Calabi-Yau structure and special Lagrangian submanifold of the complexified symmeric space

TL;DR

This paper explores Calabi-Yau structures on complexified symmetric spaces, detailing their description via Schwarz's theorem and constructing invariant special Lagrangian submanifolds using the Stenzel metric.

Contribution

It provides a detailed description of Calabi-Yau structures on complexified symmetric spaces and constructs invariant special Lagrangian submanifolds within this framework.

Findings

Calabi-Yau structures are described in terms of Schwarz's theorem.

Constructed invariant special Lagrangian submanifolds of any phase.

Applied the Stenzel metric to these structures.

Abstract

It is known that there exist Calabi-Yau structures on the complexifications of symmetric spaces of compact type. In this paper, we describe the Calabi-Yau structures of the complexified symmetric spaces in terms of the Schwarz's theorem in detail. We consider the case where the Calabi-Yau structure arises from the Riemannian metric corresponding to the Stenzel metric. In the complexified symmetric spaces equipped with such a Calabi-Yau structure, we give constructions of special Lagrangian submanifolds of any phase which are invariant under the actions of symmetric subgroups of the isometry group of the original symmetric space of compact type.

Equations

$${\omega }^n=(-1{)}^{n(n-1)/2}(\sqrt{-1}{)}^{n}c(\Omega \wedge \overline{\Omega })$$

$$\Psi (v):={\mathrm{Exp}}_p((J_0{)}_p(v))(p\in G/K,v\in T_p(G/K))$$

$$\begin{array}[]{r}\displaystyle{\iota(q):=\cosh(\|\Psi^{-1}(q)\|)\cdot\overrightarrow{Op}+\sqrt{-1}\cdot\frac{\sinh(\|\Psi^{-1}(q)\|)}{\|\Psi^{-1}(q)\|}\cdot\Psi^{-1}(q)}\\ \displaystyle{(q\in SO(n+1,\mathbb{C})/SO(\mathbb{C})),}\end{array}$$

$$\iota ({\Psi }_p(v))=\cosh (\Vert v\Vert )\cdot \overrightarrow{Op}+\sqrt{-1}\cdot \frac{\sinh (\Vert v\Vert )}{\Vert v\Vert }\cdot v(v\in T_p{S}^n(1)).$$

$${{\left(\frac{\det \nabla d\bar{\psi }}{\det {\beta }_{G^d/K}}\right)}^h|}_{G^d/K}=\frac{\det \nabla d\bar{\psi }}{\det {\beta }_{G^d/K}}.$$

$${{\left(\frac{\det {\nabla }^{0}d\overline{\stackrel{ˉ}{\psi }}}{\det {\beta }_0}\right)}^h|}_{{\mathrm{Exp}}_o(\sqrt{-1}\mathfrak{a})}\circ {\mathrm{Exp}}_o=\frac{\det {\nabla }^{0}d\overline{\stackrel{ˉ}{\psi }}}{\det {\beta }_0}.$$

$${\left(\frac{\det \nabla d\bar{\psi }}{\det {\beta }_{G^d/K}}\right)}^h={\left(\frac{\det {\nabla }^{0}d\overline{\stackrel{ˉ}{\psi }}}{\det {\beta }_0}\right)}^h.$$

$$C_W^{\infty }(\sqrt{-1}\mathfrak{a})=\{f\circ ({\rho }_1,\cdots ,{\rho }_l)|f\in C^{\infty }({\mathbb{R}}^l)\},$$

$${\rho }_1(\sqrt{-1}v):=\Vert v{\Vert }^2+1(v\in \mathfrak{a}).$$

$$({\nabla }^{0}d{\overline{\stackrel{ˉ}{\psi }}}_f)\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_j}\right)=\sum _{k=1}^l\sum _{\hat{k}=1}^l$$

$$+\left(\frac{\partial f}{\partial y_k}\circ \overrightarrow{\rho }\right)\cdot \frac{{\partial }^2{\rho }_k}{\partial x_i\partial x_j}).$$

$$\det \left(({\nabla }^{0}d{\overline{\stackrel{ˉ}{\psi }}}_f)\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_j}\right)\right)=c>0,$$

$$\det \left({\beta }_0\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_j}\right)\right)=1.$$

$$\frac{\det {\nabla }^{0}d{\overline{\stackrel{ˉ}{\psi }}}_f}{\det {\beta }_0}=c.$$

$$\det \left(2x_i{x}_j\cdot \left(\frac{{\partial }^{2}f}{\partial y_1^2}\circ \overrightarrow{\rho }\right)+\left(\frac{\partial f}{\partial y_1}\circ \overrightarrow{\rho }\right)\cdot {\delta }_{ij}\right)$$

$$=$$

$$X_p^{\ast }:={\frac{d}{dt}|}_{t=0}(\exp tX)\cdot p(p\in M)$$

$$(\mu (p))(X):=F_X(p)(p\in M,X\in \mathfrak{h}).$$

$$d\alpha (X^{\ast },Y)=X^{\ast }(\alpha (Y))-Y(\alpha (X^{\ast }))-\alpha ({\mathcal{L}}_{X^{\ast }}Y)=({\mathcal{L}}_{X^{\ast }}\alpha )(Y)-d{F}_X(Y),$$

Full text

1. Introduction

An $2n$-dimensional Riemannian manifold is called a Calabi-Yau manifold if the holonomy group is a subgroup of $SU(n)$. A Kaehler manifold is Calabi-Yau if and only if it is Ricci-flat. Let $(M,J,\omega)$ be a complex $n$-dimensional Kaehler manifold, where $J$ is the complex structure and $\omega$ is the Kaehler form. Also, let $g$ be the Kaehler metric associated to $(J,\omega)$. If there exists a non-vanishing holomorphic $(n,0)$-form $\Omega$ on $M$ (i.e., the holomorphic complex line bundle $\bigwedge_{h}^{(n,0)}(M)$ is trivial), then $(M,J,\omega)$ is called a almost Calabi-Yau manifold. In particular, if $(\omega,\Omega)$ satisfies

[TABLE]

for some positive real constant $c$, then $(M,J,\omega)$ is Ricci-flat and hence it is Calabi-Yau. By replacing $\Omega$ to a suitable positive real constant-multiple of $\Omega$ if necessary, we may assume that $c=\frac{n!}{2^{n}}$. In the sequel, the Calabi-Yau manifold (resp. the Calabi-Yau structure) means a quadruple $(M,J,\omega,\Omega)$ (resp. a triple $(J,\omega,\Omega)$) such that $(J,\omega)$ is a Kaehler structure and that $(\omega,\Omega)$ satisfies

[TABLE]

Let $(J,\omega,\Omega)$ be a Calabi-Yau structure on $M$ and $g$ the Kaehler metric associated to $(J,\omega)$. Then, for any real constant $\theta$, a $n$-form ${\rm Re}(e^{\sqrt{-1}\theta}\Omega)$ is a calibration on $(M,g)$. A submanifold calibrated by ${\rm Re}(e^{\sqrt{-1}\theta}\Omega)$ is called a special Lagrangian submanifold of phase $\theta$. According to Strominger-Yau-Zaslov’s conjecture ([SYZ]) for the Mirror symmetry in the string theory, it is important to construct special Lagrangian submanifolds in a Calabi-Yau manifold.

Let $M$ be $C^{\omega}$-Riemannian manifold and $M^{\mathbb{C}}$ its complexification. In 1991, V. Gillemin and M. Stenzel ([GS]) gave a construction of Ricci-flat metrics on a sufficiently small tubular neighborhood of $M$ in $M^{\mathbb{C}}$. Let $G/K$ be a (Reimannian) symmetric space of compact type. The complexification $(G/K)^{\mathbb{C}}$ of $G/K$ is defined as the complexified symmetric space $G^{\mathbb{C}}/K^{\mathbb{C}}$ equipped with the $G^{\mathbb{C}}$-invariant anti-Kaehler metric $\beta_{A}$. The anti-Kaheler manifold $(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$ is called an anti-Kaehler symmetric space. This space $(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$ is identified with the tangent bundle $T(G/K)$ of $G/K$ under the one-to-one correspondence $\displaystyle{\Psi:T(G/K)\mathop{\longrightarrow}_{\cong}G^{\mathbb{C}}/K^{\mathbb{C}}}$ defined by

[TABLE]

(see Figure 1), where ${\rm Exp}_{p}$ denotes the exponential map of $(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$ at $p$, $J_{0}$ denotes the natural complex structure of $G^{\mathbb{C}}/K^{\mathbb{C}}$ and $v$ is regarded as a tangent vector of the submanifold $G\cdot o(\approx G/K)$ ($o=eK^{\mathbb{C}}$) in $G^{\mathbb{C}}/K^{\mathbb{C}}$. For each $p\in G/K(\approx G\cdot o)$ set $\Psi_{p}:=\Psi|_{T_{p}(G/K)}(={\rm Exp}_{p}\circ(J_{0})_{p})$ and $(G/K)^{d}_{p}:=\Psi(T_{p}(G/K))$. Note that $(G/K)^{d}_{p}$’s equipped with the (Riemannian) metric induced from $\beta_{A}$ are isometric to the symmetric space $G^{d}/K$ of non-compact type given as the dual of $G/K$ and they are totally geodesic submanifolds in $(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$.

o$$p$$\Psi(T_{o}(G\cdot o))$$G\cdot o=G/K$$\Psi(T_{p}(G\cdot o))$$(J_{0})_{p}(v)$$v$$G^{\mathbb{C}}/K^{\mathbb{C}}$$\Psi(v)={\rm Exp}_{p}((J_{0})_{p}(v))

Figure 1.

We consider the case where $G/K$ is the sphere $SO(n+1)/SO(n)(=S^{n}(1))$. Then the complexification $SO(n+1,\mathbb{C})/SO(n,\mathbb{C})$ of $SO(n+1)/SO(n)$ is embedded into $\mathbb{C}^{n+1}$ as the complex sphere $S^{n}_{\mathbb{C}}(1):=\{(z_{1},\cdots,z_{n+1})\,|\,\sum\limits_{i=1}^{n+1}z_{i}^{2}=1\}$ of complex radius $1$. The natural embedding $\iota$ of $SO(n+1,\mathbb{C})/SO(n,\mathbb{C})$ into $\mathbb{C}^{n+1}$ is given by

[TABLE]

where $p$ is the base point of $\Psi^{-1}(q)$, $O$ is the origin of of the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$ including $S^{n}(1)(=SO(n+1)/SO(n))$, and $\overrightarrow{Op}$ and $\Psi^{-1}(q)$ are regarded as vectors of $\mathbb{R}^{n+1}$ (see Figure 2). Hence we have

[TABLE]

O$$p$$S^{n}(1)$$\sqrt{-1}\Psi^{-1}(q)$$\Psi^{-1}(q)$$S^{n}_{\mathbb{C}}(1)$$\displaystyle{\iota(q)=\cosh(\|\Psi^{-1}(q)\|)\cdot\overrightarrow{Op}+\sqrt{-1}\cdot\frac{\sinh(\{\Psi^{-1}(q)\|}{\|\Psi^{-1}(q)\|}\cdot\Psi^{-1}(q)}$$\overrightarrow{Op}$$\mathbb{C}^{n+1}$$\mathbb{R}^{n+1}$$\sqrt{-1}\mathbb{R}^{n+1}

Figure 2.

In 1993, M.B. Stenzel ([St]) gave a construction of complete Ricci-flat metrics on the cotangent bundle $T^{\ast}(G/K)$ of $G/K$ in the case where the rank of $G/K$ is equal to one, where we note that $T^{\ast}(G/K)$ is identified with $T(G/K)(\approx G^{\mathbb{C}}/K^{\mathbb{C}})$ by the metric of $G/K$. In 2004, R. Bielawski ([B2]) gave a construction of complete Ricci-flat metrics on $G^{\mathbb{C}}/K^{\mathbb{C}}$ in the case where the rank of $G/K$ is general. These complete Ricci metrics give Calabi-Yau structures on $G^{\mathbb{C}}/K^{\mathbb{C}}$ together with the natural complex structure $J_{0}$ and the natural non-vanishing closed holomorphic $(n,0)$-form $\Omega_{0}$ on $G^{\mathbb{C}}/K^{\mathbb{C}}$. H. Anciaux ([An]) constructed special Lagrangian submanifolds of some phase in the complexification $SO(n+1,{\mathbb{C}})/SO(n,{\mathbb{C}})$ of the $n$-dimensional sphere $SO(n+1)/SO(n)$ which are invariant under the natural action $SO(n)\curvearrowright SO(n+1,{\mathbb{C}})/SO(n,{\mathbb{C}})$. M. Ionel and M. Min-Oo ([IO]) constructed cohomogeneity one special Lagrangian submanifolds of some phase in $SO(4,{\mathbb{C}})/SO(3,{\mathbb{C}})$ which are invariant under the natural action $SO(2)\times SO(2)\curvearrowright SO(4,{\mathbb{C}})/SO(3,{\mathbb{C}})$. K. Hashimoto and T. Sakai ([HS]) constructed cohomogeneity one special Lagrangian submanifolds of any phase in $SO(n+1,{\mathbb{C}})/SO(n,{\mathbb{C}})$ which are invariant under the natural action $SO(p)\times SO(n+1-p)\curvearrowright SO(n+1,{\mathbb{C}})/SO(n,{\mathbb{C}})$ ($1\leq p\leq[(n+1)/2]$). Later, K. Hashimoto and K. Mashimo ([HM]) constructed cohomogeneity one special Lagrangian submanifolds of any phase in $SO(n+1,{\mathbb{C}})/SO(n,{\mathbb{C}})$ which are invariant under the natural action $K\curvearrowright SO(n+1,{\mathbb{C}})/SO(n,{\mathbb{C}})$ induced from the linear isotropy action $K\curvearrowright SO(n+1)/SO(n)(=S^{n}(1)\subset T_{eK}(G/K))$ of any irreducible rank two symmetric space $G/K$, where $n:={\rm dim}\,G/K-1$. Recently M. Arai and K. Baba ([AB]) constructed cohomogeneity one special Lagarangian submanifolds of any phase and in the complexification $SL(n+1,\mathbb{C})/(SL(1,\mathbb{C})\times SL(n,\mathbb{C}))=T(\mathbb{C}P^{n})$ of the complex projective space $\mathbb{C}P^{n}=SU(n+1)/S(U(1)\times U(n))$.

In this paper, we first construct an almost Calabi-Yau structure $(J_{0},\omega_{\psi_{f}},$

$\Omega_{0})$ on the complexification $G^{\mathbb{C}}/K^{\mathbb{C}}$, which is invariant under the natural action $G\curvearrowright G^{\mathbb{C}}/K^{\mathbb{C}}$, in terms of a $C^{\infty}$-function $f$ over $\mathbb{R}^{l}$ ($l:$ a natural number) and investigate in what case it is a Calabi-Yau structure, where $J_{0}$ and $\Omega_{0}$ are the natural complex structure and the natural non-vanishing closed holomorphic $(n,0)$-form on $G^{\mathbb{C}}/K^{\mathbb{C}}$ (Section 2). In Section 3, we investigate the [math]-level set of the moment map of a Hamiltonian action on the Calabi-Yau manifold $(G^{\mathbb{C}}/K^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$. Let $H$ be a symmetric subgroup of $G$. The natural action $H\curvearrowright G/K$ (which is called a Hermann action) is extended to the action on $G^{\mathbb{C}}/K^{\mathbb{C}}$ naturally. This extended action $H\curvearrowright G^{\mathbb{C}}/K^{\mathbb{C}}$ is a Hamiltonian action. In section 4, we investigate the orbit structure of this Hamiltonian action $H\curvearrowright G^{\mathbb{C}}/K^{\mathbb{C}}$. In Section 5, in the case where $\beta_{\psi_{f}}(\cdot,\cdot):=\omega_{\psi_{f}}(J_{0}(\cdot),\cdot)$ is the metric generalized the Stenzel metric, we first give a construction of an $H$-invariant special Lagrangian submanifold of cohomogeneity $r$ in $(G^{\mathbb{C}}/K^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$, where $r$ denotes the cohomogeneity of $H\curvearrowright G/K$ (see Theorem 5.4 and Corollary 5.5).

2. Calabi-Yau structures on complexified symmetric spaces

Let $G$ be a compact semi-simple Lie group and $\theta$ an involutive automorphism of $G$. Let $K$ be a closed subgroup of $G$ with $({\rm Fix}\,\theta)_{0}\subset K\subset{\rm Fix}\,\theta$, where ${\rm Fix}\,\theta$ is the fixed point group of $\theta$ and $({\rm Fix}\,\theta)_{0}$ is the identity component of ${\rm Fix}\,\theta$. Denote by $\mathfrak{g}$ (resp. $\mathfrak{k}$) the Lie algebra of $G$ (resp. $K$) and $B$ the Killing form of $\mathfrak{g}$. Denote by the same symbol $\theta$ the involution of $\mathfrak{g}$ induced from $\theta$. Set $\mathfrak{p}:={\rm Ker}(\theta+{\rm id}_{\mathfrak{g}})$, which is identified with the tangent space $T_{o}(G/K)$ of $G/K$ at $o:=eK$ ($e\,:\,$ the identity element of $G$), where ${\rm id}_{\mathfrak{g}}$ is the identity transformation of $\mathfrak{g}$. Since $B|_{\mathfrak{p}\times\mathfrak{p}}$ is the ${\rm Ad}_{G}(K)|_{\mathfrak{p}}$-invariant, we obtain a $G$-invariant metric $\beta$ on $G/K$ with $\beta_{eK}=B$, where ${\rm Ad}_{G}$ is adjoint representation of $G$. This Riemannian manifold $(G/K,\beta)$ is called a (Riemannian) symmetric space of compact type. The dimension of maximal flat totally geodesic submanifold in $G/K$ is called the rank of $G/K$. Denote by $\widetilde{r}$ the rank of $G/K$. Also, assume that $G$ and $K$ admit faithful real representations. Hence the complexifications $G^{\mathbb{C}}$ and $K^{\mathbb{C}}$ of $G$ and $K$ are defined. For the complexification $B^{\mathbb{C}}(:\mathfrak{p}^{\mathbb{C}}\times\mathfrak{p}^{\mathbb{C}}\to{\mathbb{C}}$) of $B$, its real part ${\rm Re}\,B^{\mathbb{C}}$ is a ${\rm Ad}_{G^{\mathbb{C}}}(K^{\mathbb{C}})|_{\mathfrak{p}^{\mathbb{C}}}$-invariant non-degenerate bilinear form (of half index) of $\mathfrak{p}^{\mathbb{C}}(=T_{eK^{\mathbb{C}}}(G^{\mathbb{C}}/K^{\mathbb{C}}))$ and hence we obtain a $G^{\mathbb{C}}$-invariant neutral metric $\beta_{A}$ on $G^{\mathbb{C}}/K^{\mathbb{C}}$ with $(\beta_{A})_{eK}={\rm Re}\,B^{\mathbb{C}}$, where ${\rm Ad}_{G^{\mathbb{C}}}$ is adjoint representation of $G^{\mathbb{C}}$. This pseudo-Riemannian manifold $(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$ is called an anti-Kaehler symmetric space, which is one of semi-simple pseudo-Riemannian symmetric spaces. Note that the terminology “anti-Kaehler” is used in [BFV] and [Koi3, Koi4] for example. Define $j:\mathfrak{p}^{\mathbb{C}}\to\mathfrak{p}^{\mathbb{C}}$ by $j(v):=\sqrt{-1}v$ ($v\in\mathfrak{p}^{\mathbb{C}}$). Since $j$ is the ${\rm Ad}_{G^{\mathbb{C}}}(K^{\mathbb{C}})|_{\mathfrak{p}^{\mathbb{C}}}$-invariant, we obtain a $G^{\mathbb{C}}$-invariant almost complex structure $J_{0}$ of $G^{\mathbb{C}}/K^{\mathbb{C}}$ with $(J_{0})_{eK^{\mathbb{C}}}=j$. Take an orthonormal base $(e_{1},\cdots,e_{n})$ of $\mathfrak{p}$ with respect to $B$ and let $(\theta^{1},\cdots,\theta^{n})$ be the dual base of $(e_{1},\cdots,e_{n})$. Set $(\theta^{1})^{\mathbb{C}}\wedge\cdots\wedge(\theta^{n})^{\mathbb{C}}$. Since $(\theta^{1})^{\mathbb{C}}\wedge\cdots\wedge(\theta^{n})^{\mathbb{C}}$ is ${\rm Ad}_{G^{\mathbb{C}}}(K^{\mathbb{C}})|_{\mathfrak{p}^{\mathbb{C}}}$-invariant, we obtain a $G^{\mathbb{C}}$-invariant holomorphic $(n,0)$-form $\Omega_{0}$ on $G^{\mathbb{C}}/K^{\mathbb{C}}$ with $(\Omega_{0})_{eK^{\mathbb{C}}}=(\theta^{1})^{\mathbb{C}}\wedge\cdots\wedge(\theta^{n})^{\mathbb{C}}$. Let $\psi$ be a strictly plurisubharmonic function over $G^{\mathbb{C}}/K^{\mathbb{C}}$, where we note that “strictly plurisubharmonicity” means that the Hermitian matrix $\displaystyle{\left(\frac{\partial^{2}\psi}{\partial z_{i}\partial\bar{z}_{j}}\right)}$ is positive (or equivalently, $(\sqrt{-1}\partial\overline{\partial}\psi)(Z,\overline{Z})>0$ holds for any nonzero $(1,0)$-vector $Z$). Then $\omega_{\psi}:=\sqrt{-1}\partial\overline{\partial}\psi$ is a real non-degenerate closed $2$-form on $G^{\mathbb{C}}/K^{\mathbb{C}}$ and the symmetric $(0,2)$-tensor field $\beta_{\psi}$ associated with $J_{0}$ and $\omega_{\psi}$ is positive definite. Hence $(J_{0},\omega_{\psi},\Omega_{0})$ is an almost Calabi-Yau structure on $G^{\mathbb{C}}/K^{\mathbb{C}}$. Thus, from each strictly plurisubharmonic function over $G^{\mathbb{C}}/K^{\mathbb{C}}$, we obtain an almost Calabi-Yau structure on $G^{\mathbb{C}}/K^{\mathbb{C}}$. Hence we suffice to construct a strictly plurisubharmonic function on $G^{\mathbb{C}}/K^{\mathbb{C}}$ to construct an almost Calabi-Yau structure on $G^{\mathbb{C}}/K^{\mathbb{C}}$. Denote by ${\rm Exp}_{p}$ the exponential map of the anti-Kaehler manifold $(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$ at $p(\in G^{\mathbb{C}}/K^{\mathbb{C}})$ and $\exp$ the exponentional map of the Lie group $G^{\mathbb{C}}$. Set $\mathfrak{g}^{d}:=\mathfrak{k}\oplus\sqrt{-1}\mathfrak{p}(\subset\mathfrak{g}^{\mathbb{C}})$ and $G^{d}=\exp(\mathfrak{g}^{d})$. Denote by $\beta_{G/K}$ the $G$-invariant (Riemannian) metric on $G/K$ induced from $B|_{\mathfrak{p}\times\mathfrak{p}}$ and $\beta_{G^{d}/K}$ the $G^{d}$-invariant (Riemannian) metric on $G^{d}/K$ induced from $-({\rm Re}\,B^{\mathbb{C}})|_{\sqrt{-1}\mathfrak{p}\times\sqrt{-1}\mathfrak{p}}$. We may assume that the metric of $G/K$ is equal to $\beta_{G/K}$ by homothetically transforming the metric of $G/K$ if necessary. On the other hand, the Riemannian manifold $(G^{d}/K,\beta_{G^{d}/K})$ is a (Riemannian) symmetric space of non-compact type. The orbit $G\cdot o$ is isometric to $(G/K,\beta_{G/K})$ and the normal umbrella ${\rm Exp}_{o}(T^{\perp}_{o}(G\cdot o))(=G^{d}\cdot o)$ is isometric to $(G^{d}/K,\beta_{G^{d}/K})$. The complexification $\mathfrak{p}^{\mathbb{C}}$ of $\mathfrak{p}$ is identified with $T_{o}(G^{\mathbb{C}}/K^{\mathbb{C}})$ and $\sqrt{-1}\mathfrak{p}$ is identified with $T_{o}({\rm Exp}_{o}(T^{\perp}_{o}(G\cdot o)))$. Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$, where we note that ${\rm dim}\,\mathfrak{a}=\widetilde{r}$. Denote by $W$ the Weyl group of $G^{d}/K$ with respect to $\sqrt{-1}\mathfrak{a}$. This group acts on $\sqrt{-1}\mathfrak{a}$. Let $C(\subset\sqrt{-1}\mathfrak{a})$ be a Weyl domain (i.e., a fundamental domain of the action $W\curvearrowright\sqrt{-1}\mathfrak{a}$). Then we have $G\cdot{\rm Exp}_{o}(\overline{C})=G^{\mathbb{C}}/K^{\mathbb{C}}$, where $\overline{C}$ is the closure of $C$. For a connected open neighborhood $D$ of [math] in $\sqrt{-1}\mathfrak{a}$, we define a neighborhood $U_{1}(D)$ of $o$ in ${\rm Exp}_{o}(\sqrt{-1}\mathfrak{a})$ by $U_{1}(D):={\rm Exp}_{o}(D)$, a neighborhood $U_{2}(D)$ of $o$ in $G^{d}/K$ by $U_{2}(D):=K\cdot U_{1}(D)$ and a tubular neighborhood $U_{3}(D)$ of $G\cdot o$ in $G^{\mathbb{C}}/K^{\mathbb{C}}$ by $U_{3}(D):=G\cdot U_{1}(D)$ and (see Figure 3). Denote by ${\rm Conv}_{W}^{+}(D)$ the space of all $W$-invariant strictly convex ($C^{\infty}$-)functions over $D$, ${\rm Conv}_{K}^{+}(U_{2}(D))$ the space of all $K$-invariant strictly convex ($C^{\infty}$-)functions

G^{d}\cdot o$$U_{2}(D)$${\rm Exp}_{o}(\sqrt{-1}\mathfrak{a})$$U_{1}(D)$$U_{3}(D)$$G\cdot o$$G^{\mathbb{C}}/K^{\mathbb{C}}

Figure 3.

over $U_{2}(D)$ and $PH_{G}^{+}(U_{3}(D))$ the space of all $G$-invariant strictly plurisubharmonic ($C^{\infty}$-)functions over $U_{3}(D)$. The restriction map from $U_{3}(D)$ to $U_{2}(D)$ gives an isomorphism of $PH_{G}^{+}(U_{3}(D))$ onto ${\rm Conv}_{K}^{+}(U_{2}(D))$ and the composition of the restriction map from $U_{3}(D)$ to $U_{1}(D)$ with ${\rm Exp}_{o}$ gives an isomorphism of $PH_{G}^{+}(U_{3}(D))$ onto ${\rm Conv}_{W}^{+}(D)$ (see [AL]). Hence we suffice to construct $W$-invariant strictly convex functions over $D$ or $K$-invariant strictly convex functions over $U_{2}(D)$ to construct strictly plurisubharmonic functions over $U_{3}(D)$. Let $\psi$ be a $G$-invariant strictly plurisubharmonic ($C^{\infty}$-)functions over $U_{3}(D)$. Denote by $\bar{\psi}$ the restriction of $\psi$ to $U_{2}(D)$ and $\bar{\bar{\psi}}$ the composition of the restriction of $\psi$ to $U_{1}(D)$ with ${\rm Exp}_{o}$. Denote by $Ric_{\psi}$ the Ricci form of $\beta_{\psi}$. By a result of R. Bielawski (Theorem 3.3 in [B2]), we have

[TABLE]

where $\nabla$ denotes the Riemannian connection of $\beta_{G^{d}/K}$, $(z_{1},\cdots,z_{n})$ is any complex coordinate of $G^{\mathbb{C}}/K^{\mathbb{C}}$ and $\displaystyle{\left(\frac{{\rm det}\,\nabla d\bar{\psi}}{{\rm det}\,\beta_{G^{d}/K}}\right)^{h}}$ is the $G$-invariant function over $G^{\mathbb{C}}/K^{\mathbb{C}}$ satisfying

[TABLE]

According to the result of [B1], for any given $K$-invariant positive $C^{\infty}$-function $\varphi$ on $G^{d}/K$, the Monge-Amp$\grave{\rm e}$re equation

[TABLE]

has a global $K$-invariant strictly convex $C^{\infty}$-solution.

Furthermore, we can derive the following fact directly.

Lemma 2.1. (i) For any $G$-invariant strictly plurisubharmonic ($C^{\infty}$-)function $\psi$ over $U_{3}(D)$, we have

[TABLE]

where $\beta_{0}$ is the Euclidean metric of $\sqrt{-1}\mathfrak{a}$ associated to $-{\rm Re}\,B^{\mathbb{C}}|_{\sqrt{-1}\mathfrak{a}\times\sqrt{-1}\mathfrak{a}}$ and $\nabla^{0}$ is the Euclidean connection of $\beta_{0}$ and $\displaystyle{\left(\frac{{\rm det}\,\nabla^{0}d\bar{\bar{\psi}}}{{\rm det}\,\beta_{0}}\right)^{h}}$ is the $G$-invariant function over $G^{\mathbb{C}}/K^{\mathbb{C}}$ satisfying

[TABLE]

(ii) For any given $W$-invariant positive $C^{\infty}$-function $\varphi$ on $\sqrt{-1}\mathfrak{a}$, the Monge

-Amp$\grave{\rm e}$re equation

[TABLE]

has a global $W$-invariant strictly convex $C^{\infty}$-solution.

Proof. Since $\bar{\psi}$ is $K$-invariant, we have

[TABLE]

Therefore the statement (i) is directly derived from the above result by R. Bielawski. The statement (ii) is trivial. ∎

From a global $W$-invariant strictly convex $C^{\infty}$-solution $\rho$ of the Monge-Amp$\grave{\rm e}$re equation

[TABLE]

we can construct a complete Ricci-flat metric $\beta_{\psi}$ on $G^{\mathbb{C}}/K^{\mathbb{C}}$, where $\psi$ is the $G$-invariant strictly plurisubharmonic $C^{\infty}$-function satisfying $\psi|_{{\rm Exp}_{o}(\sqrt{-1}\mathfrak{a})}\circ{\rm Exp}_{o}=\rho$. Hence we obtain a Calabi-Yau structure $(J_{0},\omega_{\psi},\Omega_{0})$ on $G^{\mathbb{C}}/K^{\mathbb{C}}$ by replacing $\rho$ to a suitable positive constant-multiple of $\rho$ if necessary.

We consider the case of $D=\sqrt{-1}\mathfrak{a}$. Then, according to the Schwarz’s theorem ([Sc]), the ring $C^{\infty}_{W}(\sqrt{-1}\mathfrak{a})$ of all $W$-invariant $C^{\infty}$-functions over $\sqrt{-1}\mathfrak{a}$ is given by

[TABLE]

where $\rho_{1},\cdots,\rho_{l}$ are generators of $C^{\infty}_{W}(\sqrt{-1}\mathfrak{a})$ of the ring ${\rm Pol}_{W}(\sqrt{-1}\mathfrak{a})$ of all $W$-invariant polynomials over $\sqrt{-1}\mathfrak{a}$. In the sequel, set $\overrightarrow{\rho}:=(\rho_{1},\cdots,\rho_{l})$ for simplicity. Let $\psi_{i}$ ($i=1,\cdots,l$) be the elements of $PH_{G}^{+}(G^{\mathbb{C}}/K^{\mathbb{C}})$ with $\bar{\bar{\psi}}_{i}=\rho_{i}$. In the sequel, set $\overrightarrow{\psi}:=(\psi_{1},\cdots,\psi_{l})$ for simplicity. Hence any element $\psi$ of $PH_{G}^{+}(G^{\mathbb{C}}/K^{\mathbb{C}})$ is described as $\psi=f\circ\overrightarrow{\psi}$ in terms of some $f\in C^{\infty}(\mathbb{R}^{l})$. As the first generator $\rho_{1}$ of $C^{\infty}_{W}(\sqrt{-1}\mathfrak{a})$, we take

[TABLE]

In the following, set $\psi_{f}:=f\circ\overrightarrow{\psi}$. By using Lemma 2.1, we can derive the following fact.

Theorem 2.2. (i) The triple $(J_{0},\omega_{\psi_{f}},\Omega_{0})$ is a Calabi-Yau structure of $G^{\mathbb{C}}/K^{\mathbb{C}}$ when

[TABLE]

where $c$ is a positive constant, and $(x_{1},\cdots,x_{r})$ and $(y_{1},\cdots,y_{l})$ are the natural coordinates of $\sqrt{-1}\mathfrak{a}$ and $\mathbb{R}^{l}$, respectively.

(ii) Assume that $\displaystyle{\frac{\partial f}{\partial y_{2}}=\cdots=\frac{\partial f}{\partial y_{l}}=0}$. Then $(J_{0},\omega_{\psi_{f}},\Omega_{0})$ is a Calabi-Yau structure of $G^{\mathbb{C}}/K^{\mathbb{C}}$ when

[TABLE]

where $c$ is a positive constant, and $(x_{1},\cdots,x_{r})$ and $(y_{1},\cdots,y_{l})$ are as above.

Proof. By a simple calculation, we have

[TABLE]

Hence, from $(2.6)$, we obtain

[TABLE]

that is, $\bar{\bar{\psi}}_{f}$ is convex. Also, we have

[TABLE]

Hence we have

[TABLE]

Therefore, from Lemma 2.1, we obtain ${\rm Ric}_{\psi_{f}}=0$. Thus $(J_{0},\omega_{\psi_{f}},\Omega_{0})$ is a Calabi-Yau structure of $G^{\mathbb{C}}/K^{\mathbb{C}}$. The statement (ii) follows from (i) direcctly. ∎

Remark 2.1. (i) By using the result of [B1], we can show that the Monge-Amp$\grave{\rm e}$re type equation $(2.6)$ has global solution $f$.

(ii) The Monge-Amp$\grave{\rm e}$re type equations $(2.6)$ and $(2.7)$ coincide in the case of ${\rm rank}\,G/K=1$.

From (ii) of Theorem 2.2, we can derive the following fact.

Corollary 2.3. Let $f$ be the $C^{\infty}$-function over $\mathbb{R}^{l}$ defined by

[TABLE]

where $a,b$ and $c$ are positive constants. Then $(J_{0},\omega_{\psi_{f}},\Omega_{0})$ is a Calabi-Yau structure of $G^{\mathbb{C}}/K^{\mathbb{C}}$.

Proof. By a simple calculation, we have

[TABLE]

Hence, it follows from (ii) of Theorem 2.2 that $(J_{0},\omega_{\psi_{f}},\Omega_{0})$ is a Calabi-Yau structure of $G^{\mathbb{C}}/K^{\mathbb{C}}$. ∎

Remark 2.2. For $f$ as in $(2.8)$, $\beta_{\psi_{f}}$ coincides with the Stenzel metric in the case where $G/K=SO(n+1)/SO(n)(=S^{n})$.

3. Hamiltonian actions and the moment maps

Let $(M,\omega)$ be a symplectic manifold and the action $H\curvearrowright M$ of a Lie group $H$ on $(M,\omega)$. This action $H\curvearrowright M$ is called a Hamiltonian action if it satisfies the following conditions (i)$\sim$(iii):

(i) For any $h\in H$, $h^{\ast}\omega=\omega$ holds;

(ii) For any element $X$ of the Lie algebra $\mathfrak{h}$ of $H$, $i_{X^{\ast}}\omega$ is exact, where $X^{\ast}$ denote the fundamental vector field on $M$ associated to $X$, that is,

[TABLE]

and $i_{X^{\ast}}$ denotes the inner product operator by $X^{\ast}$;

(iii) There exists a family $\{F_{X}\}_{X\in\mathfrak{h}}$ of $C^{\infty}$-functions over $M$ such that $dF_{X}=i_{X^{\ast}}\omega\,\,\,(X\in\mathfrak{h})$ and that the correspndence $X\mapsto F_{X}\,\,\,(X\in\mathfrak{h})$ is a Lie algebra homomorphism of $\mathfrak{h}$ into $C^{\infty}(M)$.

Here we note that, by the condition (ii), it is assured that there exists a family $\{F_{X}\}_{X\in\mathfrak{h}}$ of $C^{\infty}$-functions over $M$ such that $dF_{X}=i_{X^{\ast}}\omega\,\,\,(X\in\mathfrak{h})$ and that the correspndence $X\mapsto F_{X}\,\,\,(X\in\mathfrak{h})$ is linear. For a function $F$ over $(M,\omega)$, the s-gradient vector field ${\rm sgrad}\,F$ is defined by $dF(Y)=\omega({\rm sgrad}\,F,Y)\,\,\,(Y\in TM)$. Clearly we have ${\rm sgrad}\,F_{X}=X^{\ast}$. The moment map $\mu:M\to\mathfrak{h}^{\ast}$ of this Hamiltonian action is defined by

[TABLE]

Hence the level set $\mu^{-1}(0)$ is given by

[TABLE]

Let $(G^{\mathbb{C}}/K^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$ be a Calabi-Yau manifold stated in the previous section and $H$ be a closed subgroup of $G$. Denote by $\mathfrak{h}$ the Lie algebra of $H$. Let $n:={\rm dim}\,G/K$. For simplicity, set $M:=G\cdot o(\approx G/K),\,\,M^{\mathbb{C}}:=G^{\mathbb{C}}/K^{\mathbb{C}}$ and $M^{d}:=G^{d}\cdot o(\approx G^{d}/K)$. As stated in Introduction, set $\Psi_{p}={\rm Exp}_{p}\circ(J_{0})_{p}$ ($p\in M$). Set $M^{d}_{p}:=\Psi_{p}(T_{p}(G\cdot o))$ ($p\in M$), which is the normal umbrella of $M$ in $(M^{\mathbb{C}},\beta_{A})$. Note that $M^{d}_{o}=M^{d}$.

Lemma 3.1. (i) The action $H\curvearrowright(M^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$ is a Hamiltonian action and its moment map $\mu_{\psi_{f}}$ is given by

[TABLE]

where ${\rm Im}(\cdot)$ denotes the imaginary part of $(\cdot)$.

(ii) The level set $\mu_{\psi_{f}}^{-1}(0)$ is given by

[TABLE]

Proof. Since $\omega_{\psi_{f}}$ is $G$-invariant and $H$ is a closed subgroup, it is $H$-invariant. Set $\alpha:=-{\rm Im}\,\overline{\partial}\psi_{f}$. For each $X\in\mathfrak{h}$, define a function $F_{X}$ over $M^{\mathbb{C}}$ by $F_{X}(q):=\alpha_{q}(X^{\ast}_{q})$ ($q\in M^{\mathbb{C}}$). Then, for any tangent vector field $Y$ over $M^{\mathbb{C}}$, we have

[TABLE]

where ${\mathcal{L}}_{X^{\ast}}$ denotes the Lie derivative with respect to $X^{\ast}$. Since $\alpha$ is $H$-invariant, we have ${\mathcal{L}}_{X^{\ast}}\alpha=0$. Also, we have $d\alpha=-\omega$. Hence we obtain $dF_{X}=i_{X^{\ast}}\omega$. Also, it is clear that the correspndence $X\mapsto F_{X}\,\,\,(X\in\mathfrak{h})$ is a Lie algebra homomorphism of $\mathfrak{h}$ into $C^{\infty}(M)$. Therefore the action $H\curvearrowright(M^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$ is a Hamiltonian action and its moment map $\mu_{\psi_{f}}$ is given by

[TABLE]

Thus the statement (i) has been shown. The statement (ii) follows from (i) directly. ∎

By using this lemma, we obtain the following fact.

Lemma 3.2. Let $f$ be as in $(2.8)$. Then the level set $\mu_{\psi_{f}}^{-1}(0)$ is given by

[TABLE]

where $T_{p}^{\perp}(H\cdot p)$ denotes the normal space of $H\cdot p$ in $M$ at $p$. Also, if ${\rm cohom}\,(H\curvearrowright G/K)=r$, then we have ${\rm dim}\,\mu_{\psi_{f}}^{-1}(0)=n+r$.

Proof. Let $(U,(z_{1}=x_{i}+\sqrt{-1}y_{i})_{i=1}^{n})$ be a holomorphic coordinate of $M^{\mathbb{C}}$ such that ${\rm Span}\{(\frac{\partial}{\partial x_{i}})_{p}\,|\,i=1,\cdots,n\}=T_{p}M$ holds for any $p\in U\cap M$. Note that, for $q\in U\cap M_{p}^{d}$, the following relation holds:

[TABLE]

Fix $p\in M$ and $q\in U\cap M_{p}^{d}$. Take any $X\in\mathfrak{h}$. Then, by a simple calculation, we have

[TABLE]

where $g_{ij}:=g\left(\frac{\partial}{\partial x_{i}},\,\frac{\partial}{\partial x_{j}}\right)$, $X^{\ast}$ denotes the fundamental vector field on $M^{\mathbb{C}}$ associated to $X$ and $X^{\ast}_{i}$ is the function given by $X^{\ast}=\sum\limits_{i=1}^{n}\left(X^{\ast}_{i}\frac{\partial}{\partial x_{i}}+{\widehat{X}}^{\ast}_{i}\frac{\partial}{\partial y_{i}}\right)$. Hence $q\in\mu_{\psi_{f}}^{-1}(0)$ if and only if

[TABLE]

holds for any $X\in\mathfrak{h}$. On the othe hand, $X$ moves over $\mathfrak{h}$, $\sum\limits_{i=1}^{n}X^{\ast}_{i}(p)\left(\frac{\partial}{\partial x_{i}}\right)_{p}$ moves over the whole of $T_{p}(H\cdot p)$. Therefore $q\in\mu_{\psi_{f}}^{-1}(0)$ if and only if

[TABLE]

holds. From this fact, the relation $(3.4)$ follows.

Let $U$ be the open subset of $G/K$ of all regular points of $H\curvearrowright G/K$. Then $\displaystyle{\mathop{\amalg}_{p\in U}\Psi_{p}\left(T_{p}^{\perp}(H\cdot p)\right)}$ is an open subset of $\mu_{\psi_{f}}^{-1}(0)$. It is clear that the dimension of this open subset is equal to $n+r$. Hence we obtain ${\rm dim}\,\mu_{\psi_{f}}^{-1}(0)=n+r$. ∎

4. The actions of symmetric subgroups on complexified symmetric spaces

Let $(G^{\mathbb{C}}/K^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$ be a Calabi-Yau manifold stated in Section 2. As in the previous section, set $M:=G\cdot o(=G/K),\,\,M^{\mathbb{C}}:=G^{\mathbb{C}}/K^{\mathbb{C}}$, $M^{d}:=G^{d}\cdot o(=G^{d}/K)$ and $M^{d}_{p}:=\Psi_{p}(T_{p}(G\cdot o))$. Let $H$ be a symmetric subgroup of $G$ and $\sigma$ the involutive automorphism of $G$ satisfying $({\rm Fix}\,\sigma)_{0}\subset H\subset{\rm Fix}\,\sigma$. The natural action $H$ of on $G/K(=M)$ is called a Hermann action. Assume that $\sigma\circ\theta=\theta\circ\sigma$. Then the action is called a commutative Hermann action. Set $n:={\rm dim}\,M$ and denote by $r$ the cohomogeneity of the action $H\curvearrowright M$. The group $H$ acts on $M^{\mathbb{C}}$ as a subaction of the natural action $G\curvearrowright M^{\mathbb{C}}$, where we note that $G\curvearrowright M^{\mathbb{C}}$ is a Hermann type action (this terminology was used in [Koi1]). It is easy to show that the subaction $H\curvearrowright M^{\mathbb{C}}$ is a Hamiltonian action. Set $\mathfrak{q}:={\rm Ker}(\sigma+{\rm id}_{\mathfrak{g}})$. From $\sigma\circ\theta=\theta\circ\sigma$, we have

[TABLE]

Take a maximal abelian subspace $\mathfrak{b}$ of $\mathfrak{p}\cap\mathfrak{q}$ and a maximal abelian subsapce $\mathfrak{a}$ of $\mathfrak{p}$ including $\mathfrak{b}$. For $\beta\in\mathfrak{b}^{\ast}$, we define $\mathfrak{p}_{\beta}$ and $\mathfrak{k}_{\beta}$ by

[TABLE]

and

[TABLE]

Also, we define $\triangle_{\mathfrak{b}}(\subset\mathfrak{b}^{\ast})$ by

[TABLE]

which is the root system. Let $(\triangle_{\mathfrak{b}})_{+}$ be the positive root subsystem of $\triangle_{\mathfrak{b}}$ with respect to some lexicographic ordering of $\mathfrak{b}^{\ast}$. Then we have

[TABLE]

where $\mathfrak{z}_{\bullet}(\mathfrak{b})$ is the cetralizer of $\mathfrak{b}$ in $(\bullet)$. Set

[TABLE]

Note that $\Sigma_{\mathfrak{a}}$ (resp. $\Sigma^{d}_{\mathfrak{a}}$) is included by $M$ (resp. $M^{d}$) because $\sqrt{-1}\mathfrak{p}$ is identified with $T_{o}(M^{d})$. Set $H^{d}:=\exp((\mathfrak{h}\cap\mathfrak{k})\oplus\sqrt{-1}(\mathfrak{h}\cap\mathfrak{p}))$, $\theta^{d}:=\theta^{\mathbb{C}}|_{{\mathfrak{g}}^{d}}$, $\sigma^{d}:=\sigma^{\mathbb{C}}|_{{\mathfrak{g}}^{d}}$, $L:={\rm Fix}(\sigma\circ\theta)$ and $L^{d}:={\rm Fix}(\sigma^{d}\circ\theta^{d})$. The normal umbrella ${\rm Exp}_{o}(T^{\perp}_{o}(H^{d}\cdot o))$ of $H^{d}\cdot o$ in $M^{d}$ is isometric to the symmetric space $L^{d}/H\cap K$ and that the normal umbrella ${\rm Exp}_{o}(T^{\perp}_{o}(H\cdot o)\cap T_{o}M)$ of $H\cdot o$ in $M$ is isometric to the symmetric space $L/H\cap K$ (see [Koi1, Koi3, Koi4]), where $T^{\perp}_{o}(H^{d}\cdot o)$ is the normal space of $H^{d}\cdot o$ in $M^{d}$ at $o$. It is shown that $T_{o}(L^{d}/H\cap K)=\sqrt{-1}(\mathfrak{p}\cap\mathfrak{q})$, $T_{o}(H^{d}\cdot o)=\sqrt{-1}(\mathfrak{p}\cap\mathfrak{h})$ and that all orbits of $G\curvearrowright M^{\mathbb{C}}$ meet $\Sigma^{d}_{\mathfrak{a}}$ orthogonally (see [Koi1, Koi3, Koi4]). Denote by $H_{p}$ the isotropy group of $H\curvearrowright M$ at $p(\in M)$ and $\mathfrak{h}_{p}$ the Lie algebra of $H_{p}$. Also, let $\mathfrak{h}_{p}^{\perp}$ be the orthogonal complement of $\mathfrak{h}_{p}$ in $\mathfrak{h}$. Set $H_{p}^{\perp}:=\{\exp\,X\,|\,X\in\mathfrak{h}_{p}^{\perp}\}$. The group $H_{p}$ acts on the normal umbrella $M^{d}_{p}$. First we prove the following fact.

Lemma 4.1. For $p\in M$ and $q\in M^{d}_{p}$, we have

[TABLE]

Hence $H\cdot q$ has the structure of the fiber bundle over $H\cdot p$ with the standard fibre $H_{p}\cdot q$ and the structure group $H_{p}$.

Proof. Since $({\rm Exp}_{o}X)\cdot M_{p}^{d}=M_{({\rm Exp}_{o}X)\cdot p}^{d}$ holds for any $X\in\mathfrak{h}_{p}^{\perp}$, the first relation is derived. For any $X\in({\rm Exp}_{o},X)$, $H_{({\rm Exp}_{o}X)\cdot p}$ is conjugate to $H_{p}$ and $H_{({\rm Exp}_{o}X)\cdot p}\cdot(({\rm Exp}_{o}X)\cdot q)$ is diffeomorphic to $H_{p}\cdot q$. Hence the second-half part of the statement is derived. ∎

M^{d}_{p}$$({\rm Exp}_{o}X)\cdot p$$p$$M$$({\rm Exp}_{o}X)\cdot q$$q$$H\cdot q$$H_{({\rm Exp}_{o}X)\cdot p}\cdot(({\rm Exp}_{o}X)\cdot q)in factM$$p$$({\rm Exp}_{o}X)\cdot p$$H\cdot p$$o$$H_{p}\cdot q$$M^{d}_{({\rm Exp}_{o}X)\cdot p}

Figure 4.

Lemma 4.2 Let $q\in\Psi_{p}\left(T_{p}^{\perp}(H\cdot p)\right)$ and denote by ${\rm Hol}^{\perp}_{-\Psi_{p}^{-1}(q)}(H\cdot p)$ the normal holonomy bundle of the submanifold $H\cdot p$ in $M$ through $-\Psi_{p}^{-1}(q)$. Then we have

[TABLE]

Proof. It is clear that $\{({\rm Exp}_{o}X)\cdot p\,|\,X\in\mathfrak{h}_{p}^{\perp}\}=H\cdot p$. Since $H\curvearrowright G/K$ is hyperpolar, ${\rm Exp}_{({\rm Exp}_{o}X)\cdot p}$

$\displaystyle{\left(T^{\perp}_{({\rm Exp}_{o}X)\cdot p}(H\cdot p)\right)}$ is totally geodesic in $M$. From this fact, we can show that the orbit $H_{({\rm Exp}_{o}X)\cdot p}$

$(({\rm Exp}_{o}X)\cdot q)$ is equal to the image of the fibre of the normal holonomy bundle ${\rm Hol}^{\perp}_{-\Psi_{p}^{-1}(q)}(H\cdot p)$ over $({\rm Exp}_{o}X)\cdot p$ by $\Psi_{({\rm Exp}_{o}X)\cdot p}$. Hence it follows from Lemma 4.1 that $H\cdot q$ is described as in the statement. ∎

5. Special Lagrangian submanifolds in complexified symmetric spaces

Let $(G^{\mathbb{C}}/K^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$ be the Calabi-Yau manifold stated in Section 2, where $f$ is as in (i) of Theorem 2.2. As in the previous section, set $M:=G\cdot o(=G/K),\,\,M^{\mathbb{C}}:=G^{\mathbb{C}}/K^{\mathbb{C}}$, $M^{d}:=G^{d}/K(=G^{d}\cdot o)$ and $M^{d}_{p}:=\Psi_{p}(T_{p}(G\cdot o))$. Let $H$ be a symmetric subgroup of $G$ and $r$ be the cohomogeneity of the Hermann action $H\curvearrowright G/K$. The naturally extended action of $H$ on $(M^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$ is a Hamiltonian action. Denote by $\mu_{\psi_{f}}$ the moment map of this Hamiltonian action. Let $Z(\mathfrak{h}^{\ast})$ be the center of $\mathfrak{g}^{\ast}$, that is,

[TABLE]

where ${\rm Ad}^{\ast}$ denotes the coadjoint representation of $H$. It is clear that $\mu_{\psi_{f}}^{-1}(c)$ is $H$-invariant if and only if $c$ belongs to $Z(\mathfrak{h}^{\ast})$. According to Proposition 2.5 of [HS], the following fact holds.

Proposition 5.1([HS]). Assume that $L$ is a $H$-invariant connected isotropic submanifold in

$(M^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$, where “isotropic” means that $\omega_{\psi_{f}}(TL,TL)=0$ holds. Then $L\subset\mu_{\psi_{f}}^{-1}(c)$ holds for some $c\in Z(\mathfrak{h}^{\ast})$.

In the method of the proof of Proposition 2.6 of [HS], we can show the following fact.

Proposition 5.2. Let $L$ be a $H$-invariant connected submanifold in $M^{\mathbb{C}}$ and $r_{0}$ be the cohomogeneity of the action $H\curvearrowright L$. Assume that $L\subset\mu_{\psi_{f}}^{-1}(c)$ for some $c\in Z(\mathfrak{h}^{\ast})$ and that there exists a $r_{0}$-dimensional isotropic submanifold $L_{0}$ in $(M^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$ satisfying the following conditions:

(i) $L_{0}\subset L$,

(ii) $L_{0}$ is transversal to the principal orbits of the action $H\curvearrowright L$,

(iii) $H\cdot L_{0}=L$,

Then $L$ also is an isotropic submanifold in $(M^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$.

Proof. Take any $X\in\mathfrak{h}$ and any $Y\in T_{p}L$. From $L\subset\mu_{\psi_{f}}^{-1}(c)$, we have $d(\mu_{\psi_{f}})_{p}(Y)=0$. On the other hand, we have $(d(\mu_{\psi_{f}})_{p}(Y))(X)=$

$(\omega_{\psi_{f}})_{p}(Y,X^{\ast}_{p})$, where $X^{\ast}$ is the vector field on $M^{\mathbb{C}}$ associated to the one-parameter transformation group $\{\exp\,tX\}_{t\in\mathbb{R}}$ of $M^{\mathbb{C}}$ ($\exp\,:\,$ the exponential map of $H$). Hence we have $(\omega_{\psi_{f}})_{p}(Y,X^{\ast}_{p})=0$. Therefore, it follows from the arbitrariness of $X$ and $Y$ that $(\omega_{\psi_{f}})_{p}(T_{p}L,T_{p}(H\cdot p))=0$. Also, since $L_{0}$ is isotropic, we have $(\omega_{\psi_{f}})_{p}(T_{p}L_{0},T_{p}L_{0})=0$. Hence we obtain $(\omega_{\psi_{f}})_{p}(T_{p}L,T_{p}L)=0$. Therefore, it follows from the arbitariness of $p$ that $L$ is isotropic. ∎

By Proposition 2.4 of [HS], we can show the following fact.

Proposition 5.3. Let $L$ be a $n$-dimensional connected submanifold in

$(M^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$. Then $L$ is a special Lagrangian submanifold of phase $\theta$ if and only if

$\omega_{\psi_{f}}|_{TL\times TL}=0$ and ${\rm Im}\left(e^{\sqrt{-1}\theta}\Omega_{0}|_{(TL)^{n}}\right)=0$.

Let $f$ be as in $(2.8)$. We give constructions of special Lagrangian submanifolds in the Calabi-Yau manifold $(M^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$. Let $U$ be the open subset of $M$ of all regular points of $H\curvearrowright M$. Then, as stated in the proof of Lemma 3.2, $\displaystyle{\widetilde{\Sigma}:=\mathop{\amalg}_{p\in U}\Psi_{p}\left(T_{p}^{\perp}(H\cdot p)\right)}$ is an open subset of $\mu_{\psi_{f}}^{-1}(0)$. Since $H\curvearrowright M$ is a Hermann action, it is hyperpolar (see Subsections 3.1 in [HPTT]). Hence the principal orbit $H\cdot p_{0}$ ($p_{0}\in U$) is an equifocal submanifold in $M$ and its section $\displaystyle{\Sigma:={\rm Exp}_{p_{0}}\left(T^{\perp}_{p_{0}}(H\cdot p_{0})\right)}$ is an $r$-dimensional flat torus $T^{r}=S^{1}\times\cdots\times S^{1}$ ($r$-times) embedded totally geodesically into $M$. Without loss of generality, we may assume that $\Sigma$ passes through $o$. Let $C$ be the component of $U\cap\Sigma$ containing $p_{0}$. Then we have $H\cdot C=U$. Set $\displaystyle{\widehat{\Sigma}:=\mathop{\amalg}_{p\in U\cap\Sigma}\Psi_{p}\left(T^{\perp}_{p}(H\cdot p)\right)}$. It is clear that $\widehat{\Sigma}$ is a dense open subset of $\displaystyle{\Sigma^{\mathbb{C}}:=\mathop{\amalg}_{p\in\Sigma}\Psi_{p}(T_{p}\Sigma)\,(\approx(T^{r})^{\mathbb{C}}=S^{1}_{\mathbb{C}}\times\cdots\times S^{1}_{\mathbb{C}})}$, where $S^{1}_{\mathbb{C}}\times\cdots\times S^{1}_{\mathbb{C}})$ denotes the $r$-times of $S^{1}_{\mathbb{C}}$’s. We identify $T_{o}\Sigma\,(\subset\mathfrak{p})$ and $T_{o}(\Sigma^{\mathbb{C}})\,(\subset\mathfrak{p}^{\mathbb{C}})$ with $\mathbb{R}^{r}$ and $\mathbb{C}^{r}$, respectively. Let $\tau_{i}:I_{i}\to\mathbb{C}$ ($i=1,\cdots,r$) be regular curves, where $I_{i}$ is an open interval. Define an immersion $\tau:I_{1}\times\cdots\times I_{r}\hookrightarrow\mathbb{C}^{r}$ by $\tau:=\tau_{1}\times\cdots\times\tau_{r}$. Set $\overline{\tau}:={\rm Exp}_{o}\circ\tau(:I_{1}\times\cdots\times I_{r}\to S^{1}_{\mathbb{C}}\times\cdots\times S^{1}_{\mathbb{C}}(=\Sigma^{\mathbb{C}}))$. Assume that $(L_{\tau})_{0}:=\overline{\tau}(I_{1}\times\cdots\times I_{r})$ is included by $\widehat{\Sigma}$. It is clear that $(L_{\tau})_{0}$ is an isotropic submanifold in $\Sigma^{\mathbb{C}}$ (hence in $(M^{\mathbb{C}},J_{0},\omega_{\psi_{f}},\Omega_{0})$). Set $L_{\tau}:=H\cdot(L_{\tau})_{0}$. For any $p\in U$ and any $\displaystyle{q\in\Psi_{p}\left(T_{p}^{\perp}(H\cdot p)\right)\,(\subset\widetilde{\Sigma})}$, since $H\cdot p$ is an equifocal submanifold in $M$, the normal connection of the submanifold $H\cdot p$ in $M$ is flat and hence the (restricted) normal holonomy representation

[TABLE]

is trivial, where $(H_{({\rm Exp}_{o}X)\cdot p})_{0}$ denotes the identity component of $H_{({\rm Exp}_{o}X)\cdot p}$. Hence the action

[TABLE]

also is trivial. Therefore it follows from Lemma 4.1 that each component of $H\cdot q$ is diffeomorphic to $\displaystyle{\mathop{\cup}_{X\in\mathfrak{h}_{p}^{\perp}}({\rm Exp}_{o}X)\cdot q}$. From this fact, ${\rm dim}\,H\cdot q=n-r$ follows. Since $(L_{\tau})_{0}$ is included by $\widetilde{\Sigma}$, $L_{\tau}$ is an $n$-dimensional submanifold of cohomogeneity $r$ in $M^{\mathbb{C}}$. By Proposition 5.2, $L_{\tau}$ is a Lagrangian submanifold.

Here we shall explain that the cohomogeneity of the Hamiltonian action $H\curvearrowright M^{\mathbb{C}}$ is possible to be smaller than $(n+r)$. For $\displaystyle{\widehat{q}\in\Psi_{p}\left(T_{p}(H\cdot p)\right)\,(\subset\widetilde{\Sigma})}$, the (restricted) holonomy representation

[TABLE]

of the Riemannian manifold $H\cdot p$ at $({\rm Exp}_{o}X)\cdot p$ is not necessarily trivial. Hence the action

[TABLE]

also is not necessarily trivial. On the other hand, we can show that $H\cdot\widehat{q}$ is equal to the image of the holonomy bundle ${\rm Hol}_{\Psi_{p}^{-1}(\widehat{q})}(H\cdot p)$ of $H\cdot p$ throught $\Psi_{p}^{-1}(\widehat{q})$ by $\Psi$. From these facts, it follows that ${\rm dim}\,(H\cdot\widehat{q})$ is possible to be larger than $(n-r)$. That is, the cohomogeneity of the action $H\curvearrowright M^{\mathbb{C}}$ is possible to be smaller than $(n+r)$. Set $\mathfrak{b}:=T_{o}\Sigma$, which is a maximal abelian subspace of $\mathfrak{p}\cap\mathfrak{q}$. Note that $\tau$ is rearded as a regular curve in $\mathfrak{b}^{\mathbb{C}}$ under the identification of $\mathbb{C}^{r}$ with $\mathfrak{b}^{\mathbb{C}}$. Let $\triangle_{\mathfrak{b}},\,(\triangle_{\mathfrak{b}})_{+},\,\mathfrak{k}_{\beta},\,\mathfrak{p}_{\beta},\,\mathfrak{h}_{\beta}$ and $\mathfrak{q}_{\beta}$ be as in Section 4. Define $(\triangle_{\mathfrak{b}}^{V})_{+}$ and $(\triangle_{\mathfrak{b}}^{H})_{+}$ by

[TABLE]

and

[TABLE]

respectively. Note that ${\rm dim}(\mathfrak{p}_{\beta}\cap\mathfrak{q})={\rm dim}(\mathfrak{k}_{\beta}\cap\mathfrak{h})$ and ${\rm dim}(\mathfrak{p}_{\beta}\cap\mathfrak{h})={\rm dim}(\mathfrak{k}_{\beta}\cap\mathfrak{q})$. Set $m_{\beta}^{V}:={\rm dim}(\mathfrak{p}_{\beta}\cap\mathfrak{q})$ ($\beta\in(\triangle_{\mathfrak{b}}^{V})_{+}$) and $m_{\beta}^{H}:={\rm dim}(\mathfrak{p}_{\beta}\cap\mathfrak{h})$ ($\beta\in(\triangle_{\mathfrak{b}}^{H})_{+}$). Let $\{X_{\beta,i}^{V}\,|\,i=1,\cdots,m_{\beta}^{V}\}$ be a basis of $\mathfrak{k}_{\beta}\cap\mathfrak{h}$ ($\beta\in(\triangle_{\mathfrak{b}}^{V})_{+}$) and $\{X_{\beta,i}^{H}\,|\,i=1,\cdots,m_{\beta}^{H}\}$ be a basis of $\mathfrak{p}_{\beta}\cap\mathfrak{h}$ ($\beta\in(\triangle_{\mathfrak{b}}^{V})_{+}$). Also, let $Y_{\beta,i}^{V}$ be the element of $\mathfrak{k}_{\beta}\cap\mathfrak{h}$ such that ${\rm ad}(Z)(X_{\beta,i}^{V})=\beta(Z)Y_{\beta,i}^{V}$ holds for any $Z\in\mathfrak{b}$. Define a Killing vector field $(Y^{V}_{\beta,i})^{\ast}$ over $M^{\mathbb{C}}$ by

[TABLE]

and a Killing vector field $(X^{H}_{\beta,i})^{\ast}$ over $M^{\mathbb{C}}$ by

[TABLE]

’s :$$(Y^{V}_{\beta,i})^{\ast}|_{M}$$H\cdot o$$H\cdot p$$o$$p$$M

Figure 5.

H\cdot o$$H\cdot p$$p$$o’s :$$(X^{H}_{\beta,i})^{\ast}|_{M}$$M

Figure 6.

For $Z\in\mathfrak{b}^{\mathbb{C}}$, $(Y^{V}_{\beta,i})^{\ast}_{{\rm Exp}_{o}(Z)}$ and $(X^{H}_{\beta,i})^{\ast}_{{\rm Exp}_{o}(Z)}$ are described as

[TABLE]

and

[TABLE]

respectively. A basis of $T_{\overline{\tau}({\bf s})}(H\cdot\overline{\tau}({\bf s}))(=T_{\overline{\tau}({\bf s})}L_{\tau}\cap T^{\perp}_{\overline{\tau}({\bf s})}(L_{\tau})_{0})$ is given by

[TABLE]

On the other hand, a basis of $T_{\overline{\tau}({\bf s})}(L_{\tau})_{0}(\subset T_{\overline{\tau}({\bf s})}(\Sigma^{\mathbb{C}})\,(\approx\mathbb{C}^{r}))$ is given by

[TABLE]

where $\displaystyle{{\bf e}_{i}:=(0,\cdots,0,\mathop{1}^{i},0,\cdots 0)}$ ($\displaystyle{\mathop{1}^{i}}$ means that $i$-component is equal to $1$). Let $(\triangle_{\mathfrak{b}}^{V})_{+}=$

$\{\beta^{V}_{i}\,|\,i=1,\cdots,k_{V}\}$ and $(\triangle_{\mathfrak{b}}^{H})_{+}=\{\beta^{H}_{i}\,|\,i=1,\cdots,k_{H}\}$. From $(5.1)$ and $(5.2)$, we have

[TABLE]

[TABLE]

In the last equality, we used the fact that $\exp(\tau({\bf s})))_{\ast}^{-1}({\bf e}_{i})={\bf e}_{i}$ ($i=1,\cdots,r$) hold under the identification $T_{\overline{\tau}({\bf s})}\Sigma=T_{o}\Sigma=\mathbb{C}^{r}$ because $\exp(\tau({\bf s})))_{\ast}^{-1}$ is the parallel translation along the geodesic $t\mapsto{\rm Exp}_{o}(t\tau({\bf s}))$ in $\Sigma^{\mathbb{C}}$. It is clear that

[TABLE]

is a nonzero real constant independent of ${\bf s}=(s_{1},\cdots,s_{r})$. From these facts and Proposition 5.3, we obtain the following fact for $L_{\tau}$.

Theorem 5.4. The submanifold $L_{\tau}$ is a special Lagrangian submanifold of phase $\theta$ if and only if $\tau_{1},\cdots,\tau_{r}$ satisfy the following ordinary differential equation:

[TABLE]

Next we shall give solutions of the ordinary differential equation $(5.3)$. Let $\tau_{i}(s_{i})=\varphi_{i}(s_{i})+\sqrt{-1}\rho_{i}(s_{i})$ ($i=1,\cdots,r$), where $\varphi_{i}$ and $\rho_{i}$ are real-valued functions. Set

[TABLE]

Also, let

[TABLE]

and

[TABLE]

where $u_{0},\,v_{0},\,U_{0}$ and $V_{0}$ are real-valued functions. Define $u_{1}$ and $v_{1}$ by

[TABLE]

and

[TABLE]

It is clear that

[TABLE]

Let

[TABLE]

where $U_{1}$ and $V_{1}$ are real-valued functions. In the sequel, we define $u_{i},v_{i},U_{i}$ and $V_{i}$ ($i=2,\cdots,r$) by repeating the same process. Set

[TABLE]

It is easy to show that

[TABLE]

Corollary 5.5. Let $F$ be the complex-valued function over $\mathbb{R}^{2r}$ defined by

[TABLE]

If $\tau_{i}(s_{i})=\varphi_{i}(s_{i})+\sqrt{-1}\rho_{i}(s_{i})$ ($i=1,\cdots,r$) satisfy

[TABLE]

then they are a solution of $(5.3)$ and hence $L_{\tau}$ ($\tau:=\tau_{1}\times\cdots\times\tau_{r}$) is a special Lagrangian submanifold of phase $\theta$.

Proof. Since $F$ is a holomorphic function, we have $\displaystyle{\frac{\partial u_{0}}{\partial\varphi_{1}}=\frac{\partial v_{0}}{\partial\rho_{1}}}$ and $\displaystyle{\frac{\partial u_{0}}{\partial\rho_{1}}=}$

$\displaystyle{-\frac{\partial v_{0}}{\partial\varphi_{1}}}$. From these relations and the definitions of $U_{0}$ and $V_{0}$, we have

[TABLE]

Hence we obtain

[TABLE]

Since $u_{1}+\sqrt{-1}v_{1}$ is holomorphic with respect to $\tau_{2}(=\varphi_{2}+\sqrt{-1}\rho_{2})$, we have $\displaystyle{\frac{\partial u_{1}}{\partial\varphi_{2}}=\frac{\partial v_{1}}{\partial\rho_{2}}}$ and $\displaystyle{\frac{\partial u_{1}}{\partial\rho_{2}}=-\frac{\partial v_{1}}{\partial\varphi_{2}}}$. From these relations and the definitions of $U_{1}$ and $V_{1}$, we have

[TABLE]

Hence we obtain

[TABLE]

In the sequel, by repeating the same discussion, we obtain

[TABLE]

From this relation, we can derive the statement of this corollary directly. ∎

We consider the case where $N=G/K$ is an $dm$-dimensional simply connected rank one symmetirc space of compact type and constant maximal sectional curvature $4c$, that is, $N=\mathbb{F}P^{m}(4c)$ ($\mathbb{F}=\mathbb{C},\,\mathbb{Q}$ or $\mathbb{O}$), where $\mathbb{Q}$ (resp. $\mathbb{O}$) denotes the quaternionic algebra (resp. the Octonian) and $d$ is given by $d=2$ (when $\mathbb{F}=\mathbb{C}$), $d=4$ (when $\mathbb{F}=\mathbb{Q}$) or $d=8$ (when $\mathbb{F}=\mathbb{O}$). Note that

[TABLE]

In these cases, we have $\triangle_{+}=\{\sqrt{c}e^{\ast},\,2\sqrt{c}e^{\ast}\}$, where $e^{\ast}$ denotes the dual $1$-form of the unit vector $e$ of $\mathfrak{b}$ (${\rm dim}\,\mathfrak{b}=1$ in these cases). Also, we have ${\rm dim}\,\mathfrak{p}_{\sqrt{c}{\bf e}^{\ast}}=d(m-1)$ and ${\rm dim}\,\mathfrak{p}_{2\sqrt{c}{\bf e}^{\ast}}=d-1$. Hence, as a corollary of Theorem 5.4, we obtain the following fact.

Corollary 5.6. Let $H\curvearrowright\mathbb{F}P^{m}(4c)$ be a Hermann action. Then the submanifold $L_{\tau}$ is a special Lagrangian submanifold of phase $\theta$ if and only if $\tau$ satisfies the following ordinary differential equation:

[TABLE]

where $m^{V}$ (resp. $m^{H}$) denotes $m_{\sqrt{c}{\bf e}^{\ast}}^{V}$ (resp. $m_{\sqrt{c}{\bf e}^{\ast}}^{H}$).

Proof. Let $\{J_{1},\cdots,J_{d-1}\}$ be the complex structure, the canonical local basis of the quaternionic structure or the Cayley structure of $\mathbb{F}P^{m}(4c)$. Then, since $\mathbb{F}P^{dm}(4c)$ is of rank one, this action is of cohomogeneity one. It is shown that this action is commutative (i.e., $\theta\circ\sigma=\sigma\circ\theta$). In fact, Hermann actions on $\mathbb{F}P^{m}(4c)$ are classified as in Table 1 and all of Hermann actions in Table 1 are commutative. Since $H\curvearrowright\mathbb{F}P^{dm}(4c)$ is commutative, it is shown that $H\cdot o$ and the normal umbrella ${\rm Exp}_{o}(T^{\perp}_{o}(H\cdot o))$ are reflective submanifolds, that is, they are $J_{i}$-invariant ($i=1,\cdots,d-1$). This implies that $T^{\perp}_{o}(H\cdot o)=\mathfrak{p}\cap\mathfrak{q}$ includes $\mathfrak{p}_{2\sqrt{c}e^{\ast}}$. Hence we have $m_{2\sqrt{c}e^{\ast}}^{V}=d-1$ and $m_{2\sqrt{c}e^{\ast}}^{H}=0$. Therefore, the statement of this corollary follows from Theorem 5.4 directly. ∎

K. Hashimoto and K. Mashimo ([HM]) gave the ordinary differential equation corresponding to $(5.3)$ for the Hamiltonian action $K\curvearrowright TS^{n}(1)(=SO(n+1,{\mathbb{C}})/SO(n,{\mathbb{C}}))$ induced from the restricted action $K\curvearrowright S^{n}(1)(=SO(n+1)/SO(n))$ of the linear isotropy action of any irreducible rank two symmetric space $G/K$ (see Theorem 5.2 of [HM]), where $n:={\rm dim}\,G/K-1$ and $S^{n}(1)$ is the unit sphere of $T_{o}(G/K)$. Here we note that the action $K\curvearrowright S^{n}(1)$ is a non-Hermann action of cohomogeneity one (stated in (ii) of Theorem A in [Kol]).

Let $K\curvearrowright\mathbb{C}P^{n}(4)(=SU(n+1)/S(U(1)\times U(n)))$ be the action induced from the restricted action $K\curvearrowright S^{2n+1}$ of the linear isotropy action of any irreducible rank two Hermitian symmetric space $G/K$ through the Hopf fibration $\pi:S^{2n+1}(1)\to\mathbb{C}P^{n}(4)$, where $n=\frac{1}{2}\cdot{\rm dim}\,G/K-1$ and $S^{2n+1}(1)$ is the unit sphere of $T_{o}(G/K)$. Here we note that the action $K\curvearrowright\mathbb{C}P^{n}(4)$ is a non-Hermann action of cohomogeneity one (stated in (iii) of Theorem A in [Kol]). Recently, M. Arai and K. Baba ([AB]) gave the ordinary differential equation corresponding to $(5.3)$ for the Hamiltonian action $K\curvearrowright T\mathbb{C}P^{n}(4)(=SL(n+1,{\mathbb{C}})/(SL(1,{\mathbb{C}})\times SL(n,{\mathbb{C}})))$ induced from the action $K\curvearrowright\mathbb{C}P^{n}(4)$ (see Theorems 2.1-2.4 of [AB]).

According to the classification of cohomogeneity one actions on irreducible symmetric spaces $G/K$ of compact type such that $G$ is simple (i.e., $G/K$ is of type I in [H]) by A. Kollross (see Theorem B of [Kol]), any Hermann action on $\mathbb{F}P^{m}(4c)$ is orbit equivalent to one of Hermann actions in Table 1.

[TABLE]

Table 1 : Hermann actions on $\mathbb{F}P^{m}(4c)$

For all Hermann actions of cohomogeneity two on irreducible rank two symmetric spaces of compact type, we shall give the following datas:

[TABLE]

All of such Hermann actions and the above datas for the actions are given as in Table 2. By using Table 2, we can explicitly describe the ordinary differential equation $(5.3)$ for the Hermann actions of cohomogeneity two on irreducible rank two symmetric spaces of compact type. In Table 2, in the case where $(\triangle_{\mathfrak{b}})_{+}=(\triangle_{\mathfrak{b}})_{+}^{V}\cup(\triangle_{\mathfrak{b}})_{+}^{H}$ is $\{\beta_{1},\beta_{2},\beta_{1}+\beta_{2}\}$, it implies a positive root system of the root system of (${\mathfrak{a}}_{2}$)-type ($(\beta_{1}({\bf e}_{1}),\beta_{1}({\bf e}_{2}))=(2,0),(\beta_{2}({\bf e}_{1}),\beta_{2}({\bf e}_{2}))=(-1,\sqrt{3})$), in the case where $(\triangle_{\mathfrak{b}})_{+}$ is $\{\beta_{1},\beta_{2},\beta_{1}+\beta_{2},2\beta_{1}+\beta_{2}\}$, it implies a positive root system of the root system of (${\mathfrak{b}}_{2}$)(=(${\mathfrak{c}}_{2}$))-type ($(\beta_{1}({\bf e}_{1}),\beta_{1}({\bf e}_{2}))=(1,0),(\beta_{2}({\bf e}_{1}),\beta_{2}({\bf e}_{2}))=(-1,1)$) and in the case where $(\triangle_{\mathfrak{b}})_{+}$ is $\{\beta_{1},\beta_{2},\beta_{1}+\beta_{2},\beta_{1}+2\beta_{2},\beta_{1}+3\beta_{2},2\beta_{1}+3\beta_{2}\}$, it implies a positive root system of the root system of (${\mathfrak{g}}_{2}$)-type ($(\beta_{1}({\bf e}_{1}),\beta_{1}({\bf e}_{2}))=(2\sqrt{3},0),(\beta_{2}({\bf e}_{1}),\beta_{2}({\bf e}_{2}))=(-\sqrt{3},1)$). Also, $\rho_{i}$ ($i=1,\cdots,16$) imply automorphisms of $G$ whose dual actions are given as in Table 3 and $(\bullet)^{2}$ implies the product Lie group $(\bullet)\times(\bullet)$ of a Lie group $(\bullet)$, $\displaystyle{\mathop{\bullet}_{(m)}}$ in the column of $(\triangle_{\mathfrak{b}})_{+}$, $(\triangle_{\mathfrak{b}})^{V}_{+}$ and $(\triangle_{\mathfrak{b}})^{H}_{+}$ imply $m_{\bullet}=m,\,m_{\bullet}^{V}=m$ and $m_{\bullet}^{H}=m$, respectively. Note that Tables 2 and 3 are based on Tables 1 and 2 in [Koi2].

[TABLE]

Table 2 : Hermann actions on rank two symmetric spaces

[TABLE]

Table 2 : Hermann actions on rank two symmetric spaces (continued)

[TABLE]

Table 2 : Hermann actions on rank two symmetric spaces (continued2)

[TABLE]

Table 3 : The dual actions of $\rho_{i}$