# Geodesic orbit metrics in a class of homogeneous bundles over real and
complex Stiefel manifolds

Andreas Arvanitoyeorgos∗, Nikolaos Panagiotis Souris and Marina Statha
University of Patras, Department of Mathematics, GR-26500 Rion, Greece
<EMAIL_ADDRESS>University of Patras, Department of Mathematics,
GR-26500 Rion, Greece<EMAIL_ADDRESS>University of Patras, Department of
Mathematics, GR-26500 Rion and University of Thessaly, Department of
Mathematics, GR-35100 Lamia, Greece<EMAIL_ADDRESS>
###### Abstract.

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous
Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter
subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit
metric. We study the geodesic orbit spaces of the form $(G/H,g)$, such that
$G$ is one of the compact classical Lie groups $\operatorname{SO}(n)$, $U(n)$,
and $H$ is a diagonally embedded product $H_{1}\times\cdots\times H_{s}$,
where $H_{j}$ is of the same type as $G$. This class includes spheres, Stiefel
manifolds, Grassmann manifolds and real flag manifolds. The present work is a
contribution to the study of g.o. spaces $(G/H,g)$ with $H$ semisimple.

2020 Mathematics Subject Classification. Primary 53C25; Secondary 53C30.

Keywords: geodesic orbit space; geodesic orbit metric; Stiefel manifold; real
flag manifold; Grassmann manifold

##  1\. Introduction

Geodesic orbit spaces $(M=G/H,g)$ are defined by the simple property that any
geodesic $\gamma$ has the form

$\gamma(t)=\exp(tX)\cdot o,$

where $\exp$ is the exponential map on $G$, $o=\gamma(0)$ is a point in $M$
and $\cdot$ denotes the action of $G$ on $M$. These spaces were initially
considered in [20], and up to this day they have been extensively studied
within various geometric contexts, including the Riemannian ([18]), pseudo-
Riemannian ([11]), Finsler ([31]) and affine ([16]) context. The
classification of g.o. spaces remains an open problem, whereas several partial
classifications have been obtained ([2], [3], [13], [14], [15], [17], [25],
[28] to name a few).

There are diverse examples of g.o. spaces, including the classes of symmetric
spaces, weakly symmetric spaces ([10], [30]), isotropy irreducible spaces
([29]), $\delta$-homogeneous spaces ([7]) and Clifford-Wolf homogeneous spaces
([8]). The most important subclass of g.o. spaces are the _naturally reductive
spaces_ , whose complete description is also open (see the recent low-
dimensional classifications [1], [27]). Another related topic of recent
interest is the study of Einstein metrics that are not g.o. metrics ([12],
[23]). For a review about g.o. spaces we refer to [5] and in the introduction
of the article [22]. We also point out the recently published book [9].

Determining the g.o. metrics among the $G$-invariant metrics on a space $G/H$
presents some challenges. The main challenge lies in the fact that the space
of $G$-invariant metrics may have complicated structure, depending on whether
the _isotropy representation_ of $H$ on the tangent space $T_{o}(G/H)$
contains pairwise equivalent submodules. To remedy this obstruction, various
simplification results for g.o. metrics have been established (e.g. [22],
[24]). A general observation is that the existence and the form of the g.o.
metrics on $G/H$ depends to a large extent on the structure of the tangent
space $T_{o}(G/H)$ induced from the isotropy representation and on the Lie
algebraic relations between the corresponding submodules (e.g. [15]).

When $G$ is compact semisimple, the classification of the g.o. spaces
$(G/H,g)$ with $H$ abelian and $H$ simple has been obtained in the works [25]
and [15] respectively. On the other hand, the classification of compact g.o.
spaces $(G/H,g)$ with $H$ semisimple remains open, while no general results
are known for this case. As a first step towards this direction, in this paper
we study the g.o. metrics on a general family of spaces $G/H$ with $H$
semisimple, such that the isotropy representation of all of its members has a
similar description.

In particular, we consider spaces $G/H$ where $G$ is a compact classical Lie
group and $H$ is a diagonally embedded product of Lie groups of the same type
as $G$. More specifically, we study the spaces
$\operatorname{SO}(n)/\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$
and
$\operatorname{U}(n)/\operatorname{U}(n_{1})\times\cdots\times\operatorname{U}(n_{s})$
with $0<n_{1}+\cdots+n_{s}\leq n$. This class properly includes the spheres
$\operatorname{SO}(n)/\operatorname{SO}(n-1)$ and
$\operatorname{U}(n)/\operatorname{U}(n-1)$, the Stiefel manifolds
$\operatorname{SO}(n)/\operatorname{SO}(n-k)$ and
$\operatorname{U}(n)/\operatorname{U}(n-k)$, the Grassmann manifolds
$\operatorname{SO}(n)/\operatorname{SO}(k)\times\operatorname{SO}(n-k)$,
$\operatorname{U}(n)/\operatorname{U}(k)\times\operatorname{U}(n-k)$ as well
as the real flag manifolds
$\operatorname{SO}(n)/\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$
and
$\operatorname{U}(n)/\operatorname{U}(n_{1})\times\cdots\times\operatorname{U}(n_{s})$
with $n_{1}+\cdots+n_{s}=n$. If $n_{1}+\cdots+n_{s}<n$, each of these spaces
can be viewed as a total space over a Stiefel manifold, with the fiber being a
real flag manifold, e.g.

$\operatorname{SO}(m)/\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})\rightarrow\operatorname{SO}(n)/\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})\rightarrow\operatorname{SO}(n)/\operatorname{SO}(m),$

with $m=n_{1}+\cdots+n_{s}$. The first main result is the following.

######  Theorem 1.1.

Let $G/H$ be the space
$\operatorname{SO}(n)/\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$,
where $0<n_{1}+\cdots+n_{s}\leq n$, and $n_{j}>1,j=1,\dots,s$. A $G$-invariant
Riemannian metric on $G/H$ is geodesic orbit if and only if it is a normal
metric, i.e. it is induced from an $\operatorname{Ad}$-invariant inner product
on the Lie algebra $\mathfrak{so}(n)$ of $\operatorname{SO}(n)$.

######  Remark 1.2.

We remark that if $n\neq 4$ then $\mathfrak{so}(n)$ is simple, and thus any
$\operatorname{Ad}$-invariant inner product is homothetic to the negative of
the Killing form $B(X,Y)=(n-2)\operatorname{Trace}(XY)$. If $n=4$ then
$\mathfrak{so}(n)\equiv\mathfrak{so}(3)\oplus\mathfrak{so}(3)$, and thus any
$\operatorname{Ad}$-invariant inner product is homothetic to the negative of
the one-parameter family $B_{1}+\lambda B_{2}$, $\lambda>0$, where $B_{1}$
denotes the Killing form of the first simple factor $\mathfrak{so}(3)$ and
$B_{2}$ denotes the Killing form of the second simple factor
$\mathfrak{so}(3)$.

As a result of Theorem 1.1 and Proposition 3.9, we obtain the following.

######  Corollary 1.3.

Let $G/H$ be one of the spaces
$\operatorname{O}(n)/\operatorname{O}(n_{1})\times\cdots\times\operatorname{O}(n_{s})$
or
$\operatorname{SO}(n)/\operatorname{S}(\operatorname{O}(n_{1})\times\cdots\times\operatorname{O}(n_{s}))$,
where $0<n_{1}+\cdots+n_{s}\leq n$, $n_{j}>1$. A $G$-invariant Riemannian
metric on $G/H$ is geodesic orbit if and only if it is normal.

The second main result is the following.

######  Theorem 1.4.

Let $G/H$ be the space
$\operatorname{U}(n)/\operatorname{U}(n_{1})\times\cdots\times\operatorname{U}(n_{s})$,
where $n_{1}+\cdots+n_{s}\leq n$, and let $N_{G}(H)$ be the normalizer of $H$
in $G$. If $n_{1}+\dots+n_{s}=n$, then a $G$-invariant Riemannian metric on
$G/H$ is geodesic orbit if and only if it is the normal metric induced from
the $\operatorname{Ad}$-invariant inner product
$B(X,Y)=-\operatorname{Trace}(XY)$ in $\mathfrak{u}(n)$. If
$n_{1}+\cdots+n_{s}<n$, then a $G$-invariant Riemannian metric $g$ on $G/H$ is
geodesic orbit if and only if $g=g_{\mu}$, $\mu>0$, where $g_{\mu}$ denotes a
one-parameter family of deformations of the normal metric induced from the
inner product $B$, along the center of the group $N_{G}(H)/H$.

The case
$G/H=\operatorname{Sp}(n)/\operatorname{Sp}(n_{1})\times\cdots\times\operatorname{Sp}(n_{s})$
has been treated in [6].

We note that the metrics in Theorem 1.4 generalize the g.o. metrics on the
Berger spheres $\operatorname{U}(n)/\operatorname{U}(n-1)$ ([21]) and the g.o.
metrics on the complex Stiefel manifolds
$\operatorname{U}(n)/\operatorname{U}(n-k)$ ([24]). We also note that the g.o.
metrics on the related class of real flag manifolds were recently studied in
[19]. Among other results, it is shown in [19] that every g.o. metric on the
real flag manifold
$\operatorname{SO}(n)/\operatorname{S}(\operatorname{O}(n_{1})\times\cdots\times\operatorname{O}(n_{s}))$,
$n_{1}+\cdots+n_{s}=n$, is normal, which is a special case of Corollary 1.3.

The paper is structured as follows: In Sections 2 and 3, some preliminary
facts for homogeneous spaces and g.o. spaces are given respectively. Theorems
1.1 and 1.4 are proved in Sections 4.2 and 5.2 respectively. To this end, we
firstly compute the isotropy representation in terms of a suitable basis for
each of the spaces (Sections 4.1 and 5.1 respectively) and then we apply
simplification results from [22] and [24] in order to complete the proofs
(Sections 4.2 and 5.2 respectively).

Acknowledgments. This research is co-financed by Greece and the European Union
(European Social Fund- ESF) through the Operational Programme “Human Resources
Development, Education and Lifelong Learning 2014-2020” in the context of the
project “Geodesic orbit metrics on homogeneous spaces of classical Lie groups”
(MIS 5047124). Remark 4.1 emerged through a discussion of the first author
with Professor McKenzie Wang and Remark 3.5 was kindly pointed out to us by
the referee. The authors acknowledge both of them for their clarifications.

##  2\. Invariant metrics on homogeneous spaces

Let $G/H$ be a homogeneous space with origin $o=eH$ and assume that $G$ is
compact. Let $\mathfrak{g},\mathfrak{h}$ be the Lie algebras of $G,H$
respectively. Moreover, let
$\operatorname{Ad}:G\rightarrow\operatorname{Aut}(\mathfrak{g})$ and
$\operatorname{ad}:\mathfrak{g}\rightarrow\operatorname{End}(\mathfrak{g})$ be
the adjoint representations of $G$ and $\mathfrak{g}$ respectively, where
$\operatorname{ad}(X)Y=[X,Y]$. Since $G$ is compact, there exists an
$\operatorname{Ad}$-invariant (and hence $\operatorname{ad}$ skew-symmetric)
inner product $B$ on $\mathfrak{g}$, which we henceforth fix. In turn, we have
a $B$-orthogonal _reductive decomposition_

$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m},$ (1)

where the subspace $\mathfrak{m}$ is $\operatorname{Ad}(H)$-invariant (and
$\operatorname{ad}(\mathfrak{h})$-invariant) and is naturally identified with
the tangent space of $G/H$ at the origin.

A Riemannian metric $g$ on $G/H$ is called $G$-invariant if for any $x\in G$,
the left translations $\tau_{x}:G/H\rightarrow G/H$, $pH\mapsto(xp)H$, are
isometries of $(G/H,g)$. The $G$-invariant metrics are in one to one
correspondence with $\operatorname{Ad}(H)$-invariant inner products $\langle\
,\ \rangle$ on $\mathfrak{m}$. Moreover, any such product corresponds to a
unique endomorphism $A:\mathfrak{m}\rightarrow\mathfrak{m}$, called the
_corresponding metric endomorphism_ , that satisfies

$\langle X,Y\rangle=B(AX,Y)\ \ \makebox{for all}\ \ X,Y\in\mathfrak{m}.$ (2)

It follows from Equation (2) that the metric endomorphism $A$ is symmetric
with respect $B$, positive definite and $\operatorname{Ad}(H)$-equivariant,
that is $(\operatorname{Ad}(h)\circ A)(X)=(A\circ\operatorname{Ad}(h))(X)$ for
all $h\in H$ and $X\in\mathfrak{m}$. Conversely, any endomorphism on
$\mathfrak{m}$ with the above properties determines a unique $G$-invariant
metric on $G/H$.

Since $A$ is diagonalizable, there exists a decomposition
$\mathfrak{m}=\bigoplus_{j=1}^{l}\mathfrak{m}_{\lambda_{j}}$ into eigenspaces
$\mathfrak{m}_{\lambda_{j}}$ of $A$ corresponding to distinct eigenvalues
$\lambda_{j}$. Each eigenspace $\mathfrak{m}_{\lambda_{j}}$ is
$\operatorname{Ad}(H)$-invariant. When an $\operatorname{Ad}$-invariant inner
product $B$ and a $B$-orthogonal reductive decomposition (1) have been fixed,
we will make no distinction between a $G$-invariant metric $g$ and its
corresponding metric endomorphism $A$.

The form of the $G$-invariant metrics on $G/H$ depends on the _isotropy
representation_
$\operatorname{Ad}^{G/H}:H\rightarrow\operatorname{Gl}(\mathfrak{m})$, defined
by $\operatorname{Ad}^{G/H}(h)X:=(d\tau_{h})_{o}(X)$, $h\in H$,
$X\in\mathfrak{m}$. We consider a $B$-orthogonal decomposition

$\mathfrak{m}=\mathfrak{m}_{1}\oplus\cdots\oplus\mathfrak{m}_{s},$ (3)

into $\operatorname{Ad}^{G/H}$-invariant and irreducible submodules. We recall
that two submodules $\mathfrak{m}_{i}$ and $\mathfrak{m}_{j}$ are equivalent
if there exists an $\operatorname{Ad}^{G/H}$-equivariant isomorphism
$\phi:\mathfrak{m}_{i}\rightarrow\mathfrak{m}_{j}$. The simplest case occurs
when all the submodules $\mathfrak{m}_{i}$ are pairwise inequivalent. Then any
$G$-invariant metric $A$ on $G/H$ has a diagonal expression with respect to
decomposition (3). In particular,
$\left.A\right|_{\mathfrak{m}_{j}}=\lambda_{j}\operatorname{Id}$,
$j=1,\dots,s$.

The next proposition is useful for computing the isotropy representation of a
reductive homogeneous space.

######  Proposition 2.1.

([4]) Let $G/H$ be a reductive homogeneous space and let
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ be a reductive decomposition of
$\mathfrak{g}$. Let $h\in H$, $X\in\mathfrak{h}$ and $Y\in\mathfrak{m}$. Then
the adjoint representation of $G$ decomposes as
$\operatorname{Ad}^{G}(h)(X+Y)=\operatorname{Ad}^{G}(h)X+\operatorname{Ad}^{G/H}(h)Y$
that is, the restriction $\operatorname{Ad}^{G}\big{|}_{H}$ splits into the
sum $\operatorname{Ad}^{H}\oplus\operatorname{Ad}^{G/H}$. We denote by $\chi$
the representation $\operatorname{Ad}^{G/H}$.

The following lemma provides a simple condition for proving that two
$\operatorname{Ad}^{G/H}$-submodules are inequivalent.

######  Lemma 2.2.

Let $G/H$ be a homogeneous space with reductive decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ and let
$\mathfrak{m}_{i},\mathfrak{m}_{j}\subseteq\mathfrak{m}$ be submodules of the
isotropy representation $\operatorname{Ad}^{G/H}$. Assume that for any pair of
non-zero vectors $X\in\mathfrak{m}_{i}$, $Y\in\mathfrak{m}_{j}$, there exists
a vector $a\in\mathfrak{h}$ such that $[a,X]=0$ and $[a,Y]\neq 0$. Then the
submodules $\mathfrak{m}_{i},\mathfrak{m}_{j}$ are
$\operatorname{Ad}^{G/H}$-inequivalent.

###### Proof.

If $\mathfrak{m}_{i},\mathfrak{m}_{j}$ were equivalent, then there exists an
$\operatorname{Ad}^{G/H}$-equivariant isomorphism
$\phi:\mathfrak{m}_{i}\rightarrow\mathfrak{m}_{j}$. Let $X$ be a non-zero
vector in $\mathfrak{m}_{i}$ and set $Y:=\phi(X)\in\mathfrak{m}_{j}$. The
$\operatorname{Ad}^{G/H}$-equivariance of $\phi$ implies that $\phi$ is
$\operatorname{ad}_{\mathfrak{h}}$-equivariant, and hence, $\phi([a,X])=[a,Y]$
for any $a\in\mathfrak{h}$. However, $\phi$ is an isomorphism, therefore,
$[a,X]$ is non-zero if and only if $[a,Y]$ is non-zero, which contradicts the
hypothesis of the lemma. Hence, the submodules
$\mathfrak{m}_{i},\mathfrak{m}_{j}$ are inequivalent.∎

######  Remark 2.3.

For any two $\operatorname{Ad}^{G/H}$-submodules
$\mathfrak{m}_{1},\mathfrak{m}_{2}$, we denote by
$[\mathfrak{m}_{1},\mathfrak{m}_{2}]$ the space generated by the vectors
$[X_{1},X_{2}]$ where $X_{1}\in\mathfrak{m}_{1}$ and
$X_{2}\in\mathfrak{m}_{2}$. Similarly, denote by
$[\mathfrak{h},\mathfrak{m}_{1}]$ the space generated by the vectors
$[a,X_{1}]$ where $a\in\mathfrak{h}$ and $X_{1}\in\mathfrak{m}_{1}$. If
$\mathfrak{m}_{1},\mathfrak{m}_{2}$ are $B$-orthogonal then
$[\mathfrak{m}_{1},\mathfrak{m}_{2}]\subseteq\mathfrak{m}$. Indeed,
$[\mathfrak{m}_{1},\mathfrak{m}_{2}]$ is $B$-orthogonal to $\mathfrak{h}$
because $B([\mathfrak{m}_{1},\mathfrak{m}_{2}],\mathfrak{h})\subseteq
B(\mathfrak{m}_{1},[\mathfrak{m}_{2},\mathfrak{h}])\subseteq
B(\mathfrak{m}_{1},\mathfrak{m}_{2})=\\{0\\}$. Moreover,
$[\mathfrak{m}_{1},\mathfrak{m}_{2}]$ is also an
$\operatorname{Ad}^{G/H}$-submodule of $\mathfrak{m}$ and
$[\mathfrak{h},\mathfrak{m}_{1}]$ is an $\operatorname{Ad}^{G/H}$-submodule of
$\mathfrak{m}_{1}$.

##  3\. Properties of geodesic orbit spaces

######  Definition 3.1.

A $G$-invariant metric $g$ on $G/H$ is called a geodesic orbit metric (g.o.
metric) if any geodesic of $(G/H,g)$ through $o$ is an orbit of a one
parameter subgroup of $G$. Equivalently, $g$ is a geodesic orbit metric if for
any geodesic $\gamma$ of $(G/H,g)$ though $o$ there exists a non-zero vector
$X\in\mathfrak{g}$ such that $\gamma(t)=\exp(tX)\cdot o$, $t\in\mathbb{R}$.
The space $(G/H,g)$ is called a geodesic orbit space (g.o. space).

Let $G/H$ be a homogeneous space with $G$ compact. We fix an
$\operatorname{Ad}$-invariant inner product $B$ on $\mathfrak{g}$ and consider
the $B$-orthogonal reductive decomposition (1). Moreover, identify each
$G$-invariant metric on $G/H$ with the corresponding metric endomorphism
$A:\mathfrak{m}\rightarrow\mathfrak{m}$ satisfying Equation (2). We have the
following condition.

######  Proposition 3.2.

_([2], [24])_ The metric $A$ on $G/H$ is geodesic orbit if and only if for any
vector $X\in\mathfrak{m}$ there exists a vector $a\in\mathfrak{h}$ such that

$[a+X,AX]=0.$ (4)

The following result, which we will call the _normalizer lemma_ , can be used
to simplify the necessary form of the g.o. metrics on $G/H$ by using the
normalizer $N_{G}(H^{0})$.

######  Lemma 3.3.

_([22])_ The inner product $\langle\ ,\ \rangle$ in (2), generating the metric
of a geodesic orbit Riemannian space $(G/H,g)$, is not only
$\operatorname{Ad}(H)$-invariant but also
$\operatorname{Ad}(N_{G}(H^{0}))$-invariant, where $N_{G}(H^{0})$ is the
normalizer of the identity component $H^{0}$ of the group $H$ in $G$.

As a result of the normalizer lemma, the metric endomorphism $A$ of a g.o.
metric on $G/H$ is $\operatorname{Ad}(N_{G}(H^{0}))$-equivariant. We will now
state a complementary result to the normalizer lemma for compact spaces, that
characterizes the restriction of a g.o. metric to the compact Lie group
$N_{G}(H^{0})/H^{0}$.

######  Lemma 3.4.

Let $G$ be a compact Lie group and let $(G/H,g)$ be a geodesic orbit space
with the $B$-orthogonal reductive decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$, where $B$ is an
$\operatorname{Ad}$-invariant inner product on $\mathfrak{g}$. Let
$A:\mathfrak{m}\rightarrow\mathfrak{m}$ be the corresponding metric
endomorphism of $g$, and let $\mathfrak{n}\subseteq\mathfrak{m}$ be the Lie
algebra of the compact Lie group $N_{G}(H^{0})/H^{0}$. Then the restriction of
$A$ to $\mathfrak{n}$ defines a bi-invariant metric on $N_{G}(H^{0})/H^{0}$.

###### Proof.

We denote by $\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h})\subset\mathfrak{g}$
the Lie algebra of $N_{G}(H^{0})$. We have a $B$-orthogonal decomposition
$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{h}\oplus\mathfrak{n}$,
where $\mathfrak{n}$ coincides with the Lie algebra of $N_{G}(H^{0})/H^{0}$.
Moreover, we have a $B$-orthogonal decomposition
$\mathfrak{m}=\mathfrak{n}\oplus\mathfrak{p}$, where $\mathfrak{p}$ coincides
with the tangent space of $G/N_{G}(H^{0})$ at the origin. By the normalizer
lemma, the restriction of $A$ on $\mathfrak{p}$ defines an invariant metric on
$G/N_{G}(H^{0})$, and thus $A\mathfrak{p}\subseteq\mathfrak{p}$. By taking
into account the symmetry of $A$ with respect to the product $B$, we deduce
that $B(A\mathfrak{n},\mathfrak{p})=B(\mathfrak{n},A\mathfrak{p})\subseteq
B(\mathfrak{n},\mathfrak{p})=\\{0\\}$. Hence, the image $A\mathfrak{n}$ is
$B$-orthogonal to $\mathfrak{p}$ which, along with decomposition
$\mathfrak{m}=\mathfrak{n}\oplus\mathfrak{p}$, yields
$A\mathfrak{n}\subseteq\mathfrak{n}$. Therefore, the restriction
$\left.A\right|_{\mathfrak{n}}:\mathfrak{n}\rightarrow\mathfrak{n}$ defines a
left-invariant metric on $N_{G}(H^{0})/H^{0}$. Since $A$ is a g.o. metric on
$G/H$, Proposition 3.2 implies that for any $X\in\mathfrak{n}$ there exists a
vector $a\in\mathfrak{h}$ such that
$0=[a+X,AX]=[a+X,\left.A\right|_{\mathfrak{n}}X]$. Therefore, by the same
proposition, $\left.A\right|_{\mathfrak{n}}$ defines a g.o. metric on
$N_{G}(H^{0})/H^{0}$. On the other hand, any left-invariant g.o. metric on a
Lie group is necessarily bi-invariant ([3]), and hence
$\left.A\right|_{\mathfrak{n}}$ is a bi-invariant metric on
$N_{G}(H^{0})/H^{0}$.∎

######  Remark 3.5.

As an alternative to the above proof, it was observed by the referee that
Lemma 3.4 follows easily from Lemma 3.3, if we take into account that any
$\operatorname{Ad}(N_{G}(H^{0}))$-invariant Riemannian metric on the Lie group
$N_{G}(H^{0})/H^{0}$ is generated by a suitable bi-invariant Riemannian metric
on $N_{G}(H^{0})/H^{0}$.

######  Remark 3.6.

Let $\mathfrak{p}\subset\mathfrak{m}$ be the tangent space of
$G/N_{G}(H^{0})$, let $\mathfrak{n}\subset\mathfrak{m}$ be the Lie algebra of
$N_{G}(H^{0})/H^{0}$, and consider the decomposition
$\mathfrak{m}=\mathfrak{n}\oplus\mathfrak{p}$. By combining Lemmas 3.3 and
3.4, we conclude that the metric endomorphism $A$ corresponding to a g.o.
metric on $G/H$ has the block-diagonal form

$A=\begin{pmatrix}\left.A\right|_{\mathfrak{n}}&0\\\
0&\left.A\right|_{\mathfrak{p}}\end{pmatrix}.$

The next lemma describes the invariant g.o. metrics on compact Lie groups, in
terms of their metric endomorphism with respect to an
$\operatorname{Ad}$-invariant inner product $B$. Recall that the Lie algebra
$\mathfrak{g}$ of a compact Lie group $G$ has the (Lie algebra) direct sum
decomposition
$\mathfrak{g}=\mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{k}\oplus\mathfrak{z}$,
where $\mathfrak{g}_{j}$ are the simple ideals of $\mathfrak{g}$ and
$\mathfrak{z}$ is its center.

######  Lemma 3.7.

_([3], [24])_ Let $G$ be a compact Lie group with Lie algebra
$\mathfrak{g}=\mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{k}\oplus\mathfrak{z}$.
A left invariant metric $A$ on $G$ is a g.o. metric if and only if it is bi-
invariant. In particular, $A$ is a g.o. metric if and only if

$A=\begin{pmatrix}\lambda_{1}\left.\operatorname{Id}\right|_{\mathfrak{g}_{1}}&0&\cdots&0\\\
\vdots&\ddots&\cdots&\vdots\\\
0&\cdots&\lambda_{k}\left.\operatorname{Id}\right|_{\mathfrak{g}_{k}}&0\\\
0&\cdots&0&\left.A\right|_{\mathfrak{z}}\end{pmatrix},\ \lambda_{j}>0.$

We set some notation. For a subspace $W$ of a vector space $V$ we write
$V=W\oplus W^{\bot}$ with respect to some inner product $B$ on $V$. Then, for
$v\in V$ it is $v=w+w^{\prime}$, where $w\in W$ and $w^{\prime}\in W^{\bot}$.
We say that $v$ has _non zero projection on $W$_ if $w\neq 0$. Moreover, we
say that a subset $S$ of $V$ has non zero projection on $W$ if there exists a
vector $v\in S$ that has non zero projection on $W$.

The following lemma is useful, since it will enable us to equate some of the
eigenvalues of a g.o. metric.

######  Lemma 3.8.

_([24])_ Let $(G/H,g)$ be a g.o. space with $G$ compact and with corresponding
metric endomorphism $A$ with respect to an $\operatorname{Ad}$-invariant inner
product $B$. Let $\mathfrak{m}$ be the $B$-orthogonal complement of
$\mathfrak{h}$ in $\mathfrak{g}$.
1. Assume that $\mathfrak{m}_{1},\mathfrak{m}_{2}$ are $\operatorname{ad}(\mathfrak{h})$-invariant, pairwise $B$-orthogonal subspaces of $\mathfrak{m}$ such that $[\mathfrak{m}_{1},\mathfrak{m}_{2}]$ has non-zero projection on ${(\mathfrak{m}_{1}\oplus\mathfrak{m}_{2})^{\bot}}$. Let $\lambda_{1},\lambda_{2}$ be eigenvalues of $A$ such that $\left.A\right|_{\mathfrak{m}_{i}}=\lambda_{i}\operatorname{Id}$, $i=1,2$. Then $\lambda_{1}=\lambda_{2}$.
2. Assume that $\mathfrak{m}_{1},\mathfrak{m}_{2},\mathfrak{m}_{3}$ are $\operatorname{ad}(\mathfrak{h})$-invariant, pairwise $B$-orthogonal subspaces of $\mathfrak{m}$ such that $[\mathfrak{m}_{1},\mathfrak{m}_{2}]$ has non-zero projection on $\mathfrak{m}_{3}$. Let $\lambda_{1},\lambda_{2},\lambda_{3}$ be eigenvalues of $A$ such that $\left.A\right|_{\mathfrak{m}_{i}}=\lambda_{i}\operatorname{Id}$, $i=1,2,3$. Then $\lambda_{1}=\lambda_{2}=\lambda_{3}$.

Finally, the following result will be used in relating g.o. metrics on a space
$G/H$ to g.o. metrics on its universal covering space
$\widetilde{G}/\widetilde{H}$ (see also [26]). Recall that a $G$-invariant
metric is called standard if it is induced by the negative of the Killing form
on $\mathfrak{g}$.

######  Proposition 3.9.

Let $G/H$, $\widetilde{G}/\widetilde{H}$ be homogeneous spaces with $G$
compact and $\widetilde{H}$ connected, such that the Lie algebras of $G$ and
$\widetilde{G}$ coincide and the Lie algebras of $H$ and $\widetilde{H}$
coincide. If every ($\widetilde{G}$-invariant) g.o. metric on
$\widetilde{G}/\widetilde{H}$ is normal (resp. standard), then every
($G$-invariant) g.o. metric on $G/H$ is also normal (resp. standard).
Moreover, the converse is true if $H$ is connected.

###### Proof.

Let $\mathfrak{g}$ denote the Lie algebra of the groups $G$ and
$\widetilde{G}$, and let $\mathfrak{h}$ denote the Lie algebra of the groups
$H$ and $\widetilde{H}$. Let $B$ be an $\operatorname{Ad}$-invariant inner
product on $\mathfrak{g}$ and consider the $B$-orthogonal decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}_{B}$. Then $\mathfrak{m}_{B}$ can
be identified with the tangent spaces $T_{o}(G/H)$ and
$T_{o}(\widetilde{G}/\widetilde{H})$. Let $g$ be a $G$-invariant g.o. metric
on $G/H$. We will prove that $g$ is normal. The proof will be completed in
three steps.

Step 1. _The $G$-invariant metric $g$ on $G/H$ induces a
$\widetilde{G}$-invariant metric $\widetilde{g}$ on
$\widetilde{G}/\widetilde{H}$_. Indeed, let
$A_{B}:\mathfrak{m}_{B}\rightarrow\mathfrak{m}_{B}$ be the corresponding
metric endomorphism of $g$. The endomorphism $A_{B}$ is
$\operatorname{Ad}_{H}$-invariant and hence
$\operatorname{ad}_{\mathfrak{h}}$-equivariant. Given that $\widetilde{H}$ is
connected, the $\operatorname{ad}_{\mathfrak{h}}$-equivariance of $A_{B}$
yields its $\operatorname{Ad}_{\widetilde{H}}$-equivariance. Therefore,
$A_{B}$ defines a $\widetilde{G}$-invariant metric $\widetilde{g}$ on
$\widetilde{G}/\widetilde{H}$.

Step 2. _The metric $\widetilde{g}$ is also a g.o. metric on
$\widetilde{G}/\widetilde{H}$._ Indeed, since $g$ is a go. metric then the
corresponding metric endomorphism $A_{B}$ satisfies Proposition 3.2. Since
$A_{B}$ is also the metric endomorphism of $\widetilde{g}$, Proposition 3.2
implies that $\widetilde{g}$ is a g.o. metric on
$\widetilde{G}/\widetilde{H}$.

Step 3. _The metric $g$ is normal._ Indeed, since $\widetilde{g}$ is a g.o.
metric on $\widetilde{G}/\widetilde{H}$, by hypothesis it is also normal.
Hence, there exists an $\operatorname{Ad}$-invariant inner product
$B^{\prime}$ on $\mathfrak{g}$ and a $B^{\prime}$-orthogonal decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}_{B^{\prime}}$ such that the
corresponding metric endomorphism
$A_{B^{\prime}}:\mathfrak{m}_{B^{\prime}}\rightarrow\mathfrak{m}_{B^{\prime}}$
of $\widetilde{g}$ satisfies $A_{B^{\prime}}=\lambda\operatorname{Id}$,
$\lambda>0$. Since $A_{B^{\prime}}$ coincides with the metric endomorphism of
$g$, we conclude that $g$ is also normal. ∎

######  Corollary 3.10.

If every ($\widetilde{G}$-invariant) g.o. metric on the universal cover
$\widetilde{G}/\widetilde{H}$ of $G/H$ is normal (resp. standard), then every
($G$-invariant) g.o. metric on $G/H$ is also normal (resp. standard).

##  4\. The space
$M=G/H=\operatorname{SO}(n)/\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$,
$n_{1}+\cdots+n_{s}\leq n$, $n_{j}>1$.

###  4.1. Isotropy representation of
$G/H=\operatorname{SO}(n)/\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$

We set

$n_{0}:=n-(n_{1}+\cdots+n_{s}).$ (5)

We view $H=\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$
embedded diagonally in $\operatorname{SO}(n)$, so that
$H\cong\begin{pmatrix}\operatorname{Id}_{n_{0}}&0\\\ 0&H\end{pmatrix}.$ Hence

$\mathfrak{h}=\begin{pmatrix}0_{n_{0}}&&&0\\\ &\mathfrak{so}(n_{1})&&\\\
&&\ddots&\\\ 0&&&\mathfrak{so}(n_{s})\\\ \end{pmatrix},$

where $0_{n_{0}}$ is the $n_{0}\times n_{0}$ zero matrix. We remark that the
above embedding of $H$ is equivalent (via conjugation in
$\operatorname{SO}(n)$) to any block-diagonal embedding of the factors
$\operatorname{SO}(n_{j})$.

Recall that if $\pi:G\to\operatorname{Aut}(V),$
$\pi^{\prime}:G\to\operatorname{Aut}(W)$ are two representations of $G$, then
for the second exterior power the following identity is valid:
$\wedge^{2}(\pi\oplus\pi^{\prime})=\wedge^{2}\pi\oplus\wedge^{2}\pi^{\prime}\oplus(\pi\otimes\pi^{\prime}).$

Denote by
$\lambda_{n}:\operatorname{SO}(n)\to\operatorname{Aut}(\mathbb{R}^{n})$ the
standard representation of $\operatorname{SO}(n)$. Then the adjoint
representation $\operatorname{Ad}^{\operatorname{SO}(n)}$ of
$\operatorname{SO}(n)$ (and $\operatorname{Ad}^{\operatorname{O}(n)}$ of
$\operatorname{O}(n)$) is equivalent to $\wedge^{2}\lambda_{n}$.

Let
$\sigma_{n_{i}}:\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})\to\operatorname{SO}(n_{i})$
be the projection onto the $i$-factor and
$p_{i}=\lambda_{n_{i}}\circ\sigma_{n_{i}}$ be the standard representation of
$H$, i.e.

$\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})\stackrel{{\scriptstyle\sigma_{n_{i}}}}{{\longrightarrow}}\operatorname{SO}(n_{i})\stackrel{{\scriptstyle\lambda_{n_{i}}}}{{\longrightarrow}}\operatorname{Aut}(\mathbb{R}^{n_{i}}).$

Then we have:

$\displaystyle\operatorname{Ad}^{G}\big{|}_{H}$ $\displaystyle=$
$\displaystyle\wedge^{2}\lambda_{n}\big{|}_{H}=\wedge^{2}(p_{1}\oplus\cdots\oplus
p_{s}\oplus\mathbbm{1}_{n_{0}})=\wedge^{2}p_{{1}}\oplus\wedge^{2}p_{{2}}\oplus\cdots\oplus\wedge^{2}p_{s}\oplus\wedge^{2}\mathbbm{1}_{n_{0}}$
$\displaystyle\oplus\\{(p_{{1}}\otimes
p_{{2}})\oplus\cdots\oplus(p_{{1}}\otimes p_{{s}})\\}\oplus\\{(p_{{2}}\otimes
p_{{3}})\oplus\cdots\oplus(p_{2}\otimes
p_{s})\\}\oplus\cdots\oplus(p_{s-1}\otimes p_{s})$
$\displaystyle\oplus(p_{{1}}\otimes\mathbbm{1}_{n_{0}})\oplus(p_{{2}}\otimes\mathbbm{1}_{n_{0}})\oplus\cdots\oplus(p_{{s}}\otimes\mathbbm{1}_{n_{0}}),$

where $\wedge^{2}\mathbbm{1}_{n_{0}}$ is the sum of $\binom{n_{0}}{2}$ trivial
representations.

The dimension of the representation
$\wedge^{2}p_{{1}}\oplus\wedge^{2}p_{{2}}\oplus\cdots\oplus\wedge^{2}p_{s}$ is
$\binom{n_{1}}{2}+\binom{n_{2}}{2}+\cdots+\binom{n_{s}}{2}$ and is equal to
the dimension of the adjoint representation of
$H=\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$, that
is
$\operatorname{Ad}^{H}=\wedge^{2}p_{{1}}\oplus\wedge^{2}p_{{2}}\oplus\cdots\oplus\wedge^{2}p_{s}$.
Therefore, by Proposition 2.1 the isotropy representation of $G/H$ is given by

$\displaystyle\chi$ $\displaystyle=$
$\displaystyle\wedge^{2}\mathbbm{1}_{n_{0}}\oplus(p_{{1}}\otimes
p_{{2}})\oplus\cdots\oplus(p_{{1}}\otimes p_{{s}})\oplus(p_{{2}}\otimes
p_{{3}})\oplus\cdots\oplus(p_{2}\otimes
p_{s})\oplus\cdots\oplus(p_{s-1}\otimes p_{s})$ (6)
$\displaystyle\oplus(p_{{1}}\otimes\mathbbm{1}_{n_{0}})\oplus(p_{{2}}\otimes\mathbbm{1}_{n_{0}})\oplus\cdots\oplus(p_{{s}}\otimes\mathbbm{1}_{n_{0}}).$

The dimension of each of $p_{i}\otimes p_{j}$ is $n_{i}\times n_{j}$ and each
of $p_{i}\otimes\mathbbm{1}_{n_{0}}$, $i=1,2,\ldots,s$, contains $n_{0}$
equivalent representations of dimension $n_{i}$.

######  Remark 4.1.

Each of the representations $p_{i}\otimes p_{j}$ corresponds to the isotropy
representation of the Grassmannian
$\operatorname{SO}(n_{i}+n_{j})/\operatorname{SO}(n_{i})\times\operatorname{SO}(n_{j})$
and is irreducible unless $n_{i}=n_{j}=2$. If $n_{i}=n_{j}=2$, then the
summand $p_{i}\otimes p_{j}$ in (6) reduces into two 2-dimensional irreducible
non-equivalent representations. This summand corresponds to the isotropy
representation of the Grassmannian
$\operatorname{SO}(4)/\operatorname{SO}(2)\times\operatorname{SO}(2)$.
However, the reducibility does not carry to the Grassmannian
$\operatorname{O}(4)/\operatorname{O}(2)\times\operatorname{O}(2)$, which is
isotropy irreducible. Therefore, even if the isotropy subgroup
$\operatorname{O}(n_{1})\times\cdots\times\operatorname{O}(n_{s})$ contains at
least two $\operatorname{O}(2)$-factors, the isotropy representation of
$\operatorname{O}(n)/\operatorname{O}(n_{1})\times\cdots\times\operatorname{O}(n_{s})$
is still (6).

Expression (6) induces a decomposition of the tangent space $\mathfrak{m}$ of
$G/H$ as

$\mathfrak{m}=\mathfrak{n}_{1}\oplus\cdots\oplus\mathfrak{n}_{\binom{n_{0}}{2}}\bigoplus_{1\leq
i<j\leq s}\mathfrak{m}_{ij}\bigoplus_{j=1}^{s}\mathfrak{m}_{0j},$ (7)

where $\dim(\mathfrak{n}_{i})=1$,
$\mathfrak{m}_{0j}=\mathfrak{m}_{1}^{j}\oplus\mathfrak{m}_{2}^{j}\oplus\cdots\oplus\mathfrak{m}_{n_{0}}^{j}$
with $\mathfrak{m}_{\alpha}^{j}\cong\mathfrak{m}_{\beta}^{j}$,
$\alpha\neq\beta$ and $\dim(\mathfrak{m}^{j}_{\ell})=n_{j}$,
$\ell=1,2,\ldots,n_{0}$. Note that
$\mathfrak{n}_{1}\oplus\cdots\oplus\mathfrak{n}_{\binom{n_{0}}{2}}\cong\mathfrak{so}(n_{0})$.
Moreover, if $n_{0}=0$ or $n_{0}=1$ then there are no trivial submodules and
hence $\mathfrak{so}(n_{0})=\\{0\\}$.

We now give explicit matrix representations of the modules $\mathfrak{m}_{ij}$
and $\mathfrak{m}_{0j}$. We consider the
$\operatorname{Ad}(\operatorname{SO}(n))$-invariant inner product
$B:\mathfrak{so}(n)\times\mathfrak{so}(n)\to\mathbb{R}$ given by

$B(X,Y)=-{\rm Trace}(XY),\quad X,Y\in\mathfrak{so}(n),$ (8)

and obtain a $B$-orthogonal decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$, where
$\mathfrak{h}=\mathfrak{so}(n_{1})\oplus\cdots\oplus\mathfrak{so}(n_{s})$ and
$\mathfrak{m}\cong T_{o}(G/H)$. We consider a basis of
$\mathfrak{g}=\mathfrak{so}(n)$ as follows:
Let $M_{n}\mathbb{R}$ be the set of real $n\times n$ matrices and let
$E_{ab}\in M_{n}\mathbb{R}$ be the matrix with 1 in the $(a,b)$-entry and zero
elsewhere. For $1\leq a<b\leq n$ we set

$e_{ab}:=E_{ab}-E_{ba}.$ (9)

Note that $e_{ab}=-e_{ba}$. The set

$\mathcal{B}:=\left\\{{e_{ab}:1\leq a<b\leq n}\right\\},$ (10)

constitutes a basis of $\mathfrak{so}(n)$, which is orthogonal with respect to
$B$. The proof of the following lemma is immediate.

######  Lemma 4.2.

The only non zero bracket relations among the vectors (9) are
$[e_{ab},e_{bc}]=e_{ac}$, for $a,b,c$ distinct.

A choice for the modules in the decomposition (7) is the following:

$\displaystyle\mathfrak{m}_{ij}$ $\displaystyle=$
$\displaystyle\operatorname{span}\\{e_{ab}:n_{0}+n_{1}+\cdots+n_{i-1}+1\leq
a\leq n_{0}+n_{1}+\cdots+n_{i},$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \quad
n_{0}+n_{1}+\cdots+n_{j-1}+1\leq b\leq n_{0}+n_{1}+\cdots+n_{j}\\},\ \ 1\leq
i<j\leq s.$ $\displaystyle\mathfrak{m}_{0j}$ $\displaystyle=$
$\displaystyle\operatorname{span}\\{e_{ab}:1\leq a\leq n_{0},\
n_{0}+n_{1}+\cdots+n_{j-1}+1\leq b\leq n_{0}+n_{1}+\cdots+n_{j}\\},\ \ 1\leq
j\leq s.$ $\displaystyle\mathfrak{so}(n_{0})$ $\displaystyle=$
$\displaystyle\operatorname{span}\\{e_{ab}:1\leq a<b\leq n_{0}\\}.$

The equivalent modules in the decomposition of $\mathfrak{m}_{0j}$ are given
by

$\mathfrak{m}^{j}_{\ell}=\operatorname{span}\\{e_{\ell
b}:n_{0}+n_{1}+\cdots+n_{j-1}+1\leq b\leq n_{0}+n_{1}+\cdots+n_{j}\\},\
\ell=1,\dots,n_{0}.$

Also,

$\mathfrak{so}(n_{j})=\operatorname{span}\\{e_{ab}:n_{0}+n_{1}+\cdots+n_{j-1}+1\leq
a<b\leq n_{0}+n_{1}+\cdots+n_{j}\\},\quad j=1,\dots,s.$

Hence, we obtain the $B$-orthogonal decomposition

$\mathfrak{m}=\mathfrak{n}\oplus\mathfrak{p},$ (11)

where

$\mathfrak{n}=\mathfrak{so}(n_{0}),\quad\mathfrak{p}=\bigoplus_{1\leq i<j\leq
s}\mathfrak{m}_{ij}\bigoplus_{j=1}^{s}\mathfrak{m}_{0j}.$

The above decomposition can be depicted in the following matrix, which shows
the upper triangular part of $\mathfrak{so}(n)$:

$\begin{pmatrix}\mathfrak{so}(n_{0})&\mathfrak{m}_{01}&\mathfrak{m}_{02}&\mathfrak{m}_{03}&\cdots&\mathfrak{m}_{0s}\\\
&0_{n_{1}}&\mathfrak{m}_{12}&\mathfrak{m}_{13}&\cdots&\mathfrak{m}_{1s}\\\
&&0_{n_{2}}&\mathfrak{m}_{23}&\cdots&\mathfrak{m}_{2s}\\\
&\ast&&\ddots&\vdots&\vdots\\\ &&&&&0_{n_{s}}\end{pmatrix}$

The matrices $\mathfrak{m}_{0j}$ are of size $n_{0}\times n_{j}$, the matrices
$\mathfrak{m}_{ij}$ are of size $n_{i}\times n_{j}$, and the matrices
$\mathfrak{so}(n_{i})$ of size $n_{i}\times n_{i}$. We remark that if
$n_{0}=0$ or $n_{0}=1$, then $\mathfrak{n}=\\{0\\}$. In the former case, the
submodules $\mathfrak{m}_{0j}$ are zero while in the latter case they are non
zero and irreducible.

Moreover, using Lemma 4.2, we observe that

$[\mathfrak{so}(n_{i}),\mathfrak{m}_{lm}]=\left\\{\begin{array}[]{lll}\mathfrak{m}_{lm},\quad\makebox{if
$i=l$ or $i=m$}\\\ \\{0\\},\quad\makebox{otherwise}\end{array}\right.,\ \ \
0\leq i\leq s,\ \ 0\leq l<m\leq s,$ (12)

and

$[\mathfrak{m}_{ij},\mathfrak{m}_{jl}]=\mathfrak{m}_{il}\ \ \makebox{for all}\
\ 0\leq i<j<l\leq s.$ (13)

######  Remark 4.3.

In view of Remark 4.1, if the isotropy subgroup
$H=\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$
contains at least two $\operatorname{SO}(2)$-factors, say $n_{i}=n_{j}=2$,
then the module $\mathfrak{m}_{ij}$ splits into two
$\operatorname{Ad}(H)$-irreducible non equivalent summands each of dimension
2. More precisely, if we set $I=n_{0}+n_{1}+\cdots+n_{i-1}+1$,
$J=n_{0}+n_{1}+\cdots+n_{j-1}+1$, $K=n_{0}+n_{1}+\cdots+n_{i-1}+2$,
$L=n_{0}+n_{1}+\cdots+n_{j-1}+2$, then $\mathfrak{m}_{ij}=V_{ij}^{1}\oplus
V_{ij}^{2}$, where
$V_{ij}^{1}=\operatorname{span}\\{e_{IJ}-e_{KL},e_{IL}+e_{KJ}\\}$,
$V_{ij}^{2}=\operatorname{span}\\{e_{IJ}+e_{KL},e_{IL}-e_{KJ}\\}$.

For example, for
$M=G/H=\operatorname{SO}(14)/\operatorname{SO}(3)\times\operatorname{SO}(2)\times\operatorname{SO}(3)\times\operatorname{SO}(2)\times\operatorname{SO}(2)$
we have $n_{0}=2,n_{1}=3,n_{2}=2,n_{3}=3,n_{4}=2,n_{5}=2$, and

$\mathfrak{m}=\mathfrak{n}\oplus\mathfrak{p}=\mathfrak{so}(2)\bigoplus_{1\leq
i<j\leq 5}\mathfrak{m}_{ij}\bigoplus_{j=1}^{5}\mathfrak{m}_{0j}.$

The submodules $\mathfrak{m}_{24}$, $\mathfrak{m}_{25}$ and
$\mathfrak{m}_{45}$ split as above. For instance,
$\mathfrak{m}_{24}=V_{24}^{1}\oplus V_{24}^{2}$, where
$V_{24}^{1}=\operatorname{span}\\{e_{6,11}-e_{7,12},e_{6,12}+e_{7,11}\\}$,
$V_{24}^{2}=\\{e_{6,11}+e_{7,12},e_{6,12}-e_{7,11}\\}$.

###  4.2. Proof of Theorem 1.1

Any normal metric on $G/H$ induced from an $\operatorname{Ad}$-invariant inner
product $B$ on $\mathfrak{g}$ is a g.o. metric, and hence the sufficiency part
of the theorem holds trivially. For the necessity part, assume initially that
$n_{1}=\cdots=n_{s}=2$ so that $H$ is abelian. Since $G$ is semisimple, the
main results in [25] imply that any g.o. metric on $G/H$ is normal, and hence
Theorem 1.1 follows in this case.

Now assume that $n_{j}\neq 2$ for some $j=1,\dots,s$. Let $g$ be a
$G$-invariant g.o. metric on $G/H$. Assume that
$A:\mathfrak{m}\rightarrow\mathfrak{m}$ is the corresponding metric
endomorphism satisfying Equation (2). Recall the spaces
$\mathfrak{n}=\mathfrak{so}(n_{0})$ and $\mathfrak{p}=\bigoplus_{0\leq i<j\leq
s}\mathfrak{m}_{ij}$ defined in Section 4.1, and the decomposition
$\mathfrak{m}=\mathfrak{n}\oplus\mathfrak{p}$. Since $H$ is connected, we have
$H^{0}=H$. The Lie algebra of the normalizer $N_{G}(H^{0})=N_{G}(H)$ coincides
with
$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h})=\\{Y\in\mathfrak{g}:[Y,\mathfrak{h}]\subseteq\mathfrak{h}\\}$.
In our case,
$\mathfrak{h}=\mathfrak{so}(n_{1})\oplus\cdots\oplus\mathfrak{so}(n_{s})$ and
it is not hard to show using the results in Section 4.1 that

$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{so}(n_{0})\oplus\mathfrak{so}(n_{1})\oplus\cdots\oplus\mathfrak{so}(n_{s})=\mathfrak{n}\oplus\mathfrak{h}.$

Therefore, the tangent space of $G/N_{G}(H)$ coincides with $\mathfrak{p}$.
The normalizer Lemma 3.3 then implies that $\left.A\right|_{\mathfrak{p}}$
defines a $G$-invariant g.o. metric on $G/N_{G}(H)$.

By taking into account Lemma 4.2 and the expressions of the subspaces
$\mathfrak{m}_{ij},\mathfrak{so}(n_{j}),\mathfrak{so}(n_{0})$ in terms of the
basis $\mathcal{B}$, we deduce that the submodules $\mathfrak{m}_{ij}$, $0\leq
i<j\leq s$ are
$\operatorname{ad}(\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}))$-invariant. If
$n_{i}\neq 2$ or $n_{j}\neq 2$ then the submodules $\mathfrak{m}_{ij}$ are
$\operatorname{ad}(\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}))$-irreducible. On
the other hand, in view of Remark 4.3, $\mathfrak{m}_{ij}=V_{ij}^{1}\oplus
V_{ij}^{2}$ if $n_{i}=n_{j}=2$.

We now claim that the
$\operatorname{ad}(\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}))$-submodules
$\mathfrak{m}_{ij}$, $0\leq i<j\leq s$, are pairwise inequivalent. Indeed, if
$\mathfrak{m}_{ij}$ and $\mathfrak{m}_{lm}$, with $0\leq i<j\leq s$, $0\leq
l<m\leq s$, are two distinct
$\operatorname{ad}(\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}))$-submodules,
then there exists an index $i_{0}$ such that one of the following happens:

Case 1. $i_{0}=i$ or $i_{0}=j$ and $i_{0}\neq l,m$, Case 2. $i_{0}=l$ or
$i_{0}=m$ and $i_{0}\neq i,j$.

For Case 1, we take into account relation (12) and obtain that
$[\mathfrak{so}(n_{i_{0}}),\mathfrak{m}_{ij}]=\mathfrak{m}_{ij}$ and
$[\mathfrak{so}(n_{i_{0}}),\mathfrak{m}_{lm}]=\\{0\\}$. For Case 2 we have
that $[\mathfrak{so}(n_{i_{0}}),\mathfrak{m}_{ij}]=\\{0\\}$ and
$[\mathfrak{so}(n_{i_{0}}),\mathfrak{m}_{lm}]=\mathfrak{m}_{lm}$. By virtue of
Lemma 2.2, we deduce that the submodules $\mathfrak{m}_{ij}$ and
$\mathfrak{m}_{lm}$ are inequivalent in both cases, which proves the claim.

Therefore, the restriction of the g.o. metric $\left.A\right|_{\mathfrak{p}}$
of $G/N_{G}(H)$ to each of the
$\operatorname{ad}(\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}))$-submodules
$\mathfrak{m}_{ij}$, $0\leq i<j<l\leq s$, has the diagonal form

$\left.A\right|_{\mathfrak{m}_{ij}}=\left\\{\begin{array}[]{lll}\lambda_{ij}\operatorname{Id},\qquad\qquad\qquad\qquad\
\ \makebox{if $n_{i}\neq 2$ or $n_{j}\neq 2$}\\\ \\\
\begin{pmatrix}\lambda_{ij}^{1}\left.\operatorname{Id}\right|_{V_{ij}^{1}}&0\\\
0&\lambda_{ij}^{2}\left.\operatorname{Id}\right|_{V_{ij}^{2}}\end{pmatrix},\ \
\makebox{if $n_{i}=n_{j}=2$}.\end{array}\right.$ (14)

In view of the decomposition $\mathfrak{m}=\mathfrak{n}\oplus\mathfrak{p}$ and
Remark 3.6, Theorem 1.1 now follows from the following two propositions, whose
proofs we present in the next subsection.

######  Proposition 4.4.

$\left.A\right|_{\mathfrak{p}}=\lambda\operatorname{Id}$, where $\lambda>0$.

######  Proposition 4.5.

$\left.A\right|_{\mathfrak{n}}=\lambda\operatorname{Id}$, where $\lambda$ is
given by Proposition 4.4.∎

###  4.3. Proof of Propositions 4.4 and 4.5

To prove Proposition 4.4 we need the following lemmas.

######  Lemma 4.6.

Let $A$ be the metric endomorphism of a g.o. metric on
$M=\operatorname{SO}(n)/\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$.
Then for any $r=1,\dots,s$, the restriction of $A$ to the tangent space of
$\widetilde{M}:=\operatorname{SO}(n_{r}+\cdots+n_{s})/\operatorname{SO}(n_{r})\times\cdots\times\operatorname{SO}(n_{s})$
defines a $\operatorname{SO}(n_{r}+\cdots+n_{s})$-invariant g.o. metric on
$\widetilde{M}$.

###### Proof.

Let $\widetilde{\mathfrak{g}}:=\mathfrak{so}(n_{r}+\cdots+n_{s})$ and
$\widetilde{\mathfrak{h}}:=\mathfrak{so}(n_{r})\oplus\cdots\oplus\mathfrak{so}(n_{s})$
be the Lie algebras of $\operatorname{SO}(n_{r}+\cdots+n_{s})$ and
$\operatorname{SO}(n_{r})\times\cdots\times\operatorname{SO}(n_{s})$
respectively. Taking into account the notation and results of section 4.1, the
tangent space of $\widetilde{M}$ coincides with the $B$-orthogonal complement
$\widetilde{\mathfrak{m}}:=\bigoplus_{r\leq i<j\leq s}\mathfrak{m}_{ij}$ of
$\widetilde{\mathfrak{h}}$ in $\widetilde{\mathfrak{g}}$. Relation (14)
implies that $\left.A\right|_{\widetilde{\mathfrak{m}}}$ defines an
endomorphism of $\widetilde{\mathfrak{m}}$, which is
$\operatorname{ad}(\mathfrak{h})$-equivariant and hence
$\operatorname{ad}(\widetilde{\mathfrak{h}})$-equivariant. Therefore,
$\left.A\right|_{\widetilde{\mathfrak{m}}}$ defines an
$\operatorname{SO}(n_{r}+\cdots+n_{s})$-invariant metric on $\widetilde{M}$.
Since $A$ defines a g.o. metric on $G/H$, Proposition 3.2 implies that for any
$X\in\widetilde{\mathfrak{m}}$, there exists $a\in\mathfrak{h}$ such that

$0=[a+X,AX]=[a+X,\left.A\right|_{\widetilde{\mathfrak{m}}}X].$ (15)

On the other hand, relation (12) along with the definition of
$\widetilde{\mathfrak{m}}$ imply that
$[\mathfrak{h},\widetilde{\mathfrak{m}}]=[\widetilde{\mathfrak{h}},\widetilde{\mathfrak{m}}]$.
Therefore, we may assume that the vector $a$ in expression (15) lies in
$\widetilde{\mathfrak{h}}$. By Proposition 3.2 we conclude that
$\left.A\right|_{\widetilde{\mathfrak{m}}}$ defines a g.o. metric on
$\widetilde{M}$. ∎

######  Lemma 4.7.

Let $R_{s}=\\{\lambda_{ij}:\ 0\leq i<j\leq s\\}$ be a set such that
$\lambda_{ij}=\lambda_{jk}=\lambda_{ik}$ for all $0\leq i<j<k\leq s$. Then
$R_{s}$ is a singleton.

###### Proof.

We will proceed by induction on $s\geq 1$. If $s=1$ then
$R_{1}=\\{\lambda_{01}\\}$ and the result holds trivially. Assume that the
lemma holds for $s=N$ and let $s=N+1$. The assumption that
$\lambda_{ij}=\lambda_{jk}=\lambda_{ik}$ for all $0\leq i<j<k\leq s=N+1$
implies that $\lambda_{ij}=\lambda_{jk}=\lambda_{ik}$ for all $0\leq i<j<k\leq
N$. By the induction hypothesis, the set $R_{N}=\\{\lambda_{ij}:\ 0\leq
i<j\leq N\\}$ is a singleton, i.e. $R_{N}=\\{\lambda\\}$.

To prove that $R_{N+1}$ is a singleton, it remains to show that

$\lambda_{0,N+1}=\lambda_{1,N+1}=\cdots=\lambda_{N,N+1}=\lambda.$ (16)

To this end, consider two arbitrary elements $\lambda_{i,N+1}$,
$\lambda_{j,N+1}$. Without loss of generality, assume that $i<j$. Since $0\leq
i<j<N$, we have $\lambda_{ij}\in R_{N}$ and hence $\lambda_{ij}=\lambda$. On
the other hand, the assumption of the lemma yields
$\lambda=\lambda_{ij}=\lambda_{j,N+1}=\lambda_{i,N+1}$. Since the choice of
$i,j$ is arbitrary, Equation (16) is true and thus $R_{N+1}=\\{\lambda\\}$,
which concludes the induction.∎

_Proof of Proposition 4.4_. The proof consists of two steps.

Step 1. Prove that both quantities $\lambda_{ij}^{1}$, $\lambda_{ij}^{2}$ in
relation (14) are equal for all $i,j$, which will imply that

$\left.A\right|_{\mathfrak{m}_{ij}}=\lambda_{ij}\operatorname{Id}.$ (17)

Step 2. Prove that all $\lambda_{ij}$ in relation (17) are equal.

For Step 1, recall the assumption that at least one of the $n_{j}$,
$j=1,\dots,s$ is not equal to $2$. Recall also from definition (5) that
$n_{0}=n-(n_{1}+\cdots+n_{s})$. Without any loss of generality, we may apply a
conjugation $\phi\in\operatorname{Aut}(\operatorname{SO}(n))$, permuting the
position of the diagonal blocks $\operatorname{SO}(n_{j})$, $j=1,\dots,s$, in
the embedding of $H$ in $G$, so that

$n_{j}\neq 2\ \ \makebox{for}\ \ j=0,\dots,r\ \ \makebox{and}\ \
n_{r+1}=n_{r+2}=\cdots=n_{s}=2,\ \ \makebox{i.e.}$ (18)
$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{so}(n_{0})\times\cdots\times\mathfrak{so}(n_{r})\times\underbrace{\mathfrak{so}(2)\times\cdots\times\mathfrak{so}(2)}_{s-r\
\makebox{times}},\ \ r=0,\dots,s.$

For example, any diagonal embedding of
$H=\operatorname{SO}(2)\times\operatorname{SO}(3)\times\operatorname{SO}(3)$
in $\operatorname{SO}(10)$ is conjugate to

$\begin{pmatrix}\operatorname{SO}(3)&0&0&0\\\ 0&\operatorname{SO}(3)&0&0\\\
0&0&\operatorname{SO}(2)&0\\\ 0&0&0&\operatorname{Id}_{2}\end{pmatrix}$, in
which case
$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h})=\underbrace{\mathfrak{so}(3)\oplus\mathfrak{so}(3)\oplus\mathfrak{so}(2)}_{\mathfrak{h}}\oplus\underbrace{\mathfrak{so}(2)}_{\mathfrak{n}}$.

Such a conjugation leaves the space $\mathfrak{p}$ invariant by permuting the
$\operatorname{ad}(\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}))$-submodules
$\mathfrak{m}_{ij}$, $0\leq i<j\leq s$. Under this conjugation and in the case
where the number $s-r$ of $\operatorname{SO}(2)$-blocks is greater than $2$,
relation (14) becomes

$\left.A\right|_{\mathfrak{m}_{ij}}=\left\\{\begin{array}[]{lll}\lambda_{ij}\operatorname{Id},\
\ \makebox{if $0\leq i<j\leq r+1$}\\\ \\\
\begin{pmatrix}\lambda_{ij}^{1}\left.\operatorname{Id}\right|_{V_{ij}^{1}}&0\\\
0&\lambda_{ij}^{2}\left.\operatorname{Id}\right|_{V_{ij}^{2}}\end{pmatrix},\ \
\makebox{if $r+1\leq i<j\leq s$}.\end{array}\right.$ (19)

If $s-r<2$ then relation (17) is trivially true.

We now assume that $s-r\geq 2$ and consider the space
$\widetilde{M}:=\operatorname{SO}(2(s-r))/\underbrace{\operatorname{SO}(2)\times\cdots\times\operatorname{SO}(2)}_{(s-r)-\makebox{times}}$,
whose tangent space is $\widetilde{\mathfrak{m}}:=\bigoplus_{r+1\leq i<j\leq
s}\mathfrak{m}_{ij}\subset\mathfrak{p}$. By Lemma 4.6 the restriction
$\left.A\right|_{\widetilde{\mathfrak{m}}}$ defines a
$\operatorname{SO}(2(s-r))$-invariant g.o. metric on $\widetilde{M}$.

We consider the following cases:

$\makebox{{\bf Case I}}\ \ s-r>2\ \ \makebox{and \ {\bf Case II}}\ \ s-r=2.$

In Case I, it is $\operatorname{SO}(2(s-r))\neq\operatorname{SO}(4)$ and hence
$\operatorname{SO}(2(s-r))$ is simple. On the other hand, the isotropy
subgroup
$\underbrace{\operatorname{SO}(2)\times\cdots\times\operatorname{SO}(2)}_{(s-r)-\makebox{times}}$
of $\widetilde{M}$ is abelian. By the main theorem in [25], the g.o. metric
$\left.A\right|_{\widetilde{\mathfrak{m}}}$ is standard and thus
$\left.A\right|_{\widetilde{\mathfrak{m}}}=\lambda\left.\operatorname{Id}\right|_{\widetilde{\mathfrak{m}}}$.
In particular, the last relation implies that the quantities
$\lambda_{ij}^{1}$, $\lambda_{ij}^{2}$ in relation (14) are equal for all
$i,j$ with $r+1\leq i<j\leq s$, and thus relation (17) is true for Case I.

In Case II, we have
$\widetilde{M}=\operatorname{SO}(4)/\operatorname{SO}(2)\times\operatorname{SO}(2)$,
$\widetilde{\mathfrak{m}}=\mathfrak{m}_{s-1,s}=V_{s-1,s}^{1}\oplus
V_{s-1,s}^{2}$ and relation (19) yields
$\left.A\right|_{\widetilde{\mathfrak{m}}}=\begin{pmatrix}\lambda_{s-1,s}^{1}\left.\operatorname{Id}\right|_{V_{s-1,s}^{1}}&0\\\
0&\lambda_{s-1,s}^{2}\left.\operatorname{Id}\right|_{V_{s-1,s}^{2}}\end{pmatrix}$.
We consider the
$\operatorname{ad}(\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}))$-irreducible
submodules $\mathfrak{m}_{1,s-1}$, and $V_{s-1,s}^{1}$.

Relation (13) yields
$[\mathfrak{m}_{1,s-1},V_{s-1,s}^{1}]\subseteq[\mathfrak{m}_{1,s-1},\mathfrak{m}_{s-1,s}]=\mathfrak{m}_{1,s}$.
More importantly, along with the description of the submodules $V_{ij}^{l}$ in
Remark 4.3, we deduce that
$[\mathfrak{m}_{1,s-1},V_{s-1,s}^{1}]\subseteq\mathfrak{m}_{1,s}\setminus\\{0\\}$.
As a result, the space $[\mathfrak{m}_{1,s-1},V_{s-1,s}^{1}]$ has non zero
projection on $(\mathfrak{m}_{1,s-1}\oplus V_{s-1,s}^{1})^{\bot}$. By using
the first part of Lemma 3.8, the last relation, along with the facts that
$\left.A\right|_{\mathfrak{m}_{1,s-1}}=\lambda_{1,s-1}\left.\operatorname{Id}\right|_{\mathfrak{m}_{1,s-1}}$
and
$\left.A\right|_{V_{s-1,s}^{1}}=\lambda_{s-1,s}^{1}\left.\operatorname{Id}\right|_{V_{s-1,s}^{1}}$,
yield

$\lambda_{1,s-1}=\lambda_{s-1,s}^{1}.$ (20)

By using the same argument for $V_{s-1,s}^{2}$ we deduce that

$\lambda_{1,s-1}=\lambda_{s-1,s}^{2}$ (21)

Equations (20) and (21) yield $\lambda_{s-1,s}^{1}=\lambda_{s-1,s}^{2}$ and
thus relation (17) is also true for Case II. This concludes Step 1.

For Step 2 we proceed as follows: Due to relation (13), along with Equation
(17), the $\operatorname{ad}({\mathfrak{h}})$-invariance of
$\mathfrak{m}_{ij},\mathfrak{m}_{jl}$ and their $B$-orthogonality, part 2. of
Lemma 3.8 yields

$\lambda_{ij}=\lambda_{jl}=\lambda_{il}\ \ \makebox{for all}\ \ 0\leq
i<j<l\leq s.$

By Lemma 4.7 we deduce that the set $R_{s}=\\{\lambda_{ij}:\ 0\leq i<j\leq
s\\}$ of the eigenvalues of $\left.A\right|_{\mathfrak{p}}$ has only one
element, and thus Step 2 is concluded. ∎

Before we proceed to the proof of the second proposition, we note that if
$n_{0}=0$ or $n_{0}=1$, then $\mathfrak{n}=\\{0\\}$ and
$\mathfrak{m}=\mathfrak{p}$, and thus Theorem 1.1 follows directly from
Proposition 4.4.

_Proof of Proposition 4.5_. We recall the spaces
$\mathfrak{m}_{0j}=\mathfrak{m}^{j}_{1}\oplus\cdots\oplus\mathfrak{m}^{j}_{n_{0}}$
defined in Section 4.1. By Proposition 4.4 we have

$\left.A\right|_{\mathfrak{m}^{j}_{i}}=\lambda\operatorname{Id},\ \
i=1,\dots,n_{0}.$ (22)

The Lie algebra $\mathfrak{n}=\mathfrak{so}(n_{0})$ coincides with the Lie
algebra of $N_{G}(H^{0})/H^{0}=N_{G}(H)/H$. By Lemma 3.4,
$\left.A\right|_{\mathfrak{n}}$ defines a bi-invariant metric on $N_{G}(H)/H$,
which in turn corresponds to an $\operatorname{Ad}$-invariant inner product on
$\mathfrak{so}(n_{0})$. For $n_{0}=2$, $\mathfrak{n}$ is one-dimensional and
thus $\left.A\right|_{\mathfrak{n}}=\mu\operatorname{Id}$. For $2<n_{0}\neq
4$, $\mathfrak{so}(n_{0})$ is simple and the only
$\operatorname{Ad}$-invariant inner product is a scalar multiple of the
Killing form. Therefore, if $2<n_{0}\neq 4$ we also have
$\left.A\right|_{\mathfrak{n}}=\mu\operatorname{Id}$. For both cases, choose
the vectors $e_{12}\in\mathfrak{n}$ and
$e_{1,n_{0}+1}\in\mathfrak{m}^{1}_{1}$. We have

$[e_{12},e_{1n_{0}+1}]=-e_{2,n_{0}+1}\in\mathfrak{m}^{1}_{2}.$

Therefore, $[\mathfrak{n},\mathfrak{m}_{1}^{1}]$ has non-zero projection on
$(\mathfrak{n}\oplus\mathfrak{m}_{1}^{1})^{\bot}$. Along with the
$\operatorname{ad}(\mathfrak{h})$-invariance of $\mathfrak{n}$ and
$\mathfrak{m}_{1}^{1}$ and the facts that
$\left.A\right|_{\mathfrak{m}^{j}_{i}}=\lambda\operatorname{Id}$ and
$\left.A\right|_{\mathfrak{n}}=\mu\operatorname{Id}$, Lemma 3.8 yields
$\lambda=\mu$. We conclude that
$\left.A\right|_{\mathfrak{so}(n_{0})}=\lambda\operatorname{Id}$ if $2\leq
n_{0}\neq 4$.

If $n_{0}=4$, $\mathfrak{n}$ decomposes into two simple ideals as

$\mathfrak{n}=\mathfrak{so}(4)=\mathfrak{n}_{1}\oplus\mathfrak{n}_{2},$

where
$\mathfrak{n}_{1}=\operatorname{span}\\{e_{12}+e_{34},-e_{13}+e_{24},e_{23}+e_{14}\\}\approx\mathfrak{so}(3)$
and
$\mathfrak{n}_{2}=\operatorname{span}\\{e_{12}-e_{34},-e_{13}-e_{24},e_{23}-e_{14}\\}\approx\mathfrak{so}(3)$.
By Lemma 3.7 we have

$\left.A\right|_{\mathfrak{n}_{1}}=\mu_{1}\operatorname{Id}\ \ \makebox{and}\
\ \left.A\right|_{\mathfrak{n}_{2}}=\mu_{2}\operatorname{Id}.$ (23)

It remains to show that $\mu_{1}=\mu_{2}=\lambda$. To this end, choose the
vectors $e_{12}+e_{34}\in\mathfrak{n}_{1}$, $e_{12}-e_{34}\in\mathfrak{n}_{2}$
and $e_{15}\in\mathfrak{m}^{1}_{1}$. The submodules
$\mathfrak{n}_{1},\mathfrak{n}_{2}$ and $\mathfrak{m}^{1}_{1}$ are
$\operatorname{ad}(\mathfrak{h})$-invariant. Moreover, we have

$[e_{12}+e_{34},e_{15}]=-e_{25}\in\mathfrak{m}^{1}_{2}\subset(\mathfrak{n}_{1}\oplus\mathfrak{m}^{1}_{1})^{\bot}\
\ \makebox{and}\ \
[e_{12}-e_{34},e_{15}]=-e_{25}\in\mathfrak{m}^{1}_{2}\subset(\mathfrak{n}_{2}\oplus\mathfrak{m}^{1}_{1})^{\bot}.$

Hence, $[\mathfrak{n}_{1},\mathfrak{m}^{1}_{1}]$ and
$[\mathfrak{n}_{2},\mathfrak{m}^{1}_{1}]$ have non zero projections on
$(\mathfrak{n}_{1}\oplus\mathfrak{m}_{1}^{1})^{\bot}$ and
$(\mathfrak{n}_{2}\oplus\mathfrak{m}^{1}_{1})^{\bot}$ respectively. Along with
Equations (22) and (23), part 1. of Lemma 3.8 yields
$\mu_{1}=\lambda=\mu_{2}$, which concludes the proof. ∎

##  5\. The space
$M=G/H=\operatorname{U}(n)/\operatorname{U}(n_{1})\times\cdots\times\operatorname{U}(n_{s})$

###  5.1. Isotropy representation of
$G/H=\operatorname{U}(n)/\operatorname{U}(n_{1})\times\cdots\times\operatorname{U}(n_{s})$

Denote by $\mu_{n}$ the standard representation of $\operatorname{U}(n)$ in
$\mathbb{C}^{n}$. Then the complexified adjoint representation of
$\operatorname{U}(n)$ is

$\operatorname{Ad}^{\operatorname{U}(n)}\otimes\mathbb{C}=\mu_{n}\otimes_{\mathbb{C}}\bar{\mu}_{n}.$
(24)

By taking into account the assumption that $H$ is embedded diagonally in $G$,
we can identify $H$ with the subgroup

$H=\begin{pmatrix}\operatorname{Id}_{n_{0}}&&&0\\\
&\operatorname{U}(n_{1})&&\vdots\\\ &&\ddots&\\\
0&\cdots&&\operatorname{U}(n_{s})\\\ \end{pmatrix},$

of $G$, where $n_{0}:=n-(n_{1}+\cdots+n_{s})$.

Let
$\tau_{n_{i}}:\operatorname{U}(n_{1})\times\cdots\times\operatorname{U}(n_{s})\to\operatorname{U}(n_{i})$
be the projection onto the $i$-factor and $q_{i}=\mu_{n_{i}}\circ\tau_{n_{i}}$
be the standard representation of $H$, i.e.

$\operatorname{U}(n_{1})\times\cdots\times\operatorname{U}(n_{s})\stackrel{{\scriptstyle\tau_{n_{i}}}}{{\longrightarrow}}\operatorname{U}(n_{i})\stackrel{{\scriptstyle\mu_{n_{i}}}}{{\longrightarrow}}\operatorname{Aut}(\mathbb{C}^{n_{i}}).$

By using relation (24), we obtain

$\displaystyle\left.\operatorname{Ad}^{G}\otimes\mathbb{C}\right|_{H}$
$\displaystyle=$
$\displaystyle\left.\mu_{n}\otimes_{\mathbb{C}}\bar{\mu}_{n}\right|_{H}=\left.\mu_{n}\right|_{H}\otimes_{\mathbb{C}}\left.\bar{\mu}_{n}\right|_{H}=({q_{1}\oplus\cdots\oplus
q_{s}}\oplus\mathbbm{1}_{n_{0}})\otimes_{\mathbb{C}}({\bar{q}_{1}}\oplus\cdots\oplus{\bar{q}_{s}}\oplus\mathbbm{1}_{n_{0}})$
(25) $\displaystyle=$ $\displaystyle\mathbbm{1}_{{n_{0}}^{2}}\
\bigoplus_{i=1}^{s}\left\\{{q_{i}\otimes_{\mathbb{C}}\bar{q}_{i}}\right\\}\bigoplus_{j=1}^{s}\\{(q_{j}\oplus\bar{q}_{j})\otimes_{\mathbb{C}}\mathbbm{1}_{n_{0}}\\}$
$\displaystyle\bigoplus_{1\leq i<j\leq
s}\left\\{(q_{i}\otimes_{\mathbb{C}}\bar{q}_{j})\oplus(q_{j}\otimes_{\mathbb{C}}\bar{q}_{i})\right\\}.$

The summand $\bigoplus_{i=1}^{s}\\{{q_{i}\otimes_{\mathbb{C}}\bar{q}_{i}}\\}$
in (25) corresponds to $\operatorname{Ad}^{H}\otimes\mathbb{C}$, therefore, by
virtue of Proposition 2.1, $\chi\otimes\mathbb{C}$ is given by

$\displaystyle\chi\otimes\mathbb{C}$ $\displaystyle=$
$\displaystyle\mathbbm{1}_{n_{0}^{2}}\
\bigoplus_{j=1}^{s}\\{(q_{j}\oplus\bar{q}_{j})\otimes_{\mathbb{C}}\mathbbm{1}_{n_{0}}\\}\bigoplus_{1\leq
i<j\leq
s}\left\\{(q_{i}\otimes_{\mathbb{C}}\bar{q}_{j})\oplus(q_{j}\otimes_{\mathbb{C}}\bar{q}_{i})\right\\}.$
(26)

Expression (26) induces a real decomposition of the tangent space

$\mathfrak{m}=\mathfrak{n}\oplus\mathfrak{p},$ (27)

where

$\mathfrak{n}=\mathfrak{u}(n_{0}),\ \ \
\mathfrak{p}=\bigoplus_{j=1}^{s}{\mathfrak{m}_{0j}}\bigoplus_{1\leq i<j\leq
s}\mathfrak{m}_{ij}.$ (28)

In the above decomposition we have
$\mathfrak{m}_{0j}=\mathfrak{m}_{1}^{j}\oplus\mathfrak{m}_{2}^{j}\oplus\cdots\oplus\mathfrak{m}_{n_{0}}^{j}$
with $\mathfrak{m}_{\alpha}^{j}\cong\mathfrak{m}_{\beta}^{j}$,
$\alpha\neq\beta$ and
$\dim(\mathfrak{m}_{\ell}^{j})=2n_{j},\ell=1,2,\dots,n_{0}$.

We now give explicit matrix representations of the modules $\mathfrak{m}_{0j}$
and $\mathfrak{m}_{ij}$. We consider the
$\operatorname{Ad}(\operatorname{U}(n))$-invariant inner product
$B:\mathfrak{u}(n)\times\mathfrak{u}(n)\rightarrow\mathbb{R}$ given by

$B(X,Y)=-\operatorname{Trace}(XY),\quad X,Y\in\mathfrak{u}(n).$ (29)

Then there is a $B$-orthogonal decomposition

$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m},$ (30)

where $\mathfrak{h}=\mathfrak{u}(n_{1})\oplus\cdots\oplus\mathfrak{u}(n_{s})$
and $\mathfrak{m}\approx T_{o}(G/H)$. We consider a basis of
$\mathfrak{g}=\mathfrak{u}(n)$ as follows:
Let $M_{n}\mathbb{C}$ be the set of complex $n\times n$ matrices and let
$E_{ab}\in M_{n}\mathbb{C}$, $a,b=1,\dots,n$, be the matrix with 1 in the
$(a,b)$-entry and zero elsewhere. For $a,b=1,\dots,n$, we set

$e_{ab}=E_{ab}-E_{ba},\quad f_{ab}=\sqrt{-1}(E_{ab}+E_{ba}).$ (31)

Note that $e_{ab}=-e_{ba},{f}_{ab}={f}_{ba}$. Then the set

$\mathcal{B}=\left\\{e_{ab},{f}_{cd}:1\leq a<b\leq n,\ 1\leq c\leq d\leq
n\right\\},$ (32)

constitutes a basis of $\mathfrak{u}(n)$ which is orthogonal with respect to
$B$. We have the following.

######  Lemma 5.1.

The non zero bracket relations among the vectors (31) are given by

$\begin{array}[]{ccc}[e_{ab},e_{cd}]=\delta_{bc}e_{ad}-\delta_{ad}e_{cb}-\delta_{ac}e_{bd}-\delta_{bd}e_{ac},&[{f}_{ab},e_{cd}]=\delta_{bc}{f}_{ad}-\delta_{ad}{f}_{cb}+\delta_{ac}{f}_{bc}-\delta_{bd}f_{ac},\\\
\ \ \
[{f}_{ab},{f}_{cd}]=-\delta_{bc}e_{ad}+\delta_{ad}e_{cb}-\delta_{ac}e_{bd}-\delta_{bd}e_{ac}.\end{array}$

###### Proof.

We observe that $[E_{ab},E_{cd}]=\delta_{bc}E_{ad}-\delta_{ad}E_{cb}$ and the
lemma follows by direct computation. ∎

Then a choice of the modules in the decomposition (28) is the following:

$\displaystyle\mathfrak{m}_{0j}$ $\displaystyle=$
$\displaystyle\operatorname{span}\left\\{{e_{ab},f_{cd}\in\mathcal{B}:1\leq
a,c\leq n_{0},\ n_{0}+n_{1}+\cdots+n_{j-1}+1\leq b,d\leq
n_{0}+n_{1}+\cdots+n_{j}}\right\\},$ $\displaystyle\mathfrak{m}_{ij}$
$\displaystyle=$
$\displaystyle\operatorname{span}\\{{e_{ab},f_{cd}\in\mathcal{B}:n_{0}+n_{1}+\cdots+n_{i-1}+1\leq
a,c\leq n_{0}+n_{1}+\cdots+n_{i},}$ $\displaystyle
n_{0}+n_{1}+\cdots+n_{j-1}+1\leq b,d\leq n_{0}+n_{1}+\cdots+n_{j}\\},\quad
1\leq i<j\leq s,$ $\displaystyle\mathfrak{u}(n_{0})$ $\displaystyle=$
$\displaystyle\operatorname{span}\left\\{e_{ab},{f}_{cd}\in\mathcal{B}:1\leq
a<b=n,\ 1\leq c\leq d\leq n\right\\}.$

The equivalent modules in the decomposition of $\mathfrak{m}_{0j}$ are given
by

$\mathfrak{m}^{j}_{\ell}=\operatorname{span}\\{e_{\ell b},f_{\ell
d}\in\mathcal{B}:n_{0}+n_{1}+\cdots+n_{j-1}+1\leq b,d\leq
n_{0}+n_{1}+\cdots+n_{j}\\},\ \ell=1,\dots,n_{0}.$

Also,

$\mathfrak{u}(n_{j})=\operatorname{span}\\{e_{ab},f_{cd}\in\mathcal{B}:n_{0}+n_{1}+\cdots+n_{j-1}+1\leq
a,b,c,d\leq n_{0}+n_{1}+\cdots+n_{j}\\},\quad j=1,\dots,s.$

Then the $B$-orthogonal decompositions (27) and (28) can be depicted in the
following matrix, which shows the upper triangular part of $\mathfrak{u}(n)$:

$\begin{pmatrix}\mathfrak{u}(n_{0})&\mathfrak{m}_{01}&\mathfrak{m}_{02}&\mathfrak{m}_{03}&\cdots&\mathfrak{m}_{0s}\\\
&0_{n_{1}}&\mathfrak{m}_{12}&\mathfrak{m}_{13}&\cdots&\mathfrak{m}_{1s}\\\
&&0_{n_{2}}&\mathfrak{m}_{23}&\cdots&\mathfrak{m}_{2s}\\\
&\ast&&\ddots&&\vdots\\\ &&&&&0_{n_{s}}\end{pmatrix}$

The matrices $\mathfrak{m}_{0j}$ are of size $n_{0}\times n_{j}$, the matrices
$\mathfrak{m}_{ij}$ are of size $n_{i}\times n_{j}$, and the matrices
$\mathfrak{u}(n_{i})$ of size $n_{i}\times n_{i}$.

###  5.2. Proof of Theorem 1.4

Recall the spaces $\mathfrak{n}$ and $\mathfrak{p}$ defined in subsection 5.1
and the decompositions (27) and (28). We note that if $n_{0}=0$ (i.e.
$n_{1}+\cdots+n_{s}=n$), then $\mathfrak{n}=\\{0\\}$, and Theorem 1.4 follows
directly from Proposition 5.2 below. Assume henceforth that $n_{0}\geq 1$. To
prove Theorem 1.4 we need the following two propositions, which we will prove
at the end of the subsection.

######  Proposition 5.2.

$\left.A\right|_{\mathfrak{p}}=\lambda\operatorname{Id}$, where $\lambda>0$.

We further decompose
$\mathfrak{n}=\mathfrak{u}(1)\oplus\mathfrak{su}(n_{0})=\mathfrak{z}(\mathfrak{n})\oplus\mathfrak{su}(n_{0})$
into its simple and abelian ideals. With respect to the basis $\mathcal{B}$,
we have
$\mathfrak{z}(\mathfrak{n})=\operatorname{span}\\{f_{11}+\cdots+f_{n_{0}n_{0}}\\}$
and $\mathfrak{su}(n_{0})$ is the $B$-orthogonal complement of
$\mathfrak{z}(\mathfrak{n})$ in $\mathfrak{u}(n_{0})$.

######  Proposition 5.3.

Let
$\mathfrak{n}=\mathfrak{u}(1)\oplus\mathfrak{su}(n_{0})=\mathfrak{z}(\mathfrak{n})\oplus\mathfrak{su}(n_{0})$
be the decomposition of $\mathfrak{n}=\mathfrak{u}(n_{0})$ into its simple and
abelian ideals. If $n_{0}=1$ then
$\left.A\right|_{\mathfrak{n}}=\left.\mu\operatorname{Id}\right|_{\mathfrak{n}}$
for some $\mu>0$. If $n_{0}\geq 2$, then
$\left.A\right|_{\mathfrak{n}}=\begin{pmatrix}\left.\mu\operatorname{Id}\right|_{\mathfrak{z}(\mathfrak{n})}&0\\\
0&\left.\lambda\operatorname{Id}\right|_{\mathfrak{su}(n_{0})}\end{pmatrix}$,
where $\lambda$ is given by Proposition 5.2.

From Propositions 5.2 and 5.3, the decomposition
$\mathfrak{m}=\mathfrak{n}\oplus\mathfrak{p}=\mathfrak{u}(1)\oplus\mathfrak{su}(n_{0})\oplus\mathfrak{p}$,
and after normalizing the metric, we conclude that any g.o. metric on $G/H$
has necessarily the form (up to homothety)

$A=\begin{pmatrix}\left.\mu\operatorname{Id}\right|_{\mathfrak{z}(\mathfrak{n})}&0\\\
0&\left.\operatorname{Id}\right|_{\mathfrak{su}(n_{0})\oplus\mathfrak{p}}\end{pmatrix}.$
(33)

To conclude the proof of Theorem 1.4, it remains to prove that the above form
is also sufficient, i.e. the metrics $A$ are g.o. metrics. Let
$X\in\mathfrak{m}$. By Proposition 3.2, we need to find a vector
$a\in\mathfrak{h}$ such that

$[a+X,AX]=0.$ (34)

Let $X_{\mathfrak{p}}$, $X_{\mathfrak{su}(n_{0})}$ and
$X_{\mathfrak{z}(\mathfrak{n})}$ denote the projections of $X$ on
$\mathfrak{p}$, $\mathfrak{su}(n_{0})$ and $\mathfrak{z}(\mathfrak{n})$
respectively. Then

$X_{\mathfrak{z}(\mathfrak{n})}=r\sum_{i=1}^{n_{0}}{f_{ii}},\quad\mbox{for
some}\ r\in\mathbb{R},$

and $X_{\mathfrak{p}}=X_{1}+X_{2}$, where $X_{1}$ is the projection of
$X_{\mathfrak{p}}$ on $\mathfrak{m}_{01}\oplus\cdots\oplus\mathfrak{m}_{0s}$
and $X_{2}$ is the projection of $X_{\mathfrak{p}}$ on $\bigoplus_{1\leq
i<j\leq s}\mathfrak{m}_{ij}$. Moreover, we write

$X_{1}=\sum_{i=1}^{n_{0}}\sum_{j=n_{0}+1}^{n}{(a_{ij}e_{ij}+b_{ij}f_{ij})},\quad
a_{ij},b_{ij}\in\mathbb{R}.$

We also consider the vector

$\widetilde{X}_{1}:=\sum_{i=1}^{n_{0}}\sum_{j=n_{0}+1}^{n}{(b_{ij}e_{ij}-a_{ij}f_{ij})}.$

Finally, we choose the vector

$a=r(1-\mu)\sum_{i=n_{0}+1}^{n}{f_{ii}}\in\mathfrak{h}=\mathfrak{u}(n_{1})\oplus\cdots\oplus\mathfrak{u}(n_{s}).$

By using Lemma 5.1 it is straightforward to check the following relations:

$[X_{\mathfrak{z}(\mathfrak{n})},X_{1}]=-2r\widetilde{X}_{1},\ \
[X_{\mathfrak{z}(\mathfrak{n})},X_{2}]=0,\quad[a,X_{1}]=2r(1-\mu)\widetilde{X}_{1}\
\ \makebox{and}\ \ [a,X_{2}]=0.$ (35)

More specifically, the last relation can be verified by viewing both vectors
$a,X_{2}$ as elements of the Lie algebra
$\mathfrak{k}:=\mathfrak{u}(n_{1}+\cdots+n_{s})$, embedded diagonally in
$\mathfrak{g}$ as $\begin{pmatrix}0_{n_{0}\times n_{0}}&0\\\
0&\mathfrak{k}\end{pmatrix}$, and observing that $a$ lies in the center of
$\mathfrak{k}$. Finally, since $[\mathfrak{h},\mathfrak{n}]=0$ and
$[\mathfrak{z}(\mathfrak{n}),\mathfrak{n}]=0$, we obtain

$[a,X_{\mathfrak{z}(\mathfrak{n})}]=[a,X_{\mathfrak{su}(n_{0})}]=[X_{\mathfrak{z}(\mathfrak{n})},X_{\mathfrak{su}(n_{0})}]=0.$
(36)

We can now verify condition (34). Indeed, by taking into account relations
(35) and (36), as well as Equation (33), we obtain

$\displaystyle[a+X,AX]$ $\displaystyle=$
$\displaystyle[a+X_{\mathfrak{p}}+X_{\mathfrak{su}(n_{0})}+X_{\mathfrak{z}(\mathfrak{n})},X_{\mathfrak{p}}+X_{\mathfrak{su}(n_{0})}+\mu
X_{\mathfrak{z}(\mathfrak{n})}]$ $\displaystyle=$
$\displaystyle[a,X_{\mathfrak{p}}]+(1-\mu)[X_{\mathfrak{z}(\mathfrak{n})},X_{\mathfrak{p}}]$
$\displaystyle=$
$\displaystyle[a,X_{1}+X_{2}]+(1-\mu)[X_{\mathfrak{z}(\mathfrak{n})},X_{1}+X_{2}]$
$\displaystyle=$
$\displaystyle[a,X_{1}]+(1-\mu)[X_{\mathfrak{z}(\mathfrak{n})},X_{1}]$
$\displaystyle=$ $\displaystyle
2r(1-\mu)\widetilde{X}_{1}-2r(1-\mu)\widetilde{X}_{1}=0,$

and this concludes the proof of the theorem.∎

We now give the proofs of the above propositions.

_Proof of Proposition 5.2_. Since $H$ is connected, the Lie algebra of the
normalizer $N_{G}(H)$ coincides with
$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h})=\\{Y\in\mathfrak{g}:[Y,\mathfrak{h}]\subseteq\mathfrak{h}\\}$.
In our case,
$\mathfrak{h}=\mathfrak{u}(n_{1})\oplus\cdots\oplus\mathfrak{u}(n_{s})$ and

$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{u}(n_{0})\oplus\mathfrak{u}(n_{1})\oplus\cdots\oplus\mathfrak{u}(n_{s})=\mathfrak{n}\oplus\mathfrak{h}.$

Therefore, the tangent space of $G/N_{G}(H)$ coincides with $\mathfrak{p}$.
The normalizer Lemma 3.3 then implies that $\left.A\right|_{\mathfrak{p}}$
defines a $G$-invariant g.o. metric on $G/N_{G}(H)$.

Similarly to the space
$\operatorname{SO}(n)/\operatorname{SO}(n_{1})\times\cdots\times\operatorname{SO}(n_{s})$,
by taking into account Lemma 5.1 and the expressions of the subspaces
$\mathfrak{m}_{ij},\mathfrak{u}(n_{j}),\mathfrak{u}(n_{0})$ in terms of the
basis $\mathcal{B}$, we deduce that the submodules $\mathfrak{m}_{ij}$, $0\leq
i<j\leq s$ are
$\operatorname{ad}(\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}))$-invariant,
$\operatorname{ad}(\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}))$-irreducible and
pairwise inequivalent. Therefore,

$\left.A\right|_{\mathfrak{m}_{ij}}=\lambda_{ij}\operatorname{Id}.$ (37)

It remains to prove that all $\lambda_{ij}$ are equal. To this end, we will
use similar arguments as in Step 2 in Proposition 4.4. More specifically,
Lemma 5.1 yields

$[\mathfrak{m}_{ij},\mathfrak{m}_{jl}]=\mathfrak{m}_{il}\ \ \makebox{for all}\
\ 0\leq i<j<l\leq s.$

Using the above relation, along with Equation (37), the
$\operatorname{ad}({\mathfrak{h}})$-invariance of
$\mathfrak{m}_{ij},\mathfrak{m}_{jl}$ and their $B$-orthogonality, part 2. of
Lemma 3.8 yields that

$\lambda_{ij}=\lambda_{jl}=\lambda_{il}\ \ \makebox{for all}\ \ 0\leq
i<j<l\leq s.$

By Lemma 4.7, we deduce that the set $R_{s}=\\{\lambda_{ij}:\ 0\leq i<j\leq
s\\}$ of the eigenvalues of $\left.A\right|_{\mathfrak{p}}$ has only one
element, therefore $\left.A\right|_{\mathfrak{p}}=\lambda\operatorname{Id}$.∎

_Proof of Proposition 5.3_. The Lie algebra $\mathfrak{n}=\mathfrak{u}(n_{0})$
coincides with the Lie algebra of $N_{G}(H)/H$. By Lemma 3.4,
$\left.A\right|_{\mathfrak{n}}$ defines a bi-invariant (and hence g.o.) metric
on $U(n_{0})$, which in turn corresponds to an $\operatorname{Ad}$-invariant
inner product on $\mathfrak{u}(n_{0})$. Since the center
$\mathfrak{z}(\mathfrak{n})$ of $\mathfrak{n}$ is one-dimensional, Lemma 3.7
yields

$\left.A\right|_{\mathfrak{n}}=\begin{pmatrix}\left.\mu\operatorname{Id}\right|_{\mathfrak{z}(\mathfrak{n})}&0\\\
0&\left.\widetilde{\lambda}\operatorname{Id}\right|_{\mathfrak{su}(n_{0})}\end{pmatrix},\
\ \widetilde{\lambda}>0.$

If $n_{0}=1$, then $\mathfrak{n}=\mathfrak{u}(1)=\mathfrak{z}(\mathfrak{n})$,
verifying Proposition 5.3 for this case. Assume that $n_{0}\geq 2$. It remains
to show that $\widetilde{\lambda}$ is equal to the eigenvalue $\lambda$ given
in Proposition 5.2. We recall the spaces
$\mathfrak{m}_{0j}=\mathfrak{m}^{j}_{1}\oplus\cdots\oplus\mathfrak{m}^{j}_{n_{0}}$
defined in Section 5.1. By Proposition 5.2, we have

$\left.A\right|_{\mathfrak{m}^{j}_{i}}=\lambda\operatorname{Id}.$

Choose the vectors $e_{12}\in\mathfrak{su}(n_{0})$ and
$e_{1,n_{0}+1}\in\mathfrak{m}^{1}_{1}$. By Lemma 5.1, we have

$[e_{12},e_{1,n_{0}+1}]=-e_{2,n_{0}+1}\in\mathfrak{m}^{1}_{2}.$

Therefore, $[\mathfrak{su}(n_{0}),\mathfrak{m}_{1}^{1}]$ has non-zero
projection on $(\mathfrak{su}(n_{0})\oplus\mathfrak{m}_{1}^{1})^{\bot}$. Along
with the $\operatorname{ad}(\mathfrak{h})$-invariance of
$\mathfrak{su}(n_{0})$ and $\mathfrak{m}_{1}^{1}$ and the facts that
$\left.A\right|_{\mathfrak{m}^{j}_{i}}=\lambda\operatorname{Id}$ and
$\left.A\right|_{\mathfrak{su}(n_{0})}=\widetilde{\lambda}\operatorname{Id}$,
Lemma 3.8 yields $\lambda=\widetilde{\lambda}$. ∎

## References

  * [1] Agricola I., Ferreira A. C. and Friedrich, T.: The classification of naturally reductive homogeneous spaces in dimensions $n\leq 6$, Differential Geom. Appl. 39 (2015) 59–92.
  * [2] Alekseevsky D. and Arvanitoyeorgos A.: Metrics with homogeneous geodesics on flag manifolds, Trans. Amer. Math. Soc. 359(8) (2007) 3769–3789.
  * [3] Alekseevsky D.V. and Nikonorov Yu. G.: Compact Riemannian manifolds with homogeneous geodesics, SIGMA Symmetry Integrability Geom. Methods Appl. 5(93) (2009) 16 pages.
  * [4] Arvanitoyeorgos A.: An Introduction to Lie groups and the Geometry of homogenous spaces, American Mathematical Society, 2003.
  * [5] Arvanitoyeorgos A.: Homogeneous manifolds whose geodesics are orbits. Recent results and some open problems, Irish Math. Soc. Bull. 79 (2017) 5–29.
  * [6] Arvanitoyeorgos A., Souris N.P. and Statha M.: Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds, J. Geom. Phys. 165 (2021) 104223.
  * [7] Berstovskii V. N. and Nikonorov Y. G.: On $\delta$-homogeneous Riemannian manifolds, Differential Geom. Appl. 26 (2008) 514–535.
  * [8] Berstovskii V. N. and Nikonorov Y. G.: Clifford-Wolf homogeneous Riemannian manifolds, J. Differential Geom. 82(3) (2009) 467–500.
  * [9] Berstovskii V. N. and Nikonorov Y. G.: Riemannian Manifolds and Homogeneous Geodesics, Springer 2020.
  * [10] Berndt J., Kowalski O. and Vanhecke L.: Geodesics in weakly symmetric spaces, Ann. Global Anal. Geom. 15 (1997) 153–156.
  * [11] Calvaruso G. and Zaeim, A.: Four-dimensional pseudo-Riemannian g.o. spaces and manifolds, J. Geom. Phys. 130 (2018) 63–80.
  * [12] Chen H., Chen Z. and Deng S.: Compact simple Lie groups admitting left invariant Einstein metrics that are not geodesic orbit C. R. Math. Acad. Sci. Paris 356 (2018) 81–84.
  * [13] Chen H., Chen Z. and Wolf, J.: Geodesic orbit metrics on compact simple Lie groups arising from flag manifolds, C. R. Math. Acad. Sci. Paris 356 (2018) 846–851.
  * [14] Chen Z. and Nikonorov Y. G.: Geodesic orbit Riemannian spaces with two isotropy summands I, Geom. Dedicata 203 (2019) 163–178.
  * [15] Chen Z., Nikolayevski Y. and Nikonorov Y. G.: Compact geodesic orbit spaces with a simple isotropy group, arXiv:2009.06329 (2020).
  * [16] Dušek Z.: Homogeneous geodesics and g.o. manifolds, Note Mat. 38(1) (2018) 1–15.
  * [17] Gordon C.S.: Homogeneous Riemannian manifolds whose geodesics are orbits, Topics in Geometry: In Memory of Joseph D’Atri (Ed. S. Gindikin), Progress in Nonlinear Differential Equations 20, Birkhauser- Verlag, Boston, Basel, Berlin, 1996, 155–174.
  * [18] Gordon C. S. and Nikonorov Y. G.: Geodesic orbit Riemannian structures on $R^{n}$, J. Geom. Phys. 134 (2018) 235–243.
  * [19] Grajales B., Grama L. and Negreiros C.J.: Geodesic orbit spaces in real flag manifolds, Comm. Anal. Geom. 28(8) (2020) 1933–2003.
  * [20] Kowalski O. and Vanhecke L.: Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. B 7(5) (1991) 189–246.
  * [21] Nikonorov Yu. G.: Geodesic orbit Riemannian metrics on spheres, Vladikavkaz. Mat. Zh. 15(3) (2013) 67–76.
  * [22] Nikonorov Y. G.: On the structure of geodesic orbit Riemannian spaces, Ann. Global Anal. Geom. 52 (2017) 289–311.
  * [23] Nikonorov Y. G.: On left invariant Einstein Riemannian metrics that are not geodesic orbit, Transform. Groups 24 (2019) 511-530.
  * [24] Souris N. P.: Geodesic orbit metrics in compact homogeneous manifolds with equivalent isotropy submodules, Transform. Groups 23 (2018) 1149–1165.
  * [25] Souris N. P.: On a class of geodesic orbit spaces with abelian isotropy subgroup, Manuscripta Math. (2020). https://doi.org/10.1007/s00229-020-01236-9
  * [26] Souris N. P.: Geodesic orbit spaces of compact Lie groups of rank two, arXiv:2008.13588v2 (2020).
  * [27] Storm R.: The classification of 7- and 8-dimensional naturally reductive spaces, Canad. J. Math. 72(5) (2020) 1246–1274.
  * [28] Tamaru H.: Riemannian g.o. spaces fibered over irreducible symmetric spaces, Osaka J. Math. 36 (1999) 835–851.
  * [29] Wolf J. A.: The geometry and structure of isotropy irreducible homogeneous spaces, Acta Math. 120 (1968) 59-148.
  * [30] Wolf J.A.: Harmonic analysis on commutative spaces, Mathematical Surveys and Monographs, Vol. 142, American Mathematical Society, Providence, RI, 2007.
  * [31] Yan Z. and Deng S.: Finsler spaces whose geodesics are orbits, Differential Geom. Appl. 36 (2014) 1–23.