2x12(A112 − d2)(d1 + 2d2 + d3 + d4) +A112x2

2(d1 + d3 + d4)

+2d1d2x1x2) + 2x1(−A123x12(d1 + 2d2 + d3 + d4) +A123x2

2(d1 + d3 + d4) + d2d3x1x2)

−2A123x1x32(d1 + 2d2 + d3 + d4))− 2A134d2x1x2

2(x12 + x3

2 + x42)−A224x1

2x3x42(2d1

+3d2 + 2(d3 + d4)) + 2d2x12x2

2x3(d4 −A224))

x3 = −d1d2d4x3(x2x4(d3x3(A112(2x12 + x2

2)− 2x1(d1x2 + d2x1)) + 2x1(A123(x12 + x2

2)

−d3x1x2)(d1 + d2 + 2d3 + d4)− 2A123x1x32(d1 + d2 + d4)) + 2x1x2

2(A134x12(d1 + d2

+2d3 + d4)−A134x32(d1 + d2 + d4) + d3x1x3(A224 − d4)) + x1x4

2(2A134x22(d1 + d2

+2d3 + d4) +A224d3x1x3))

x4 = d1d2d3x4(−d4x2x4(x3(2x12(A112 − d2) +A112x22 − 2d1x1x2) + 2A123x1(x1

2 + x22 + x3

2)

−2d3x12x2)− 2x1x2

2(d1 + d2 + d3 + 2d4)(A134(x12 + x3

2) + x1x3(A224 − d4))

+x1x42(d1 + d2 + d3)(2A134x2

2 +A224x1x3)). (44)


If we denote by Pi(x1, x2, x3, x4) the xi, for i = 1, 2, 3, 4 respectively then the degree of vector field

X =∑4

i=1 Pi(x1, x2, x3, x4)∂

∂xiis 6. Next, it is easy to see the following lemma

Lemma 7.11. The coordinate planes along with the straight line γ(t) = (t, 2t, 3t, 4t), remain invariant under

the normalized Ricci flow defined by the system (44).

In order to study the singularities at infinity of (44), we should write this system in the local charts ofthe Poincare compactification. Since we are interested for positive values of x1, x2, x3 and x4, we study thebehavior of the previous system only in to the chart (U1, ϕ1). Therefore from (30) we have:

z1 = d3d4z1(d1 + d2 + d3 + d4)(z3(A112z1z2(z12(d1 + 2d2)− 2d1) + 2A123z1((z1

2− 1)(d1 + d2)

+z22(d2 − d1))− 2d1z2(A224z3 + d2(z1 − 1)z1)) + 2A134d2z1

2(z22 + z3

2− 1))

z2 = 2d2d4z1z2(d1 + d2 + d3 + d4)(z3(d3z1(A112z1z2 + d1(−z2) + d1)− A123(z12(d1 − d3) + d1 + d3)

+A123z22(d1 + d3)) + A134z1((z2

2− 1)(d1 + d3) + z3

2(d3 − d1)))

z3 = −d2d3z3(d1 + d2 + d3 + d4)(−2d4z1z3(z1z2(A112z1 − d1) + A123(z12 + z2

2− 1))

+2z12(A134(z2

2(d1 − d4) + d1 + d4) + d1z2(A224 − d4))− z32(2A134z1

2(d1 + d4) + A224d1z2))

z4 = d2d3d4z4(z1z3(z2(A112z12(3d1 + 2(d2 + d3 + d4)) + 2A112d1 − 2d1z1(2d1 + d2 + d3 + d4)− 2d1d2)

+2A123z12(2d1 + d2 + d3 + d4) + 2A123z2

2(2d1 + d2 + d3 + d4)− 2A123(d2 + d3 + d4)− 2d1d3z1)

+2z12(A134z2

2(2d1 + d2 + d3 + d4)− A134(d2 + d3 + d4) + d1z2(A224 − d4))

+z32(2A134z1

2(2d1 + d2 + d3 + d4) + A224d1z2)). (45)

We set z4 = 0 in order to obtain the behavior at infinity of the system (45). Next we will study the

system {z1 = 0, z2 = 0, z3 = 0}, in any case of flag manifolds with b2(G/K) separately. First we substitute

the values of the dimensions di, i = 1, 2, 3, 4 and the numbers A224, A112, A123 and A134 from Table 4. It is

easy to see that system (45) has always a singularity located at (2, 3, 4), which corresponds to the Kahler

metric (1, 2, 3, 4). For the flag manifolds which correspond to the exceptional Lie groups F4,E7 and E8(α3)

we found two more fixed points and for E8(α6) four more. Actually we have:

◮ F4 / SU(3)× SU(2)×U(1)

z1 = 11520z1(z13(8z2 + 5)z3 + 2z1

2(z22 − 18z2z3 + z3

2 − 1) + z1(z2(z2 + 32)− 5)z3 − 4z2z32)

z2 = 23040z1z2(3(z2 − 1)z3(z12 − 6z1 + 2z2 + 2) + 4z1(z2

2 − 1)− 2z1z32)

z3 = −2880z3(−12z1z3(z12 + 2(z1− 6)z1z2 + z2

2 − 1)− 24z32(z1

2 + z2) + 8z12((z2 − 12)z2 + 3))

The solutions are: (0.970488, 0.229171, 1.0097) and (1.27614, 1.95786, 2.31788)

◮ E7 / SU(4)× SU(3)× SU(2)×U(1)

z1 = 976896z1(z13(10z2 + 7)z3 + z1

2(z22 − 36z2z3 + z3

2 − 1)− z1((z2 − 28)z2 + 7)z3 − 2z2z32)

z2 = 1953792z1z2(3(z2 − 1)z3(z12 − 6z1 + 2z2 + 2) + 2z1(z2

2 − 1)− z1z32)

z3 = 976896z3(3z1z3(z12 + 2(z1 − 6)z1z2 + z2

2 − 1) + 3z32(3z1

2 + 2z2) + z12(−(z2 − 3))(7z2 − 3))


◮ E8(α6)/ SU(7)× SU(2)×U(1)

z1 = 15059072z1(z13(16z2 + 11)z3 + 2z1

2(z22 − 30z2z3 + z3

2 − 1)− z1((z2 − 48)z2 + 11)z3 − 4z2z32)

z2 = 30118144z1z2(5(z2 − 1)z3(z12 − 6z1 + 2z2 + 2) + 4z1(z2

2 − 1)− 2z1z32)

z3 = 15059072z3(5z1z3(z12 + 2(z1 − 6)z1z2 + z2

2 − 1) + 2z32(7z1

2 + 5z2)− 2z12(5(z2 − 4)z2 + 7))


◮ E8(α3)/ SO(10)× SU(3)×U(1)

z1 = 7151616z1(z13(18z2 + 13)z3 + z1

2(z22 − 60z2z3 + z3

2 − 1) + z1((44− 3z2)z2 − 13)z3 − 2z2z32)

z2 = 14303232z1z2(5(z2 − 1)z3(z12 − 6z1 + 2z2 + 2) + 2z1(z2

2 − 1)− z1z32)

z3 = 7151616z3(5z1z3(z12 + 2(z1 − 6)z1z2 + z2

2 − 1) + z32(17z1

2 + 10z2) + z12(5(8− 3z2)z2 − 17))

24 Marina Statha

The solutions are: (0.649612, 1.10943, 1.06103), (0.763357, 1.00902, 0.191009), (1.15607, 1.01783, 0.214618)

and (1.09705, 0.770347, 1.29696)

We work on the chart (U1, ϕ1) that corresponds to the plane y1 = 1, so we consider metrics whose are the

form (1, α, β, γ), where α, β and γ are the solution of the system (45) for z4 = 0. These metrics are invariant

Einstein metrics, and the one with coefficients (1, 2, 3, 4) is the unique Kahler-Einstein that admits M . We

have thus proved the following.

Theorem 7.12. Let M = G/K be a generalized flag manifold with four isotropy summands and b2(M) = 1.

The normalized Ricci flow, on the space of invariant Riemannian metrics on M , possesses exactly three

singularities at infinity in case of F4, E7, E8(α6) and exactly five in case of E8(α3). These fixed points

correspond to the G-invariant Einstein metrics on M (cf. Theorem 5.6).

Acknowledgement. This work was developed and completed while the author had a research grant from

DAAD at Philipps-Universitat Marburg, during Fall 2019.

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University of Patras, Department of Mathematics, GR-26500 Rion, Greece

E-mail address: [email protected]

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