# Progress on homogeneous Einstein manifolds and some open probrems

Andreas Arvanitoyeorgos  University of Patras, Department of Mathematics,
GR-26500 Patras, Greece<EMAIL_ADDRESS>
###### Abstract.

We give an overview of progress on homogeneous Einstein metrics on large
classes of homogeneous manifolds, such as generalized flag manifolds and
Stiefel manifolds. The main difference between these two classes of
homogeneous spaces is that their isotropy representation does not
contain/contain equivalent summands. We also discuss a third class of
homogeneous spaces that falls into the intersection of such dichotomy, namely
the generalized Wallach spaces. We give new invariant Einstein metrics on the
Stiefel manifold $V_{5}\mathbb{R}^{n}$ ($n\geq 7$) and through this example we
show how to prove existence of invariant Einstein metrics by manipulating
parametric systems of polynomial equations. This is done by using Gröbner
bases techniques. Finally, we discuss some open problems.

2010 Mathematics Subject Classification. Primary 53C25; Secondary 53C30,
22C05, 22E60, 13P05, 13P10, 13P15

Keywords: Homogeneous space; Einstein metric; isotropy representation;
generalized flag manifold; Stiefel manifold; generalized Wallach space;
algebraic system of equations; Gröbner basis

##  1\. Introduction

A Riemannian manifold $(M,g)$ is called Einstein if it has constant Ricci
curvature, i.e. $\operatorname{Ric}_{g}=\lambda\cdot g$ for some
$\lambda\in\mathbb{R}$. Due to an old result of Hilbert, for $M$ compact,
Einstein metrics are precisely the critical points of the scalar curvature
functional over the set of Riemannian metrics of volume $1$. If $\lambda>0$
then $M=G/H$ is compact, if $\lambda=0$, $M$ is Ricci flat, and if $\lambda<0$
then $G/H$ non-compact. For results on Einstein manifolds before 1987 we refer
to the book by A. Besse [Be]. The two articles [Wa1], [Wa2] of M. Wang contain
results up to 1999 and 2013 respectively. General existence results are
difficult to obtain and some methods are described in [Bö], [BöWaZi] and
[WaZi].

For homogeneous spaces $M=G/K$ the problem is to find and classify all
$G$-invariant Einstein metrics, or decide if the set of $G$-invariant Einstein
metrics is finite or not. A conjecture by W. Ziller ([BöWaZi]) states that if
${\rm rk}(G)={\rm rk}(H)$, or if the isotropy representation of $G/K$ contains
only non equivalent summands, then the number of $G$-invariant Einstein
metrics is finite.

The aim of the present article is to give an overview of recent progress on
homogeneous Einstein metrics on large classes of homogeneous spaces, namely
generalized flag manifolds, Stiefel manifolds and generalized Wallach spaces.
The first two classes of homogeneous spaces fall into a certain dichotomy
between homogeneous spaces whose isotropy representation decomposes into non
equivalent summands and those whose isotropy representation contain equivalent
summands. Generalized Wallach spaces lie in both classes of spaces.
Furthermore, interesting and non trivial problems arise in both classes of
spaces, which are discussed at the end of the paper as open problems.

The article also contains some new results concerning existence of new
invariant Einstein metrics on the Stiefel manifold $V_{5}\mathbb{R}^{n}$
$(n\geq 7)$, thus extending a result in [ArSaSt]. Through this class of
homogeneous spaces we have the opportunity to show how to prove existence on
invariant Einstein metrics by manipulating parametric systems of polynomial
equations. This is done by using Gröbner bases techniques.

##  2\. The Ricci tensor for reductive homogeneous spaces

In this section we recall an expression for the Ricci tensor for an
$G$-invariant Riemannian metric on a reductive homogeneous space whose
isotropy representation is decomposed into a sum of non equivalent irreducible
summands.

Let $G$ be a compact semisimple Lie group, $K$ a connected closed subgroup of
$G$ and let $\mathfrak{g}$ and $\mathfrak{k}$ be the corresponding Lie
algebras. The Killing form $B$ of $\mathfrak{g}$ is negative definite, so we
can define an $\mbox{Ad}(G)$-invariant inner product $-B$ on $\mathfrak{g}$.
Let $\mathfrak{g}$ = $\mathfrak{k}\oplus\mathfrak{m}$ be a reductive
decomposition of $\mathfrak{g}$ with respect to $-B$ so that
$\left[\,\mathfrak{k},\,\mathfrak{m}\,\right]\subset\mathfrak{m}$ and
$\mathfrak{m}\cong T_{o}(G/K)$. We assume that ${\mathfrak{m}}$ admits a
decomposition into mutually non equivalent irreducible $\mbox{Ad}(K)$-modules
as

${\mathfrak{m}}={\mathfrak{m}}_{1}\oplus\cdots\oplus{\mathfrak{m}}_{q}.$ (1)

Then any $G$-invariant metric on $G/K$ can be expressed as

$\displaystyle\langle\ ,\
\rangle=x_{1}(-B)|_{\mbox{\footnotesize$\mathfrak{m}$}_{1}}+\cdots+x_{q}(-B)|_{\mbox{\footnotesize$\mathfrak{m}$}_{q}},$
(2)

for positive real numbers $(x_{1},\dots,x_{q})\in\mathbb{R}^{q}_{+}$. Note
that $G$-invariant symmetric covariant 2-tensors on $G/K$ are of the same form
as the Riemannian metrics (although they are not necessarilly positive
definite). In particular, the Ricci tensor $r$ of a $G$-invariant Riemannian
metric on $G/K$ is of the same form as (2), that is

$r=y_{1}(-B)|_{\mbox{\footnotesize$\mathfrak{m}$}_{1}}+\cdots+y_{q}(-B)|_{\mbox{\footnotesize$\mathfrak{m}$}_{q}},$

for some real numbers $y_{1},\ldots,y_{q}$.

Let $\\{e_{\alpha}\\}$ be a $(-B)$-orthonormal basis adapted to the
decomposition of $\mathfrak{m}$, i.e. $e_{\alpha}\in{\mathfrak{m}}_{i}$ for
some $i$, and $\alpha<\beta$ if $i<j$. We put
${A^{\gamma}_{\alpha\beta}}=-B\left(\left[e_{\alpha},e_{\beta}\right],e_{\gamma}\right)$
so that
$\left[e_{\alpha},e_{\beta}\right]=\displaystyle{\sum_{\gamma}A^{\gamma}_{\alpha\beta}e_{\gamma}}$
and set $\displaystyle{k\brack{ij}}=\sum(A^{\gamma}_{\alpha\beta})^{2}$, where
the sum is taken over all indices $\alpha,\beta,\gamma$ with
$e_{\alpha}\in{\mathfrak{m}}_{i},\ e_{\beta}\in{\mathfrak{m}}_{j},\
e_{\gamma}\in{\mathfrak{m}}_{k}$ (cf. [WaZi]). Then the positive numbers
$\displaystyle{k\brack{ij}}$ are independent of the $B$-orthonormal bases
chosen for ${\mathfrak{m}}_{i},{\mathfrak{m}}_{j},{\mathfrak{m}}_{k}$, and
$\displaystyle{k\brack{ij}}\ =\ \displaystyle{k\brack{ji}}\ =\
\displaystyle{j\brack{ki}}.$

Let $d_{k}=\dim{\mathfrak{m}}_{k}$. Then we have the following:

######  Lemma 2.1.

([PaSa]) The components ${r}_{1},\dots,{r}_{q}$ of the Ricci tensor ${r}$ of
the metric $\langle\,\,\,,\,\,\,\rangle$ of the form (2) on $G/K$ are given by

${r}_{k}=\frac{1}{2x_{k}}+\frac{1}{4d_{k}}\sum_{j,i}\frac{x_{k}}{x_{j}x_{i}}{k\brack{ji}}-\frac{1}{2d_{k}}\sum_{j,i}\frac{x_{j}}{x_{k}x_{i}}{j\brack{ki}}\quad(k=1,\
\dots,q),$ (3)

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

Since by assumption the submodules $\mathfrak{m}_{i},\mathfrak{m}_{j}$ in the
decomposition (1) are mutually non equivalent for any $i\neq j$, it is
$r(\mathfrak{m}_{i},\mathfrak{m}_{j})=0$. If
$\mathfrak{m}_{i}\cong\mathfrak{m}_{j}$ for some $i\neq j$ then we need to
check whether $r(\mathfrak{m}_{i},\mathfrak{m}_{j})=0$. This is not an easy
task in general. Once the condition $r(\mathfrak{m}_{i},\mathfrak{m}_{j})=0$
is confirmed we can use Lemma 2.1. Then $G$-invariant Einstein metrics on
$M=G/K$ are exactly the positive real solutions
$g=(x_{1},\ldots,x_{q})\in\mathbb{R}^{q}_{+}$ of the polynomial system
$\\{r_{1}=\lambda,\ r_{2}=\lambda,\ \ldots,\ r_{q}=\lambda\\}$, where
$\lambda\in\mathbb{R}_{+}$ is the Einstein constant. If some of the submodules
in (1) are equivalent as $\operatorname{Ad}(K)$-modules, then the computation
of the Ricci tensor is more laborious (e.g. using basic formulas from [Be] or
proving other variations of these).

##  3\. Homogeneous spaces with non equivalent isotropy summands

A generalized flag manifold is a a homogeneous space $M=G/K$ where $G$ is a
compact semisimple Lie group and $K$ is the cetralizer of a torus in $G$.
Equivalently, it is diffeomorphic to the adjoint orbit
$\operatorname{Ad}(G)w$, for some $w\in\mathfrak{g}$, the Lie algebra of $G$.
Typical examples are the manifolds of partial flags
$\operatorname{SU}(n)/\operatorname{S}(U(n_{1})\times\cdots\times\operatorname{U}(n_{k}))$
($n=n_{1}+\cdots+n_{k}$) and full flags $\operatorname{SU}(n)/T$ in
$\mathbb{C}^{n}$, where
$T=\operatorname{S}(\operatorname{U}(1)\times\cdots\operatorname{U}(1))$ is a
maximal torus in $\operatorname{SU}(n)$.

Next, we will present the Lie theoretic description of a generalized flag
manifold.

###  3.1. Decomposition associated to generalized flag manifolds

Let $G$ be a compact semisimple Lie group, $\mathfrak{g}$ the Lie algebra of
$G$ and $\mathfrak{h}$ a maximal abelian subalgebra of $\mathfrak{g}$. We
denote by ${\mathfrak{g}}^{\mathbb{C}}$ and ${\mathfrak{h}}^{\mathbb{C}}$ the
complexification of $\mathfrak{g}$ and $\mathfrak{h}$ respectively. We
identify an element of the root system $\Delta$ of
${\mathfrak{g}}^{\mathbb{C}}$ relative to the Cartan subalgebra
${\mathfrak{h}}^{\mathbb{C}}$ with an element of
${\mathfrak{h}}_{0}=\sqrt{-1}\mathfrak{h}$ by the duality defined by the
Killing form of ${\mathfrak{g}}^{\mathbb{C}}$. Let $\Pi$ =
$\\{\alpha_{1},\dots,\alpha_{l}\\}$ be a fundamental system of $\Delta$ and
$\\{\Lambda_{1},\dots,\Lambda_{l}\\}$ the fundamental weights of
${\mathfrak{g}}^{\mathbb{C}}$ corresponding to $\Pi$, that is
$\frac{2(\Lambda_{i},\alpha_{j})}{(\alpha_{j},\alpha_{j})}=\delta_{ij},\
(1\leq i,j\leq l).$ Let $\Pi_{0}$ be a subset of $\Pi$ and $\Pi-\Pi_{0}$ =
$\\{\alpha_{i_{1}},\cdots,\alpha_{i_{r}}\\}$, where
$1\leq{i_{1}}<\cdots<{i_{r}}\leq l$. We put
$[\Pi_{0}]=\Delta\cap\\{\Pi_{0}\\}_{\mathbb{Z}}$, where
$\\{\Pi_{0}\\}_{\mathbb{Z}}$ denotes the subspace of ${\mathfrak{h}}_{0}$
generated by $\Pi_{0}$. Consider the root space decomposition of
${\mathfrak{g}}^{\mathbb{C}}$ relative to ${\mathfrak{h}}^{\mathbb{C}}$ as
${\mathfrak{g}}^{\mathbb{C}}={\mathfrak{h}}^{\mathbb{C}}+\sum_{\alpha\in\Delta}{\mathfrak{g}}^{\mathbb{C}}_{\alpha}.$
We define a parabolic subalgebra ${\mathfrak{u}}$ of
${\mathfrak{g}}^{\mathbb{C}}$ by
$\displaystyle{\mathfrak{u}}={\mathfrak{h}}^{\mathbb{C}}+\sum_{\alpha\in[\Pi_{0}]\cup\Delta^{+}}{\mathfrak{g}}^{\mathbb{C}}_{\alpha},$
where $\Delta^{+}$ is the set of all positive roots relative to $\Pi$. Note
that the nilradical ${\mathfrak{n}}$ of ${\mathfrak{u}}$ is given by
${\mathfrak{n}}=\sum_{\alpha\in\Delta^{+}-[\Pi_{0}]}{\mathfrak{g}}^{\mathbb{C}}_{\alpha}.$
We put $\Delta_{\mathfrak{m}}^{+}=\Delta^{+}-[\Pi_{0}]$.

Let $G^{\mathbb{C}}$ be a simply connected complex semi-simple Lie group whose
Lie algebra is ${\mathfrak{g}}^{\mathbb{C}}$ and $U$ the parabolic subgroup of
$G^{\mathbb{C}}$ generated by ${\mathfrak{u}}$. Then the complex homogeneous
manifold $G^{\mathbb{C}}/U$ is compact simply connected and $G$ acts
transitively on $G^{\mathbb{C}}/U$. Note also that $K=G\cap U$ is a connected
closed subgroup of $G$, $G^{\mathbb{C}}/U$ = $G/K$ as $C^{\infty}$-manifolds,
and $G^{\mathbb{C}}/U$ admits a $G$-invariant Kähler metric. Let
$\mathfrak{k}$ be the Lie algebra of $K$ and ${\mathfrak{k}}^{\mathbb{C}}$ the
complexification of $\mathfrak{k}$. Then we have direct decompositions
$\displaystyle{\mathfrak{u}}={\mathfrak{k}}^{\mathbb{C}}\oplus{\mathfrak{n}},\quad\displaystyle{\mathfrak{k}}^{\mathbb{C}}={\mathfrak{h}}^{\mathbb{C}}+\sum_{\alpha\in[\Pi_{0}]}{\mathfrak{g}}^{\mathbb{C}}_{\alpha}.$
Take a Weyl basis
$E_{-\alpha}\in{\mathfrak{g}}^{\mathbb{C}}_{\alpha}\,\,(\alpha\in\Delta)$ with

$\begin{array}[]{ll}\left[E_{\alpha},E_{-\alpha}\right]&=-\alpha\,(\alpha\in\Delta)\\\
{}\left[E_{\alpha},E_{\beta}\right]&=\left\\{\begin{array}[]{ll}N_{\alpha,\,\beta}E_{\alpha+\beta}&\mbox{if
\quad}\alpha+\beta\in\Delta\\\ 0&\mbox{if
\quad}\alpha+\beta\not\in\Delta,\end{array}\right.\end{array}$

where $N_{\alpha,\,\beta}$ = $N_{-\alpha,\,-\beta}\in{\mathbb{R}}.$ Then we
have

${\mathfrak{g}}={\mathfrak{h}}+\sum_{\alpha\in\Delta}\left\\{{\mathbb{R}}(E_{\alpha}+E_{-\alpha})+{\mathbb{R}}\sqrt{-1}(E_{\alpha}-E_{-\alpha})\right\\}$

and the Lie subalgebra $\mathfrak{k}$ is given by

${\mathfrak{k}}={\mathfrak{h}}+\sum_{\alpha\in\left[\Pi_{0}\right]}\left\\{{\mathbb{R}}(E_{\alpha}+E_{-\alpha})+{\mathbb{R}}\sqrt{-1}(E_{\alpha}-E_{-\alpha})\right\\}.$

For integers $j_{1},\cdots,j_{r}$ with $(j_{1},\cdots,j_{r})\neq(0,\cdots,0)$
, we put

$\Delta(j_{1},\cdots,j_{r})=\left\\{\
\sum_{j=1}^{l}m_{j}\alpha_{j}\in\Delta^{+}\ \ \Big{|}\ \
m_{i_{1}}=j_{1},\cdots,m_{i_{r}}=j_{r}\ \right\\}.$

Note that
$\displaystyle\Delta_{\mathfrak{m}}^{+}=\Delta^{+}-[\Pi_{0}]=\bigcup_{j_{1},\cdots,\,j_{r}}\Delta(j_{1},\cdots,j_{r})$.

For $\Delta(j_{1},\cdots,j_{r})\neq\emptyset$, we define an
$\mbox{Ad}_{G}(K)$-invariant subspace ${\mathfrak{m}}(j_{1},\cdots,j_{r})$ of
$\mathfrak{g}$ by

${\mathfrak{m}}(j_{1},\cdots,j_{r})=\sum_{\alpha\in\Delta(j_{1},\cdots,\,j_{r})}\left\\{{\mathbb{R}}(E_{\alpha}+E_{-\alpha})+{\mathbb{R}}\sqrt{-1}(E_{\alpha}-E_{-\alpha})\right\\}.$

Then we have a decomposition of $\mathfrak{m}$ into mutually non-equivalent
irreducible $\mbox{Ad}_{G}(K)$-modules ${\mathfrak{m}}(j_{1},\cdots,j_{r})$ as
$\mathfrak{m}=\sum_{j_{1},\cdots,\,j_{r}}{\mathfrak{m}}(j_{1},\cdots,j_{r}).$
We put ${\mathfrak{t}}=\Big{\\{}H\in{\mathfrak{h}}_{0}\ |\ (H,\
\Pi_{0})=0\Big{\\}}$. Then $\\{\Lambda_{i_{1}},\cdots,\Lambda_{i_{r}}\\}$ is a
basis of ${\mathfrak{t}}$. Put $\mathfrak{s}=\sqrt{-1}{\mathfrak{t}}$. Then
the Lie algebra ${\mathfrak{k}}$ is given by
${\mathfrak{k}}={\mathfrak{z}}({\mathfrak{s}})$ (the Lie algebra of
centralizer of a torus $S$ in $G$).

We consider the restriction map
$\kappa:{\mathfrak{h}}_{0}^{*}\to{\mathfrak{t}}^{*},\quad\alpha\mapsto\alpha|_{\mathfrak{t}}$
and set $\Delta_{T}=\kappa(\Delta)$. The elements of $\Delta_{T}$ are called
$T$-roots.

There exists ([AlPe]) a 1-1 correspondence between $T$-roots $\xi$ and
irreducible submodules $\displaystyle{\mathfrak{m}}_{\xi}$ of the
$\mbox{Ad}_{G}(K)$-module ${\mathfrak{m}}^{\mathbb{C}}$ given by

$\displaystyle\Delta_{T}\ni\xi\mapsto{\mathfrak{m}}_{\xi}=\sum_{\kappa(\alpha)=\xi}{\mathfrak{g}}^{\mathbb{C}}_{\alpha}.$

Thus we have a decomposition of the $\mbox{Ad}_{G}(K)$-module
${\mathfrak{m}}^{\mathbb{C}}$ as
${\mathfrak{m}}^{\mathbb{C}}=\sum_{\xi\in\Delta_{T}}{\mathfrak{m}}_{\xi}.$
Denote by $\Delta_{T}^{+}$ the set of all positive $T$-roots, that is, the
restricton of the system $\Delta^{+}$. Then we have
$\displaystyle{\mathfrak{n}}=\sum_{\xi\in\Delta_{T}^{+}}{\mathfrak{m}}_{\xi}$.
Denote by $\tau$ the complex conjugation of ${\mathfrak{g}}^{\mathbb{C}}$ with
respect to $\mathfrak{g}$ (note that $\tau$ interchanges
${\mathfrak{g}}^{\mathbb{C}}_{\alpha}$ and
${\mathfrak{g}}^{\mathbb{C}}_{-\alpha}$) and by ${\mathfrak{v}}^{\tau}$ the
set of fixed points of $\tau$ in a complex vector subspace ${\mathfrak{v}}$ of
${\mathfrak{g}}^{\mathbb{C}}$. Thus we have a decomposition of
$\mbox{Ad}_{G}(K)$-module ${\mathfrak{m}}$ into irreducible submodules as
${\mathfrak{m}}=\sum_{\xi\in\Delta_{T}^{+}}\left({\mathfrak{m}}_{\xi}+{\mathfrak{m}}_{-\xi}\right)^{\tau}.$

There exists a natural 1-1 correspondence between $\Delta_{T}^{+}$ and the set
$\left\\{\Delta(j_{1},\cdots,j_{r})\neq\emptyset\right\\}$. For a generalized
flag manifold $G/K$, we have a decomposition of $\mathfrak{m}$ into mutually
non-equivalent irreducible $\mbox{Ad}_{G}(H)$-modules as
$\displaystyle\mathfrak{m}=\sum_{\xi\in\Delta_{T}^{+}}\left({\mathfrak{m}}_{\xi}+{\mathfrak{m}}_{-\xi}\right)^{\tau}=\sum_{j_{1},\cdots,\,j_{r}}{\mathfrak{m}}(j_{1},\cdots,j_{r}).$
Thus a $G$-invariant metric $g$ on $G/K$ can be written as

$\displaystyle
g=\sum_{\xi\in\Delta_{T}^{+}}x_{\xi}B|_{\left({\mathfrak{m}}_{\xi}+{\mathfrak{m}}_{-\xi}\right)^{\tau}}=\sum_{j_{1},\cdots,j_{r}}x_{j_{1}\cdots
j_{r}}B|_{{\mathfrak{m}}(j_{1},\cdots,j_{r})}$ (4)

for positive real numbers $x_{\xi}$, $x_{j_{1}\cdots j_{r}}$.

Put $\displaystyle Z_{\mathfrak{t}}=\Bigg{\\{}\Lambda\in{\mathfrak{t}}\
\Big{|}\ \frac{2(\Lambda,\ \alpha)}{(\alpha,\ \alpha)}\in{\mathbb{Z}}\ \mbox{
for }\mbox{ each }\alpha\in\Delta\Bigg{\\}}$. Then $Z_{\mathfrak{t}}$ is a
lattice of ${\mathfrak{t}}$ generated by
$\\{\Lambda_{i_{1}},\dots,\Lambda_{i_{r}}\\}$. For each $\Lambda\in
Z_{\mathfrak{t}}$ there exists a unique holomorphic character $\chi_{\Lambda}$
of $U$ such that $\chi_{\Lambda}(\exp H)=\exp\Lambda(H)$ for each
$H\in{\mathfrak{h}}^{\mathbb{C}}$. Then the correspondence
$\Lambda\to\chi_{\Lambda}$ gives an isomorphism of $Z_{\mathfrak{t}}$ to the
group of holomorphic characters of $U$.

Let $F_{\Lambda}$ denote the holomorphic line bundle on $G^{\mathbb{C}}/U$
associated to the principal bundle $U\to G^{\mathbb{C}}\to G^{\mathbb{C}}/U$
by the holomorphic character $\chi_{\Lambda}$, and
$H(G^{\mathbb{C}}/U,\underline{\mathbb{C}}^{*})$ the group of isomorphism
classes of holomorphic line bundles on $G^{\mathbb{C}}/U$.

The correspondence $\Lambda\mapsto F_{\Lambda}:Z_{\mathfrak{t}}\to
H(G^{\mathbb{C}}/U,\underline{\mathbb{C}}^{*})$ induces a homomorphism. Also
the correspondence $F\mapsto c_{1}(F)$ defines a homomorphism of
$H(G^{\mathbb{C}}/U,\underline{\mathbb{C}}^{*})$ to $H^{2}(M,\mathbb{Z})$.
Then it is known that homomorphisms
$Z_{\mathfrak{t}}\overset{F}{\to}H(G^{\mathbb{C}}/U,\underline{\mathbb{C}}^{*})\overset{c_{1}}{\to}H^{2}(M,\mathbb{Z})$
are in fact isomorphisms. In particular, the second Betti number $b_{2}(M)$ of
$M$ is given by $b_{2}(M)=\dim{\mathfrak{t}}$, the cardinality of
$\Pi-\Pi_{0}$.

###  3.2. Kähler-Einstein metrics on a generalized flag manifold

We set
$Z_{\mathfrak{t}}^{+}=\left\\{\lambda\in Z_{\mathfrak{t}}\ \big{|}\
(\lambda,\alpha)>0\ \ \mbox{for}\ \alpha\in\Pi-\Pi_{0}\right\\}$. Then we have
$\displaystyle
Z_{\mathfrak{t}}^{+}=\sum_{\alpha\in\Pi-\Pi_{0}}{\mathbb{Z}}^{+}\Lambda_{\alpha}$.
We define an element $\delta_{\mathfrak{m}}\in\sqrt{-1}\mathfrak{h}$ by
$\displaystyle\delta_{\mathfrak{m}}=\frac{1}{2}\sum_{\alpha\in\Delta_{\mathfrak{m}}^{+}}\alpha$.
Let $c_{1}(M)$ be the first Chern class of $M$. Then we have that
$2\delta_{\mathfrak{m}}\in Z_{\mathfrak{t}}^{+}$ and
$c_{1}(M)=c_{1}(F_{2\delta_{\mathfrak{m}}})$. Put $\displaystyle
k_{\alpha}=\frac{2(2\delta_{\mathfrak{m}},\alpha)}{(\alpha,\alpha)}$ for
$\alpha\in\Pi-\Pi_{0}$. Then $\displaystyle
2\delta_{\mathfrak{m}}=\sum_{\alpha\in\Pi-\Pi_{0}}k_{\alpha}\Lambda_{\alpha}=k_{\alpha_{i_{1}}}\Lambda_{\alpha_{i_{1}}}+\cdots+k_{\alpha_{i_{r}}}\Lambda_{\alpha_{i_{r}}}$
and each $k_{\alpha_{i_{s}}}$ is a positive integer. The $G$-invariant metric
$g_{2\delta_{\mathfrak{m}}}$ on $G/K$ corresponding to
$2\delta_{\mathfrak{m}}$, which is a Kähler Einstein metric, is given by

$g_{2\delta_{\mathfrak{m}}}=\sum_{j_{1},\dots,j_{r}}\left(\sum_{\ell=1}^{r}k_{\alpha_{i_{\ell}}}j_{\ell}\frac{(\alpha_{j_{\ell}},\alpha_{j_{\ell}})}{2}\right)B|_{{\mathfrak{m}}(j_{1},\dots,j_{r})}.$

Example. For the generalized flag manifold $G/K=Sp(n)/(U(p)\times U(q)\times
Sp(n-p-q))$ where $n\geq 3,\ p,\,q\geq 1$, we see that
$\Pi-\Pi_{0}=\\{\alpha_{p},\ \alpha_{p+q}\\}$, where $\alpha_{p}$ and
$\alpha_{p+q}$ are two simple roots in the Dynkin diagram of
$\mathfrak{sp}(n)$ painted black. The painted Dynkinh diagram is shown below

$\alpha_{1}$2$\ldots$2$\alpha_{p-1}$$\alpha_{p}$22$\ldots$$\alpha_{p+q}$2$\ldots$$\alpha_{n-1}$2$<$$\alpha_{n}$1.

It is $\displaystyle
2\delta_{\mathfrak{m}}=(p+q)\Lambda_{\alpha_{p}}+(2n-2p-q+1)\Lambda_{\alpha_{p+q}}$.
Then the Kähler-Einstein metric $g_{2\delta_{\mathfrak{m}}}$ on $G/K$ is given
by

$\displaystyle
g_{2\delta_{\mathfrak{m}}}=(p+q)B|_{{\mathfrak{m}}(1,0)}+(2n-2p-q+1)B|_{{\mathfrak{m}}(0,1)}$
$\displaystyle+(2n-p+1)B|_{{\mathfrak{m}}(1,1)}+2(2n-2p-q+1)B|_{{\mathfrak{m}}(0,2)}$
$\displaystyle+(4n-3p-q+2)B|_{{\mathfrak{m}}(1,2)}+2(2n-p+1)B|_{{\mathfrak{m}}(2,2)}.$
(5)

###  3.3. Einstein metrics on generalized flag manifolds with two isotropy
summands

We assume that the Lie group $G$ is simple. For a generalized flag manifold
$G/K$, we denote by $q$ the number of mutually non equivalent irreducible
$\mbox{Ad}_{G}(K)$-modules ${\mathfrak{m}}(j_{1},\dots,j_{r})$ with
$\displaystyle\mathfrak{m}=\sum_{j_{1},\cdots,\,j_{r}}{\mathfrak{m}}(j_{1},\dots,j_{r}).$

If $q=1$, it is known that $G/K$ is an irreducible Hermitian symmetric space
with the symmetric pair $(\mathfrak{g},\mathfrak{k})$. If $q=2$, we have two
$G$-invariant Einstein metrics on $G/K$. One is Kähler-Einstein metric and the
other is non Kähler Einstein metric ([ArCh1]). In fact, we see that the case
$q=2$ occurs only in the case $r=b_{2}(G/K)=1$ and
$\displaystyle\mathfrak{m}={\mathfrak{m}}(1)\oplus{\mathfrak{m}}(2)$ (cf.
[ArChSa1]). Note that only $\displaystyle{2\brack{11}}$ is non-zero. Put
$d_{1}=\dim{\mathfrak{m}}(1)$ and $d_{2}=\dim{\mathfrak{m}}(2)$. For a
$G$-invariant metric $\langle\,\,\,,\,\,\,\rangle=x_{1}\cdot
B|_{{\mathfrak{m}}(1)}+x_{2}\cdot B|_{{\mathfrak{m}}(2)}$, the components
${r}_{1},{r}_{2}$ of Ricci tensor ${r}$ of the metric
$\langle\,\,\,,\,\,\,\rangle$ are given by

$\left\\{\begin{array}[]{ll}r_{1}&=\displaystyle{\frac{1}{2x_{1}}-\frac{x_{2}}{2\,d_{1}\,{x_{1}}^{2}}{2\brack
11}}\\\ &\\\
r_{2}&=\displaystyle{\frac{1}{2x_{2}}-\frac{1}{2\,d_{2}\,x_{2}}{1\brack
21}+\frac{x_{2}}{4\,d_{2}\,{x_{1}}^{2}}{2\brack 11}\;.}\end{array}\right.$ (6)

Since the metric $(\,\,,\,\,)=1\cdot B|_{{\mathfrak{m}}(1)}+2\cdot
B|_{{\mathfrak{m}}(2)}$ is Kähler Einstein, we see that $\displaystyle{2\brack
11}=\frac{d_{1}d_{2}}{d_{1}+4d_{2}}$. Note that a $G$-invariant metric
$\langle\,\,\,,\,\,\,\rangle=x_{1}\cdot B|_{{\mathfrak{m}}(1)}+x_{2}\cdot
B|_{{\mathfrak{m}}(2)}$ is Einstein if and only if $r_{1}=r_{2}$. We normalize
the equation $r_{1}=r_{2}$ by putting $x_{1}=1$. Then we see that the equation
$r_{1}=r_{2}$ is reduced to a quadratic equation of $x_{2}$ and we have
solutions $x_{2}=2$ and $\displaystyle x_{2}=\frac{4d_{2}}{d_{1}+2d_{2}}$.
Since $\displaystyle x_{2}=\frac{4d_{2}}{d_{1}+2d_{2}}\neq 2$, the Einstein
metric $\displaystyle 1\cdot
B|_{{\mathfrak{m}}(1)}+\frac{4d_{2}}{d_{1}+2d_{2}}\cdot
B|_{{\mathfrak{m}}(2)}$ is not Kähler.

###  3.4. Einstein metrics on generalized flag manifolds with three isotropy
summands

The case $q=3$ was studied by the author and Kimura independently in [Ar1] and
[Ki]. We see that either $r=b_{2}(G/K)=1$ or $r=b_{2}(G/K)=2$.

For the case of $r=b_{2}(G/K)=1$ we denote the $T$-roots system of type
$A_{1}(3)$, that is $\Delta_{T}^{+}=\\{\,\xi,2\xi,3\xi\,\\}$. There are seven
cases. The components ${r}_{1},{r}_{2},{r}_{3}$ of the Ricci tensor ${r}$ of
the metric $\langle\,\,\,,\,\,\,\rangle=x_{1}\cdot
B|_{{\mathfrak{m}}(1)}+x_{2}\cdot B|_{{\mathfrak{m}}(2)}+x_{1}\cdot
B|_{{\mathfrak{m}}(3)}$ are given by

$\left\\{\begin{array}[]{l}r_{1}\displaystyle=\frac{1}{2{x_{1}}}-\frac{{x_{2}}}{2{d_{1}}{x_{1}}^{2}}{2\brack
11}+\frac{1}{2{d_{1}}}\left(\frac{{x_{1}}}{{x_{2}}{x_{3}}}-\frac{{x_{3}}}{{x_{1}}{x_{2}}}-\frac{{x_{2}}}{{x_{1}}{x_{3}}}\right){3\brack
12}\\\
r_{2}\displaystyle=\frac{1}{2{x_{2}}}+\frac{1}{4{d_{2}}}\left(\frac{x_{2}}{{x_{1}}^{2}}-\frac{2}{x_{2}}\right){2\brack
11}\displaystyle+\frac{1}{2{d_{2}}}\left(\frac{x_{2}}{{x_{1}}{x_{3}}}-\frac{x_{1}}{{x_{2}}{x_{3}}}-\frac{x_{3}}{{x_{1}}{x_{2}}}\right){3\brack
12}\\\
r_{3}\displaystyle=\frac{1}{2{x_{3}}}+\frac{1}{2{d_{3}}}\left(\frac{{x_{3}}}{{x_{1}}{x_{2}}}-\frac{{x_{1}}}{{x_{2}}{x_{3}}}-\frac{{x_{2}}}{{x_{1}}{x_{3}}}\right){3\brack
12}.\end{array}\right.$

The system of equations $r_{1}=r_{2}=r_{3}$ reduces to a polynomial equation
of degree 5. There is a unique Kähler-Einstein metric and two non Kähler
Einstein metrics. These were explicitely found in [AnCh].

For the case $q=3$ and $b_{2}(G/K)=2$ we denote the case of $r=b_{2}(G/K)=2$
that $T$-roots system is of type $A_{2}$, that is
$\Delta_{T}^{+}=\\{\,\xi_{1},\xi_{2},\xi_{1}+\xi_{2}\,\\}$. There are three
cases here. The space
$\operatorname{SU}(n)/\operatorname{S}(\operatorname{U}(\ell)\\!\times\\!\operatorname{U}(m)\\!\times\\!\operatorname{U}(k))$
($n=\ell+m+k$) admits three Kähler-Einstein metrics (for $\ell,m,l$ distinct)
and one non Kähler Einstein metric. The space
$\operatorname{SO}(2n)/(\operatorname{U}(n-1)\\!\times\\!\operatorname{U}(1))$
admits two Kähler-Einstein metrics and one non-Kähler Einstein metric.
Finally, the space
$E_{6}/(\operatorname{SO}(8)\\!\times\\!\operatorname{U}(1)\\!\times\\!\operatorname{U}(1))$
admits a unique Kähler-Einstein and a unique non Kähler Einstein metric. This
metric is normal.

###  3.5. Einstein metrics on generalized flag manifolds with four isotropy
summands

We see that in this case either $r=b_{2}(G/K)=1$ or $r=b_{2}(G/K)=2$. For the
case $r=b_{2}(G/K)=1$ we denote the $T$-root system of type $A_{1}(4)$, that
is $\Delta_{T}^{+}=\\{\,\xi,2\xi,3\xi,\,4\xi\,\\}$. There are four flag
manifolds of this type and $G$ is always an exceptional Lie group. It follows
that they admit one Kähler-Einstein and two non Kähler Einstein metrics.

For the case of $r=b_{2}(G/K)=2$ we denote the $T$-root system of type
$B_{2}$, that is
$\Delta_{T}^{+}=\\{\,\xi_{1},\xi_{2},\xi_{1}+\xi_{2},\,2\xi_{1}+\xi_{2}\,\\}$.
There are six flag manifolds of this type. We divide the cases of type $B_{2}$
into $B_{2}(a)$ and $B_{2}(b)$.

The case of $B_{2}(a)$ is one of
$\operatorname{SO}(2\ell+1)/\operatorname{SO}(2\ell-3)\times\operatorname{U}(1)\times\operatorname{U}(1)$,
$\operatorname{SO}(2\ell)/\operatorname{SO}(2\ell-4)\times\operatorname{U}(1)\times\operatorname{U}(1)$,
$E_{6}/\operatorname{SU}(5)\times\operatorname{U}(1)\times\operatorname{U}(1)$
or
$E_{7}/\operatorname{SO}(10)\times\operatorname{U}(1)\times\operatorname{U}(1)$.
The case of $B_{2}(b)$ is one of
$\operatorname{Sp}(\ell)/\operatorname{U}(p)\times\operatorname{U}(\ell-p)$ or
$\operatorname{SO}(2\ell)/\operatorname{U}(p)\times\operatorname{U}(\ell-p)$.
These results are presented in [ArCh2].

For the case of $B_{2}(a)$, there exist exactly eight $G$-invariant Einstein
metrics on $G/K$, four of them are Kähler-Einstein and the others are non-
Kähler Einstein. All above results are presented in [ArCh2].

The case of $B_{2}(b)$ is the most difficult one. For
$\operatorname{SO}(2\ell)/(\operatorname{U}(p)\times\operatorname{U}(\ell-p))$
( $\ell\geq 4$ and $2\leq p\leq\ell-2$), there exist four non-Kähler Einstein
metrics for the pairs $(\ell,p)$ $=(12,6)$, $(10,5)$, $(8,4)$, $(7,4)$,
$(7,3)$, $(6,4)$, $(6,3)$, $(6,2)$, $(5,3)$, $(5,3)$, $(4,2)$. For the other
pairs $(\ell,p)$, there exit two non-Kähler Einstein metrics. The complete
study if this space as well as the isometry problem is presented in the paper
[ArChSa1].

Finally, the space
$\operatorname{Sp}(\ell)/(\operatorname{U}(p)\times\operatorname{U}(\ell-p))$
admits four isometric Kähler-Einstein metrics and two non Kähler Einstein
metrics (cf. [ArChSa1]). These metrics are isometric, as it was proved in
[ArChSa5].

In order to get an idea of the difficulty of such parametric systems of
equations, we mention that the Einstein equation for
$\operatorname{Sp}(n)/(\operatorname{U}(p)\times\operatorname{U}(n-p))$
reduces to the system $r_{1}-r_{3}=0,r_{1}-r_{2}=0,r_{3}-r_{4}=0$, which is
equivalent to the parametric system

$\left.\begin{tabular}[]{l}$(x_{1}-x_{3})(x_{1}x_{2}+px_{1}x_{2}+x_{2}x_{3}+px_{2}x_{3}+x_{1}x_{4}+nx_{1}x_{4}$\\\
$-px_{1}x_{4}-2x_{2}x_{4}-2nx_{2}x_{4}+x_{3}x_{4}+nx_{3}x_{4}-px_{3}x_{4})=0,$\\\
$4(n+1)x_{3}x_{4}(x_{2}-x_{1})+(n+p+1)x_{4}(x_{1}^{2}-x_{2}^{2})-(n-3p+1)x_{3}^{2}x_{4}$\\\
$+(p+1)x_{2}(x_{1}^{2}-x_{3}^{2}-x_{4}^{2})=0,$\\\
$4(n+1)x_{1}x_{2}(x_{4}-x_{3})+(2n-p+1)x_{2}(x_{3}^{2}-x_{4}^{2})+(2n-3p-1)x_{1}^{2}x_{2}$\\\
$+(n-p+1)x_{4}(x_{3}^{2}-x_{1}^{2}-x_{2}^{2})=0$\\\ \end{tabular}\right\\}$

for the unknowns $x_{1},x_{2},x_{3},x_{4}>0$.

###  3.6. Einstein metrics on generalized flag manifolds with five isotropy
summands

The classification of generalized flag manifolds $M=G/K$ with five isotropy
summands was obtained in [ArChSa4]. They are obtained in the following ways:
paint black one simple root of Dynkin mark 5, that is
$\Pi\backslash\Pi_{0}=\\{\alpha_{p}:\operatorname{Mrk}(\alpha_{p})=5\\}$, or
paint black two simple roots, one of Dynkin mark 1 and one of Dynkin mark 2,
that is
$\Pi\backslash\Pi_{0}=\\{\alpha_{i},\alpha_{j}:\operatorname{Mrk}(\alpha_{i})=1,\
\operatorname{Mrk}(\alpha_{j})=2\\}$, or paint black two simple roots both of
Dynkin mark 2, that is
$\Pi\backslash\Pi_{0}=\\{\alpha_{i},\alpha_{j}:\operatorname{Mrk}(\alpha_{i})=\operatorname{Mrk}(\alpha_{j})=2\\}$.
They correspond to the spaces
$E_{8}/\operatorname{U}(1)\times\operatorname{SU}(4)\times\operatorname{SU}(5)$,
$\operatorname{SO}(2\ell+1)/\operatorname{U}(1)\times\operatorname{U}(p)\times\operatorname{SO}(2(\ell-p-1)+1)$,
$\operatorname{SO}(2\ell)/\operatorname{U}(1)\times\operatorname{U}(p)\times\operatorname{SO}(2(\ell-p-1))$,
$E_{6}/\operatorname{SU}(4)\times\operatorname{SU}(2)\operatorname{U}(1)\times\operatorname{U}(1)$
and
$E_{6}/\operatorname{SU}(6)\times\operatorname{U}(1)\times\operatorname{U}(1)$.

The main difficulty in constructing the Einstein equation for a $G$-invariant
metric on one of the above homogeneous spaces is the calculation of the non
zero structure constants $\displaystyle{k\brack{ij}}$ of $G/K$ with respect to
the decomposition
$T_{o}M\cong\mathfrak{m}=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\mathfrak{m}_{3}\oplus\mathfrak{m}_{4}\oplus\mathfrak{m}_{5}$
of the tangent space of $M$. A first step towards this procedure, is to use
the known Kähler-Einstein metric (cf. Section 4). Secondly, and this is was
the main contribution of the papers [ArChSa4] and [ChSa], we can take
advantage of the fibration of a flag manifold over another flag manifold, and
by using methods of Riemannian submersions it possible to compare the Ricci
tensors of the total and base spaces, respectively. Such fibration is a an
extension of the well known twistor fibration of a generalized flag manifold
over a symmetric space (cf. [BuRa]). In this way we can calculate
$\displaystyle{k\brack{ij}}$ in terms of the dimension of the associated
submodules $\mathfrak{m}_{i}$.

The Einstein equation reduces to a polynomial system of four equations in four
unknowns. For the exceptional flag manifolds it is possible to classify all
homogeneous Einstein metrics. For the classical flag manifolds a complete
classification of homogeneous Einstein metrics in the general case is a
difficult task, because the corresponding systems of equations depend on four
positive parameters (which define the invariant Riemannian metric), the
Einstein constant $\lambda>0$ and the positive integers $\ell$ and $p$.
However, by using Gröbner bases we can show that the equations are reduced to
a polynomial equation of one variable and then we can prove the existence of
non Käler Einstein metrics. In fact, this is another contribution of
[ArChSa4], because we show existence of real solutions for polynomial
equations whose coefficients depend on parameters ($\ell$ and $p$). Finally
the isometry question for the Einstein metrics found, is also answered in this
work. The results of [ArChSa4] and [ChSa] can be summarized as follows:

######  Theorem 3.1.

Let $M=G/K$ be one of the flag manifolds
$\operatorname{E}_{6}/(\operatorname{SU}(4)\times\operatorname{SU}(2)\times\operatorname{U}(1)\times\operatorname{U}(1))$
or $\operatorname{E}_{7}/(\operatorname{U}(1)\times\operatorname{U}(6))$. Then
$M$ admits exactly seven $G$-invariant Einstein metrics up to isometry. There
are two Kähler-Einstein metrics and five non Kähler Einstein metrics (up to
scalar).

######  Theorem 3.2.

Let $M=G/K$ be one of the flag manifolds
$\operatorname{SO}(2\ell+1)/(\operatorname{U}(1)\times\operatorname{U}(p)\times\operatorname{SO}(2(\ell-p-1)+1))$
$(\ell\geq 3,\ 3\leq p\leq\ell-1)$ or
$\operatorname{SO}(2\ell)/(\operatorname{U}(1)\times\operatorname{U}(p)\times\operatorname{SO}(2(\ell-p-1)))$
$(\ell\geq 5,\ 3\leq p\leq\ell-3)$. Then $M$ admits at least two $G$-invariant
non Kähler Einstein metrics.

######  Theorem 3.3.

Let $M=G/K$ be one of the flag manifolds
$\operatorname{SO}(2\ell+1)/(\operatorname{U}(1)\times\operatorname{U}(2)\times\operatorname{SO}(2\ell-5))$
$(\ell\geq 6)$ or
$\operatorname{SO}(2\ell)/(\operatorname{U}(1)\times\operatorname{U}(2)\times\operatorname{SO}(2(\ell-3)))$
$(\ell\geq 7)$. Then $M$ admits at least four $G$-invariant non Kähler
Einstein metrics.

######  Theorem 3.4.

The flag manifold
$\operatorname{E}_{8}/\operatorname{U}(1)\times\operatorname{SU}(4)\times\operatorname{SU}(5)\times\operatorname{U}(1)$
admits exactly six $E_{8}$-invariant Einstein metrics up to isometry. One is
Kähler-Einstein metric and five are non Kähler Einstein metrics (up to
scalar).

###  3.7. Einstein metrics on generalized flag manifolds with six isotropy
summands

Invariant Einstein metrics on generalized flag manifolds with six isotropy
summands have not been completely classified. A typical example is the flag
manifold $G_{2}/T$, where $T$ is a maximal torus in $G_{2}$. The painted
Dynking diamgram is shown below

$\alpha_{1}$3$>$$\alpha_{2}$2

The highest root $\widetilde{\alpha}$ of ${\mathfrak{g}_{2}^{\mathbb{C}}}$ is
given by $\widetilde{\alpha}=2\alpha_{1}+3\alpha_{2}$. Thus we have a
decomposition of $\mathfrak{m}$ into six mutually non-equivalent irreducible
$\mbox{Ad}_{G}(H)$-modules ${\mathfrak{m}}(j_{1},j_{2})$ :
$\displaystyle\mathfrak{m}={\mathfrak{m}}(1,0)\oplus{\mathfrak{m}}(0,1)\oplus{\mathfrak{m}}(1,1)\oplus{\mathfrak{m}}(1,2)\oplus{\mathfrak{m}}(1,3)\oplus{\mathfrak{m}}(2,3).$
There is only one complex strucure and thus, up to isometry, there exist only
one Kähler-Einstein metric. There exist two non-Kähler Einstein metrics up to
isometry. These are obtained from solutions of polynomial of degree 14 (cf.
[ArChSa6]). There are four other generalized flag manifolds (all determined by
the exceptional Lie groups, $F_{4}$, $E_{6}$, $E_{7}$, $E_{8}$) with $T$-roots
being of $G_{2}$ type. For all cases, there is only one Kähler-Einstein metric
and at least six non Kähler Einstein metrics up to isometry.

Other examples are the flag manifold $\operatorname{SU}(4)/T$ and
$\operatorname{SU}(10)/\operatorname{S}(\operatorname{U}(1)\times\operatorname{U}(2)\times\operatorname{U}(3)\times\operatorname{U}(4))$.
Note that for these cases $q=6$ and the system of $T$-roots is of type
$A_{3}$. For the case $\operatorname{SU}(4)/T$ there is only one complex
strucure and thus, up to isometry, there exists only one Kähler-Einstein
metric. There exist three non Kähler Einstein metrics up to isometry, one of
them is normal (cf. [Sak]). For the case
$\operatorname{SU}(10)/\operatorname{S}(\operatorname{U}(1)\times\operatorname{U}(2)\times\operatorname{U}(3)\times\operatorname{U}(4))$
there are twelve complex strucure and thus, up to isometry, there exist twelve
Kähler-Einstein metrics. There exist twelve non Kähler Einstein metrics up to
isometry. These are obtained from solutions of polynomial of degree 68.

In [ArChSa2] all invariant Einstein metrics were found for the generalized
flag manifolds
$\operatorname{Sp}(3)/(\operatorname{U}(1)\times\operatorname{U}(1)\times\operatorname{Sp}(1))$,
$\operatorname{Sp}(4)/(\operatorname{U}(1)\times\operatorname{U}(1)\times\operatorname{Sp}(2))$
and
$\operatorname{Sp}(4)/(\operatorname{U}(2)\times\operatorname{U}(1)\times\operatorname{Sp}(1))$,
and in [ArChSa3] generalized flag manifolds with $G_{2}$-type
$\mathfrak{t}$-roots (apart from the full flag manifold $G_{2}/T$), were
classified. The main result is the following:

######  Theorem 3.5.

A generalized flag manifold $G/K$ with $G_{2}$-type $\mathfrak{t}$-roots,
which is not the full flag manifold $G_{2}/T$, admits exactly one invariant
Kähler Einstein metric and six non Kähler invariant Einstein metrics up to
isometry and scalar. These are the spaces
$F_{4}/\operatorname{U}(3)\times\operatorname{U}(1)$,
$E_{6}/\operatorname{U}(3)\times\operatorname{U}(3)$,
$E_{7}/\operatorname{U}(6)\times\operatorname{U}(1)$ and
$E_{8}/E_{6}\times\operatorname{U}(1)\times\operatorname{U}(1)$. The full flag
manifold $G_{2}/T$ admits exactly one invariant Kähler Einstein metric and two
non Kähler invariant Einstein metrics up to isometry and scalar.

Also, in [ChSa] all $E_{8}$-invariant Einstein metrics on
$E_{8}/\operatorname{U}(1)\times\operatorname{SU}(2)\times\operatorname{SU}(3)\times\operatorname{SU}(5)$
were classified and in [WaZh] all invariant Einstein metrics were classified
on $F_{4}/\operatorname{U}(1)\times\operatorname{U}(1)\operatorname{SO}(5)$,
$E_{6}/\operatorname{U}(1)\times\operatorname{U}(1)\times\operatorname{SU}(2)\times\operatorname{SU}(3)\times\operatorname{SU}(2)$
and
$E_{8}/\operatorname{SO}(12)\times\operatorname{U}(1)\times\operatorname{U}(1)$.

##  4\. Homogeneous spaces with equivalent isotropy summands

If the isotropy representation of a homogeneous space $G/K$
${\mathfrak{m}}={\mathfrak{m}}_{1}\oplus\cdots\oplus{\mathfrak{m}}_{q}$
contains equivalent summands, then the description of all $G$-invariant
metrics is more complicated, hence the problem of finding all invariant
Einstein metrics is quite difficult. We refer to [St] for a study of invariant
metrics on homogeneous spaces with equivalent isotropy summands.

An important class of homogeneous spaces with such a property are the real
Steifel manifolds
$V_{k}\mathbb{R}^{n}=\operatorname{SO}(n)/\operatorname{SO}(n-k)$ of
orthonormal $k$-frames in $\mathbb{R}^{n}$ (as well as the complex and
quaternionic Stiefel manifolds
$V_{k}\mathbb{C}^{n}=\operatorname{SU}(n)/\operatorname{SU}(n-k)$ and
$V_{k}\mathbb{H}^{n}=\operatorname{Sp}(n)/\operatorname{Sp}(n-k)$ of
orthonormal $k$-frames in $\mathbb{C}^{n}$ and $\mathbb{H}^{n}$ respectively).
The simplest case is the sphere
$\mathbb{S}^{n-1}=\operatorname{SO}(n)/\operatorname{SO}(n-1)$, which is an
irreducible symmetric space, therefore it admits up to scale a unique
invariant Einstein metric. Concerning Einstein metrics on other Stiefel
manifolds we recall the following: In [Ko] S. Kobayashi proved existence of an
invariant Einstein metric on the unit tangent bundle
$T_{1}S^{n}=\operatorname{SO}(n)/\operatorname{SO}(n-2)$. In [Sa] A. Sagle
proved that the Stiefel manifolds
$V_{k}\mathbb{R}^{n}=\operatorname{SO}(n)/\operatorname{SO}(n-k)$ admit at
least one homogeneous Einstein metric. For $k\geq 3$ G. Jensen in [Je2] found
a second metric. For $n=3$ the Lie group $\operatorname{SO}(3)$ admits a
unique Einstein metric. For $n\geq 5$ A. Back and W.Y. Hsiang in [BaHs] proved
that $\operatorname{SO}(n)/\operatorname{SO}(n-2)$ admits exactly one
homogeneous Einstein metric. The same result was obtained by M. Kerr in [Ke]
by proving that the diagonal metrics are the only invariant metrics on
$V_{2}\mathbb{R}^{n}$ (see also [Ar2], [AbKo]). The Stiefel manifold
$\operatorname{SO}(4)/\operatorname{SO}(2)$ admits exactly two invariant
Einstein metrics ([AlDoFe]). One is Jensen’s metric and the other one is the
product metric on $S^{3}\times S^{2}$.

Finally, in [ArDzNi1] the author, V.V. Dzhepko and Yu. G. Nikonorov proved
that for $s>1$ and $\ell>k\geq 3$ the Stiefel manifold
$\operatorname{SO}(sk+\ell)/\operatorname{SO}(\ell)$ admits at least four
$\operatorname{SO}(sk+\ell)$-invariant Einstein metrics, two of which are
Jensen’s metrics. The special case
$\operatorname{SO}(2k+\ell)/\operatorname{SO}(\ell)$ admitting at least four
$\operatorname{SO}(2k+\ell)$-invariant Einstein metrics was treated in
[ArDzNi2]. Corresponding results for the quaternionic Stiefel manifolds
$\operatorname{Sp}(sk+\ell)/\operatorname{Sp}(\ell)$ were obtained in
[ArDzNi3].

In the recent work [ArSaSt] it was shown that the Stiefel manifold
$V_{4}\mathbb{R}^{n}=\operatorname{SO}(n)/\operatorname{SO}(n-4)$ admits two
more $\operatorname{SO}(n)$-invariant Einstein metrics and that
$\operatorname{SO}(7)/\operatorname{SO}(2)$ admits four more
$\operatorname{SO}(7)$-invariant Einstein metrics (in addition to the ones
obtained in [ArDzNi1]). This was achieved by making appropriate symmetry
assumptions on the set of $\operatorname{SO}(n)$-invariant metrics on
$V_{4}\mathbb{R}^{n}$. We refer to [ArDzNi1] and [St] for a more general
presentation of such method.

###  4.1. The Stiefel manifolds
$V_{k_{1}+k_{2}}\mathbb{R}^{k_{1}+k_{2}+k_{3}}=\operatorname{SO}(k_{1}+k_{2}+k_{3})/\operatorname{SO}(k_{3})$

We first consider the homogeneous space
$G/K=\operatorname{SO}(k_{1}+k_{2}+k_{3})/(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$,
where the embedding of $K$ in $G$ is diagonal. This is an example of a
generalized Wallach space. These spaces have been recently classified by Yu.G.
Nikonorov ([Ni2]) and Z. Cheng, Y. Kang and K. Liang ([ChKaLi]). If
$\lambda_{n}$ denotes the standard representation of $\operatorname{SO}(n)$,
then $\operatorname{Ad}^{\operatorname{SO}(n)}=\wedge^{2}\lambda_{n}$. Let
$\sigma_{i}:\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3})\to\operatorname{SO}(k_{i})$
be the projection onto the factor $\operatorname{SO}(k_{i})$ ($i=1,2,3$) and
let $p_{k_{i}}=\lambda_{k_{i}}\circ\sigma_{i}$. Then a calculation shows that

$\left.\operatorname{Ad}^{G}\right|_{K}=\operatorname{Ad}^{K}\oplus(p_{k_{1}}\otimes
p_{k_{2}})\oplus(p_{k_{1}}\otimes p_{k_{3}})\oplus(p_{k_{2}}\otimes
p_{k_{3}}).$

Thus the isotropy representation of $G/K$ is $(p_{k_{1}}\otimes
p_{k_{2}})\oplus(p_{k_{1}}\otimes p_{k_{3}})\oplus(p_{k_{2}}\otimes
p_{k_{3}})$ and this induces a decomposition of the tangent space
$\mathfrak{m}$ of $G/K$ into three $\operatorname{Ad}(K)$-submodules

$\mathfrak{m}=\mathfrak{m}_{12}\oplus\mathfrak{m}_{13}\oplus\mathfrak{m}_{23}.$

In fact, $\mathfrak{m}$ is given by $\mathfrak{k}^{\perp}$ in
$\mathfrak{g}=\mathfrak{so}(k_{1}+k_{2}+k_{3})$ with respect to $-B$. If we
denote by $M(p,q)$ the set of all $p\times q$ matrices, then we see that
$\mathfrak{m}$ is given by

$\mathfrak{m}=\left\\{\begin{pmatrix}0&{A}_{12}&{A}_{13}\\\
-{}^{t}\\!{A}_{12}&0&{A}_{23}\\\
-{}^{t}\\!{A}_{13}&-{}^{t}\\!{A}_{23}&0\end{pmatrix}\ \Big{|}\ {A}_{12}\in
M(k_{1},k_{2}),{A}_{13}\in M(k_{1},k_{3}),{A}_{23}\in M(k_{2},k_{3})\right\\}$

and we have

$\mathfrak{m}_{12}=\begin{pmatrix}0&{A}_{12}&0\\\ -{}^{t}\\!{A}_{12}&0&0\\\
0&0&0\end{pmatrix},\quad\mathfrak{m}_{13}=\begin{pmatrix}0&0&{A}_{13}\\\
0&0&0\\\
-{}^{t}\\!{A}_{13}&0&0\end{pmatrix},\quad\mathfrak{m}_{23}=\begin{pmatrix}0&0&0\\\
0&0&{A}_{23}\\\ 0&-{}^{t}\\!{A}_{23}&0\end{pmatrix}.$

Note that the action of $\operatorname{Ad}(k)$ ($k\in K$) on $\mathfrak{m}$ is
given by

$\operatorname{Ad}(k)\begin{pmatrix}0&{A}_{12}&{A}_{13}\\\
-{}^{t}\\!{A}_{12}&0&{A}_{23}\\\
-{}^{t}\\!{A}_{13}&-{}^{t}\\!{A}_{23}&0\end{pmatrix}=\begin{pmatrix}0&{}^{t}h_{1}{A}_{12}h_{2}&{}^{t}h_{1}{A}_{13}h_{3}\\\
-{}^{t}h_{2}{}^{t}\\!{A}_{12}h_{1}&0&{}^{t}h_{2}{A}_{23}h_{3}\\\
-{}^{t}h_{3}{}^{t}\\!{A}_{13}h_{1}&-{}^{t}h_{3}{}^{t}\\!{A}_{23}h_{2}&0\end{pmatrix},$

where $\begin{pmatrix}h_{1}&0&0\\\ 0&h_{2}&0\\\ 0&0&h_{3}\end{pmatrix}\in K$.
Thus the irreducible submodules $\mathfrak{m}_{12}$, $\mathfrak{m}_{13}$ and
$\mathfrak{m}_{23}$ are mutually non equivalent.

We now consider the Stiefel manifold
$G/H=\operatorname{SO}(k_{1}+k_{2}+k_{3})/\operatorname{SO}(k_{3})$ and we
take into account the diffeomorphism
$G/H=(G\times\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2}))/((\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2}))\times\operatorname{SO}(k_{3}))=\widetilde{G}/\widetilde{H}.$
Let
$\mathfrak{p}=\mathfrak{so}(k_{1})\oplus\mathfrak{so}(k_{2})\oplus\mathfrak{m}_{12}\oplus\mathfrak{m}_{13}\oplus\mathfrak{m}_{23}$
be an
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$-invariant
decomposition of the tangent space $\mathfrak{p}$ of $G/H$ at $eH$, where the
corresponding submodules are non equivalent. Then we consider a subset of all
$G$-invariant metrics on $G/H$ determined by the
$\operatorname{Ad}(\operatorname{SO}(k_{1})\times\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$-invariant
scalar products on $\mathfrak{p}$ given by

$\langle\ ,\
\rangle=x_{1}\,(-B)|_{\mathfrak{so}(k_{1})}+x_{2}\,(-B)|_{\mathfrak{so}(k_{2})}+x_{12}\,(-B)|_{\mathfrak{m}_{12}}+x_{13}\,(-B)|_{\mathfrak{m}_{13}}+x_{23}\,(-B)|_{\mathfrak{m}_{23}},$
(7)

for $k_{1}\geq 2$, $k_{2}\geq 2$ and $k_{3}\geq 1$. Since the submodules are
non equivalent we can use Lemma [PaSa] and obtain the following:

######  Lemma 4.1.

The components of the Ricci tensor ${r}$ for the invariant metric $\langle\ ,\
\rangle$ on $G/H$ defined by (7) are given as follows:

$\left.\small{\begin{array}[]{lll}r_{1}&=&\displaystyle{\frac{k_{1}-2}{4(n-2)x_{1}}+\frac{1}{4(n-2)}\biggl{(}k_{2}\frac{x_{1}}{{x_{12}}^{2}}}+k_{3}\frac{x_{1}}{{x_{13}}^{2}}\biggr{)},\\\
\\\
r_{2}&=&\displaystyle{\frac{k_{2}-2}{4(n-2)x_{2}}+\frac{1}{4(n-2)}\biggl{(}k_{1}\frac{x_{2}}{{x_{12}}^{2}}+k_{3}\frac{x_{2}}{{x_{23}}^{2}}\biggr{)},}\\\
\\\
r_{12}&=&\displaystyle{\frac{1}{2x_{12}}+\frac{k_{3}}{4(n-2)}\biggl{(}\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}}\\\
\\\
&&\displaystyle{-\frac{1}{4(n-2)}\biggl{(}(k_{1}-1)\frac{x_{1}}{{x_{12}}^{2}}+(k_{2}-1)\frac{x_{2}}{{x_{12}}^{2}}\biggr{)}},\\\
\\\
r_{13}&=&\displaystyle{\frac{1}{2x_{13}}+\frac{k_{2}}{4(n-2)}\biggl{(}\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}-\frac{1}{4(n-2)}\biggl{(}(k_{1}-1)\frac{x_{1}}{{x_{13}}^{2}}\biggr{)}}\\\
\\\
r_{23}&=&\displaystyle{\frac{1}{2x_{23}}+\frac{k_{1}}{4(n-2)}\biggl{(}\frac{x_{23}}{x_{13}x_{12}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{23}x_{13}}\biggr{)}-\frac{1}{4(n-2)}\biggl{(}(k_{2}-1)\frac{x_{2}}{{x_{23}}^{2}}\biggr{)}}.\end{array}}\right\\}$
(8)

where $n=k_{1}+k_{2}+k_{3}$.

For $k_{1}=1$ we have the Stiefel manifold
$G/H=\operatorname{SO}(1+k_{2}+k_{3})/\operatorname{SO}(k_{3})$ with
corresponding decomposition
$\mathfrak{p}=\mathfrak{so}(k_{2})\oplus\mathfrak{m}_{12}\oplus\mathfrak{m}_{13}\oplus\mathfrak{m}_{23}.$
We then consider $G$-invariant metrics on $G/H$ determined by the
$\operatorname{Ad}(\operatorname{SO}(k_{2})\times\operatorname{SO}(k_{3}))$-invariant
scalar products on $\mathfrak{p}$ given by

$\langle\ ,\
\rangle=x_{2}\,(-B)|_{\mathfrak{so}(k_{2})}+x_{12}\,(-B)|_{\mathfrak{m}_{12}}+x_{13}\,(-B)|_{\mathfrak{m}_{13}}+x_{23}\,(-B)|_{\mathfrak{m}_{23}}$
(9)

######  Lemma 4.2.

The components of the Ricci tensor ${r}$ for the invariant metric $\langle\ \
,\ \ \rangle$ on $G/H$ defined by (9), are given as follows:

$\left.{\small\begin{array}[]{l}r_{2}=\displaystyle{\frac{k_{2}-2}{4(n-2)x_{2}}+\frac{1}{4(n-2)}\biggl{(}\frac{x_{2}}{{x_{12}}^{2}}+k_{3}\frac{x_{2}}{{x_{23}}^{2}}\biggr{)}},\\\
\\\
r_{12}=\displaystyle{\frac{1}{2x_{12}}+\frac{k_{3}}{4(n-2)}\biggl{(}\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}-\frac{1}{4(n-2)}\biggl{(}(k_{2}-1)\frac{x_{2}}{{x_{12}}^{2}}\biggr{)}},\\\
\\\
r_{23}=\displaystyle{\frac{1}{2x_{23}}+\frac{1}{4(n-2)}\biggl{(}\frac{x_{23}}{x_{13}x_{12}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{23}x_{13}}\biggr{)}-\frac{1}{4(n-2)}\biggl{(}(k_{2}-1)\frac{x_{2}}{{x_{23}}^{2}}\biggr{)}},\\\
\\\
r_{13}=\displaystyle{\frac{1}{2x_{13}}+\frac{k_{2}}{4(n-2)}\biggl{(}\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}}.\end{array}}\right\\}$
(10)

where $n=1+k_{2}+k_{3}$.

###  4.2. The Stiefel manifold $V_{5}\mathbb{R}^{n}$

For the Stiefel manifold
$V_{5}\mathbb{R}^{n}=\operatorname{SO}(n)/\operatorname{SO}(n-5)$ we can let
$k_{1}=2,k_{2}=3,k_{3}=n-5$ and consider
$\operatorname{Ad}(\operatorname{SO}(2)\times\operatorname{SO}(3)\times\operatorname{SO}(n-5))$-invariant
scalar products of the form (7), or let $k_{1}=1,k_{2}=4,k_{3}=n-5$ and
consider
$\operatorname{Ad}(\operatorname{SO}(4)\times\operatorname{SO}(n-5))$-invariant
scalar products of the form (9).

######  Theorem 4.3.

The Stiefel manifold
$V_{5}\mathbb{R}^{n}=\operatorname{SO}(n)/\operatorname{SO}(n-5)$ ($n\geq 7$)
admits at least six invariant Einstein metrics. Two of them are Jensen’s
metrics, two are given by
$\operatorname{Ad}(\operatorname{SO}(4)\times\operatorname{SO}(n-5))$-invariant
scalar products of the form (9), and the other two are given by
$\operatorname{Ad}(\operatorname{SO}(2)\times\operatorname{SO}(3)\times\operatorname{SO}(n-5))$-invariant
scalar products of the form (7).

###### Proof.

We consider
$\operatorname{Ad}(\operatorname{SO}(4)\times\operatorname{SO}(n-5))$-invariant
inner products of the form 9. Then from Lemma 4.2 we see that the components
of the Ricci tensor $r$ are given by:

$\left.{\small\begin{array}[]{l}r_{2}=\displaystyle{\frac{1}{4(n-2)x_{2}}+\frac{1}{4(n-2)}\biggl{(}\frac{x_{2}}{{x_{12}}^{2}}+(n-5)\frac{x_{2}}{x_{23}^{2}}\biggr{)}},\\\
\\\
r_{12}=\displaystyle{\frac{1}{2x_{12}}+\frac{n-4}{4(n-2)}\biggl{(}\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}-\frac{3}{4(n-2)}\frac{x_{2}}{{x_{12}}^{2}}},\\\
\\\
r_{23}=\displaystyle{\frac{1}{2x_{23}}+\frac{1}{4(n-2)}\biggl{(}\frac{x_{23}}{x_{13}x_{12}}-\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{23}x_{13}}\biggr{)}-\frac{3}{4(n-2)}\frac{x_{2}}{{x_{23}}^{2}}},\\\
\\\
r_{13}=\displaystyle{\frac{1}{2x_{13}}+\frac{1}{(n-2)}\biggl{(}\frac{x_{13}}{x_{12}x_{23}}-\frac{x_{12}}{x_{13}x_{23}}-\frac{x_{23}}{x_{12}x_{13}}\biggr{)}}.\end{array}}\right\\}$
(11)

We consider the system of equations

$r_{2}=r_{12},\,\,r_{12}=r_{23},\,\,r_{23}=r_{13}.$ (12)

We put $x_{23}=1$ and from system (12) we have:

$\left.{\begin{array}[]{lll}f_{1}&=&-nx_{2}x_{12}^{3}+5x_{2}x_{12}^{3}+nx_{13}x_{2}^{2}x_{12}^{2}-5x_{13}x_{2}^{2}x_{12}^{2}+2x_{13}x_{12}^{2}+nx_{13}^{2}x_{2}x_{12}\\\
&&-5x_{13}^{2}x_{2}x_{12}+nx_{2}x_{12}-2nx_{13}x_{2}x_{12}+4x_{13}x_{2}x_{12}-5x_{2}x_{12}+4x_{13}x_{2}^{2}=0\\\
f_{2}&=&nx_{12}^{3}-4x_{12}^{3}-2nx_{13}x_{12}^{2}+4x_{13}x_{12}^{2}+3x_{13}x_{2}x_{12}^{2}-nx_{13}^{2}x_{12}+6x_{13}^{2}x_{12}\\\
&&-nx_{12}+2nx_{13}x_{12}-4x_{13}x_{12}+4x_{12}-3x_{13}x_{2}=0\\\
f_{3}&=&3x_{12}^{2}-2nx_{12}+2nx_{13}x_{12}-4x_{13}x_{12}-3x_{13}x_{2}x_{12}+4x_{12}-5x_{13}^{2}+5=0.\end{array}}\right\\}$
(13)

We consider a polynomial ring $R={\mathbb{Q}}[z,x_{2},x_{12},x_{13}]$ and an
ideal $I$ generated by $\\{f_{1},\,f_{2},\,f_{3},$
$\,z\,x_{2}\,x_{12}\,x_{13}-1\\}$ to find non zero solutions of the above
equations. We take a lexicographic order $>$ with $z>x_{2}>x_{12}>x_{13}$ for
a monomial ordering on $R$. Then by the aid of computer, we see that a Gröbner
basis for the ideal $I$ contains the polynomial $(x_{13}-1)\,h_{1}(x_{13}),$
where $h_{1}(x_{13})$ is a polynomial of $x_{13}$ given by

$\begin{array}[]{l}h_{1}(x_{13})=(-1+n)^{3}(-7+3n)^{2}(-81+78n-22n^{2}+2n^{3})(-1+22n-14n^{2}+2n^{3})x_{13}^{10}\\\
-2(-1+n)^{2}(-7+3n)(73599-286463n+441002n^{2}-361584n^{3}+175526n^{4}-52216n^{5}\\\
+9376n^{6}-936n^{7}+40n^{8})x_{13}^{9}+(-1+n)(11395069-47604434n+87540578n^{2}\\\
-93088081n^{3}+63294239n^{4}-28742935n^{5}+8835132n^{6}-1817860n^{7}\\\
+239764n^{8}-18192n^{9}+572n^{10}+4n^{11})x_{13}^{8}\\\
-2(-22336092+102242734n-207451554n^{2}+246088757n^{3}-189448366n^{4}+99318803n^{5}\\\
-36163892n^{6}+9130532n^{7}-1555886n^{8}+166276n^{9}-9008n^{10}+16n^{11}\\\
+16n^{12})x_{13}^{7}+(-28792736+134723232n-268245792n^{2}+301569274n^{3}\\\
-211579655n^{4}+95750323n^{5}-27336050n^{6}+4215160n^{7}+516n^{8}-146852n^{9}\\\
+30868n^{10}-2928n^{11}+112n^{12})x_{13}^{6}-2(65620080-227005040n+365106888n^{2}\\\
-368314424n^{3}+261802867n^{4}-137589389n^{5}+54260010n^{6}-16033372n^{7}\\\
+3504636n^{8}-550600n^{9}+58748n^{10}-3800n^{11}+112n^{12})x_{13}^{5}\\\
+(59501248-288258784n+627812916n^{2}-804675780n^{3}+674941445n^{4}-391428642n^{5}\\\
+161902013n^{6}-48456452n^{7}+10471948n^{8}-1597772n^{9}+163380n^{10}-10040n^{11}\\\
+280n^{12})x_{13}^{4}-4(-24508224+70221360n-72172288n^{2}+18690688n^{3}\\\
+27533464n^{4}-33792759n^{5}+19444979n^{6}-7070399n^{7}+1733171n^{8}\\\
-287260n^{9}+30980n^{10}-1968n^{11}+56n^{12})x_{13}^{3}+4(-1+n)(4378624-15912576n\\\
+22754384n^{2}-15980992n^{3}+4953656n^{4}+505552n^{5}-1046519n^{6}+426038n^{7}\\\
-93847n^{8}+12221n^{9}-889n^{10}+28n^{11})x_{13}^{2}-8(-6+n)(-4+n)(-1+n)^{2}(2+n)(31664\\\
-44256n+19472n^{2}-1636n^{3}-1423n^{4}+535n^{5}-76n^{6}+4n^{7})x_{13}\\\
+4(-6+n)^{2}(-4+n)^{2}(-1+n)^{3}(2+n)^{2}(124-24n-5n^{2}+n^{3})\end{array}$

For the case $x_{13}=1$ system (13) gives:

$f_{3}=x_{12}(x_{12}-x_{2})=0\Longleftrightarrow x_{12}=x_{2}$

Thus we obtain the system

$x_{12}=x_{2},\ \ \ \ \ 3x_{2}+2x_{2}^{2}(2-n)-x_{2}^{3}(1-n)=0$

The above system has three solutions:

$x_{2}=0,\ \ x_{2}=\frac{-2+n-\sqrt{7-7n+n^{2}}}{n-1}\ \
x_{2}=\frac{-2+n+\sqrt{7-7n+n^{2}}}{n-1}.$

The first solution is rejected, so we get the following

$x_{12}=x_{2}=\frac{-2+n-\sqrt{7-7n+n^{2}}}{n-1},\ \ x_{13}=x_{23}=1$

and

$x_{12}=x_{2}=\frac{-2+n+\sqrt{7-7n+n^{2}}}{n-1},\ \ x_{13}=x_{23}=1.$

These two solutions of system (13) are Jensen’s Einstein metrics on Stiefel
manifolds pictured as follows:

$\begin{pmatrix}0&\alpha&1\\\ \alpha&\alpha&1\\\ 1&1&*\end{pmatrix}.$

Next we consider the case $x_{13}\neq 1$. Then $h_{1}(x_{13})=0$ and we will
prove that the equation $h_{1}(x_{13})=0$ has at least two positive roots. We
observe that

$h_{1}(1)=988524-7380396n+9224766n^{2}-3877551n^{3}+671409n^{4}-40824n^{5}$

is negative for $n\geq 7$ and

$\displaystyle
h_{1}(0)=-1142784+3459072n-3064576n^{2}-222464n^{3}+1556672n^{4}$
$\displaystyle-558592n^{5}-114256n^{6}+106352n^{7}-19036n^{8}-1072n^{9}+776n^{10}-96n^{11}+4n^{12}$

is positive for $n\geq 7$, hence we obtain one solution $x_{13}=\alpha_{13}$
between $0<\alpha_{13}<1$. In fact, it is possible to show that
$1-4/n<\alpha_{13}<1-3/n$. In the same way we observe that for $n\geq 7$

$\displaystyle
h_{1}(2)=-1077260544+2404260096n-1787496640n^{2}+192056128n^{3}+482885648n^{4}$
$\displaystyle-324418304n^{5}+97443600n^{6}-15521168n^{7}+1168004n^{8}$
$\displaystyle+8544n^{9}-8968n^{10}+528n^{11}+4n^{12}$

is always positive, hence we have a second solution $x_{13}=\beta_{13}$
between $1<\beta_{13}<2$. In fact, it is possible to show that
$1<\beta_{13}<1+10/n^{2}$.

Because for $x_{13}=1$ we take Jensen’s metrics, we consider a Gröbner basis
(take a lexicographic order $>$ with $z>x_{2}>x_{12}>x_{13}$) for the ideal
$J$ generated by the polynomials $\\{f_{1},\,f_{2},\,f_{3},$
$\,z\,x_{2}\,x_{12}\,x_{13}\,(x_{13}-1)-1\\}.$ This basis contains the
polynomial $h_{1}(x_{13})$ and the polynomials

$6(-6+n)(-2+n)(-1+n)^{3}(2+n)(124-24n-5n^{2}+n^{3})a(n)x_{2}-w_{2}(x_{13},n),$
$12(-4+n)(-2+n)(-1+n)^{2}a(n)x_{12}-w_{12}(x_{13},n),$

where the first polynomial is obtained by the lexicographic order
$z>x_{12}>x_{2}>x_{13}$ and the second by the lexicographic order
$z>x_{2}>x_{12}>x_{13}$. The polynomial $a(n)$ of $n$ is of degree $51$ and
can be written as

$\begin{array}[]{l}a(n)=1603489(n-7)^{51}+331629(n-7)^{50}+3330235(n-7)^{49}+21654178(n-7)^{48}+\\\
10256996(n-7)^{47}+37749148(n-7)^{46}+1124243(n-7)^{45}+278618(n-7)^{44}+\\\
586377(n-7)^{43}+10642404(n-7)^{42}+16857968(n-7)^{41}+23530271(n-7)^{40}+\\\
29165722(n-7)^{39}+32307839(n-7)^{38}+32153102(n-7)^{37}+28875189(n-7)^{36}+\\\
23485774(n-7)^{35}+17353528(n-7)^{34}+11678023(n-7)^{33}+71720699(n-7)^{32}+\\\
40265082(n-7)^{31}+20690586(n-7)^{30}+97403757(n-7)^{29}+42033064(n-7)^{28}+\\\
16631548(n-7)^{27}+60336057(n-7)^{26}+20061477(n-7)^{25}+61092907(n-7)^{24}+\\\
17022304(n-7)^{23}+43336722(n-7)^{22}+10063783(n-7)^{21}+21273147(n-7)^{20}+\\\
40831997(n-7)^{19}+70963420(n-7)^{18}+11131153(n-7)^{17}+15703148(n-7)^{16}+\\\
19849915(n-7)^{15}+22400611(n-7)^{14}+22495760(n-7)^{13}+20062962(n-7)^{12}+\\\
15890989(n-7)^{11}+11209194(n-7)^{10}+70758073(n-7)^{9}+40094148(n-7)^{8}+\\\
20263187(n-7)^{7}+89201697(n-7)^{6}+32751057(n-7)^{5}+94463613(n-7)^{4}+\\\
19935475(n-7)^{3}+28717417(n-7)^{2}+27565510(n-7)+16817600.\end{array}$

Hence, we see that for $n\geq 7$ the polynomial $a(n)$ is positive. Thus for
the positive values $x_{13}=\alpha_{13},\beta_{13}$ found above, we obtain
real values $x_{2}=\alpha_{2},\beta_{2}$ and $x_{12}=\alpha_{12},\beta_{12}$
as solutions of the system (13). We claim that
$\alpha_{2},\beta_{2},\alpha_{12},\beta_{12}$ are positive. We consider the
ideal $J$ generated by $\\{f_{1},\,f_{2},\,f_{3},$
$\,z\,x_{2}\,x_{12}\,x_{13}\,(x_{13}-1)-1\\}$ and now take a lexicographic
order $>$ with $z>x_{2}>x_{13}>x_{12}$ for a monomial ordering on $R$. Then we
see that a Gröbner basis for the ideal $J$ contains the polynomial
$h_{2}(x_{12})$

$\begin{array}[]{l}h_{2}(x_{12})=(-1+n)^{3}(-81+78n-22n^{2}+2n^{3})(-1+22n-14n^{2}+2n^{3})x_{12}^{10}\\\
-2(-2+n)(-1+n)^{2}(-5001+9686n-6508n^{2}+1740n^{3}-96n^{4}-32n^{5}+4n^{6})x_{12}^{9}\\\
+(-1+n)(52393-76703n-6162n^{2}+73700n^{3}-56664n^{4}+19352n^{5}-3176n^{6}+204n^{7})x_{12}^{8}\\\
-4(-2+n)(46818-149187n+191207n^{2}-123037n^{3}+41605n^{4}-6890n^{5}\\\
+394n^{6}+10n^{7})x_{12}^{7}+(2108590-5378218n+5585179n^{2}-2957751n^{3}+809684n^{4}\\\
-92936n^{5}-1860n^{6}+912n^{7})x_{12}^{6}-2(-2+n)(829930-1018838n\\\
+348263n^{2}+24142n^{3}-31870n^{4}+4152n^{5}+16n^{6})x_{12}^{5}+(-150922+2293544n\\\
-3272491n^{2}+1786572n^{3}-423924n^{4}+33984n^{5}+712n^{6})x_{12}^{4}-20(-2+n)(-104374\\\
+109331n-35890n^{2}+2645n^{3}+312n^{4})x_{12}^{3}+25(-76485+58035n-4052n^{2}\\\
-5488n^{3}+1072n^{4})x_{12}^{2}-250(-2+n)(2051-1298n+224n^{2})x_{12}+5625(107-56n+8n^{2})\end{array}$

The polynomial $h_{2}(x_{12})$ can be written as

$\begin{array}[]{l}h_{2}(x_{12})=(2412504+5213484(-7+n)+4966002(-7+n)^{2}+2728881(-7+n)^{3}\\\
+950664(-7+n)^{4}+217336(-7+n)^{5}+32580(-7+n)^{6}+3088(-7+n)^{7}\\\
+168(-7+n)^{8}+4(-7+n)^{9})x_{12}^{10}\\\
-(15481800+30522960(-7+n)+26336250(-7+n)^{2}+13013966(-7+n)^{3}+4047916(-7+n)^{4}\\\
+819792(-7+n)^{5}+107792(-7+n)^{6}+8840(-7+n)^{7}+408(-7+n)^{8}+8(-7+n)^{9})x_{12}^{9}\\\
+(48245892+85904548(-7+n)+66240641(-7+n)^{2}+28829438(-7+n)^{3}+7737264(-7+n)^{4}\\\
+1310572(-7+n)^{5}+136796(-7+n)^{6}+8044(-7+n)^{7}+204(-7+n)^{8})x_{12}^{8}\\\
-(97049440+154931088(-7+n)+105623840(-7+n)^{2}+39883780(-7+n)^{3}\\\
+9037872(-7+n)^{4}+1239340(-7+n)^{5}+97472(-7+n)^{6}+3736(-7+n)^{7}+40(-7+n)^{8})x_{12}^{7}\\\
+(137946250+196719755(-7+n)+117648788(-7+n)^{2}+38055081(-7+n)^{3}\\\
+7138384(-7+n)^{4}+767392(-7+n)^{5}+42828(-7+n)^{6}+912(-7+n)^{7})x_{12}^{6}\\\
-(141888350+179760770(-7+n)+93307270(-7+n)^{2}+25366350(-7+n)^{3}\\\
+3804144(-7+n)^{4}+298660(-7+n)^{5}+9808(-7+n)^{6}+32(-7+n)^{7})x_{12}^{5}\\\
+(105439675+117258450(-7+n)+51819665(-7+n)^{2}+11453180(-7+n)^{3}\\\
+1288836(-7+n)^{4}+63888(-7+n)^{5}+712(-7+n)^{6})x_{12}^{4}\\\
-(55868000+53548600(-7+n)+19613300(-7+n)^{2}+3365760(-7+n)^{3}\\\
+258820(-7+n)^{4}+6240(-7+n)^{5})x_{12}^{3}\\\
+(20567500+16633875(-7+n)+4896700(-7+n)^{2}+613200(-7+n)^{3}\\\
+26800(-7+n)^{4})x_{12}^{2}-(+4926250+3282750(-7+n)+739500(-7+n)^{2}+56000(-7+n)^{3})x_{12}\\\
+601875+315000(-7+n)+45000(-7+n)^{2}\end{array}$

Then we have that, for $n\geq 7$ the coefficients of the polynomial
$h_{2}(x_{12})$ is positive for even degree and negative for odd degree. Thus
if the equation $h_{2}(x_{12})=0$ has real solutions, then these is are all
positive.

Next we take the lexicographic order $z>x_{12}>x_{13}>x_{2}$ for the monomial
ordering $R$. Then the Gröbner basis for the ideal $J$ contains the polynomial
$h_{3}(x_{2})$ give by

$\begin{array}[]{l}h_{3}(x_{2})=81(-1+n)^{3}(-7+3n)^{2}(124-24n-5n^{2}+n^{3})x_{2}^{10}-54(-2+n)(-1+n)^{2}(-7\\\
+3n)(-12620+10066n-1796n^{2}-63n^{3}+20n^{4}+n^{5})x_{2}^{9}\\\
+9(-1+n)(5444348-12985508n+12566667n^{2}-6115028n^{3}+1510310n^{4}\\\
-165004n^{5}+6142n^{6}-961n^{7}+153n^{8}+n^{9})x_{2}^{8}-6(-2+n)(-2979568\\\
+8238150n-7797806n^{2}+4055755n^{3}-1567204n^{4}+409230n^{5}\\\
-37660n^{6}-4903n^{7}+798n^{8}+8n^{9})x_{2}^{7}+(664693040-1960669464n+2468056527n^{2}\\\
-1726283201n^{3}+735938803n^{4}-196313137n^{5}+31516618n^{6}-2509926n^{7}\\\
+9408n^{8}+8308n^{9}+64n^{1}0)x_{2}^{6}-4(-2+n)(-59469494+203982575n-258060568n^{2}\\\
+163886493n^{3}-58390388n^{4}+11911514n^{5}-1270014n^{6}+44728n^{7}+1634n^{8})x_{2}^{5}\\\
+(-1148666992+2483878280n-2018846305n^{2}+682058094n^{3}+1140539n^{4}-74249792n^{5}\\\
+24038016n^{6}-3335112n^{7}+180072n^{8})x_{2}^{4}-20(-2+n)(7338836-31781095n+35343394n^{2}\\\
-17774551n^{3}+4654668n^{4}-624056n^{5}+34040n^{6})x_{2}^{3}+100(8425328-17876816n\\\
+15076868n^{2}-6558218n^{3}+1569983n^{4}-197660n^{5}+10272n^{6})x_{2}^{2}\\\
-1000(-4+n)(-2+n)(-41149+30677n-7916n^{2}+700n^{3})x_{2}\\\
+22500(-4+n)^{2}(107-56n+8n^{2})\end{array}$

The above polynomial can be rewritten as follows:

$\begin{array}[]{l}h_{3}(x_{2})=(185177664+353699136(-7+n)+287249328(-7+n)^{2}+128537280(-7+n)^{3}\\\
+34614540(-7+n)^{4}+5746788(-7+n)^{5}+574533(-7+n)^{6}+31590(-7+n)^{7}+729(-7+n)^{8})x_{2}^{10}\\\
-(1776660480+3383889696(-7+n)+2750137920(-7+n)^{2}+1239055920(-7+n)^{3}\\\
+338884128(-7+n)^{4}+57943890(-7+n)^{5}+6131268(-7+n)^{6}+381348(-7+n)^{7}\\\
+12420(-7+n)^{8}+162(-7+n)^{9})x_{2}^{9}\\\
+(7529406192+14385411216(-7+n)+11794825368(-7+n)^{2}+5412553560(-7+n)^{3}\\\
+1528684479(-7+n)^{4}+275559957(-7+n)^{5}+31789143(-7+n)^{6}+2286063(-7+n)^{7}\\\
+96003(-7+n)^{8}+1998(-7+n)^{9}+9(-7+n)^{1}0)x_{2}^{8}\\\
-(19003144320+36647336304(-7+n)+30550728648(-7+n)^{2}+14409983892(-7+n)^{3}\\\
+4241037702(-7+n)^{4}+809574732(-7+n)^{5}+100606440(-7+n)^{6}+7901580(-7+n)^{7}\\\
+362442(-7+n)^{8}+8052(-7+n)^{9}+48(-7+n)^{1}0)x_{2}^{7}\\\
+(31751845080+61819273140(-7+n)+52334228666(-7+n)^{2}+25252421727(-7+n)^{3}\\\
+7654138784(-7+n)^{4}+1509829133(-7+n)^{5}+193077556(-7+n)^{6}+15306474(-7+n)^{7}\\\
+673932(-7+n)^{8}+12788(-7+n)^{9}+64(-7+n)^{1}0)x_{2}^{6}\\\
-(35544821200+69080883040(-7+n)+58454603500(-7+n)^{2}+28210577088(-7+n)^{3}\\\
+8532473268(-7+n)^{4}+1666191528(-7+n)^{5}+207197760(-7+n)^{6}+15378664(-7+n)^{7}\\\
+577608(-7+n)^{8}-6536(-7+n)^{9})x_{2}^{5}\\\
+(25602766500+48488384800(-7+n)+39762417135(-7+n)^{2}+18448573770(-7+n)^{3}\\\
+5297021059(-7+n)^{4}+962339608(-7+n)^{5}+107676312(-7+n)^{6}+6748920(-7+n)^{7}\\\
+180072(-7+n)^{8})x_{2}^{4}\\\
-(11214712000+19967074400(-7+n)+15143699900(-7+n)^{2}+6348086000(-7+n)^{3}\\\
+1588873060(-7+n)^{4}+237204560(-7+n)^{5}+19516480(-7+n)^{6}+680800(-7+n)^{7})x_{2}^{3}\\\
+(2853346500+4609869000(-7+n)+3090157200(-7+n)^{2}+1101382600(-7+n)^{3}\\\
+220180300(-7+n)^{4}+23376400(-7+n)^{5}+1027200(-7+n)^{6})x_{2}^{2}\\\
-(387090000+547743000(-7+n)+309590000(-7+n)^{2}+87525000(-7+n)^{3}\\\
+12384000(-7+n)^{4}+700000(-7+n)^{5})x_{2}\\\
+21667500+25785000(-7+n)+11587500(-7+n)^{2}+2340000(-7+n)^{3}+180000(-7+n)^{4}\end{array}$

Then we see that for $n\geq 7$ the coefficients of $h_{3}(x_{2})$ are positive
for even degree and negative for odd degree. Thus if the equation
$h_{3}(x_{2})=0$ has real solutions then these are all positive. Hence the
numbers $\alpha_{12},\alpha_{2},\beta_{12},\beta_{2}$ are positive. In
particular the positive solutions of (13) are

$\\{x_{2}=\alpha_{2},x_{12}=\alpha_{12},x_{13}=\alpha_{13},x_{23}=1\\}\ \
\mbox{and}\ \
\\{x_{2}=\beta_{2},x_{12}=\beta_{12},x_{13}=\beta_{13},x_{23}=1\\}$

and satisfy $\alpha_{13},\beta_{13}\neq 1$. Thus, these solutions are
different from Jensen’s Einstein metrics, and can be pictured as

$\begin{pmatrix}0&\beta&\gamma\\\ \beta&\alpha&1\\\ \gamma&1&*\end{pmatrix}\ \
\ (\alpha,\beta,\gamma\ \ \mbox{are all different and}\ \ \gamma\neq 1).$

Similarly we can show that the Stiefel manifold
$V_{5}\mathbb{R}^{n}=\operatorname{SO}(n)/\operatorname{SO}(n-5)$ with
$\operatorname{Ad}(\operatorname{SO}(2)\times\operatorname{SO}(3)\times\operatorname{SO}(n-5))$-invariant
scalar products of the form (7) admits at least four invariant Einstein
metrics. Two of them are Jensen’s Einstein metrics obtained before, and the
other two are different from the Einstein metrics given by
$\operatorname{Ad}(\operatorname{SO}(4)\times\operatorname{SO}(n-5))$-invariant
scalar products. The computations are quite long, so we present the proof very
briefly.

We consider the system of equations

$r_{1}=r_{2},\,r_{2}=r_{12},\,r_{12}=r_{23},\,r_{23}=r_{13}.$ (14)

We put $x_{23}=1$

$\left.\begin{array}[]{lll}g_{1}&=&n{x_{1}}{x_{12}}^{2}{x_{2}}-n{x_{12}}^{2}{x_{13}}^{2}{x_{2}}^{2}-5{x_{1}}{x_{12}}^{2}{x_{2}}+3{x_{1}}{x_{13}}^{2}{x_{2}}+5{x_{12}}^{2}{x_{13}}^{2}{x_{2}}^{2}\\\
&&+{x_{12}}^{2}-{x_{13}}^{2}-2{x_{13}}^{2}{x_{2}}^{2},\\\
g_{2}&=&n{x_{1}}{x_{12}}^{2}-n{x_{12}}^{3}{x_{13}}+n{x_{12}}{x_{13}}^{3}-2n{x_{12}}{x_{13}}^{2}+n{x_{12}}{x_{13}}-5{x_{1}}{x_{12}}^{2}\\\
&&+4{x_{1}}{x_{13}}^{2}+5{x_{12}}^{3}{x_{13}}-5{x_{12}}{x_{13}}^{3}+4{x_{12}}{x_{13}}^{2}-5{x_{12}}{x_{13}}+2{x_{13}}^{2}{x_{2}}\\\
g_{3}&=&n{x_{1}}{x_{12}}^{2}-2n{x_{12}}^{2}{x_{13}}-4{x_{1}}{x_{12}}^{2}+3{x_{1}}{x_{13}}^{2}+3{x_{12}}^{3}{x_{13}}+4{x_{12}}^{2}{x_{13}}\\\
&&-3{x_{12}}{x_{13}}^{3}+3{x_{12}}{x_{13}},\\\
g_{4}&=&2n{x_{12}}{x_{13}}^{2}-2n{x_{12}}{x_{13}}+{x_{1}}{x_{12}}+{x_{12}}^{2}{x_{13}}-2{x_{12}}{x_{13}}^{2}{x_{2}}-4{x_{12}}{x_{13}}^{2}\\\
&&+4{x_{12}}{x_{13}}-5{x_{13}}^{3}+5{x_{13}}.\end{array}\right\\}$ (15)

We consider a polynomial ring $R={\mathbb{Q}}[z,x_{1},x_{2},x_{12},x_{13}]$
and an ideal $I$ generated by $\\{g_{1},g_{2},g_{3},g_{4},$
$zx_{1}x_{2}x_{12}x_{13}-1\\}$ to find non zero solutions of the above
equations. We take a lexicographic order $>$ with
$z>x_{1}>x_{2}>x_{12}>x_{13}$ for a monomial ordering on $R$. Then, by the aid
of computer, we see that a Gröbner basis for the ideal $I$ contains the
polynomial $(x_{13}-1)p_{1}(x_{13})$, where $p_{1}(x_{13})$ is a polynomial of
$x_{13}$ with degree 22. By the same method as in the case
$\operatorname{Ad}(\operatorname{SO}(4)\times\operatorname{SO}(n-5))$-invariant
metrics, we can show that there are two positive solutions of (14)

$\\{x_{1}=\alpha_{1},\,x_{2}=\alpha_{2},\,x_{12}=\alpha_{12}\,x_{13}=\alpha_{13},x_{23}=1\\},\\{x_{1}=\beta_{1},\,x_{2}=\beta_{2},\,x_{12}=\beta_{12}\,x_{13}=\beta_{13},x_{23}=1\\},$

with $\alpha_{13}\neq 1,\beta_{13}\neq 1$ by considering the solutions
$p_{1}(x_{13})=0$. In fact, we see that

$\displaystyle p_{1}(1)$ $\displaystyle=$
$\displaystyle-51984\left(5n^{3}-44n^{2}+130n-75\right)^{2}\left(56n^{5}-921n^{4}+5319n^{3}-12654n^{2}+10124n-1356\right)$
$\displaystyle=$
$\displaystyle-51984\left(5(n-7)^{3}+61(n-7)^{2}+249(n-7)+394\right)^{2}\times$
$\displaystyle(56(n-7)^{5}+1039(n-7)^{4}+6971(n-7)^{3}+20351(n-7)^{2}+23529(n-7)+3754)<0,$

and also we see that $p_{1}(1-5/n)>0$ and $p_{1}(1+10/n^{2})>0$ for $n\geq 7$.
Thus we see $1-5/n<\alpha_{13}<1$ and $1<\beta_{13}<1+10/n^{2}$.

These solutions can be pictured as

$\begin{pmatrix}\alpha&\beta&\gamma\\\ \beta&\delta&1\\\
\gamma&1&*\end{pmatrix}\ \ \ (\alpha,\beta,\gamma,\delta\ \ \mbox{are all
different and}\ \ \gamma\neq 1).$

Note that in this case Jensen’s metrics are pictured as

$\begin{pmatrix}\alpha&\alpha&1\\\ \alpha&\alpha&1\\\ 1&1&*\end{pmatrix}.$

∎

##  5\. Generalized Wallach spaces

A generalised Wallach space is a homogeneous space $M=G/K$ whose isotropy
representation $\mathfrak{m}$ decomposes into three
$\operatorname{Ad}(K)$-invariant irreducible and pairwise orthogonal
submodules as
$\mathfrak{m}=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\mathfrak{m}_{3},$
which satisfy the relations
$[\mathfrak{m}_{i},\mathfrak{m}_{i}]\subset\mathfrak{k}$ $(i=1,2,3)$. The
original terminology of these spaces was three-locally-symmetric spaces
([LoNiFi]) since they generalize the defining property of classical symmetric
spaces.

Some examples of generalised Wallach spaces are the Wallach spaces
$\operatorname{SU}(3)/T_{\mbox{max}}$,
$\operatorname{Sp}(3)/(\operatorname{SU}(2)\times\operatorname{SU}(2)\times\operatorname{SU}(2))$
and $F_{4}/\operatorname{Spin}(8)$, the generalized flag manifolds
$\operatorname{SU}(l+m+n)/\operatorname{S}(\operatorname{U}(l)\times\operatorname{U}(m)\times\operatorname{U}(n))$,
$\operatorname{SO}(2l)/(\operatorname{U}(1)\times\operatorname{U}(l-1))$,
$E_{6}/(\operatorname{U}(1)\times\operatorname{U}(1)\times\operatorname{Spin}(8))$,
the homogeneous spaces
$\operatorname{SO}(l+m+n)/(\operatorname{SO}(l)\times\operatorname{SO}(m)\times\operatorname{SO}(n))$,
$\operatorname{Sp}(l+m+n)/(\operatorname{Sp}(l)\times\operatorname{Sp}(m)\times\operatorname{Sp}(n))$,
and (as special case) the Stiefel manifolds
$\operatorname{SO}(n+2)/\operatorname{SO}(2)$. As a consequence of their
definition if follows that
$[\mathfrak{m}_{i},\mathfrak{m}_{j}]\subset\mathfrak{m}_{k}$, for $i,j,k$
distinct. Despite their simple description, a complete classification of
generalized Wallach spaces was given only recently by Yu.G. Nikonorov in [Ni2]
and Z. Chen, Y. Kang and K. Liang in [ChKaLi].

Invariant Einstein metrics on generalized Wallach spaces were been originally
studied in [Ni1]. In that paper it is proved that every generalized Wallach
space admits at least one invariant Einstein metric, and in [LoNiFi] that it
admits at most four invariant Einstein metrics (up to a homothety). A good
survey about them can be found in [ChKaLi] and [ChNi].

The most subtle example is the homogeneous space
$\operatorname{SO}(l+m+n)/(\operatorname{SO}(l)\times\operatorname{SO}(m)\times\operatorname{SO}(n))$,
and in [ChNi] there is serious progress towards the classification of Einstein
metrics in this space. It turns out that the number of invariant Einstein
metrics on
$\operatorname{SO}(l+m+n)/(\operatorname{SO}(l)\times\operatorname{SO}(m)\times\operatorname{SO}(n))$
could be estimated by using special properties of the normalized Ricci flow on
generalized Wallach spaces (cf. [AbiArNiSi], [ChNi]). The main results in
[ChNi] are the following:

######  Theorem 5.1.

Assume that $l\geq m\geq n\geq 1$ and $m\geq 2$. Then the number of invariant
Einstein metrics on the space
$G/K=\operatorname{SO}(l+m+n)/(\operatorname{SO}(l)\times\operatorname{SO}(m)\times\operatorname{SO}(n))$
is four for $n>\sqrt{2l+2m-4}$, and two for $n<\sqrt{l+m}$ (up to a
homothety).

######  Theorem 5.2.

Let $q=2,3$ or $4$. Then there are infinitely many homogeneous spaces
$\operatorname{SO}(l+m+n)/(\operatorname{SO}(l)\times\operatorname{SO}(m)\times\operatorname{SO}(n))$
that admit exactly $q$ invariant Einstein metrics up to a homothety.

##  6\. Some open problems

There is no doubt that there has been a lot of progress towards the
classification of invariant Einstein metrics on generalized flag manifolds. It
seems that Ziller’s finiteness conjecture is in fact true in this case, that
is the number of Einstein metrics (up to isometry) is finite. However, as the
number of isotropy summands increases then determining the total number of
Einstein metrics is getting a difficult task, especially for flag manifolds
determined by classical Lie groups. The Gröbner bases techniques used can
contribute a lot, but seem to be unable to shed more light in the set of
invariant Einstein metrics. It seems that new tools and techniques have to be
used. The deep works [Bö] by C. Böhm and [BöWaZi] by C. Böhm, M. Wang and W.
Ziller constitute an alternative point of view. Also, the recent works by M.M.
Graev [Gr1], [Gr2] and [Gr3] are important contributions towards the
understanding of the number of complex Einstein metrics on flag manifolds.

Concerning invariant Einstein metrics on Stiefel manifolds (or other
homogeneous spaces where the isotropy representation contains equivalent
summnands) the picture is more foggy. The set of invariant metrics is quite
vast in this case, so searching for invariant Einstein metrics is not an easy
matter. One has to search in certain subsets of the set of invariant metrics
(“symmetry assumption”) so that computations of the Ricci tensor get slightly
simpler, and then hope to find Einstein metrics in these subsets. Even though
there is considerable success by such approach, there is no particular
evidence on whether the finiteness conjecture is true or not. There might even
exist one-parameter families of invariant Einstein metrics.

Finally some open problems related to the generalized Wallach space
$\operatorname{SO}(l+m+n)/(\operatorname{SO}(l)\times\operatorname{SO}(m)\times\operatorname{SO}(n))$
are listed in [ChNi].

Acknowledgements. The work was supported by Grant $\\#E.037$ from the Research
Committee of the University of Patras (Programme K. Karatheodori). I would
like to take the opportunity to express my gratitude to Professor M. Guest for
directing me in various areas of geometry including homogeneous geometry.
Through the years I had (and still have) beneficial collaborations with D.V.
Alekseevsky, Yu. G. Nikonorov, Y. Sakane, I. Chrysikos, M. Statha and N.P.
Souris. It is a pleasure to still have fruitful discussions with them.

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