Equi-harmonic maps and Plücker formulae for horizontal-holomorphic curves on flag manifolds

Abstract

In this paper, we derive Plücker formulae for holomorphic maps into the maximal flag manifolds of the complex semi-simple Lie groups. Holomorphy is taken with respect to either an invariant complex structure or an invariant almost complex structure that takes part of a $(1,2)$-symplectic Hermitian structure. The maps are assumed to be horizontal, in the case of a complex structure or to satisfy a generalization of this hypothesis in the $(1,2)$-symplectic case. We also provide a relationship between holomorphic-horizontal curves and equiharmonic maps.

1 Introduction

The purpose of this paper is to compute some formulae relating topological and geometrical properties of manifolds. These are the Plücker formulae for holomorphic curves in a generalized flag manifold. These formulae relate the genus $g$ of the curve with the volume and the number of singularities of singular metrics on the curve.

Plücker formulae for curves in projective spaces give a relationship between an intrinsic invariant (the genus of the curve) and a set of extrinsic invariants like the associated degrees and the ramification indices (see Griffiths–Harris [12]). In this paper, we consider, instead of projective spaces, generalized flag manifolds of a complex semi-simple Lie group.

Let $\mathfrak{g }$ be a complex semi-simple Lie algebra and take a Lie group $G$ with Lie algebra $\mathfrak{g }$. Our manifolds are generalized flag manifolds $\mathbb{F }_{\Theta }=G/P_{\Theta }$, where $P_{\Theta }$ is a parabolic subgroup of $G$, which is determined by the subset $\Theta$ of the simple root system $\Sigma$ for $\mathfrak{g }$. The Lie algebra of $P_{\Theta }$ is denoted by $\mathfrak{p }_{\Theta }$ and is a parabolic subalgebra of $\mathfrak{g }$.

It is known that if $U$ is a compact real form of $G$ then $\mathbb{F }_{\Theta }=U/K_{\Theta }$ where $K_{\Theta }=P_{\Theta }\cap U$ is the centralizer of a torus. When $\Theta =\emptyset$ we write simply $\mathbb{F }_{\Theta }=\mathbb{F }$, which is the maximal flag manifold.

An integrable almost complex structure $J$ on $\mathbb{F }$ defines the holomorphic horizontal sub-bundle $\mathcal H$ of the tangent bundle of $\mathbb{F }$ which is spanned by the root spaces of the simple system of roots $\Sigma$ defined by $J$.

A holomorphic map tangent to the distribution $\mathcal H$ is called holomorphic-horizontal. The study of holomorphic-horizontal maps in a maximal flag manifolds was introduced by Bryant [4] in order to obtain a better understanding of the harmonic maps into symmetric spaces via twistorial methods. These maps are deeply connected with the study of harmonic and minimal surfaces in $S^{n}$, $C\mathbb P ^{n}$, $H\mathbb P ^{n}$, Twistor Theory and so on (see Burstall–Rawnsley [5], Calabi [6], Chern–Wolfson [7], Eells–Wood [10, 17]).

The point is that an interesting approach that makes use of the method of moving frames allows us to think the projective curves as horizontal curves in the complex flag manifold $\mathrm SU (n+1)/T$. This fact provides a new interpretation of the classical Plucker formulae in terms of horizontal-holomorphic curves. Another interpretation is due to Yang [23] where these formulae were computed for the flag manifold $\mathbb{F }=\mathrm SO (n)/T$ (see also [22]).

In this paper, we weaken the hypothesis on the almost complex structure $J$ by assuming that a maximal flag manifold $(\mathbb{F },J,g)$ has an invariant $(1,2)$-structure, that is, the $(1,2)$-component of the exterior derivative of the Kähler form $\omega (\cdot ,\cdot )=g(\cdot ,J\cdot )$ is zero.

For the invariant $(1,2)$-symplectic structures, we can replace the horizontal distribution $\mathcal H$ by a similar distribution spanned by the indecomposable roots, in the sense of [20]. In case $J$ is integrable, the indecomposable roots are just simple roots. With this in mind, we define a $\mathcal P$-distribution by the indecomposable root spaces and we say that a map $f:M^{2}\rightarrow (\mathbb{F },J,g)$ is subordinate to $\mathcal P$ if it is $J$-holomorphic and tangent to the $\mathcal P$-distribution.

Using the method of moving frames combined with algebraic properties of the indecomposable roots, we obtain Plücker formulae for curves subordinate to $\mathcal P$ in a generalized maximal flag manifold $\mathbb{F }=U/T$. When $J$ is integrable, our formulae specialize to those of [23].

According to [20], the understanding of the $(1,2)$-symplectic Hermitian structures is the key point for deriving the classification of all invariant Hermitian structures on maximal flag manifolds. Using [20] in a essential way, the authors study in [18] variational aspects of the same holomorphic-horizontal maps discussed in this paper. Stability and non-stability results for these maps were obtained with respect to a large number of invariant Hermitian structures including the main one namely: the $(1,2)$-symplectic invariant Hermitian structure.

In the last section of this paper, we study the relationship between equi-harmonic maps and holomorphic-horizontal curves. By a equi-harmonic map on a flag manifold, we mean a map which is harmonic for each $U$-invariant Hermitian metric. A natural question is concerned with the existence of examples of equi-harmonic maps. The main result of this section provides a class of examples of equi-harmonic maps, namely any holomorphic-horizontal curve is equi-harmonic.

2 Flag manifolds and Riemann surfaces

The purpose of this section is to fix notations and to state general results concerning flag manifolds and Riemann surfaces. The references for this section are [1, 3, 12–14].

2.1 Flag manifolds

We shall work only with simple Lie algebras and groups. The results in the semi-simple case are easily obtained by piecing together the simple components.

Let $\mathfrak{g }$ be a complex simple Lie algebra and take a Lie group $G$ with Lie algebra $\mathfrak{g }$. Given a Cartan subalgebra $\mathfrak h$ of $\mathfrak{g }$, denote by $\Pi$ the set of roots of the pair $\left( \mathfrak{g },\mathfrak h \right)$, so that

$$\begin{aligned} \mathfrak{g }=\mathfrak h \oplus \sum _{\alpha \in \Pi }\mathfrak{g }_{\alpha }, \end{aligned}$$

(1)

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

We denote by $\langle \cdot ,\cdot \rangle$ the Cartan-Killing form of $\mathfrak{g }$ and fix once and for all a Weyl basis of $\mathfrak{g }$ which amounts to take $X_{\alpha }\in \mathfrak{g }_{\alpha }$ such that $\langle X_{\alpha },X_{-\alpha }\rangle =1$, and $[X_{\alpha },X_{\beta }]=m_{\alpha ,\beta }X_{\alpha +\beta }$ with $m_{\alpha ,\beta }\in \mathbb R$, $m_{-\alpha ,-\beta }=-m_{\alpha ,\beta }$ and $m_{\alpha ,\beta }=0$ if $\alpha +\beta$ is not a root (see Helgason [13], chapter IX).

Recall that $\langle \cdot ,\cdot \rangle$ is nondegenerate on $\mathfrak h$. Given $\alpha \in \mathfrak h ^{*}$ we let $H_{\alpha }$ be given by $\alpha (\cdot )=\langle H_{\alpha },\cdot \rangle$, and denote by $\mathfrak h _{R}$ the real subspace spanned by $H_{\alpha }$, $\alpha \in \Pi$. Accordingly $\mathfrak h _{R}^{*}$ stands for the real subspace of the dual $\mathfrak{g }^{*}$ spanned by the roots.

Let $\Pi ^{+}$ be a choice of positive roots and $\Sigma$ the corresponding set of simple roots. If $\Theta$ is a subset of $\Sigma$ we denote by $\langle \Theta \rangle$ for the set of roots spanned by $\Theta$, and set $\langle \Theta \rangle ^{\pm }:=\langle \Theta \rangle \cap \Pi ^{\pm }$. We have

$$\begin{aligned} \mathfrak{g }=\mathfrak h \oplus \sum _{\alpha \in {\langle \Theta \rangle } ^{+}}\mathfrak{g }_{\alpha }\oplus \sum _{\alpha \in {\langle \Theta \rangle } ^{+}}\mathfrak{g }_{-\alpha }\oplus \sum _{\beta \in \Pi ^{+}\setminus { \langle \Theta \rangle }}\mathfrak{g }_{\beta }\oplus \sum _{\beta \in \Pi ^{+}\setminus {\langle \Theta \rangle }}\mathfrak{g }_{-\beta }. \end{aligned}$$

(2)

Let

$$\begin{aligned} \mathfrak{p }_{\Theta }=\mathfrak h \oplus \sum _{\alpha \in {\langle \Theta \rangle }^{-}}\mathfrak{g }_{\alpha }\oplus \sum _{\alpha \in {\Pi }^{+}} \mathfrak{g }_{\alpha } \end{aligned}$$

(3)

be the standard parabolic subalgebra determined by $\Theta$. Set

$$\begin{aligned} \mathfrak{q }_{\Theta }=\sum _{\beta \in \Pi ^{+}-{\langle \Theta \rangle }} \mathfrak{g }_{-\beta } \end{aligned}$$

(4)

so that $\mathfrak{g }=\mathfrak{q }_{\Theta }\oplus \mathfrak{p }_{\Theta }$.

The generalized flag manifold $\mathbb{F }_{\Theta }$ associated with $\mathfrak{p }_{\Theta }$ is defined as the homogeneous space

$$\begin{aligned} \mathbb{F }_{\Theta }=G/P_{\Theta }, \end{aligned}$$

(5)

where $P_{\Theta }$ is the normalizer of $\mathfrak{p }_{\Theta }$ in $G$.

We take as a compact real form of $\mathfrak{g }$ the real subalgebra

$$\begin{aligned} \mathfrak u =\mathrm span _\mathbb{R }\{i\mathfrak h _\mathbb{R },A_{\alpha },iS_{\alpha }:\alpha \in \Pi \} \end{aligned}$$

where $A_{\alpha }=X_{\alpha }-X_{-\alpha }$ and $S_{\alpha }=X_{\alpha }+X_{-\alpha }$. Denote by $U=\exp \mathfrak u$ the corresponding compact real form of $G$ and write $K_{\Theta }=P_{\Theta }\cap U$. It is well known that $U$ acts transitively on each $\mathbb{F }_{\Theta }$, which is identified with $U/K_{\Theta }$.

Let $\mathfrak{k }_{\Theta }$ be the Lie algebra of $K_{\Theta }$ and write $\mathfrak{k }_{\Theta }^\mathbb{C }$ for its complexification. We have $\mathfrak{k }_{\Theta }=\mathfrak u \cap \mathfrak{p }_{\Theta }$ and

$$\begin{aligned} \mathfrak{k }_{\Theta }^\mathbb{C }=\mathfrak h \oplus \sum _{\alpha \in \langle \Theta \rangle }\mathfrak{g }_{\alpha }. \end{aligned}$$

Denote by $o=eK_{\Theta }$ the origin of $\mathbb{F }_{\Theta }$. The tangent space $T_{o}\mathbb{F }_{\Theta }$ can be now identified with the orthogonal complement of $\mathfrak{k }_{\Theta }$ in $\mathfrak u$, namely

$$\begin{aligned} T_{o}\mathbb{F }_{\Theta }= \mathfrak{m }_{\Theta }=\mathrm span _\mathbb{R }\{A_{\alpha },iS_{\alpha }:\alpha \notin \langle \Theta \rangle \}=\sum _{\alpha \in \Pi \setminus \langle \Theta \rangle }\mathfrak u _{\alpha }, \end{aligned}$$

where $\mathfrak u _{\alpha }=\left( \mathfrak{g }_{\alpha }\oplus \mathfrak g _{-\alpha }\right)~\cap \mathfrak u =\mathrm span _\mathbb{R }\{A_{\alpha },iS_{\alpha }\}$. By complexifying $\mathfrak{m }_{\Theta }$, we obtain the complex tangent space of $T_{o}^\mathbb{C }\mathbb{F }_{\Theta }$, which can be identified with

$$\begin{aligned} \mathfrak{m }_{\Theta }^\mathbb{C }=\mathfrak{q }_{\Theta }=\sum _{\beta \in \Pi \setminus \langle \Theta \rangle }\mathfrak{g }_{\beta }. \end{aligned}$$

An irreducible component for the representation of $K_{\Theta }$ in $\mathfrak{q }_{\Theta }$ has the form

$$\begin{aligned} \mathfrak{g }_{\sigma }=\sum _{\alpha \in \sigma }\mathfrak{g }_{\alpha } \end{aligned}$$

where $\sigma$ is a set of roots.

The adjoint representations of $\mathfrak{k }_{\Theta }$ and $K_{\Theta }$ leave $\mathfrak{m }_{\Theta }$ invariant, so that we get a well-defined representation of both $\mathfrak{k }_{\Theta }$ and $K_{\Theta }$ in $\mathfrak{m }_{\Theta }$. Analogously the complex tangent space $\mathfrak{q } _{\Theta }$ is invariant under the adjoint representation of $\mathfrak{k } _{\Theta }^\mathbb{C }$. This representation is semi-simple and hence decomposes into irreducible components, each one is a sum of root spaces. In the sequel, we write an irreducible component as $\mathfrak{g }_{\sigma }$, where $\sigma$ is the set of roots $\alpha$ such that $\mathfrak{g } _{\alpha }\subset \mathfrak{g }_{\sigma }$, so that $\mathfrak{g }_{\sigma }=\Sigma _{\alpha \in \sigma }\mathfrak{g }_{\alpha }$. We also write $\Pi (\Theta )$ for the collection of the sets $\sigma$ giving rise to an irreducible component. In this notation, we have

$$\begin{aligned} \mathfrak{q }_{\Theta }=\bigoplus _{\sigma \in \Pi (\Theta )}\mathfrak{g } _{\sigma }. \end{aligned}$$

It is a standard fact that the roots appearing in the irreducible component $\sigma \in \Pi (\Theta )$ are either all positive or all negative. Hence, it makes sense to write $\Pi (\Theta )^{+}$ and $\Pi (\Theta )^{-}$ for the set of those irreducible components containing only positive roots or negative roots respectively. We denote by $\sum (\Theta )$=$\left\{ \sigma \in \Pi (\Theta );\text{ the} \text{ height} \text{ of} \sigma \text{ in} \Pi (\Theta ) \text{ is} \text{1} \right\}$

Each $\sigma \in \Pi (\Theta )$ defines a complex plane field on $\mathbb{F } _{\Theta }$ by

$$\begin{aligned} E_{\sigma }\left( k\cdot o\right) =k_{*}\left( \mathfrak{g }_{\sigma }\right) , \end{aligned}$$

which is well defined since $\mathrm Ad \left( k\right) \left( \mathfrak{g } _{\sigma }\right) =\mathfrak{g }_{\sigma }$, $\sigma \in \Pi (\Theta )$. For any $x\in \mathbb{F }_{\Theta }$, we have

$$\begin{aligned} T_{x}^\mathbb{C }\mathbb{F }_{\Theta }=\sum _{\sigma }E_{\sigma }\left( x\right) . \end{aligned}$$

2.2 Almost complex structures and $f$-structures

A $U$-invariant almost complex structure $J$ on $\mathbb{F }_{\Theta }$ is completely determined by its value $J:\mathfrak{m }_{\Theta }\rightarrow \mathfrak{m }_{\Theta }$ in its tangent space at the origin. The map $J$ satisfies $J^{2}=-1$ and commutes with the adjoint action of $K_{\Theta }$ on $\mathfrak{m }_{\Theta }$. We also denote by $J$ its complexification to $\mathfrak{q }_{\Theta }$.

The invariance of $J$ entails that $J(\mathfrak{g }_{\sigma })=\mathfrak{g } _{\sigma }$ for all $\sigma \in \Pi (\Theta )$. The eigenvalues of $J$ are $\pm \sqrt{-1}$ and the eigenvectors in $\mathfrak{q }_{\Theta }$ are $X_{\alpha }$, $\alpha \in \Pi _{\Theta }$. Hence, in each irreducible component $\mathfrak{g }_{\sigma }$, we have $J=\sqrt{-1}\epsilon _{\sigma } \mathrm id$ with $\epsilon _{\sigma }=\pm 1$ satisfying $\epsilon _{-\sigma }=-\epsilon _{\sigma }$. A $U$-invariant structure on $\mathbb{F }_{\Theta }$ is completely determined by the numbers $\varepsilon _{\sigma }=\pm 1$, $\sigma \in \Pi (\Theta )$.

As usual, the eigenvectors associated with $\sqrt{-1}$ are said to be of type $(1,0)$ while the $-\sqrt{-1}$ eigenvectors are of type $(0,1)$. Thus the $(1,0)$ vectors are multiples of $X_{\alpha }$, $\epsilon _{\alpha }=1$, and the $(0,1)$ vectors are also multiples of $X_{\alpha }$, $\epsilon _{\alpha }=-1$ (see [16, 20]).

Since $\mathbb{F }_{\Theta }$ is a homogeneous space of a complex Lie group, it has a natural structure of a complex manifold. The associated integrable almost complex structure $J_{C}$ is given by $\epsilon _{\sigma }=1$ if $\sigma <0$. The conjugate structure $-J_{C}$ is also integrable.

It will be useful, later, to introduce the generalization of $J$ to an $f$-structure in the sense of Yano [24] (see also [8, 19]).

Definition 2.1

An $f$-structure $\mathcal{F }$ in $\mathbb{F }{_{\Theta }}$ is a section of ${\text{ End}}\bigl (T({\mathbb{F }_{\Theta }})\bigr )$ such that $\mathcal{F }^{3}+ \mathcal{F }=0$.

We will restrict ourselves to the case of invariant $f$-structures. As in the case of the almost complex structures, an $f$-structure is given by the data $(f_{\alpha })_{\alpha \in \Pi _{\Theta }}$ with $f_{\alpha }=1$, $-1$ or $0$ , according to the eigenvalues of $\mathcal{F }$.

2.3 Invariant metrics

A $U$-invariant Riemannian metric $\mathrm{d}s_{{\Lambda }}^{2}$ on $\mathbb{F } _{\Theta }$ is completely determined by its value at the origin, that is, by an inner product $\langle \cdot ,\cdot \rangle$ in $\mathfrak{m }_{\Theta }$ which is invariant under the adjoint action of $K_{\Theta }$. Any such inner product has the form

$$\begin{aligned} \langle X,Y\rangle _{{\Lambda }}:=-\langle \Lambda X,Y\rangle \end{aligned}$$

(6)

with $\Lambda :\mathfrak{m }_{\Theta }\rightarrow \mathfrak{m }_{\Theta }$ positive-definite with respect to the Cartan-Killing form. The inner product $\langle \cdot ,\cdot \rangle _{{\Lambda }}$ admits a natural extension to a symmetric bilinear form on $\mathfrak{q }_{\Theta }=\mathfrak{m }_{\Theta }^{ \mathbb C }$. We do not change notation for objects in $\mathfrak{m }_{\Theta }$ and $\mathfrak{q }_{\Theta }$ either for the bilinear form $\langle \cdot ,\cdot \rangle _{{\Lambda }}$ or for the corresponding complexified map ${ \Lambda _{\Theta }}$. The $K_{\Theta }$-invariance of $\langle \cdot ,\cdot \rangle _{{\Lambda }}$ is equivalent to the fact that the elements of the standard basis $A_{\alpha }$, $\sqrt{-1}S_{\alpha }$, $\alpha \in \Pi _{\Theta }$, are eigenvectors of ${\Lambda }$, for the same eigenvalue. Thus, in each irreducible component of $\mathfrak{q }_{\Theta }$, we have $\Lambda =\lambda _{\sigma }\mathrm id$ with $\lambda _{-\sigma }=\lambda _{\sigma }>0$.

We denote either by $\langle X,Y\rangle _{\Lambda }$ or by $\mathrm{d}s_{\Lambda }^{2}$ the invariant metric given by ${\Lambda }$. In what follows with an abuse of notation, we say that ${\Lambda }$ itself is an invariant metric.

Let $\tau$ be the conjugation of $\mathfrak{g }$ with respect to $\mathfrak u$. Then $\langle \langle X,Y\rangle \rangle _{\Lambda }=\langle X,\tau Y\rangle _{\Lambda }$ is a Hermitian form on $\mathfrak{g }$ which restricts to a $U$-invariant Hermitian form on each $\mathbb{F }_{\Theta }$.

2.4 Riemann surfaces

We recall some basic concepts on Riemann surfaces. We suggest [12] for more details.

A Riemann surface is a $1$-dimensional complex manifold. In this paper, $M$ always denotes a compact connected Riemann surface.

A Hermitian metric on $M$ is given by a positive definite hermitian inner product

$$\begin{aligned} (\cdot ,\cdot )_{z}:T^{(1,0)}\otimes \overline{T_{z}^{(1,0)}}\rightarrow \mathbb C \end{aligned}$$

on the holomorphic tangent space at $z$, depending smoothly on $z$. Writing $(\cdot ,\cdot )_{z}$ in terms of the basis $\{\mathrm{d}z\otimes \mathrm{d}\bar{z}\}$, the hermitian metric is given by

$$\begin{aligned} \mathrm{d}s^{2}=h(z)\mathrm{d}z\otimes \mathrm{d}\bar{z}, \end{aligned}$$

where $h(z)=(\frac{\partial }{\partial z},\frac{\partial }{\partial z})_{z}$.

A coframe for the hermitian metric $\mathrm{d}s^{2}$ is a 1-form $\varphi$ of type $(1,0)$ such that

$$\begin{aligned} \mathrm{d}s^{2}=\varphi \otimes \bar{\varphi }. \end{aligned}$$

It is clear that coframes always exist locally. For more details, see [12].

There exist a natural $\mathbb R$-linear isomorphism between $T_\mathbb{R ,z}(M)$ and $T^{(1,0)}M$. We consider an Hermitian metric $\mathrm{d}s^2$ on $M$, and via this isomorphism, we can show that $\text{ Re}\,\mathrm{d}s^2$ induces a Riemannian metric on $M$, called the induced Riemannian metric. The notions of distance, area or volume on a complex manifold with hermitian metric are always given with respect to the induced Riemannian metric.

The alternating quadratic form $\Phi =-\text{ Im}\,\mathrm{d}s^2$ provides a real form of degree 2 called the associated $(1,1)$-form (or Kähler form).

Explicitly, if $\varphi$ is a coframe for the hermitian metric $\mathrm{d}s^2$ we have

$$\begin{aligned} \Phi =\frac{1}{2}\sqrt{-1} \varphi \wedge \bar{\varphi }. \end{aligned}$$

We say that a complex manifold is a Kähler manifold if the associated $(1,1)$-form is closed. Therefore, any Riemann surface is a Kähler manifold because $\dim _\mathbb{R }M=2$.

We just mention the well-known fact on connections and curvature forms on the holomorphic tangent bundle of a Riemann surface.

Proposition 2.2

[12] Let $M$ be a Riemann surface with Hermitian metric $\mathrm{d}s^{2}=\varphi \otimes \bar{\varphi }$. Then

• (a) There exists an unique $1$-form $\psi$ such that $\psi +\bar{\psi }=0$ and

  $$\begin{aligned} \mathrm{d}\varphi =-\psi \wedge \varphi . \end{aligned}$$

  The form $\psi$ is called the connection $1$-form on the tangent bundle.

• (b) Let $\psi$ be the connection $1$-form on $M$. Then

  $$\begin{aligned} \sqrt{-1}\mathrm{d}\psi =K\cdot \Phi , \end{aligned}$$

  where $K$ is the Gaussian curvature of $\mathrm{d}s^{2}$ and $\Phi$ is the associated $(1,1)$-form. We call $\mathrm{d}\psi$ the curvature $2$-form on the tangent bundle.

3 Plücker formulae on maximal flag manifolds

In this section, we compute the Plücker formulae for maps $f:M\rightarrow \mathbb{F }$, defined on a Riemann surface with values in a maximal flag manifold. These formulae relate the genus $g$ of $M$ with the volume and the number of singularities of singular metrics on $M$ induced by $f$.

As in Sect. 2 let $G$ be a complex simple Lie group and $P$ be a Borel (parabolic minimal) subgroup of $G$. Then $\mathbb{F }=G/P=U/T$, where $U$ is a compact real form of $G$ and $T=U\cap P$ is a maximal torus of $U$.

Let $\{X_{\alpha },H_{\beta }\}$ be a Weyl basis of $\mathfrak u ^\mathbb{C }=\mathfrak{g }$ as defined on Sect. 2 and write $A_{\alpha }=X_{\alpha }-X_{-\alpha }$ and $S_{\alpha }=\sqrt{-1}(X_{\alpha }+X_{-\alpha })$. The vectors

$$\begin{aligned} A_{\alpha },\,\,\,\,S_{\alpha },\,\,\,\,\sqrt{-1}H_{\beta } \end{aligned}$$

(7)

form a basis of $\mathfrak u$ where $\alpha \in \Pi ^{+}$ and $\beta \in \Sigma$. The Lie brackets between these elements are

$$\begin{aligned} \begin{array}{llll} [\sqrt{-1}H_{\alpha },A_{\beta }]=\beta (H_{\alpha })S_{\beta }&\,&[A_{\alpha },A_{\beta }]=m_{\alpha ,\beta }A_{\alpha +\beta }+m_{-\alpha ,\beta }A_{\alpha -\beta } \\ \left[ \sqrt{-1}H_{\alpha },S_{\beta }\right] =-\beta (H_{\alpha })A_{\beta }&\,&[S_{\alpha },S_{\beta }]=-m_{\alpha ,\beta }A_{\alpha +\beta }-m_{\alpha ,-\beta }A_{\alpha -\beta } \\ \left[ A_{\alpha },S_{\alpha }\right] =2\sqrt{-1}H_{\alpha }&\,&[A_{\alpha },S_{\beta }]=m_{\alpha ,\beta }S_{\alpha +\beta }+m_{\alpha ,-\beta }S_{\alpha -\beta } \end{array} , \end{aligned}$$

(8)

where $m_{\alpha ,\beta }$ is a real number defined by $[X_{\alpha },X_{\beta }]=m_{\alpha ,\beta }X_{\alpha +\beta }$ with $m_{\alpha ,\beta }=0$ if $\alpha +\beta$ is not a root. Denote by $\{\mu _{\alpha },\nu _{\alpha },\xi _{\beta }:\alpha \in \Pi ^{+},\,\,\beta \in \Sigma \}$ the dual basis of (7). For easy reference, we write the basis as

$$\begin{aligned} \begin{array}{lllllllllllll} A_{\alpha _{1}}&S_{\alpha _{1}}&\ldots&A_{\alpha _{k}}&S_{\alpha _{k}}&A_{\alpha _{k+1}}&S_{\alpha _{k+1}}&\ldots&A_{\alpha _{n}}&S_{\alpha _{n}}&\sqrt{-1}H_{\alpha _{1}}&\ldots&\sqrt{-1}H_{\alpha _{k}} \\ \mu _{\alpha _{1}}&\nu _{\alpha _{1}}&\ldots&\mu _{\alpha _{k}}&\nu _{\alpha _{k}}&\mu _{\alpha _{k+1}}&\nu _{\alpha _{k+1}}&\ldots&\mu _{\alpha _{n}}&\nu _{\alpha _{n}}&\xi _{\alpha _{1}}&\ldots&\xi _{k} \end{array} \end{aligned}$$

(9)

The first $2k$ vectors are associated with the vectors of the Weyl basis on the set of simple roots. The forms $\mu _{i},\nu _{i},\xi _{i}$ are the Maurer-Cartan forms. Their exterior differentials are computed using the following general basic fact (see eg. Sternberg [21]).

Proposition 3.1

Let $\mathfrak u$ be a Lie algebra with basis $\{v_{i}\}$ and dual basis $\{\rho _{i}\}$. We have

$$\begin{aligned} d\rho _{i}=\sum _{p<q}{C_{pq}^{i}\,\rho _{p}\wedge \rho _{q}}, \end{aligned}$$

(10)

where $C_{pq}^{i}$ are the structure constants of the Lie algebra $\mathfrak u$ defined by $[v_{p},v_{q}]=\sum _{k=1}^{m}{C_{pq}^{k}v_{k}}$.

Define the $\mathfrak{g }$-valued $\mathbb C$-linear 1-form $\Psi _{i}$ by

$$\begin{aligned} \Psi _{i}=\mu _{\alpha _{i}}-\sqrt{-1}\nu _{\alpha _{i}}, \end{aligned}$$

where $1\le i\le n$.

Let $f:M\rightarrow (\mathbb{F },J)$, where $M$ is a Riemann surface, $\mathbb{F }$ the full flag manifold of $\mathfrak{g }$ and $J$ is an invariant almost complex structure on $\mathbb{F }$.

We consider a local $U$-frame $e:V\subset M\rightarrow U$ such that $e\circ \pi =f$, where $\pi :U\rightarrow \mathbb{F }$ is the cannonical projection. The 1-form $\varphi _{i}=e^{*}\Psi _{i}$ is a $\mathfrak{g }$-valued $\mathbb C$-linear 1-form on $M$ (see [11]).

Definition 3.2

The map $\psi :M^{2}\rightarrow {\mathbb{F }}$ is said to be subordinate to $\mathcal P \subset \Pi$ if $\psi _{\sigma }=0$ when $\sigma \in \Pi \setminus \mathcal P$. Here, as before, $\psi _{\sigma }$ denotes the $E_{\sigma }$-component of the derivative of $\psi$.

We remark that in case $f$ is subordinated to a subset $\mathcal P$ then $\varphi _{i}=0$, if $\alpha _{i}\notin \mathcal P$.

The holomorphy of $f$ is reflected by the fact that the forms $\varphi _{i}$ are all of type $(1,0)$ on $M$. Define $\tilde{\mu }_{\alpha _{i}}=e^{*}\mu _{\alpha _{i}}$, $\tilde{\nu }_{\alpha _{i}}=e^{*}\nu _{\alpha _{i}}$, $\tilde{\xi }_{\alpha _{i}}=e^{*}\xi _{\alpha _{i}}$. These are $\mathfrak u$-valued real $1$-forms on $M$.

Theorem 3.3

Let $\mathbb{F }=U/T$ be a full flag manifold and let $M$ be a compact Riemann surface. Suppose that

• 1. $f:M\rightarrow (\mathbb{F },\Lambda ,J)$ is a $J$-holomorphic map subordinate to $\mathcal P$.

• 2. Each $1$-form $\varphi _{i}$ has only isolated zeros of finite multiplicity.

• 3. If $\alpha \in \mathcal P$ and $\alpha =\beta +\gamma$ with $\beta ,\gamma \in \Pi$ then $\beta$ or $\gamma$ is not in $\mathcal P$.

Then the Plücker formulae for $f$ are given by

$$\begin{aligned} 2g-2-\sharp _{i}=-\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})d_{j}} ,\,\,\,\,\,\,\,1\le i\le k, \end{aligned}$$

(11)

where $g$ is the genus of the Riemann surface $M$, $k$ is the number of simple roots of $\mathfrak{g }=\mathfrak u ^\mathbb{C }$, $d_{j}$ and $\sharp _{j}$ $(1\le j\le k)$ are the volume and the number of zeros of a pseudometric on $M$ induced by the $1$-forms $\varphi _{i}$, respectively.

The proof of this theorem is done in several steps. The idea is to use the $1$-forms $\varphi _{i}$ to get singular metrics $\varphi _{i}\otimes \bar{ \varphi }_{i}$ on $M$. According to Proposition 2.2, away from the zeros of $\varphi _{i}$ we have

$$\begin{aligned} \mathrm{d}\varphi _{i}&= -\psi _{i}\wedge \varphi _{i}, \end{aligned}$$

(12)

$$\begin{aligned} \mathrm{d}\psi _{i}&= -\sqrt{-1}K_{i}\Phi _{i}, \end{aligned}$$

(13)

where $\psi _{i}$ is the connection $1$-form, $\mathrm{d}\psi _{i}$ is the curvature $2$-form, $K_{i}$ and $\Phi _{i}$ are the Gaussian curvature and a Kähler form of $(M,\varphi _{i}\bar{\varphi }_{i})$, respectively. Then an application of the generalized Gauss–Bonet theorem (see Griffiths–Harris [12]) to $\mathrm{d}\psi _{i}$ yields the desired formulae.

1. Exterior differential of $\varphi _{i}$:

By definition we have

$$\begin{aligned} \varphi _{i}=e^{*}\Psi _{i}=e^{*}\left(\mu _{\alpha _{i}}-\sqrt{-1}\nu _{\alpha _{i}}\right)=e^{*}\mu _{\alpha _{i}}-\sqrt{-1}e^{*}\nu _{\alpha _{i}}=\tilde{\mu }_{\alpha _{i}}-\sqrt{-1}\tilde{\nu }_{\alpha _{i}}, \end{aligned}$$

where $1\le i\le k$. Therefore in order to compute $\mathrm{d}\varphi _{i}$, we need to compute $d\mu _{\alpha _{i}}$ and $d\nu _{\alpha _{i}}$ using the structure constants of the Lie algebra $\mathfrak{g }$.

Here we use the assumptions that $f$ is subordinate to the set $\mathcal P$ satisfying the stated property. In fact, if $\alpha _{i}\in \mathcal P$ then

$$\begin{aligned} \mathrm{d}\varphi _{i}=\sum _{\alpha _{p}+\alpha _{q}=\alpha _{i}}{C_{pq}^{i}\,\varphi _{p}\wedge \varphi _{q}+R,} \end{aligned}$$

(14)

where $R$ is a sum envolving ${\xi _{\alpha _{j}}\wedge \mu _{\alpha _{i}}}$ and ${\xi _{\alpha _{j}}\wedge \nu _{\alpha _{i}}}$.

If $\alpha _{i}=\alpha _{p}+\alpha _{q}$ then either $\alpha _{p}$ or $\alpha _{q}$ is not in $\mathcal P$, say $\alpha _{p}\notin \mathcal P$. Then $f_{\alpha _{p}}=0$ (the $E_{\alpha _{p}}$-part of the derivative of $f$). Therefore, the first summand of the right-hand side of () does not contribute to the computation of $\mathrm{d}\varphi _{i}$.

To get $R$ we note

$$\begin{aligned}{}[\sqrt{-1}H_{\alpha _{j}},S_{\alpha _{i}}]&= -\alpha _{i}(H_{\alpha _{j}})A_{\alpha _{i}}, \end{aligned}$$

(15)

$$\begin{aligned}{}[\sqrt{-1}H_{\alpha _{j}},A_{\alpha _{i}}]&= \alpha _{i}(H_{\alpha _{j}})S_{\alpha _{i}}, \end{aligned}$$

(16)

with $1\le i,j\le k$. Hence

$$\begin{aligned} d\mu _{\alpha _{i}}&= \sum _{j=1}^{k}{\sqrt{-1}\alpha _{i}(H_{\alpha _{j}})\xi _{\alpha _{j}}\wedge -\sqrt{-1}\nu _{\alpha _{i}}},\\ d\nu _{\alpha _{i}}&= -\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})\xi _{\alpha _{j}}\wedge \mu _{\alpha _{i}}}, \end{aligned}$$

and since $d\Psi _{i}=d(\mu _{\alpha _{i}}-\sqrt{-1}\nu _{\alpha _{i}})$ we have

$$\begin{aligned} d\Psi _{i}=\sqrt{-1}\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})\xi _{\alpha _{j}}}\wedge \Psi _{i}. \end{aligned}$$

(17)

Now $\mathrm{d}\varphi _{i}$ is obtained just by pull-back:

$$\begin{aligned} \mathrm{d}\varphi _{i}&= de^{*}\Psi _{i} \\&= e^{*}(d\Psi _{i}) \\&= e^{*}\left(\sqrt{-1}\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})\xi _{\alpha _{j}}}\wedge \Psi _{i}\right) \\&= \sqrt{-1}\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})\,e^{*}\xi _{\alpha _{j}}}\wedge e^{*}\Psi _{i} \\&= \sqrt{-1}\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})\tilde{\xi }_{\alpha _{j}}}\wedge \varphi _{i}. \end{aligned}$$

2. Connection form: By the expression obtained for $\mathrm{d}\varphi _{i}$, we get the connection $1$-forms as

$$\begin{aligned} \psi _{i}=-\sqrt{-1}\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})\tilde{\xi } _{\alpha _{j}}}. \end{aligned}$$

Using the structural equations, we compute $d\tilde{\xi }_{\alpha _{j}}$:

$$\begin{aligned} d\tilde{\xi }_{\alpha _{j}}=e^{*}(d\xi _{\alpha _{j}})=-2e^{*}(\mu _{\alpha _{j}}\wedge \nu _{\alpha _{j}})=-2e^{*}\mu _{\alpha _{j}}\wedge e^{*}\nu _{\alpha _{j}}=-2\tilde{\mu }_{\alpha _{j}}\wedge \tilde{\nu }_{\alpha _{j}}, \end{aligned}$$

because $[A_{\alpha },S_{\alpha }]=2\sqrt{-1}H_{\alpha }$.

In order to obtain a relationship between $d\tilde{\xi }_{\alpha _{j}}$ and the Kähler form $\Phi _{j}$ we compute

$$\begin{aligned} \varphi _{j}\wedge \bar{\varphi }_{j}=\tilde{\mu }_{\alpha _{j}}-\sqrt{-1} \tilde{\nu }_{\alpha _{j}}\wedge \tilde{\mu }_{\alpha _{j}}+\sqrt{-1}\tilde{\nu }_{\alpha _{j}}=2\tilde{\mu }_{\alpha _{j}}\wedge \sqrt{-1}\tilde{\nu } _{\alpha _{j}}, \end{aligned}$$

and therefore

$$\begin{aligned} d\tilde{\xi }_{\alpha _{j}}=-2\tilde{\mu }_{\alpha _{j}}\wedge \tilde{\nu } _{\alpha _{j}}=\left(\sqrt{-1}\right)^{2}2\tilde{\mu }_{\alpha _{j}}\wedge \tilde{\nu } _{\alpha _{j}}=\sqrt{-1}\,.\,2\tilde{\mu }_{\alpha _{j}}\wedge \sqrt{-1} \tilde{\nu }=\sqrt{-1}\varphi _{j}\wedge \bar{\varphi }_{j}. \end{aligned}$$

Now $\Phi _{j}=\frac{\sqrt{-1}}{2}\varphi _{j}\wedge \bar{\varphi }_{j}$. Then we have

$$\begin{aligned} d\tilde{\xi }_{\alpha _{j}}=2\Phi _{j}\qquad 1\le j\le k. \end{aligned}$$

(18)

Using (18) we can compute the curvature $2$-form:

$$\begin{aligned} \mathrm{d}\psi _{i}=-\sqrt{-1}\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})d\tilde{\xi } _{\alpha _{j}}}=-2\sqrt{-1}\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})\Phi _{j}}. \end{aligned}$$

(19)

3. Application of the generalized Gauss–Bonet theorem: This theorem asserts that

$$\begin{aligned} \int _{M}{K_{i}\cdot \Phi _{i}}=2\pi (\chi (M)+\sharp _{i}), \end{aligned}$$

(20)

where $\chi (M)=2-2g$ is the Euler characteristic of the Riemann surface with genus $g$ and $\sharp _{i}$ is the number of zeros of $\varphi _{i}$ counted with multiplicities.

Since $\mathrm{d}\psi _{i}=-\sqrt{-1}K_{i}\cdot \Phi _{i}$ thus

$$\begin{aligned} \int _{M}{\mathrm{d}\psi _{i}}=-\sqrt{-1}\int _{M}{K_{i}\cdot \Phi _{i}}=-2\pi \sqrt{-1} (\chi (M)+\sharp _{i})=2\pi \sqrt{-1}(2g-2-\sharp _{i}). \end{aligned}$$

(21)

On the other hand

$$\begin{aligned} \int _{M}{\mathrm{d}\psi _{i}}=-2\sqrt{-1}\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})\int _{M}{\Phi _{j}}}. \end{aligned}$$

(22)

Set $d_{i}=\frac{1}{\pi }\int _{M}{\Phi _{i}}$= area of $(M,\varphi _{i}\bar{ \varphi }_{i})$. Hence, Eqs. (21) and (22) give

$$\begin{aligned} 2g-2-\sharp _{i}=-\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})d_{j}} ,\,\,\,\,\,\,\,1\le i\le k, \end{aligned}$$

(23)

that are the desired Plücker formulae for the map $f$, concluding the proof of Theorem 3.3.

Now we describe a class of invariant almost complex structures $J$ having subsets $\mathcal P$ satisfying the condition of Theorem 3.3.

Definition 3.4

[20] Let $J$ be an invariant almost complex structure on $\mathbb{F }$. A root $\alpha$ is said to be $J$-decomposable (or simply decomposable) if there are roots $\beta ,\gamma$ such that $\alpha =\beta +\gamma$ with $\varepsilon _\alpha =\varepsilon _\beta =\varepsilon _\gamma$. The sum $\beta +\gamma$ is a $J$-decomposition of $\alpha$. A root is $J$ -indecomposable otherwise.

We denote by $\mathcal I$ the set of indecomposable roots. In principle, $\mathcal I$ can be empty. However, the presence of a $(1,2)$-symplectic metric $\Lambda$ allows a treatment of $\mathcal I$ analogous to the usual construction of a simple system of roots. In fact, set

$$\begin{aligned} \mathcal I ^{+}=\{\alpha \in \mathcal I :\varepsilon _{\alpha }=+1\}. \end{aligned}$$

(24)

Then, there is the following result of [20], which was central in the description of the $\left( 1,2\right)$-symplectic metrics on $\mathbb{F }$.

Theorem 3.5

[20] Let $(\mathbb{F },\Lambda ,J)$ be a $(1,2)$-symplectic structure on the full flag manifold $\mathbb{F }$. Then there exists a simple system of roots $\Sigma \subset \Pi$ such that either $\mathcal I ^{+}=\Sigma$ or

$$\begin{aligned} \mathcal I ^{+}=\Sigma \cup \{-\mu \} \end{aligned}$$

where $\mu$ is the highest root with respect to $\Sigma$. In the first case $J$ is integrable (complex structure) and $(\mathbb{F },\Lambda ,J)$ is K ähler.

In both cases stated in this theorem, the set $\mathcal P =\mathcal I ^{+}=\Sigma$ or $\Sigma \cup \{-\mu \}$ satisfies the condition of Theorem 3.3. In the integrable case ($\mathcal I ^{+}=\Sigma$), holomorphic maps subordinate to $\mathcal I ^{+}$ are called holomorphic-horizontal maps (following [4]).

Therefore, we have the following result that generalize the result due to Yang [23].

Corollary 3.6

Let $(\mathbb{F },\Lambda ,J)$ be a full flag manifold equipped with a $\left( 1,2\right)$-symplectic invariant structure. Let $f:M\rightarrow (\mathbb{F },\Lambda ,J)$ be a $J$-holomorphic map subordinated to $\mathcal I ^{+}$ and suppose that each $1$-form $\varphi _{i}$ has only isolated zeros of finite multiplicity. Then the Plücker formulae for $f$ are given by

$$\begin{aligned} 2g-2-\sharp _{i}=-\sum _{j=1}^{k}{\alpha _{i}(H_{\alpha _{j}})d_{j}} ,\,\,\,\,\,\,\,1\le i\le k, \end{aligned}$$

(25)

where $g$ is the genus of the Riemann surface $M$, $k$ is the number of simple roots of $\mathfrak{g }=\mathfrak u ^\mathbb{C }$, $d_{j}$ and $\sharp _{j}$ $(1\le j\le k)$ are the volume and the number of zeros of the singular metric on $M$ induced by the $1$-forms $\varphi _{i}$, respectively.

The condition about the zeros of the $1$-forms $\varphi _{i}$ is handled with the following definition.

Definition 3.7

[12, 23] Let $f:M\rightarrow (\mathbb{F },\Lambda ,J)$ be a map subordinate to $\mathcal P$ and let $\text{ rank} U=n$. We say $f$ is nondegenerate if $f(M)$ does not lie in any $U^{\prime }/(T\cap U^{\prime })$, where $U^{\prime }$ is a closed subgroup of $U$ with $\text{ rank} U^{\prime }<n$.

It can be proved (see [23]) that when $(\mathbb{F },J,\Lambda )$ is K ähler and $f:M\rightarrow (\mathbb{F },J,\Lambda )$ is a holomorphic map (in the usual sense), non-degenerated and subordinate to $\mathcal P = \mathcal I ^{+}=\Sigma$ then the $1$-forms $\varphi _{i}$ have only isolated zeros of finite multiplicity, and hence Corollary 3.6 applies to $f$.

We now compute the Plücker formulae in some examples. The Cartan matrix of the Lie algebra $\mathfrak{g }^\mathbb{C }$ is used in order to compute $\alpha _i(H_{\alpha _j})$. Example 3.8 was already studied in [23]. In the following examples, we consider $J$ to be a complex structure.

Example 3.8

[Flags of $D_{l}$-type] Consider the full flag manifold $\mathrm SO (12)/T$. The Cartan matrix of $\mathfrak so (12,\mathbb C )=D_{6}$ is

$$\begin{aligned} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 2&-1&0&0&0&0 \\ -1&2&-1&0&0&0 \\ 0&-1&2&-1&0&0 \\ 0&0&-1&2&-1&-1 \\ 0&0&0&-1&2&0 \\ 0&0&0&-1&0&2 \end{array} \right) . \end{aligned}$$

Then the Eq. (11) are given by

$$\begin{aligned} 2g-2-\sharp _{1}&= -2d_{1}+d_{2} \\ 2g-2-\sharp _{2}&= d_{1}-2d_{2}+d_{3} \\ 2g-2-\sharp _{3}&= d_{2}-2d_{3}+d_{4} \\ 2g-2-\sharp _{4}&= d_{3}-2d_{4}+d_{5}+d_{6} \\ 2g-2-\sharp _{5}&= d_{4}-2d_{5} \\ 2g-2-\sharp _{6}&= d_{4}-2d_{6}. \end{aligned}$$

Example 3.9

(Flags of $A_l$-type, [12]) Consider the full flag manifold $SU(4)/T$. The Cartan matrix of $\mathfrak sl (4,\mathbb C )$ is

$$\begin{aligned} \left( \begin{array}{ccc} 2&-1&0 \\ -1&2&-1 \\ 0&-1&2 \end{array} \right), \end{aligned}$$

and therefore the Eq. (11) are given by

$$\begin{aligned} 2g-2-\sharp _1&= -2d_1+d_2 \\ 2g-2-\sharp _2&= d_1-2d_2+d_3 \\ 2g-2-\sharp _3&= d_2-2d_3 . \end{aligned}$$

Example 3.10

[Case $G_2$] Consider the full flag manifold $G_2/T$. The Cartan matrix of $\mathfrak{g } _2$ is

$$\begin{aligned} \left( \begin{array}{cc} 2&-1 \\ -3&2 \end{array} \right), \end{aligned}$$

hence, the Eq. (11) are given by

$$\begin{aligned} 2g-2-\sharp _1&= -2d_1+d_2 \\ 2g-2-\sharp _2&= 3d_1-2d_2. \end{aligned}$$

4 Equi-harmonic maps and generalized horizontal: holomorphic maps on ${\mathbb{F }_\Theta }$

Let $(M,g)$ and $(N,h)$ be Riemannian manifolds. A smooth map $\phi :M\rightarrow N$ is said to be a harmonic if it is a critical point of the energy functional. We refer to [9, 15] for details about the theory of harmonic maps.

As before $M=M^{2}$ denotes a Riemann surface and $N={\mathbb{F }_{\Theta }}$ is a generalized flag manifold. We remark that the harmonicity of a smooth map $\phi$ depends on the metric on the manifold $N$. In this paper, we study a special class of harmonic maps, the so-called equi-harmonic maps.

Definition 4.1

A equi-harmonic map is a map $\phi :M^{2}\rightarrow \mathbb{F }_{\Theta }$ which is harmonic for each invariant metric $\mathrm{d}s_{\Lambda }^{2}$ on $\mathbb F _{\Theta }$.

Equi-harmonic maps were studied in [17] for flag manifolds of $\mathrm SU (n)$ type in [2] for general flag manifolds. One of the tools of [2] are the Yano $f$-structures, whose basic facts are recalled in the sequel.

Let $\mathcal{F }$ be an invariant $f$-structure on ${\mathbb{F }_\Theta }$.

Definition 4.2

A map $\phi :\left( M^{2},J\right) \rightarrow \left( \mathbb{F }{_{\Theta }}, \mathcal{F }\right)$ is said to be subordinate to $\mathcal{F }$ if for $\sigma \in \Pi ({\Theta })$ and $\mathcal{F }_{\sigma }\ne 1$ we have $\phi _{\sigma }\equiv 0$.

We select a special class of $f$-structures which will be used later.

Definition 4.3

Consider an invariant $f$-structure $\mathcal{F }$ in $\mathbb{F _{\Theta }}$. Let

$$\begin{aligned} \mathcal{F }_{+}:=\sum _{\begin{array}{c} \alpha \in \Pi ({\Theta }) \\ \mathcal{F } _{\alpha }=1 \end{array}}\mathfrak{g }_{\alpha } \end{aligned}$$

and

$$\begin{aligned} \mathcal{F }_{-}:=\sum _{\begin{array}{c} \alpha \in \Pi ({\Theta }) \\ \mathcal{F } _{\alpha }=-1 \end{array}}\mathfrak{g }_{\alpha }. \end{aligned}$$

We say that $\mathcal{F }$ is horizontal if $[\mathcal{F }_{+},\mathcal{F } _{-}]\subset \mathfrak{k }_{\Theta }^\mathbb{C }$.

The following theorem proved by Black [2] provides a necessary condition for a smooth map to be equi-harmonic.

Theorem 4.4

Let $\phi :(M^{2},J,g)\rightarrow ({\mathbb{F }_{\Theta }}, \mathcal{F }, \mathrm{d}s_{{\Lambda }}^{2})$ be subordinate to a horizontal $f$-structure $\mathcal{F }$. Then $\phi$ is equi-harmonic.

Now we recall another class of holomorphic maps on flag manifolds. They appear as twistor lifts of isotropic harmonic maps on Riemannian symmetric spaces, see for example [4] and [5].

Definition 4.5

A generalized holomorphic-horizontal map is a map $\phi :M^{2}\rightarrow ( \mathbb{F }_{\theta },J)$ which is $J$-holomorphic and satisfies $\phi _{\sigma }=0$ if $\sigma \in \Pi (\Theta )\setminus \Sigma (\Theta )$. Here, as before, $\phi _{\sigma }$ denotes the $E_{\sigma }$-component of the derivative of $\phi$.

Remark

The generalized holomorphic-horizontal maps are also called super-horizontal maps, see [5]. Here we follow the notation of Bryant in [4].

Using the techniques introduced by Black, we prove that these maps are in fact equiharmonic. This is the main result of this section and provides a large class of examples of equiharmonic maps.

Theorem 4.6

If $\phi :M^{2}\rightarrow \mathbb{F }_{\Theta }$ is a generalized holomorphic-horizontal map then $\phi$ is equiharmonic.

Proof

First we need to describe the irreducible components of $\mathfrak{q }_{\Theta }$ associated with roots of $\mathfrak{g }^\mathbb{C }$ which have height one modulo $\Theta$.

If $\alpha \in (\Pi \setminus \left\langle \Theta \right\rangle )\cap \Sigma$ (simple roots complementary to $\langle \Theta \rangle$), we define the set

$$\begin{aligned} \Pi (\Theta )_{\alpha }^{+}:=\{\beta \in (\Pi \setminus \left\langle \Theta \right\rangle )\cap \Pi ^{+}:\beta =\alpha +\left\langle \Theta \right\rangle ^{+}\}. \end{aligned}$$

(26)

On the other hand, if $\alpha$ is the negative of a simple root (that is, $-\alpha \in (\Pi \setminus \left\langle \Theta \right\rangle )\cap \Sigma$) then we put

$$\begin{aligned} \Pi (\Theta )_{\alpha }^{-}:=\{\beta \in (\Pi \setminus \left\langle \Theta \right\rangle )\cap \Pi ^{-}:\beta =\alpha +\left\langle \Theta \right\rangle ^{-}\}. \end{aligned}$$

(27)

The sets $\Pi (\Theta )_{\alpha }^{+}$ and $\Pi (\Theta )_{\alpha }^{-}$ contain the roots with height one modulo $\left\langle \Theta \right\rangle$ . Now consider, for each $\alpha \in (\Pi \setminus \left\langle \Theta \right\rangle )\cap \Sigma$,

$$\begin{aligned} \mathfrak{m }_{\alpha }=\sum _{\beta \in \Pi (\Theta )_{\alpha }^{+}}{ \mathfrak{g }_{\beta }}, \end{aligned}$$

(28)

and

$$\begin{aligned} \mathfrak{m }_{-\alpha }=\sum _{\beta \in \Pi (\Theta )_{\alpha }^{-}}{ \mathfrak{g }_{\beta }}. \end{aligned}$$

(29)

It is well known that both $\mathfrak{m }_{\alpha }$ and $\mathfrak{m } _{-\alpha }$ are irreducible components of the (complexified) isotropic representation. These irreducible components satisfy

$$\begin{aligned}{}[\mathfrak{m }_{\alpha },\mathfrak{m }_{-\beta }]\left\{ \begin{array}{ll} =0,&\text{ if} \alpha \ne \beta , \\ \subset \mathfrak{k }_{\Theta }^\mathbb{C },&\text{ if} \alpha =\beta \end{array} \right. \end{aligned}$$

(30)

where $\alpha ,\beta \in (\Pi \setminus \left\langle \Theta \right\rangle )\cap \Sigma$.

Now given a generalized horizontal-holomorphic map $\phi :M\rightarrow ( \mathbb{F }_{\Theta },J)$, we can associate an $f$-structure $\mathcal{F } ^{\phi }$, called associated $f$-structure, to the holomorphic-horizontal map $\phi$ in the following way:

$$\begin{aligned} \mathcal{F }^{\phi }=\{\varepsilon _{\alpha }\}_{\alpha \in \Pi }=\left\{ \begin{array}{ll} +1,&\text{ if} \alpha \in \Pi (\Theta )_{\beta }^{+} \text{ for} \text{ some} \beta \in (\Pi \setminus \left\langle \Theta \right\rangle )\cap \Sigma ; \\ -1,&\text{ if} \alpha \in \Pi (\Theta )_{\beta }^{-} \text{ for} \text{ some} -\beta \in (\Pi \setminus \left\langle \Theta \right\rangle )\cap \Sigma ; \\ 0,&\text{ otherwise}. \end{array} \right. \end{aligned}$$

Using the linearity of the Lie bracket and Eq. (30), we have $[\mathcal{F }_{+}^{\phi },\mathcal{F }_{-}^{\phi }]\subset \mathfrak{k } _{\Theta }$ and the $f$-structure is horizontal. On the other hand, if $\varepsilon _{\alpha }\ne 1$ then $\alpha$ is not root of height one modulo $\Theta$ and $\phi _{\alpha }=0$ because $\phi$ is horizontal. Hence, $\phi$ is subordinate to $\mathcal{F }^{\phi }$ and according Theorem 4.4, $\phi$ is an equi-harmonic map.$\square$