Real Hypersurfaces in Hermitian Symmetric Space of Rank Two with Killing Shape Operator

Abstract

We have considered a new notion of the shape operator A satisfies Killing tensor type for real hypersurfaces M in complex Grassmannians of rank two. With this notion we prove the non-existence of real hypersurfaces M in complex Grassmannians of rank two.

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1 Introduction

A typical example of Hermitian symmetry spaces of rank two is the complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ defined by the set of all complex two-dimensional linear subspaces in ${\mathbb C}^{m+2}$. Another one is complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_{2}{\cdot }U_{m})$ the set of all complex two-dimensional linear subspaces in indefinite complex Euclidean space ${\mathbb C}_2^{m+2}$.

Characterizing model spaces of real hypersurfaces under certain geometric conditions is one of our main interests in the classification theory in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$ or complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_{2}{\cdot }U_{m})$. In this paper, we use the same geometric condition on real hypersurfaces in $SU_{2,m}/S(U_{2}{\cdot }U_{m})$ as used in $G_2({\mathbb C}^{m+2})$ to compare the results.

$G_2({\mathbb C}^{m+2})=SU_{2+m}/S(U_2{\cdot }U_m)$ has a compact transitive group $SU_{2+m}$, however $SU_{2,m}/S(U_{2}{\cdot }U_{m})$ has a noncompact indefinite transitive group $SU_{2,m}$. This distinction gives various remarkable results. Riemannian symmetric space $SU_{2,m}/S(U_{2}{\cdot }U_{m})$ has a remarkable geometrical structure. It is the unique noncompact, Kähler, irreducible, quaternionic Kähler manifold with negative curvature.

Suppose that M is a real hypersurface in $G_2({\mathbb C}^{m+2})$ (or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$). Let N be a local unit normal vector field of M in $G_2({\mathbb C}^{m+2})$ (or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$). Since $G_2({\mathbb C}^{m+2})$(or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$) has the Kähler structure J, we may define the Reeb vector field $\xi =-{ JN}$ and a one dimensional distribution $[\xi ]={\mathscr {C}}^{\bot }$ where $\mathscr {C}$ denotes the orthogonal complement in $T_xM$, $x\in M$, of the Reeb vector field $\xi$. The Reeb vector field $\xi$ is said to be Hopf if $\mathscr {C}$ (or ${\mathscr {C}}^{\bot }$) is invariant under the shape operator A of M. The one dimensional foliation of M defined by the integral curves of $\xi$ is said to be a Hopf foliation of M. We say that M is a Hopf hypersurface if and only if the Hopf foliation of M is totally geodesic. By the formulas in [5, Sect. 2], it can be checked that $\xi$ is Hopf vector field if and only if M is Hopf hypersurface.

From the quaternionic Kähler structure $\mathfrak J$ of $G_2({\mathbb C}^{m+2})$ (or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$), there naturally exist almost contact 3-structure vector fields $\xi _{\nu }=-J_{\nu }N$, $\nu =1,2,3$. Put ${\mathscr {Q}}^{\bot }= \text {Span}\{\,\xi _1, \xi _2, \xi _3\}$. It is a 3-dimensional distribution in the tangent bundle ${ TM}$ of M. In addition, we denoted by $\mathscr {Q}$ the orthogonal complement of ${\mathscr {Q}}^{\bot }$ in ${ TM}$. It is the quaternionic maximal subbundle of ${ TM}$. Thus the tangent bundle of M is expressed as a direct sum of $\mathscr {Q}$ and ${\mathscr {Q}}^{\bot }$.

For any geodesic $\gamma$ in M, a (1,1) type tensor field T is said to be Killing if $T\dot{\gamma }$ is parallel displacement along $\gamma$, which gives $0=\nabla _{\dot{\gamma }}(T\dot{\gamma })=(\nabla _{\dot{\gamma }}T)\dot{\gamma }+T(\nabla _{\dot{\gamma }}\dot{\gamma })=(\nabla _{\dot{\gamma }}T)\dot{\gamma }$. That is, $(\nabla _X T)X=0$ for any tangent vector field X on M (see [2]).

$$\begin{aligned} \begin{aligned} 0&=(\nabla _{X+Y}T)(X+Y)\\&=(\nabla _{X}T)X+(\nabla _{X}T)Y+(\nabla _{Y}T)X+(\nabla _{Y}T)Y\\&=(\nabla _{X}T)Y+(\nabla _{Y}T)X \end{aligned} \end{aligned}$$

for any vector fields X and Y on M.

Thus the Killing tensor field T is equivalent to $(\nabla _{X}T)Y+(\nabla _{Y}T)X=0$.

From this notion, in this paper we consider a new condition related to the shape operator A of M defined in such a way that

$$\begin{aligned} (\nabla _XA)Y+(\nabla _YA)X=0 \end{aligned}$$

(C-1)

for any vector fields X on M.

In this paper, we give a complete classification for real hypersurfaces in $\bar{M}$ ($G_2({\mathbb C}^{m+2})$ or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$) with Killing shape operator. In order to do it, we consider a problem related to the following:

Theorem 1

There does not exist any real hypersurface in $\bar{M}$ complex Grassmannians of rank two, $m \ge 3$, with Killing shape operator.

Since the notion of Killing tensor field is weaker than the notion of parallel tensor field, by Theorem 1, we naturally have the following:

quotation There does not exist any real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with parallel shape operator (see [11]).

On the other hand, by virtue of Theorem 2 we can assert the following:

Corollary 1

There does not exist any hypersurface in $SU_{2,m}/S(U_{2}{\cdot }U_{m})$, $m \ge 3$ with parallel shape operator.

2 Riemannian Geometry of $G_2({\mathbb C}^{m+2})$ and $SU_{2,m}/S(U_{2}{\cdot }U_{m})$

In this section we summarize basic material about $G_2({\mathbb C}^{m+2})$, for details we refer to [5, 6, 11, 12]. The complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ is defined by the set of all complex two-dimensional linear subspaces in ${\mathbb C}^{m+2}.$ The special unitary group $G = SU(m+2)$ acts transitively on $G_2({\mathbb C}^{m+2})$ with stabilizer isomorphic to $K = S(U(2) \times U(m))\subset G$. Then $G_2({\mathbb C}^{m+2})$ can be identified with the homogeneous space G / K, which we equip with the unique analytic structure for which the natural action of G on $G_2({\mathbb C}^{m+2})$ becomes analytic. Denote by ${\mathfrak g}$ and ${\mathfrak k}$ the Lie algebra of G and K, respectively, and by ${\mathfrak m}$ the orthogonal complement of ${\mathfrak k}$ in ${\mathfrak g}$ with respect to the Cartan-Killing form B of ${\mathfrak g}$. Then ${\mathfrak g} = {\mathfrak k} \oplus {\mathfrak m}$ is an Ad(K)-invariant reductive decomposition of ${\mathfrak g}$. We put $o = eK$ and identify $T_oG_2({\mathbb C}^{m+2})$ with ${\mathfrak m}$ in the usual manner. Since B is negative definite on ${\mathfrak g}$, its negative restricted to ${\mathfrak m} \times {\mathfrak m}$ yields a positive definite inner product on ${\mathfrak m}$. By Ad(K)-invariance of B, this inner product can be extended to a G-invariant Riemannian metric g on $G_2({\mathbb C}^{m+2})$. In this way, $G_2({\mathbb C}^{m+2})$ becomes a Riemannian homogeneous space, even a Riemannian symmetric space. For computational reasons we normalize g such that the maximal sectional curvature of $(G_2({\mathbb C}^{m+2}),g)$ is eight.

When $m=1$, $G_2({\mathbb C}^3)$ is isometric to the two-dimensional complex projective space ${\mathbb C}P^2$ with constant holomorphic sectional curvature eight.

When $m=2$, we note that the isomorphism $Spin(6) \simeq SU(4)$ yields an isometry between $G_2({\mathbb C}^4)$ and the real Grassmann manifold $G_2^+({\mathbb R}^6)$ of oriented two-dimensional linear subspaces in ${\mathbb R}^6$. In this paper, we will assume $m{\ge }3$.

The Lie algebra ${\mathfrak k}$ of K has the direct sum decomposition ${\mathfrak k} = {\mathfrak su}(m) \oplus {\mathfrak su}(2) \oplus \mathfrak R$, where $\mathfrak R$ denotes the center of ${\mathfrak k}$. Viewing ${\mathfrak k}$ as the holonomy algebra of $G_2({\mathbb C}^{m+2})$, the center $\mathfrak R$ induces a Kähler structure J and the ${\mathfrak su}(2)$-part a quaternionic Kähler structure ${\mathfrak J}$ on $G_2({\mathbb C}^{m+2})$. If $J_\nu$ is any almost Hermitian structure in ${\mathfrak J}$, then $JJ_\nu = J_\nu J$, and $JJ_\nu$ is a symmetric endomorphism with $(JJ_\nu )^2 = I$ and $\text{ tr }(JJ_\nu ) =0$ for $\nu =1,2,3$.

A canonical local basis $\{J_1,J_2,J_3\}$ of ${\mathfrak J}$ consists of three local almost Hermitian structures $J_\nu$ in ${\mathfrak J}$ such that $J_\nu J_{\nu +1} = J_{\nu +2}=-J_{\nu +1}J_\nu$, where the index $\nu$ is taken modulo three. Since ${\mathfrak J}$ is parallel with respect to the Riemannian connection $\bar{\nabla }$ of $(G_2({\mathbb C}^{m+2}),g)$, there exist for any canonical local basis $\{J_1,J_2,J_3\}$ of ${\mathfrak J}$ three local one-forms $q_1,q_2,q_3$ such that

$$\begin{aligned} \bar{\nabla }_XJ_\nu = q_{\nu +2}(X)J_{\nu +1} - q_{\nu +1}(X)J_{\nu +2} \end{aligned}$$

for all vector fields X on $G_2({\mathbb C}^{m+2})$.

The Riemannian curvature tensor $\bar{R}$ of $G_2({\mathbb C}^{m+2})$ is locally given by

$$\begin{aligned} \begin{aligned} \widetilde{R}(X,Y)Z&= g(Y,Z)X - g(X,Z)Y + \ g(JY,Z)JX \\&\quad - g(J X,Z)JY- 2g(JX,Y)JZ \\&\quad + \ \sum _{\nu =1}^3\Big \{g(J_{\nu } Y,Z)J_\nu X - g(J_\nu X,Z)J_{\nu }Y - 2g(J_\nu X,Y)J_\nu Z\Big \}\\&\quad + \ \sum _{\nu =1}^3\Big \{g(J_\nu JY,Z)J_\nu JX- g(J_\nu JX,Z)J_{\nu }JY\Big \}, \end{aligned} \end{aligned}$$

(2.1)

where $\{J_1,J_2,J_3\}$ denotes a canonical local basis of ${\mathfrak J}$.

Now we summarize basic material about complex hyperbolic two-plane Grassmann manifolds $SU_{2,m}/S(U_{2}{\cdot }U_{m})$, for details we refer to [14, 16].

The Riemannian symmetric space $SU_{2,m}/S(U_2{\cdot }U_m)$, which consists of all complex two-dimensional linear subspaces in indefinite complex Euclidean space ${\mathbb C}_2^{m+2}$, is a connected, simply connected, irreducible Riemannian symmetric space of noncompact type and with rank 2. Let $G = SU_{2,m}$ and $K = S(U_2{\cdot }U_m)$, and denote by ${\mathfrak {g}}$ and ${\mathfrak {k}}$ the corresponding Lie algebra of the Lie group G and K, respectively. Let B be the Killing form of ${\mathfrak {g}}$ and denote by ${\mathfrak {p}}$ the orthogonal complement of ${\mathfrak {k}}$ in ${\mathfrak {g}}$ with respect to B. The resulting decomposition ${\mathfrak {g}} = {\mathfrak {k}} \oplus {\mathfrak {p}}$ is a Cartan decomposition of ${\mathfrak {g}}$. The Cartan involution $\theta \in {\text{ A }ut}({\mathfrak {g}})$ on ${\mathfrak {s}}{\mathfrak {u}}_{2,m}$ is given by $\theta (A) = I_{2,m} A I_{2,m}$, where

$$\begin{aligned} \begin{aligned} I_{2,m} = \begin{pmatrix} -I_{2} &{} 0_{2,m} \\ 0_{m,2} &{} I_{m} \end{pmatrix} \end{aligned} \end{aligned}$$

$I_2$ (resp., $I_m$) denotes the identity $2 \times 2$-matrix (resp., $m \times m$-matrix). Then $< X , Y > = -B(X,\theta Y)$ becomes a positive definite ${\text{ A }d}(K)$-invariant inner product on ${\mathfrak {g}}$. Its restriction to ${\mathfrak {p}}$ induces a metric g on $SU_{2,m}/S(U_2{\cdot }U_m)$, which is also known as the Killing metric on $SU_{2,m}/S(U_2{\cdot }U_m)$. Throughout this paper we consider $SU_{2,m}/S(U_2{\cdot }U_m)$ together with this particular Riemannian metric g.

The Lie algebra ${\mathfrak {k}}$ decomposes orthogonally into ${\mathfrak {k}} = {\mathfrak {s}}{\mathfrak {u}}_2 \oplus {\mathfrak {s}}{\mathfrak {u}}_m \oplus {\mathfrak {u}}_1$, where ${\mathfrak {u}}_1$ is the one-dimensional center of ${\mathfrak {k}}$. The adjoint action of ${\mathfrak {s}}{\mathfrak {u}}_2$ on ${\mathfrak {p}}$ induces the quaternionic Kähler structure ${\mathfrak {J}}$ on $SU_{2,m}/S(U_2{\cdot }U_m)$, and the adjoint action of

$$\begin{aligned} \begin{aligned} Z = \begin{pmatrix} \frac{mi}{m+2}I_2 &{} 0_{2,m} \\ 0_{m,2} &{} \frac{-2i}{m+2}I_m \end{pmatrix} \in {\mathfrak {u}}_1 \end{aligned} \end{aligned}$$

induces the Kähler structure J on $SU_{2,m}/S(U_2{\cdot }U_m)$. By construction, J commutes with each almost Hermitian structure $J_{\nu }$ in ${\mathfrak {J}}$ for ${\nu }=1,2,3$. Recall that a canonical local basis $\{J_1,J_2,J_3\}$ of a quaternionic Kähler structure ${\mathfrak {J}}$ consists of three almost Hermitian structures $J_1,J_2,J_3$ in ${\mathfrak {J}}$ such that $J_{\nu } J_{{\nu }+1} = J_{{\nu } + 2} = - J_{{\nu }+1} J_{\nu }$, where the index ${\nu }$ is to be taken modulo 3. The tensor field $JJ_{\nu }$, which is locally defined on $SU_{2,m}/S(U_2{\cdot }U_m)$, is self-adjoint and satisfies $(JJ_{\nu })^2 = I$ and ${\text{ t }r}(JJ_{\nu }) = 0$, where I is the identity transformation. For a nonzero tangent vector X we define ${\mathbb R}X = \{\lambda X \vert \lambda \in {\mathbb R}\}$, ${\mathbb C}X = {\mathbb R}X \oplus {\mathbb R}JX$, and ${\mathbb H}X = {\mathbb R}X \oplus {\mathfrak {J}}X$.

We identify the tangent space $T_oSU_{2,m}/S(U_2{\cdot }U_m)$ of $SU_{2,m}/S(U_2{\cdot }U_m)$ at o with ${\mathfrak {p}}$ in the usual way. Let ${\mathfrak {a}}$ be a maximal abelian subspace of ${\mathfrak {p}}$. Since $SU_{2,m}/S(U_2{\cdot }U_m)$ has rank 2, the dimension of any such subspace is two. Every nonzero tangent vector $X \in T_oSU_{2,m}/S(U_2{\cdot }U_m) \cong {\mathfrak {p}}$ is contained in some maximal abelian subspace of ${\mathfrak {p}}$. Generically this subspace is uniquely determined by X, in which case X is called regular. If there exists more than one maximal abelian subspaces of ${\mathfrak {p}}$ containing X, then X is called singular. There is a simple and useful characterization of the singular tangent vectors: A nonzero tangent vector $X \in {\mathfrak {p}}$ is singular if and only if $JX \in {\mathfrak {J}}X$ or $JX \perp {\mathfrak {J}}X$.

Up to scaling there exists a unique $SU_{2,m}$-invariant Riemannian metric g on complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_{2}{\cdot }U_{m})$. Equipped with this metric $SU_{2,m}/S(U_{2}{\cdot }U_{m})$ is a Riemannian symmetric space of rank 2 which is both Kähler and quaternionic Kähler. For computational reasons we normalize g such that the minimal sectional curvature of $({SU_{2,m}/S(U_{2}{\cdot }U_{m})},g)$ is $-4$. The sectional curvature K of the noncompact symmetric space $SU_{2,m}/S(U_2{\cdot }U_m)$ equipped with the Killing metric g is bounded by $-4{\le }K{\le }0$. The sectional curvature $-4$ is obtained for all 2-planes ${\mathbb C}X$ when X is a non-zero vector with $JX \in {\mathfrak {J}}X$.

When $m=1$, $G_2^{*}({\mathbb C}^3)=SU_{1,2}/S(U_1{\cdot }U_2)$ is isometric to the two-dimensional complex hyperbolic space ${\mathbb C}H^2$ with constant holomorphic sectional curvature $-4$.

When $m=2$, we note that the isomorphism $SO(4,2)\simeq SU_{2,2}$ yields an isometry between $G_2^{*}({\mathbb C}^4)=SU_{2,2}/S(U_2{\cdot }U_2)$ and the indefinite real Grassmann manifold $G_2^{*}({\mathbb R}_2^6)$ of oriented two-dimensional linear subspaces of an indefinite Euclidean space ${\mathbb R}_2^6$. For this reason we assume $m \ge 3$ from now on, although many of the subsequent results also hold for $m = 1,2$.

Hereafter X,Y and Z always stand for any tangent vector fields on M.

The Riemannian curvature tensor $\bar{R}$ of $SU_{2,m}/S(U_{2}{\cdot }U_{m})$ is locally given by

$$\begin{aligned} \begin{aligned} \bar{R}(X,Y)Z =&- \frac{1}{2}\Big [ g(Y,Z)X - g(X,Z)Y + g(JY,Z)JX\\&\quad - g(JX,Z)JY- 2g(JX,Y)JZ \\&\quad + \sum _{\nu =1}^3\{g(J_{\nu } Y,Z)J_{\nu } X - g(J_{\nu } X,Z)J_{\nu }Y\\&\quad \quad - 2g(J_{\nu } X,Y)J_{\nu } Z\}\\&\quad + \sum _{\nu =1}^3\{g(J_{\nu } JY,Z)J_{\nu } JX - g(J_{\nu } JX,Z)J_{\nu }JY\}\Big ], \end{aligned} \end{aligned}$$

(2.2)

where $\{J_1,J_2,J_3\}$ is any canonical local basis of ${\mathfrak {J}}$.

3 Basic Formulas

In this section we derive some basic formulas and the Codazzi equation for a real hypersurface in $G_2({\mathbb C}^{m+2})$ (or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$) (see [3, 5, 7, 10,11,12, 18]).

Let M be a real hypersurface in $G_2({\mathbb C}^{m+2})$ (or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$). The induced Riemannian metric on M will also be denoted by g, and $\nabla$ denotes the Riemannian connection of (M, g). Let N be a local unit normal vector field of M and A the shape operator of M with respect to N.

Now let us put

$$\begin{aligned} JX={\phi }X+{\eta }(X)N,\quad J_{\nu }X={\phi }_{\nu }X+{\eta }_{\nu }(X)N \end{aligned}$$

(3.1)

for any tangent vector field X of a real hypersurface M in $G_2({\mathbb C}^{m+2})$, where N denotes a unit normal vector field of M in $G_2({\mathbb C}^{m+2})$. From the Kähler structure J of $G_2({\mathbb C}^{m+2})$ (or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$) there exists an almost contact metric structure $(\phi ,\xi ,\eta ,g)$ induced on M in such a way that

$$\begin{aligned} \phi ^{2}X=-X+\eta (X)\xi ,\quad \eta (\xi )=1,\quad \phi \xi =0, \quad \eta (X)=g(X,\xi ) \end{aligned}$$

(3.2)

for any vector field X on M. Furthermore, let $\{J_1,J_2,J_3\}$ be a canonical local basis of ${\mathfrak J}$. Then the quaternionic Kähler structure $J_\nu$ of $G_2({\mathbb C}^{m+2})$ (or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$), together with the condition $J_{\nu }J_{\nu +1} = J_{\nu +2} = -J_{\nu +1}J_{\nu }$ in Sect. 1, induces an almost contact metric 3-structure $(\phi _{\nu }, \xi _{\nu }, \eta _{\nu }, g)$ on M as follows:

$$\begin{aligned} \begin{aligned}&\phi _{\nu }^{2}X=-X+\eta _{\nu }(X)\xi _{\nu },\quad \eta _{\nu }(\xi _{\nu })=1, \quad \phi _{\nu } \xi _{\nu } =0, \\&{\phi }_{\nu +1}{\xi }_{\nu }=-{\xi }_{{\nu }+2},\quad {\phi }_{\nu }{\xi }_{{\nu }+1}={\xi }_{{\nu }+2},\\&{\phi }_{\nu }{\phi }_{{\nu }+1}X = {\phi }_{{\nu }+2}X+{\eta }_{{\nu }+1}(X){\xi }_{\nu },\\&{\phi }_{{\nu }+1}{\phi }_{\nu }X=-{\phi }_{{\nu }+2}X+{\eta }_{\nu }(X) {\xi }_{{\nu }+1} \end{aligned} \end{aligned}$$

(3.3)

for any vector field X tangent to M. Moreover, from the commuting property of $J_{\nu }J=JJ_{\nu }$, ${\nu }=1,2,3$ in Sect. 2 and (3.1), the relation between these two almost contact metric structures $(\phi , \xi , \eta , g)$ and $(\phi _{\nu }, \xi _{\nu }, \eta _{\nu }, g)$, $\nu =1,2,3$, can be given by

$$\begin{aligned} \begin{aligned}&{\phi }{\phi }_{\nu }X ={\phi }_{\nu }{\phi }X+{\eta }_{\nu }(X){\xi }-{\eta }(X) {\xi }_{\nu },\\&\eta _{\nu }(\phi X) = \eta (\phi _{\nu }X), \quad {\phi }{\xi }_{\nu }={\phi }_{\nu }{\xi }. \end{aligned} \end{aligned}$$

(3.4)

On the other hand, from the parallelism of Kähler structure J, that is, $\bar{\nabla } J=0$ and the quaternionic Kähler structure $\mathfrak J$, together with Gauss and Weingarten formulas it follows that

$$\begin{aligned} ({\nabla }_X{\phi })Y={\eta }(Y)AX-g(AX,Y){\xi },\quad {\nabla }_X{\xi }={\phi }AX, \end{aligned}$$

(3.5)

$$\begin{aligned} {\nabla }_X{\xi }_{\nu }=q_{{\nu }+2}(X){\xi }_{{\nu }+1}-q_{{\nu }+1}(X) {\xi }_{{\nu }+2}+{\phi }_{\nu }AX, \end{aligned}$$

(3.6)

$$\begin{aligned} \begin{aligned} ({\nabla }_X{\phi }_{\nu })Y&=-q_{{\nu }+1}(X){\phi }_{{\nu }+2}Y+q_{{\nu }+2}(X) {\phi }_{{\nu }+1}Y +{\eta }_{\nu }(Y)AX-g(AX,Y){\xi }_{\nu }. \end{aligned} \end{aligned}$$

(3.7)

Combining these formulas, we find the following:

$$\begin{aligned} \begin{aligned} {\nabla }_X({\phi }_{\nu }{\xi })&={\nabla }_X({\phi }{\xi }_{\nu })\\&=({\nabla }_X{\phi }){\xi }_{\nu }+{\phi }({\nabla }_X{\xi }_{\nu })\\&=q_{{\nu }+2}(X){\phi }_{{\nu }+1}{\xi }-q_{{\nu }+1}(X){\phi }_{{\nu }+2} {\xi }+{\phi }_{\nu }{\phi }AX\\&\ \ \ -g(AX,{\xi }){\xi }_{\nu }+{\eta }({\xi }_{\nu })AX. \end{aligned} \end{aligned}$$

(3.8)

Using the above expression (2.1) for the curvature tensor $\bar{R}$ of $G_2({\mathbb C}^{m+2})$ (or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$), the equations of Codazzi is given by

$$\begin{aligned} \begin{aligned} k\big \{ (\nabla _XA)Y - (\nabla _YA)X\big \}&= \eta (X)\phi Y - \eta (Y)\phi X - 2g(\phi X,Y)\xi \\&\quad + \sum _{\nu =1}^3 \Big \{\eta _\nu (X)\phi _\nu Y - \eta _\nu (Y)\phi _\nu X - 2g(\phi _\nu X,Y)\xi _\nu \Big \} \\&\quad + \sum _{\nu =1}^3 \Big \{\eta _\nu (\phi X)\phi _\nu \phi Y - \eta _\nu (\phi Y)\phi _\nu \phi X\Big \} \\&\quad + \sum _{\nu =1}^3 \Big \{\eta (X)\eta _\nu (\phi Y) - \eta (Y)\eta _\nu (\phi X)\Big \}\xi _\nu , \end{aligned} \end{aligned}$$

(3.9)

where in the case of $G_2({\mathbb C}^{m+2})$ (resp., $SU_{2,m}/S(U_{2}{\cdot }U_{m})$), the constant $k=1$ and $SU_{2,m}/S(U_{2}{\cdot }U_{m})$ (resp., $k=-2$).

4 Proof of Theorems

In this section, we classify real hypersurfaces in $\bar{M}$ ($G_2({\mathbb C}^{m+2})$ or $SU_{2,m}/S(U_{2}{\cdot }U_{m})$) whose shape operator has Killing tensor field.

From (C-1) and the Codazzi equation (3.9), we have

$$\begin{aligned} \begin{aligned} -2k(\nabla _YA)X&= \eta (X)\phi Y - \eta (Y)\phi X - 2g(\phi X,Y)\xi \\&\quad + \sum _{\nu =1}^3 \Big \{\eta _\nu (X)\phi _\nu Y - \eta _\nu (Y)\phi _\nu X - 2g(\phi _\nu X,Y)\xi _\nu \Big \} \\&\quad + \sum _{\nu =1}^3 \Big \{\eta _\nu (\phi X)\phi _\nu \phi Y - \eta _\nu (\phi Y)\phi _\nu \phi X\Big \} \\&\quad + \sum _{\nu =1}^3 \Big \{\eta (X)\eta _\nu (\phi Y) - \eta (Y)\eta _\nu (\phi X)\Big \}\xi _\nu \end{aligned} \end{aligned}$$

(4.1)

Putting $Y=\xi$ into (4.1),

$$\begin{aligned} -2k(\nabla _{\xi } A)X=-\phi X+\sum _{\nu =1}^3\big \{{\eta }_{\nu }(X)\phi _{\nu }\xi -{\eta }_{\nu }(\xi )\phi _{\nu }X-3{\eta }_{\nu }(\phi X)\xi _{\nu }\big \}. \end{aligned}$$

(4.2)

Lemma 1

Let M be a real hypersurface in complex Grassmannians of rank two $\bar{M}$, $m \ge 3$ with Killing shape operator. Then the Reeb vector field $\xi$ belongs to either the distribution $\mathscr {Q}$ or the distribution ${\mathscr {Q}}^{\bot }$.

Proof

Without loss of generality, $\xi$ is written as

$$\begin{aligned} \xi = \eta (X_{0})X_{0}+ \eta ({\xi }_1){\xi }_1, {(**)} \end{aligned}$$

where $X_{0}$ (resp., ${\xi }_1$) is a unit vector in $\mathscr {Q}$ (resp., ${\mathscr {Q}}^{\bot }$).

Taking the inner product of (4.2) with $\xi$, we have

$$\begin{aligned} -2kg\big ((\nabla _{\xi } A)X,\xi \big )=-4\sum _{\nu =1}^3{\eta }_{\nu }(\xi ){\eta }_{\nu }(\phi X). \end{aligned}$$

(4.3)

Since $(\nabla _{\xi } A)$ is self-adjoint, it follows from (C-1) that $-4\sum _{\nu =1}^3{\eta }_{\nu }(\xi ){\eta }_{\nu }(\phi X)=0$. By putting $X=\phi X_{0}$ and using (**), we have $-4\eta _1^2(\xi )\eta (X_{0})=0$.

Thus we have only two cases: $\xi \in {\mathscr {Q}}^{\bot }$ or $\xi \in \mathscr {Q}$.

• Case 1. $\xi \in {\mathscr {Q}}^{\bot }$.

Without loss of generality, we may put $\xi ={\xi }_1\in {\mathscr {Q}}^{\bot }$. Then (4.2) is reduced into

$$\begin{aligned} -2k(\nabla _{\xi } A)X=-\phi X-\phi _{1}X+2\eta _{3}(X)\xi _{2}-2\eta _{2}(X)\xi _{3}. \end{aligned}$$

(4.4)

The symmetric endomorphism of (4.4) with respect to the metric g, we have

$$\begin{aligned} -2k(\nabla _{\xi } A)X=\phi X+\phi _{1}X-2\eta _{3}(X)\xi _{2}+2\eta _{2}(X)\xi _{3}. \end{aligned}$$

(4.5)

Combining (4.4) with (4.5), we have $\phi X+\phi _{1}X-2\eta _{3}(X)\xi _{2}+2\eta _{2}(X)\xi _{3}=0$. By putting $X={\xi }_3$ into the equation above, we have $2{\xi }_3=0$. This is a contradiction.

Thus, there does not exist any hypersurface in $\bar{M}$, $m\ge 3$, with Killing shape operator and $\xi \in {\mathscr {Q}}^{\bot }$ everywhere.

• Case 2. $\xi \in \mathscr {Q}$.

Equation (4.2) becomes

$$\begin{aligned} -2k(\nabla _{\xi } A)X=-\phi X+\sum _{\nu =1}^3\big \{{\eta }_{\nu }(X)\phi _{\nu }\xi -3{\eta }_{\nu }(\phi X)\xi _{\nu }\big \}. \end{aligned}$$

(4.6)

The symmetric endomorphism of (4.6) with respect to the metric g, we have

$$\begin{aligned} -2k(\nabla _{\xi } A)X=\phi X+\sum _{\nu =1}^3\big \{-{\eta }_{\nu }(\phi X)\xi _{\nu }+3{\eta }_{\nu }(X)\phi \xi _{\nu }\big \}. \end{aligned}$$

(4.7)

Combining (4.6) with (4.7), we have $2\phi X+2\sum _{\nu =1}^3\big \{{\eta }_{\nu }(X)\phi \xi _{\nu }+{\eta }_{\nu }(\phi X)\xi _{\nu }\big \}=0$. By putting $X={\xi }_1$ into above equation, we have $4\phi {\xi }_1=0$. This is a contradiction, too. Thus, there does not exist any hypersurface in $\bar{M}$, $m\ge 3$, with Killing shape operator and $\xi \in \mathscr {Q}$ everywhere.

Accordingly, we complete the proof of Theorem 1 in the introduction.

Usually, the notion of parallel is stronger than the notion of Killing, we also have a non-existence of parallel hypersurface in $SU_{2,m}/S(U_{2}{\cdot }U_{m})$, $m\ge 3$. Then Corollary 1 in the introduction is naturally proved.