Scalar Curvatures of Invariant Almost Hermitian Structures

Scalar Curvatures of Invariant Almost Hermitian
Structures on Generalized Flag Manifolds

Lino GRAMA a and Ailton R. OLIVEIRA b

L. Grama and A.R. Oliveira

a) IMECC - Universidade Estadual de Campinas (Unicamp), Departamento de
Matemática,
a) Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz.
a) 13083-859 Campinas - SP, Brazil<EMAIL_ADDRESS>https://sites.google.com/unicamp.br/linograma/

b) UEMS - Universidade Estadual de Mato Grosso do Sul - MS, Cidade
Universitária
b) de Dourados, Rodovia Itahum, Km 12 s/n - Jardim Aeroporto, Dourados - MS,
Brazil<EMAIL_ADDRESS>
Received August 02, 2021, in final form December 11, 2021; Published online
December 21, 2021

In this paper we study invariant almost Hermitian geometry on generalized flag
manifolds. We will focus on providing examples of Kähler like scalar curvature
metric, that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_{\rm
C}$, where $s$ is Riemannian scalar curvature and $s_{\rm C}$ is the Chern
scalar curvature.

curvature of almost Hermitian structures; generalized flag manifolds; Kähler
like scalar curvature

53C55; 53C21; 14M15

##  1 Introduction

In the present paper we will study curvature properties of invariant almost
Hermitian structures on homogeneous spaces. Special attention will be given to
providing examples of Kähler like scalar curvature $($Klsc$)$ metric, that is,
almost Hermitian structures $(g,J)$ satisfying $s=2s_{\rm C}$, where $s$ is
the Riemannian scalar curvature and $s_{\rm C}$ is the Chern scalar curvature
on generalized flag manifolds.

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:

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

where $\omega_{n}$ is the volume of the unity ball in $\mathbb{R}^{n}$, see
[6, 9]. 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. If the metric is not Kähler, there is not an immediate geometric
interpretation for the Chern scalar curvature $s_{\rm C}$. However if the
Hermitian structure is non-Kähler but satisfies $s=2s_{\rm C}$, a curvature
$s_{\rm C}$ has an interpretation like $s$ in the expression (1.1).

Dabkowski and Lock in [9] call the Hermitian metrics satisfying the equation

$\displaystyle s=2s_{\rm C}$

by Kähler like scalar curvature $($Klsc$)$ metric, and the authors exhibit
examples of non-compact Hermitian manifolds satisfying $s=2s_{\rm C}$.

According to [2] and [18] nearly-Kähler manifolds and almost-Kähler manifolds
satisfying $2s_{\rm C}-s=0$ are Kähler. Fu and Zhou show in [12] that if the
pair $(g,J)$ belongs to the Gray–Hervella class
$\mathcal{W}_{2}\oplus\mathcal{W}_{3}\oplus\mathcal{W}_{4}$ with $2s_{\rm
C}=s$ then $(g,J)$ is Kähler. Recently Lejmi–Upmeier propose the following
question [18, Remark 3.3]:

######  Question 1.1.

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

In this paper we will investigate the Question 1.1 in a class of high
dimensional homogeneous spaces called generalized flag manifolds, equipped
with an invariant almost Hermitian structure. It is well know that this class
of homogeneous spaces have a very rich almost Hermitian geometry, see for
instance [20, 21]. We will proceed by computing explicitly the Hermitian
scalar curvatures, using the tools of Lie theory. The approach to compute
explicitly the Hermitian scalar curvatures on flag manifolds is by analyzing
the decomposition of the covariant derivative of the Kähler form with respect
to the Levi-Civita connection, the codifferential of the Lee form, and the
scalar Riemannian curvature. See Section 3 for details.

Let us denote by $\nabla^{t}$ be the 1-parameter family of connection
introduced by Gauduchon in [14]:

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

where $D$ is the Levi-Civita connection. It is well know that $\nabla^{1}$
coincides with the Chern connection. According to [12], 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
$\\{u_{i}\\}_{i=1,2,\dots,n}$ is a unitary frame. We will call $s_{1}(t)$ and
$s_{2}(t)$ the first and second Hermitian scalar curvature. One of the main
features of the generalized flag manifolds is that the first scalar curvature
$s_{1}(t)$ is constant in $t$ and we will denote this curvature just by
$s_{1}$ and it coincides with the Chern scalar curvature $s_{\rm C}$. 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?

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

######  Theorem 1.3.

Consider the flag manifold ${\rm SU}(3)/T^{2}$, equipped with an invariant
almost Hermitian structure $(g,J)$, with invariant metric $g$ parametrized by
a triple of positive numbers $(x,y,z)$ and $J$ non-integrable. Denote by
$s_{1}$ the first Hermitian scalar curvature and by $s$ the Riemannian scalar
curvature. The pair satisfies the equation $2s_{1}-s=0$ if, and only if,
$(g,J)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$, and the metric $g$ satisfies
one of the following relations:

  1. $1.$

$z=3(x+y)-2\sqrt{2}\sqrt{x^{2}+3xy+y^{2}}$, for $y>\big{(}2\sqrt{2}+3\big{)}x$
and $x>0$; or $0<y<\big{(}3-2\sqrt{2}\big{)}x$ and $x>0$.

  2. $2.$

$z=2\sqrt{2}\sqrt{x^{2}+3xy+y^{2}}+3(x+y)$, for $x>0$ and $y>0$.

######  Remark 1.4.

The solutions of the equation $2s_{1}-s=0$ for the integrable invariant almost
complex structure $J$ on ${\rm SU}(3)/T^{2}$ are already know:
$(g,J)\in\mathcal{W}_{3}$, being $J$ integrable and the metric $g$ satisfies
$z=x+y$. In this case, the pair $(g,J)$ is Kähler, see [11].

######  Remark 1.5.

It is worth to point out that in the case of ${\rm SU}(3)/T^{2}$ if $J$ is
non-integrable and the metric $g$ is parametrized by $(x,x,x)$ then
$(g,J)\in\mathcal{W}_{1}$ (nearly-Kähler) and there is no solution to the
equation $2s_{1}-s=0$.

######  Theorem 1.6.

Consider the complex projective plane $\mathbb{CP}^{3}={\rm Sp}(2)/{\rm
Sp}(1)\times{\rm U}(1)$, equipped with an invariant almost Hermitian structure
$(g,J)$, with invariant metric $g$ parametrized by a pair of positive numbers
$(x,y)$. Denote by $s_{1}$ the first Hermitian scalar curvature and by $s$ the
Riemannian scalar curvature. The pair $(g,J)$ satisfies the equation
$2s_{1}-s=0$ if, and only if,

  1. $1.$

$(g,J)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$, being $J$ non-integrable and
the metric $g$ satisfies $y=2x\big{(}\sqrt{10}+3\big{)}$.

  2. $2.$

$(g,J)\in\mathcal{W}_{3}$, being $J$ integrable and the metric $g$ satisfies
$y=2x$. In this case, the pair $(g,J)$ is Kähler.

######  Remark 1.7.

It is worthwhile to point out that in the case of ${\rm Sp}(2)/{\rm
Sp}(1)\times{\rm U}(1)$ if $J$ is non-integrable and the metric $g$ is
parametrized by $(x,x)$ then $(g,J)\in\mathcal{W}_{1}$ (nearly-Kähler) and
there is no solution to the equation $2s_{1}-s=0$.

######  Theorem 1.8.

Consider the $10$-dimensional flag manifold $G_{2}/{\rm U}(2)$, where ${\rm
U}(2)$ is represented by the short root of $G_{2}$, equipped with an invariant
almost Hermitian structure $(g,J)$, with invariant metric $g$ parametrized by
a pair of positive numbers $(x,y)$. Denote by $s_{1}$ the first Hermitian
scalar curvature and by $s$ the Riemannian scalar curvature. The pair $(g,J)$
satisfies the equation $2s_{1}-s=0$ if, and only if,

  1. $1.$

$(g,J)\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$, being $J$ non-integrable and
the metric $g$ satisfies $y=2x\big{(}\sqrt{10}+3\big{)}$.

  2. $2.$

$(g,J)\in\mathcal{W}_{3}$, being $J$ integrable and the metric $g$ satisfies
$y=2x$. In this case, the pair $(g,J)$ is Kähler.

######  Remark 1.9.

It is worth to point out that in the case of $G_{2}/{\rm U}(2)$, where ${\rm
U}(2)$ is represented by the short root of $G_{2}$, if $J$ is non-integrable
and the metric $g$ is parametrized by $(x,x)$ then $(g,J)\in\mathcal{W}_{1}$
(nearly-Kähler) and there is no solution to the equation $2s_{1}-s=0$.

Let $G$ be a compact simple Lie group and $T$ be a maximal torus in $G$. In
the next two results we will consider the full flag manifolds $G/T$ as a total
space of a homogeneous fibration over a symmetric space. We will restrict
ourselves to a family of invariant metrics on $G/T$ such that this fibration
becomes a Riemannian submersion with totally geodesic fibers. These metrics
appear naturally in the study of bifurcation theory for the Yamabe problem in
homogeneous space, see for instance [15]. We will work out the Hermitian
geometry and Hermitian scalar curvatures of the following homogeneous space:
the (real) $12$-dimensional flag manifold ${\rm SU}(4)/T^{3}$ as total space
of

$\displaystyle{\rm SU}(3)/T^{2}\,\cdots{\rm
SU}(4)/T^{3}\,\mapsto\mathbb{CP}^{3},$

and the (real) $12$-dimensional flag manifold $G_{2}/T^{2}$ as total space

$\displaystyle S^{2}\times S^{2}\,\cdots G_{2}/T^{2}\,\mapsto G_{2}/{\rm
SO}(4).$

In both homogeneous spaces we will consider a 1-parameter family of invariant
metrics $g_{x}$ obtained by re-scaling the normal metric in the direction of
the fibers of the homogeneous fibration by the factor $x^{2}$.

######  Theorem 1.10.

Let us consider the flag manifold ${\rm SU}(4)/T^{3}$, equipped with an
invariant almost Hermitian structure $(g_{x},J_{i})$, $i=1,\ldots,4$, where
$g_{x}$ is 1-parameter family of invariant metric parametrized by the
$6$-tuple $\big{(}x^{2},x^{2},1,x^{2},1,1\big{)}$ , $x>0$ and $J_{i}$,
$i=1,\ldots,4$ is an invariant almost complex structure. Denote by $s_{1}$ the
first Hermitian scalar curvature and by $s$ the Riemannian scalar curvature.
Then solution of the equation $2s_{1}-s=0$ is summarized in Table 1.

Table 1: ${\rm SU}(4)/T^{3}$. | GH Class | Condition | $2s_{1}-s=0$
---|---|---|---
$(J_{1},g_{x})$ | $\mathcal{W}_{3}$ | $\forall x$ | $\nexists x$
$(J_{2},g_{x})$ | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $\forall x$ | $x=\sqrt[4]{5}$
$(J_{3},g_{x})$ | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ |  $\vphantom{\dfrac{1}{3}^{2}}x=\sqrt{\frac{1}{3}\big{(}8-\sqrt{61}\big{)}}$ or $x=\sqrt{\frac{1}{3}\big{(}\sqrt{61}+8\big{)}}$
| $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(J_{4},g_{x})$ | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $\vphantom{\dfrac{1}{3}^{2}}x=\sqrt{\frac{1}{3}\big{(}\sqrt{165}+12\big{)}}$
| $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$

######  Theorem 1.11.

Let us consider the flag manifold $G_{2}/T^{2}$, equipped with an invariant
almost Hermitian structure $(g_{x},J_{i})$, $i=1,\ldots,32$, where $g_{x}$ is
the $1$-parameter family of invariant metric parametrized by the $6$-tuple
$\big{(}1,1,x^{2},1,x^{2},1\big{)}$ and $J_{i}$, $i=1,\ldots,32$ is an
invariant almost complex structure. Denote by $s_{1}$ the first Hermitian
scalar curvature and by $s$ the Riemannian scalar curvature. Then solution of
the equation $2s_{1}-s=0$ is summarized in Table 2.

Table 2: $G_{2}/T^{2}$. | $J_{i}$ | GH Class | Condition | $2s_{1}-s=0$
---|---|---|---|---
$(J_{i},g_{x})$ | $i=1,2,3,10,21,30$ | $\mathcal{W}_{3}$ | $\forall x$ | $\nexists x$
$(J_{i},g_{x})$ | $i=4,\ldots,9,11,\ldots,20,$ | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | at least one solution
| $22,\ldots,29,31,32.$ | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$

######  Remark 1.12.

The explicit solutions for the equation $2s_{1}-s=0$ for the invariant
Hermitian structures $(J_{i},g_{x})$, where
$i=4,\ldots,9,11,\ldots,20,22,\ldots,29,31,32$ and $x\neq 1$, are described in
Section 3.6.2.

##  2 Preliminaries: generalized flag manifolds
and invariant almost Hermitian structures

In this section we recall some well know results about the geometry of
generalized flag manifolds, from a Lie theoretical point of view. References
about Riemannian and Hermitian geometry of flag manifolds are [1, 3, 6, 20,
21].

###  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
$\big{(}\mathfrak{g}^{\mathbb{C}},\mathfrak{h}\big{)}$. We have the following
decomposition

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

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

The Cartan–Killing form is defined by

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

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
$\mathfrak{h}_{\mathbb{R}}=\operatorname{span}_{\mathbb{R}}\\{H_{\alpha}\colon\alpha\in\Pi\\}$
and $\mathfrak{h}_{\mathbb{R}}^{*}$ being the real subspace of
$\mathfrak{h}^{*}$ spanned by the roots.

Let us fix a Weyl basis of $\mathfrak{g}^{\mathbb{C}}$ given by

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

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 $\Sigma$. Let
us fix 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

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

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

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

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

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

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

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

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

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

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

When $\Theta=\varnothing$ we have the following decomposition

$\displaystyle\mathfrak{g}^{\mathbb{C}}=\mathfrak{h}\oplus\sum_{\beta\in\Pi^{+}}\mathfrak{g}_{\beta}\oplus\sum_{\beta\in\Pi^{+}}\mathfrak{g}_{-\beta},$

and therefore

$\displaystyle\mathfrak{p}_{\Theta}=\mathfrak{p}=\mathfrak{h}\oplus\sum_{\beta\in\Pi^{+}}\mathfrak{g}_{\beta}$

is a minimal parabolic sub-algebra (Borel sub-algebra) of
$\mathfrak{g}^{\mathbb{C}}$ and

$\displaystyle\mathbb{F}=G^{\mathbb{C}}/P=G/T$

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

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

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:

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

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

$\displaystyle\mathfrak{m}=\sum_{\beta\in\Pi_{M}}\mathfrak{u}_{\beta}.$

An important ingredient 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

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

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

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

where each component $\mathfrak{m}_{i}$ satisfies
$\operatorname{Ad}(K_{\Theta})(\mathfrak{m}_{i})\subset\mathfrak{m}_{i}$. We
have also that 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.

In the sequel 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
$\Lambda\colon\mathfrak{m}\longrightarrow\mathfrak{m}$ commuting with
$\operatorname{Ad}(k)\left|{}_{\mathfrak{m}}\right.$ for all $k\in K$ such
that

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

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 bilinear 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 the 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 number $(\lambda_{1},\ldots,\lambda_{n})$ representing
the eigenvalues of the operator $\Lambda$ and parametrized 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 $J\colon
T_{x}\mathbb{F}\longrightarrow T_{x}\mathbb{F}$ 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

$\displaystyle J_{ux}={\rm d}E_{u}J_{x}{\rm d}E_{u^{-1}},\qquad\text{for
all}\quad u\in G,$

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

$\displaystyle{\rm d}E_{u}J_{x}X=J_{ux}{\rm d}E_{u}X.$

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 $J_{x_{0}}\colon
T_{x_{0}}\mathbb{F}\to T_{x_{0}}\mathbb{F}$ satisfying
$J_{x_{0}}^{2}=-\operatorname{id}$ and commute with the isotropy
representation, that is,

$\displaystyle\operatorname{Ad}^{G/K}(k)J_{x_{0}}=J_{x_{0}}\operatorname{Ad}^{G/K}(k)\qquad\text{for
all}\quad k\in K.$

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

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

######  Proposition 2.5 ([7, Proposition 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 [14] as follow:

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

where $D$ is the Levi-Civita connection.

There are three special cases:

  * $(i)$

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

  * $(ii)$

$t=1$, $\nabla^{1}$ is the second canonical Hermitian connection, also know as
Chern connection (this connection was used by Chern in the integrable case,
see [8]).

  * $(iii)$

$t=-1$, $\nabla^{-1}$ is the Bismut connection. In the integrable case,
$\nabla^{-1}$ is characterized by its anti-symmetric torsion, see for instance
[10, 17].

According to [12], let us define the Hermitian scalar curvatures $s_{1}(t)$
and $s_{2}(t)$ by

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

where $R^{t}$ is the curvature tensor associated to $\nabla^{t}$ and
$\\{u_{i}\\}_{i=1,2,\dots,n}$ is a unitary frame. We call $s_{1}(t)$ and
$s_{2}(t)$ the first and second Hermitian scalar curvature, 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 study, see for
instance [13, 19].

Let $\\{e_{1},e_{2},\dots,e_{2n}\\}$ be a local orthonormal frame of
$(M,g,J)$. Recall the $J$-twisted version of the Ricci tensor, called
$J$-Ricci tensor and denoted by $\operatorname{Ric}_{J}$ (also called the
$*$-Ricci tensor [12, 16, 22]) defined by

$\displaystyle\operatorname{Ric}_{J}(X,Y)=R(e_{A},X,Je_{A},JY).$

The corresponding $J$-scalar curvature, denoted by $s_{J}$ is given by
$s_{J}=\operatorname{Ric}_{J}(e_{A},e_{A})$.

The Nijenhuis tensor $N$ is given by

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

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

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

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

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

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

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

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

where $({\rm d}F)_{0}$ is the primitive part of ${\rm d}F$.

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

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

Moreover,

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

The following expression of $DF$ will be very useful for our purposes (see
[12, 14])

$\displaystyle(DF)(X,Y,Z)={}$ $\displaystyle({\rm
d}F)^{-}(X,Y,Z)-\dfrac{1}{2}N(JX,Y,Z)$ $\displaystyle+\dfrac{1}{2}[({\rm
d}F)^{+}(X,Y,Z)-({\rm d}F)^{+}(X,JY,JZ)],$

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

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

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

According to [14], we have

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

The four components described above $({\rm d}F)^{-}$, $N^{0}$, $({\rm
d}F)^{+}_{0}$ 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 [16] correspond to the vanishing of the some
subset of

$\displaystyle\big{\\{}({\rm d}F)^{-},N^{0},({\rm
d}F)^{+}_{0},\alpha_{F}\big{\\}}.$

Let us describe some remarkable classes of almost Hermitian structures:

  * •

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

  * •

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

  * •

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

  * •

$\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}=$ 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

$\displaystyle\|DF\|^{2}=\|{\rm
d}F\|^{2}+\dfrac{1}{4}\big{\|}N^{0}\big{\|}^{2}-\dfrac{2}{3}\|({\rm
d}F)^{-}\|^{2}=\|({\rm
d}F)^{+}\|^{2}+\dfrac{1}{4}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{3}\|({\rm
d}F)^{-}\|^{2}.$

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

######  Theorem 3.1 ([12, Theorem 4.3]).

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

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

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

Note that

$\displaystyle 2s_{1}(t)-s={}$ $\displaystyle-\dfrac{5}{6}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{8}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{2}\|({\rm
d}F)_{0}^{+}\|^{2}+\bigg{[}\dfrac{1}{2(n-1)}+(t-1)\bigg{]}\|\alpha_{F}\|^{2}$
$\displaystyle+(t-2)\delta\alpha_{F}.$

######  Proposition 3.2 ([12]).

The $J$-scalar curvature of $(M,g,J)$ is given by

$\displaystyle s_{J}=s-\dfrac{2}{3}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{4}\big{\|}N^{0}\big{\|}^{2}-\|\alpha_{F}\|^{2}-2\delta\alpha_{F},$

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

###  3.2 Computations on generalized 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 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,
$\mathfrak{m}=\mathfrak{m}_{1}\oplus\dots\oplus\mathfrak{m}_{n}$. 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 parametrized 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 [21]. We have $N=0$,
except in the following situation:

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

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

We also have

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

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

The next result was initially proved by San Martin-Negreiros [21] in the case
of full flag manifolds and by R. de Jesus in her Ph.D. Thesis for the case of
generalized flag manifold. We include the proof here for the convenience of
the reader.

######  Lemma 3.3.

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

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

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
$\\{X_{i}\\}=\big{\\{}A_{\alpha},\sqrt{-1}S_{\alpha};\alpha\in\Pi^{+}_{M}\big{\\}}$
and
$\\{Y_{i}\\}=\big{\\{}\sqrt{-1}S_{\alpha},A_{\alpha};\allowbreak\alpha\in\Pi^{+}_{M}\big{\\}}$.
So,

$\displaystyle{\rm d}F(X,X_{\alpha}-X_{-\alpha},{\rm i}X_{\alpha}+{\rm
i}X_{-\alpha})$ $\displaystyle\qquad{}={\rm i}\,{\rm
d}F(X,X_{\alpha},X_{\alpha})+{\rm i}\,{\rm d}F(X,X_{\alpha},X_{-\alpha})-{\rm
i}\,{\rm d}F(X,X_{-\alpha},X_{\alpha})-{\rm i}\,{\rm
d}F(X,X_{-\alpha},X_{-\alpha})$ $\displaystyle\qquad{}=2{\rm i}({\rm
d}F(X,X_{\alpha},X_{\alpha})+{\rm d}F(X,X_{\alpha},X_{-\alpha})).$

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

$\displaystyle\begin{cases}{\rm
d}F(X_{\beta},X_{\alpha},X_{\alpha})=0,&\text{because
if}\quad\alpha+\beta+\alpha=0\quad\text{then}\quad\beta=-2\alpha,\\\ {\rm
d}F(X_{\beta},X_{\alpha},X_{-\alpha})=0,&\text{because
if}\quad\alpha+\beta-\alpha=0\quad\text{then}\quad\beta=0,\end{cases}$

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

$\displaystyle\alpha_{F}(X)=\dfrac{1}{2n-1}\sum_{\alpha>0}{\rm
d}F(X_{\gamma},X_{\alpha}-X_{-\alpha},{\rm i}X_{\alpha}+{\rm
i}X_{-\alpha})=0.$ ∎

######  Remark 3.4.

  1. 1.

According to Lemma 3.3, the Lee form vanishes identically for every
generalized flag manifold. Therefore we have ${\rm d}F=({\rm d}F)_{0}$ and
hence $({\rm d}F)^{+}=({\rm d}F)_{0}^{+}$. From now on, we will use $({\rm
d}F)^{+}$ instead of $({\rm d}F)_{0}^{+}$.

  2. 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 $s_{\rm C}$.

  3. 3.

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

The next definition was introduced in [21] in order to obtain the
classification of invariant almost Hermitian structures on full flag
manifolds, and we will use this concept widely.

######  Definition 3.5.

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.

Let us consider an orthonormal basis on the tangent space at the origin with
respect to the invariant metric, induced by the Weyl basis of the Lie algebra
of $G$:

$\displaystyle\bigg{\\{}E_{\alpha}=\dfrac{X_{\alpha}}{\sqrt{\lambda_{\alpha}}}\colon\alpha\in\Pi_{M}^{+}\bigg{\\}}.$

In the next lemmas we will compute the exterior derivatives and covariant
derivatives of the Kähler form $F$ (with respect to the Levi-Civita
connection) as well its $(p,q)$-parts, by using the Lie theoretical elements
of the description of flag manifolds. Since the proofs are straightforward
computation we will omit them.

######  Lemma 3.6.

The expression
$(\epsilon_{\alpha}\epsilon_{\beta}+\epsilon_{\alpha}\epsilon_{\gamma}+\epsilon_{\beta}\epsilon_{\gamma}+1)$
is not zero if, and only if, $\\{\alpha,\beta,\gamma\\}$ is a $(0,3)$-triple,
that is, $\epsilon_{\alpha}=\epsilon_{\beta}=\epsilon_{\gamma}$.

###### Proof.

  1. $(i)$

$\epsilon_{\alpha}=\epsilon_{\beta}\neq\epsilon_{\gamma}$:
$\epsilon_{\alpha}\epsilon_{\beta}+\epsilon_{\alpha}\epsilon_{\gamma}+\epsilon_{\beta}\epsilon_{\gamma}+1=+1-1-1+1=0$,

  2. $(ii)$

$\epsilon_{\alpha}=\epsilon_{\gamma}\neq\epsilon_{\beta}$:
$\epsilon_{\alpha}\epsilon_{\beta}+\epsilon_{\alpha}\epsilon_{\gamma}+\epsilon_{\beta}\epsilon_{\gamma}+1=-1+1-1+1=0$,

  3. $(iii)$

$\epsilon_{\beta}=\epsilon_{\gamma}\neq\epsilon_{\alpha}$:
$\epsilon_{\alpha}\epsilon_{\beta}+\epsilon_{\alpha}\epsilon_{\gamma}+\epsilon_{\beta}\epsilon_{\gamma}+1=-1-1+1+1=0$,

  4. $(iv)$

$\epsilon_{\alpha}=\epsilon_{\beta}=\epsilon_{\gamma}$:
$\epsilon_{\alpha}\epsilon_{\beta}+\epsilon_{\alpha}\epsilon_{\gamma}+\epsilon_{\beta}\epsilon_{\gamma}+1=+1+1+1+1=4$.
∎

######  Lemma 3.7.

The covariant derivative of the Kähler form $F$ with respect to the Levi-
Civita connection $D$ on a flag manifold $G/K$ is given by

$\displaystyle
DF(E_{\alpha},E_{\beta},E_{\gamma})=\begin{cases}-\dfrac{\sqrt{-1}m_{\alpha,\beta}(\varepsilon_{\beta}+\varepsilon_{\gamma})(-\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})}{2\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}&\text{if}\quad\alpha+\beta+\gamma=0,\\\
0&\text{otherwise}.\end{cases}$

###### Proof.

$\displaystyle DF(E_{\alpha},E_{\beta},E_{\gamma})$
$\displaystyle=\dfrac{g((D_{X_{\alpha}}J)X_{\beta},X_{\gamma})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$
$\displaystyle=\dfrac{g(D_{X_{\alpha}}JX_{\beta},X_{\gamma})+g(D_{X_{\alpha}}X_{\beta},JX_{\gamma})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$
$\displaystyle=\dfrac{{\rm
i}m_{\alpha,\gamma}(\lambda_{\gamma}-\lambda_{\alpha}+\lambda_{\beta})(\varepsilon_{\beta}+\varepsilon_{\gamma})}{2\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$
$\displaystyle=-\dfrac{\sqrt{-1}m_{\alpha,\beta}(\varepsilon_{\beta}+\varepsilon_{\gamma})(-\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})}{2\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}.$
∎

######  Lemma 3.8.

Let ${\rm d}F$ be the exterior derivative of the Kähler form $F$ on a flag
manifold $G/K$. We have ${\rm d}F(E_{\alpha},E_{\beta},E_{\gamma})=0$, unless
$\alpha+\beta+\gamma=0$. In this case we have

$\displaystyle{\rm
d}F(E_{\alpha},E_{\beta},E_{\gamma})=-\dfrac{\sqrt{-1}m_{\alpha,\beta}(\varepsilon_{\alpha}\lambda_{\alpha}+\varepsilon_{\beta}\lambda_{\beta}+\varepsilon_{\gamma}\lambda_{\gamma})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}.$
(3.1)

###### Proof.

The differential is given by

$\displaystyle{\rm
d}F(E_{\alpha},E_{\beta},E_{\gamma})=\dfrac{g((D_{X_{\alpha}}J)X_{\beta},X_{\gamma})-g((D_{X_{\beta}}J)X_{\alpha},X_{\gamma})+g((D_{X_{\gamma}}J)X_{\alpha},X_{\beta})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}.$

Moreover

$\displaystyle g((D_{X_{\alpha}}J)X_{\beta},X_{\gamma})$
$\displaystyle=g(D_{X_{\alpha}}JX_{\beta},X_{\gamma})+g(D_{X_{\alpha}}X_{\beta},JX_{\gamma})$
$\displaystyle=\dfrac{\sqrt{-1}}{2}m_{\alpha,\gamma}(\lambda_{\gamma}-\lambda_{\alpha}+\lambda_{\beta})(\varepsilon_{\beta}+\varepsilon_{\gamma}).$

The other pieces of ${\rm d}F$ are obtained in a similar way and result
follows. ∎

######  Lemma 3.9.

The $(0,2)$-component of the exterior derivative $DF$ is given by

$\displaystyle(DF)^{0,2}(E_{\alpha},E_{\beta},E_{\gamma})=\begin{cases}\dfrac{\sqrt{-1}m_{\alpha,\beta}(\lambda_{\alpha}\\!-\\!\lambda_{\beta}\\!-\\!\lambda_{\gamma})(\varepsilon_{\alpha}\\!+\\!\varepsilon_{\beta}\\!+\\!\varepsilon_{\gamma}\\!+\\!\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\gamma})}{4\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}&\text{if}\
\ \alpha\\!+\\!\beta\\!+\\!\gamma=0,\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

By a direct computation we have

$\displaystyle(DF)^{0,2}(E_{\alpha},E_{\beta},E_{\gamma})={}$
$\displaystyle\dfrac{g((D_{X_{\alpha}}J)X_{\beta},X_{\gamma})+g(J(D_{JX_{\alpha}}J)X_{\beta},X_{\gamma})}{2\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$
$\displaystyle={}$
$\displaystyle\dfrac{1}{2\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}[g(D_{X_{\alpha}}JX_{\beta},X_{\gamma})+g(D_{X_{\alpha}}X_{\beta},JX_{\gamma})$
$\displaystyle\hphantom{\dfrac{1}{2\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}[}+g(D_{JX_{\alpha}}X_{\beta},X_{\gamma})-g(D_{JX_{\alpha}}J{X_{\beta}},JX_{\gamma})]$
$\displaystyle={}$
$\displaystyle\dfrac{\sqrt{-1}m_{\alpha,\beta}(\lambda_{\alpha}-\lambda_{\beta}-\lambda_{\gamma})(\varepsilon_{\alpha}+\varepsilon_{\beta}+\varepsilon_{\gamma}+\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\gamma})}{4\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}},$

and the result follows. ∎

Since the $(2,0)$-component of $DF$ is defined by

$\displaystyle(DF)^{2,0}(E_{\alpha},E_{\beta},E_{\gamma})=\dfrac{g((D_{X_{\alpha}}J)X_{\beta},X_{\gamma})-g(J(D_{JX_{\alpha}}J)X_{\beta},X_{\gamma})}{2\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$

we obtain the next result in similar way.

######  Lemma 3.10.

The $(2,0)$-component of the exterior derivative $DF$ is given by

$\displaystyle(DF)^{2,0}(E_{\alpha},E_{\beta},E_{\gamma})$
$\displaystyle\qquad{}=\begin{cases}-\dfrac{\sqrt{-1}m_{\alpha,\beta}(-\varepsilon_{\alpha}+\varepsilon_{\beta}+\varepsilon_{\gamma}-\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\gamma})(-\lambda_{\alpha}+\lambda_{\gamma}+\lambda_{\beta})}{4\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}&\text{if}\quad\alpha+\beta+\gamma=0,\\\
0&\text{otherwise.}\end{cases}$

######  Lemma 3.11.

The $(0,3)+(3,0)$-component of ${\rm d}F$, denoted by $({\rm d}F)^{-}$, is
given by

$\displaystyle({\rm d}F)^{-}(E_{\alpha},E_{\beta},E_{\gamma})$
$\displaystyle\qquad{}=\begin{cases}-\dfrac{\sqrt{-1}m_{\alpha,\beta}(\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})(\varepsilon_{\alpha}+\varepsilon_{\beta}+\varepsilon_{\gamma}+\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\gamma})}{4\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}&\text{if}\quad\alpha+\beta+\gamma=0,\\\
0&{otherwise.}\end{cases}$

###### Proof.

According to [14], the $(0,3)+(3,0)$-component of ${\rm d}F$ is defined by

$\displaystyle({\rm d}F)^{-}(E_{\alpha},E_{\beta},E_{\gamma})$
$\displaystyle=\dfrac{\mathfrak{b}(DF)^{0,2}(X_{\alpha},X_{\beta},X_{\gamma})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$
$\displaystyle=\dfrac{(DF)^{0,2}(X_{\alpha},X_{\beta},X_{\gamma})\\!+\\!(DF)^{0,2}(X_{\beta},X_{\gamma},X_{\alpha})\\!+\\!(DF)^{0,2}(X_{\gamma},X_{\alpha},X_{\beta})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$
$\displaystyle=-\dfrac{\sqrt{-1}m_{\alpha,\beta}(\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})(\varepsilon_{\alpha}+\varepsilon_{\beta}+\varepsilon_{\gamma}+\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\gamma})}{4\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}},$

and the result follows. ∎

######  Lemma 3.12.

The $(1,2)+(2,1)$-component of ${\rm d}F$, denoted by $({\rm d}F)^{+}$, is
given by

$\displaystyle({\rm d}F)^{+}(E_{\alpha},E_{\beta},E_{\gamma})$
$\displaystyle\qquad{}=\begin{cases}-\dfrac{\sqrt{-1}m_{\alpha,\beta}\\{4(\varepsilon_{\alpha}\lambda_{\alpha}\\!+\varepsilon_{\beta}\lambda_{\beta}\\!+\varepsilon_{\gamma}\lambda_{\gamma})-(\varepsilon_{\alpha}\\!+\varepsilon_{\beta}\\!+\varepsilon_{\gamma}\\!+\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\gamma})(\lambda_{\alpha}\\!+\lambda_{\beta}\\!+\lambda_{\gamma})\\}}{4\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}\\\\[-4.30554pt]
\hskip 278.837pt\text{if}\quad\alpha+\beta+\gamma=0,\\\
0\qquad\text{otherwise.}\end{cases}$

###### Proof.

The $(1,2)+(2,1)$-component of ${\rm d}F$ is defined by

$\displaystyle({\rm d}F)^{+}(E_{\alpha},E_{\beta},E_{\gamma})$
$\displaystyle\qquad{}=\dfrac{3\mathfrak{b}(DF)^{2,0}(X_{\alpha},X_{\beta},X_{\gamma})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$
$\displaystyle\qquad{}=\dfrac{(DF)^{2,0}(X_{\alpha},X_{\beta},X_{\gamma})+(DF)^{2,0}(X_{\beta},X_{\gamma},X_{\alpha})+(DF)^{2,0}(X_{\gamma},X_{\alpha},X_{\beta})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$
$\displaystyle\qquad{}=-\dfrac{\sqrt{-1}m_{\alpha,\beta}\\{4(\varepsilon_{\alpha}\lambda_{\alpha}+\varepsilon_{\beta}\lambda_{\beta}+\varepsilon_{\gamma}\lambda_{\gamma})-(\varepsilon_{\alpha}+\varepsilon_{\beta}+\varepsilon_{\gamma}+\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\gamma})(\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})\\}}{4\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}},$

and the result follows. ∎

######  Remark 3.13.

It is worth to point out an alternative description of $DF^{0,2}$ and
$DF^{2,0}$ that will be very useful in our work:

$\displaystyle DF^{0,2}(E_{\alpha},E_{\beta},E_{\gamma})=\dfrac{2({\rm
d}F)^{-}(X_{\alpha},X_{\beta},X_{\gamma})-N(JX_{\alpha},X_{\beta},X_{\gamma})}{2\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}},$

and

$\displaystyle DF^{2,0}(E_{\alpha},E_{\beta},E_{\gamma})=\dfrac{({\rm
d}F)^{+}(X_{\alpha},X_{\beta},X_{\gamma})-({\rm
d}F)^{+}(X_{\alpha},JX_{\beta},JX_{\gamma})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}.$

We also point out a description of $({\rm d}F)^{-}$ in terms of the Nijenhius
tensor $N$ given by (see [14]):

$\displaystyle({\rm d}F)^{-}(X,Y,Z)=-N(JX,Y,Z)+N(JY,X,Z)-N(JZ,X,Y).$

The description of $N^{0}$ in terms of the Nijenhius tensor $N$ is given in
[14]):

$\displaystyle
N^{0}(X,Y,Z)=\frac{2}{3}N(X,Y,Z)-\frac{1}{3}N(Y,Z,X)-\frac{1}{3}N(Z,X,Y).$

######  Lemma 3.14.

We have

$\displaystyle
N^{0}(E_{\alpha},E_{\beta},E_{\gamma})=\begin{cases}-\dfrac{\sqrt{-1}\,m_{\alpha,\beta}(\epsilon_{\alpha}\epsilon_{\beta}\epsilon_{\gamma}\\!+\epsilon_{\alpha}\\!+\epsilon_{\beta}\\!+\epsilon_{\gamma})(\lambda_{\alpha}\\!+\lambda_{\beta}\\!+\lambda_{\gamma})}{4\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}&\text{if}\quad\alpha\\!+\beta\\!+\gamma=0,\\\
0&\text{otherwise}.\end{cases}$

######  Remark 3.15.

When $\alpha+\beta+\gamma=0$ is a $(1,2)$-triple we have

$\displaystyle({\rm
d}F)^{-}(E_{\alpha},E_{\beta},E_{\gamma})=N^{0}(E_{\alpha},E_{\beta},E_{\gamma})=0$

and

$\displaystyle({\rm d}F)^{+}(E_{\alpha},E_{\beta},E_{\gamma})=\dfrac{-{\rm
i}m_{\alpha,\beta}(\varepsilon_{\alpha}\lambda_{\alpha}+\varepsilon_{\beta}\lambda_{\beta}+\varepsilon_{\gamma}\lambda_{\gamma})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}.$

Otherwise when $\alpha+\beta+\gamma=0$ is a $(0,3)$-triple we have $({\rm
d}F)^{+}(E_{\alpha},E_{\beta},E_{\gamma})=0$,

$\displaystyle({\rm d}F)^{-}(E_{\alpha},E_{\beta},E_{\gamma})=-\dfrac{{\rm
i}m_{\alpha,\beta}(\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})}{\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}$

and

$\displaystyle
N^{0}(E_{\alpha},E_{\beta},E_{\gamma})=\dfrac{4m_{\alpha,\beta}(-2\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})}{3\sqrt{\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}}.$

######  Remark 3.16.

Let us recall some useful relations using the derivatives of Kähler form. See
[14] for further details.

  * •

The pair $(g,J)$ is integrable if, and only if, $D_{JX}J=JD_{X}J$. Note that

$\displaystyle(g,J)\ \text{is integrable}\Leftrightarrow N\equiv
0\Leftrightarrow DF^{0,2}=0\Leftrightarrow D_{JX}J=JD_{X}J.$

  * •

The pair $(g,J)$ is $(1,2)$-symplectic if, and only if, $D_{JX}J=-JD_{X}J$. We
have

$\displaystyle(g,J)\ \text{is}\ \text{(1,2)-symplectic}\Leftrightarrow({\rm
d}F)^{+}\equiv 0\Leftrightarrow DF^{2,0}=0\Leftrightarrow D_{JX}J=-JD_{X}J.$

  * •

Note that

$\displaystyle\begin{cases}DF=DF^{0,2}+DF^{2,0},\\\ {\rm d}F=({\rm
d}F)^{-}+({\rm d}F)^{+}.\end{cases}$

In order to calculate the Hermitian scalar curvatures, the first step is
computing the norms of $N^{0}$, $({\rm d}F)^{-}$, $({\rm d}F)^{+}$ and
$\alpha_{F}$, as described in [12]. Since these computations are standard we
omit the details.

######  Proposition 3.17.

We have

$\displaystyle\big{\|}N^{0}\big{\|}^{2}=\sum_{\alpha+\beta+\gamma=0}(m_{\alpha,\beta})^{2}\dfrac{(\epsilon_{\alpha}\epsilon_{\beta}\epsilon_{\gamma}+\epsilon_{\alpha}+\epsilon_{\beta}+\epsilon_{\gamma})^{2}}{54\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}$
$\displaystyle\hphantom{\big{\|}N^{0}\big{\|}^{2}=}{}\times\big{[}(-2\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})^{2}+(-2\lambda_{\gamma}+\lambda_{\alpha}+\lambda_{\beta})^{2}+(-2\lambda_{\beta}+\lambda_{\alpha}+\lambda_{\gamma})^{2}\big{]},$
$\displaystyle\|({\rm
d}F)^{-}\|^{2}=\sum_{\alpha+\beta+\gamma=0}\dfrac{(m_{\alpha,\beta})^{2}(\epsilon_{\alpha}\epsilon_{\beta}\epsilon_{\gamma}+\epsilon_{\alpha}+\epsilon_{\beta}+\epsilon_{\gamma})^{2}(\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})^{2}}{96\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}},$
$\displaystyle\|DF\|^{2}=\sum_{\alpha+\beta+\gamma=0}\dfrac{m_{\alpha,\beta}^{2}}{3\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}\bigg{[}\dfrac{1}{4}(\varepsilon_{\beta}+\varepsilon_{\gamma})^{2}(-\lambda_{\alpha}+\lambda_{\beta}+\lambda_{\gamma})^{2}+\dfrac{1}{4}(\varepsilon_{\alpha}+\varepsilon_{\gamma})^{2}(\lambda_{\alpha}-\lambda_{\beta}+\lambda_{\gamma})^{2}$
$\displaystyle\hphantom{\|DF\|^{2}=}{}+\dfrac{1}{4}(\varepsilon_{\alpha}+\varepsilon_{\beta})^{2}(\lambda_{\alpha}+\lambda_{\beta}-\lambda_{\gamma})^{2}\bigg{]},$
$\displaystyle\|({\rm
d}F)^{+}\|^{2}=\\!\sum_{\alpha\\!+\beta\\!+\gamma=0}\\!\\!\\!\\!\dfrac{m_{\alpha,\beta}^{2}\\{4(\varepsilon_{\alpha}\lambda_{\alpha}\\!+\varepsilon_{\beta}\lambda_{\beta}\\!+\varepsilon_{\gamma}\lambda_{\gamma})\\!-(\varepsilon_{\alpha}\\!+\varepsilon_{\beta}\\!+\varepsilon_{\gamma}\\!+\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\gamma})(\lambda_{\alpha}\\!+\lambda_{\beta}\\!+\lambda_{\gamma})\\}}{96\lambda_{\alpha}\lambda_{\beta}\lambda_{\gamma}}^{2}\\!.$

###  3.3 The flag manifold $\boldsymbol{{\rm SU}(3)/T^{2}}$

Let us consider the 6-dimensional full flag manifold $\mathbb{F}(3)={\rm
SU}(3)/T^{2}$. Recall the Cartan subalgebra $\mathfrak{h}$ of
$\mathfrak{su}(3)$ is given by

$\displaystyle\mathfrak{h}=\\{\operatorname{diag}(x_{1},x_{2},x_{3})\colon
x_{1}+x_{2}+x_{3}=0,\ x_{1},x_{2},x_{3}\in\mathbb{C}\\}.$

The set of positive roots is given by $\alpha_{12}=x_{1}-x_{2}$,
$\alpha_{23}=x_{2}-x_{3}$ and $\alpha_{13}=x_{1}-x_{3}$. Recall that on full
flag manifolds, the set of complementary roots $\Pi_{M}$ coincides with set of
roots $\Pi$. The isotropy representation of $\mathbb{F}(3)$ admits three
irreducible components, and each component corresponds to a unique positive
root $\Pi^{+}$, that is,

$\displaystyle\mathfrak{m}$
$\displaystyle=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\mathfrak{m}_{3}$
$\displaystyle=\mathfrak{u}_{12}\oplus\mathfrak{u}_{23}\oplus\mathfrak{u}_{13},$

where
$\mathfrak{u}_{ij}=\mathfrak{su}(3)\cap(\mathfrak{g}_{ij}\oplus\mathfrak{g}_{ji})$,
where $\mathfrak{g}_{ji}$ is the root space associated to the root
$\alpha_{ij}$. Therefore there exist two invariant almost complex structures,
up to conjugation and equivalence:

$\displaystyle J_{1}=(+,+,+)\qquad\text{and}\qquad J_{2}=(+,+,-),$

where the almost complex structure $J_{1}$ is given by
$\varepsilon_{\alpha_{12}}=+1$, $\varepsilon_{\alpha_{21}}=-1$,
$\varepsilon_{\alpha_{23}}=+1$, $\varepsilon_{\alpha_{32}}=-1$,
$\varepsilon_{\alpha_{13}}=+1$, $\varepsilon_{\alpha_{31}}=-1$, and so on. We
remark that $J_{1}$ is integrable and $J_{2}$ is non-integrable, see for
instance [21].

We will denote an invariant metric $g$ by the triple of positive numbers
$(x,y,z)$. To be consistent with Section 3.2, we set $\lambda_{12}=x$,
$\lambda_{23}=y$ and $\lambda_{13}=z$. We will consider the orthonormal basis
of $TM_{\mathbb{C}}$ given by

$\displaystyle\bigg{\\{}\dfrac{X_{12}}{\sqrt{x}},\dfrac{X_{23}}{\sqrt{y}},\dfrac{X_{13}}{\sqrt{z}},\dfrac{X_{21}}{\sqrt{x}},\dfrac{X_{32}}{\sqrt{y}},\dfrac{X_{31}}{\sqrt{z}}\bigg{\\}},$

where $X_{ij}\in\mathfrak{g}_{ij}$ represents the elements of Weyl basis of
$\mathfrak{sl}(3,\mathbb{C})$.

######  Proposition 3.18 ([3]).

Let us consider the full flag manifold ${\rm SU}(3)/T$, with invariant metric
$g$ parametrized by $(x,y,z)$. Then the Riemannian scalar curvature of $({\rm
SU}(3)/T,g)$ is given by

$\displaystyle s=-\dfrac{x^{2}+y^{2}-6yz+z^{2}-6x(y+z)}{6xyz}.$

######  Lemma 3.19.

The invariant Hermitian structure $(g,J_{2})$ does not admit $(1,2)$-triple.
The $(0,3)$-triples are given by $\alpha_{12}+\alpha_{23}+\alpha_{31}=0$ and
$\alpha_{21}+\alpha_{32}+\alpha_{13}=0$.

######  Proposition 3.20.

Let us consider the flag manifold ${\rm SU}(3)/T^{2}$, equipped with invariant
Hermitian structure $(g,J_{2})$. Denote by $s$ the Riemannian scalar curvature
and $s_{1}$ the first Hermitian scalar curvature. Then the invariant metric
$g$ that are solution of the equation $2s_{1}-s=0$ are given by:

  * $(i)$

$z=3(x+y)-2\sqrt{2}\sqrt{x^{2}+3xy+y^{2}}$, for $y>2\sqrt{2}\sqrt{x^{2}}+3x$
and $x>0$; or $0<y<3x-2\sqrt{2}\sqrt{x^{2}}$ and $x>0$,

  * $(ii)$

$z=2\sqrt{2}\sqrt{x^{2}+3xy+y^{2}}+3(x+y)$, for $x>0$ and $y>0$.

###### Proof.

By Proposition 3.17 we have $\|({\rm d}F)^{+}\|^{2}=0$,

$\displaystyle\|({\rm d}F)^{-}\|^{2}=\dfrac{(x+y+z)^{2}}{3xyz},$
$\displaystyle\big{\|}N^{0}\big{\|}^{2}=\dfrac{16\big{[}(-2x\\!+\\!y\\!+\\!z)^{2}\\!+\\!(x\\!-\\!2y\\!+\\!z)^{2}\\!+\\!(x\\!+\\!y\\!-\\!2z)^{2}\big{]}}{27xyz}=\dfrac{32\big{(}x^{2}\\!+\\!y^{2}\\!-\\!yz\\!+\\!z^{2}\\!-\\!x(y\\!+\\!z)\big{)}}{9xyz},$
$\displaystyle\|DF\|^{2}=\dfrac{3x^{2}+3y^{2}-2yz+3z^{2}-2x(y+z)}{3xyz}.$

The first Hermitian scalar curvature is

$\displaystyle s_{1}=\dfrac{s}{2}-\dfrac{5}{12}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{16}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{4}\|({\rm
d}F)^{+}\|^{2}=0.$

Therefore in order to solve the equation $2s_{1}-s=0$ we need to find the
zeros of the Riemannian scalar curvature.

We have $s=0$ if, and only if,

  * $(i)$

$z=3(x+y)-2\sqrt{2}\sqrt{x^{2}+3xy+y^{2}}$, for $y>2\sqrt{2}\sqrt{x^{2}}+3x$
and $x>0$; or $0<y<3x-2\sqrt{2}\sqrt{x^{2}}$ and $x>0$,

  * $(ii)$

$z=2\sqrt{2}\sqrt{x^{2}+3xy+y^{2}}+3(x+y)$, for $x>0$ and $y>0$. ∎

An immediate consequence of the proof of Proposition 3.20, by analyzing the
vanishing of the components $({\rm d}F)^{-}$, $({\rm d}F)^{+}$, $N^{0}$ is the
following result:

######  Proposition 3.21.

With the notation above, the invariant almost Hermitian structure $(g,J_{2})$
on ${\rm SU}(3)/T^{2}$ belongs to the Gray–Hervella class
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. The pair $(g,J_{2})$ belongs to the
class $\mathcal{W}_{1}$ $($nearly Kähler$)$ if, and only if, the metric $g$ is
parametrized by $x=y=z$.

For the sake of completeness we will compute the scalar curvatures
$s_{2}=s_{2}(t)$ and $s_{J}$. We will use the ingredients computed in the
Proposition 3.20.

######  Proposition 3.22.

Let us consider the flag manifold ${\rm SU}(3)/T^{2}$, equipped with invariant
Hermitian structure $(g,J_{2})$. The second Hermitian scalar curvature $s_{2}$
and the $J$-scalar curvature are given by

$\displaystyle
s_{2}=\frac{1}{3}\bigg{(}\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\bigg{)},\qquad
s_{J}=\frac{3x^{2}-2x(y+z)+3y^{2}-2yz+3z^{2}}{6xyz}.$

Let us consider now the invariant Hermitian structure $(g,J_{1})$.

######  Lemma 3.23.

The invariant Hermitian structure $(g,J_{1})$ does not admit $(0,3)$-triple.
The $(1,2)$-triples are given by $\alpha_{12}+\alpha_{23}+\alpha_{31}=0$ and
$\alpha_{21}+\alpha_{32}+\alpha_{13}=0$.

######  Proposition 3.24.

Let us consider the flag manifold ${\rm SU}(3)/T^{2}$, equipped with invariant
Hermitian structure $(g,J_{1})$. Denote by $s$ the Riemannian scalar curvature
and $s_{1}$ the first Hermitian scalar curvature. Then the invariant metric
$g$ that is solution of the equation $2s_{1}-s=0$ is given by $z=x+y$. In this
case, the pair $(g,J_{1})$ is a Kähler structure.

###### Proof.

We have $\|({\rm d}F)^{-}\|^{2}=\|N^{0}\|^{2}=0$ since the $(1,2)$-triples are
exactly the zero-sum triples of roots.

By Proposition 3.17 we have

$\displaystyle\|({\rm
d}F)^{+}\|^{2}=\dfrac{(\varepsilon_{12}\lambda_{12}+\varepsilon_{23}\lambda_{23}+\varepsilon_{31}\lambda_{13})^{2}}{3\lambda_{12}\lambda_{23}\lambda_{13}}=\dfrac{(\lambda_{12}+\lambda_{23}-\lambda_{13})^{2}}{3xyz}=\dfrac{(x+y-z)^{2}}{3xyz},$
$\displaystyle\|DF\|^{2}=\dfrac{(-\lambda_{12}+\lambda_{23}+\lambda_{13})^{2}+(\lambda_{12}-\lambda_{23}+\lambda_{13})^{2}+(\lambda_{12}+\lambda_{23}-\lambda_{13})^{2}}{12\lambda_{12}\lambda_{23}\lambda_{13}}=\dfrac{(x+y-z)^{2}}{3xyz}.$

The first Hermitian scalar curvature is given by

$\displaystyle s_{1}=\dfrac{s}{2}-\dfrac{5}{12}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{16}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{4}\|({\rm
d}F)^{+}\|^{2}=\dfrac{1}{3}\bigg{(}\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\bigg{)}.$

In this case, we have

$\displaystyle 2s_{1}-s=\dfrac{(x+y-z)^{2}}{6xyz}.$

Therefore,

$\displaystyle 2s_{1}-s=0\Leftrightarrow z=x+y,$

that is, $2s_{1}-s=0$ if, and only if, $(M,g,J_{1})$ is Kähler. ∎

An immediate consequence of the proof of Proposition 3.24, by analyzing the
vanishing of the components $({\rm d}F)^{-}$, $({\rm d}F)^{+}$, $N^{0}$ is the
following result:

######  Proposition 3.25.

The invariant almost Hermitian structure $(g,J_{1})$ on ${\rm SU}(3)/T^{2}$
belongs to the Gray–Hervella class $\mathcal{W}_{3}$.

######  Remark 3.26.

It is well know that the invariant Hermitian structure $(g,J_{1})$ with $g$
satisfying $z=x+y$ is a Kähler structure, see [23]. It is worth to point out
that the results of Propositions 3.21, 3.24 and 3.25 are well-known, see [11].
We include the proofs of these propositions for illustrate our methods.

For the sake of completeness we will compute the scalar curvatures $s_{2}$ and
$s_{J}$ for the complex structure $J_{1}$. We will use the ingredients
computed in the Proposition 3.24.

######  Proposition 3.27.

Let us consider the flag manifold ${\rm SU}(3)/T^{2}$, equipped with invariant
Hermitian structure $(g,J_{1})$. The second Hermitian scalar curvature $s_{2}$
and the $J$-scalar curvature are given by

$\displaystyle
s_{2}(t)=\frac{(t-2)t(x+y-z)^{2}+x^{2}-6x(y+z)+y^{2}-6yz+z^{2}}{12xyz},$
$\displaystyle s_{J}=-\frac{x^{2}-6x(y+z)+y^{2}-6yz+z^{2}}{6xyz}.$

###  3.4 The complex projective space $\boldsymbol{\mathbb{CP}^{3}={\rm
Sp}(2)/{\rm Sp}(1)\times{\rm U}(1)}$

Let us consider another 6-dimensional homogeneous space, the partial flag
manifold $\mathbb{CP}^{3}={\rm Sp}(2)/{\rm Sp}(1)\times{\rm U}(1)$. Let
$\mathfrak{h}=\left(\begin{smallmatrix}\Lambda&\\\
&-\Lambda\end{smallmatrix}\right)$, with
$\Lambda=\left(\begin{smallmatrix}x&\\\ &y\end{smallmatrix}\right)$, be the
Cartan subalgebra of $\mathfrak{sp}(2)$.

One can describe precisely each one of the irreducible components
$\mathfrak{m}_{1},\mathfrak{m}_{2}$ in terms of the root space decomposition
of the Lie algebra $\mathfrak{sp}(2)$. We will use the following basis of
$\mathfrak{sp}(2)$: let $E_{ij}$ $(1\leq i,j\leq 4)$ be the $4\times 4$ matrix
with $1$ in the $ij$-position and zero otherwise. We define

$\displaystyle X_{12}=\dfrac{\sqrt{3}}{6}(E_{12}-E_{43}),\qquad
X_{-12}=\dfrac{\sqrt{3}}{6}(E_{21}-E_{34}),\qquad
X_{12}^{+}=\dfrac{\sqrt{3}}{6}(E_{14}+E_{23}),$ $\displaystyle
X_{-12}^{+}=\dfrac{\sqrt{3}}{6}(E_{32}+E_{41}),\qquad
X_{11}=\dfrac{\sqrt{6}}{6}(E_{13}),\qquad
X_{-11}=\dfrac{\sqrt{6}}{6}(E_{31}).$

The matrices defined above are the Weyl basis of $\mathfrak{sp}(4,\mathbb{C})$
associated to the Cartan subalgebra of diagonal matrices.

Let us consider the linear functional given by

$\displaystyle\lambda_{1}\colon\
\Lambda=\operatorname{diag}\\{x,y\\}\longrightarrow
x\qquad\text{and}\qquad\lambda_{2}\colon\
\Lambda=\operatorname{diag}\\{x,y\\}\longrightarrow y.$

We have the following positive roots of $\mathfrak{sp}(2)$:

$\displaystyle\alpha_{1}=\lambda_{1}-\lambda_{2},\qquad\alpha_{2}=\lambda_{1}+\lambda_{2}\qquad\text{and}\qquad\alpha_{3}=2\lambda_{1}.$

The isotropy representation decomposes in the following way

$\displaystyle\mathfrak{m}$
$\displaystyle=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}$
$\displaystyle=(\mathfrak{g}_{\alpha_{1}}\oplus\mathfrak{g}_{\alpha_{2}})\oplus\mathfrak{g}_{\alpha_{3}}.$

We will denote an invariant metric $g$ by the pair of positive numbers
$(x,y)$. To be consistent with Section 3.2, we set $\lambda_{1}=x$ and
$\lambda_{2}=y$. We will consider the orthonormal basis of $TM_{\mathbb{C}}$:

$\displaystyle\bigg{\\{}\dfrac{\sqrt{x}}{x\sqrt{2}}X_{12},\,\dfrac{\sqrt{x}}{x\sqrt{2}}X_{12}^{+},\,\dfrac{\sqrt{y}}{y}X_{11},\,\dfrac{\sqrt{x}}{x\sqrt{2}}X_{-12},\,\dfrac{\sqrt{x}}{x\sqrt{2}}X_{-12}^{+},\,\dfrac{\sqrt{y}}{y}X_{-11}\bigg{\\}}.$

######  Proposition 3.28 ([3]).

Let us consider $\mathbb{CP}^{3}$, with invariant metric $g$ parametrized by
$(x,y)$. Then the Riemannian scalar curvature of $(\mathbb{CP}^{3},g)$ is
given by

$\displaystyle s=\frac{2}{x}+\frac{2}{3y}-\frac{y}{6x^{2}}.$

The flag manifold $\mathbb{CP}^{3}$ admits two invariant almost complex
structures (up to conjugation and equivalence) parametrized in the following
way: the integrable structure $J_{1}=(+,+)$ and the non-integrable structure
$J_{2}=(+,-)$.

######  Lemma 3.29.

The invariant Hermitian structure $(g,J_{2})$ does not admit $(1,2)$-triple.
The $(0,3)$-triples are given by $\alpha_{1}+\alpha_{2}-\alpha_{3}=0$ and
$-\alpha_{1}-\alpha_{2}+\alpha_{3}=0$.

######  Proposition 3.30.

Let us consider the flag manifold $\mathbb{CP}^{3}$, equipped with invariant
Hermitian structure $(g,J_{2})$. Denote by $s$ the Riemannian scalar curvature
and $s_{1}$ the first Hermitian scalar curvature. Then the invariant metric
$g$ that is solution of the equation $2s_{1}-s=0$ is given by

$\displaystyle y=2x\big{(}\sqrt{10}+3\big{)}.$ (3.2)

###### Proof.

By Proposition 3.17, we have $\|({\rm d}F)^{+}\|^{2}=0$,

$\displaystyle\|({\rm d}F)^{-}\|^{2}=\frac{(2x+y)^{2}}{3x^{3}},$
$\displaystyle\big{\|}N^{0}\big{\|}^{2}=\frac{32(x-y)^{2}}{9x^{3}}$

and the covariant derivative of the Kähler form is

$\displaystyle\|DF\|^{2}=\frac{4x^{2}-4xy+3y^{2}}{3x^{3}}.$

The first Hermitian scalar curvature is given by

$\displaystyle s_{1}=\dfrac{s}{2}-\dfrac{5}{12}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{16}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{4}\|({\rm
d}F)^{+}\|^{2}=\frac{4x^{3}+8x^{2}y-13xy^{2}+y^{3}}{12x^{3}y}.$

Thus we obtain

$\displaystyle 2s_{1}-s=\frac{-4x^{2}-12xy+y^{2}}{6x^{3}}.$

Therefore,

$\displaystyle 2s_{1}-s=0\Leftrightarrow y=2x\big{(}\sqrt{10}+3\big{)}.$ ∎

An immediate consequence of the proof of Proposition 3.30, by analyzing the
vanishing of the components $({\rm d}F)^{-}$, $({\rm d}F)^{+}$, $N^{0}$ is the
following result:

######  Proposition 3.31.

With the notation above, the invariant almost Hermitian structure $(g,J_{2})$
on $\mathbb{CP}^{3}$ belongs to the Gray–Hervella class
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. The pair $(g,J_{2})$ belongs to the
class $\mathcal{W}_{1}$ $($nearly Kähler$)$ if, and only if, the parameters of
the metric $g$ satisfies $x=y$.

For the sake of completeness we will compute the scalar curvatures
$s_{2}=s_{2}(t)$ and $s_{J}$ for the complex structure $J_{2}$. We will use
the ingredients computed in the Proposition 3.30.

######  Proposition 3.32.

Let us consider the flag manifold $\mathbb{CP}^{3}$, equipped with invariant
Hermitian structure $(g,J_{2})$. The second Hermitian scalar curvature $s_{2}$
and the $J$-scalar curvature are given by

$\displaystyle s_{2}=\frac{4x^{3}+12x^{2}y-5xy^{2}+y^{3}}{12x^{3}y},\qquad
s_{J}=\frac{1}{6}\bigg{(}\frac{4y^{2}}{x^{3}}-\frac{17y}{x^{2}}+\frac{12}{x}+\frac{4}{y}\bigg{)}.$

Let us deal now with the invariant almost Hermitian structure $(g,J_{1})$.

######  Lemma 3.33.

The invariant Hermitian structure $(g,J_{1})$ does not admit $(0,3)$-triple.
The $(1,2)$-triples are given by $\alpha_{1}+\alpha_{2}-\alpha_{3}=0$ and
$-\alpha_{1}-\alpha_{2}+\alpha_{3}=0$.

######  Proposition 3.34.

Let us consider the flag manifold $\mathbb{CP}^{3}$, equipped with invariant
Hermitian structure $(g,J_{1})$. Denote by $s$ the Riemannian scalar curvature
and $s_{1}$ the first Hermitian scalar curvature. Then the invariant metric
$g$ that is solution of the equation $2s_{1}-s=0$ is given by $y=2x$. In this
case, the pair $(g,J_{1})$ is a Kähler structure.

###### Proof.

We get $\|({\rm d}F)^{-}\|^{2}=\big{\|}N^{0}\big{\|}^{2}=0$ and the norm of
the $(1,2)+(2,1)$-part of ${\rm d}F$ satisfies

$\displaystyle\|({\rm d}F)^{+}\|^{2}=\dfrac{(2x-y)^{2}}{6x^{4}y}.$

The first Hermitian scalar curvature is

$\displaystyle s_{1}=\dfrac{s}{2}-\dfrac{5}{12}\|({\rm
d}F)^{-}\|^{2}\\!+\dfrac{1}{16}\big{\|}N^{0}\big{\|}^{2}\\!+\dfrac{1}{4}\|({\rm
d}F)^{+}\|^{2}=\frac{8x^{4}+24x^{3}y-2x^{2}\\!\big{(}y^{2}-2\big{)}-4xy+y^{2}}{24x^{4}y}.$

Therefore,

$\displaystyle 2s_{1}-s=\frac{(y-2x)^{2}}{12x^{4}y}.$

and

$\displaystyle 2s_{1}-s=0\Leftrightarrow y=2x\qquad(\text{the pair}\
(g,J_{1})\ \text{is K\"{a}hler}).$ ∎

An immediate consequence of the proof of Proposition 3.34, by analyzing the
vanishing of the components $({\rm d}F)^{-}$, $({\rm d}F)^{+}$, $N^{0}$ is the
following result:

######  Proposition 3.35.

The invariant almost Hermitan structure $(g,J_{1})$ on $\mathbb{CP}^{3}$
belongs to the Gray–Hervella class $\mathcal{W}_{3}$. The invariant Hermitian
structure $(g,J_{1})$ with $g$ satisfying $y=2x$ is a Kähler structure

For the sake of completeness we will compute the scalar curvatures $s_{2}(t)$
and $s_{J}$ for the complex structure $J_{1}$. We will use the ingredients
computed in the Proposition 3.34.

######  Proposition 3.36.

Let us consider the flag manifold $\mathbb{CP}^{3}$, equipped with invariant
Hermitian structure $(g,J_{1})$. The second Hermitian scalar curvature
$s_{2}(t)$ and the $J$-scalar curvature are given by

$\displaystyle
s_{2}(t)=\frac{-t^{2}(y-2x)^{2}+2t(y-2x)^{2}+2x^{2}\big{(}4x^{2}+12xy-y^{2}\big{)}}{24x^{4}y},$
$\displaystyle s_{J}=-\frac{y}{6x^{2}}+\frac{2}{x}+\frac{2}{3y}.$

###  3.5 Full flag manifold $\boldsymbol{{\rm SU}(4)/T^{3}}$

Let us consider now the 12-dimensional full flag manifold ${\rm SU}(4)/T^{3}$.
Recall that a Cartan sub-algebra $\mathfrak{h}$ of $\mathfrak{su}(4)$ is given
by

$\displaystyle\mathfrak{h}=\\{\operatorname{diag}(x_{1},x_{2},x_{3},x_{4})\colon
x_{1}+x_{2}+x_{3}+x_{4}=0,\ x_{1},x_{2},x_{3},x_{4}\in\mathbb{C}\\}.$

The set of roots is given by $\lambda_{ij}=x_{i}-x_{j}$, and the positive
roots are $\lambda_{ij}$, $i<j$.

The isotropy representation of ${\rm SU}(4)/T^{3}$ splits into six
irreducibles components of the isotropy representation, where each component
is associated to a positive root:

$\displaystyle\mathfrak{m}$
$\displaystyle=\mathfrak{m}_{1}\oplus\mathfrak{m}_{2}\oplus\mathfrak{m}_{3}\oplus\mathfrak{m}_{4}\oplus\mathfrak{m}_{5}\oplus\mathfrak{m}_{6}$
$\displaystyle=\mathfrak{u}_{12}\oplus\mathfrak{u}_{13}\oplus\mathfrak{u}_{14}\oplus\mathfrak{u}_{23}\oplus\mathfrak{u}_{24}\oplus\mathfrak{u}_{34},$

where
$\mathfrak{u}_{ij}=\mathfrak{su}(4)\cap(\mathfrak{g}_{ij}\oplus\mathfrak{g}_{ji})$,
and $\mathfrak{g}_{ij}$ is the root space associated to the root
$\lambda_{ij}$.

There exist four invariant almost complex structures, up to conjugation and
equivalence:

$\displaystyle J_{1}=(+,+,+,+,+,+),\qquad$ $\displaystyle
J_{2}=(-,+,+,-,+,+),$ $\displaystyle J_{3}=(+,+,+,-,+,-),\qquad$
$\displaystyle J_{4}=(+,+,-,+,+,+).$

The zero sum triples of ${\rm SU}(4)/T^{3}$ are given by
$\alpha_{12}+\alpha_{23}+\alpha_{31}=0$,
$\alpha_{12}+\alpha_{24}+\alpha_{41}=0$,
$\alpha_{13}+\alpha_{34}+\alpha_{41}=0$,
$\alpha_{23}+\alpha_{34}+\alpha_{42}=0$,
$\alpha_{21}+\alpha_{32}+\alpha_{13}=0$,
$\alpha_{21}+\alpha_{42}+\alpha_{14}=0$,
$\alpha_{31}+\alpha_{43}+\alpha_{14}=0$ and
$\alpha_{32}+\alpha_{43}+\alpha_{24}=0$.

An invariant metric $g$ on ${\rm SU}(4)/T^{3}$ depends on six parameters. We
will restrict our analysis to a family of invariant metric induced by the
deformation of the normal metric ${\rm SU}(4)/T^{3}$ in the direction of the
fibers of the fibration

$\displaystyle{\rm SU}(3)/T^{2}\,\cdots{\rm
SU}(4)/T^{3}\,\mapsto\mathbb{CP}^{3}.$

The resulting invariant metric $g$ is the 1-parameter family of metrics
parametrized by

$\displaystyle
g=(\lambda_{12},\lambda_{13},\lambda_{14},\lambda_{23},\lambda_{24},\lambda_{34})=\big{(}x^{2},x^{2},1,x^{2},1,1\big{)}.$
(3.3)

######  Proposition 3.37.

The Riemannian scalar curvature $\big{(}{\rm SU}(4)/T^{3},g\big{)}$ is given
by

$\displaystyle s=\frac{3}{8}\bigg{(}{-}x^{2}+\frac{5}{x^{2}}+8\bigg{)}.$

In the sequel we will prove Theorem 1.10 stated in the Introduction. We will
use extensively the results of Section 3.2. Since these computations are
extensive and laborious we will omit some details. We will deal case-by-case
for each pair of invariant almost Hermitian structure $(g,J_{i})$,
$i=1,\ldots,4$, where the metric $g$ is given in equation (3.3).

$\boldsymbol{(g,J_{1}).}$ Using the results of Section 3.2 we have

$\displaystyle\big{\|}N^{0}\big{\|}^{2}=0,$ $\displaystyle\|({\rm
d}F)^{-}\|^{2}=0,$ $\displaystyle\|({\rm
d}F)^{+}\|^{2}=x^{2}+\frac{1}{3x^{2}},$
$\displaystyle\|DF\|^{2}=x^{2}+\frac{1}{3x^{2}}.$

Therefore the invariant almost Hermitian structure $(g,J_{1})$ belongs in the
class $\mathcal{W}_{3}$ for all $x$. The first Hermitian scalar curvature is

$\displaystyle s_{1}=\dfrac{s}{2}-\dfrac{5}{12}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{16}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{4}\|({\rm
d}F)^{+}\|^{2}=\frac{1}{48}\bigg{(}3x^{2}+\frac{49}{x^{2}}+72\bigg{)}.$

Therefore the equation $2s_{1}-s$ reads:

$\displaystyle 2s_{1}-s=-\dfrac{5}{6}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{8}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{2}\|({\rm
d}F)^{+}\|^{2}=\frac{3x^{4}+1}{6x^{2}},$

and there is no $x$ such that $2s_{1}-s=0$.

$\boldsymbol{(g,J_{2}).}$ Using the results of Section 3.2 we have

$\displaystyle\|({\rm d}F)^{-}\|^{2}=\dfrac{3}{x^{2}},$
$\displaystyle\big{\|}N^{0}\big{\|}^{2}=0,$ $\displaystyle\|({\rm
d}F)^{+}\|^{2}=x^{2},$ $\displaystyle\|DF\|^{2}=x^{2}+\frac{1}{x^{2}}.$

Therefore the invariant almost Hermitian structure $(g,J_{2})$ belongs in the
class $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ for all $x$. The first Hermitian
scalar curvature is

$\displaystyle s_{1}=\dfrac{s}{2}-\dfrac{5}{12}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{16}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{4}\|({\rm
d}F)^{+}\|^{2}=\frac{x^{4}+24x^{2}-5}{16x^{2}}.$

Therefore the equation $2s_{1}-s$ reads:

$\displaystyle 2s_{1}-s=-\dfrac{5}{6}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{8}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{2}\|({\rm
d}F)^{+}\|^{2}=\frac{x^{4}-5}{2x^{2}},$

and $2s_{1}-s=0$ if, and only if, $x=\sqrt[4]{5}$.

$\boldsymbol{(g,J_{3}).}$ Using the results of Section 3.2 we have

$\displaystyle\|({\rm
d}F)^{-}\|^{2}=\frac{\big{(}x^{2}+2\big{)}^{2}}{3x^{2}},$
$\displaystyle\big{\|}N^{0}\big{\|}^{2}=\frac{32\big{(}x^{2}-1\big{)}^{2}}{9x^{2}},$
$\displaystyle\|({\rm
d}F)^{+}\|^{2}=\frac{1}{3}\bigg{(}2x^{2}+\frac{5}{x^{2}}-4\bigg{)},$
$\displaystyle\|DF\|^{2}=\frac{5x^{2}}{3}+\frac{3}{x^{2}}-\frac{8}{3}.$

Therefore the invariant almost Hermitian structure $(g,J_{2})$ belongs in the
class $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ for all
$x\neq 1$. When $x=1$ we have $N^{0}=0$ and in this case
$(g,J_{2})\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$. The first Hermitian scalar
curvature is

$\displaystyle s_{1}=\dfrac{s}{2}-\dfrac{5}{12}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{16}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{4}\|({\rm
d}F)^{+}\|^{2}=\frac{3x^{4}+8x^{2}+49}{48x^{2}}.$

Therefore the equation $2s_{1}-s$ reads:

$\displaystyle 2s_{1}-s=-\dfrac{5}{6}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{8}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{2}\|({\rm
d}F)^{+}\|^{2}=\frac{1}{6}\bigg{(}3x^{2}+\frac{1}{x^{2}}-16\bigg{)},$

and $2s_{1}-s=0$ if, and only if,
$x=\sqrt{\frac{1}{3}\big{(}8-\sqrt{61}\big{)}}$ or
$x=\sqrt{\frac{1}{3}\big{(}\sqrt{61}+8\big{)}}$.

$\boldsymbol{(g,J_{4}).}$ Using the results of Section 3.2 we have

$\displaystyle\|({\rm
d}F)^{-}\|^{2}=\frac{2\big{(}x^{2}+2\big{)}^{2}}{3x^{2}},$
$\displaystyle\big{\|}N^{0}\big{\|}^{2}=\frac{64\big{(}x^{2}-1\big{)}^{2}}{9x^{2}},$
$\displaystyle\|({\rm d}F)^{+}\|^{2}=\frac{x^{4}+1}{3x^{2}},$
$\displaystyle\|DF\|^{2}=\frac{7x^{2}}{3}+\frac{3}{x^{2}}-\frac{8}{3}.$

Therefore the invariant almost Hermitian structure $(g,J_{3})$ belongs in the
class $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ for all
$x\neq 1$. When $x=1$ we have $N^{0}=0$ and in this case
$(g,J_{3})\in\mathcal{W}_{1}\oplus\mathcal{W}_{3}$. The first Hermitian scalar
curvature is

$\displaystyle s_{1}=\dfrac{s}{2}-\dfrac{5}{12}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{16}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{4}\|({\rm
d}F)^{+}\|^{2}=\frac{1}{48}\bigg{(}3x^{2}+\frac{17}{x^{2}}-24\bigg{)}.$

Therefore the equation $2s_{1}-s$ reads:

$\displaystyle 2s_{1}-s=-\dfrac{5}{6}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{8}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{2}\|({\rm
d}F)^{+}\|^{2}=\frac{x^{2}}{2}-\frac{7}{6x^{2}}-4,$

and $2s_{1}-s=0$ if, and only if,
$x=\sqrt{\frac{1}{3}\big{(}\sqrt{165}+12\big{)}}$.

For the sake of completeness we will finish this section by computing the
Hermitan scalar curvatures $s_{2}(t)$ and $s_{J}$ for almost Hermitian
structures $(g,J_{i})$, $i=1,\ldots,4$ on ${\rm SU}(4)/T^{3}$. We summarize
the computations in the next result.

######  Proposition 3.38.

Let us consider the flag manifold ${\rm SU}(4)/T^{3}$. The second Hermitian
scalar curvature $s_{2}(t)$ and the $J$-scalar curvature $s_{J}$ for each
invariant Hermitian structure $(g,J_{i})$, $i=1,\ldots,4$ are given by

Invariant Hermitian structure | $s_{2}(t)$ | $s_{J}$
---|---|---
$(g,J_{1})$ | $-\dfrac{4(t-2)t(3x^{4}+1)+9(x^{4}-8x^{2}-5)}{48x^{2}}$ | $\dfrac{3}{8}\bigg{(}{-}x^{2}+\dfrac{5}{x^{2}}+8\bigg{)}$
$(g,J_{2})$ | $\dfrac{(-4t^{2}+8t-3)x^{4}+24x^{2}+11}{16x^{2}}$ | $-\dfrac{3x^{2}}{8}-\dfrac{1}{8x^{2}}+3$
$(g,J_{3})$ | $\dfrac{-4t^{2}(2x^{4}\\!-4x^{2}\\!+5)+8t(2x^{4}\\!-4x^{2}\\!+5)-5x^{4}+56x^{2}+45}{48x^{2}}$ | $\dfrac{1}{24}\bigg{(}7x^{2}+\dfrac{45}{x^{2}}+8\bigg{)}$
$(g,J_{4})$ | $\dfrac{-4t^{2}(x^{4}+1)+8t(x^{4}+1)-x^{4}+40x^{2}+45}{48x^{2}}$ | $\dfrac{1}{24}\bigg{(}23x^{2}+\dfrac{45}{x^{2}}-56\bigg{)}$

###  3.6 Flag manifolds of the exceptional Lie group $\boldsymbol{G_{2}}$

Let us recall some well known facts about the Lie algebra
$\mathfrak{g}_{2}=\operatorname{Lie}(G_{2})$.

Let us consider the Cartan sub-algebra $\mathfrak{h}$ of the
$\mathfrak{g_{2}}$ as a subalgebra of the diagonal matrices of
$\mathfrak{sl}(3)$ and let $\lambda_{i}$ be the linear functional of
$\mathfrak{h}$ defined by

$\displaystyle\lambda_{i}\colon\operatorname{diag}\\{a_{1},a_{2},a_{3}\\}\longmapsto
a_{i}.$

The simple roots of $G_{2}$ are $\alpha_{1}=\lambda_{1}-\lambda_{2}$ and
$\alpha_{2}=\lambda_{2}$. The set of positive roots is given by

$\displaystyle\Pi^{+}=\\{\alpha_{1},\alpha_{2},\alpha_{1}+\alpha_{2},\alpha_{1}+2\alpha_{2},\alpha_{1}+3\alpha_{2},2\alpha_{1}+3\alpha_{2}\\}.$

The maximal root of $G_{2}$ is $\mu=2\alpha_{1}+3\alpha_{2}$.

####  3.6.1 Flag of $\boldsymbol{G_{2}}$ with two isotropy components

Let us consider the 10-dimensional flag manifold $G_{2}/{\rm U}(2)$, where the
isotropy group ${\rm U}(2)$ is represented by the short root. The isotropy
representation of this homogeneous space decomposes into two irreducible
components: $\mathfrak{m}_{1}$ and $\mathfrak{m}_{2}$, described explicitly as

$\displaystyle\mathfrak{m}_{1}=R^{+}(\alpha_{1},1)=\\{\alpha_{1},\alpha_{1}+\alpha_{2},\alpha_{1}+2\alpha_{2},\alpha_{1}+3\alpha_{2}\\},$
$\displaystyle\mathfrak{m}_{2}=R^{+}(\alpha_{1},2)=\\{2\alpha_{1}+3\alpha_{2}\\}.$

Therefore $G_{2}/{\rm U}(2)$ admits $2$ invariant almost complex structures,
up to conjugation:

$\displaystyle J_{1}=(+,+)\qquad\text{and}\qquad J_{2}=(+,-).$

We will denote an invariant metric $g$ by the pair of positive numbers
$(x,y)$. To be consistent with Section 3.2, we set $\lambda_{1}=x$ and
$\lambda_{2}=y$. Let us consider the orthonormal basis of $TM_{\mathbb{C}}$:

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

######  Proposition 3.39 ([4]).

Let us consider $G_{2}/{\rm U}(2)$, with invariant metric $g$ parametrized by
$(x,y)$. Then the Riemannian scalar curvature of $(G_{2}/{\rm U}(2),g)$ is
given by

$\displaystyle s=-\frac{y}{4x^{2}}+\frac{4}{x}+\frac{1}{2y}.$

######  Lemma 3.40.

The invariant Hermitian structure $(g,J_{2})$ does not admit $(1,2)$-triple.
The $(0,3)$-triples are given by
$\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$,
$-\alpha_{1}+(-\alpha_{1}-3\alpha_{2})+(2\alpha_{1}+3\alpha_{2})=0$,
$(\alpha_{1}+\alpha_{2})+(\alpha_{1}+2\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$,
$(-\alpha_{1}-\alpha_{2})+(-\alpha_{1}-2\alpha_{2})+(2\alpha_{1}+3\alpha_{2})=0$.

######  Proposition 3.41.

Let us consider the flag manifold $G_{2}/{\rm U}(2)$, equipped with invariant
Hermitian structure $(g,J_{2})$. Denote by $s$ the Riemannian scalar curvature
and $s_{1}$ the first Hermitian scalar curvature. Then the invariant metric
$g$ that is solution of the equation $2s_{1}-s=0$ is given by

$\displaystyle y=2x\big{(}\sqrt{10}+3\big{)}.$

###### Proof.

Note that $\|({\rm d}F)^{+}\|^{2}=0$. The squared norm of the
$(0,3)+(3,0)$-part of the Kähler form is given by

$\displaystyle\|({\rm d}F)^{-}\|^{2}=\frac{(2x+y)^{2}}{12x^{2}y}$

and the form $N^{0}$ is

$\displaystyle\big{\|}N^{0}\big{\|}^{2}=\frac{8(x-y)^{2}}{9x^{2}y}.$

The covariant derivative of $F$ satisfies

$\displaystyle\|DF\|^{2}=\frac{1}{12}\bigg{(}\frac{3y}{x^{2}}-\frac{4}{x}+\frac{4}{y}\bigg{)},$

and the first Hermitian scalar curvature is given by

$\displaystyle
s_{1}=\dfrac{1}{48}\bigg{(}\dfrac{84}{x}+\dfrac{8}{y}-\dfrac{5y}{x^{2}}\bigg{)}.$

Therefore, we have

$\displaystyle 2s_{1}-s=-\dfrac{5}{6}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{8}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{2}\|({\rm
d}F)^{+}\|^{2}=\frac{-4x^{2}-12xy+y^{2}}{24x^{2}y},$

and $2s_{1}-s=0$ if, and only if, $y=2x(\sqrt{10}+3)$. ∎

An immediate consequence of the proof of Proposition 3.41, by analyzing the
vanishing of the components $({\rm d}F)^{-}$, $({\rm d}F)^{+}$, $N^{0}$ is the
following result:

######  Proposition 3.42.

With the notation above, the invariant almost Hermitan structure $(g,J_{2})$
on $G_{2}/{\rm U}(2)$ belongs to the Gray–Hervella class
$\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. The pair $(g,J_{2})$ belongs to the
class $\mathcal{W}_{1}$ $($nearly Kähler$)$ if, and only if, the parameters of
the metric $g$ satisfies $x=y$.

For the sake of completeness we will compute the scalar curvatures
$s_{2}=s_{2}(t)$ and $s_{J}$ for the complex structure $J_{2}$. We will use
the ingredients computed in the Proposition 3.41.

######  Proposition 3.43.

Let us consider the flag manifold $G_{2}/{\rm U}(2)$, equipped with invariant
Hermitian structure $(g,J_{2})$. The second Hermitian scalar curvature $s_{2}$
and the $J$-scalar curvature is given by

$\displaystyle
s_{2}=\frac{1}{48}\bigg{(}{-}\frac{5y}{x^{2}}+\frac{92}{x}+\frac{12}{y}\bigg{)},$
$\displaystyle
s_{J}=\frac{1}{12}\bigg{(}{-}\frac{y}{x^{2}}+\frac{40}{x}+\frac{6}{y}\bigg{)}.$

Let us work now with the invariant almost Hermitian structure $(g,J_{1})$.

######  Lemma 3.44.

The invariant Hermitian structure $(g,J_{1})$ does not admit $(0,3)$-triple.
The $(1,2)$-triples are given by
$\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$,
$-\alpha_{1}+(-\alpha_{1}-3\alpha_{2})+(2\alpha_{1}+3\alpha_{2})=0$,
$(\alpha_{1}+\alpha_{2})+(\alpha_{1}+2\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$,
$(-\alpha_{1}-\alpha_{2})+(-\alpha_{1}-2\alpha_{2})+(2\alpha_{1}+3\alpha_{2})=0$.

######  Proposition 3.45.

Let us consider the flag manifold $G_{2}/{\rm U}(2)$, equipped with invariant
Hermitian structure $(g,J_{1})$. Denote by $s$ the Riemannian scalar curvature
and $s_{1}$ the first Hermitian scalar curvature. Then the invariant metric
$g$ that is solution of the equation $2s_{1}-s=0$ is given by $y=2x$. In this
case, the pair $(g,J_{1})$ is a Kähler structure.

###### Proof.

By Proposition 3.17, we obtain $\|({\rm
d}F)^{-}\|^{2}=\big{\|}N^{0}\big{\|}^{2}=0$, the squared norm of the
$(1,2)+(2,1)$-part of $F$ is

$\displaystyle\|({\rm d}F)^{+}\|^{2}=\dfrac{(2x-y)^{2}}{12x^{2}y}$

and the covariant derivative of $F$ is

$\displaystyle\|DF\|^{2}=\dfrac{(y-2x)^{2}}{12x^{2}y}.$

The first Hermitian scalar curvature is

$\displaystyle
s_{1}=\dfrac{1}{48}\bigg{(}{-}\dfrac{5y}{x^{2}}+\dfrac{92}{x}+\dfrac{16}{y}\bigg{)}.$

Therefore,

$\displaystyle 2s_{1}-s=-\dfrac{5}{6}\|({\rm
d}F)^{-}\|^{2}+\dfrac{1}{8}\big{\|}N^{0}\big{\|}^{2}+\dfrac{1}{2}\|({\rm
d}F)^{+}\|^{2}=\dfrac{(y-2x)^{2}}{24x^{2}y}.$

We have $2s_{1}-s=0$ if, and only if, $y=2x$. In other words $2s_{1}-s=0$ if,
and only if, $(J,\Lambda)$ is Kähler. ∎

An immediate consequence of the proof of Proposition 3.45, by analyzing the
vanishing of the components $({\rm d}F)^{-}$, $({\rm d}F)^{+}$, $N^{0}$ is the
following result:

######  Proposition 3.46.

The invariant almost Hermitan structure $(g,J_{1})$ on $G_{2}/{\rm U}(2)$
belongs to the Gray–Hervella class $\mathcal{W}_{3}$. The invariant Hermitian
structure $(g,J_{1})$ with $g$ satisfying $y=2x$ is a Kähler structure.

For the sake of completeness we will compute the scalar curvatures $s_{2}(t)$
and $s_{J}$ for the complex structure $J_{1}$. We will use the ingredients
computed in the Proposition 3.45.

######  Proposition 3.47.

Let us consider the flag manifold $G_{2}/{\rm U}(2)$, equipped with invariant
Hermitian structure $(g,J_{1})$. The second Hermitian scalar curvature
$s_{2}(t)$ and the $J$-scalar curvature are given by

$\displaystyle
s_{2}(t)=-\frac{4\big{(}t^{2}-2t-3\big{)}x^{2}-4\big{(}t^{2}-2t+24\big{)}xy+\big{(}t^{2}-2t+6\big{)}y^{2}}{48x^{2}y},$
$\displaystyle s_{J}=-\frac{y}{4x^{2}}+\frac{4}{x}+\frac{1}{2y}.$

####  3.6.2 Full flag manifold $\boldsymbol{G_{2}/T^{2}}$

Let us consider now the 12-dimensional full flag manifold $G_{2}/T^{2}$. Since
on flag manifolds the irreducible components of the isotropy representation
are the real root space, we have 6 isotropy components given by

$\displaystyle\mathfrak{m}$
$\displaystyle=\mathfrak{m_{1}}\oplus\mathfrak{m_{2}}\oplus\mathfrak{m_{3}}\oplus\mathfrak{m_{4}}\oplus\mathfrak{m_{5}}\oplus\mathfrak{m_{6}}$
$\displaystyle=\mathfrak{g}_{\alpha_{1}}\oplus\mathfrak{g}_{\alpha_{2}}\oplus\mathfrak{g}_{\alpha_{1}+\alpha_{2}}\oplus\mathfrak{g}_{\alpha_{1}+2\alpha_{2}}\oplus\mathfrak{g}_{\alpha_{1}+3\alpha_{2}}\oplus\mathfrak{g}_{2\alpha_{1}+3\alpha_{2}}.$

There exist 32 invariant almost complex structure on $G_{2}/T^{2}$, up to
conjugation and equivalence. These invariant almost complex structures are
parametrized by a set of 6 signals, as described in Table LABEL:tab:long.

The zero sum triples of $G_{2}/T^{2}$ (up to sign) are given by

$\displaystyle\begin{cases}\alpha_{1}+\alpha_{2}-(\alpha_{1}+\alpha_{2})=0,\\\
\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0,\\\
\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0,\\\
\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0,\\\
(\alpha_{1}+\alpha_{2})+(\alpha_{1}+2\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0.\end{cases}$
(3.4)

An invariant metric $g$ on $G_{2}/T^{2}$ depends on six parameters. We will
restrict our analysis to a family of invariant metric induced by the
deformation of the normal metric $G_{2}/T^{2}$ in the direction of the fibers
of the fibration

$\displaystyle S^{2}\times S^{2}\cdots G_{2}/T^{2}\mapsto G_{2}/{\rm SO}(4).$
(3.5)

The resulting invariant metric $g$ is the 1-parameter family of metrics
parametrized by

$\displaystyle
g=(\lambda_{\alpha_{1}},\lambda_{\alpha_{2}},\lambda_{\alpha_{1}+\alpha_{2}},\lambda_{\alpha_{1}+2\alpha_{2}},\lambda_{\alpha_{1}+3\alpha_{2}},\lambda_{2\alpha_{1}+3\alpha_{2}})=\big{(}1,1,x^{2},1,x^{2},1\big{)}.$
(3.6)

One can compute the Riemannian scalar curvature by using the Ricci curvature
of an invariant metric on $G_{2}/T^{2}$ computed in [5].

######  Proposition 3.48.

The Riemannian scalar curvature of $\big{(}G_{2}/T^{2},g\big{)}$, where the
invariant metric $g$ is defined in (3.6), is given by

$\displaystyle s=\frac{2+12x^{2}-2x^{4}}{3x^{2}}.$

In the sequel we will summarize the computation used in the proof of Theorem
1.11 stated in the Introduction. We will deal case-by-case for each pair of
invariant almost Hermitian structure $(g,J_{i})$, $i=1,\ldots,32$, where the
metric $g$ is given in equation (3.6). Since the details of the proof are
similar to the previous section we will omit them. The solutions for the
equation $2s_{1}-s=0$ and all other information about the invariant almost
Hermitian structures $(g,J_{i})$ concerning to this question are summarized in
Table LABEL:tab:long. We remark that we list just $(0,3)$-triple of the pair
$(g,J_{i})$. The set of the corresponding $(1,2)$-triples is the complementary
set of the $(0,3)$-triples inside the set of all zero sum triples listed in
(3.4).

### Acknowledgements

L. Grama is partially supported by 2018/13481-0; 2021/04003-0 (FAPESP) and
305036/2019-0 (CNPq).

Table 3: Full flag manifold $G_{2}/T^{2}$. Invariant Hermitian structure |  Parametrization of $J_{i}$ |  $(0,3)$-triples (up to sign) | $({\rm d}F)^{-}$ | $N^{0}$ | $({\rm d}F)^{+}$ | $\alpha_{F}$ | Class | Remark | $2s_{1}-s=0$
---|---|---|---|---|---|---|---|---|---
$(g,J_{1})$ | $(+,+,+,+,+,+)$ | $0$ | $0$ | $0$ | $\frac{1}{3}(5x^{2}+\frac{8}{x^{2}}-8)$ | 0 | $\mathcal{W}_{3}$ | $\forall x$ | $\nexists x$
$(g,J_{2})$ | $(-,+,+,+,+,+)$ | $0$ | 0 | $0$ | $\frac{1}{3}(5x^{2}+\frac{8}{x^{2}}-8)$ | 0 | $\mathcal{W}_{3}$ | $\forall x$ | $\nexists x$
$(g,J_{3})$ | $(+,-,+,+,+,+)$ | $0$ | 0 | 0 | $\frac{1}{3}(5x^{2}+\frac{4}{x^{2}}-4)$ | 0 | $\mathcal{W}_{3}$ | $\forall x$ | $\nexists x$
$(g,J_{4})$ | $(+,+,-,+,+,+)$ | $\alpha_{1}+\alpha_{2}-(\alpha_{1}+\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-x^{2}+1)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{4}{\sqrt{5}}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{5})$ | $(+,+,+,-,+,+)$ | $\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-2x^{2}+2)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ |  $x=\frac{23}{50}$ or $x=\frac{39}{20}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{6})$ | $(+,+,+,+,-,+)$ | $\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-x^{2}+1)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{4}{\sqrt{5}}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{7})$ | $(+,+,+,+,+,-)$ | $\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$ | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}+\frac{8}{3x^{2}}-\frac{8}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{4\sqrt{2}}{\sqrt{5}}$
|  | $(\alpha_{1}\\!+\\!\alpha_{2})\\!+\\!(\alpha_{1}\\!+\\!2\alpha_{2})\\!-\\!(2\alpha_{1}\\!+\\!3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{8})$ | $(+,-,-,+,+,+)$ | $\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4x^{2}}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{41}{25}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{9})$ | $(+,-,+,-,+,+)$ | $\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-x^{2}+1)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{4}{\sqrt{5}}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{10})$ | $(+,-,+,+,-,+)$ | $0$ | 0 | 0 | $\frac{1}{3}(5x^{2}+\frac{4}{x^{2}}-4)$ | 0 | $\mathcal{W}_{3}$ | $\forall x$ | $\nexists x$
$(g,J_{11})$ | $(+,-,+,+,+,-)$ | $\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$ | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}+\frac{4}{3x^{2}}-\frac{4}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{12}{5}$
|  | $(\alpha_{1}\\!+\\!\alpha_{2})\\!+\\!(\alpha_{1}\\!+\\!2\alpha_{2})\\!-\\!(2\alpha_{1}\\!+\\!3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{12})$ | $(+,+,-,-,+,+)$ |  $\alpha_{1}+(\alpha_{2})-(\alpha_{1}+\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(-1+x^{2})^{2}}{9x^{2}}$ | $x^{2}+\frac{4}{3x^{2}}-\frac{4}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{12}{5}$
|  | $(\alpha_{1}\\!+\\!\alpha_{2})\\!+\\!(\alpha_{1}\\!+\\!2\alpha_{2})\\!-\\!(2\alpha_{1}\\!+\\!3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{13})$ | $(+,+,-,+,-,+)$ |  $\alpha_{1}+(\alpha_{2})-(\alpha_{1}+\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{113}{50}$
|  | $\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{14})$ | $(+,+,-,+,+,-)$ |  $\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}+\frac{8}{3x^{2}}-\frac{8}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{4\sqrt{2}}{\sqrt{5}}$
|  | $\alpha_{1}+\alpha_{2}-(\alpha_{1}+\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{15})$ | $(+,+,+,-,-,+)$ | $\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-2x^{2}+2)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ |  $x=\frac{23}{50}$ or $x=\frac{39}{20}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{16})$ | $(+,+,+,-,+,-)$ |  $\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}+\frac{4}{3x^{2}}-\frac{4}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{12}{5}$
|  | $\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{17})$ | $(+,+,+,+,-,-)$ |  $\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}+\frac{8}{3x^{2}}-\frac{8}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{4\sqrt{2}}{\sqrt{5}}$
|  | $(\alpha_{1}\\!+\\!\alpha_{2})\\!+\\!(\alpha_{1}\\!+\\!2\alpha_{2})\\!-\\!(2\alpha_{1}\\!+\\!3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{18})$ | $(+,-,-,-,+,+)$ |  $\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{113}{50}$
|  | $(\alpha_{1}\\!+\\!\alpha_{2})\\!+\\!(\alpha_{1}\\!+\\!2\alpha_{2})\\!-\\!(2\alpha_{1}\\!+\\!3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{19})$ | $(+,-,-,+,-,+)$ | $\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4x^{2}}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{41}{25}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{20})$ | $(+,-,-,+,+,-)$ |  $\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}+\frac{4}{3x^{2}}-\frac{4}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{12}{5}$
|  | $\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{21})$ | $(+,-,+,-,-,+)$ | $0$ | 0 | 0 | $\frac{1}{3}(5x^{2}+\frac{8}{x^{2}}-8)$ | 0 | $\mathcal{W}_{3}$ | $\forall x$ | $\nexists x$
$(g,J_{22})$ | $(+,-,+,-,+,-)$ | $\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$ | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{113}{50}$
|  | $\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{23})$ | $(+,-,+,+,-,-)$ | $(\alpha_{1}\\!+\\!\alpha_{2})\\!+\\!(\alpha_{1}\\!+\\!2\alpha_{2})\\!-\\!(2\alpha_{1}\\!+\\!3\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-2x^{2}+2)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ |  $x=\frac{23}{50}$ or $x=\frac{39}{20}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{24})$ | $(+,+,-,-,-,+)$ |  $\alpha_{1}+(\alpha_{2})-(\alpha_{1}+\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}+\frac{4}{3x^{2}}-\frac{4}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{12}{5}$
|  | $(\alpha_{1}\\!+\\!\alpha_{2})\\!+\\!(\alpha_{1}\\!+\\!2\alpha_{2})\\!-\\!(2\alpha_{1}\\!+\\!3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{25})$ | $(+,+,-,+,-,-)$ |  $\alpha_{1}+(\alpha_{2})-(\alpha_{1}+\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}+\frac{8}{3x^{2}}-\frac{8}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{4\sqrt{2}}{\sqrt{5}}$
|  | $\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{26})$ | $(+,+,+,-,-,-)$ | $\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-2x^{2}+2)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ |  $x=\frac{23}{50}$ or $x=\frac{39}{20}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{27})$ | $(+,-,-,-,-,+)$ | $(\alpha_{1}\\!+\\!\alpha_{2})\\!+\\!(\alpha_{1}\\!+\\!2\alpha_{2})\\!-\\!(2\alpha_{1}\\!+\\!3\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-x^{2}+1)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{4}{\sqrt{5}}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{28})$ | $(+,-,-,-,+,-)$ |  $\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{113}{50}$
|  | $\alpha_{2}+(\alpha_{1}+2\alpha_{2})-(\alpha_{1}+3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{29})$ | $(+,-,-,+,-,-)$ | $\alpha_{2}+(\alpha_{1}+\alpha_{2})-(\alpha_{1}+2\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-2x^{2}+2)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ |  $x=\frac{23}{50}$ or $x=\frac{39}{20}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{30})$ | $(+,-,+,-,-,-)$ | $0$ | 0 | 0 | $\frac{1}{3}(5x^{2}+\frac{8}{x^{2}}-8)$ | 0 | $\mathcal{W}_{3}$ | $\forall x$ | $\nexists x$
$(g,J_{31})$ | $(+,+,-,-,-,-)$ | $\alpha_{1}+(\alpha_{2})-(\alpha_{1}+\alpha_{2})=0$ | $\frac{(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{32(x^{2}-1)^{2}}{9x^{2}}$ | $\frac{4(x^{4}-2x^{2}+2)}{3x^{2}}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ |  $x=\frac{23}{50}$ or $x=\frac{39}{20}$
|  |  |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$
$(g,J_{32})$ | $(+,+,-,-,+,-)$ |  $\alpha_{1}+(\alpha_{2})-(\alpha_{1}+\alpha_{2})=0$, | $\frac{2(x^{2}+2)^{2}}{3x^{2}}$ | $\frac{64(x^{2}-1)^{2}}{9x^{2}}$ | $x^{2}+\frac{4}{3x^{2}}-\frac{4}{3}$ | 0 | $\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{3}$ | $x\neq 1$ | $x=\frac{12}{5}$
|  | $\alpha_{1}+(\alpha_{1}+3\alpha_{2})-(2\alpha_{1}+3\alpha_{2})=0$ |  |  |  |  | $\mathcal{W}_{1}\oplus\mathcal{W}_{3}$ | $x=1$ | $\nexists x$

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