Levi-Civita and generalized Tanaka–Webster covariant derivatives for real hypersurfaces in complex two-plane Grassmannians

Abstract

It is known that submanifolds in Kaehler manifolds have many kinds of connections. Among them, we consider two connections, that is, Levi-Civita and Tanaka–Webster connections for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. When they are equal to each other, we give some characterizations in $G_2({\mathbb C}^{m+2})$.

1 Introduction

The study of real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$ was initiated by Berndt and Suh [1]. Let us denote by $G_{2}(\mathbb C ^{m+2})$ the set of all complex two-dimensional linear subspaces in $\mathbb C^{m+2}$. This set can be identified with the homogeneous space $SU(m+2)/S(U(2) \times U(m))$. From this, we know that $G_2({\mathbb C}^{m+2})$ becomes the unique compact, irreducible, Riemannian manifold being equipped with both a Kaehler structure $J$ and a quaternionic Kaehler structure $\mathfrak {J}$ not containing $J$. In other words, $G_{2}(\mathbb C ^{m+2})$ is the unique compact, irreducible, Kaehler, quaternionic Kaehler manifold which is not a hyper-Kaehler manifold [1, 2].

For real hypersurfaces $M$ in $G_{2}(\mathbb C^{m+2})$, we have the following two natural geometric conditions: 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 $M$. Here the almost contact structure vector field $\xi$ defined by $\xi = -JN$ 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 $\xi _{1},\xi _{2},\xi _{3}$ spanning 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 the quaternionic Kaehler structure $\mathfrak J$ such that $T_{x}M = \mathfrak D \oplus \mathfrak D^{\bot }$, $x \in M$.

By using these two invariant conditions and the result in Alekseevskii [3], Berndt and Suh [1] proved the following:

Theorem A

Let $M$ be a connected 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 $G_2({\mathbb C}^{m+2})$.

The Reeb vector field $\xi$ is said to be Hopf if it is invariant under the shape operator $A$. The 1-dimensional foliation of $M$ by the integral curves 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 almost contact metric structure $(\phi , \xi , \eta , g)$ and the formula $\nabla _{X}\xi =\phi AX$ for any $X \in TM$ in Sect. 2, it can be easily checked that $M$ is Hopf if and only if the Reeb vector field $\xi$ is Hopf. We will give a brief review of $(\phi , \xi , \eta , g)$ on $M$ in Sect. 2.

On the other hand, when the distribution $\mathfrak D^{\bot }$ of a hypersurface $M$ in $G_2({\mathbb C}^{m+2})$ is invariant under the shape operator, we say that $M$ is a $\mathfrak D^{\bot }$-invariant hypersurface. Moreover, we say that the Reeb flow on $M$ in $G_2({\mathbb C}^{m+2})$ is isometric, when the Reeb vector field $\xi$ on $M$ is Killing. This means that the metric tensor $g$ is invariant under the Reeb flow of $\xi$ on $M$.

In [4], Berndt and Suh gave some equivalent conditions of isometric Reeb flow. They gave a characterization of real hypersurfaces of Type $(A)$ in Theorem A in terms of the Reeb flow on $M$ 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 $G_2({\mathbb C}^{m+2})$.

In the proof of our Main Theorems, we will use that the Reeb flow on $M$ is isometric if and only if the shape operator $A$ commutes with the structure tensor field $\phi$, that is, $A\phi = \phi A$. Related to this commuting property, recently, the authors gave many characterizations of model spaces of Type $(A)$ in $G_2({\mathbb C}^{m+2})$ mentioned in Theorems A and B (see [5, 6]).

On the other hand, Suh [7] gave a characterization of real hypersurfaces of Type $(B)$ in $G_2({\mathbb C}^{m+2})$ in terms of the contact hypersurface. Moreover, as another characterization of one of Type $(B)$ in $G_2({\mathbb C}^{m+2})$ related to the Reeb vector field $\xi$ Lee and Suh [8] obtained the following:

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 $G_2({\mathbb C}^{m+2})$, where $m=2n$.

Usually, any submanifold in Kaehler manifolds has many kinds of connections. Among them, we consider two connections, namely, Levi-Civita and Tanaka–Webster connections for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$. In fact, $G_2({\mathbb C}^{m+2})$ is a Riemannian symmetric space with Riemannian metric and Levi-Civita connection. Using the induced connection from the Levi-Civita connection, many geometers gave some characterizations for real hypersurfaces in $G_2({\mathbb C}^{m+2})$ related to the covariant derivative $\nabla$ of the shape operator on $M$ ([9, 10], etc). For real hypersurfaces in a Kaehler manifold, we consider a new affine connection $\widehat{\nabla }^{(k)}$ different from the Levi-Civita connection $\nabla$, namely, the generalized Tanaka–Webster connection (in short, the g-Tanaka–Webster connection). It becomes a generalization of the well-known connection defined by Tanno [11]. Besides, it coincides with Tanaka–Webster connection if a real hypersurface in Kaehler manifolds satisfies $\phi A + A \phi = 2k \phi$ for a nonzero real number $k$. The Tanaka–Webster connection is defined as the canonical affine connection on a non-degenerate, pseudo-Hermitian CR-manifold [12–14]. Using the generalized Tanaka–Webster connection, $\widehat{\nabla }^{(k)}$ defined in such a way that

$$\begin{aligned} \widehat{\nabla }^{(k)}_{X}Y = \nabla _{X}Y + g(\phi AX, Y) \xi - \eta (Y) \phi AX - k \eta (X) \phi Y \end{aligned}$$

(*)

for any $X$, $Y$ tangent to $M$, where $\nabla$ denotes the Levi-Civita connection on $M$, $A$ is the shape operator on $M$ and $k$ is a nonzero real number, the authors studied some characterizations of real hypersurfaces in $G_2({\mathbb C}^{m+2})$  ([15, 16], etc). The latter part of the generalized Tanaka–Webster connection $g(\phi AX, Y) \xi - \eta (Y) \phi AX - k \eta (X) \phi Y$ is denoted by $F_{X}Y$. Here the operator $F_{X}$ is a kind of (1,1)-type tensor and said to be the Tanaka–Webster operator.

On the other hand, there are many results for the classification problem of real hypersurfaces in $G_2({\mathbb C}^{m+2})$ related to the structure Jacobi operator and Ricci tensor, for example, [17–24] and so on. Recently, Pérez and Suh [25] investigated the Levi-Civita and g-Tanaka–Webster covariant derivatives for the shape operator or the structure Jacobi operator of real hypersurfaces in complex projective space $\mathbb C P^{m}$. In particular, the authors  [25] gave the result about the shape operator as follows:

Theorem D

There exist no real hypersurfaces $M$ in $\mathbb C P^{m}$, $m \ge 2$ such that $\nabla A = \widehat{\nabla }^{(k)}A$.

Motivated by Theorem D, in this paper, we study a real hypersurface $M$ in $G_2({\mathbb C}^{m+2})$ whose Levi-Civita covariant derivative coincides with generalized Tanaka–Webster derivative for the shape operator of $M$, that is,

$$\begin{aligned} \left( \nabla _{X}A\right) Y = \left( \widehat{\nabla }^{(k)}_{X}A\right) Y \end{aligned}$$

(C-1)

for arbitrary tangent vector fields $X$ and $Y$ on $M$.

The condition (C-1) has a geometric meaning such that the shape operator $A$ commutes with the Tanaka–Webster operator $F_{X}$, that is, $A \cdot F_{X} = F_{X} \cdot A$. This meaning gives any eigenspaces of the shape operator $A$ are invariant by the Tanaka–Webster operator $F_{X}$.

From such a point of view, in Sect. 3, we prove that a real hypersurface in Kaehler manifolds satisfying (C-1) must be Hopf. Then from this result, we assert the following:

Theorem 1

There does not exist any real hypersurface in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$, $m \ge 3$, satisfying (C-1).

First, if we restrict $X=\xi$ in (C-1), then the following condition (C-2) along the Reeb vector field $\xi$ becomes a generalized condition weaker than the condition (C-1). This also has a geometric meaning that any eigenspaces of the shape operator $A$ are invariant by the restricted Tanaka–Webster operator $F_{\xi }$ in the direction of the Reeb vector field $\xi$. Thus, we assert the following:

Theorem 2

Let $M$ be a Hopf hypersurface in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$, $m \ge 3$. If $M$ satisfies

$$\begin{aligned} \left( \nabla _{\xi }A\right) Y = \left( \widehat{\nabla }^{(k)}_{\xi }A\right) Y \end{aligned}$$

(C-2)

for any tangent vector field $Y$ on $M$, then $M$ is locally congruent to a tube of radius $r$ over a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.

As a second, let us consider a distribution $\mathfrak D^{\bot }$ spanned by $\{{\xi }_1, \xi _{2}, {\xi }_3\}$. Accordingly, if we consider the condition (C-1) to the distribution $\mathfrak D^{\bot }$, the derivatives of the shape operator $A$ of $M$ along the distribution $\mathfrak D^{\bot }$ becomes a condition more weaker than (C-1). Obviously, this has a geometric meaning that any eigenspaces of the shape operator $A$ are invariant by the restricted Tanaka–Webster operator $F_{\xi _{\nu }}$, $\nu =1,2,3$, along the distribution $\mathfrak D^{\bot }$. Then we have the following:

Theorem 3

There does not exist a Hopf hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, satisfying

$$\begin{aligned} \left( \nabla _{\xi _{\nu }}A\right) Y = \left( \widehat{\nabla }^{(k)}_{\xi _{\nu }}A\right) Y, \quad \nu =1,2,3 \end{aligned}$$

(C-3)

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

Finally, we consider a distribution $\mathfrak D$ which is an orthogonal complement of $\mathfrak D^{\bot }$ in $TM$. Then by restricting the condition (C-1) to the distribution $\mathfrak D$, we get the following condition (C-4), which becomes another condition more weaker than (C-1). Using this geometric notion, we get:

Theorem 4

There does not exist a Hopf hypersurface in $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with

$$\begin{aligned} \left( \nabla _{X}A\right) Y = \left( \widehat{\nabla }^{(k)}_{X}A\right) Y \end{aligned}$$

(C-4)

for all vector fields $X \in \mathfrak D$ and $Y$ on $M$.

In this paper, we refer to [1, 2, 4, 26] for Riemannian geometric structures of $G_2({\mathbb C}^{m+2})$, and [11, 13–16, 27] for generalized Tanaka–Webster connection of real hypersurfaces in Kaehler manifolds.

2 Key Lemmas

Let $M$ be a real hypersurface in Kaehler manifolds $(\tilde{M}, \tilde{g})$. The induced Riemannian metric on $M$ is denoted by $g$. In addition, $\tilde{\nabla }$ and $\nabla$ denote the Levi-Civita connections of $\tilde{M}$ and $M$, respectively. Let $N$ be a local unit normal vector field of $M$ and $A$ the shape operator of $M$ with respect to $N$.

From the Kaehler structure $J$ of $\tilde{M}$, we have a tensor field $\phi$ of type (1,1) on $M$, given by

$$\begin{aligned} g(\phi X, Y) = \tilde{g}(JX, Y) \end{aligned}$$

for all tangent vector fields $X$ of $M$. Moreover, we obtain the unit tangent vector field $\xi$ and the 1-form $\eta$ of $M$ defined by

$$\begin{aligned} \xi =-JN \ \ \text{ and }\ \ \eta (X)=g(X, \xi )=\tilde{g}(JX, N), \end{aligned}$$

respectively. It implies that $\phi ^{2}X=-X+\eta (X)\xi$, $\eta (\xi )=1$, $\phi \xi =0$ and

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

together with Gauss and Weingarten formulas. Thus, the Kaehler structure $J$ of $\tilde{M}$ induces an almost contact metric structure $(\phi , \xi , \eta , g)$ on $M$.

Now let us assume that a real hypersurface $M$ in $\tilde{M}$ satisfies

$$\begin{aligned} \left( \nabla _{X}A\right) Y = \left( \widehat{\nabla }^{(k)}_{X}A\right) Y \end{aligned}$$

(C-1)

for all tangent vector fields $X$ and $Y$ on $M$.

From the definition of the g-Tanaka–Webster connection (*), we have

$$\begin{aligned} (\widehat{\nabla }^{(k)}_{X} A)Y&= \widehat{\nabla }^{(k)}_{X}(AY) - A (\widehat{\nabla }^{(k)}_{X} Y) \\&= (\nabla _{X}A)Y + g(\phi AX, AY) \xi - \eta (AY) \phi AX - k \eta (X) \phi AY \\&\quad - g(\phi AX, Y) A \xi + \eta (Y) A \phi AX + k \eta (X) A \phi Y. \end{aligned}$$

Therefore, the condition (C-1) can be written as

$$\begin{aligned}&g(\phi AX, AY) \xi - \eta (AY) \phi AX - k \eta (X) \phi AY \nonumber \\&\quad - g(\phi AX, Y) A \xi + \eta (Y) A \phi AX + k \eta (X) A \phi Y=0 \end{aligned}$$

(2.1)

for all tangent vector fields $X$ and $Y$ on $M$.

In a situation like this, we prove

Lemma 2.1

Let $M$ be a real hypersurface in a Kaehler manifold $\tilde{M}$ with the condition (C-1). Then $M$ becomes a Hopf hypersurface.

Proof

The purpose of this lemma is to show that the structure vector field $\xi$ is principal. In order to prove this, let us suppose that there is a point where the Reeb vector field $\xi$ is not principal. Then there exists a neighborhood $\mathfrak U$ of this point, on which we can define a unit vector field $U$ orthogonal to $\xi$ in such a way that

$$\begin{aligned} \beta U = A\xi - g(A\xi , \xi ) \xi = A\xi - \alpha \xi \end{aligned}$$

where $\beta$ denotes the length of vector filed $A\xi - \alpha \xi$ and $\beta (x) \ne 0$ for any point $x$ in $\mathfrak U$. Hereafter, unless otherwise stated, let us continue our discussion on this neighborhood $\mathfrak U$.

Taking $X=Y=\xi$ in (2.1), we get $\beta (\alpha + k) \phi U = \beta A \phi U$. Since $\beta \ne 0$, it follows that

$$\begin{aligned} A \phi U = (\alpha + k) \phi U. \end{aligned}$$

(2.2)

Moreover, putting $X=Y=U$ in (2.1), we have $-\beta \phi AU =0$. It implies that

$$\begin{aligned} AU=\beta \xi , \end{aligned}$$

(2.3)

together with $\beta \ne 0$ and $\phi ^{2}AU = -AU + \eta (AU) \xi = -AU + \beta \xi$.

Replacing $Y$ by $U$ in (2.1), we have

$$\begin{aligned} -\beta \phi AX-g(\phi AX, U)A\xi + k \eta (X) A\phi U =0 \end{aligned}$$

(2.4)

for any tangent vector field $X$ on $M$. Substituting $X=\xi$ in the above equation, we get

$$\begin{aligned} \big (-\beta ^{2} + k (\alpha + k)\big ) \phi U =0 \end{aligned}$$

together with $\phi A \xi = \beta \phi U$ and (2.2). Taking the inner product with $\phi U$, it turns to

$$\begin{aligned} \alpha + k = \frac{\beta ^{2}}{k} \end{aligned}$$

(2.5)

because $k$ is nonzero real number from the definition of g-Tanaka–Webster connection on real hypersurfaces in Kaehler manifolds.

On the other hand, putting $X=\phi U$ in (2.4), we get

$$\begin{aligned} 2\beta (\alpha + k) U + \alpha (\alpha +k)\xi =0 \end{aligned}$$

(2.6)

from (2.2) and $\phi ^{2}U=-U$. Taking the inner product with $\xi$, we obtain $\alpha (\alpha +k)=0$. By (2.5), this equation is written as $\frac{\alpha \beta ^{2}}{k}=0$. Since $k \ne 0$ and $\beta \ne 0$, we have $\alpha =0$. Moreover, taking the inner product of (2.6) with $U$, we have $\beta (\alpha + k)=0$. It follows that $\beta =0$, together with $\alpha =0$ and $k \ne 0$, which gives a contradiction. This is, the set $\mathfrak U$ should be empty. Thus, there does not exist such an open neighborhood $\mathfrak U$ in $M$, which means that the structure vector field $\xi$ is principle. Hence, $M$ must be Hopf under our assumption.

By means of Lemma 2.1, the condition (C-1) implies

$$\begin{aligned}&g(\phi AX, AY) \xi - \alpha \eta (Y) \phi AX - k \eta (X) \phi AY \nonumber \\&\quad - \alpha g(\phi AX, Y) \xi + \eta (Y) A \phi AX + k \eta (X) A \phi Y=0 \end{aligned}$$

(2.7)

for all tangent vector fields $X$ and $Y$ on $M$. Moreover, putting $Y=\xi$ in the above equation, we obtain $A \phi AX = \alpha \phi AX$ for any tangent vector field $X$ on $M$. From this, the Eq. (2.7) is reduced to

$$\begin{aligned} k \eta (X)(A \phi - \phi A)Y=0 \end{aligned}$$

for all tangent vector fields $X$ and $Y$ on $M$. By the definition of generalized Tanaka–Webster connection for real hypersurfaces in a Kaehler manifold, it follows that

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

for all tangent vector fields $X$ and $Y$ on $M$.

Summing up above discussions, we assert the following

Lemma 2.2

Let $M$ be a real hypersurface in a Kaehler manifold $\tilde{M}$ with the condition (C-1). Then we have

$$\begin{aligned}&A\phi AX = \alpha \phi AX, \end{aligned}$$

(2.8)

$$\begin{aligned}&\eta (X)(A \phi - \phi A)Y=0 \end{aligned}$$

(2.9)

for all tangent vector fields $X, Y$ on $M$.

3 Proof of Theorem 1

From now on, we will prove Theorem 1 in the introduction by using the above two Lemmas which are induced from our condition (C-1).

In fact, since $M$ is a real hypersurface in $G_2({\mathbb C}^{m+2})$ with the property (C-1), $M$ becomes a Hopf hypersurface (Lemma 2.1). From this, we have

$$\begin{aligned} \eta (X) (A\phi -\phi A)Y =0, \end{aligned}$$

(3.1)

because $k$ is a nonzero constant (Lemma 2.2).

Putting $X=\xi$ in (3.1), it follows that $A \phi - \phi A =0$. On the other hand, Berndt and Suh [4] gave a characterization of real hypersurfaces of Type $(A)$ in $G_2({\mathbb C}^{m+2})$ when the shape operator $A$ of $M$ commutes with the structure tensor $\phi$ of $M$. By virtue of this result, we assert that if $M$ is a real hypersurface in $G_2({\mathbb C}^{m+2})$ satisfying (C-1), then $M$ is locally congruent to an open part of a tube around a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.

Let us check that whether the model space $M_{A}$ of Type $(A)$ satisfies the condition (C-1). In order to do this, let us assume that the shape operator $A$ of $M_{A}$ satisfies the condition (C-1). According to Proposition $3$ given in  [1], the Eq. (2.8) implies

$$\begin{aligned} \beta (\beta - \alpha )=0 \end{aligned}$$

(3.2)

if $X=\xi _{2}$. But it does not hold, because $\beta (\beta - \alpha )=2$ where $\alpha = \sqrt{8} \cot (2\sqrt{2}r)$ and $\beta = \sqrt{2} \cot (\sqrt{2}r)$, $r \in (0, \pi /{2\sqrt{2}}\,)$. It completes the proof of Theorem 1. $\square$

4 Proofs of Theorems 2 and 3

In this section, we investigate Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ satisfying the property (C-2) and (C-3) which are weaker than (C-1), respectively. On the other hand, $G_2({\mathbb C}^{m+2})$ is equipped with both a Kaehler and a quaternionic Kaehler structure. By applying these two structures to the normal vector field $N$ of $M$ in $G_2({\mathbb C}^{m+2})$, we get 1- and 3-dimensional distributions on $M$. For the sake of convenience, we denote $[\xi ]=\text {Span}\{\xi \}$ or $\mathfrak D^{\bot }=\text {Span}\{{\xi }_1, \xi _{2}, {\xi }_3\}$, respectively. For these two distributions, we define a new distribution $\mathfrak F$ given by $\mathfrak F= [\xi ] \cup \mathfrak D^{\bot }$. If we restrict $X \in \mathfrak F$ in (C-1), then it becomes a new weaker condition for (C-1). Accordingly, we also consider this case.

First, we assume that $M$ is a Hopf hypersurface in $G_2({\mathbb C}^{m+2})$ satisfying

$$\begin{aligned} (\nabla _{\xi }A) Y = (\widehat{\nabla }^{(k)}_{\xi }A)Y \end{aligned}$$

(C-2)

for any vector field $Y \in TM$.

Under our assumptions, this condition means that the structure tensor field $\phi$ commutes with the shape operator $A$ of $M$. In fact, putting $X=\xi$ in (2.1), it follows that for any tangent vector field $Y$ on $M$

$$\begin{aligned} \phi AY - A \phi Y =0, \end{aligned}$$

because $M$ is Hopf and $k$ is a nonzero real number. By Theorem B, we assert our Theorem $2$ in the introduction. $\square$

Next, we observe the following condition of covariant derivatives with respect to the Levi-Civita and g-Tanaka–Webster connections for shape operator $A$ on Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ given by

$$\begin{aligned} (\nabla _{\xi _{\nu }}A) Y = (\widehat{\nabla }^{(k)}_{\xi _{\nu }}A)Y, \quad \nu =1,2,3 \end{aligned}$$

(C-3)

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

According to (2.1), the condition (C-3) is equal to

$$\begin{aligned}&g(\phi A \xi _{\nu }, AY) \xi - \alpha \eta (Y) \phi A\xi _{\nu }- k \eta (\xi _{\nu }) \phi AY \nonumber \\&\quad - \alpha g(\phi A\xi _{\nu }, Y) \xi + \eta (Y) A \phi A\xi _{\nu }+ k \eta (\xi _{\nu }) A \phi Y=0 \end{aligned}$$

(4.1)

where $Y$ is any tangent vector field on $M$ and $\nu =1,2,3$.

Putting $Y=\xi$ in (4.1), we have that

$$\begin{aligned} A \phi A \xi _{\nu }= \alpha \phi A \xi _{\nu }, \quad \nu =1,2,3. \end{aligned}$$

(4.2)

From this, (4.1) can be written as

$$\begin{aligned} \eta (\xi _{\nu }) (A \phi - \phi A ) Y =0 \end{aligned}$$

for any vector field $Y \in TM$ and $\nu =1,2,3$.

By virtue of this equation, we have the following two cases:

• Case 1 $\eta (\xi _{\nu })=0$, $\nu =1,2,3$   and

• Case 2 $A \phi = \phi A$.

First, we consider the case $\eta (\xi _{\nu }) =0$ for any $\nu =1,2,3$. It means that the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D$. By Theorem C, it implies that $M$ is of Type $(B)$ in Theorem A given in the introduction.

On the other hand, due to Berndt and Suh’s classification [1], all the principal curvatures on a model space of Type $(B)$ are given as follows: $\alpha =-2\tan (2r)$, $\beta =2\cot (2r)$, $\gamma =0$, $\lambda =\cot (r)$ and $\mu =-\tan (r)$ for some $r \in (0, \pi /4)$. Since $\gamma =0$, we get

$$\begin{aligned} \left( \widehat{\nabla }^{(k)}_{\xi _{\nu }}A\right) \xi - (\nabla _{\xi _{\nu }}A)\xi&= A \phi A \xi _{\nu }- \alpha \phi A \xi _{\nu }\\&= - \alpha \beta \phi \xi _{\nu }\end{aligned}$$

for $\nu =1,2,3$. In fact, since $\alpha =-2\tan (2r)$, $\beta =2\cot (2r)$ for some $r \in (0, \pi /4)$, the constant $\alpha \beta$ must be nonzero. It means that the model space of Type $(B)$ does not satisfy our condition (C-3).

Next we consider the remain case that the structure tensor $\phi$ commutes with the shape operator $A$ of $M$. By virtue of Theorem B, we see that $M$ must be a real hypersurface of Type $(A)$ in $G_2({\mathbb C}^{m+2})$.

From now on, let us check the converse problem, that is, whether a model space $M_{A}$ of Type $(A)$ satisfies the condition (C-3) or not. In fact, we suppose that $M_{A}$ has the condition (C-3), that is, $M_{A}$ satisfies (4.2). For $\nu =2$, it becomes $\beta (\beta -\alpha )=0$. In the proof of Theorem 1, we get $\beta (\beta -\alpha )=2$, because $\alpha =\sqrt{8}\cot (2\sqrt{2}r)$ and $\beta =\sqrt{2}\cot {\sqrt{2}r}$ where $r \in (0, \pi / 2 \sqrt{2})$. Hence, we assert that $M_{A}$ does not satisfy the condition (C-3).

Summing up these subcases, we give a complete proof of Theorem 3. $\square$

As mentioned above, the distribution $\mathfrak F$ is defined by $\mathfrak F= [\xi ] \cup \mathfrak D^{\bot }$. From the structure of $\mathfrak F$ and the proofs of Theorems 2 and 3, we naturally obtain

Corollary 4.1

There does not exist a Hopf hypersurface in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$, $m \ge 3$, with

$$\begin{aligned} \left( \nabla _{X}A\right) Y = \left( \widehat{\nabla }^{(k)}_{X}A\right) Y \end{aligned}$$

for any $X\in \mathfrak F$ and $Y \in TM$.

5 Proof of Theorem 4

In this section, we observe the condition

$$\begin{aligned} \left( \nabla _{X}A\right) Y = \left( \widehat{\nabla }^{(k)}_{X}A\right) Y \end{aligned}$$

(C-4)

for all tangent vector fields $X \in \mathfrak D$ and $Y \in TM$. Putting $Y=\xi$ in (2.1) and using the assumption that $M$ is Hopf, we obtain

$$\begin{aligned} A \phi AX = \alpha \phi AX \end{aligned}$$

(5.1)

for any tangent vector field $X \in \mathfrak D$. Thus, the condition (C-4) is equal to

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

(5.2)

for any $X \in \mathfrak D$ and $Y \in TM$. From this, we have the following two cases:

• Case 1    $A \phi = \phi A$ and

• Case 2    $\eta (X) =0$ for any $X \in \mathfrak D$.

For the first case $A \phi = \phi A$, we know that $M$ becomes a model space of Type $(A)$ by Theorem B in the introduction.

Now let us consider the remaining case $\eta (X) =0$ for any $X \in \mathfrak D$. It means that the Reeb vector field $\xi$ belongs to the distribution $\mathfrak D^{\bot }$. Thus, without loss of generality we may put $\xi ={\xi }_1$. Under these assumptions, we now prove that $M$ becomes to be a $\mathfrak D^{\bot }$-invariant hypersurface, that is, $g(A\mathfrak D, \mathfrak D^{\bot })=0$.

Since $M$ is Hopf, we have the following formula given by Berndt and Suh [4]:

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

for any tangent vector field $X$ on $M$. It can be written as

$$\begin{aligned} 2 A\phi AX =\alpha A\phi X + \alpha \phi A X + 2 \phi X + 2 \phi _{1}X \end{aligned}$$

(5.3)

for any $X \in \mathfrak D$ and $\xi ={\xi }_1$. Substituting (5.1) into (5.3), we get

$$\begin{aligned} \alpha ( A\phi - \phi A) X = -2 (\phi X + \phi _{1}X) \end{aligned}$$

(5.4)

for any $X \in \mathfrak D$.

Let $\{e_{1}, e_{2}, \cdots , e_{4m-4}, e_{4m-3}=\xi , e_{4m-2}=\xi _{2}, e_{4m-1}={\xi }_3\}$ be an orthonormal basis for $T_{x}M$, $x\in M$. Then for any tangent vector field $Y$ on $M$ it follows that

$$\begin{aligned} \alpha (A \phi - \phi A)Y&= \sum _{i=1}^{4m-1} g(\alpha (A \phi -\phi A) Y, e_{i})e_{i}\\&= \sum _{i=1}^{4m-4} g(\alpha (A \phi -\phi A) Y, e_{i})e_{i} + \sum _{\nu =1}^{3} g(\alpha (A \phi -\phi A) Y, \xi _{\nu })\xi _{\nu }\\&= \sum _{i=1}^{4m-4} g(\alpha (A \phi -\phi A) e_{i}, Y )e_{i} + \sum _{\nu =1}^{3} g(\alpha (A \phi -\phi A) Y, \xi _{\nu })\xi _{\nu }. \end{aligned}$$

Putting $Y=e_{k} \in \mathfrak D\,(k=1,2, \cdots , 4m-4)$, this equation can be changed

$$\begin{aligned} \alpha (A \phi - \phi A)e_{k} = \sum _{i=1}^{4m-4} g(\alpha (A \phi -\phi A) e_{i}, e_{k} )e_{i} + \sum _{\nu =1}^{3} g(\alpha (A \phi -\phi A) e_{k}, \xi _{\nu })\xi _{\nu }. \end{aligned}$$

From (5.4), it follows that

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

where in the fourth and sixth equalities, we have used $g(\phi e_{k}, \xi _{\nu }) = g(\phi _{1} e_{k}, \xi _{\nu })=0$ for any $\nu \pmod 3$ and nonzero real number $k$. Thus, we get

$$\begin{aligned} \phi X = -\phi _{1}X \end{aligned}$$

(5.5)

for any tangent vector field $X\in \mathfrak D$. Differentiating this equation covariantly in the direction of $Y$, we have

$$\begin{aligned} g(AX, Y)=0 \end{aligned}$$

for all tangent vector fields $X \in \mathfrak D$ and $Y \in TM$, where we have used the formulas about the covariant derivative of structure tensors $\phi$ and $\phi _{\nu }$ $(\nu =1,2,3)$. It implies that $M$ must be a $\mathfrak D^{\bot }$-invariant hypersurface, if we restrict to $Y \in \mathfrak D^{\bot }$. Accordingly, for this case we can assert that $M$ is locally congruent to model spaces of Type $(A)$ by virtue of Theorem A in the introduction.

Summing up these cases, we consequently know that any Hopf hypersurface $M$ in $G_2({\mathbb C}^{m+2})$ satisfying the condition (C-4) is of Type $(A)$.

Now it remains only to show that whether a real hypersurface $M_{A}$ of Type $(A)$ satisfies the condition (C-4) or not. To check this, let us assume that $M_{A}$ has the condition $(\nabla _{X}A)Y = (\widehat{\nabla }^{(k)}_{X}A)Y$ for any $X \in \mathfrak D$ and $Y \in TM_{A}$. It is equivalent that

$$\begin{aligned} A \phi AX = \alpha \phi AX, \end{aligned}$$

(5.6)

for $X \in \mathfrak D$ as observed in this section.

From the structure of the tangent vector space $T_{x}M_{A}$ for a model space of Type $(A)$ at any point $x$ on $M_{A}$, we see that the distribution $\mathfrak D$ is composed with two eigenspaces $T_{\lambda }$ and $T_{\mu }$. In addition, since the eigenspace $T_{\lambda }$ is given by $T_{\lambda }=\{ X |\ X \bot \mathbb H \xi , JX = J_{1}X\}$ where $\mathbb H \xi$ denotes quaternionic span of $\xi$, we see that $\phi X \in T_{\lambda }$ for any $X \in T_{\lambda }$. Using these facts, the Eq. (5.6) is reformed as

$$\begin{aligned} (\lambda ^{2} - \alpha \lambda ) \phi X = 0 \end{aligned}$$

for any $X \in T_{\lambda }\subset \mathfrak D$. From this, we get $\lambda ^{2} - \alpha \lambda =0$.

On the other hand, from Proposition $3$ in [1], we know that

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

where $\lambda =-\sqrt{2}\tan (\sqrt{2}r)$ and $\alpha = 2 \sqrt{2} \cot (2\sqrt{2}r)$ for some $r \in (0, \pi /2\sqrt{2})$. This makes a contradiction, and therefore, we have Theorem 4 in the introduction. $\square$