# Scalar curvatures of invariant almost Hermitian structures on flag manifolds
with two and three isotropic summands

Lino Grama  Imecc - Unicamp, Departamento de Matemática. Rua Sérgio Buarque de
Holanda, 651, Cidade Universitária Zeferino Vaz. 13083-859 Campinas - SP,
Brazil<EMAIL_ADDRESS>and  Ailton R. Oliveira  UEMS - Universidade
Estadual de Mato Grosso do Sul - MS, Cidade Universitária de Dourados, Rodovia
Itahum, Km 12 s/n - Jardim Aeroporto, Dourados - MS, Brazil
<EMAIL_ADDRESS>
###### Abstract.

In this paper we study invariant almost Hermitian geometry on generalized flag
manifolds which the isotropy representation decompose into two or three
irreducible components. We will provide a classification of such flag
manifolds admitting Kähler like scalar curvature metric, that is, almost
Hermitian structures $(g,J)$ satisfying $s=2s_{C}$ where $s$ is Riemannian
scalar curvature and $s_{C}$ is the Chern scalar curvature.

##  1\. Introduction

In the present paper we will study curvature properties of invariant almost
Hermitian structures on homogeneous spaces. We search for Kähler like scalar
curvature (Klsc) metrics, that is, almost Hermitian structures $(g,J)$
satisfying $s=2s_{C}$ where $s$ is the Riemannian scalar curvature and $s_{C}$
is the Chern scalar curvature on generalized flag manifolds with two or three
isotropy summands.

Recall the geometric meaning of the Riemannian scalar curvature $s$: given a
Riemannian manifold $(M,g)$, the volume of the geodesic ball of radius $r$
with center at $p\in M$ has the asymptotic expansion as follows

(1.1)
$\operatorname{vol}(B(p,r))=\omega_{n}r^{n}\left(1-\dfrac{s(p)}{6(n+2)}r^{2}+\mathcal{O}(r^{4})\right),$

where $\omega_{n}$ is the volume of the unity ball in $\mathbb{R}^{n}$, see
[4], [6]. Therefore the Riemannian scalar curvature $s(p)$ is positive or
negative at a point $p$, if the volume of a small geodesic ball at $p$ is
respectively smaller or larger than the corresponding Euclidean ball of the
same radius. It is well known that if $(g,J)$ is Kähler, then $s=2s_{C}$ (see
[17]). On the other hand, if the metric is non-Kähler, there is no immediate
geometric interpretation for the Chern scalar curvature $s_{C}$. However if
the Hermitian structure is non-Kähler but satisfies $s=2s_{C}$, has the
standard representation such as the one provided by equation (1.2). Another
motivation is that the existence of Klsc metric allow us to ask about a
Kazdan–Warner–type theorem (see [13]) for admissible functions of scalar
curvature, in the case of Chern scalar curvature, such as in [8]. For results
on geometric flows related to complex manifolds (e.g. Chern-Ricci flow) we
suggest [20].

Dabkowski and Lock in [6] call the Hermitian metrics satisfying

(1.2) $s=2s_{C}$

by Kähler like scalar curvature (Klsc) metric, and the authors exhibit
examples of non-compact Hermitian manifolds satisfying the equation 1.2.

According to [16] and [2] nearly-Kähler manifolds and almost-Kähler manifolds
satisfying $s=2s_{C}$ are Kähler. Fu and Zhou showed in [7] that, if the pair
$(g,J)$ belongs to the Gray-Hervella class
$\mathcal{W}_{2}\oplus\mathcal{W}_{3}\oplus\mathcal{W}_{4}$ with $s=2s_{c}$,
then $(g,J)$ is Kähler. Recently, Lejmi-Upmeier proposed the following
Question (Remark 3.3 of [16]):

######  Question 1.1.

Do higher-dimensional closed almost Hermitian non-Kähler manifolds with
$2s_{C}=s$ exist?

On the other hand, let us denote by $\nabla^{t}$ be the 1-parameter family of
connections introduced by Gauduchon in [10]:

$\displaystyle g(\nabla^{t}_{X}Z,Y)$ $\displaystyle=$ $\displaystyle
g(D_{X}Z,Y)-\frac{1}{2}g(J(D_{X}J)Z,Y)$ $\displaystyle+$
$\displaystyle\frac{t}{4}g((D_{JY}J+JD_{Y}J)X,Z)-\frac{t}{4}g((D_{JZ}J+JD_{Z}J)X,Y),$

where $D$ is the Levi-Civita connection. It is well know that $\nabla^{1}$
coincides with the Chern connection. According to [7], let us define the
Hermitian scalar curvatures

$s_{1}(t)=R^{t}(u_{\overline{i}},u_{i},u_{j},u_{\overline{j}})$

and

$s_{2}(t)=R^{t}(u_{\overline{i}},u_{j},u_{i},u_{\overline{j}}),$

where $R^{t}$ is the curvature tensor associated to $\nabla^{t}$ and
$\\{u_{i}\\}_{i=1,2,\cdots,n}$ is a unitary frame. We will call $s_{1}(t)$ and
$s_{2}(t)$ the first and second Hermitian scalar curvatures. One of the main
features of the generalized flag manifolds relies in the fact that the scalar
curvature $s_{1}(t)$ is constant. This manner, thereon we shall denote it
solely by $s_{1}$. Furthermore, it coincides with the Chern scalar curvature
$s_{C}$. See Section 3 for details about the comments above.

With the discussion above in mind, let us rephrase the Question 1.1 in our
context:

######  Question 1.2.

Do higher-dimensional generalized flag manifolds $G/K$ equipped with
$G$-invariant almost Hermitian structure with $2s_{1}=s$ exist?

In our previous [11] we study the Question 1.2 in a class of generalized flag
manifolds equipped with an invariant almost Hermitian structure. In this work
we provided a complete answer to Question 1.2 in the context of invariant
Hermitian structures for a large class of homogeneous space, namely flag
manifolds with two and three isotropy summands. As we have done in [11], we
proceed by computing explicitly the Hermitian scalar curvatures using the
tools of Lie Theory. We recall the description of these manifolds in Sections
4 and 5.

We obtain the following results (for a description of invariant metrics and
almost complex structures on flag manifolds, see Section 2):

######  Theorem 1.3 (= Theorem 4.1).

Let $G/K$ be a generalized flag manifold with two isotropy summands, and let
$g=(\lambda_{1},\lambda_{2})$ be an invariant Riemannian metric. Then
$2s_{1}-s=0$ holds for $G/K$ if the pair $(g,J_{i})$ satisfies the following
conditions:

  * •

For the invariant complex struture $J=(+,+)$, the metric $g$ is parameterized
by $\lambda_{2}=2\lambda_{1}$. In this case, $(g,J)$ is Kähler.

  * •

For the invariant almost complex structure $J_{1}=(+,-)$, the metric $g$ is
parameterized by $\lambda_{2}=2\lambda_{1}(\sqrt{10}+3)$. In this case,
$(g,J_{1})\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$.

The analysis of the equation $2s_{1}-s=0$ in the case of flag manifolds with
three isotropy summands is a little bit trickier, so we deal with the question
working case-by-case. The description of the flag manifolds with three
summands splits into two sub-families: Type I listed in Table 4 and Type II
listed in Table 5. Theorem 1.4 below encodes our results in a simple fashion.
The details and explicit computations can be found in Section 5. It is worth
to point out that when $(g,J)$ is Kähler, the equation $2s_{1}-s=0$ is
satisfied. In this case, by non-trivial solution of $2s_{1}-s=0$ we mean that
the pair $(g,J)$ is non-Kähler.

######  Theorem 1.4.

Let $G/K$ be a generalized flag manifold with three isotropy summands,
equipped with an invariant almost complex structure (IACS) $J_{i}$,
$i=1,\ldots,4$. Then there exists an invariant Riemannian metric $g$ such that
the almost Hermitian structure $(g,J_{i})$ admits a non-trivial solution of
the equation $2s_{1}-s=0$ in the cases listed in the Table 1. Moreover, for
each IACS admiting non-trivial solution we provide explicity families of
invariant Kähler-like scalar metrics $g$ on $G/K$.

Table 1. Kähler-like scalar curvature metric on flag manifolds with 3 isotropy summands. IACS |  | Flag manifold
---
$G/K$
Integrable ? |  | non-trivial solution
---
$2s_{1}-s=0$
Details
$J_{1}=(+,+,+)$ | type I | ✓ | ✗ |  Thm 5.2
type II | ✓ | ✗ |  Thm 5.2
$J_{2}=(-,+,+)$ | type I | ✗ | ✓ |  Thm 5.3
type II | ✓ | ✗ |  Thm 5.3
$J_{3}=(+,-,+)$ | type I | ✗ | ✓ |  Thm 5.4
type II | ✓ | ✗ |  Thm 5.4
$J_{4}=(+,+,-)$ | type I | ✗ | ✓ |  Thm 5.5
type II | ✗ | ✓ |  Thm 5.5

In the last Section of the paper we work out three examples of flag manifolds
with three isotropy summands: $G_{2}/U(2)$, $SU(n+2)/S(U(n)\times U(1)\times
U(1))$ and $F_{4}/SU(3)\times SU(2)\times U(1)$, providing family of example
of Kähler-like scalar curvature metrics by using the techniques we have
developed in this work.

Acknowledgments:  We would like to thank Leonardo Cavenaghi and Eder Correa
for for useful discussions. LG is partially supported by FAPESP grants
2018/13481-0, 2021/04003-0, 2021/04065-6 and CNPq grant no. 305036/2019-0.

##  2\. Invariant geometry of flag manifolds

In this section we recall some well known results about the almost Hermitian
geometry of generalized flag manifolds, from a Lie theoretical point of view.

###  2.1. Generalized flag manifolds

Let $\mathfrak{g}^{\mathbb{C}}$ be a complex semi-simple Lie algebra. Given a
Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}^{\mathbb{C}}$, denotes by
$\Pi$ the set of roots with respect to the pair
$(\mathfrak{g}^{\mathbb{C}},\mathfrak{h})$. We have the following root space
decomposition

$\mathfrak{g}^{\mathbb{C}}=\mathfrak{h}\oplus\sum_{\alpha\in\Pi}\mathfrak{g}_{\alpha},$

where $\mathfrak{g}_{\alpha}=\\{X\in\mathfrak{g}:\forall
H\in\mathfrak{h},[H,X]=\alpha(H)X\\}$ denote the corresponding $1$-dimensional
(complex) root space.

The Cartan-Killing form is defined by

$\langle
X,Y\rangle=\operatorname{tr}(\operatorname{ad}(X)\operatorname{ad}(Y))$

and its restriction to $\mathfrak{h}$ is non-degenerate. Given a root
$\alpha\in\mathfrak{h}^{*}$ we define $H_{\alpha}$ by $\alpha(\cdot)=\langle
H_{\alpha},\cdot\rangle$. Moreover we denote
$\mathfrak{h}_{\mathbb{R}}=\operatorname{span}_{\mathbb{R}}\\{H_{\alpha}:\alpha\in\Pi\\}$
and $\mathfrak{h}_{\mathbb{R}}^{*}$ being the real subspace of
$\mathfrak{h}^{*}$ spanned by the roots.

We will fix a Weyl basis of $\mathfrak{g}^{\mathbb{C}}$ as follows:

$X_{\alpha}\in\mathfrak{g}_{\alpha},\ \mbox{ such that }\ \langle
X_{\alpha},X_{-\alpha}\rangle=1\ \mbox{and}\
[X_{\alpha},X_{\beta}]=m_{\alpha,\beta}X_{\alpha+\beta},$

with $m_{\alpha,\beta}\in\mathbb{R}$, $m_{-\alpha,-\beta}=-m_{\alpha,\beta}$
and $m_{\alpha,\beta}=0$ if $\alpha+\beta$ is not a root.

Let $\Pi^{+}\subset\Pi$ be a choice of positive roots, $\Sigma$ be the
corresponding system of simple roots and $\Theta$ be a subset of $\Sigma$. We
will use the following notation: $\langle\Theta\rangle$ is the set of roots
spanned by $\Theta$, $\Pi_{M}=\Pi\setminus\langle{\Theta}\rangle$ be the set
of complementary roots and $\Pi_{M}^{+}$ be the set of complementary positive
roots.

Let

$\mathfrak{p}_{\Theta}=\mathfrak{h}\oplus\sum_{\alpha\in\langle\Theta\rangle^{+}}\mathfrak{g}_{\alpha}\oplus\sum_{\alpha\in\langle\Theta\rangle^{+}}\mathfrak{g}_{-\alpha}\oplus\sum_{\beta\in\Pi_{M}^{+}}\mathfrak{g}_{\beta}$

be a parabolic sub-algebra of $\mathfrak{g}^{\mathbb{C}}$ determined by
$\Theta$.

The generalized flag manifold $\mathbb{F}_{\Theta}$ is the homogeneous space

$\mathbb{F}_{\Theta}=G^{\mathbb{C}}/P_{\Theta},$

where $G^{\mathbb{C}}$ is a complex connected Lie group with Lie algebra
$\mathfrak{g}^{\mathbb{C}}$ and $P_{\Theta}$ is the normalizer of
$\mathfrak{p}_{\Theta}$ in $G^{\mathbb{C}}$.

Let $\mathfrak{g}$ be the compact real form of $\mathfrak{g}^{\mathbb{C}}$. We
have

$\mathfrak{g}=\operatorname{span}_{\mathbb{R}}\\{i\mathfrak{h}_{\mathbb{R}},A_{\alpha},iS_{\alpha};\alpha\in\Pi\\},$

where $A_{\alpha}=X_{\alpha}-X_{-\alpha}$ and
$S_{\alpha}=X_{\alpha}+X_{-\alpha}$. We remark that the Lie algebra
$\mathfrak{g}$ is semi-simple.

Denote by $G$ the compact real form of $G^{\mathbb{C}}$ with
$\operatorname{Lie}(G)=\mathfrak{g}$ and let $\mathfrak{k}_{\Theta}$ the Lie
algebra of $K_{\Theta}:=P_{\Theta}\cap G$. It is well known that the Lie group
$K_{\Theta}\subset G$ is a centralizer of a torus.

We have

$\mathfrak{k}_{\Theta}^{\mathbb{C}}=\mathfrak{h}\oplus\sum_{\alpha\in\langle\Theta\rangle^{+}}\mathfrak{g}_{\alpha}\oplus\sum_{\alpha\in\langle\Theta\rangle^{+}}\mathfrak{g}_{-\alpha},$

where $\mathfrak{k}_{\Theta}^{\mathbb{C}}$ denotes the complexification of the
real Lie algebra
$\mathfrak{k}_{\Theta}=\mathfrak{g}\cap\mathfrak{p}_{\Theta}$.

The Lie group $G$ also acts transitively on $\mathbb{F}_{\Theta}$ and we have

$\mathbb{F}_{\Theta}=G^{\mathbb{C}}/P_{\Theta}=G/(P_{\Theta}\cap
G)=G/K_{\Theta}.$

When $\Theta=\emptyset$ the parabolic sub-algebra is a Borel sub-algebra of
$\mathfrak{g}^{\mathbb{C}}$ and $\mathbb{F}=G^{\mathbb{C}}/P=G/T$ is called
full flag manifold, where $T=P\cap G$ is a maximal torus of $G$.

Recall that the flag manifold $\mathbb{F}_{\Theta}=G/K_{\Theta}$ is a
reductive homogeneous space, that is, there exists a subspace $\mathfrak{m}$
of $\mathfrak{g}$ such that

$\mathfrak{g}=\mathfrak{k}_{\Theta}\oplus\mathfrak{m}\ \ \mbox{and}\ \
\operatorname{Ad}(k)\mathfrak{m}\subseteq\mathfrak{m},\forall k\in
K_{\Theta},$

and one can identify the tangent space $T_{x_{0}}\mathbb{F}_{\Theta}$ with
$\mathfrak{m}$, where $x_{0}=eK_{\Theta}$ is the origin of the
$\mathbb{F}_{\Theta}$ (trivial coset).

Let us give a description of the tangent space $\mathfrak{m}$ in terms of the
Lie algebra structure of $\mathfrak{g}$ as follows:

$\mathfrak{g}=\mathfrak{k}_{\Theta}\oplus\sum_{\beta\in\Pi_{M}}{\mathfrak{u}_{\beta}},$

with
$\mathfrak{u}_{\beta}=\mathfrak{g}\cap(\mathfrak{g}_{\beta}\oplus\mathfrak{g}_{-\beta})$
and for each root $\beta\in\Pi_{M}$, $\mathfrak{u}_{\beta}$ has real dimension
two and it is spanned by $A_{\beta}$ and $\sqrt{-1}S_{\beta}$, and we have the
following identification
$\mathfrak{m}=\sum_{\beta\in\Pi_{M}}\mathfrak{u}_{\beta}$.

An important tool to study invariant tensors on homogeneous space is the
isotropy representation. We will restrict ourselves to the situation of flag
manifolds. In this case, since the flag manifolds are reductive homogeneous
spaces it is well know that the isotropy representation is equivalent to the
following representation

$\operatorname{Ad}(k)\mid_{\mathfrak{m}}:\mathfrak{m}\longrightarrow\mathfrak{m}.$

The isotropy representation decomposes $\mathfrak{m}$ into irreducible
components, that is,

$\mathfrak{m}=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\cdots\oplus\mathfrak{m}_{n},$

where each component $\mathfrak{m}_{i}$ satisfies
$\operatorname{Ad}(K_{\Theta})(\mathfrak{m}_{i})\subset\mathfrak{m}_{i}$.
Moreover, each component $\mathfrak{m}_{i}$ is irreducible, that is, the only
invariant sub-spaces of $\mathfrak{m}_{i}$ by
$\operatorname{Ad}(K_{\Theta})|_{\mathfrak{m}_{i}}$ are the trivial sub-
spaces. We will call the sub-spaces $\mathfrak{m}_{i}$ by isotropic summands
of the isotropy representation.

######  Remark 2.1.

From now on we will omit the symbol $\Theta$ whenever there is no risk of
confusion. We will denote a flag manifold just by $\mathbb{F}=G/K$.

###  2.2. Invariant Metrics

Let us denote by $(\cdot,\cdot)$ the Cartan-Killing form of $\mathfrak{g}$.
For each $\operatorname{Ad}(K)-$invariant inner product
$(\cdot,\cdot)_{\Lambda}$ on $\mathfrak{m}$, there exists a unique
$(\cdot,\cdot)-$self-adjoint, positive operator
$\Lambda:\mathfrak{m}\longrightarrow\mathfrak{m}$ commuting with
$\operatorname{Ad}(k)\left|{}_{\mathfrak{m}}\right.$ for all $k\in K$ such
that

$(X,Y)_{\Lambda}=(\Lambda X,Y),\,\,X,Y\in\mathfrak{m}.$

Therefore an invariant Riemannian metric $g$ on $\mathbb{F}$ is completely
determined by the invariant inner product $(X,Y)_{\Lambda}$, and the inner
product is determined by $\Lambda$.

The vectors $A_{\alpha}$, $\sqrt{-1}S_{\alpha}$, $\alpha\in\Pi$, are
eigenvectors of $\Lambda$ associated to the same eigenvalue
$\lambda_{\alpha}$.

The invariant inner product $(X,Y)_{\Lambda}$ admits a natural extension to a
symmetric bi-linear form on $\mathfrak{m}^{\mathbb{C}}$. On the complexified
tangent space we have $\Lambda(X_{\alpha})=\lambda_{\alpha}X_{\alpha}$ with
$\lambda_{\alpha}>0$ and $\lambda_{-\alpha}=\lambda_{\alpha}$.

Notation: In the sequence of this work, we will abuse notation and will denote
the invariant metric $g$ just by the operator $\Lambda$ associated to the
invariant inner product. We also denote the invariant metric $g$ just by a
$n$-tuple of positive numbers $(\lambda_{1},\ldots,\lambda_{n})$ representing
the eigenvalues of the operator $\Lambda$ and is parameterized by the number
of irreducible components. ∎

###  2.3. Invariant Almost Complex Structures

######  Definition 2.2.

An almost complex structure on the flag manifold $\mathbb{F}$ is a tensor $J$
such that for every point $x\in\mathbb{F}$, there is an endomorphism
$J:T_{x}\mathbb{F}\longrightarrow T_{x}\mathbb{F}$ such that
$J^{2}=-\operatorname{id}$.

######  Definition 2.3.

A $G$-invariant almost complex structure $J$ on $\mathbb{F}=G/K$ is an almost
complex structure that satisfies

$J_{ux}=dE_{u}J_{x}dE_{u^{-1}},\ \mbox{for all }\ u\in G,$

where $dE_{u}:T(G/K)\to T(G/K)$ denotes the differential of the left
translation by $u$, that is, for all $X\in T_{x}(G/K)$ we have

$dE_{u}J_{x}X=J_{ux}dE_{u}X.$

The following result allows us to describe invariant almost complex structures
on flag manifolds in terms of complex structure in a simple vector space,
namely the tangent space at the origin (trivial coset) of the homogeneous
space.

######  Proposition 2.4.

There exist an $1-1$ correspondence between a $G$-invariant almost complex
structure $J$ and a linear endomorphism $J_{x_{0}}:T_{x_{0}}\mathbb{F}\to
T_{x_{0}}\mathbb{F}$ satisfying $J_{x_{0}}^{2}=-\operatorname{id}$ and commute
with the isotropy representation, that is,

$\operatorname{Ad}^{G/K}(k)J_{x_{0}}=J_{x_{0}}\operatorname{Ad}^{G/K}(k),\
\mbox{for all }\ k\in K.$

An interesting consequence of the Proposition 2.4 is that
$J(\mathfrak{g}_{\alpha})=\mathfrak{g}_{\alpha}$, where
$\mathfrak{g}_{\alpha}$ is the root space associated to the root
$\alpha\in\Pi$. The eigenvalue of $J$ are $\pm\sqrt{-1}$ and the eigenvectors
on $\mathfrak{m}^{\mathbb{C}}$ are $X_{\alpha}$, $\alpha\in\Pi$. Therefore
$J(X_{\alpha})=\varepsilon_{\alpha}\sqrt{-1}X_{\alpha}$, with
$\varepsilon_{\alpha}=\pm 1$ and
$\varepsilon_{\alpha}=-\varepsilon_{-\alpha}$.

As a consequence of the discussion above we conclude that an invariant almost
complex structure on $\mathbb{F}$ is completely described by a set of signals

$\left\\{\varepsilon_{\alpha}=\pm 1,\ \alpha\in\Pi_{M},\mbox{ satisfying
}\varepsilon_{\alpha}=-\varepsilon_{-\alpha}\right\\}.$

######  Proposition 2.5 ([5],13.4).

Consider the almost complex homogeneous space $M=G/K$ and assume that the
isotropy representation admits a decomposition into irreducible and pairwise
non-equivalent components, namely,
$\mathfrak{m}=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\cdots\oplus\mathfrak{m}_{s}$.
Then $M$ admits $2^{s}$ invariant almost complex structures.

If we identify the conjugated invariant almost complex structures, then $M$
admits $2^{s-1}$ invariant almost complex structures, up to conjugation.

##  3\. Scalar Curvatures of invariant almost Hermitian structures

###  3.1. Review: general results about curvatures of almost Hermitian
structures

Let $(M,J,g)$ be an almost Hermitian manifold with real dimension $2n$, with
$J$ being an almost complex structure orthogonal with respect to the
Riemannian metric $g$. A linear connection $\nabla$ on $M$ is Hermitian if it
preserves the metric $g$ and the almost complex structure $J$, that is,
$\nabla g=0$ and $\nabla J=0$ (we are not assuming that $J$ is integrable).

Lets us recall the $1$-parameter family of Hermitian connection defined by
Gauduchon in [10] as follow:

$\displaystyle g(\nabla^{t}_{X}Z,Y)$ $\displaystyle=$ $\displaystyle
g(D_{X}Z,Y)-\frac{1}{2}g(J(D_{X}J)Z,Y)$ $\displaystyle+$
$\displaystyle\frac{t}{4}g((D_{JY}J+JD_{Y}J)X,Z)-\frac{t}{4}g((D_{JZ}J+JD_{Z}J)X,Y),$

where $D$ is the Levi-Civita connection.

There are three special cases:

  * i)

$t=0$, $\nabla^{0}$ is the first canonical Hermitian connection, also known as
Lichnerowicz connection or minimal connection.

  * ii)

$t=1$, $\nabla^{1}$ is the second canonical Hermitian connection, also known
as Chern connection (this connection was used by Chern in the integrable case.

  * iii)

$t=-1$, $\nabla^{-1}$ is the Bismut connection. In the integrable case,
$\nabla^{-1}$ is characterized by its anti-symmetric torsion.

According to [7], let us define the Hermitian scalar curvatures $s_{1}(t)$ and
$s_{2}(t)$ by

(3.1) $s_{1}(t)=R^{t}(u_{\overline{i}},u_{i},u_{j},u_{\overline{j}})$ (3.2)
$s_{2}(t)=R^{t}(u_{\overline{i}},u_{j},u_{i},u_{\overline{j}})$

where $R^{t}$ is the curvature tensor associated to $\nabla^{t}$ and
$\\{u_{i}\\}_{i=1,2,\cdots,n}$ is a unitary frame. We call $s_{1}(t)$ and
$s_{2}(t)$ the first and second Hermitian scalar curvatures, respectively.

For an Hermitian manifold equipped with the Chern connection $\nabla^{1}$ or
the Bismut connection $\nabla^{-1}$, the relations between the Hermitian
scalar curvatures and Riemannian scalar curvatures were widely studied, see
for instance [9] e [15].

The Nijenhuis tensor $N$ is given by

$N(X,Y)=-[JX,JY]+J[JX,Y]+J[X,JY]+[X,Y],$

where $X,Y\in\Gamma(TM)$.

Let us consider the fundamental (or Kähler) 2-form

$F(X,Y)=g(JX,Y).$

The Lee form $\alpha_{F}$ of $(M,J,g)$ is defined by

$\alpha_{F}=J\delta F,$

where $\delta=-*d*$ is the codifferential with respect to $g$. The Lee form is
also defined by

$dF=(dF)_{0}+\dfrac{1}{n-1}\alpha_{F}\wedge F,$

where $(dF)_{0}$ is the primitive part of $dF$.

The covariant derivative of $F$ with respect to the Levi-Civita connection $D$
is

$(DF)(X,Y,Z)=\dfrac{1}{2}\left[dF(X,Y,Z)-dF(X,JY,JZ)-N(JX,Y,Z)\right].$

Moreover,

$(DF)(X,Y,Z)=-(DF)(X,Z,Y)=-(DF)(X,JY,JZ).$

The following expression of $DF$ will be very useful for our purposes (see
[10], [7]).

$(DF)(X,Y,Z)=(dF)^{-}(X,Y,Z)-\dfrac{1}{2}N(JX,Y,Z)+\dfrac{1}{2}\left[(dF)^{+}(X,Y,Z)-(dF)^{+}(X,JY,JZ)\right],$

where $(dF)^{+}$ is the $(1,2)+(2,1)$-part of $dF$ and $(dF)^{-}$ is the
$(0,3)+(3,0)$-part of $dF$.

Consider $N^{0}=N-\mathfrak{b}N$, where $\mathfrak{b}$ is the Bianchi
projector, $\mathfrak{b}N^{0}=0$ and $\mathfrak{b}N$ is the anti-symmetric
part of $N$ defined by

$\mathfrak{b}N(X,Y,Z)=\dfrac{1}{3}\left[N(X,Y,Z)+N(Y,Z,X)+N(Z,X,Y)\right].$

According to [10], we have

$3\mathfrak{b}N(X,Y,Z)=(d^{c}F)^{-}(X,Y,Z)=(dF)^{-}(JX,JY,JZ).$

The four components described above $(dF)^{-},N^{0},(dF)^{+}_{0}$ and
$\alpha_{F}$ provide us several geometric information about an almost
Hermitian manifold. The $16$ classes of almost Hermitian structures described
by Gray-Hervella in [12] correspond to the vanishing of the some subset of

$\\{(dF)^{-},N^{0},(dF)^{+}_{0},\alpha_{F}\\}.$

Let us describe some remarkable classes of almost Hermitian structures:

  * •

$\\{0\\}=$ Kähler manifolds: all components vanish,
$(dF)^{-}=N^{0}=(dF)^{+}_{0}=\alpha_{F}=0$.

  * •

$\mathcal{W}_{1}=$ nearly-Kähler manifolds: $N^{0}=(dF)^{+}_{0}=\alpha_{F}=0$.

  * •

$\mathcal{W}_{1}\oplus\mathcal{W}_{2}=$ $(1,2)$-symplectic manifolds (or
quasi-Kähler): $(dF)^{+}_{0}=\alpha_{F}=0$.

  * •

$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}=$ cosymplectic
manifolds: $\alpha_{F}=0$.

The Hermitian metric $g$ induces a natural inner product on $\wedge^{k}M$, the
bundle of real $k$-forms, and also on $TM\otimes\wedge^{k}M$ the bundle of
$TM$-valuated $k$-forms. The norm of the covariant derivative of the Kähler
form is given by:

(3.3) $\displaystyle\|DF\|^{2}$ $\displaystyle=$
$\displaystyle\|dF\|^{2}+\dfrac{1}{4}\|N^{0}\|^{2}-\dfrac{2}{3}\|(dF)^{-}\|^{2}$
(3.4) $\displaystyle=$
$\displaystyle\|(dF)^{+}\|^{2}+\dfrac{1}{4}\|N^{0}\|^{2}+\dfrac{1}{3}\|(dF)^{-}\|^{2}.$

In particular, if $J$ is integrable, then $\|DF\|^{2}=\|dF\|^{2}$.

######  Theorem 3.1 (Theorem 4.3, [7]).

Let $(M,g,J)$ be an almost Hermitian manifold of real dimension $2n$. Then

$\displaystyle s_{1}(t)$ $\displaystyle=$
$\displaystyle\dfrac{s}{2}-\dfrac{5}{12}\|(dF)^{-}\|^{2}+\dfrac{1}{16}\|N^{0}\|^{2}+\dfrac{1}{4}\|(dF)_{0}^{+}\|^{2}$
$\displaystyle+$
$\displaystyle\left[\dfrac{1}{4(n-1)}+\dfrac{t-1}{2}\right]\|\alpha_{F}\|^{2}+\dfrac{t-2}{2}\delta\alpha_{F}$
$\displaystyle s_{2}(t)$ $\displaystyle=$
$\displaystyle\frac{s}{2}-\frac{1}{12}\|(dF)^{-}\|^{2}+\frac{1}{32}\|N^{0}\|^{2}-\dfrac{t^{2}-2t}{4}\|(dF)_{0}^{+}\|^{2}$
$\displaystyle-$
$\displaystyle\left[\dfrac{t^{2}-2t}{4(n-1)}+\dfrac{(t+1)^{2}}{8}\right]\|\alpha_{F}\|^{2}-\dfrac{t+1}{2}\delta\alpha_{F},$

where $s$ denotes the Riemannian scalar curvature of $(M,g)$.

Note that

$\displaystyle 2s_{1}(t)-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)_{0}^{+}\|^{2}$
$\displaystyle+$
$\displaystyle\left[\dfrac{1}{2(n-1)}+(t-1)\right]\|\alpha_{F}\|^{2}+(t-2)\delta\alpha_{F}$

###  3.2. Curvature of invariant almost Hermitian structures on flag manifolds

Let us consider a flag manifold $G/K$. We will consider on $G/K$ an invariant
metric $g$ and an invariant almost complex structure $J$. Recall the notation
introduced in the Section 2: we will represent an invariant metric $g$ by an
$n$-tuple $(\lambda_{1},\ldots,\lambda_{n})$, where $n$ is the number of
irreducible components of the isotropy representation of
$\mathfrak{m}=T_{o}(G/K)$, that is,
$\mathfrak{m}=\mathfrak{m}_{1}\oplus\ldots\oplus\mathfrak{m}_{n}$. It is worth
to point out that if $\\{X_{\alpha}\\}$ is a basis of $\mathfrak{m}$ induced
by the Weyl basis then $X_{\alpha}$ is an eigenvector of the operator
$\Lambda$ associated to the metric $g$, with same eigenvalue. This means that
every vector in the irreducible component $\mathfrak{m}_{i}$ has length
$\lambda_{i}$. The invariant almost complex structure $J$ is parameterized by
a set of sign $\\{\varepsilon_{\alpha}\\}=\pm 1$, with
$\varepsilon_{-\alpha}=-\varepsilon_{\alpha}$, where $\alpha$ is in the set of
roots $\Pi$. Each vector $X_{\alpha}$ is an eigenvector of $J$ with eigenvalue
$\varepsilon_{\alpha}\sqrt{-1}$ and every vector in an irreducible component
$\mathfrak{m}_{i}$ is associated to the same $\varepsilon_{i}$.

We will write the Nijenhuis tensor in a similar way as [18]. We have $N=0$,
except in the following situation:

(3.6)
$g(N(X_{\alpha},X_{\beta}),X_{\gamma}))=-\lambda_{\gamma}m_{\alpha,\beta}(\epsilon_{\alpha}\epsilon_{\beta}+\epsilon_{\alpha}\epsilon_{\gamma}+\epsilon_{\beta}\epsilon_{\gamma}+1),$

where $\alpha+\beta+\gamma=0$.

We also have

(3.7)
$g((N(X_{\alpha},X_{\beta}),JX_{\gamma})_{=}-\sqrt{-1}\lambda_{\gamma}m_{\alpha,\beta}(\epsilon_{\alpha}\epsilon_{\beta}\epsilon_{\gamma}+\epsilon_{\alpha}+\epsilon_{\beta}+\epsilon_{\gamma}),$

where $\alpha+\beta+\gamma=0$.

The next result was initially proved by San Martin-Negreiros [18] in the case
of full flag manifolds and by R. de Jesus in her PhD. Thesis for the case of
generalized flag manifold (see also [19]). We include the proof here for the
convenience of the reader.

######  Lemma 3.2.

Every invariant almost Hermitian structure $(g,J)$ on a generalized flag
manifolds is cosymplectic.

###### Proof.

An almost Hermitian structure is cosymplectic if and only if

$\alpha_{F}(X)=\dfrac{1}{2n-1}\sum_{i}dF(X,X_{i},Y_{i})=0,$

where $\\{X_{i}\\}$ is a basis of the tangent space, $\\{Y_{i}\\}$ is a basis
of the dual space with relation a nondegenerate form $F$. We take
$\\{X_{i}\\}=\\{A_{\alpha},\sqrt{-1}S_{\alpha};\alpha\in\Pi^{+}_{M}\\}$ and
$\\{Y_{i}\\}=\\{\sqrt{-1}S_{\alpha},A_{\alpha};\alpha\in\Pi^{+}_{M}\\}$. So,

$\displaystyle dF(X,X_{\alpha}-X_{-\alpha},iX_{\alpha}+iX_{-\alpha})$
$\displaystyle=$ $\displaystyle
idF(X,X_{\alpha},X_{\alpha})+idF(X,X_{\alpha},X_{-\alpha})$ $\displaystyle-$
$\displaystyle idF(X,X_{-\alpha},X_{\alpha})-idF(X,X_{-\alpha},X_{-\alpha})$
$\displaystyle=$ $\displaystyle
2i(dF(X,X_{\alpha},X_{\alpha})+dF(X,X_{\alpha},X_{-\alpha})).$

We know that $dF(X_{\alpha},X_{\beta},X_{\gamma})=0$ unless
$\alpha+\beta+\gamma=0$. Thus, we take $X=X_{\beta}$ and

$\left\\{\begin{array}[]{l}dF(X_{\beta},X_{\alpha},X_{\alpha})=0,\ \mbox{
because if}\ \alpha+\beta+\alpha=0\ \mbox{then}\ \beta=-2\alpha.\\\
dF(X_{\beta},X_{\alpha},X_{-\alpha})=0,\ \mbox{because if}\
\alpha+\beta-\alpha=0\ \mbox{then}\ \beta=0\end{array}\right\\},$

and it is a contradiction. Therefore, for every root $\gamma$,

$\alpha_{F}(X)=\dfrac{1}{2n-1}\sum_{\alpha>0}dF(X_{\gamma},X_{\alpha}-X_{-\alpha},iX_{\alpha}+iX_{-\alpha})=0.$

∎

######  Remark 3.3.

1\. According to Lemma 3.2, the Lee form vanishes identically for every
generalized flag manifold. Therefore we have $dF=(dF)_{0}$ and hence
$(dF)^{+}=(dF)_{0}^{+}$. From now on, we will use $(dF)^{+}$ instead of
$(dF)_{0}^{+}$.

2\. We have $\delta\alpha_{F}=0$, where $\delta$ is the codifferential. In
this case $s_{1}(t)$ is independent of $t$ (see Theorem 3.1), and we will
denote $s_{1}(t)$ simply by $s_{1}$. In this case, $s_{1}$ coincides with the
Chern scalar curvature $s_{C}$

3\. In Theorem 3.1 we will omit the terms which $\alpha_{F}$ appears.

The next definition was introduced in [18] in order to obtain the
classification of invariant almost Hermitian structures on full flag
manifolds, and we will use this concept widely.

######  Definition 3.4.

Let $J$ be an invariant almost complex structure on the flag manifold $G/K$.
The triple of roots $\alpha,\beta,\gamma\in\Pi^{+}_{M}$, with
$\alpha+\beta+\gamma=0$ is said to be a $(0,3)$-triple if
$\varepsilon_{\alpha}=\varepsilon_{\beta}=\varepsilon_{\gamma}$. It is a
$(1,2)$-triple otherwise.

In order to calculate the Hermitian scalar curvatures, the first step is
computing the norms of $N^{0}$, $(dF)^{-}$, $(dF)^{+}$ e $\alpha_{F}$, as
described in [7]. Since these computations are standard we omit the details.

######  Proposition 3.5 ([11]).

Let $(g,J)$ be an invariant almost Hermitian structure in the generalized flag
manifold $G/K$. We have

$\displaystyle\|N^{0}\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{\alpha+\beta+\gamma=0}\left(m_{\alpha,\beta}\right)^{2}\dfrac{(\epsilon_{\alpha}\epsilon_{\beta}\epsilon_{\gamma}+\epsilon_{\alpha}+\epsilon_{\beta}+\epsilon_{\gamma})^{2}}{54\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}[(-2\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})^{2}$
$\displaystyle+(-2\lambda_{\gamma}+\lambda_{\alpha}+\lambda_{\beta})^{2}+(-2\lambda_{\beta}+\lambda_{\alpha}+\lambda_{\gamma})^{2}],$
$\displaystyle\|(dF)^{-}\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{\alpha+\beta+\gamma=0}\dfrac{\left(m_{\alpha,\beta}\right)^{2}(\epsilon_{\alpha}\epsilon_{\beta}\epsilon_{\gamma}+\epsilon_{\alpha}+\epsilon_{\beta}+\epsilon_{\gamma})^{2}\left(\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma}\right)^{2}}{96\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}},$
$\displaystyle\|DF\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{\alpha+\beta+\gamma=0}\dfrac{m_{\alpha,\beta}^{2}}{3\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}\left[\dfrac{1}{4}(\varepsilon_{\beta}+\varepsilon_{\gamma})^{2}(-\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})^{2}\right.$
$\displaystyle+\dfrac{1}{4}(\varepsilon_{\alpha}+\varepsilon_{\gamma})^{2}(\lambda_{\alpha}-\lambda_{\beta}+\lambda_{\gamma})^{2}+\left.\dfrac{1}{4}(\varepsilon_{\alpha}+\varepsilon_{\beta})^{2}(\lambda_{\alpha}+\lambda_{\beta}-\lambda_{\gamma})^{2}\right],$
$\displaystyle\|(dF)^{+}\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{\alpha+\beta+\gamma=0}\dfrac{m_{\alpha,\beta}^{2}\\{4(\varepsilon_{\alpha}\lambda_{\alpha}+\varepsilon_{\beta}\lambda_{\beta}+\varepsilon_{\gamma}\lambda_{\gamma})-(\varepsilon_{\alpha}+\varepsilon_{\beta}+\varepsilon_{\gamma}+\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\gamma})(\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})\\}}{96\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}^{2}.$

##  4\. Flag manifolds with two isotropy summands

In this section we will discuss the equation $2s_{1}-s=0$ for generalized flag
manifolds $G/K$ whose the isotropy representation splits into two irreducible
and non-equivalent components, namely
$\mathfrak{m}=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}$. The classification of
this family of homogeneous space is given as follows: recall the height of a
simple root $\alpha_{p}$ is the integer that appear as its coefficient in the
highest root, that is, if
$\mu=c_{1}{\alpha_{1}}+\ldots+c_{p}\alpha_{p}+\ldots+c_{n}\alpha_{n}$ is the
highest root then $\operatorname{ht}(\alpha_{p})=c_{p}$. A generalized flag
manifold has two istropy components iff it is parameterized by the set
$\Theta\subset\Sigma$ such that $\Theta=\Sigma\setminus\\{\alpha_{p}\\}$ with
$\operatorname{ht}(\alpha_{p})=2$ (see Section 2). This class of homogeneous
spaces is listed in Table 2 (see [1]).

Table 2. Flag manifolds with two isotropy summnads
$SO(2\ell+1)/U(\ell-m)\times SO(2m+1),\quad\ell>0,m\geq 0,\ell-m\neq 1$
---
$Sp(\ell)/U(\ell-m)\times Sp(m),\quad\ell>0,m>0$
$SO(2\ell)/U(\ell-m)\times SO(2m),\quad\ell>0,m>0,\ell-m\neq 1$
$G_{2}/U(2)$, $U(2)$ represented by the short root of $G_{2}$
$F_{4}/SO(7)\times U(1)$
$F_{4}/Sp(3)\times U(1)$
$E_{6}/SU(6)\times U(1)$
$E_{6}/SU(2)\times SU(5)\times U(1)$
$E_{7}/SU(7)\times U(1)$
$E_{7}/SU(2)\times SO(10)\times U(1)$
$E_{7}/SO(12)\times U(1)$
$E_{8}/E_{7}\times U(1)$
$E_{8}/SO(14)\times U(1)$

The isotropy summands $\mathfrak{m}_{1}$ and $\mathfrak{m}_{2}$ satisfy the
following relations:

$[\mathfrak{m}_{1},\mathfrak{m}_{1}]\subset\mathfrak{k}\oplus\mathfrak{m}_{2},\
\ \ \ [\mathfrak{m}_{1},\mathfrak{m}_{2}]\subset\mathfrak{m}_{1}\ \ \ \
\mbox{and}\ \ \ \ [\mathfrak{m}_{2},\mathfrak{m}_{2}]\subset\mathfrak{k}.$

Note that the zero sum triples can be parameterized by:

$\begin{array}[]{ll}\alpha+\beta+\gamma=0&\mbox{where}\
\alpha\in\mathfrak{m}_{1},\ \beta\in\mathfrak{m}_{1}\ \mbox{and}\
\gamma\in\mathfrak{m}_{2}.\end{array}$

For flag manifolds $G/K$ with two isotropy summands we have two invariant
almost complex structures (up to conjugation), namely $J=(+,+)$ and
$J_{1}=(+,-)$. Let us consider our analysis for $J$ and $J_{1}$ separately.

Consider complex (integrable) structure $J=(+,+)$. The (1,2)-triples are given
by $\mathfrak{m}_{1}+\mathfrak{m}_{1}-\mathfrak{m}_{2}=0$ and there is no
(0,3)-triples. We define

$L=\sum_{\begin{subarray}{c}\\{\delta,\sigma,\eta\\}\ \rm{is}\\\
(1,2)-\rm{triple}\end{subarray}}m_{\delta,\sigma}^{2},$

where the real numbers $m_{\delta,\sigma}$ are the structural constants
induced by the Weyl basis.

Let us consider an invariant metric $g$ parameterized by
$g=(\lambda_{1},\lambda_{2})$. We get

$\|(dF)^{-}\|^{2}=\|N^{0}\|^{2}=0,\quad\|(dF)^{+}\|^{2}=L\dfrac{(2\lambda_{1}-\lambda_{2})^{2}}{6\lambda_{1}^{2}\lambda_{2}}.$

If the metric $g$ satisfies $\lambda_{2}=2\lambda_{1}$ then $(J,g)$ is Kähler,
otherwise $(J,g)\in\mathcal{W}_{3}$. The expression $2s_{1}-s$ reads:

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$ $\displaystyle
L\dfrac{(2\lambda_{1}-\lambda_{2})^{2}}{12\lambda_{1}^{2}\lambda_{2}}.$

Thus, $2s_{1}-s=0$ if, and only if, $\lambda_{2}=2\lambda_{1}$ and $(J,g)$ is
Kähler.

Let us consider now the invariant almost complex structure $J_{1}=(+,-)$ (non-
integrable). The (0,3)-triples are given by
$\mathfrak{m}_{1}+\mathfrak{m}_{1}-\mathfrak{m}_{2}=0$, and there is no
(1,2)-triples. We define

$K=\sum_{\begin{subarray}{c}\\{\alpha,\beta,\gamma\\}\ \rm{is}\\\
(0,3)-\rm{triple}\end{subarray}}m_{\alpha,\beta}^{2},$

where the real numbers $m_{\alpha,\beta}$ are the structural constants induced
by the Weyl basis.

We have

$\|(dF)^{+}\|^{2}=0,\quad\|(dF)^{-}\|^{2}=K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{6\lambda_{1}^{2}\lambda_{2}},\quad\|N^{0}\|^{2}=8K\dfrac{2(-\lambda_{1}+\lambda_{2})^{2}+(2\lambda_{1}-2\lambda_{2})^{2}}{27\lambda_{1}^{2}\lambda_{2}}.$

If the metric $g$ satisfies $\lambda_{2}=\lambda_{1}$ then
$(J_{1},g)\in\mathcal{W}_{1}$, otherwise
$(J_{1},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. The expression $2s_{1}-s$
reads:

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{36\lambda_{1}^{2}\lambda_{2}}+K\dfrac{(-2\lambda_{1}+\lambda_{1}+\lambda_{2})^{2}+(\lambda_{1}-2\lambda_{1}+\lambda_{2})^{2}+(\lambda_{1}+\lambda_{1}-2\lambda_{2})^{2}}{27\lambda_{1}^{2}\lambda_{2}}$
$\displaystyle=$
$\displaystyle\frac{K\left(-4\lambda_{1}^{2}-12\lambda_{1}\lambda_{2}+\lambda_{2}^{2}\right)}{12\lambda_{1}^{2}\lambda_{2}}.$

Therefore, if $(J_{1},g)\in\mathcal{W}_{1}$ there is not solution for equation
$2s_{1}-s=0$. If $(J_{1},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$ then
$\lambda_{2}=2\lambda_{1}(\sqrt{10}+3)$ is a solution of $2s_{1}-s=0$.

Summing up we obtain the following result:

######  Theorem 4.1.

Let $G/K$ be a generalized flag manifold with two isotropy summands, and let
$g=(\lambda_{1},\lambda_{2})$ be an invariant Riemannian metric. Then
$2s_{1}-s=0$ holds for $G/K$ if the pair $(g,J_{i})$ satisfies the following
conditions:

  * •

For the invariant complex struture $J=(+,+)$, the metric $g$ is parameterized
by $\lambda_{2}=2\lambda_{1}$. In this case $(g,J)$ is Kähler.

  * •

For the invariant almost complex structure $J_{1}=(+,-)$, the metric $g$ is
parameterized by $\lambda_{2}=2\lambda_{1}(\sqrt{10}+3)$. In this case,
$(g,J_{1})\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$.

##  5\. Flag manifolds with three isotropy summands

Generalized flag manifolds whose the isotropy representation decompose into
three pairwise non-isomorphic irreducible components, i.e.,
$\mathfrak{m}=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\mathfrak{m}_{3}$
are classified by Kimura in [14] and described in terms of the choice of
$\Theta\subset\Sigma$ in Table 3. It is worth to point out there are two sub-
families of flags with three isotropy components, namely Type I and II, see
Tables 4 and 5.

Table 3. Types of flag manifolds with three summands Type | $\Theta\subset\Sigma$
---|---
$I$ | $\Theta=\Sigma\setminus\\{\alpha_{p}:\operatorname{ht}(\alpha_{p})=3\\}$
$II$ | $\Theta=\Sigma\setminus\\{\alpha_{p},\alpha_{q}:\operatorname{ht}(\alpha_{p})=\operatorname{ht}(\alpha_{q})=1\\}$
Table 4. The dimensions $d_{i}=\dim(\mathfrak{m}_{i})$ for any $M=G/K$ of type I Flag manifold $G/K$ | $d_{1}$ | $d_{2}$ | $d_{3}$
---|---|---|---
$G_{2}/U(2)$ | $4$ | $2$ | $4$
$F_{4}/SU(3)\times SU(2)\times U(1)$ | $24$ | $12$ | $4$
$E_{6}/SU(3)\times SU(3)\times SU(2)\times U(1)$ | $36$ | $18$ | $4$
$E_{7}/SU(5)\times SU(3)\times U(1)$ | $60$ | $30$ | $8$
$E_{7}/SU(6)\times SU(2)\times U(1)$ | $60$ | $30$ | $4$
$E_{8}/E_{6}\times SU(2)\times U(1)$ | $108$ | $54$ | $4$
$E_{8}/SU(8)\times U(1)$ | $112$ | $56$ | $16$
Table 5. Flag manifolds with three summands of type II Flag manifold $G/K$ | $d_{1}$ | $d_{2}$ | $d_{3}$
---|---|---|---
$SU(l+m+n)/S(U(l)\times U(m)\times U(n)\ (l,m,n\in\mathbb{Z}^{+})$ | $2mn$ | $2mp$ | $2np$
$SO(2l)/U(1)\times U(l-1)\ (l\geq 4)$ | $2(l-1)$ | $2(l-1)$ | $(l-1)(l-2)$
$E_{6}/SO(8)\times U(1)\times U(1)$ | 16 | 16 | 16

Flag manifolds with three isotropic summands of type I the summands satisfy
the following relations with respect to $\mathfrak{m}_{1}$, $\mathfrak{m}_{2}$
and $\mathfrak{m}_{3}$:

$\displaystyle[\mathfrak{m}_{1},\mathfrak{m}_{1}]\subset\mathfrak{k}\oplus\mathfrak{m}_{2},\
\ \
[\mathfrak{m}_{1},\mathfrak{m}_{2}]\subset\mathfrak{m}_{1}\oplus\mathfrak{m}_{3}\
\ \ [\mathfrak{m}_{1},\mathfrak{m}_{3}]\subset\mathfrak{m}_{2}$
$\displaystyle[\mathfrak{m}_{2},\mathfrak{m}_{2}]\subset\mathfrak{k},\ \ \
[\mathfrak{m}_{2},\mathfrak{m}_{3}]\subset\mathfrak{m}_{1}\ \ \ \mbox{and}\ \
\ [\mathfrak{m}_{3},\mathfrak{m}_{3}]\subset\mathfrak{k}.$

In the case of flag manifolds with three isotropic summands of type II the
components $\mathfrak{m}_{1}$, $\mathfrak{m}_{2}$ and $\mathfrak{m}_{3}$
satisfy the following inclusions:

$\displaystyle[\mathfrak{m}_{1},\mathfrak{m}_{1}]\subset\mathfrak{k},\ \ \
[\mathfrak{m}_{1},\mathfrak{m}_{2}]=\mathfrak{m}_{3}\ \ \
[\mathfrak{m}_{1},\mathfrak{m}_{3}]=\mathfrak{m}_{2}$
$\displaystyle[\mathfrak{m}_{2},\mathfrak{m}_{2}]\subset\mathfrak{k},\ \ \
[\mathfrak{m}_{2},\mathfrak{m}_{3}]=\mathfrak{m}_{1}\ \ \ \mbox{and}\ \ \
[\mathfrak{m}_{3},\mathfrak{m}_{3}]\subset\mathfrak{k}.$

Consequently, there are two groups of triples with zero sum:

$\begin{array}[]{ll}\alpha+\beta+\gamma=0&\mbox{where}\
\alpha\in\mathfrak{m}_{1},\ \beta\in\mathfrak{m}_{1}\ \mbox{and}\
\gamma\in\mathfrak{m}_{2};\\\ \alpha+\beta+\gamma=0&\mbox{where}\
\alpha\in\mathfrak{m}_{1},\ \beta\in\mathfrak{m}_{2}\ \mbox{and}\
\gamma\in\mathfrak{m}_{3}.\end{array}$

######  Definition 5.1.

Let $G/K$ be a generalized flag manifold equipped with an invariant almost
complex structure $J$ and Riemannian metric $g$. We define

(5.1) $K=\sum_{\begin{subarray}{c}\\{\alpha,\beta,\gamma\\}\ \rm{is}\\\
(0,3)-\rm{triple}\end{subarray}}m_{\alpha,\beta}^{2},\hskip 14.22636pt{\rm
and}\hskip 14.22636ptL=\sum_{\begin{subarray}{c}\\{\delta,\sigma,\eta\\}\
\rm{is}\\\ (1,2)-\rm{triple}\end{subarray}}m_{\delta,\sigma}^{2},\\\ $

where the real numbers $m_{\alpha,\beta}$, $m_{\delta,\sigma}$ are the
structure constants induced by the Weyl basis.

Recall we have four invariant almost complex structure (up to conjugation)
$J_{1}=(+,+,+)$, $J_{2}=(-,+,+)$, $J_{3}=(+,-,+)$, $J_{4}=(+,+,-)$ on a flag
manifold $G/K$ with three isotropic summands (cf. [14]). We will analyze the
solution of the equation $2s_{1}-s=0$ separately for each invariant almost
complex structure $J_{i}$, $i=1,\ldots,4$.

Table 6. Invariant almost complex structure $J_{1}=(+,+,+)$ Flag Manifold |  $(1,2)$-triples $(\alpha+\beta+\gamma)=0$ |  $(0,3)$-triples $(\alpha+\beta+\gamma)=0$ |  | Gray-Hervella
---
Classification
Class | Condition on the metric
Type I |  | $\alpha\in m_{1},\beta\in\mathfrak{m}_{1},\gamma\in\mathfrak{m}_{2}$
---
or
$\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{2},\gamma\in\mathfrak{m}_{3}$
✗ |  | Kähler
---
$\mathcal{W}_{3}$
|  $\lambda_{2}=2\lambda_{1}$ and $\lambda_{3}=3\lambda_{1}$
---
Otherwise
Type II | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{2},\gamma\in\mathfrak{m}_{3}$ | ✗ |  | Kähler
---
$\mathcal{W}_{3}$
| $\lambda_{3}=\lambda_{1}+\lambda_{2}$
---
Otherwise

######  Theorem 5.2.

Let $G/K$ be a generalized flag manifold with three isotropy summands equipped
with the invariant almost complex structure $J_{1}$ described in Table 6 and
invariant Riemannian metric $g=(\lambda_{1},\lambda_{2},\lambda_{3})$. Then
$2s_{1}-s=0$ holds for $(G/K,g,J_{1})$ if the metric $g$ satisfies the
following conditions:

  1. (1)

$G/K$ is type I

    1. (a)

$g=(\lambda_{1},2\lambda_{1},3\lambda_{1})$. In this case, $(g,J_{1})$ is
Kähler.

  2. (2)

$G/K$ is type II

    1. (a)

$g=(\lambda_{1},\lambda_{2},\lambda_{3})$ with
$\lambda_{3}=\lambda_{1}+\lambda_{2}$. In this case, $(g,J_{1})$ is Kähler.

###### Proof.

Let us split our analysis for flag manifolds of type I and II:

  1. (a)

Flag manifolds of type I:

Let us define

$L_{1}=\sum_{\begin{subarray}{c}\\{\delta,\sigma,\eta\\}\ \rm{is}\\\
(1,2)-\rm{triple}\end{subarray}}m_{\delta,\sigma}^{2},\ \ \mbox{where}\ \
\delta\in\mathfrak{m}_{1},\ \sigma\in\mathfrak{m}_{1}\ \mbox{and}\
\eta\in\mathfrak{m}_{2}\\\ $

$L_{2}=\sum_{\begin{subarray}{c}\\{\delta,\sigma,\eta\\}\ \rm{is}\\\
(1,2)-\rm{triple}\end{subarray}}m_{\delta,\sigma}^{2},\ \ \mbox{where}\ \
\delta\in\mathfrak{m}_{1},\ \sigma\in\mathfrak{m}_{2}\ \mbox{and}\
\eta\in\mathfrak{m}_{3}\\\ $

We obtain $\|(dF)^{-}\|^{2}=\|N^{0}\|^{2}=0$.

$\|(dF)^{+}\|^{2}=L_{1}\dfrac{(\lambda_{1}+\lambda_{1}-\lambda_{2})^{2}}{6\lambda_{1}^{2}\lambda_{3}}+L_{2}\dfrac{(\lambda_{1}+\lambda_{2}-\lambda_{3})^{2}}{6\lambda_{1}\lambda_{2}\lambda_{3}}$.

It follows that,

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$ $\displaystyle
L_{1}\dfrac{(\lambda_{1}+\lambda_{1}-\lambda_{2})^{2}}{12\lambda_{1}^{2}\lambda_{2}}+L_{2}\dfrac{(\lambda_{1}+\lambda_{2}-\lambda_{3})^{2}}{12\lambda_{1}\lambda_{2}\lambda_{3}}\geq
0$

Therefore, the equation $2s_{1}-s=0$ there is no solution at least the pair
$(g,J)$ is Kähler what we already well known.

  2. (b)

Flag manifolds of type II:

We start by computing the components of the covariant derivative of the Kähler
form. By a direct computation we get

(5.2) $\|(dF)^{-}\|^{2}=\|N^{0}\|^{2}=0,\quad{\rm
and}\quad\|(dF)^{+}\|^{2}=L\dfrac{(\lambda_{1}+\lambda_{2}-\lambda_{3})^{2}}{6\lambda_{1}\lambda_{2}\lambda_{3}},$

where the non-zero constant $L$ is defined by equation 5.1 and depends on the
flag manifold.

By equation 3.1 we find

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$ $\displaystyle
L\dfrac{(\lambda_{1}+\lambda_{2}-\lambda_{3})^{2}}{12\lambda_{1}\lambda_{2}\lambda_{3}}\geq
0$

Hence, there is no solution for the equation $2s_{1}-s=0$ unless the pair
$(g,J)$ is Kähler, and this solution is already well known.

∎

Table 7. Invariant almost complex structure $J_{2}=(-,+,+)$ Flag Manifold |  $(1,2)$-triples $(\alpha+\beta+\gamma)=0$ |  $(0,3)$-triples $(\alpha+\beta+\gamma)=0$ |  | Gray-Hervella
---
Classification
Class | Condition on the metric
Type I | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{1},\gamma\in\mathfrak{m}_{2}$ | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{2},\gamma\in\mathfrak{m}_{3}$ |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$
---
$\mathcal{W}_{1}\oplus\mathcal{W}_{3}$
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$
|  $\lambda_{3}=-\lambda_{1}+\lambda_{2}$ and $\lambda_{2}>\lambda_{1}$
---
$\lambda_{1}=\lambda_{2}$
Otherwise
Type II | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{2},\gamma\in\mathfrak{m}_{3}$ | ✗ |  | Kähler
---
$\mathcal{W}_{3}$
| $\lambda_{3}=\lambda_{1}+\lambda_{2}$
---
Otherwise

######  Theorem 5.3.

Let $G/K$ be a generalized flag manifold with three isotropy summands equipped
with the invariant almost complex structure $J_{2}$ described in Table 7 and
invariant Riemannian metric $g=(\lambda_{1},\lambda_{2},\lambda_{3})$. Then
$2s_{1}-s=0$ holds for $(G/K,g,J_{2})$ if the metric $g$ satisfies the
following conditions:

  1. (1)

$G/K$ is type I

    1. (a)

$\lambda_{1}<\lambda_{2}<2(\sqrt{10}+3)\lambda_{1}$ and
$\lambda_{3}=\frac{-\sqrt{K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)\left(K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)+4L\lambda_{1}(\lambda_{2}-\lambda_{1})\right)}+K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)-2L\lambda_{1}^{2}+2L\lambda_{1}\lambda_{2}}{2L\lambda_{1}}$.
In this case, $(g,J_{2})$ belongs to the Gray-Hervella class
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$.

    2. (b)

$g=(\lambda_{1},\lambda_{2},\lambda_{3})$ with
$\lambda_{2}=(2\sqrt{10}+6)\lambda_{1}$ and
$\lambda_{3}=\lambda_{2}-\lambda_{1}$. In this case, $(g,J_{2})$ belongs to
the Gray-Hervella class $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$.

    3. (c)

$g=(\lambda_{1},\lambda_{2},\lambda_{3})$ with
$\lambda_{1}=\lambda_{2}=\frac{L\lambda_{3}}{15K}$. In this case, $(g,J_{2})$
belongs to the Gray-Hervella class $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$.

  2. (2)

$G/K$ is type II

    1. (a)

$g=(\lambda_{1},\lambda_{2},\lambda_{3})$ with
$\lambda_{2}=\lambda_{1}+\lambda_{3}$. In this case, $(g,J_{2})$ is Kähler.

###### Proof.

We will proceed as in the proof of Theorem 5.2. Let us split our analysis for
flag manifolds of type I and II:

  1. a)

Flag manifolds of type I:

We get:

(5.3)
$\|(dF)^{-}\|^{2}=K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{6\lambda_{1}^{2}\lambda_{2}},\quad\quad\|(dF)^{+}\|^{2}=L\dfrac{(\lambda_{1}-\lambda_{2}+\lambda_{3})^{2}}{6\lambda_{1}\lambda_{2}\lambda_{3}}$

and

(5.4)
$\|N^{0}\|^{2}=8K\dfrac{(-2\lambda_{1}+\lambda_{1}+\lambda_{2})^{2}+(\lambda_{1}-2\lambda_{1}+\lambda_{2})^{2}+(2\lambda_{1}-2\lambda_{2})^{2}}{27\lambda_{1}^{2}\lambda_{2}}$

, where the non-zero constants $K$ and $L$ are defined by equation 5.1 and
depends on the flag manifold.

Let us analyze case-by-case the Gray-Hervella class with respect to the almost
complex structure $J_{2}$:

    1. (I)

For $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$:

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{36\lambda_{1}^{2}\lambda_{2}}+K\dfrac{(-2\lambda_{1}+\lambda_{1}+\lambda_{2})^{2}+(\lambda_{1}-2\lambda_{1}+\lambda_{2})^{2}+(2\lambda_{1}-2\lambda_{2})^{2}}{27\lambda_{1}^{2}\lambda_{2}}$
$\displaystyle+$ $\displaystyle
L\dfrac{(\lambda_{1}-\lambda_{2}+\lambda_{3})^{2}}{12\lambda_{1}\lambda_{2}\lambda_{3}}$

Thus, $2s_{1}-s=0$ if, and only if

$K(-4\lambda_{1}^{2}-12\lambda_{1}\lambda_{2}+\lambda_{2}^{2})\lambda_{3}+L\,\lambda_{1}(\lambda_{1}-\lambda_{2}+\lambda_{3})^{2}=0.$

Examples of solutions: $g=(\lambda_{1},\lambda_{2},\lambda_{3})$ parameterized
by

      1. (i)

$\lambda_{1}<\lambda_{2}<2(\sqrt{10}+3)\lambda_{1}$ and
$\lambda_{3}=\frac{-\sqrt{K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)\left(K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)+4L\text{l1}(\lambda_{2}-\lambda_{1})\right)}+K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)-2L\lambda_{1}^{2}+2L\lambda_{1}\lambda_{2}}{2L\lambda_{1}}$

      2. (ii)

$\lambda_{1}=1,\lambda_{2}=2$ and
$\lambda_{3}=\frac{2\sqrt{6}\sqrt{K(24K+4L)}+24K+2L}{2L}$

    2. (II)

For $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$: Condition:
$\lambda_{3}=-\lambda_{1}+\lambda_{2}$, with $\lambda_{2}>\lambda_{1}$.

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{36\lambda_{1}^{2}\lambda_{2}}+K\dfrac{(-2\lambda_{1}+\lambda_{1}+\lambda_{2})^{2}+(\lambda_{1}-2\lambda_{1}+\lambda_{2})^{2}+(2\lambda_{1}-2\lambda_{2})^{2}}{27\lambda_{1}^{2}\lambda_{2}}$
$\displaystyle=$
$\displaystyle\frac{K\left(-4\lambda_{1}^{2}-12\lambda_{1}\lambda_{2}+\lambda_{2}^{2}\right)}{12\lambda_{1}^{2}\lambda_{2}}$

We conclude that $2s_{1}-s=0$ if and only if
$\lambda_{2}=(2\sqrt{10}+6)\lambda_{1}$.

    3. (III)

For $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$: Condition
$\lambda_{1}=\lambda_{2}$.

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{36\lambda_{1}^{2}\lambda_{2}}+L\dfrac{(\lambda_{1}-\lambda_{2}+\lambda_{3})^{2}}{12\lambda_{1}\lambda_{2}\lambda_{3}}$

.

In this case, we conclude that $2s_{1}-s=0$ if and only if
$\lambda_{1}=\frac{L\lambda_{3}}{15K}$.

  2. (b)

Flag manifolds of type II:

We start by computing the components of the covariant derivative of the Kähler
form. By a direct computation we have

(5.5) $\|(dF)^{-}\|^{2}=\|N^{0}\|^{2}=0,\quad{\rm
and}\quad\|(dF)^{+}\|^{2}=L\dfrac{(\lambda_{1}-\lambda_{2}+\lambda_{3})^{2}}{6\lambda_{1}\lambda_{2}\lambda_{3}}$

where the non-zero constant $L$ is defined by equation 5.1 and depends on the
flag manifold.

By equation 3.1 we have

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$ $\displaystyle
L\dfrac{(\lambda_{1}-\lambda_{2}+\lambda_{3})^{2}}{12\lambda_{1}\lambda_{2}\lambda_{3}}$

In this case, $2s_{1}-s=0$ if and only if
$\lambda_{2}=\lambda_{1}+\lambda_{3}$, i.e., $(g,J_{2})$ is Kähler.

∎

Table 8. Invariant almost complex structure $J_{3}=(+,-,+)$ Flag Manifold |  $(1,2)$-triples $(\alpha+\beta+\gamma)=0$ |  $(0,3)$-triples $(\alpha+\beta+\gamma)=0$ |  | Gray-Hervella
---
Classification
Class | Condition on the metric
Type I | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{2},\gamma\in\mathfrak{m}_{3}$ | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{1},\gamma\in\mathfrak{m}_{2}$ |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$
---
$\mathcal{W}_{1}\oplus\mathcal{W}_{3}$
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$
|  $\lambda_{3}=-\lambda_{1}+\lambda_{2}$ and $\lambda_{2}>\lambda_{1}$
---
$\lambda_{1}=\lambda_{2}$
Otherwise
Type II | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{2},\gamma\in\mathfrak{m}_{3}$ | ✗ |  | Kähler
---
$\mathcal{W}_{3}$
| $\lambda_{3}=\lambda_{1}+\lambda_{2}$
---
Otherwise

######  Theorem 5.4.

Let $G/K$ be a generalized flag manifold with three isotropy summands equipped
with the invariant almost complex structure $J_{3}$ described in Table 8 and
invariant Riemannian metric $g=(\lambda_{1},\lambda_{2},\lambda_{3})$. Then
$2s_{1}-s=0$ holds for $(G/K,g,J_{3})$ if the metric $g$ satisfies the
following conditions:

  1. (1)

$G/K$ is type I

    1. (a)

$\lambda_{2}=\lambda_{1}\left(6-\frac{2L}{K}+\frac{2}{K}\sqrt{10K^{2}-5KL+L^{2}}\right)$,
$\lambda_{3}=\lambda_{1}\left(5-\frac{2L}{K}+\frac{2}{K}\sqrt{10K^{2}-5KL+L^{2}}\right)$

    2. (b)

$\lambda_{1}<\lambda_{2}$ and
$\lambda_{3}=\frac{1}{2}\left(\frac{1}{L\lambda_{1}}\sqrt{K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)\left(K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)+4L\lambda_{1}(\lambda_{1}-\lambda_{2})\right)}+2\lambda_{1}-2\lambda_{2}\right)+\frac{1}{2L\lambda_{1}}K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right).$

    3. In the case of items $a)$ and $b)$ above, $(g,J_{3})$ belongs to the Gray-Hervella class $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$.

    4. (c)

$\lambda_{1}=\lambda_{2}=\frac{L\lambda_{3}}{15K}$. In this case, $(g,J_{3})$
belongs to the Gray-Hervella class $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$.

  2. (2)

$G/K$ is type II

    1. (a)

$g=(\lambda_{1},\lambda_{2},\lambda_{3})$ with
$\lambda_{1}=\lambda_{2}+\lambda_{3}$. In this case, $(g,J_{3})$ is Kähler.

###### Proof.

We will proceed as in the proof of Theorem 5.2. Let us split our analysis for
flag manifolds of type I and II:

  1. (a)

Flag manifolds of type I: We obtain

$\|(dF)^{-}\|^{2}=K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{6\lambda_{1}^{2}\lambda_{2}},\quad\quad\|(dF)^{+}\|^{2}=L\dfrac{(-\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{6\lambda_{1}\lambda_{2}\lambda_{3}},$

$\|N^{0}\|^{2}=8K\dfrac{(-2\lambda_{1}+\lambda_{1}+\lambda_{2})^{2}+(\lambda_{1}-2\lambda_{1}+\lambda_{2})^{2}+(2\lambda_{1}-2\lambda_{2})^{2}}{27\lambda_{1}^{2}\lambda_{2}},$

where the non-zero constants $K$ and $L$ are defined by equation 5.1 and
depend on the flag manifold.

Let us analyze case-by-case the Gray-Hervella class with respect to the almost
complex structure $J_{3}$:

    1. (I)

For $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$:

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{36\lambda_{1}^{2}\lambda_{2}}+K\dfrac{(-2\lambda_{1}+\lambda_{1}+\lambda_{2})^{2}+(\lambda_{1}-2\lambda_{1}+\lambda_{2})^{2}+(2\lambda_{1}-2\lambda_{2})^{2}}{27\lambda_{1}^{2}\lambda_{2}}$
$\displaystyle+$ $\displaystyle
L\dfrac{(-\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{12\lambda_{1}\lambda_{2}\lambda_{3}}$

Therefore we have $2s_{1}-s=0$ if, and only if

$K(-4\lambda_{1}^{2}-12\lambda_{1}\lambda_{2}+\lambda_{2}^{2})\lambda_{3}+L\lambda_{1}(-\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}=0.$

Examples of solutions:

      1. (i)

$\lambda_{2}=\lambda_{1}\left(6-\frac{2L}{K}+\frac{2}{K}\sqrt{10K^{2}-5KL+L^{2}}\right)$,
$\lambda_{3}=\lambda_{1}\left(5-\frac{2L}{K}+\frac{2}{K}\sqrt{10K^{2}-5KL+L^{2}}\right)$

      2. (ii)

$\lambda_{1}>0,0<\lambda_{2}<\lambda_{1}$ and
$\lambda_{3}=\frac{1}{2}\left(\frac{1}{L\lambda_{1}}\sqrt{K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)\left(K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)+4L\lambda_{1}(\lambda_{1}-\lambda_{2})\right)}+2\lambda_{1}-2\lambda_{2}\right)$
$+\frac{K\left(4\lambda_{1}^{2}+12\lambda_{1}\lambda_{2}-\lambda_{2}^{2}\right)}{2L\lambda_{1}}$

    2. (II)

For $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$: Condition:
$\lambda_{3}=\lambda_{1}-\lambda_{2}$, with $\lambda_{1}>\lambda_{2}$.

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{36\lambda_{1}^{2}\lambda_{2}}+K\dfrac{(-2\lambda_{1}+\lambda_{1}+\lambda_{2})^{2}+(\lambda_{1}-2\lambda_{1}+\lambda_{2})^{2}+(2\lambda_{1}-2\lambda_{2})^{2}}{27\lambda_{1}^{2}\lambda_{2}}$
$\displaystyle=$
$\displaystyle\frac{K\left(-4\lambda_{1}^{2}-12\lambda_{1}\lambda_{2}+\lambda_{2}^{2}\right)}{12\lambda_{1}^{2}\lambda_{2}}$

There isn’t solution of the equation $2s_{1}-s=0$.

    3. (III)

For $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$: Condition
$\lambda_{1}=\lambda_{2}$

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(2\lambda_{1}+\lambda_{2})^{2}}{36\lambda_{1}^{2}\lambda_{2}}+L\dfrac{(-\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{12\lambda_{1}\lambda_{2}\lambda_{3}}$

In this case, we conclude that $2s_{1}-s=0$ if and only if
$\lambda_{3}=\frac{15K\lambda_{1}}{L}$.

  2. (b)

Flag manifolds of type II:  We start by computing the components of the
covariant derivative of the Kähler form. By a direct computation we have

(5.6) $\|(dF)^{-}\|^{2}=\|N^{0}\|^{2}=0,\quad{\rm
and}\quad\|(dF)^{+}\|^{2}=L\dfrac{(-\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{6\lambda_{1}\lambda_{2}\lambda_{3}}.$

where the non-zero constant $L$ is defined by equation 5.1 and depends on the
flag manifold.

By equation 3.1 we have

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$ $\displaystyle
L\dfrac{(-\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{12\lambda_{1}\lambda_{2}\lambda_{3}}.$

Hence $2s_{1}-s=0$ if and only if $\lambda_{1}=\lambda_{2}+\lambda_{3}$, i.e.,
$(g,J_{3})$ is Kähler.

∎

Table 9. Invariant almost complex structure $J_{4}=(+,+,-)$ Flag Manifold |  $(1,2)$-triples $(\alpha+\beta+\gamma)=0$ |  $(0,3)$-triples $(\alpha+\beta+\gamma)=0$ |  | Gray-Hervella
---
Classification
Class | Condition on the metric
Type I | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{1},\gamma\in\mathfrak{m}_{2}$ | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{2},\gamma\in\mathfrak{m}_{3}$ |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$
---
$\mathcal{W}_{1}\oplus\mathcal{W}_{3}$
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$
| $\lambda_{2}=2\lambda_{1}$
---
$\lambda_{1}=\lambda_{2}=\lambda_{3}$
Otherwise
Type II | ✗ | $\alpha\in\mathfrak{m}_{1},\beta\in\mathfrak{m}_{2},\gamma\in\mathfrak{m}_{3}$ |  | $\mathcal{W}_{1}$
---
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}$
| $\lambda_{1}=\lambda_{2}=\lambda_{3}$
---
Otherwise

######  Theorem 5.5.

Let $G/K$ be a generalized flag manifold with three isotropy summands equipped
with the invariant almost complex structure $J_{4}$ described in Table 9 and
invariant Riemannian metric $g=(\lambda_{1},\lambda_{2},\lambda_{3})$. Then
$2s_{1}-s=0$ holds for $(G/K,g,J_{4})$ if the metric $g$ satisfies the
following conditions:

  1. (1)

$G/K$ is type I

    1. (a)

$g=(\lambda_{1},\lambda_{2},\lambda_{3})$ with $\lambda_{1}=1$,
$\lambda_{2}=1$ and
$\lambda_{3}=\frac{1}{2}\sqrt{\frac{160K^{2}-24KL+L^{2}}{K^{2}}}+\frac{12K-L}{2K}$.
In this case, $(g,J_{4})$ belongs to the Gray-Hervella class
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$.

    2. (b)

$g=(\lambda_{1},2\lambda_{1},(9+2\sqrt{22})\lambda_{1})$. In this case,
$(g,J_{4})$ belongs to the Gray-Hervella class
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}$.

  2. (2)

$G/K$ is type II

    1. (a)

$g=(\lambda_{1},\lambda_{2},\lambda_{3})$ with
$\lambda_{3}=2\sqrt{2}\sqrt{\lambda_{1}^{2}+3\lambda_{1}\lambda_{2}+\lambda_{2}^{2}}+3(\lambda_{1}+\lambda_{2})$.
In this case, $(g,J_{4})$ belongs to the Gray-Hervella class
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}$.

###### Proof.

Let us split our analysis for flag manifolds of type I and II:

  1. (a)

Flag manifolds of type I: We get

$\|(dF)^{-}\|^{2}=K\dfrac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{6\lambda_{1}\lambda_{2}\lambda_{3}},\quad\|(dF)^{+}\|^{2}=L\dfrac{(-\lambda_{1}-\lambda_{1}+\lambda_{2})^{2}}{6\lambda_{1}^{2}\lambda_{2}},$

$\|N^{0}\|^{2}=8K\dfrac{(-2\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}-2\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}+\lambda_{2}-2\lambda_{3})^{2}}{27\lambda_{1}\lambda_{2}\lambda_{3}}.$

where the non-zero constants $K$ and $L$ are defined by equation 5.1 and
depends on the flag manifold.

Let us analyze case-by-case the Gray-Hervella class with respect to the almost
complex structure $J_{4}$:

    1. (I)

For $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$:

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{36\lambda_{1}\lambda_{2}\lambda_{3}}+K\dfrac{(-2\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}-2\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}+\lambda_{2}-2\lambda_{3})^{2}}{27\lambda_{1}\lambda_{2}\lambda_{3}}$
$\displaystyle+$ $\displaystyle
L\dfrac{(-\lambda_{1}-\lambda_{1}+\lambda_{2})^{2}}{12\lambda_{1}^{2}\lambda_{2}}$

Therefore, $2s_{1}-s=0$ if, and only if,

$K\lambda_{1}\left(\lambda_{1}^{2}-6\lambda_{1}(\lambda_{2}+\lambda_{3})+\lambda_{2}^{2}-6\lambda_{2}\lambda_{3}+\lambda_{3}^{2}\right)+L\lambda_{3}(\lambda_{2}-2\lambda_{1})^{2}=0.$

Note that $\lambda_{1}=1$, $\lambda_{2}=1$ and
$\lambda_{3}=\frac{1}{2}\sqrt{\frac{160K^{2}-24KL+L^{2}}{K^{2}}}+\frac{12K-L}{2K}$
is a solution of the equation $2s_{1}-s=0$.

    2. (II)

For $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$: Condition:
$\lambda_{2}=2\lambda_{1}$

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{36\lambda_{1}\lambda_{2}\lambda_{3}}+K\dfrac{(-2\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}-2\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}+\lambda_{2}-2\lambda_{3})^{2}}{27\lambda_{1}\lambda_{2}\lambda_{3}}$

Therefore, $2s_{1}-s=0$ if, and only if,

$K\left(5x\lambda_{1}^{2}-12\lambda_{1}\lambda_{3}-6\lambda_{1}(2\lambda_{1}+\lambda_{3})+\lambda_{3}^{2}\right)=0$

In this case, we conclude that $2s_{1}-s=0$ if and only if the metric
satisfies $(\lambda_{1},2\lambda_{1},(9+2\sqrt{22})\lambda_{1})$

    3. (III)

For $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$: Condition:
$\lambda_{1}=\lambda_{2}=\lambda_{3}$

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{36\lambda_{1}\lambda_{2}\lambda_{3}}+L\dfrac{(-\lambda_{1}-\lambda_{1}+\lambda_{2})^{2}}{12\lambda_{1}^{2}\lambda_{2}}$

Therefore, $2s_{1}-s=0$ if, and only if,

$-45K\lambda_{3}^{3}+3L\lambda_{3}^{3}=0,$

and there is not solution of the equation $2s_{1}-s=0$ independent of $L$ and
$K$. Remark: It is worth to point out that $L=15K$ is a solution of the
equation $2s_{1}-s=0$, but it solution depends on case-by-case examination and
we do not include it as a general solution.

  2. (b)

Flag manifolds type II: We find

$\|(dF)^{+}\|^{2}=0,\quad\quad\|(dF)^{-}\|^{2}=K\dfrac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{6\lambda_{1}\lambda_{2}\lambda_{3}},$

$\|N^{0}\|^{2}=8K\dfrac{(-2\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}-2\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}+\lambda_{2}-2\lambda_{3})^{2}}{27\lambda_{1}\lambda_{2}\lambda_{3}}$

where the non-zero constant $K$ is defined by equation 5.1 and depends on the
flag manifold.

By equation 3.1 we obtain

$\displaystyle 2s_{1}-s$ $\displaystyle=$
$\displaystyle-\dfrac{5}{6}\|(dF)^{-}\|^{2}+\dfrac{1}{8}\|N^{0}\|^{2}+\dfrac{1}{2}\|(dF)^{+}\|^{2}$
$\displaystyle=$
$\displaystyle-5K\dfrac{(\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}}{36\lambda_{1}\lambda_{2}\lambda_{3}}+K\dfrac{(-2\lambda_{1}+\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}-2\lambda_{2}+\lambda_{3})^{2}+(\lambda_{1}+\lambda_{2}-2\lambda_{3})^{2}}{27\lambda_{1}\lambda_{2}\lambda_{3}}$

Therefore, $2s_{1}-s=0$ if, and only if,

$K\left(\lambda_{1}^{2}-6\lambda_{1}(\lambda_{2}+\lambda_{3})+\lambda_{2}^{2}-6\lambda_{2}\lambda_{3}+\lambda_{3}^{2}\right)=0.$

and the positive solutions are given by $\lambda_{1}>0$, $\lambda_{2}>0$ and

$\lambda_{3}=2\sqrt{2}\sqrt{\lambda_{1}^{2}+3\lambda_{1}\lambda_{2}+\lambda_{2}^{2}}+3(\lambda_{1}+\lambda_{2}).$

∎

##  6\. Examples

In this section we will provide some explicit computations on some generalized
flag manifolds. Here one can obtain precisely the $(1,2)$ and $(0,3)$ triples
and consequently the constants $K$ and $L$ defined by equation 5.1. Using this
facts we will provide some solutions for the equation $s=2s_{1}$ for specific
flag manifolds.

###  6.1. $G_{2}/U(2)$

We consider the Type I flag manifold $G_{2}/U(2)$ of real dimension $10$,
where $U(2)$ is represented by the long root of $G_{2}$ (the algebraic
structure of the $G_{2}$ can be seen in [3]). This manifold has three
isotropic summands given by $\mathfrak{m}_{1}$, $\mathfrak{m}_{2}$ and
$\mathfrak{m}_{3}$ given by

$\begin{array}[]{l}\mathfrak{m}_{1}=\mathfrak{g}_{\alpha_{1}+\alpha_{2}}\oplus\mathfrak{g}_{\alpha_{2}},\\\
\mathfrak{m}_{2}=\mathfrak{g}_{\alpha_{1}+2\alpha_{2}},\\\
\mathfrak{m}_{3}=\mathfrak{g}_{\alpha_{1}+3\alpha_{2}}\oplus\mathfrak{g}_{2\alpha_{1}+3\alpha_{2}}.\end{array}$

From proposition 2.5, $G_{2}/U(2)$ admits $4$ invariant almost complex
structures, up to conjugation:

$\begin{array}[]{ccccccc}J_{1}=(+,+,+)&&&&&&J_{3}=(-,+,-)\\\
J_{2}=(-,+,+)&&&&&&J_{4}=(+,+,-),\end{array}$

where $J_{1}$ is integrable.

The triples of zero sum of $G_{2}/U(2)$ are (up to sign):

$\begin{array}[]{cl}1)&\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0\\\
2)&\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0\\\
3)&(\alpha_{1}+\alpha_{2})+(\alpha_{1}+2\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0.\end{array}$

We also compute the value of the constants $K$ and $L$ (equations 5.1) for
each specific invariant almost complex structure. We summarize the computation
in the Table 10.

Table 10. Constants $K$ and $L$ Structure | $K$ | $L$
---|---|---
$(+,+,+)$ | $0$ | $5/12$
$(-,+,+)$ | $1/6$ | $1/4$
$(+,-,+)$ | $1/6$ | $1/4$
$(+,+,-)$ | $1/4$ | $1/6$

Consider the orthonormal basis of $TM_{\mathbb{C}}$:

$\displaystyle\left\\{\dfrac{X_{\alpha_{2}}}{\sqrt{x}},\dfrac{X_{\alpha_{1}+\alpha_{2}}}{\sqrt{x}},\dfrac{X_{\alpha_{1}+2\alpha_{2}}}{\sqrt{y}},\dfrac{X_{\alpha_{1}+3\alpha_{2}}}{\sqrt{z}},\dfrac{X_{2\alpha_{1}+3\alpha_{2}}}{\sqrt{z}},\dfrac{X_{-\alpha_{2}}}{\sqrt{x}},\right.$
$\displaystyle\left.\dfrac{X_{-(\alpha_{1}+\alpha_{2})}}{\sqrt{x}},\dfrac{X_{-(\alpha_{1}+2\alpha_{2})}}{\sqrt{y}},\dfrac{X_{-(\alpha_{1}+3\alpha_{2})}}{\sqrt{z}},\dfrac{X_{-(2\alpha_{1}+3\alpha_{2})}}{\sqrt{z}}\right\\}.$

One can classified Gray-Hervella (GH) classes for $G_{2}/U(2)$ and study the
equation $2s_{1}=s$ for each GH-class.

  * •

Let us consider the invariant almost complex structure $J_{1}=(+,+,+)$
(integrable) and Riemannian metric parameterized by $g=(x,y,z)$. We find

$\|(dF)^{-}\|^{2}=\|N^{0}\|^{2}=0,$

and

$\|(dF)^{+}\|=\frac{1}{36x^{2}y}\left(\frac{3x(x+y-z)^{2}}{z}+2(y-2x)^{2}\right).$

The covariant derivative of Kähler form is

$\displaystyle\|DF\|^{2}=\frac{3x^{3}+2x^{2}(3y+z)+x\left(3y^{2}-14yz+3z^{2}\right)+2y^{2}z}{36x^{2}yz}.$

We have $(J_{1},g)\in\mathcal{W}_{3}$ unless $y=2x$ and $z=3x$ which
corresponds to the Kähler class. The solution for the equation $2s_{1}-s=0$ is
$y=2x$ and $z=3x$, that is, the unique solution is given by the Kähler metric.

  * •

Let us consider the invariant almost complex structure $J_{2}=(-,+,+)$ (non-
integrable) and Riemannian metric parameterized by $g=(x,y,z)$. We get

$\|(dF)^{-}\|^{2}=\dfrac{(2x+y)^{2}}{18x^{2}y},\quad\quad\|N^{0}\|^{2}=\frac{16(x-y)^{2}}{27x^{2}y},\quad\quad\|(dF)^{+}\|=\frac{(x-y+z)^{2}}{12xyz}.$

The covariant derivative of the Kähler form is

$\|DF\|^{2}=\frac{3x^{3}+x^{2}(14z-6y)+x\left(3y^{2}-14yz+3z^{2}\right)+6y^{2}z}{36x^{2}yz}.$

    1. (1)

If the metric $g$ satisfies $x=y$ then
$(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$. In this case

$2s_{1}-s=0\Leftrightarrow z=10x.$

    2. (2)

If the metric $g$ satisfies $z=-x+y$ and $y>x$ then
$(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. In this case

$2s_{1}-s=0\Leftrightarrow y=(2\sqrt{10}+6)x.$

    3. (3)

Otherwise the pair $(J_{2},g)$ belongs to
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$. In this case

$2s_{1}-s=\frac{3x^{3}-2x^{2}(3y+z)+3x\left(y^{2}-10yz+z^{2}\right)+2y^{2}z}{72x^{2}yz}$

The triples $x=1$, $y=2$ and $z=9-\sqrt{5}$ is an example of solution of the
equation $2s_{1}-s=0$.

  * •

Let us consider the invariant almost complex structure $J_{3}=(+,-,+)$ (non-
integrable) and Riemannian metric parameterized by $g=(x,y,z)$. We have

$\|(dF)^{-}\|^{2}=\frac{(2x+y)^{2}}{18x^{2}y},\quad\quad\|N^{0}\|^{2}=\frac{16(x-y)^{2}}{27x^{2}y},\quad\quad\|(dF)^{+}\|=\frac{(-x+y+z)^{2}}{12xyz}.$

The covariant derivative of Kähler form is

$\|DF\|^{2}=\frac{3x^{3}+x^{2}(2z-6y)+x\left(3y^{2}-2yz+3z^{2}\right)+6y^{2}z}{36x^{2}yz}.$

    1. (1)

If the metric $g$ satisfies $x=y$ then
$(J_{3},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$. In this case

$2s_{1}-s=0\Leftrightarrow z=10x.$

    2. (2)

If the metric $g$ satisfies $z=x-y$ and $x>y$ then
$(J_{3},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. In this case there is no
solution for equation $2s_{1}-s=0$.

    3. (3)

Otherwise the pair $(J_{3},g)$ belongs to
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$. One can provide
an example of non-trivial solution of the equation $2s_{1}-s=0$ given by
$x=1$, $y=2$ and $z=7+4\sqrt{3}$.

  * •

Let us consider the invariant almost complex structure $J_{4}=(+,+,-)$ (non-
integrable) and Riemannian metric parameterized by $g=(x,y,z)$.

The $(0,3)+(3,0)$-part exterior derivative of the Kähler form is given by

$\displaystyle\|(dF)^{-}\|^{2}=\dfrac{(x+y+z)^{2}}{12xyz}\ \ \mbox{and}$

$N^{0}$ is defined by

$\displaystyle\|N^{0}\|^{2}=\dfrac{8\left(x^{2}-x(y+z)+y^{2}-yz+z^{2}\right)}{9xyz}.$

Moreover, the $(1,2)+(2,1)$-part of the Kähler form is

$\displaystyle\|(dF)^{+}\|^{2}=\dfrac{(y-2x)^{2}}{18x^{2}y}.$

The covariant derivative of the Kähler form is

$\displaystyle\|DF\|^{2}=\frac{9x^{3}+x^{2}(2z-6y)+x\left(9y^{2}-14yz+9z^{2}\right)+2y^{2}z}{36x^{2}yz}.$

Note that
$\|DF\|^{2}=\|(dF)^{+}\|^{2}+\dfrac{1}{4}\|N^{0}\|^{2}+\dfrac{1}{3}\|(dF)^{-}\|^{2}$.

    1. (1)

The metric $g$ satisfies $x=y=z$. In this case $(J_{4},g)$ belongs to
$\mathcal{W}_{1}\oplus\mathcal{W}_{3}$.

In this case there is not solution for equation $2s_{1}-s=0$.

    2. (2)

The metric $g$ satisfies the condition $y=2x$. In this case $(J_{4},g)$
belongs to $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. We obtain

$2s_{1}-s=0\Leftrightarrow z=\dfrac{9y}{2}+\sqrt{22}y.$

    3. (3)

Otherwise the pair $(J_{4},g)$ belongs to
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$. We get

(6.1) $\displaystyle
2s_{1}-s=\frac{3x^{3}-2x^{2}(9y+5z)+x\left(3y^{2}-26yz+3z^{2}\right)+2y^{2}z}{72x^{2}yz}.$

One example of solution of Equation 6.1 is given by $x=1$, $y=1$ and
$z=\frac{1}{3}\left(5\sqrt{13}+17\right)$ and therefore this metric satisfies
the equation $2s_{1}-s=0$.

###  6.2. $SU(n+2)/S(U(n)\times U(1)\times U(1)$)

Let us consider the Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{su}(n+2)$
described by

$\mathfrak{h}=\\{\mbox{diag}(x_{1},\cdots,x_{n},x_{n+1},x_{n+2}):x_{1}+\cdots+x_{n}+x_{n+1}+x_{n+2}=0,\
x_{i}\in\mathbb{C},\ i=1,\cdots,n\\}.$

Moreover, the isotropy representation of $\mathbb{F}_{n+2}$ admits three
isotropic summands:
$\mathfrak{m}=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\mathfrak{m}_{3}$,
where

$\begin{array}[]{l}\mathfrak{m}_{1}=\displaystyle\bigoplus_{i=1}^{n}\mathfrak{u}_{\alpha_{i,n+1}},\quad\quad\mathfrak{m}_{2}=\mathfrak{u}_{\alpha_{n+1,n+2}},\quad\quad\mathfrak{m}_{3}=\displaystyle\bigoplus_{i=1}^{n}\mathfrak{u}_{\alpha_{i,n+2}},\\\
\end{array}$

with
$\mathfrak{u}_{ij}=\mathfrak{su}(n+2)\cap(\mathfrak{g}_{ij}\oplus\mathfrak{g}_{ji})$.

The matrix representation of $\mathfrak{m}$ is:

$\left(\begin{array}[]{cccc|c|c}x_{1}&&&&*&*\\\ &x_{2}&&&*&*\\\
&&\ddots&&\vdots&\vdots\\\ &&&x_{n}&*&*\\\ \hline\cr&&&&x_{n+1}&*\\\
\hline\cr&&&&&x_{n+2}\end{array}\right).$

The flag manifold $SU(n+2)/S(U(n)\times U(1)\times U(1))$ admits $3$ invariant
almost complex structures, up to conjugation and equivalence:

$\begin{array}[]{ccccccccccc}J_{1}=(+,+,+)&&&&&J_{2}=(-,+,+)&&&&&J_{3}=(+,+,-).\end{array}$

where $J_{1}$ and $J_{2}$ are integrable.

It is easy to see the triples of roots are given by:

$\mbox{2n
triples}:\left\\{\begin{array}[]{l}\alpha_{1,n+1}+\alpha_{n+1,n+2}+\alpha_{n+2,1}=0\\\
\alpha_{2,n+1}+\alpha_{n+1,n+2}+\alpha_{n+2,2}=0\\\ \ \ \ \ \ \ \ \vdots\\\
\alpha_{n,n+1}+\alpha_{n+1,n+2}+\alpha_{n+2,n}=0\\\
-\alpha_{1,n+1}-\alpha_{n+1,n+2}-\alpha_{n+2,1}=0\\\
-\alpha_{2,n+1}-\alpha_{n+1,n+2}-\alpha_{n+2,2}=0\\\ \ \ \ \ \ \ \ \vdots\\\
-\alpha_{n,n+1}-\alpha_{n+1,n+2}-\alpha_{n+2,n}=0\\\ \end{array}\right.$

We can also classify the invariant almost Hermitian structures as follow:

  * •

$J_{1}=(+,+,+)$ integrable and $g=(x,y,z)$ invariant Riemannian metric: If
$z=x+y$ then $(J_{1},g)$ is Kähler, otherwise $(J_{1},g)\in\mathcal{W}_{3}$.
We have

$\|(dF)^{-}\|^{2}=\|N^{0}\|^{2}=0,\quad\quad\|(dF)^{+}\|^{2}=\dfrac{n(x+y-z)}{3xyz},$

and $2s_{1}-s=0$ if and only if $z=x+y$, i.e., $(J_{1},g)$ is Kähler.

  * •

$J_{2}=(-,+,+)$ integrable and $g=(x,y,z)$ invariant Riemannian metric: if
$z=-x+y$ with $y>x$ then $(J_{2},g)$ is Kähler, otherwise
$(J_{2},g)\in\mathcal{W}_{3}$. We have

$\|(dF)^{-}\|^{2}=\|N^{0}\|^{2}=0\quad\quad\|(dF)^{+}\|^{2}=\dfrac{n(x-y+z)}{3xyz}$

and $2s_{1}-s=0$ if and only if $z=-x+y$, i.e., $(J_{2},g)$ is Kähler.

  * •

$J_{3}=(+,+,-)$ non-integrable and $g=(x,y,z)$ invariant Riemannian metric: if
$x=y=z$ then $(J_{3},g)\in\mathcal{W}_{1}$, otherwise
$(J_{3},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. We have

$\|(dF)^{-}\|^{2}=\frac{n(x+y+z)^{2}}{3xyz},\quad\quad\|N^{0}\|^{2}=\frac{32n\left(x^{2}-x(y+z)+y^{2}-yz+z^{2}\right)}{9xyz},\quad\quad\|(dF)^{+}\|^{2}=0.$

If $(J_{3},g)\in\mathcal{W}_{1}$ there is no solution of the equation
$2s_{1}-s=0$. If $(J_{3},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$ we find
the following non-trivial solutions (non-Kähler) for the equation
$2s_{1}-s=0$: $x>0,y>0$ and $z=3(x+y)+2\sqrt{2}\sqrt{x^{2}+3xy+y^{2}}$ ; or
$x>0,y>(3+2\sqrt{2})x$ and $z=3(x+y)-2\sqrt{2}\sqrt{x^{2}+3xy+y^{2}}$.

###  6.3. $F_{4}/SU(3)\times SU(2)\times U(1)$

Let $\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$ be the simple roots of
the Lie algebra of $F_{4}$, with maximal root
$\mu=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$. The description of the
components $\mathfrak{m}_{1}$, $\mathfrak{m}_{2}$, $\mathfrak{m}_{3}$ of the
isotropy representation is given in Table 11. The coordinates in the table are
given in terms of the simple roots $\alpha_{1},\ldots,\alpha_{4}$. For
instance $[1,2,4,2]$ means $\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{2}$.
The triples of roots with zero sum are listed in Table 12. We also compute the
value of the constants $K$ and $L$ (equations 5.1) for each specific invariant
almost complex structure. We summarize the computation in the Table 13.

Let us analyze the solution for equation $2s_{1}-s=0$ for each invariant
almost Hermitian structure $(g,J)$, being $g$ an invariant Riemannian metric
parametrized by $g=(x,y,z)$.

Table 11. Positive roots of G/K in coordinates $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ Roots in $\mathfrak{m}_{1}$ | Roots in $\mathfrak{m}_{2}$ | Roots in $\mathfrak{m}_{3}$
---|---|---
[0,1,0,0] | [1,2,2,0] | [1,3,4,2]
[1,1,0,0] | [1,2,2,1] | [2,3,4,2]
[0,1,1,0] | [1,2,3,1] |
[1,1,1,0] | [1,2,2,2] |
[0,1,2,0] | [1,2,3,2] |
[0,1,1,1] | [1,2,4,2] |
[1,1,2,0] |  |
[1,1,1,1] |  |
[0,1,2,1] |  |
[1,1,2,1] |  |
[0,1,2,2] |  |
[1,1,2,2] |  |
Table 12. Triples with zero sum: $\alpha+\beta+\gamma=0$ $\alpha\in\mathfrak{m}_{1}$ | $\beta\in\mathfrak{m}_{1}$ | $-\gamma\in\mathfrak{m}_{2}$ | $m_{\alpha,\beta}^{2}$
---|---|---|---
[0,1,0,0] | [1,1,2,0] | [1,2,2,0] | $1$
[0,1,0,0] | [1,1,2,1] | [1,2,2,1] | $1$
[0,1,0,0] | [1,1,2,2] | [1,2,2,2] | $1$
[1,1,0,0] | [0,1,2,0] | [1,2,2,0] | $1$
[1,1,0,0] | [0,1,2,1] | [1,2,2,1] | $1$
[1,1,0,0] | [0,1,2,2] | [1,2,2,2] | $1$
[0,1,1,0] | [1,1,1,0] | [1,2,2,0] | $1$
[0,1,1,0] | [1,1,1,1] | [1,2,2,1] | $1/2$
[0,1,1,0] | [1,1,2,1] | [1,2,3,1] | $1/2$
[0,1,1,0] | [1,1,2,2] | [1,2,3,2] | $1$
[1,1,1,0] | [0,1,1,1] | [1,2,2,1] | $1/2$
[1,1,1,0] | [0,1,2,1] | [1,2,3,1] | $1/2$
[1,1,1,0] | [0,1,2,2] | [1,2,3,2] | $1$
[0,1,2,0] | [1,1,1,1] | [1,2,3,1] | $1$
[0,1,2,0] | [1,1,2,2] | [1,2,4,2] | $1$
[0,1,1,1] | [1,1,2,0] | [1,2,3,1] | $1$
[0,1,1,1] | [1,1,1,1] | [1,2,2,2] | $1$
[0,1,1,1] | [1,1,2,1] | [1,2,3,2] | $1/2$
[1,1,2,0] | [0,1,2,2] | [1,2,4,2] | $1$
[1,1,1,1] | [0,1,2,1] | [1,2,3,2] | $1/2$
[0,1,2,1] | [1,1,2,1] | [1,2,4,2] | $1$
– | – | – | –
$\alpha\in\mathfrak{m}_{1}$ | $\beta\in\mathfrak{m}_{2}$ | $-\gamma\in\mathfrak{m}_{3}$ | $m_{\alpha,\beta}^{2}$
[0,1,0,0] | [1,2,4,2] | [1,3,4,2] | $1$
[1,1,0,0] | [1,2,4,2] | [2,3,4,2] | $1$
[0,1,1,0] | [1,2,3,2] | [1,3,4,2] | $1$
[1,1,1,0] | [1,2,3,2] | [2,3,4,2] | $1$
[0,1,2,0] | [1,2,2,2] | [1,3,4,2] | $1$
[0,1,1,1] | [1,2,3,1] | [1,3,4,2] | $1$
[1,1,2,0] | [1,2,2,2] | [2,3,4,2] | $1$
[1,1,1,1] | [1,2,3,1] | [2,3,4,2] | $1$
[0,1,2,1] | [1,2,2,1] | [1,3,4,2] | $1$
[1,1,2,1] | [1,2,2,1] | [2,3,4,2] | $1$
[0,1,2,2] | [1,2,2,0] | [1,3,4,2] | $1$
[1,1,2,2] | [1,2,2,0] | [2,3,4,2] | $1$
Table 13. Constants $K$ and $L$ Structure | $K$ | $L$
---|---|---
$(+,+,+)$ | $0$ | $30$
$(-,+,+)$ | $18$ | $12$
$(+,-,+)$ | $18$ | $12$
$(+,+,-)$ | $12$ | $18$

  * •

Invariant complex structure $J=(+,+,+)$ (integrable): if $y=2x$ and $z=3x$ the
pair $(g,J)$ is Kähler, otherwise $(J,g)\in\mathcal{W}_{3}$. We have

$\|(dF)^{-}\|^{2}=\|N^{0}\|^{2}=0,\quad\|(dF)^{+}\|^{2}=\frac{2}{x^{2}y}\left(\frac{2x(x+y-z)^{2}}{z}+3(y-2x)^{2}\right),$

and $2s_{1}-s=0$ if and only if $y=2x$ and $z=3x$, i.e., when $(J,g)$ is
Kähler.

  * •

Invariant almost complex structure $J_{1}=(-,+,+)$:  if $z=-x+y$ and $y>x$
then $(J_{1},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$, if $x=y$ then
$(J_{1},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$, otherwise
$(J_{1},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$. We
obtain

$\|(dF)^{-}\|^{2}=\frac{6(2x+y)^{2}}{x^{2}y},\quad\|N^{0}\|^{2}=\frac{64(x-y)^{2}}{x^{2}y},\quad\|(dF)^{+}\|^{2}=\frac{4(x-y+z)^{2}}{xyz}.$

If $(J_{1},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$ we have solution for the
equation $2s_{1}-s=0$ given by the metric $y=(2\sqrt{10}+6)x$ and $x>0$. For
$(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ we get the metric given by
$x=y>0$, $z=45y/2$ is a solution for the equation $2s_{1}-s=0$. If
$(J_{1},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ then
we have the families of solutions of $2s_{1}-s=0$:

    1. (1)

$g=(x,y,z)$ parameterized by $x>0,0<y<x$ and
$z=\frac{8x^{2}+\sqrt{3}\sqrt{16x^{4}+224x^{3}y+512x^{2}y^{2}-80xy^{3}+3y^{4}}+40xy-3y^{2}}{4x}$.

    2. (2)

$g=(x,y,z)$ parameterized by $x=1$, $y=2$ and $z=19+6\sqrt{10}$.

  * •

Invariant almost complex structure $J_{2}=(+,-,+)$:  if $z=x-y$ and $x>y$ then
$(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$, if $x=y$ then
$(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$, otherwise
$(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$. We
find

$\|(dF)^{-}\|^{2}=\frac{6(2x+y)^{2}}{x^{2}y},\quad\|N^{0}\|^{2}=\frac{64(x-y)^{2}}{x^{2}y},\quad\|(dF)^{+}\|^{2}=\frac{4(-x+y+z)^{2}}{xyz}.$

If $(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$ we do not have solution
for the equation $2s_{1}-s=0$. For
$(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ we have the metric given by
$x=y>0$, $z=45y/2$ is a solution for the equation $2s_{1}-s=0$. If
$(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ then
we have the families of solutions of $2s_{1}-s=0$:

    1. (1)

$g=(x,y,z)$ parameterized by $0<y<x$ and
$z=\frac{16x^{2}+\sqrt{3}\sqrt{80x^{4}+352x^{3}y+304x^{2}y^{2}-64xy^{3}+3y^{4}}+32xy-3y^{2}}{4x}$.

    2. (2)

$g=(x,y,z)$ parameterized by $x>0$, $y=10x$ and $z=9x$.

  * •

Invariant almost complex structure $J_{3}=(+,+,-)$:  if $y=2x$ then
$(J_{3},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$, if $x=y=z$ then
$(J_{3},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$, otherwise
$(J_{3},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$. We
have

$\|(dF)^{-}\|^{2}=\frac{4(x+y+z)^{2}}{xyz},\quad\|N^{0}\|^{2}=\frac{128\left(x^{2}-x(y+z)+y^{2}-yz+z^{2}\right)}{3xyz}$

,

$\|(dF)^{+}\|^{2}=\frac{6(y-2x)^{2}}{x^{2}y}.$

If $(J_{3},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ there isn’t solution for
the equation $2s_{1}-s=0$. For
$(J_{2},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$ we have the metric given by
$y=2x$, $z=(2\sqrt{22}+9)x$, $x>0$ is a solution for the equation
$2s_{1}-s=0$. If
$(J_{3},g)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ then
we have the families of solutions of $2s_{1}-s=0$:

    1. (1)

$g=(x,y,z)$ parameterized by
$z=\frac{\sqrt{-16x^{4}+96x^{3}y+560x^{2}y^{2}-144xy^{3}+9y^{4}}+24xy-3y^{2}}{4x}$,
for $x,y>0$ and $3x-2\sqrt{2}\sqrt{x^{2}}\leq y\leq 2\sqrt{2}\sqrt{x^{2}}+3x$

    2. (2)

$g=(x,y,z)$ parameterized by $x=1$, $y=1$ and $z=\frac{1}{4}(21+\sqrt{505})$.

## References

  * [1] ANASTASSIOU, S.; CHRYSIKOS, I. The Ricci flow approach to homogeneous Einstein metrics on flag manifolds. Journal of Geometry and Physics, v. 61, n. 8, p. 1587-1600, 2011.
  * [2] APOSTOLOV, V.; DRAGHICI, T. Almost Kähler 4-manifolds with J-invariant Ricci tensor and special Weyl tensor. Q. J. Math., 51(3):275–294, 2000. 2
  * [3] ARVANITOYEORGOS, A. et al. Homogeneous Einstein metrics on $G_{2}/T$. Proceedings of the American Mathematical Society, v. 141, n. 7, p. 2485-2499, 2013.
  * [4] BESSE, A. L.. Einstein manifolds, volume 10 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1987.1
  * [5] BOREL, A; HIRZEBRUCH, F. Characteristic classes and homogeneous spaces. I. Amer. J. Math., 80, 1958\. 7.
  * [6] DABKOWSKI, M.; LOCK, M. An equivalence of scalar curvatures on Hermitian manifolds. J. Geom. Anal., 27(1):239–270, 2017. 1, 2
  * [7] FU, J.; ZHOU, X. Scalar curvatures in almost Hermitian geometry and some applications. Science China Mathematics, p. 1-18, 2022.
  * [8] FUSI, E. The Prescribed Chern Scalar Curvature Problem. J Geom Anal 32, 187 (2022).
  * [9] GAUDUCHON, P. La 1-forme de torsion d’une variété hermitienne compacte. Math. Ann., 267(4):495–518, 1984. 7
  * [10] GAUDUCHON, P. Hermitian connections and Dirac operators. Bollettino della Unione Matematica Italiana-B, n. 2, p. 257-288, 1997.
  * [11] GRAMA, L.; OLIVEIRA, A. Scalar Curvatures of invariant almost Hermitian structures on generalized flag manifolds. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, v. 17, p. 109, 2021.
  * [12] GRAY, A.; HERVELLA, L. M. The sixteen classes of almost Hermitian manifolds and their linear invariants. Annali di Matematica pura ed applicata, v. 123, n. 1, p. 35-58, 1980.
  * [13] KAZDAN,J; WARNER, F, Scalar curvature and conformal deformation of Riemannian structure. J. Differential Geom. 10(1): 113–134 (1975).
  * [14] KIMURA, M. Homogeneous Einstein metrics on certain Kähler C-spaces. In: Recent topics in differential and analytic geometry. Academic Press, 1990. p. 303-320.
  * [15] LIU, K.; YANG, X. Ricci curvatures on Hermitian manifolds. Transactions of the American Mathematical Society, v. 369, n. 7, p. 5157-5196, 2017.
  * [16] MEHDI L.; UPMEIER, M. Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometry. Comm. Anal. Geom., 28(7):1603–1645, 2020. 2
  * [17] MOROIANU, A. Lectures on Kähler Geometry, Cambridge University Press (2007).
  * [18] SAN MARTIN, L.; NEGREIROS, C. Invariant almost Hermitian structures on flag manifolds. Adv. Math. 178, 2003, 277–310.
  * [19] SAN MARTIN, L.; de JESUS SILVA, R. Invariant nearly-Kähler structures. Geom. Dedicata 121, 2006, 143–154.
  * [20] TOSATTI,V., WEINKOVE,B. The Chern–Ricci flow. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 33 (2022), no. 1, pp. 73–107.