Ricci curvature and normalized Ricci flow on generalized Wallach spaces
Nurlan Abiev


Ricci curvature and normalized Ricci flow ... …]
Ricci curvature and normalized Ricci flow on generalized Wallach spaces


N. A. Abiev Institute of Mathematics NAS KR, Bishkek, prospect Chui, 265a, 720071, Kyrgyzstan
abievn@mail.ru


§ ABSTRACT 
We proved that  the normalized
Ricci flow does not preserve the positivity of Ricci curvature of  Riemannian metrics
on every generalized Wallach space with a_1+a_2+a_3≤ 1/2,
in particular on the spaces SU(k+l+m)/SU(k)×SU(l)
×SU(m)
and
Sp(k+l+m)/Sp(k)×Sp(l)
×Sp(m) independently on k,l and m.
The positivity of Ricci curvature is preserved  for  all original metrics with
Ric>0 on generalized Wallach spaces  a_1+a_2+a_3> 1/2
if the conditions 4(a_j+a_k)^2≥ (1-2a_i)(1+2a_i)^-1 hold
for all  {i,j,k}={1,2,3}.
We also established that the spaces
SO(k+l+m)/SO(k)×SO(l)×SO(m) satisfy the above conditions for max{k,l,m}≤ 11, moreover,
additional conditions were found to keep Ric>0
in cases when max{k,l,m}≤ 11 is violated.
Similar questions have also been studied for all other generalized Wallach spaces given in the classification of Yuriĭ Nikonorov.


 Key words and phrases:
generalized Wallach space,
invariant Riemannian metric, normalized Ricci flow, sectional curvature,
Ricci curvature, dynamical system.



2010 Mathematics Subject Classification: 53C30, 53C44, 53E20, 37C10.


[
    N. A. Abiev
    September 9, 2024
=====================







§ INTRODUCTION




In this paper we continue our studies <cit.>  related to the evolution of invariant Riemannian metrics on a class of homogeneous Riemannian spaces called generalized Wallach spaces (GWS) under the normalized Ricci flow (NRF)

    ∂∂ tg(t) = -2 Ric_g+ 2g(t)S_g/n,

introduced  in <cit.> by Hamilton, where Ric_g is the Ricci tensor and S_g is the scalar curvature of an one-parametric family of Riemannian metrics g(t) on a given Riemannian manifold of dimension n.

According to <cit.>
a generalized Wallach space is
a homogeneous almost effective compact space G/H with  a  compact semisimple
connected Lie group G and its closed subgroup H
(with corresponding Lie algebras 𝔤 and 𝔥)
such that
the isotropy representation of G/H  admits a decomposition into a direct sum
𝔭=𝔭_1⊕𝔭_2⊕𝔭_3 of three
Ad(H)-invariant irreducible modules
𝔭_1, 𝔭_2 and 𝔭_3
satisfying
[𝔭_i,𝔭_i]⊂𝔥 for each i∈{1,2,3},
where 𝔭 is an orthogonal complement of 𝔥 in 𝔤 with respect to a bi-invariant inner product ⟨· ,·⟩=-B(· ,·)
on 𝔤 defined by the Killing form
B(· ,·) of 𝔤.
For any fixed bi-invariant inner product
⟨·, ·⟩ on the Lie algebra 𝔤 of the Lie group G,
any G-invariant Riemannian metric g on G/H can be determined by an Ad (H)-invariant inner product

    (·, ·)=.x_1⟨·, ·⟩|_𝔭_1+
    .x_2⟨·, ·⟩|_𝔭_2+
    .x_3⟨·, ·⟩|_𝔭_3,

where x_1,x_2,x_3 are positive real numbers (see <cit.> and references therein for details).
We complete a brief description of these spaces recalling that one
of their important characteristics is that
every generalized Wallach space can be described by a triple of real numbers
a_i:=A/d_i∈ (0,1/2], i=1,2,3, where d_i=(𝔭_i) and A is some positive constant (see <cit.> for details).
In <cit.>  classifications of singular (equilibria) points of (<ref>) and their parametric bifurcations are studied on generalized Wallach spaces
and other classes of homogeneous spaces.
The interesting facts are
that  equilibria points are exactly Einstein metrics and vice versa, moreover,
the number of singular points and their types (hyperbolic, semi-hyperbolic, nilpotent or linearly zero) are most responsible for evolution of Riemannian metrics under NRF.  In <cit.> we initiated the study of new aspects related to evolution of positively curved Riemannian metrics on GWS.


Historically the classification of homogeneous spaces admitting
homogeneous metrics with positive sectional curvature
were obtained as a result of a joint effort of
Aloff and Wallach <cit.>,
Bérard Bergery <cit.>, Berger <cit.>,
Wallach <cit.> and Wilking <cit.>.
Note  that Bérard Bergery's classification odd-dimensional case
was recently revisited by Wolf and Xu in <cit.> and
a new version of the classification with refined proofs was
suggested by Wilking and Ziller in <cit.>.
The classification of homogeneous spaces which admit
metrics with positive Ricci curvature was obtained by Berestovskiĭ in <cit.>,
see also <cit.>
where (in particular) homogeneous Riemannian manifolds of positive sectional curvature
and positive Ricci curvature are described.


Novadays an interesting question is to understand whether
the positivity of  sectional or (and) Ricci curvature of  Riemannian metrics
on a given Riemannian manifold is preserved or not when metrics are evolved by the Ricci flow.
Cheung and Wallach
showed  in Theorem 2 in <cit.> non-maintenance of the sectional curvature for
certain metrics on the Wallach spaces

    W_6:=SU(3)/T_max,     
    W_12:=Sp(3)/Sp(1)×Sp(1)×Sp(1),     
    W_24:=F_4/Spin(8)

which are the only even-dimensional homogeneous spaces
admitting metrics with positive sectional curvature according to <cit.>.
Böhm and Wilking gave an example of metrics of positive sectional curvature
which can lose even the positivity of the Ricci curvature.
They proved in Theorem 3.1 in <cit.>  that
the (normalized) Ricci flow deforms some  metrics of positive sectional curvature
 on W_12  into metrics with mixed Ricci curvature. An analogous result was obtained for W_24 in Theorem 3 of <cit.>.
In <cit.> we considered similar questions
on a class of generalized Wallach spaces with coincided parameters a_1=a_2=a_3:=a∈ (0,1/2)
(see Section <ref> for definitions).
Note that a complete classification of generalized Wallach spaces
was obtained by Nikonorov <cit.>.
Chen et al. in <cit.> obtained similar results with simple G.

It should be also noted that
the Wallach spaces  W_6, W_12 and W_24 are exactly
partial cases of  generalized Wallach spaces with a_1=a_2=a_3=a
equal to 1/6, 1/8 and a=1/9 respectively.

The following theorems were proved involving the apparatus of dynamical systems and
asymptotic analysis (by generic metrics we meant metrics satisfying  x_i  x_j x_k x_i,  i,j,k∈{1,2,3}):


On  the   Wallach spaces W_6, W_12, and W_24,  the normalized Ricci flow
evolves all generic metrics with positive sectional curvature into metrics with mixed sectional curvature.



Using the same tools the following results were obtained about the Ricci curvature:



Let G/H be a generalized Wallach space with a_1=a_2=a_3=:a, where a∈ (0,1/4)∪(1/4,1/2).
If a<1/6, then
the normalized Ricci flow
evolves all generic metrics  with positive Ricci curvature  into metrics with mixed Ricci curvature.
If a∈ (1/6,1/4)∪(1/4,1/2), then the normalized Ricci flow evolves all generic metrics into metrics with positive Ricci curvature.




Let G/H be a generalized Wallach space with a_1=a_2=a_3=1/6.
Suppose that it is supplied with the invariant Riemannian metric (<ref>) such that x_k<x_i+x_j for all indices with {i,j,k}={1,2,3},
then the normalized Ricci flow on G/H with this metric as the initial point, preserves the positivity of the Ricci curvature.




The normalized Ricci flow on a generalized Wallach space G/H
a_1=a_2=a_3=1/4
evolves all generic metrics into metrics  with positive Ricci curvature.


Recent time there are some new results on evolution of positively curved Riemannian metrics via Ricci flow. Among them Bettiol and Krishnan showed in <cit.> that Ricci flow does not preserve positive sectional curvature of Riemannian metrics on S^4 and ℂP^2.
Some extended results on positive intermediate curvatures for an infinite series of homogeneous spaces
Sp(n+1)/Sp(n-1)×Sp(1)×Sp(1)
was obtained in <cit.>
by González-Álvaro and  Zarei,
where the space W_12 is included as the first member corresponding to n=2.
The group of authors Cavenaghi et al. proved in <cit.> that SU(3)/T_max admits metrics of positive intermediate
Ricci curvature losing positivity under Ricci flow.
They established similar properties  for
a family of homogeneous spaces
SU(m+2p)/SU(m)×SU(p)
×SU(p)
as well.
In this paper we consider
generalized Wallach spaces
SO(k+l+m)/SO(k)×SO(l)
×SO(m),
SU(k+l+m)/SU(k)×SU(l)
×SU(m)
and
Sp(k+l+m)/Sp(k)×Sp(l)
×Sp(m)
and other ones listed in Table 1
in accordance with <cit.>,
where  GWS n means a generalized Wallach space with (𝔤, 𝔥) positioned in line n.
From the geometrical point of view only a countable number of natural triples (k,l,m),
which correspond to actual generalized Wallach spaces,
can be interesting for us.
Knowing that  not every  triple (a_1, a_2, a_3)∈ (0,1/2)^3
can correspond to some  generalized Wallach spaces,
nevertheless we will consider the problem in a more global context
for any reals  a_1, a_2, a_3∈ (0,1/2), using an opportunity
to involve fruitful methods of continuous mathematics.
Such an approach  justified itself introducing in <cit.>
a special algebraic surface defined by a polynomial of degree 12 in three real variables
a_1, a_2 and a_3, which provides degenerate singular points to a three-dimensional
dynamical system obtained as a reduction of the normalized Ricci flow on generalized Wallach spaces.
Topological characteristics of that surface such as the number of connected components was found by <cit.>, and a deeper structural analysis was given in <cit.>.
It should be also noted that in <cit.>
results of Theorem <ref> were generalized
to the case a∈ (0,1/2) considering (<ref>) as an abstract dynamical system.
In the present paper we extend Theorems <ref> – <ref>  to the case
of arbitrary  a_1, a_2, a_3∈ (0,1/2).
Our main results are contained in Theorems <ref> – <ref> and Corollary <ref>.



Let G/H be a generalized Wallach space with a_1+a_2+a_3≤ 1/2.
Then the normalized Ricci flow
evolves some  metrics (<ref>)  with positive Ricci curvature  into metrics with mixed Ricci curvature.




Let G/H be a generalized Wallach space with  a_1+a_2+a_3> 1/2.


  * The normalized Ricci flow (<ref>)  evolves all
metrics (<ref>)  with positive Ricci curvature  into metrics with positive Ricci curvature if 
θ≥max{θ_1, θ_2, θ_3},

  * At least some metrics (<ref>)  with positive Ricci curvature  can be evolved into metrics with positive Ricci curvature, if  θ≥max{θ_1, θ_2, θ_3} fails,

where  θ:=a_1+a_2+a_3-1/2∈ (0,1),  
θ_i:=a_i-1/2+√((1-2a_i)(1+2a_i)^-1)/2,  
i=1,2,3.


Note that  θ≥max{θ_1, θ_2, θ_3} is equivalent to  4(a_j+a_k)^2≥ (1-2a_i)(1+2a_i)^-1 announced in abstract
for all {i,j,k}={1,2,3}.

For the spaces
SO(k+l+m)/SO(k)×SO(l)×SO(m) (GWS 1 in Table 1)
which satisfy a_1+a_2+a_3>1/2 we proved
the following special theorem under the assumptions k≥ l≥  m>1
based on symmetry.








The following assertions hold  for SO(k+l+m)/SO(k)×SO(l)×SO(m),
 k≥ l≥  m>1 (see also Table 2):



  * The normalized Ricci flow (<ref>)  evolves all metrics (<ref>)  with Ric>0
into metrics with Ric>0   if either  k≤ 11 or one of
the conditions  2<l+m≤ X(k) or  l+m≥ Y(k) is satisfied for
k∈{12, 13,14,15,16},

  * The normalized Ricci flow (<ref>)  evolves some metrics (<ref>) with Ric>0  into metrics with Ric>0 if
X(k)<l+m<Y(k)  for  k≥ 12.

  * There exists a finite number  of GWS 1 on which any metric with Ric>0 maintains  Ric>0  under NRF (<ref>).
But there are infinitely (countably)
many GWS 1 on which  Ric>0 can be kept
at least for some metrics with Ric>0,

where
X(k)=2k(k-2)/k+2+√(k^2-12k+4)-k+2,
Y(k)=2k(k-2)/k+2-√(k^2-12k+4)-k+2 and
2<X(k)<Y(k) for all k≥ 12.









Theorems <ref> and <ref> imply


Under the normalized Ricci flow (<ref>) the positivity of the Ricci curvature


  * is not preserved  on
            GWS  2, 3, 5, 7, 15;

  * is preserved on
            GWS  4, 6, 8, 9, 10, 11, 12, 13, 14.








§ PRELIMINARIES



In <cit.> the following explicit expressions
 Ric_g=. r_1 Id|_𝔭_1+
. r_2 Id|_𝔭_2+
. r_3 Id|_𝔭_3
and
S_g=d_1 r_1+d_2 r_2+d_3 r_3
were found for the Ricci tensor Ric_g and
the scalar curvature  S_g  of the metric (<ref>)
on  a generalized Wallach space G/H,
where
r_i:=1/2x_i+a_i/2(x_i/x_jx_k-x_k/x_ix_j-
x_j/x_ix_k)
are the principal Ricci curvatures,
d_i are  the dimensions of the modules 𝔭_i such that d_1+d_2+d_3=n and
a_i are real numbers in the interval (0,1/2] for {i,j,k}={1,2,3}.
Substituting the above mentioned expressions for Ric_g and
 S_g  into (<ref>) it can be reduced
 to the following system of three   ordinary differential equations
studied in <cit.>:

    dx_1dt = f_1(x_1,x_2,x_3),   dx_2dt=f_2(x_1,x_2,x_3),   dx_3dt=f_3(x_1,x_2,x_3),

where x_i=x_i(t)>0 (i=1,2,3), are parameters of the invariant metric (<ref>) and

    f_1(x_1,x_2,x_3)   =   -1-a_1x_1 ( x_1x_2x_3-  x_2x_1x_3- x_3x_1x_2)+x_1B,
    
    f_2(x_1,x_2,x_3)   =   -1-a_2x_2 ( x_2x_1x_3- x_3x_1x_2 -  x_1x_2x_3)+x_2B,
    
    f_3(x_1,x_2,x_3)   =   -1-a_3x_3 ( x_3x_1x_2-  x_1x_2x_3- x_2x_1x_3)+x_3B,


    B:=(∑_i=1^3 1a_ix_i- ∑_{i,j,k}={1,2,3}
     j< kx_ix_jx_k)
    (∑_i=1^3a_i^-1)^-1.





Recall that the function  Vol=c is a first integral of the system (<ref>) for any c>0,
where
Vol:=x_1^1/a_1x_2^1/a_2x_3^1/a_3
(see <cit.>). The surface  Vol=1 is  important
which corresponds to parameters of unite volume metrics. Denote it Σ.
The case Vol=c could actually be reduced to the case  c=1
using homogeneity of the functions f_i and autonomy property in (<ref>).
This could be made by introducing new variables x_i=x_i√(c) and t=τ√(c), then
we get a new system of ODEs in  new functions x_i(τ) but of the same form as the original one (<ref>) (see also <cit.>).
Therefore we can consider metrics with Vol=1 without loss of generality.
Being  a first integral of the system (<ref>)
the identity Vol=1 allows to reduce (<ref>)  to the following system of two ODEs
(see details in <cit.>):
 
    dx_1dt = f_1(x_1,x_2),     dx_2dt=f_2(x_1,x_2),

where
f_i(x_1,x_2):=f_i(x_1,x_2, φ(x_1,x_2)), 
φ(x_1,x_2):=x_1^-a_3/a_1x_2^-a_3/a_2,  i=1,2.


Introduce   surfaces Λ_i in (0,+∞)^3 defined  by the equations
λ_i:=a_i(x_i^2-x_j^2-x_k^2)+x_jx_k=0 (equivalent to r_i=0)
for    {i,j,k}={1,2,3}. Denote by R the set
of points (x_1,x_2,x_3)∈ (0,+∞)^3 satisfying the inequalities λ_i>0 for all i=1,2,3.
Then Ric_g>0 is equivalent to  (x_1,x_2,x_3)∈ R.





§ AUXILIARY RESULTS



Introduce the parameter
ω:=(a_i^-1+a_j^-1+a_k^-1)^-1.
We will often use real roots

    m_i=m(a_i):=1-√(1-4a_i^2)/2a_i
    M_i=M(a_i):=1+√(1-4a_i^2)/2a_i

 of the quadratic equation
t^2-a_i^-1t+1=0 which depend on the parameter a_i.
Clearly M_i=m_i^-1.
We also need the following obvious inequalities for all a_i∈ (0,1/2):

    0<a_i<m_i<2a_i<1<1/2a_i<M_i<1/a_i.






 §.§ Structural properties of some surfaces and curves
related to the set of metrics Ric>0














For an arbitrary generalized Wallach space with a_1,a_2,a_3∈ (0,1/2) the set R
is bounded by the conic surfaces
Λ_1, Λ_2 and Λ_3,
each consisting of two connected components.
Moreover,  for all a_i,a_j∈ (0,1/2)
the cones  Λ_i and Λ_j
intersect each other along two straight lines:
the first of them  is defined  by x_i=x_j, x_k=0,
and the second one has an equation

    x_i=up,      x_j=vp,      x_k=p>0,     {i,j,k}={1,2,3},

where

    [ u=Φ(a_i,a_j):=
    ϕ(a_i,a_j),        a_i  a_j,
    
    a_j,        a_i=a_j,;  v=Ψ(a_i,a_j):=
    ψ(a_i,a_j),        a_i a_j,
    
    a_i,        a_i=a_j, ]

with  ϕ(a_i,a_j):=124a_i^2-1+√(Δ_ij)a_i^2-a_j^2  a_j>0,  
ψ(a_i,a_j):=124a_j^2-1+√(Δ_ij)a_j^2-a_i^2  a_i>0 and
Δ_ij:=(1-4a_i^2)(1-4a_j^2)>0.
See Fig. <ref> for illustrations.




The inequality  λ_i=a_i(x_i^2-x_j^2-x_k^2)+x_jx_k>0 is equivalent to
x_i^2>T:=x_j^2-a_i^-1x_jx_k+x_k^2.
Any x_i>0 satisfies λ_i>0 if  T≤ 0.
If  T>0 then  λ_i>0  is equivalent to x_i> √(T).
Considering T>0 as a quadratic inequality with respect to x_j
which admits solutions   0<x_j<m_ix_k or x_j>M_ix_k
we conclude that the set R is bounded by the coordinate planes x_1=0, x_2=0, x_3=0
and by two components of the cone Λ_i,
defined by the same equation x_i=√(T) in two different domains
{x_k>0,   0<x_j<m_ix_k} and
{x_k>0,   x_j>M_ix_k} of the coordinate plane (x_k,x_j).


Due to symmetry analogous assertions can be obtained for the surfaces
Λ_j and Λ_k  using permutations of indices.
Thus  we established that R is bounded by the cones Λ_1, Λ_2 and Λ_3.
To find common points of the surfaces Λ_i and Λ_j
consider the system of equations
λ_i=a_i(x_i^2-x_j^2-x_k^2)+x_jx_k=0 and
λ_j=a_j(x_j^2-x_i^2-x_k^2)+x_ix_k=0.
We can pass to a new system
a_i(u^2-v^2-1)+v=0, a_j(v^2-u^2-1)+u=0,
where x_i=ux_k, x_j=vx_k.
It follows then  a_iu+a_jv-2a_ia_j=0 or
v=a_ia_j^-1(2a_j-u).
Substituting this expression into one of the  previous equations,
we obtain a quadratic equation with respect to u:

    (a_i^2-a_j^2)u^2-(4a_i^2-1)a_ju+(4a_i^2-1)a_j^2=0.


The case  a_i=a_j leads  to a solution u=a_j. It follows then v=a_i.

Assume that  a_i a_j.
The case a_i>a_j. Then 4a_i^2-1+√(Δ_ij)>0 and
the latter quadratic equation admits a positive root u^(1),
equal to  ϕ(a_i,a_j) shown in (<ref>).
Another root u^(2), associated with the summand -√(Δ_ij)
is out of our interest because of
4a_i^2-1-√(Δ_ij)=-(1-4a_i^2+√(Δ_ij))<0
independently on a_i and a_j.
Substituting u^(1)=ϕ(a_i,a_j)
into
v^(1)=a_ia_j^-1(2a_j-u^(1))
gives v^(1)=ψ(a_i,a_j) in (<ref>).

The case a_i<a_j. In this case  u^(1)>0 since 4a_i^2-1+√(Δ_ij)<0.
The second root u^(2) we reject again although it is positive,
because the corresponding v^(2)  occurs negative.

Summing   a_jλ_i=0 and  a_iλ_j=0
we can find another family of solutions x_i=x_j, x_k=0
of the system of λ_i=0 and λ_j=0
which will also be useful in sequel.
It should be noted that there are also negative solutions of the system of λ_i=0 and λ_j=0
non permissible for our aims.
Lemma <ref> is proved.





Assume that 0<a_j<a_i<1/2. Then  u and
v in (<ref>)
satisfy 0<u < 1 < v,
where u:=u a_j^-1 and v:=v a_i^-1.



For all a_i a_j both of the inequalities
u<1 and
v>1 are equivalent to
√(Δ_ij)<1-2(a_i^2+a_j^2) or the same
(a_i-a_j)^2(a_i+a_j)^2>0 after raising to square
based on   1-2(a_i^2+a_j^2)>0.
Lemma <ref> is proved.





Denote by Λ_ij and Λ_ji  connected components of the cones  Λ_i and Λ_j respectively,
which contain the common straight line (<ref>).
For  components of Λ_i and Λ_j meeting each other along  x_i=x_j, x_k=0
accept  notations Λ_ik and Λ_jk respectively.
Then each curve r_i formed as   Σ∩Λ_i
consists of two connected  components
r_ij:=Σ∩Λ_ij and r_ik:=Σ∩Λ_ik.
By analogy  r_j:=Σ∩Λ_j=r_ji∪ r_jk, where
r_ji:=Σ∩Λ_ji and r_jk:=Σ∩Λ_jk.




For an arbitrary generalized Wallach space with a_1,a_2,a_3∈ (0,1/2)
each curve r_i=Λ_i ∩Σ
consists of two connected components r_ij and r_ik, which are smooth curves
and can be represented by parametric  equations

    [ x_i=ϕ_i (t):=t^-ω a_j^-1(√(t^2-a_i^-1t+1))^
    ω(a_j^-1+a_k^-1),;                                          x_j= ϕ_j (t):= tx_k,;                        x_k= ϕ_k (t):=(√(t^2-a_i^-1t+1))^-1x_i ]

defined for t∈ (0,m_i) and t∈ (M_i,+∞) respectively,
where ω=(a_i^-1+a_j^-1+a_k^-1)^-1 and {i,j,k}={1,2,3}.
In addition,

    [ lim_t→ 0+ x_i=lim_t→ 0+ x_k = +∞,  lim_t→ 0+ x_j =0,
     lim_t→ m_i- x_j= lim_t→ m_i- x_k = +∞,  lim_t→ m_i- x_i=0,; lim_t→ M_i+ x_j= lim_t→ M_i+ x_k = +∞,  lim_t→ M_i+ x_i=0,
     lim_t→ +∞ x_i=lim_t→ +∞ x_j = +∞,  lim_t→ +∞ x_k =0. ]





Substituting  x_j=tx_k into the equation a_i(x_i^2-x_j^2-x_k^2)+x_jx_k=0
of Λ_i and solving the quadratic equation
(t^2-a_i^-1t+1)x_k^2=x_i^2
with respect to x_k>0, we obtain the third expression in (<ref>).
Then the expression for x_i is obtained  using
Vol=x_i^1/a_ix_j^1/a_jx_k^1/a_k=1.

As it follows from Lemma <ref>, the curve  r_i=Σ∩Λ_i
consists of two components r_ij=Σ∩Λ_ij and  r_ik=Σ∩Λ_ik. This fact  is also clear from
t^2-a_i^-1t+1=(t-m_i)(t-M_i)>0,
t∈ (0,m_i)∪ (M_i,+∞),
where m_i=m(a_i) and M_i=M(a_i) are functions given in (<ref>).
The components r_ij and r_ik are connected curves
being continuous images of the connected sets (0,m_i) and  (M_i, +∞).
Moreover since x_i=ϕ_i(t), x_j=ϕ_j(t) and x_k=ϕ_k(t)
are differentiable functions of t∈ (0,m_i) ∪ (M_i,+∞),
the components  r_ij and r_ik are smooth curves.
Recall that analogous parametric equations
can be obtained for the curves
r_j=Σ∩Λ_j and r_k=Σ∩Λ_k
using permutations of the indices  i,j and k in (<ref>).


Let us study behavior of the curves r_ij and r_ik
in neighborhoods of the boundaries of the intervals (0,m_i) and  (M_i, +∞).
Clearly x_i(t)→ +∞ as t→ 0+  and x_i(t)→ 0
as  t→ m_i- or t→ M_i+.
Analogously
x_i=ϕ_i(t)∼ t^-ω a_j^-1 t^ω(a_j^-1+a_k^-1)
=t^ω a_k^-1→ +∞
as t→ +∞.
Since 1-ω(a_j^-1+a_k^-1)>0,
 we
easily can find the limits in (<ref>) for
x_j=ϕ_j(t) = t^1-ω a_j^-1(√((t-m_i)(t-M_i)))^-1+ω(a_j^-1+a_k^-1)
and
x_k=ϕ_k(t)= t^-ω a_j^-1(√((t-m_i)(t-M_i)))^-1+ω(a_j^-1+a_k^-1).
Lemma <ref> is proved.








For each pair of the curves r_i and r_j
corresponding to an arbitrary generalized Wallach space with a_1,a_2,a_3∈ (0,1/2)
the curve I_k defined by the equations

    x_i=x_j=τ>0,      x_k=τ^-a_k(a_i^-1+ a_j^-1),
        {i,j,k}={1,2,3},

does not intersect their  components r_ik and r_jk,
but  approximates them as accurately as desired at infinity,
at the same time I_k intersects both  r_i and r_j on their components
 r_ij and r_ji. In addition, I_k can not intersect the curve r_k,
 moreover, r_k moves away from I_k in both directions
as τ→ 0+ and τ→ +∞.
See Fig. <ref> for illustrations.




 I_k is exactly the curve that belongs to the surface Σ and have a projection x_i=x_j on the coordinate plane x_k=0. Substituting x_i=x_j=τ into the equation
x_i^1/a_ix_j^1/a_jx_k^1/a_k=1 of Σ  we find x_k given in (<ref>).
Substituting the expressions (<ref>) into the functions
λ_i=a_i(x_i^2-x_j^2-x_k^2)+x_jx_k
and
λ_j=a_j(x_j^2-x_i^2-x_k^2)+x_ix_k
(recall that the cones Λ_i and Λ_j are defined by λ_i=0 and λ_j=0
respectively) we obtain

    λ_i=λ_i(τ)=1-a_iτ^-1-a_k(a_i^-1+ a_j^-1)/2τ,     λ_j=λ_j(τ)=1-a_jτ^-1-a_k(a_i^-1+ a_j^-1)/2τ.

In what follows that I_k intersects the curve r_i=Σ∩Λ_i at the unique point with coordinates
x_i^0=x_j^0=τ_0, x_k^0=τ_0^-a_k(a_i^-1+ a_j^-1)
where
τ_0=a_i^(1+a_k(a_i^-1+a_j^-1))^-1
is the unique root
of the equation λ_i(τ)=0.
Using Lemma <ref>, write out the equation
τ_0=x_j^0=tx_k^0=tτ_0^-a_k(a_i^-1+ a_j^-1)
admitting the unique root
t=t_k:=τ_0^1+a_k(a_i^-1+a_j^-1)=a_i,
at which the mentioned point I_k∩ r_i  belongs to r_i.
The inequalities 0<t_k=a_i<m_i (see (<ref>)) imply
that I_k must intersect r_i on its  component r_ij.
Since such an intersection with r_i  happens only once,
we conclude that I_k  can not intersect the component r_ik.
Moreover, the value of the limit
lim_τ→ +∞λ_i=1/2lim_τ→ +∞(τ^-1-a_iτ^-2-a_k(a_i^-1+ a_j^-1))=0
shows that r_ik can be approximated by I_k at infinity as accurately as desired.

From the analysis of the function λ_j=λ_j(τ), similar conclusions will follow regarding the components r_ji and r_jk of the curve r_j.
For the curve  r_k we have
λ_k(τ)=a_kτ^-2a_k(a_i^-1+ a_j^-1)
+(1-2a_k)τ^22τ^2-a_k(a_i^-1+ a_j^-1).
The equation λ_k(τ)=0 can not admit any solution
due to positiveness of the numerator of the  latter fraction.
Therefore I_k never cross r_k.
Moreover, they move away from each other as much as desired in both directions:

    lim_τ→ +∞λ_k(τ)=
    a_k/2lim_τ→ +∞τ^-2-a_k(a_i^-1+ a_j^-1)+
    1-2a_k/2lim_τ→ +∞τ^a_k(a_i^-1+ a_j^-1)=0+∞=+∞,
    lim_τ→ 0+λ_k(τ)=
    a_k/2lim_τ→ 0+τ^-2-a_k(a_i^-1+ a_j^-1)+
    1-2a_k/2lim_τ→ 0+τ^a_k(a_i^-1+ a_j^-1)=+∞+0=+∞.

Lemma <ref> is proved.






For any generalized Wallach space with a_1,a_2,a_3∈ (0,1/2) the following assertions hold:


  * For each pair of the curves r_i  and r_j (respectively r_k)
their components r_ij and r_ji (respectively r_ik and r_ki)
admits  exactly one common point P_ij (respectively P_ik)
with coordinates x_i=α p^0, x_j=β p^0,  x_k=p^0
(respectively x_i=γ q^0,  x_k=δ q^0, x_j= q^0),
where

    [ p^0:=α^-ω a_i^-1β^-ω a_j^-1,   α:=Φ(a_i,a_j),    β:=Ψ(a_i,a_j),;   q^0:=γ^-ω a_i^-1δ^-ω a_k^-1,   γ:=Φ(a_i,a_k),   δ:=Ψ(a_i,a_k) ]

with the functions Φ and Ψ defined in (<ref>),
{i,j,k}={1,2,3},
whereas their other components r_ik and r_jk
(respectively r_ij and r_kj)
are disjoint approximating each other at infinity as close as desired
(see Fig. <ref>).


  * The values  t_ij and t_ik of the parameter t,
which correspond to the points  P_ij
and P_ik at the parametrization (<ref>),
can be found by the formula

    t_ij=β,      t_ik=δ^-1.

In addition a_i≤ t_ij<m(a_i) and  M(a_i)<t_ik≤ a_i^-1
for all a_i, a_j, a_k∈ (0,1/2).



 (1)
By definition  r_i=Λ_i ∩Σ and r_j=Λ_j ∩Σ.
It follows then  r_i∩ r_j=(Λ_i∩Λ_j)∩Σ, and hence,
the problem of finding  r_i∩ r_j
reduces to searching for possible common points
of the straight line Λ_i∩Λ_j with the surface Σ.
As it was proved in Lemma <ref>,
the intersection Λ_i∩Λ_j equal to Λ_ij∩Λ_ji is the straight line
 x_i=α p, x_j=β p,  x_k=p.
Substituting these expressions into the equation
x_i^1/a_ix_j^1/a_jx_k^1/a_k=1 of Σ we obtain
the following equation regarding to the parameter p:

    α^1/a_iβ^1/a_j p^1/a_i+1/a_j+1/a_k=1.

This yields coordinates
x_i=α p^0, x_j=β p^0,  x_k=p^0 of an unique point
which belong to r_ij∩ r_ji=(Λ_ij∩Λ_ji)∩Σ
for  α, β and p^0 exposed in (<ref>). Let us denote that point by P_ij.
Thus  {P_ij}=r_ij∩ r_ji=r_i∩ r_j.
Repeating the same reasoning for the curves r_i and r_k,  we find coordinates
x_i=γ q^0,  x_k=δ q^0, x_j= q^0
of the unique point P_ik∈ r_ik∩ r_ki=r_i∩ r_k.


In order to highlight disjoint components of the curves r_i and r_j,
recall another  common line x_i=x_j, x_k=0
of the cones Λ_i  and Λ_j
(more precisely, the common line of their components Λ_ik and Λ_jk),
mentioned in Lemma <ref>.
The  components r_ik and r_jk are situated on opposite sides of I_k, therefore
they can not admit a common point. Moreover, r_ik and r_jk approximate
the same curve I_k at infinity by Lemma <ref>, therefore at infinity they
approximate each other.



(2) Assume that r_i  is parameterized as in Lemma <ref>.
Let  t=t_ij be a value of t which corresponds to the point P_ij
on  r_ij.
The coordinates of P_ij are x_i=α p^0, x_j=β p^0, x_k=p^0
as we established.
Then the equalities  ϕ_j(t)=β p^0,  ϕ_k(t)=p^0 and
ϕ_j(t)=tϕ_k(t) in (<ref>) easily imply
t_ij=β:=Ψ(a_i,a_j).


Now we claim that   t_ij∈ [a_i, m_i) for all a_i, a_j∈ (0,1/2).

Indeed if a_i=a_j for  i j then t_ij=a_i obviously.


If a_i a_j then assume  a_i>a_j without loss of generality.
Then   Lemma <ref> implies t_ij>a_i.
The inequality  t_ij<m_i is equivalent to

    (1-4a_j^2)a_i^2-a_i^2√(Δ_ij)<(a_i^2-a_j^2)
    (1-√(1-4a_i^2)) 
    ⇔  a_i^2-a_j^2<a_i^2√(1-4a_j^2)-a_j^2√(1-4a_i^2) ⇔ (a_i^2-a_j^2)^2<(a_i^2√(1-4a_j^2)-a_j^2√(1-4a_i^2))^2
     ⇔ √(Δ_ij)<1-2(a_i^2+a_j^2)
     ⇔ 
    (a_i-a_j)^2(a_i+a_j)^2>0.


Thus t_ij∈ (a_i, m_i) for a_i a_j.
Coordinates x_i=γ q^0,  x_k=δ q^0, x_j= q^0 of the point P_ik
can be clarified by the same way.
The value
t_ik=δ^-1=1/Ψ(a_i,a_k)
is obtained as an unique root of the equation  q^0=ϕ_j(t)=tϕ_k(t)=tδ q^0 (see (<ref>)).


Analogously  t_ik∈(M_i,  a_i^-1].
Indeed if a_i=a_k then t_ik=Ψ(a_i,a_k)^-1=a_i^-1>M_i
due to (<ref>).

If a_i a_k then assume a_i>a_k.
The inequality  t_ik<a_i^-1 is clear from Lemma <ref>.
The inequality t_ik>M_i
can be established in similar way as above.
Lemma <ref> is proved.









It is advisable to remove extra pieces of the curves r_1, r_2 and r_3,
retaining after their pairwise intersections.
In fact those extra pieces do not belong to the boundary of the domain R.
Let us consider the example of the curve r_1 which
intersects r_2 on the unique point
P_12(α p^0, β p^0, p^0) known from Lemma <ref>, where   α, β and p^0
are defined by formulas (<ref>) and (<ref>) at i=1, j=2.
An analysis of (<ref>) and (<ref>) shows that
the interval   [t_12, m_1) corresponds to the extra piece
of r_12 behind the point P_12,
where t_12 can be found by (<ref>).
Therefore (0,t_12] should be taken as an interval of parametrization
for the component r_12
instead of the original interval (0,m_1).
Similarly, the interval (M_1,+∞)  corresponding to r_13
 before its intersection with r_31 at the point
P_13(γ q^0, q^0, δ q^0), will be reduced to [t_13, +∞),
where  γ, δ and q^0 are defined by formulas (<ref>)
and (<ref>) at i=1, j=3.








 §.§ Normals to cones
and the vector field associated with the differential system


To study the behavior of trajectories of the system (<ref>)
 we need the sign of the standard inner product

    ( V, ∇λ_i)=f_i ∂λ_i/∂ x_i+f_j ∂λ_i/∂ x_j+f_k ∂λ_i/∂ x_k

on an arbitrarily chosen  curve r_i,
where  ∇λ_i is the normal  to the surface Λ_i evaluated
along r_i and
V=(f_1,f_2,f_3) is the vector field associated with the differential system (<ref>).







For any generalized Wallach space with a_1,a_2,a_3∈ (0,1/2)
for every i=1,2,3 the normal ∇λ_i of the surface Λ_i
is directed inside R
at every point of the  curve r_i.



Consider    Λ_i ⊂∂(R).
It suffices to consider points of  r_i.
Using Lemma <ref> we obtain

    ∂λ_i/∂ x_i=
    a_i(t-m_i)(t-M_i)/tx_i^2>0

for all a_1,a_2,a_3∈ (0,1/2) and all t∈ (0,m_i)∪ (M_i, +∞),
where m_i and M_i are roots of t^2-a_i^-1t+1=0 known from (<ref>).
This means that the normal
∇λ_i:=(∂λ_i∂ x_1, ∂λ_i∂ x_2, ∂λ_i∂ x_3)
is directed into R at every point of the curve r_i on the surface Λ_i⊂∂(R),
because of points of R lie above the surface Λ_i in the coordinate system (i,j,k)
according to Lemma <ref>.
Note also that

    ∂λ_i/∂ x_j=
    -√((t-m_i)(t-M_i))/2tx_i^2   (2a_it-1),     ∂λ_i/∂ x_k=
    √((t-m_i)(t-M_i))/2tx_i^2   (t-2a_i).

Since the inequalities
0<m_i<2a_i<1/2a_i<M_i hold
for all  a_i∈ (0,1/2) (see (<ref>)),
we have ∂λ_i/∂ x_j>0 and
∂λ_i/∂ x_k<0 for t∈ (0,m_i).
Respectively, ∂λ_i/∂ x_j<0 and
∂λ_i/∂ x_k>0
on the component r_ik corresponding to the interval (M_i, +∞).

Similar reasonings can be repeated for Λ_j and Λ_k.

Everything that was established here
also holds for the corresponding working“ subintervals
(0,t_ij]⊂ (0,m_i) and
[t_ik, +∞)⊂ (M_i,+∞)
for t_ij and t_ik evaluated in Lemma <ref>.
Lemma <ref> is proved.












For  an arbitrary generalized Wallach space with a_1,a_2,a_3∈ (0,1/2)
for every i=1,2,3 and for all t∈ (0,m_i) ∪ (M_i,+∞)
the sign of the inner product ( V, ∇λ_i)
evaluated at an arbitrary point of the curve r_i
coincides with the sign of the polynomial

    h(t)=c_0t^4+c_1t^3+c_2t^2+c_1t+c_0

with coefficients

    [            c_0:=a_i^2(1-2a_i+2a_j+2a_k)(2a_i+2a_j+2a_k-1),;              c_1:=a_i(1-2a_i)^3-4a_i(4a_i^2+1)(a_j+a_k)^2,; c_2:=(16a_i^4+16a_i^2+1)(a_j+a_k)^2+2a_i(1-a_i)(1-2a_i)^2. ]




As calculations show ( V, ∇λ_i) can be represented by
the following expression:

    [                                            ( V, ∇λ_i)=f_i ∂λ_i∂ x_i + f_j ∂λ_i∂ x_j + f_k ∂λ_i∂ x_k;  = a_i2 (a_ia_j + a_ia_k + a_ja_k)x_i^2 x_j^2 x_k^2[
    (a_i+a_j)(a_j+a_k)(a_k+a_i)(x_j^4+x_k^4-x_i^4); + a_k(a_i+a_j)(x_i^2-x_j^2)x_ix_j+a_j(a_i+a_k)(x_i^2-x_k^2)x_ix_k
    -2a_i(a_j+a_k)(x_j^2+x_k^2)x_jx_k;                          +{(2a_ia_k+a_ia_j+a_ja_k-a_j)x_j+(2a_ia_j+a_ia_k+a_ja_k-a_k)x_k} x_ix_jx_k;                                          + (a_j+a_k)(2a_i^2-2a_ia_j-2a_ia_k-2a_j a_k+1)x_j^2x_k^2]. ]

Substituting  the parametric equations
x_i=ϕ_i(t), x_j=ϕ_j(t), x_k=ϕ_k(t),  t∈ (0,m_i)∪ (M_i, +∞),
of the curve r_i  found in Lemma <ref>
into the latter expression for ( V, ∇λ_i) we obtain
( V, ∇λ_i)=12a_itx_i^2 (F-G),
where
F := (a_j+a_k)(t-2a_i)(2a_it-1) and
G := (1-2a_i)  a_i  (t+1)√(t^2-a_i^-1t+1).
The inequalities (<ref>) imply F>0 and  G>0
for all  t∈ (0,m_i)∪ (M_i, +∞) and all a_i, a_j, a_k∈ (0,1/2).
Therefore for t∈ (0,m_i)∪ (M_i, +∞)
the sign of ( V, ∇λ_i)
coincides with the sign of the following fourth degree polynomial
in t:

    h(t):=F^2-G^2=(a_j+a_k)^2(t-2a_i)^2(2a_it-1)^2 -
    (1-2a_i)^2a_i(t+1)^2(a_it^2-t+a_i).

Collecting its similar terms
we obtain the polynomial (<ref>) with coefficients in (<ref>).
Lemma <ref> is proved.





Since  t=0 is not a root of h(t)=0
introducing a new variable y:=t+t^-1 the equation h(t)=0
can be reduced to a quadratic equation

    p(y):=c_0 y^2+c_1 y+(c_2-2c_0)=0.



Two different real roots
t_1=y_0-√(y_0^2-4)2∈ (0,m_i) and
t_2=t_1^-1=y_0+√(y_0^2-4)2∈ (M_i, +∞)
of the equation h(t)=0 correspond to a given root y_0∈(a_i^-1, +∞)
of the equation p(y)=0.



The proof follows from the properties of the functions
x ↦1-√(1-4x^2)2x (increases) and
x ↦1+√(1-4x^2)2x (decreases) for 0<x<1/2.




In an arbitrary generalized Wallach space with a_1,a_2,a_3∈ (0,1/2)
the following assertions hold for the coefficients c_n=c_n(a_1,a_2,a_3), (n=0,1,2), of the polynomial p(y) in (<ref>):

c_2>0 and  c_2-2c_0>0 for all a_1,a_2,a_3∈ (0,1/2);

c_0=0 and c_1<0  if   a_1+a_2+a_3=1/2;

c_0<0 if  a_1+a_2+a_3<1/2;

c_0>0 and c_1<0  if  a_1+a_2+a_3>1/2.





The inequality c_2>0 follows from 16a_i^4+16a_i^2+1>0 (see (<ref>)).
Introducing a new variable  μ:= a_j+a_k we obtain the inequality
 
    c_2-2c_0=(1+4a_i^2)μ^2+2a_i(1-2a_i)^2>0

true for all a_i,a_j,a_k∈ (0,1/2).

The case a_1+a_2+a_3=1/2. Replacing  a_j+a_k by 1/2-a_i in (<ref>),
we obtain
c_0=0 and c_1=-2(1+2a_i)(1-2a_i)^2a_i^2<0.

The case a_1+a_2+a_3<1/2. Clearly c_0<0.

The case a_1+a_2+a_3>1/2. Obviously  c_0>0.
The coefficient c_1  can be considered as a function

    c_1=W(μ):=a_i(1-2a_i)^3-4a_i(1+4a_i^2)μ^2

of the independent variable μ= a_j+a_k.
Clearly W(μ) admits two zeros μ=-μ_0 and μ=μ_0, where
μ_0:=1-2a_i2√(1-2a_i1+4a_i^2).
Since  0<μ_0<12-a_i
we have W(μ)|_μ=1/2-a_i<0.
Then  W(μ)<0 for all  μ>1/2-a_i meaning that
c_1<0 at a_i+a_j+a_k>1/2. Lemma <ref> is proved.






§ PROOFS OF THEOREMS <REF> – <REF> AND COROLLARY <REF>




It suffices to consider the system (<ref>) on its invariant surface Σ
defined by the equation  Vol:=x_1^1/a_1x_2^1/a_2x_3^1/a_3=1.
Hence we are interested in parameters of metrics which belong to the domain Σ∩ R bounded by
the curves r_i=Σ∩Λ_i for i=1,2,3.
In order to establish behavior of trajectories of the system (<ref>)
with respect to that domain  we will study their interrelations (namely, the number of intersection points) with curves forming the boundary of that domain.



 §.§ The case a_1+a_2+a_3≤ 1/2

Examples <ref> and <ref>
illustrate Theorem <ref>.




It suffices to consider the border curve r_i only.
Similar reasoning can be repeated for the curves r_j and r_k
by simple permutations of indices {i,j,k}={1,2,3} in (<ref>).
Let us consider the cases  a_1+a_2+a_3<1/2  and  a_1+a_2+a_3=1/2 separately.



The case a_1+a_2+a_3<1/2.
For a_1+a_2+a_3<1/2  the discriminant

    c_1^2+8c_0^2-4c_0c_2

of the quadratic equation p(y)=c_0 y^2+c_1 y+(c_2-2c_0)=0 is positive,
because c_0<0 and c_2>0 by Lemma <ref>.
Hence  p(y)=0 admits two different real roots.
We claim that actually a single root of p(y)=0 can belong to (a_i^-1, +∞).
Indeed

    p(a_i^-1)=(a_k+a_j)^2(1-2a_i)^2(1+2a_i)^2>0

independently on a_i,a_j,a_k.
It is clear also
p(+∞):=lim_y → +∞ p(y)=-∞
due to  c_0<0.
Therefore p(y_0)=0 for some unique y_0∈ (a_i^-1, +∞).

No root of the quadratic polynomial p(y)
can be contained in the interval [0,a_i^-1]
since  p(0)=c_2-2c_0>0 by the same Lemma <ref>.
Then another root of p(y)=0 distinct from y_0 must be negative
and is not of interest providing  negative roots to h(t)=0.
Therefore the polynomial h(t)=c_0t^4+c_1t^3+c_2t^2+c_1t+c_0 of degree 4 admits two
positive roots t_1∈ (0,m_i) and t_2∈ (M_i, +∞)
which correspond to y_0 according to Lemma <ref>.
Note also that h(t) has a ∩“-shaped graph for t>0 and
h(t)<0 for t∈ (0,t_1)∪ (t_2,+∞)
and h(t)>0 for  t∈ (t_1, m_i)∪ (M_i,t_2).
Then according to Lemma <ref> we have

    sign( V, ∇λ_i)=
    
    -1,       t∈ (0,t_1),
      0,       t=t_1,
    
    +1,        t∈ (t_1, m_i)
    sign( V, ∇λ_i)=
    
    +1,        t∈ (M_i, t_2), 
      0,       t=t_2,
    
    -1,       t∈  (t_2, +∞)

respectively on r_ij and r_ik.
To reach  our aims it is quite enough to show that
trajectories of the system (<ref>)
will leave the domain R
across some part of some component of the curve r_i.
Take, for example, the second component r_ik of r_i
(a similar analysis can be given for the component
r_ij of r_i).

As it follows from the formulas above
( V, ∇λ_i)>0 for  t∈ (M_i,t_2).
This means that  on every point of a part of the curve  r_ik
which corresponds to (M_i,t_2) under the parametrization (<ref>)
the associated vector field V of the system (<ref>)
forms  an acute angle with the normal  ∇λ_i
of the surface Λ_i⊂∂(R)
(in fact its component Λ_ik).
Since  the normal ∇λ_i is directed inside R at every point of  r_i
due to Lemma <ref>, we conclude that
trajectories of (<ref>) attend R at t∈ (M_i,t_2).


For t∈ (t_2, +∞)  clearly
( V, ∇λ_i)<0 meaning
that V forms  an obtuse angle with the normal ∇λ_i.
Thus we established that trajectories of the system (<ref>)
originated in R will leave R at least through a part of the curve r_ik and never cross or touch r_ik again for all t>t_2.
 This means that there exist metrics that lose the positivity of the Ricci curvature  in finite time.
Theorem <ref> is proved for the case  a_1+a_2+a_3<1/2.




The case a_1+a_2+a_3=1/2.
Obviously we deal with the linear function  p(y)=c_1 y+c_2 since
c_0=0, c_1<0 and c_2>0 by Lemma <ref>.
Moreover, the only root y_0 of  p(y)=0 is positive.
Since p(a_i^-1)>0 due to (<ref>)
and

    p(+∞):=lim_y → +∞ p(y)=-∞

due to  c_1<0 we have y_0∈ (a_i^-1, +∞).
Lemma <ref> implies that the polynomial h(t)=(c_1t^2+c_2t+c_1) t of degree 3 has two roots t_1∈ (0,m_i) and  t_2∈ (M_i, +∞) which correspond to y_0.
We are in the same situation as in the considered case a_1+a_2+a_3<1/2.
Further reasoning is similar.
Theorem <ref> is proved.






According to Remark <ref> we had to consider
 intervals (0, t_ij]⊂  (0,m_i)
and [t_ik, +∞)⊂(M_i,+∞) in Theorem <ref>
instead of (0, m_i) and (M_i,+∞)
cutting of the extra pieces of the curves r_i, r_j and r_k
that remain after their pairwise intersections.
Although such a replacing   does not cancel the existence
of trajectories leaving R in any way, and therefore does not  affect
the proof of Theorem <ref>,
we get a little clarification to the qualitative picture of
incoming trajectories.
Such a picture depends on the relative position of t_1 and t_ij within the interval (0,m_i), for instance, if the component r_ij is considered.

If  t_ij< t_1 then h(t)<0 on all useful“
part of r_ij corresponding to t∈ (0,t_ij], hence h(t) changes its sign only at the
 tail“ of the curve r_ij behind the point P_ij.
Therefore ( V, ∇λ_i)<0 for all  t∈ (0,t_ij]
and hence every point on r_ij
emits trajectories towards the exterior of R. The interval  (0,t_ij] above
should be replaced by (0,t_ij) if t_ij=t_1.
In the case t_ij>t_1 trajectories
come into R through the part of  r_ij
corresponding to t∈ (t_1,t_ij]⊂ (t_1, m_i)
and leave  R through its  part corresponding to
t∈ (0, t_1).

Analogously, if t_ik<t_2 then trajectories attend R for
 t∈ [t_ik, t_2)⊂ (M_i,t_2) and  leave R for
t>t_2.
If t_ik>t_2 (respectively t_ik=t_2) then outgoing of trajectories from R
happens through every point of r_ik for t≥ t_ik
(respectively t> t_ik).

Thus analyzing all possible outcomes we conclude
that trajectories of (<ref>)
will leave R in any case whenever a_1+a_2+a_3≤ 1/2:
either through entire curve r_i either through some its part.






In the case a_1+a_2+a_3≤ 1/2
no metric (<ref>)  can evolve again into a metric with positive Ricci curvature,
being transformed once into a metric with mixed Ricci curvature,
but some metrics with mixed Ricci curvature can be transformed into metrics with positive Ricci curvature, and then again into metrics with mixed Ricci curvature.
This follows  from the proof of Theorem <ref> according to which
no trajectory of (<ref>)
returns back to R leaving R once,
but some trajectories attend R and come back if initiated in the exterior of R.








 §.§ The case a_1+a_2+a_3> 1/2

Recall the specific parameters θ_i:=a_i-1/2+1/2√(1-2a_i/1+2a_i)
exposed in the text of Theorem <ref>.
Clearly θ_i∈ (0,1)  for all i=1,2,3 and all a_i∈ (0,1/2).





It suffices to consider the curve r_i only.
In formulas (<ref>) for c_0, c_1 and c_2
replace the sum a_j+a_k  by 1/2-a_i+θ:

    [                                         c_0=4a_i^2 θ(1-2a_i+θ),;                  c_1:=a_i(1-2a_i)^3-a_i(4a_i^2+1)(1-2a_i+2θ)^2,; c_2:=14(16a_i^4+16a_i^2+1)(1-2a_i+2θ)^2+
    2a_i(1-a_i)(1-2a_i)^2. ]

Since each  coefficient c_0,c_1 and c_2 depends on the parameters a_i and θ only
so does the discriminant  of the quadratic equation
p(y)=0 in (<ref>).
By this reason denote them by D_i:

    D_i:=c_1^2-4c_0(c_2-2c_0)=c_1^2+8c_0^2-4c_0c_2
    
    =-4a_i^2(1+2a_i)^2(1-2a_i)^3
    {(1+2a_i)θ^2+(1-4a_i^2)θ-(1-2a_i)a_i^2}.


Since c_0>0 for a_i+a_j+a_k>1/2 and c_2>0 by Lemma <ref>,
the polynomial  D_i can accept  values of any sign.
Firstly, we find roots of D_i=0.
Observe that the value θ=θ_i
is exactly the unique root of the quadratic equation

    (1+2a_i)θ^2+(1-4a_i^2)θ-(1-2a_i)a_i^2= 0.

Its another root is negative.


(1)
The case D_i<0. The inequality D_i<0 is equivalent to

    (1+2a_i)θ^2+(1-4a_i^2)θ-(1-2a_i)a_i^2>0

which has solutions θ>θ_i.
Therefore the equation p(y)=0, and hence the equation h(t)=0
has no real roots if θ∈(θ_i,1).
Then  h(t)>0 for all t∈ (0,m_i)∪ (M_i,+∞) due to c_0>0,
in other words, by Lemma <ref> the sign of
( V, ∇λ_i) is positive along the curve r_i,
that means there is no arc or even a point on the curve r_i,
across which trajectories of the system (<ref>) could leave the domain R,
because of  the normal ∇λ_i of the
cone Λ_i is directed inside R at every point of r_i
independently on values of a_1, a_2 and a_3 due to Lemma <ref>.




The case D_i= 0. Since  D_i=0 happens at θ=θ_i only,
for the sum a_j+a_k we have

    a_j+a_k=1/2-a_i+θ_i=1/2√(1-2a_i1+2a_i).

The coefficients (<ref>) take the following forms at θ=θ_i:


    [     c_0:=-a_i^2((1-2a_i)^2-4(a_j+a_k)^2)
    =4a_i^4 ·1-2a_i1+2a_i,;                  c_1:=a_i(1-2a_i)^3-a_i(4a_i^2+1)·1-2a_i1+2a_i,; c_2:=14(16a_i^4+16a_i^2+1)
    ·1-2a_i1+2a_i+2a_i(1-a_i)(1-2a_i)^2. ]



In what follows an estimation

    y_0-a_i^-1=-c_1/2c_0-a_i^-1
    =(1-2a_i)^2(1+2a_i)^2/4a_i^2(1-4a_i^2)=
    1-4a_i^2/4a_i^2>0

 for the unique root
y_0=-c_12c_0 of the equation
P_2(y)=c_0y^2+c_1y+
c_2-2c_0=0.

Since  y_0∈ (a_i^-1, +∞) is the single root  of p(y) of multiplicity 2
the polynomial h(t) of degree 4 has two zeros t_1  and t_2 each of multiplicity 2
such that  t_1∈ (0,m_i) and t_2∈ (M_i,+∞).
Since c_0>0  then h(t)>0 anywhere on (0,m_i)∪ (M_i,+∞) excepting
the points t_1 and t_2 (the shape of the graph of h(t) is similar to the letter W“ touching the t-axis at the points t_1 and t_2).
Then by Lemma <ref> no trajectory of (<ref>)
can leave R through its boundary curve r_i.
Note that in the case D_i=0 we are able to find the mentioned   multiple roots of h(t) explicitly:

    t_1,2=1/8a_i^2[-4a_i^2+4a_i+1∓(1+2a_i)√((1+6a_i)(1-2a_i))],

since h(t)=(1-2a_i)(1+2a_i)^-1(a_i^2t^2+(a_i^2-a_i-0.25)t+a_i^2)^2.

Thus uniting the cases  θ>θ_i (D_i<0) and  θ=θ_i (D_i=0)
we conclude that no trajectory of (<ref>) can leave R through r_i.
The same will hold on the curves  r_j and r_k, if
θ≥max{θ_i, θ_j, θ_k}
(all metrics maintain the positivity of the Ricci curvature).
This finishes the proof of the first assertion in Theorem <ref>.
See also Example <ref>.



(2)
To prove the second assertion in Theorem <ref>
assume that θ< θ_i for some i.
Such an assumption equivalent to  D_i> 0.
We claim  that  both roots of p(y)=0
belong to (a_i^-1, +∞) in this case.
We are in a situation where a quadratic function with
a positive discriminant takes positive values at the ends of the interval:

    p(a_i^-1)>0,        p(+∞):=lim_y → +∞ p(y)=+∞

due to  c_0>0.
Therefore, the analysis used in the proof of Theorem <ref>
is not applicable here: such an interval can contain either both roots of the equation
or none.



No root can belong to (-∞, 0).
Suppose by contrary  p(y)=0 has roots y_1,y_2∈ (-∞, 0).
Then -c_12c_0=y_1+y_22<0 despite the facts
c_0>0 and c_1<0  established
in Lemma <ref> in the case a_1+a_2+a_3>1/2.



No root can belong to (0, a_i^-1).
There is a similar situation at the ends of the interval:
p(0)=c_2-2c_0>0 by Lemma <ref>
and p(a_i^-1)>0 by (<ref>).
Introduce a function

    U(μ):=-c_12c_0-a_i^-1,

where the sum a_j+a_k is replaced by  μ=a_j+a_k in  c_0 and c_1.
Then U(μ)>0 is equivalent to the inequality
4μ^2-(1-2a_i)<0
admitting solutions μ∈(0,μ),
where μ:=√(1-2a_i)2 is the positive root of U(μ)=0.
Since μ=1/2-a_i+θ the following value of θ
corresponds to μ:
θ_i=2a_i-1+√(1-2a_i)2.
It follows then U>0 for θ∈(0,θ_i)
and U<0  for θ∈(θ_i, 1).
It is not difficult to see that

    θ_i=
    4a_i^2-1+√((1-4a_i^2)(1+2a_i))2(1+2a_i)>
    4a_i^2-1+√(1-4a_i^2)2(1+2a_i)=θ_i.

Therefore, U=-c_12c_0-a_i^-1>0 for values
θ∈(0, θ_i)⊂(0,θ_i),
which correspond to the case D_i>0.
It follows then p(y)=0 has no roots in (0, a_i^-1).
Supposing the contrary we would conclude -c_12c_0=y_1+y_22<a_i^-1 from
  0<y_1<a_i^-1 and 0<y_2<a_i^-1 despite U>0.



Thus we showed that both roots of p(y)=0
belong to (a_i^-1, +∞) in the case θ<θ_i.
Denote them by  y_1 and y_2. Without loss of generality assume that  y_1<y_2.
By Lemma <ref>
two different roots t_1∈ (0, m_i) and t_2=t_1^-1∈ (M_i, +∞) of h(t)=0
correspond to y_1.
Analogously there are roots t_3∈ (0, m_i) and t_4=t_3^-1∈ (M_i, +∞) of
h(t)=0 corresponding to y_2.
It is easy to see that

    t_3<t_1<m_i<M_i<t_2<t_4.


Thus each of the intervals (0,m_i) and (M_i,+∞) contains
exactly two roots of the polynomial h(t) which has a W“-shaped graph.
It follows then exactly
two points are expected  on each component of the curve r_i  at which
the vector field  V=(f_1,f_2,f_3) changes its direction
in relation to the domain R (into R“ or from R“).
Consider the component r_ik only with t∈ (M_i, +∞).
Then M_i<t_2<t_4 implies that
trajectories of (<ref>) come into R through r_ik
and never cross or touch r_ik again  for all t>t_4.
For the component r_ij with  t∈ (0,m_i)
a similar analysis can be carried out
by mirroring of M_i<t_2<t_4.


Thus in the case θ<θ_i
trajectories of (<ref>) can intersect the border curve r_i twice according to the scenario
towards R – from R – towards R again“ (see also Remark <ref>).
A similar scenario will take place on  the border curves r_j or (and) r_k,
if θ<θ_j or (and) θ< θ_k
returning trajectories to R even they ever left R once.
Clearly if  θ≥θ_j or θ≥θ_k at θ<θ_i
then the curve r_j or r_k is locked for trajectories to leave R as proved before.
After such an analysis we conclude that at least some metrics with positive Ricci curvature can preserve the positivity of the Ricci curvature. The second assertion in Theorem <ref> is also proved.
Theorem <ref> is proved.





Let us  comment the behavior of trajectories in more detail
 in the case θ<θ_i in Theorem <ref>. Consider the example of r_ik.
As it was  noted in Remark <ref>
we are actually interested in the useful“ part of the component r_ik
parameterized by t∈ [t_ik, +∞)⊂ (M_i,+∞),
where t_ik>M_i by Lemma <ref>.
The dynamics of trajectories of (<ref>) depends on which link of the chain
M_i<t_2< t_4<+∞ the value  t_ik will be situated.
In the case θ<θ_i the following
situations may happen on the component r_ik of r_i:


Case 1.  t_ik∈ [t_4, +∞).   Trajectories move towards R through
a part of  r_ik which corresponds to t≥ t_ik
(respectively t> t_ik if t_ik=t_4).
No trajectory can leave R.


Case  2.  t_ik∈ (t_2, t_4).
Trajectories leave  R  for t∈[t_ik, t_4) and
return back to  R for t>t_4.

Case 3.  t_ik∈ (M_i, t_2].
Trajectories come into R
for t∈[t_ik,t_2),
leave R for t∈(t_2, t_4) and return back to R
for t>t_4.

The analysis of all possible outcomes shows that
even if the strong Case 1  never can occur at θ<θ_i,
some trajectories of (<ref>) must return back to R and remain there
 for all t>t_4 in any case.







In the case a_1+a_2+a_3> 1/2
some metrics with mixed Ricci curvature can be transformed into metrics with Ric>0, after then lose Ric>0 and restore
Ric>0 again.
This follows from the proof of Theorem <ref> according to which
trajectories of (<ref>) can attend R twice and then remain in R forever.















 Clearly   θ=a_1+a_2+a_3-12=1k+l+m-2>0 for  GWS 1.





(1)
Case A. The condition max{k,l,m}≤ 11 holds.
Without loss of generality consider the difference θ-θ_1 only.
Observe that a_1=kθ/2 and hence

    θ-θ_1=θ-a_1+12-12√(1-2a_11+2a_1)=
    θ-kθ/2+12-12√(1-kθ1+kθ).

It follows then  1-kθ≥ 0 necessarily to be well defined.
Therefore  permissible values of  θ∈ (0,1)
must belong to a narrower interval  (0, 1/k]⊆ (0,1) for each natural number k.
Under  such restrictions the inequality  θ-θ_1≥ 0 is equivalent to
the following system of inequalities

    T(θ):=k(k-2)^2θ^2-(k^2-4)θ+4≥ 0 
    0<θ≤ 1/k.



Obviously T(θ)=4>0 is trivial at k=2.
Assume that k 2.
The discriminant

    D(k):=(k^2-12k+4)(k-2)^2=(k-(6-4√(2)))
    (k-2)^2(k-(6+4√(2)))

of the polynomial T(θ), where 6-4√(2)≈ 0.3431 and 6+4√(2)≈ 11.6569,
is negative for all k∈{1, 3, 4, …, 11},
and hence  for k∈{1,2, …, 11}
we have T(θ)>0 that means  θ>θ_1.
Then by Theorem <ref>
no point on the curve r_1 can emit a trajectory towards the exterior of R.
The same will hold for the curves r_2 and r_3 if max{k,l,m}≤ 11
that imply θ≥max{θ_1, θ_2, θ_3}.
Example <ref> illustrates the case max{k,l,m}≤ 11.




Case B. The condition max{k,l,m}≤ 11 fails.
Recall our assumption k≥max{l,m} due to symmetry.
Suppose that  k≥ 12. For such values of k the discriminant  D(k) is positive.
However it does not mean that  T(θ)≥ 0.
It may be T(θ)< 0.
Firstly we show that all roots of T belong to (0, 1/k]
for such values of  k.
Since there is no double roots, we can apply Sturm's method.
For this aim we construct on [0,1/k]  the Sturm's polynomial system of T(θ):

    f_0(θ)   :=   T(θ), 
    
    f_1(θ)   :=   dT/dθ=2k(k-2)^2θ-(k^2-4), 
    
    f_2(θ)   :=   -rem(f_0,f_1)=D(k)/4lc(T)=k^2-12k+4/4k(k-2),

where the symbol rem(f_0,f_1)  means the remainder of dividing  f_0 by f_1 and  lc means the leading coefficient of a polynomial.
At the point θ=0
the number of sign changes in the Sturm system equals 2 since  k≥ 12:

    f_0(0)=4>0,      f_1(0)=4-k^2<0,      f_2(0)>0.


At θ=1/k   the corresponding number is 0:

    f_0(1/k)=8/k>0,       f_1(1/k)=(k-2)(k-6)>0,       f_2(1/k)>0.


Therefore  both  roots  of the polynomial
T(θ) belong to  the interval (0,1/k) in cases k≥ 12.
Denote them Θ^(1)=Θ^(1)(k) and
Θ^(2)=Θ^(2)(k).
Actually we are able to find them explicitly:
Θ^(1)(k)=k+2-√(k^2-12k+4)2k(k-2)
and
Θ^(2)(k)=k+2+√(k^2-12k+4)2k(k-2).
Note that

    0<Θ^(1)(k)<Θ^(2)(k)<1/k.

Now the sign  of θ-θ_1 will depend on the fact
what link in the chain of these inequalities
 will contain θ.


Subcase B1.   θ≤Θ^(1) or Θ^(2)≤θ<1/k.
Then  θ≥θ_1.
According to Theorem <ref>
no trajectory of (<ref>) can leave R via the curve r_1.
Examples <ref> and <ref> are relevant to Subcase B1.


Subcase B2.   Θ^(1)<θ<Θ^(2) is equivalent to θ<θ_1.
By Theorem <ref>  there exists a trajectory
of (<ref>) that  can leave R but return back to R.
Example <ref> is relevant to Subcase B2.


For further analysis the inequalities should be rewritten  in more convenient form in
Subcases B1 and B2
in terms of the parameters k,l and m.
For instance, Subcase B1 can be described by the union of solutions
of the inequalities 2<l+m≤ X(k) and l+m≥ Y(k),
where
 
    X(k):=2k(k-2)/k+2+√(k^2-12k+4)-k+2,      Y(k):=2k(k-2)/k+2-√(k^2-12k+4)-k+2

are  functions  appeared in the text of Theorem <ref> and Table 2.



Imagine  X(k) and Y(k) as  functions of real independent variable k to apply methods of calculus (see Fig. <ref> for their graphs).
It is easy to establish X(k)>2 for all k≥ 12
(observe that 2 is a horizontal asymptote since lim_k →∞ X(k)/k=0).
Moreover, X(k) decreases  and satisfies 2<X(k)≤ 5 for k≥ 12
since X(12)=5 and

    X'(k)=-16k^2-5k+2-(k-1)√(k^2-12k+4)/(k+2+√(k^2-12k+4))^2√(k^2-12k+4)<0

due to (k^2-5k+2)^2-(k-1)^2(k^2-12k+4)=4k^3>0.


As for Y(k) it increases since

    Y'(k)=16k^2-5k+2+(k-1)√(k^2-12k+4)/(k+2-√(k^2-12k+4))^2√(k^2-12k+4)>0

for all k≥ 12.
In addition,  Y(k)≥ 10 for all k≥ 12 with Y(12)=10.





We claim that  only the values k∈{12, …,  16} can
satisfy θ≥max{θ_1, θ_2, θ_3}  in Subcase B1.
Indeed since X(k) decreases and  X(17)≈ 2.93 with  the critical case l+m≤ 2
it looses any sense to consider 2<l+m≤ X(k) for k≥ 17: we would obtain l=m=1 for all k≥ 17 contradicting l+m>2.
Analogously due to increasing of  Y(k)  the numbers l and m in the sum l+m≥ Y(k) will be out
of the range max{l,m}≤ 11 in a short time.
Then the inequalities

    Y(k)≤ l+m+k-2

are required to be true for every permutation of the triple (k,l,m)
in order
to keep the condition θ≥max{θ_1, θ_2, θ_3}
for k,l,m≥ 12,
where Y(k):=2k(k-2)k+2-√(k^2-12k+4).
It follows than  permissible values of  l,m,k  can not be large than 16.
Indeed   k≥max{l,m} by assumption.
Since the function Y(k)=Y(k)+(k-2) increases for all k≥ 12
we obtain the following inequality

    max{Y(k),Y(l),Y(m)}= Y(k)≤ 3k-2.

Introduce the function

    Z(k):=Y(k)- (3k-2)=k^2+8k-4-(3k-2)√(k^2-12k+4)/k+2-√(k^2-12k+4).

Its  derivative Z'(k) has the numerator
k^3-6k^2-40k+16-(k^2-8k+8)√(k^2-12k+4)>0
due to
(k^3-6k^2-40k+16)^2-(k^2-8k+8)^2(k^2-12k+4)=16k^2(k^3-20k^2+104k-32)>0
for all k≥ 12 (the unique real root of k^3-20k^2+104k-32 is less than 1).
Therefore Z(k) increases for k≥ 12.
Note also that Z(16)≈ -0.0691<0 and Z(17)≈ 4.3137>0
implying that the inequality  Y(k)≤ 3k-2 can not be satisfied if k≥ 17.

Collecting all values of (k,l,m)  which provide the condition θ≥max{θ_1, θ_2, θ_3} of Theorem <ref> in  both Cases A and B
we conclude that on GWS 1 all metrics with Ric>0 maintain Ric>0
under NRF (<ref>)
if (k,l,m) satisfies either max{k,l,m}≤ 11  or one of  the inequalities
2<l+m≤ X(k) or l+m≥ Y(k) at 12≤ k≤ 16.




(2) The  system of inequalities
X(k)<l+m<Y(k) and k≥ 12 corresponding to Subcase B2
is equivalent to the negation of θ≥max{θ_1, θ_2, θ_3}
and guaranties the existence of some metrics preserving Ric>0
by the same Theorem <ref>.




(3)
Let us estimate  the number of GWS 1 on which
all metrics with Ric>0 maintain Ric>0.



It is obvious that there exists a finite number of GWS 1  which satisfy
max{k,l,m}≤ 11.
As we saw above  the inequality  2<l+m≤ X(k)  corresponding to Subcase B1 can  provide only a finite set of solutions in (k,l,m) due to decreasing of X(k).
For instance, X(12)=5 implies l+m≤ 5 etc.
Analogously, another inequality  Y(k)≤ l+m corresponding to Subcase B1   admits a finite number of solutions (k,l,m) as well
according to  assumed max{l,m}≤ k and established  12≤ k≤ 16.
Therefore the number of GWS 1
on which any metric with Ric>0 remains Ric>0
must be finite.

Obviously, there is no restriction to k
in the inequalities X(k)<l+m<Y(k) besides supposed k≥ 12 in Case B.
Indeed they admit infinitely many solutions in (k,l,m)
since the range (X(k),Y(k))  expands as k grows
providing an increasing number of pairs (l,m) which can satisfy X(k)<l+m<Y(k)
for every fixed k≥ 12.
Therefore there are  infinitely (countably) many GWS 1 on which at least some metrics can keep Ric>0.
Theorem <ref> is proved.











All GWS 1
on which any metric with Ric>0 remains Ric>0
can easily be found.
Using values of X(k) and Y(k) given in Table 3
all triples (k, l, m) are found in Tables 4 and 5 which
satisfy  2<l+m≤ X(k) or l+m≥ Y(k)
at fixed k∈{12,… 16} and k≥max{l,m},
where N=[Y(k)] if Y(k) is integer, else N=[Y(k)]+1
(symbol [x] means the integral part of x),
analogously  ν(l):=10-l if  10-l>0 else ν(l):=1.
See also Fig. <ref> and <ref>.
The single pair (l,m)=(16,16) corresponds to k=16.
No permissible pair (l,m) at k≥ 17.

















Recall that we are using the list of generalized Wallach spaces
given in Table 1 in accordance with <cit.>.



(1)  GWS 2, 5, 7.
The proof follows from Theorem <ref>
since  a_1+a_2+a_3=1/2 for spaces 2, 5, 7.
An illustration is given in Example <ref>.



(2) GWS 3, 15.
Clearly θ=a_1+a_2+a_3-12=-12(k+l+m+1)<0  for
 GWS 3.
Analogously θ=-1/6<0 for GWS 15.
The proof follows from  Theorem <ref>.




(3) GWS 4, 6, 8, 9, 10, 11, 12, 13, 14.
For all them θ>0. Desired conclusions follow by  checking the conditions
max{θ_1, θ_2, θ_3}<θ<1 directly.
We demonstrate the proof only for GWS 4 which depend
on parameters.
It is  easy to show that  θ=a_1+a_2+a_3-12=14  is greater than
each of
θ_1=2l√((3l+1)(l-1))-3l^2+2l+1/4l(3l+1),
θ_2=2l√((l+1)(3l-1))-3l^2-2l+1/4l(3l-1) and
θ_3=√(3)/6-1/4
for all l=2,3,….
Corollary <ref> is proved.







§ PLANAR ILLUSTRATIONS OF RESULTS















 §.§ The case a_1+a_2+a_3≤ 1/2



Ric>0 is not preserved on every GWS 3.
Take for instance (k,l,m)=(5, 4, 3).
Then a_1=5/26≈ 0.1923, a_2=2/13≈ 0.1538 and a_3=3/26≈ 0.1154 with a_1+a_2+a_3<1/2.
Results are depicted in Fig. <ref> and Fig. <ref>.




In Example <ref> we demonstrate all evaluations in detail.
In sequel we will restrict ourselves to concise comments.
The curve r_1=Σ∩Λ_1 defined by the system
of equations 5(x_1^2-x_2^2-x_3^2)+26x_2x_3=0 and
x_1^1/5 x_2^1/4 x_3^1/3=1
has the following parametric representation according to Lemma <ref>:
x_1=ϕ_1(t)=ψ^35/94/t^15/47,
x_2=ϕ_2(t)=t^32/47/ψ^6/47 and
x_3=ϕ_3(t)=t^-15/47/ψ^6/47,
where ψ:=t^2-26t/5+1, t∈ (0,m_1)∪ (M_1,+∞), m_1=1/5, M_1=5.

Choose the component r_13 corresponding to  t∈ (5,+∞).
Coordinates of the point P_13=r_13∩ r_31  can be found using formulas (<ref>) and (<ref>):
x_1=γ q^0≈ 0.3916, x_2=q^0≈ 3.4857, x_3=δ q^0≈ 17.4285,
where
γ=Φ(a_1,a_3)=-27+9√(10)/13≈ 0.1123, 
δ=Ψ(a_1,a_3)=50-15√(10)/13≈ 0.1974,  
q^0=γ^-ω a_1^-1δ^-ω a_3^-1=
γ^-12/47δ^-15/47≈ 3.4857.
Due to formula (<ref>)   the following value
of the parameter t corresponds to the point P_13:
 t=t_13=δ^-1=13/50-15√(10)≈ 5.0666.

The polynomial h(t) in (<ref>) takes the form
h(t)= -63375t^4-370500t^3+4529574t^2-370500t-63375.
Respectively
p(y) = -63375y^2-370500y+4656324
has a unique positive root
y_0=-38/13+192/325√(235)≈ 6.1332>a_1^-1=5.2.
The corresponding roots of h(t) are the following:
t_1≈ 0.1676<m_1=0.2 and t_2≈ 5.9656>M_1=5.

The vector field V changes its direction at the unique
point K_2 with coordinates
x_1=ϕ_1(t_2)≈ 1.0717,
x_2=ϕ_2(t_2)≈ 2.7097,
x_3=ϕ_3(t_2)≈ 0.4542.

For P_13 we have M_1 <t_13<t_2.
Therefore h(t)>0  on r_13 if  t∈ [t_13,t_2).
This means that trajectories originated in the exterior of R can reach R
across the arc of the curve r_13 with endpoints at P_13 and K_2.
In sequel the mentioned incoming trajectories
and trajectories  originated in R,
will leave R at once through an unbounded part of the curve r_13,
which extends from K_2 to infinity
as the image of the interval (t_2, +∞).
Such a conclusion is confirmed by the values of the limits in (<ref>) as well:
lim_t→ M_1+ x_1=0,
lim_t→ M_1+ x_2= lim_t→ M_1+ x_3 = +∞,
lim_t→ +∞ x_1=lim_t→ +∞ x_2 = +∞,
lim_t→ +∞ x_3 =0.

Using coordinates of the normal found in Lemma <ref>,
we can analyze values of the angle

    α(t):=180/πarccos ( V, ∇λ_1)/ V ∇λ_1

between V=(f,g,h) and λ_1 on every point of the curve r_13, in degrees.
As calculations show  73.16^∘<α(t)< 90^∘  for the incoming flow at t∈ (t_13, t_2)
and 90^∘<α(t)<93.36^∘ for the outgoing flow at  t>t_2.
Therefore, trajectories  leave R under small angles not exceeding 4^∘.


Ric>0 is not preserved on every  GWS 2.
Take for instance k=5, l=4 and m=3.
Then a_1=5/24≈ 0.2083, a_2=4/24≈ 0.1667 and a_3=3/24=0.125 with a_1+a_2+a_3=1/2.




All necessary evaluations can  be repeated in the same order as in Example <ref>.
Observe that 71.62<α(t)≤ 90 for incoming trajectories
at
4.6533≈64/5(45-√(595)√(3))=t_13≤ t≤ t_2=3713+17√(42721)/1200≈ 6.0223
and 90<α(t)<91.43 for outgoing trajectories  at  t>t_2.
Trajectories leave R under angles not greater than 1.5^∘,
that means they are almost parallel to tangent vectors to r_13.




 §.§ The case a_1+a_2+a_3>1/2










The values a_1=5/14 ≈ 0.3571,  a_2=3/7≈ 0.4286 and a_3=1/14+√(6)/12≈ 0.2756
satisfy a_1+a_2+a_3>1/2 and θ≥max{θ_1, θ_2, θ_3}.
Therefore trajectories of (<ref>) are expected only incoming into R by Theorem <ref> (see also Fig. <ref>).


Indeed θ=5/14+√(6)/12≈ 0.5613,
θ_1=-1/7+√(6)/12≈ 0.0613,
θ_2=-1/14+√(13)/26≈ 0.0672,
θ_3=1/14-239+14√(6)+7√(1434-84√(6))/48+7√(6)≈ 0.0445.
It should be noted that in general such artificial values of the parameters a_1,a_2,a_3
need not correspond to certain generalized Wallach space  as noted in Introduction.











On GWS 1 with  k=l=m=2 (a_1=a_2=a_3=a=1/4)
all metrics with Ric>0 preserve Ric>0.
Results are depicted in Fig. <ref> and Fig. <ref>.



Since max{k,l,m}≤ 11 the sufficient condition
θ≥max{θ_1, θ_2, θ_3}
is satisfied.
Indeed θ=0.25 and
θ_1=θ_2=θ_3≈ 0.0387.
Therefore we have only trajectories incoming into R.
Note that  this example confirms also Theorem <ref> (Theorem 3 in <cit.>),
because  a=1/4>1/6.









On GWS 1 with k=12, l=3, m=2 (a_1=2/5,
a_2=1/10, a_3=1/15≈ 0.0667)
 all metrics with Ric>0  preserve Ric>0.



The point (3,2) lies on the straight line l+m=X(12)=5.
Therefore θ=θ_1  meaning that  trajectories  can only touch
each component of r_1 at some unique point  of that component
remaining inside of R (see Fig. <ref>).
Also trajectories of (<ref>) can not leave R via r_2 or r_3 since
θ>max{θ_2,θ_3} due to max{l,m}<11.
Therefore they remain in R.

In detail, the polynomial p(y)=(16y-49)^2 corresponds to r_1
that admits the single root y_0=49/16=3.0625 of multiplicity 2.
Then the corresponding  polynomial
h(t)=(16t^2-49t+16)^2
admits two roots t_1=49-9√(17)/32≈ 0.3716 and
t_2=49+9√(17)/32≈ 2.6909 each of multiplicity 2.

Actual values of the parameters are
θ=θ_1=1/15≈ 0.0667,
θ_2=-2/5+√(6)/6≈ 0.0082,
θ_3=-13/30+3√(221)/34≈ 0.0039.




On GWS 1 with k=15, l=14, m=13
(a_1=3/16=0.1875, a_2=7/40=0.175, a_3=13/80=0.1625)
all metrics with Ric>0 preserve Ric>0.


The triple (15, 14,13) satisfies l+m>Y(15)=26.
Therefore θ>θ_1.
Analogously the inequalities k+m>Y(14) and l+k>Y(13)  imply
θ>θ_2 and θ>θ_3,
where  Y(14)≈ 20.49 and Y(13)≈ 15.29
are known from Table <ref>.
Therefore no trajectory of (<ref>) can leave R.




Supplementary, for
each  r_1, r_2 and r_3 their polynomials of the kind h(t) admit no real root
preserving positive sign.
Actual values of the parameters are
θ=1/40= 0.025,  θ_1=-5/16+√(55)/22≈ 0.0246,
θ_2=-13/40+√(39)/18≈ 0.0219,
θ_3=-27/80+3√(159)/106≈ 0.0194.









On GWS 1 with  k=14, l=7, m=4
(a_1=7/23≈ 0.3043, a_2=7/46≈ 0.1522, a_3=2/23≈ 0.0869)
some metrics  with Ric>0  preserve Ric>0.




Since l, m<11 and  (k,l,m)=(14,7,4) satisfies X(k)≈ 3.51 <l+m<Y(k)≈ 20.49
then θ>θ_2, θ>θ_3 and θ<θ_1
avoiding direct computations
(actually
θ=1/23≈ 0.0435,  θ_1=-9/46+3√(37)/74≈ 0.0509,
θ_2=-8/23+√(30)/15≈ 0.0173,
θ_3=-19/46+√(57)/18≈ 0.0064).
Therefore no trajectory can leave R via r_2, r_3
but  there expected exactly two points on each component of r_1
at which trajectories of (<ref>) can change directions relatively to R in accordance
with the scenario towards R - from R - towards R again“
(see Fig. <ref>).

Supplementary, the polynomial h(t) corresponding to r_1 admits four roots
t_1≈ 0.2532, t_2≈ 3.9488,
t_3≈ 0.15, t_4≈ 6.6663 such that
t_3<t_1<t_12=320/161-8√(1110)/161≈ 0.3321
and
t_13=230/7(√(2109)-57)≈ 2.9665<t_2<t_4.







§ FINAL REMARKS


1) Let us return to generalized Wallach spaces with a_1=a_2=a_3=a,
in particular, let us consider the cases
a=1/4, a=1/6, a=1/8 and a=1/9 studied in <cit.>.
As mentioned in Introduction the Wallach spaces W_6, W_12 and W_24
are examples of GWS which correspond to a=1/6, a=1/8 and a=1/9 respectively.
The value a=1/4 is very special at which four different singular points of (<ref>) merge to its unique linearly zero saddle, see <cit.>. Moreover,  on the
algebraic surface of degree 12 mentioned in Introduction,
the point (1/4,1/4,1/4) is a singular point of elliptic umbilic type in the sense of Darboux (see <cit.>).


Theorem <ref> generalizes Theorem <ref>.
Indeed generalized Wallach spaces with a=1/4
can be obtained in the special case of GWS 1 with k=l=m=2.
The condition max{k,l,m}≤ 11 is satisfied.
Therefore by Theorem <ref> all  initial metrics
with Ric>0
maintain Ric>0.

Although Theorem <ref> covers Theorem <ref>
 we have stronger statements in Theorem <ref> that all  initial metrics with Ric>0
lose Ric>0 on the narrow class of GWS with a∈ (0,1/6),
in particular on the spaces W_12 and W_24,
whereas Theorem <ref> states such a property only for some initial metrics with Ric>0, but for a wider class of GWS with a_1+a_2+a_3≤ 1/2,
in particular, for the mentioned a=1/8 and a= 1/9 (a_1+a_2+a_3<1/2),
which correspond to GWS 3 with k=l=m=1 and GWS 15 respectively.

Analogously, for a∈ (1/6,1/4)∪ (1/4,1/2) all metrics become
Ric>0 according to Theorem <ref>,
whereas by Theorem <ref> (even by Theorem <ref>)
the similar property can be guarantied
for some metrics only, but again on a wider class of GWS
which satisfy a_1+a_2+a_3>1/2.


Theorem <ref> is consistent with Theorem <ref> and extends it.
According to Theorem <ref> not every metric lose Ric>0 in the case a=1/6 (they are exactly metrics  contained in the domain bounded by parameters of Kähler metrics x_k=x_i+x_j,  {i,j,k}={1,2,3}, see <cit.>).
This fact does not contradict the conclusions of Theorem <ref>,
where spaces with a=1/6 are contained as a special class of GWS 2 with k=l=m (a_1+a_2+a_3=1/2).
Moreover, Theorem <ref> implies the departure of some metrics
from the domain with Ric>0 at a=1/6, that was not previously stated in Theorem <ref>.




2) In general there is infinitely (uncountably) many triples
(a_1,a_2,a_3)∈ (0,1/2)^3 satisfying a_1+a_2+a_3>1/2
such that trajectories of (<ref>) all remain in R originating there.
Indeed each  function
θ_i=θ_i(a_i)=a_i-1/2+1/2√(1-2a_i/1+2a_i), i=1,2,3,
attains its largest value θ^∗_i=θ_i(a^∗)≈ 0.067442248
at the point
a^∗=μ^2-2μ+4/6μ≈ 0.4196433778
where μ=√(19+3√(33))≈ 3.30905648 (see Fig. <ref>).
Then we always have an opportunity to get θ >θ_i for each i
choosing (uncountably many) a_i  at least from the interval [a^∗, 1/2)
because

    θ=a_1+a_2+a_3-0.5≥ 3a^∗-0.5>1.2-0.5=0.7>θ_i.

Therefore the number of appropriated generalized Wallach spaces, where all metrics with Ric>0  maintain  Ric>0,
might be unbounded too if they exist for such (a_1,a_2,a_3)
(compare with the third assertion of Theorem <ref>).





The author expresses his gratitude to Professor Yuriĭ Nikonorov for helpful discussions on the topic of this paper.






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