Hopf Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Shape Operator

Abstract

In this paper, we consider a new notion of Reeb parallel shape operator for real hypersurfaces $M$ in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. When $M$ has Reeb parallel shape operator and non-vanishing geodesic Reeb flow, it becomes a real hypersurface of Type $(A)$ with exactly four distinct constant principal curvatures. Moreover, if $M$ has vanishing geodesic Reeb flow and Reeb parallel shape operator, then $M$ is model space of Type $(A)$ with the radius $r = \frac{\pi }{4\sqrt{2}}$.

1 Introduction

We denote by $G_{2}(\mathbb C ^{m+2})$ the set of all complex two-dimensional linear subspaces in $\mathbb C^{m+2}$. This Riemannian symmetric space $G_{2}(\mathbb C ^{m+2})$ has a remarkable geometric structure. It is the unique compact irreducible Riemannian manifold with both a Kähler structure $J$ and a quaternionic Kähler structure $\mathfrak {J}$ not containing $J$. Namely, $G_{2}(\mathbb C ^{m+2})$ is a unique compact, irreducible, Kähler, quaternionic Kähler manifold which is not a hyper-Kähler manifold. Accordingly, in $G_{2}(\mathbb C ^{m+2})$ we have two natural geometric conditions for real hypersurfaces $M$: that the $1$-dimensional distribution $[\xi ]= \text {Span} \{\xi \}$ and the $3$-dimensional distribution $\mathfrak {D}^{\bot } = \text {Span}\{\xi _{1},\xi _{2}, \xi _{3}\}$ are invariant under the shape operator $A$ of $M$ (see [2, 3] and [4]).

The almost contact structure vector field $\xi$ is defined by $\xi = -JN$ and is said to be a Reeb vector field, where $N$ denotes a local unit normal vector field of $M$ in $G_{2}(\mathbb C ^{m+2})$. The almost contact 3-structure vector fields $\{\xi _{1},\xi _{2},\xi _{3}\}$ for the 3-dimensional distribution $\mathfrak {D}^{\bot }$ of $M$ in $G_{2}(\mathbb C ^{m+2})$ are defined by $\xi _{\nu } = - J_{\nu } N$ $(\nu =1, 2, 3$), where $J_{\nu }$ denotes a canonical local basis of a quaternionic Kähler structure $\mathfrak J$ and $T_{x}M = \mathfrak D \oplus \mathfrak D^{\bot }$, $x \in M$.

Using two invariant conditions mentioned above and the result in Alekseevskii [1], Berndt and Suh [3] proved the following:

Theorem A

Let $M$ be a connected orientable real hypersurface in $G_{2}(\mathbb C ^{m+2})$, $m \ge 3$. Then both $[\xi ]$ and $\mathfrak D ^{\bot }$ are invariant under the shape operator of $M$ if and only if

• (A) $M$ is an open part of a tube around a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$, or

• (B) $m$ is even, say $m = 2n$, and $M$ is an open part of a tube around a totally geodesic ${\mathbb H}P^n$ in $G_2({\mathbb C}^{m+2})$.

Furthermore, the Reeb vector field $\xi$ is said to be Hopf if it is invariant under the shape operator $A$. The one dimensional foliation of $M$ by the integral manifolds of the Reeb vector field $\xi$ is said to be a Hopf foliation of $M$. We say that $M$ is a Hopf hypersurface in $G_2({\mathbb C}^{m+2})$ if and only if the Hopf foliation of $M$ is totally geodesic. By the formulas in Sect. 3, it can be easily checked that $M$ is Hopf if and only if the Reeb vector field $\xi$ is Hopf. In particular, $M$ is said to be a real hypersurface with non-vanishing geodesic Reeb flow in $G_2({\mathbb C}^{m+2})$ if it has a nonzero principal curvature for the Reeb vector field $\xi$, that is, $A\xi =\alpha \xi$ where $\alpha =g(A\xi , \xi ) \ne 0$.

Using Theorem $\mathrm A$, many geometers have given characterizations for Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ under certain assumption for various geometry quantities, for instance, shape operator, normal (or structure) Jacobi operator, structure tensor, and so on.

In [4], Berndt and Suh considered some equivalent conditions of isometric Reeb flow. Here, the Reeb flow on $M$ in $G_2({\mathbb C}^{m+2})$ is isometric means the Reeb vector field $\xi$ on $M$ is Killing. Using this notion, they gave a characterization of real hypersurfaces of Type $(A)$ in Theorem $\mathrm A$ as follows:

Theorem B

Let $M$ be a connected orientable real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$. Then the Reeb flow on $M$ is isometric if and only if $M$ is an open part of a tube around a totally geodesic $G_{2}(\mathbb {C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.

Among the equivalent conditions of isometric Reeb flow in [4], it is very useful to our proof in Sect. 5 that the Reeb flow on $M$ is isometric if and only if the shape operator $A$ and the structure tensor field $\phi$ commute with each other, that is, $A\phi = \phi A$.

Moreover, Lee and Suh [9] gave a characterization of real hypersurfaces of Type $(B)$ in $G_2({\mathbb C}^{m+2})$ in terms of the Reeb vector field $\xi$ as follows:

Theorem C

Let $M$ be a connected orientable Hopf hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$. Then the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D$ if and only if $M$ is locally congruent to an open part of a tube around a totally geodesic ${\mathbb H}P^n$ in $G_2({\mathbb C}^{m+2})$, $m=2n$, where the distribution $\mathfrak D$ denotes the orthogonal complement of $\mathfrak D^{\bot }=\mathrm{Span}\{{\xi }_1, {\xi }_2, {\xi }_3\}$.

In [11], Suh proved the non-existence of real hypersurfaces in $G_2({\mathbb C}^{m+2})$ with parallel shape operator, that is, $(\nabla _{X} A)Y =0$, where $X$ and $Y$ are any tangent vector field on $M$. Moreover, he [12] also considered a new condition which is to restrict $X$ to a distribution $\mathfrak F=[\xi ] \cup \mathfrak D^{\bot }$, namely $\mathfrak {F}$ -parallel shape operator, and gave two non-existence theorems to the following two cases of real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with $\mathfrak F$-parallel shape operator: One is when $M$ is a Hopf hypersurface. Another is when $M$ is a real hypersurface in $G_2({\mathbb C}^{m+2})$ satisfying $\mathfrak D^{\bot }$-invariance under the shape operator, that is, $A \mathfrak D^{\bot } \subset \mathfrak D^{\bot }$. As regards a weaker condition for a real hypersurface in $G_2({\mathbb C}^{m+2})$ with parallel shape operator, in [6] and [8] Kim, Yang and the first author considered recurrent and $\eta$ -parallel shape operator and gave non-existence theorems of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ satisfying such weaker parallelism conditions, respectively.

Motivated by these notions, it is natural to consider a condition weaker than parallel shape operator for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$. From such a point of view, the authors in  [5] studied a generalized parallelness for the shape operator of $M$ in $G_2({\mathbb C}^{m+2})$, namely $\eta$-parallel shape operator of $M$. They defined the $\eta$ -parallel shape operator of $M$ in $G_2({\mathbb C}^{m+2})$ if the shape operator $A$ of $M$ satisfies $g((\nabla _{X}A)Y, Z) =0$ for any tangent vectors $X,Y,Z \in \mathfrak {h}$, where $\mathfrak h$ denotes the set of all tangent vectors being orthogonal to the Reeb vector $\xi$ in $T_{x}M$, $x \in M$. From this definition, we see that it becomes a weaker condition than parallel shape operator.

Accordingly, we consider a new notion weaker than parallel shape operator, that is, Reeb parallel shape operator which is defined by

$$\begin{aligned} (\nabla _{\xi }A)Y=0 \end{aligned}$$

(*)

for any tangent vector field $Y$ on $M$.

In this paper, we give a classification of real hypersurfaces in $G_2({\mathbb C}^{m+2})$ with Reeb parallel shape operator as follows:

Theorem 1

Let $M$ be a connected orientable real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with non-vanishing geodesic Reeb flow. Then the shape operator of $M$ is Reeb parallel if and only if $M$ is an open part of a tube of some radius $r \in (0, \frac{\pi }{4\sqrt{2}})\cup (\frac{\pi }{4\sqrt{2}}, \frac{\pi }{2\sqrt{2}})$ around a totally geodesic $G_{2}(\mathbb C^{m+1})$ in $G_2({\mathbb C}^{m+2})$.

Actually, when the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D^{\bot }$, the shape operator $A$ for a Hopf hypersurface $M$ in $G_2({\mathbb C}^{m+2})$ with vanishing geodesic Reeb flow satisfies automatically the Reeb parallelness (see Sect. 6). Using this fact, we give:

Theorem 2

Let $M$ be a connected orientable real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with Reeb parallel shape operator and vanishing geodesic Reeb flow. If the squared norm of the shape operator satisfies $\text {Tr}A^{2}=||A||^{2} \le 4m$, then $M$ is locally congruent to an open part of a tube of radius $r= \frac{\pi }{4\sqrt{2}}$ around a totally geodesic $G_{2}(\mathbb C^{m+1})$ in $G_2({\mathbb C}^{m+2})$.

In order to give the proof of our theorems, in Sect. 2 we recall Riemannian geometry of complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. In Sect. 3, some fundamental formulas including the Codazzi and Gauss equations for real hypersurfaces in $G_2({\mathbb C}^{m+2})$ will be also recalled. In Sect. 4, we will prove that the Reeb vector field $\xi$ of a Hopf hypersurface in $G_2({\mathbb C}^{m+2})$ with Reeb parallel shape operator belongs to either the distribution $\mathfrak D$ or the distribution $\mathfrak D^{\bot }$. And in the same section, we will check whether real hypersurfaces of Type $(A)$ or Type $(B)$ in Theorem $A$ satisfy the condition (*) or not. In Sect. 5, we will give a complete proof of our Theorem 1 according to the non-vanishing geodesic Reeb flow. Finally, we will give the proof of Theorem 2 in Sect. 6.

2 Riemannian Geometry of $G_2({\mathbb C}^{m+2})$

In this section, we summarize basic material about $G_2({\mathbb C}^{m+2})$, for details we refer to [2, 3], and [4]. By $G_2({\mathbb C}^{m+2})$ we denote 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 $\widetilde{\nabla }$ 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

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

(2.1)

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

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

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

(2.2)

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

3 Some Fundamental Formulas

In this section, we derive some basic formulas and the Codazzi equation for a real hypersurface in $G_2({\mathbb C}^{m+2})$ (see [9–12] and [7]).

Let $M$ be a real hypersurface of $G_2({\mathbb C}^{m+2})$, that is, a submanifold of $G_2({\mathbb C}^{m+2})$ with real codimension one. 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 $G_2({\mathbb C}^{m+2})$. From the Kähler structure $J$ of $G_2({\mathbb C}^{m+2})$ 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 $G_2({\mathbb C}^{m+2})$, 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:

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

(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 contact metric structures $(\phi , \xi , \eta , g)$ and $(\phi _{\nu }, \xi _{\nu }, \eta _{\nu }, g)$, $\nu =1,2,3$, can be given by

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

(3.4)

On the other hand, from the parallelism of Kähler structure $J$, that is, $\widetilde{\nabla }J=0$ and the quaternionic Kähler structure $\mathfrak J$ (see (2.1)), 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)

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

(3.7)

Combining these formulas, we find the following:

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

(3.8)

Using the above expression (2.2) for the curvature tensor $\widetilde{R}$ of $G_2({\mathbb C}^{m+2})$, the equations of Codazzi and Gauss are respectively given by

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

(3.9)

and

$$\begin{aligned} R (X,Y)Z&= g(Y,Z)X - g(X,Z)Y\nonumber \\&+\, g(\phi Y,Z)\phi X - g(\phi X, Z)\phi Y -2g(\phi X, Y)\phi Z\nonumber \\&+\, \ \sum _{\nu =1}^{3} \Big \{g(\phi _{\nu }Y, Z) \phi _{\nu }X - g(\phi _{\nu }X, Z) \phi _{\nu }Y -2g(\phi _{\nu }X, Y) \phi _{\nu } Z \Big \}\nonumber \\&+\, \ \sum _{\nu =1}^3\Big \{g(\phi _{\nu }\phi Y, Z) \phi _{\nu }\phi X - g(\phi _{\nu }\phi X, Z) \phi _{\nu }\phi Y \Big \}\nonumber \\&-\, \ \sum _{\nu =1}^3\Big \{\eta (Y) \eta _{\nu }(Z) \phi _{\nu }\phi X -\eta (X) \eta _{\nu }(Z) \phi _{\nu }\phi Y \Big \}\nonumber \\&-\,\sum _{\nu =1}^{3} \Big \{\eta (X) g(\phi _{\nu }\phi Y, Z) - \eta (Y) g(\phi _{\nu }\phi X, Z)\Big \}\xi _{\nu }\nonumber \\&+\,g(AY,Z)AX -g(AX, Z)AY, \end{aligned}$$

(3.10)

where $R$ denotes the curvature tensor of a real hypersurface $M$ in $G_2({\mathbb C}^{m+2})$.

4 Hopf Hypersurfaces in $G_2({\mathbb C}^{m+2})$ with Reeb Parallel Shape Operator

From now on, we assume that $M$ is a Hopf hypersurface in $G_2({\mathbb C}^{m+2})$ with Reeb parallel shape operator, that is, the shape operator $A$ of $M$ satisfies:

$$\begin{aligned} (\nabla _{\xi }A)Y=0 \end{aligned}$$

(*)

for any tangent vector field $Y$ on $M$.

Then from the equation of Codazzi (3.9), we have

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

for any tangent vector field $Y$ on $M$.

Since $(\nabla _{Y}A)\xi = (Y\alpha ) \xi + \alpha \phi AY -A\phi A Y$, the condition (*) can be written as

$$\begin{aligned}&(Y\alpha ) \xi + \alpha \phi AY -A\phi A Y + \phi Y\nonumber \\&\quad \quad + \sum _{\nu =1}^{3} \big \{ \eta _{\nu }(\xi ) \phi _{\nu } Y - \eta _{\nu }(Y) \phi _{\nu }\xi +3\eta _{\nu }(\phi Y) \xi _{\nu } \big \}=0 \end{aligned}$$

(4.1)

for any tangent vector field $Y$ on $M$.

Substituting $Y=\xi$ in above equation, we have $(\xi \alpha )\xi =0$. From this, we obtain the following result:

Lemma 4.1

Let $M$ be a Hopf hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with Reeb parallel shape operator. Then the principal curvature $\alpha$ is constant along the direction of $\xi$, that is, $\xi \alpha =0$.

In this section, our main purpose is to show that the Reeb vector field $\xi$ belongs to either the distribution $\mathfrak D$ or the orthogonal complement $\mathfrak D^{\bot }$ such that $T_{x}M=\mathfrak D \oplus \mathfrak D^{\bot }$ for any point $x \in M$.

To show this fact, unless otherwise stated in this section, we consider that the Reeb vector field $\xi$ satisfies

$$\begin{aligned} \xi = \eta (X_{0})X_{0}+\eta (\xi _{1})\xi _{1} \end{aligned}$$

(**)

for some unit vectors $X_{0} \in \mathfrak D$ and $\xi _{1} \in \mathfrak D^{\bot }$ and $\eta (X_{0}) \eta (\xi _{1})\ne 0$.

Remark 4.2

Under this situation, in [7] the authors proved that $\mathfrak D$ and $\mathfrak D^{\bot }$-components of the Reeb vector field $\xi$ are invariant under the shape operator when a Hopf hypersurface in $G_2({\mathbb C}^{m+2})$ satisfies the condition $\xi \alpha =0$.

On the other hand, using the notion of the geodesic Reeb flow, Berndt and Suh ([3, 4]) proved the following:

Lemma A

If $M$ is a connected orientable real hypersurface in $G_2({\mathbb C}^{m+2})$ with geodesic Reeb flow, then we have the following two equations:

$$\begin{aligned} Y\alpha = (\xi \alpha ) \eta (Y) - 4 \sum _{\nu =1}^{3}\eta _{\nu }(\xi )\eta _{\nu }(\phi Y), \end{aligned}$$

(4.2)

and

$$\begin{aligned} \alpha A\phi Y \!+\! \alpha \phi A Y \!-\!2 A\phi AY \!+\! 2 \phi Y&= 2\displaystyle \sum _{\nu =1}^3 \Big \{\!-\!\eta _\nu (Y) \phi \xi _{\nu } \!-\! \eta _\nu (\phi Y) \xi _{\nu }\nonumber \\&\!-\! \eta _\nu (\xi ) \phi _{\nu }Y \!+\! 2\eta (Y) \eta _\nu (\xi ) \phi \xi _{\nu } \!+\! 2 \eta _\nu (\phi Y)\eta _\nu (\xi ) \xi \Big \}\nonumber \\ \end{aligned}$$

(4.3)

for any tangent vector field $Y$ on $M$.

Remark 4.3

Assume that the $\mathfrak D$-component of $\xi$ is invariant under the shape operator $A$, that is, $AX_{0}=\alpha X_{0}$. By putting $Y=X_{0}$ in (4.3) and using the fact $\phi X_{0}=-\eta (\xi _{1})\phi _{1}X_{0}$ which is induced by $\phi \xi =0$, we see that

$$\begin{aligned} \alpha A\phi X_0 =\big (\alpha ^2 +4\eta ^2(X_0)\big )\,\phi X_0. \end{aligned}$$

(4.4)

Now, using these facts, we prove the following proposition:

Proposition 4.4

Let $M$ be a real hypersurface in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$, $m \ge 3$ with Reeb parallel shape operator. Then the Reeb vector field $\xi$ belongs to either the distribution $\mathfrak D$ or the distribution $\mathfrak D^{\bot }$.

Proof

Under our assumption, $M$ is a Hopf hypersurface with Reeb parallel shape operator, we see that $\xi \alpha =0$ (Lemma 4.1). Moreover, we know that $\mathfrak D$ and $\mathfrak D^{\bot }$-components of the Reeb vector field $\xi$ are invariant under the shape operator $A$ of $M$, that is, $A\xi _{1}=\alpha \xi _{1}$ and $AX_{0}=\alpha X_{0}$ (see Remark 4.2).

Actually, when the smooth function $\alpha =g(A\xi , \xi )$ identically vanishes, this proposition can be verified directly from (4.2).

Thus, in this proof, we consider only the case that the function $\alpha$ is non-vanishing. In order to prove our proposition, we put $Y=X_{0}$ in (4.1). It follows

$$\begin{aligned} (X_{0}\alpha ) \xi + \alpha \phi AX_{0}-A\phi A X_{0}+ \phi X_{0}+ \sum _{\nu =1}^{3} \big \{ \eta _{\nu }(\xi ) \phi _{\nu } X_{0}+3\eta _{\nu }(\phi X_{0}) \xi _{\nu } \big \}=0. \end{aligned}$$

Since $\phi X_{0}\in \mathfrak {D}$ and $AX_{0}=\alpha X_{0}$, it becomes

$$\begin{aligned} (X_{0}\alpha ) \xi + \alpha ^{2} \phi X_{0}- \alpha A\phi X_{0}+ \phi X_{0}+ \sum _{\nu =1}^{3}\eta _{\nu }(\xi ) \phi _{\nu } X_{0}=0. \end{aligned}$$

Moreover, using (**), we have

$$\begin{aligned} \eta (X_{0})(X_{0}\alpha ) X_{0}+ \eta ({\xi }_1) (X_{0}\alpha ) {\xi }_1+ \alpha ^{2} \phi X_{0}- \alpha A\phi X_{0}+ \phi X_{0}+ \eta ({\xi }_1) \phi _{1} X_{0}=0. \end{aligned}$$

From (4.4), we obtain

$$\begin{aligned} \eta (X_{0})(X_{0}\alpha ) X_{0}+ \eta ({\xi }_1) (X_{0}\alpha ) {\xi }_1- 4 \eta ^{2}(X_{0})\phi X_{0}+ \phi X_{0}+ \eta ({\xi }_1) \phi _{1} X_{0}=0. \end{aligned}$$

Using $\phi \xi =0$ and (**), we see that $\phi X_{0}= -\eta ({\xi }_1)\phi _{1}X_{0}$. It implies that

$$\begin{aligned} \eta (X_{0})(X_{0}\alpha ) X_{0}+ \eta ({\xi }_1) (X_{0}\alpha ) {\xi }_1+ 4 \eta ^{2}(X_{0})\eta ({\xi }_1)\phi _{1} X_{0}=0. \end{aligned}$$

(4.5)

Taking the inner product with $\phi _{1}X_{0}$ in (4.5), we get $4\eta ^{2}(X_{0})\eta ({\xi }_1)=0$. It contradicts our assumption $\eta (X_{0})\eta ({\xi }_1)\ne 0$. Accordingly, we get a complete proof of our Proposition 4.4. $\square$

Before giving the proofs of our Theorems in the introduction, let us check whether the shape operator $A$ of real hypersurfaces of Type $(A)$ and of Type $(B)$ in Theorem $\mathrm A$ satisfies the condition (*) or not for any tangent vector field $Y \in TM$.

First, let us check our problem for the case that $M$ is locally congruent to a real hypersurface of Type $(A)$, that is, an open part of a tube around a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$ with some radius $r \in (0, \frac{\pi }{2\sqrt{2}})$. In order to do this, we recall a proposition due to Berndt and Suh [3] as follows :

Proposition A

Let $M$ be a connected real hypersurface of $G_2({\mathbb C}^{m+2})$. Suppose that $A{\mathfrak D} \subset {\mathfrak D}$, $A\xi = \alpha \xi$, and $\xi$ is tangent to ${\mathfrak D}^{\bot }$. Let $J_1 \in \mathfrak {J}$ be the almost Hermitian structure such that $JN=J_1 N$. Then $M$ has three (if $r=\pi /2\sqrt{8})$ or four (otherwise) distinct constant principal curvatures

$$\begin{aligned} \alpha = \sqrt{8}\cot (\sqrt{8}r),\quad \beta = \sqrt{2}\cot (\sqrt{2}r),\quad \lambda =-\sqrt{2}\tan (\sqrt{2}r),\quad \mu = 0 \end{aligned}$$

with some $r \in (0,\pi /\sqrt{8})$. The corresponding multiplicities are

$$\begin{aligned} m(\alpha ) = 1,\quad m(\beta ) = 2, \quad m(\lambda )=2m-2=m(\mu ), \end{aligned}$$

and the corresponding eigenspaces are

$$\begin{aligned} T_\alpha&= {\mathbb R}\xi ={\mathbb R}JN={\mathbb R}\xi _1=\mathrm{Span}\big \{ \xi \big \}=\mathrm{Span}\big \{ \xi _{1} \big \},\\ T_\beta&= {\mathbb C}^{\bot }\xi ={\mathbb C}^{\bot }N={\mathbb R} \xi _2 \oplus {\mathbb R} \xi _3=\mathrm{Span}\big \{ \xi _{2},\ \xi _{3} \big \},\\ T_\lambda&= \{X| \, X \perp {\mathbb H}\xi ,\ JX=J_1 X\,\},\\ T_\mu&= \{X | \, X\perp {\mathbb H}\xi ,\ JX=-J_1 X\,\}, \end{aligned}$$

where ${\mathbb R}\xi$, ${\mathbb C}\xi$, and ${\mathbb H}\xi$, respectively, denote real, complex, and quaternionic span of the structure vector field $\xi$, and ${\mathbb C}^{\bot }\xi$ denotes the orthogonal complement of ${\mathbb C}\xi$ in ${\mathbb H}\xi$.

For our convenience, let $M_{A}$ be a real hypersurface of Type $(A)$ in $G_2({\mathbb C}^{m+2})$. Using the equation of Codazzi (3.9) and the fact that the principal curvature $\alpha$ of $\xi$ is a constant, we have the following equation.

$$\begin{aligned} (\nabla _{\xi } A) Y = \alpha \phi AY - A\phi AY + \phi Y + \phi _{1}Y + 2\eta _{3}(Y) \xi _{2}- 2\eta _{2}(Y) \xi _{3}\end{aligned}$$

(4.6)

for any tangent vector field $Y$ on $M$.

From now on, using (4.6), let us check whether each eigenspace, $T_{\alpha }$, $T_{\beta }$, $T_{\lambda }$, and $T_{\mu }$ of $M_{A}$ in $G_2({\mathbb C}^{m+2})$, has Reeb parallel shape operator or not.

Case A-1 :   $Y=\xi (=\xi _1) \in T_\alpha$.

By putting $Y=\xi$ into (4.6), we know that the shape operator $A$ becomes Reeb parallel, that is, $(\nabla _{\xi }A) \xi =0$.

Case A-2 :   $Y \in T_{\beta }$ where $T_{\beta }=\text {Span}\{ \xi _{2}, \xi _{3}\}$.

Since $T_{\beta }$ is spanned by $\xi _{2}$ and $\xi _{3}$, we put $Y =\xi _{2}$ and $Y=\xi _{3}$ in (4.6). Then we have

$$\begin{aligned} (\nabla _{\xi }A)\xi _{2}=(\beta ^2 -\alpha \beta -2)\xi _3 \end{aligned}$$

and

$$\begin{aligned} (\nabla _{\xi }A) \xi _{3}=-(\beta ^2 -\alpha \beta -2)\xi _2, \end{aligned}$$

respectively. On the other hand, we know that

$$\begin{aligned} \beta ^2 -\alpha \beta -2=0, \end{aligned}$$

because $\alpha = \sqrt{8} \cot (\sqrt{8}r)$ and $\beta = \sqrt{2} \cot (\sqrt{2}r)$ in Proposition $\mathrm A$. So we conclude that the shape operator of $M_{A}$ also satisfies $(\nabla _{\xi }A)Y=0$ for any eigenvector $Y \in T_{\beta }$.

Case A-3 :   $Y \in T_\lambda =\{\,Y\,| \ Y \in \mathfrak D,\ JY=J_1 Y\,\}$.

We naturally see that if $Y \in T_{\lambda }$ then $\phi Y = \phi _{1} Y$. Moreover, the vector $\phi Y$ also belong to the eigenspace $T_{\lambda }$ for any $Y\in T_{\lambda }$, that is, $\phi T_{\lambda } \subset T_{\lambda }$. Putting $Y \in T_{\lambda }$ in (4.6) and together with these facts, we obtain

$$\begin{aligned} (\nabla _{\xi } A)Y=-(\lambda ^2 -\alpha \lambda -2)\phi Y. \end{aligned}$$

But in Proposition $\mathrm A$, since the principal curvatures $\alpha$ and $\lambda$ are given by $\alpha =\sqrt{8} \cot (\sqrt{8}r)$ and $\lambda = -\sqrt{2} \tan (\sqrt{2}r)$ for $r\in (0,\pi /\sqrt{8})$, respectively, we get

$$\begin{aligned} \lambda ^2 -\alpha \lambda -2=0. \end{aligned}$$

It implies that $(\nabla _{\xi } A)Y=0$ for any tangent vector field $Y \in T_{\lambda }$.

Case A-4 :   $Y \in T_\mu =\{\,Y\,| \ Y \in {\mathfrak D},\ JY=-J_1 Y\,\}$.

In this case, if $Y\in T_{\mu }$ then $\phi Y = -\phi _{1}Y$. Moreover, we see $\phi T_{\mu } \subset T_{\mu }$. So, (4.6) is reduced to $(\nabla _{\xi } A)X=0$, because $\mu =0$.

Summing up all cases mentioned above, we can assert that:

Remark 4.5

The shape operator $A$ of real hypersurfaces of Type $(A)$ in $G_2({\mathbb C}^{m+2})$ is Reeb parallel.

Next, let us check whether the shape operator $A$ of real hypersurfaces of Type $(B)$ satisfies the condition (*) for any tangent vector field $Y \in TM$. From now on, we will denote such real hypersurfaces by $M_{B}$ for the sake of convenience. As is well known, $M_{B}$ has five distinct constant principal curvatures as follows [3]:

Proposition B

Let $M$ be a connected real hypersurface of $G_2({\mathbb C}^{m+2})$. Suppose that $A \mathfrak {D} \subset \mathfrak {D}$, $A\xi =\alpha \xi$, and $\xi$ is tangent to $\mathfrak {D}$. Then the quaternionic dimension $m$ of $G_2({\mathbb C}^{m+2})$ is even, say $m=2n$, and $M$ has five distinct constant principal curvatures

$$\begin{aligned} \alpha = -2 \tan (2r), \quad \beta =2 \cot (2r), \quad \gamma =0, \quad \lambda =\cot (r), \quad \mu =-\tan (r) \end{aligned}$$

with some $r \in ( 0, \pi /4)$. The corresponding multiplicities are

$$\begin{aligned} m(\alpha )=1, \quad m(\beta )=3=m(\gamma ), \quad m(\lambda )=4n-4=m(\mu ) \end{aligned}$$

and the corresponding eigenspaces are

$$\begin{aligned} T_{\alpha }&= \mathbb {R}\xi =\mathrm{Span}\big \{ \xi \big \},\nonumber \\ T_{\beta }&= \mathfrak {J}J\xi = \mathrm{Span}\big \{ \xi _{\nu }\,|\ \nu =1,2,3 \, \big \},\nonumber \\ T_{\gamma }&= \mathfrak {J}\xi = \mathrm{Span}\big \{ \phi _{\nu }\xi \,|\ \nu =1,2,3 \,\big \},\nonumber \\&T_{\lambda },\quad T_{\mu }, \end{aligned}$$

where

$$\begin{aligned} T_{\lambda }\oplus T_{\mu } = ( \mathbb {H}\mathbb {C}\xi )^{\bot }, \quad \mathfrak {J}T_{\lambda }=T_{\lambda },\quad \mathfrak {J}T_{\mu }= T_{\mu },\quad JT_{\lambda }=T_{\mu }. \end{aligned}$$

The distribution $(\mathbb {H}\mathbb {C}\xi )^{\bot }$ is the orthogonal complement of $\mathbb {H}\mathbb {C}\xi$ where

$$\begin{aligned} \mathbb {H}\mathbb {C}\xi =\mathbb {R}\xi \oplus \mathbb {R}J\xi \oplus \mathfrak {J}\xi \oplus \mathfrak {J}J \xi . \end{aligned}$$

Now, to prove the claim, we suppose that $M_{B}$ has Reeb parallel shape operator. Then, since $\xi \in \mathfrak D$, $M_{B}$ satisfies the following equation:

$$\begin{aligned} \alpha \phi AY - A \phi AY + \phi Y - \sum _{\nu =1}^{3} \big \{\eta _{\nu }(Y) \phi _{\nu }\xi - 3 \eta _{\nu } (\phi Y) \xi _{\nu }\big \}=0, \quad \forall Y \in TM , \end{aligned}$$

from (4.1).

If we put $Y=\xi _{2}\in T_{\beta }$ in above equation, it becomes

$$\begin{aligned} \alpha \beta \phi \xi _{2}=0 \end{aligned}$$

because $A\phi _{2}\xi = \gamma \phi _{2}\xi$ and $\gamma =0$. From this, it follows that

$$\begin{aligned} \alpha \beta =0. \end{aligned}$$

But, from Proposition $\mathrm B$, we see that $\alpha \beta =-4$ for some $r \in (0, \pi / 4)$. This is a contradiction. So this case can not occur.

Therefore we also have the following:

Remark 4.6

The shape operator $A$ of real hypersurfaces of Type $(B)$ in $G_2({\mathbb C}^{m+2})$ does not satisfy the Reeb parallel condition (*).

5 The Proof of Theorem 1

In this section, let $M$ be a real hypersurface in $G_2({\mathbb C}^{m+2})$ with Reeb parallel shape operator and non-vanishing geodesic Reeb flow.

In [4], Berndt and Suh proved that

Lemma B

Let $M$ be a connected orientable real hypersurface in Kähler manifolds. Then the following statements are equivalent:

• (a) The Reeb flow on $M$ is geodesic;

• (b) The Reeb vector field $\xi$ is a principal curvature vector of $M$ everywhere;

• (c) The maximal complex subbundle $\mathfrak {B}$ of $TM$ is invariant under the shape operator $A$ of $M$.

From this, we see that a real hypersurface $M$ satisfying our condition becomes Hopf, since our real hypersurface $M$ has non-vanishing geodesic Reeb flow. Thus by Proposition 4.4, we consider the following two cases:

• Case I :   the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D^{\bot }$,

• Case II :   the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D$.

First of all, let us consider the Case I, that is, $\xi \in \mathfrak D^{\bot }$. Accordingly, we may put $\xi =\xi _1$. Under these assumptions, we prove the following:

Proposition 5.1

Let $M$ be a real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with Reeb parallel shape operator and non-vanishing geodesic Reeb flow. If the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D^{\bot }$, then the shape operator $A$ commutes with the structure tensor field $\phi$.

Proof

From our assumptions, (4.1) can be written as

$$\begin{aligned} (Y \alpha ) \xi + \alpha \phi AY - A\phi AY + \phi Y + \phi _{1}Y + 2\eta _{3}(Y) \xi _{2}-2\eta _{2}(Y)\xi _{3}=0 \end{aligned}$$

for any tangent vector field $Y$ on $M$. It follows that

$$\begin{aligned} 2 A\phi AY = 2(Y \alpha ) \xi + 2 \alpha \phi AY + 2 \phi Y + 2 \phi _{1}Y + 4\eta _{3}(Y) \xi _{2}- 4\eta _{2}(Y)\xi _{3}. \end{aligned}$$

On the other hand, from (4.3) we also obtain

$$\begin{aligned} 2 A \phi AY = \alpha A\phi Y + \alpha \phi AY + 2 \phi Y + 2 \phi _{1}Y + 4\eta _{3}(Y) \xi _{2}- 4\eta _{2}(Y) \xi _{3}. \end{aligned}$$

Thus from the preceding two equations, we have finally

$$\begin{aligned} 2 (Y\alpha )\xi + \alpha \phi AY - \alpha A \phi Y =0 \end{aligned}$$

(5.1)

for any tangent vector field $Y$ on $M$.

But, under our assumptions, we have already seen that $\xi \alpha =0$ (see Lemma 4.1). From this fact, (4.2) can be written as

$$\begin{aligned} Y \alpha = - 4 \sum _{\nu =1}^{3} \eta _{\nu }(\xi )\eta _{\nu }(\phi Y) \end{aligned}$$

for any $Y \in TM$. Therefore since $\xi = {\xi }_1$, it follows that $Y \alpha =0$. Substituting this result into (5.1), it follows that

$$\begin{aligned} \alpha (\phi A - A \phi )Y=0 \end{aligned}$$

for any tangent vector field $Y$ on $M$. It means that the shape operator $A$ commutes with the structure tensor field $\phi$ on $M$ in $G_2({\mathbb C}^{m+2})$, since $M$ has non-vanishing geodesic Reeb flow. It completes the proof of our Proposition 5.1.

Remark 5.2

As mentioned in the introduction, the structure tensor field $\phi$ and the shape operator $A$ of $M$ commute with each other if and only if the Reeb flow on $M$ is isometric (see [4]).

Therefore from Theorem $\mathrm B$ and Remark 4.5, we have the following:

Theorem 5.3

Let $M$ be a connected real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$ with non-vanishing geodesic Reeb flow. The shape operator $A$ of $M$ is Reeb parallel and the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D^{\bot }$ if and only if $M$ is locally congruent to an open part of a tube around radius $r$ on a totally geodesic $G_{2}(\mathbb C^{m+1})$ in $G_2({\mathbb C}^{m+2})$ where $r \in (0, \frac{\pi }{4\sqrt{2}}) \cup (\frac{\pi }{4\sqrt{2}}, \frac{\pi }{2\sqrt{2}})$.

Next, we consider the case $\xi \in \mathfrak D$. By Theorem $\mathrm C$, we see that $M$ is locally congruent to a real hypersurface of Type $(B)$ under our assumptions. But as mentioned in Sect. 4, a real hypersurface of Type $(B)$ does not have Reeb parallel shape operator (see Remark 4.6). From these facts, we obtain the following theorem:

Theorem 5.4

There does not exist any real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with Reeb parallel shape operator and non-vanishing geodesic Reeb flow when the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D$.

Combining Proposition 4.4, and Theorems 5.3 and 5.4, this completes the proof of our Theorem 1 in the introduction.

6 The Proof of Theorem 2

From now on, let $M$ be a real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with Reeb parallel shape operator and vanishing geodesic Reeb flow. By virtue of Lemma $\mathrm B$ given in the previous section, $M$ must be Hopf, that is, $A\xi =\alpha \xi$ where $\alpha =g(A\xi , \xi )=0$. Then by Proposition 4.4, we consider the following two cases:

• Case I :   the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D^{\bot }$,

• Case II :   the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D$.

By virtue of Theorem $\mathrm C$ and Proposition $\mathrm B$, we assert that when the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D$, there does not exist any real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with Reeb parallel shape operator and vanishing geodesic Reeb flow. In fact, a real hypersurface of Type $(B)$ in Theorem $A$ due to Berndt and Suh [3] does not have vanishing geodesic Reeb flow (see Proposition $\mathrm B$ in Sect. 4).

From such a point of view, from now on we only consider the Case I, that is, $\xi \in \mathfrak D^{\bot }$. Accordingly, we may put $\xi =\xi _1$. Under these assumptions, we prove the following:

Proposition 6.1

Let $M$ be a real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with Reeb parallel shape operator and vanishing geodesic Reeb flow. If the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D^{\bot }$ and the squared norm of the shape operator satisfies $||A||^{2} \le 4m$, then the Reeb flow on $M$ is isometric.

Proof

Since $M$ has vanishing geodesic Reeb flow, that is, $A\xi =0$, we obtain

$$\begin{aligned} (\nabla _{X}A)\xi = - A \phi AX \end{aligned}$$

for any tangent vector field $X$ on $M$. Using the equation of Codazzi (3.9), we get

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

together with our assumptions that $M$ has Reeb parallel shape operator and $\xi ={\xi }_1$ (since we now consider the case $\xi \in \mathfrak D^{\bot }$, we may put $\xi ={\xi }_1$). Hence the above two equations give us

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

(6.1)

for any vector field $X \in TM$.

Moreover, applying the structure tensor $\phi$ to (6.1), it can be written as

$$\begin{aligned} \phi A \phi AX&= \phi ^{2} X + \phi \phi _{1}X + 2 \eta _{3}(X) \phi \xi _{2}- 2 \eta _{2}(X) \phi \xi _{3}\nonumber \\&= -X + \eta (X)\xi + \phi \phi _{1}X - 2 \eta _{3}(X) \xi _{3}- 2 \eta _{2}(X) \xi _{2}\end{aligned}$$

for any tangent vector field $X$ on $M$.

Let $\{e_{1}, e_{2}, \cdots , e_{4m-1} \}$ be an orthonormal basis for $T_{x}M$ where $x$ is any point of $M$. Then we get

$$\begin{aligned} \phi A \phi A e_{i}= -e_{i}+ \eta (e_{i})\xi + \phi \phi _{1}e_{i}- 2 \eta _{3}(e_{i}) \xi _{3}- 2 \eta _{2}(e_{i}) \xi _{2}\end{aligned}$$

(6.2)

for $i=1,2, \cdots , 4m-1$. From this, we calculate the trace of the matrix $\phi A \phi A$, that is,

$$\begin{aligned} \text {Tr}(\phi A \phi A)&= \sum _{i=1}^{4m-1} g(\phi A \phi A e_{i}, e_{i})\nonumber \\&= -\sum _{i=1}^{4m-1} g(e_{i}, e_{i}) + \sum _{i=1}^{4m-1} \eta (e_{i})g(\xi , e_{i}) + \sum _{i=1}^{4m-1} g(\phi \phi _{1}e_{i}, e_{i})\nonumber \\&- 2 \sum _{i=1}^{4m-1}\eta _{3}(e_{i}) g(\xi _{3}, e_{i}) - 2 \sum _{i=1}^{4m-1}\eta _{2}(e_{i}) g(\xi _{2}, e_{i})\nonumber \\&= -(4m-1) + g(\xi , \xi ) + \text {Tr}(\phi \phi _{1}) -2g(\xi _{3}, \xi _{3}) -2g(\xi _{2}, \xi _{2})\nonumber \\&= -4m, \end{aligned}$$

(6.3)

together with $\text {Tr}(\phi \phi _{\nu })=2\eta _{\nu }(\xi )$, $\nu =1,2,3$ (see [13]).

On the other hand, we are able to calculate the following:

$$\begin{aligned} ||\phi A - A \phi ||^{2}&= \sum _{i=1}^{4m-1} g\big ((\phi A - A \phi )e_{i}, (\phi A - A \phi )e_{i}\big )\nonumber \\&= - \sum _{i=1}^{4m-1}g(A \phi ^{2} A e_{i}, e_{i}) + \sum _{i=1}^{4m-1} g(\phi A \phi A e_{i}, e_{i})\nonumber \\&+ \sum _{i=1}^{4m-1}g(A \phi A \phi e_{i}, e_{i}) - \sum _{i=1}^{4m-1} g(\phi A^{2} \phi e_{i}, e_{i})\nonumber \\&= \sum _{i=1}^{4m-1}g(A^{2} e_{i}, e_{i}) - \sum _{i=1}^{4m-1}\eta (Ae_{i})g(A \xi , e_{i})\nonumber \\&+ 2\sum _{i=1}^{4m-1}g(A \phi A \phi e_{i}, e_{i}) - \sum _{i=1}^{4m-1} g(\phi A^{2} \phi e_{i}, e_{i})\nonumber \\&= \text {Tr}A^{2} + 2 \text {Tr} (A \phi A \phi ) - \text {Tr}(\phi A^{2} \phi )\nonumber \\&= \text {Tr}A^{2} + 2 \text {Tr} (\phi A \phi A) - \text {Tr}(A^{2} \phi ^{2})\nonumber \\&= 2\text {Tr}A^{2} + 2 \text {Tr} (\phi A \phi A), \end{aligned}$$

(6.4)

using the facts, $A\xi =0$ and $\text {Tr}(AB)=\text {Tr}(BA)$ for any two matrices $A, B$ with same size.

From this, together with (6.3) and our assumption for the squared norm of shape operator $A$ of $M$, the left side of (6.4) should vanish for a real hypersurface $M$ in $G_2({\mathbb C}^{m+2})$ with $\alpha =0$ and $\nabla _{\xi }A=0$. This gives that the shape operator $A$ commutes with the structure tensor $\phi$, that is, $A\phi = \phi A$. According to the result due to Berndt and Suh [4], the Reeb flow on $M$ becomes isometric. It completes a proof of our proposition. $\square$

Hence from Theorem $\mathrm B$, we can assert that if a real hypersurface $M$ in $G_2({\mathbb C}^{m+2})$ satisfies the conditions in Proposition 6.1, then $M$ becomes a model space of Type $(A)$ in Theorem $\mathrm A$. To serve the convenience of notation, a model space of Type $(A)$ with radius $r$ is denoted by $M_{A}$ or $M_{A}(r)$. From this let us now check if the model space $M_{A}$ satisfies the assumptions in Proposition 6.1 or not.

First, we can state that $M_{A}$ has Reeb parallel shape operator from the observation given in Sect. 4. Moreover, we see that a model space $M_A$ becomes an open part of a tube around a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$ with radius $r=\frac{\pi }{4\sqrt{2}}$, because the principal curvature $\alpha$ of $\xi$ on $M_{A}$ must be zero. From this and Proposition $\mathrm A$ in Sect. 4, we have the following three distinct principal curvatures and the corresponding multiplicities with respect to the eigenspaces of $M_{A}(\frac{\pi }{4\sqrt{2}})$:

Principal curvature	Multiplicity	Eigenspace
$\alpha =0$	$1$	$T_{\alpha }=\text {Span}\{\xi \}$
$\beta =\sqrt{2}$	$2$	$T_{\beta }=\text {Span}\{\xi _{2}, \xi _{3}\}$
$\lambda =-\sqrt{2}$	$2(m-1)$	$T_{\lambda }=\{X\,|\ X\bot \mathbb {H}\xi ,\ JX=J_{1}X \}$
$\mu =0$	$2(m-1)$	$T_{\mu }=\{X\,|\ X \bot \mathbb {H}\xi ,\ JX= -J_{1}X \}$

By this table and a straightforward calculation we have the squared norm of the shape operator $A$ on $M_{A}(\frac{\pi }{4\sqrt{2}})$ as follows.

$$\begin{aligned} ||A^{2}||&= \sum _{i=1}^{4m-1} g(Ae_{i}, Ae_{i})\nonumber \\&= \sum _{i=1}^{2m-2} g(Ae_{i}, Ae_{i}) + \sum _{i=2m-1}^{4m-4} g(Ae_{i}, Ae_{i}) + g(A e_{4m-3}, A e_{4m-3})\nonumber \\&+ g(A e_{4m-2}, A e_{4m-2})+ g(A e_{4m-1}, A e_{4m-1})\nonumber \\&= \sum _{i=1}^{2m-2} \lambda ^{2}g(e_{i}, e_{i}) + \sum _{i=2m-1}^{4m-4} \mu ^{2}g(e_{i}, e_{i}) + g(A \xi , A \xi )\nonumber \\&+ g(A \xi _{2}, A \xi _{2})+ g(A \xi _{3}, A \xi _{3})\nonumber \\&= 2(m-1) \lambda ^{2} + 2(m-1) \mu ^{2} + \alpha ^{2} + 2 \beta ^{2}\nonumber \\&= 4(m-1) + 4\nonumber \\&= 4m, \end{aligned}$$

where $e_{1}, e_{2}, \cdots , e_{2m-2}\in T_{\lambda }$, $e_{2m-1}, \cdots , e_{4m-4}\in T_{\mu }$, $e_{4m-3}=\xi ={\xi }_1$, $e_{4m-2}=\xi _{2}$, $e_{4m-1}=\xi _{3}$. From this calculation, we see that $M_{A}(\frac{\pi }{4\sqrt{2}})$ also satisfies our assumption in Proposition 6.1.

Summing up these discussions, we obtain our Theorem 2 mentioned in the introduction. $\square$

Lastly, we will give a proof for our assertion given in the introduction as follows.

Lemma 6.2

Let $M$ be a real hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with vanishing geodesic Reeb flow. If the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D^{\bot }$, then the shape operator $A$ of $M$ is Reeb parallel.

Proof

Using the equation of Codazzi (3.9), we obtain that

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

for any tangent vector field $Y$ on $M$.

From our assumptions, $A\xi = 0$ and $\xi = {\xi }_1$, we have

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

(6.5)

Moreover, since $M$ is Hopf, (4.3) implies that

$$\begin{aligned} A\phi AY = \phi Y + \phi _{1}Y + 2 \eta _{2}(Y) \xi _{2}-2 \eta _{3}(Y)\xi _{3}, \end{aligned}$$

(6.6)

together with $\alpha =0$ and $\xi ={\xi }_1$.

From (6.5) and (6.6), we get $(\nabla _{\xi }A)Y=0$ for any tangent vector field $Y$ on $M$. That is, a real hypersurface $M$ in $G_2({\mathbb C}^{m+2})$ with vanishing geodesic Reeb flow, that is, $\alpha =g(A\xi , \xi )=0$ and $\xi \in \mathfrak D^{\bot }$ has automatically Reeb parallel shape operator.