Some Conditions Concerning the Shape Operator of a Real Hypersurface in Complex Projective Space

Abstract

Let M be a real hypersurface of a complex projective space. For any operator B on M and any nonnull real number k, we can define two tensor fields of type (1,2) on M, $B_F^{(k)}$ and $B_T^{(k)}$. We will classify real hypersurfaces in complex projective space for which $B_F^{(k)}$ and $B_T^{(k)}$ either take values in the maximal holomorphic distribution $\mathbb {D}$ or are parallel to the structure vector field $\xi$, in the particular case of $B=A$, where A denotes the shape operator of M. We also introduce the concept of $A_F^{(k)}$ and $A_T^{(k)}$ being $\mathbb {D}$-recurrent and classify real hypersurfaces such that either $A_F^{(k)}$ or $A_T^{(k)}$ are $\mathbb {D}$-recurrent.

1 Introduction

Consider the complex projective space $\mathbb {C}P^{m}$, $m \ge 2$, endowed with the Kaehlerian structure (J, g), where J denotes the complex structure and g the Fubini-Study metric of constant holomorphic sectional curvature 4 on $\mathbb {C}P^{m}$. Let M be a connected real hypersurface in $\mathbb {C}P^{m}$ without boundary. Denote also by g the induced metric on M and by N a local unit normal vector field on M. The Reeb (or structure) vector field on M is defined by $\xi =-JN$. Let $\nabla$ be the Levi-Civita connection on M and A the shape operator associated to N. For any vector field X tangent to M write $JX=\phi X+\eta (X)N$, where $\phi X$ is the tangential component of JX and $\eta (X)=g(X,\xi )$. Then $(\phi , \xi , \eta , g)$ defines an almost contact metric structure on M [1].

The existence of such a structure allows us to define, for any nonnull real number k, the so-called kth Generalized Tanaka–Webster connection on M, ${\hat{\nabla }}^{(k)}$ [2, 3], given by

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

for any X, Y tangent to M.

This connection on M is a metric one and any element of the almost contact metric structure is parallel for such a connection. If A satisfies $\phi A+A\phi =2k\phi$, M becomes a contact manifold and this connection coincides with the Tanaka–Webster connection on M, [14,15,16].

The tensor field of type (1, 2) obtained as the difference of both connections is called the kth Cho tensor on M (see [6, Proposition 7.10]) and it is given by $F^{(k)}(X,Y)=g(\phi AX,Y)\xi -\eta (Y)\phi AX-k\eta (X)\phi Y$, for any X, Y tangent to M. From this, for any X tangent to M and any nonnull real number k, we define the kth Cho operator corresponding to X, as $F_X^{(k)}Y=F^{(k)}(X,Y)$ for any Y tangent to M.

The torsion of the connection ${\hat{\nabla }}^{(k)}$ is given by $T^{(k)}(X,Y)=F_X^{(k)}Y-F_Y^{(k)}X$ for any X, Y tangent to M, [2]. Thus we define the kth torsion operator associated to X, for any nonnull real number k and any X tangent to M, by $T_X^{(k)}Y={\hat{T}}^{(k)}(X,Y)$, for any Y tangent to M.

If ${\mathcal {L}}$ denotes the Lie derivative on M, we now that it is given by ${{\mathcal {L}}}_XY=\nabla _XY-\nabla _YX$, for any X, Y tangent to M. If we consider the kth Generalized Tanaka–Webster connection we can also define on M a differential operator of first order, that we call the derivative of Lie type associated to such a connection, ${{\mathcal {L}}}^{(k)}$, given by

$$\begin{aligned} {{\mathcal {L}}}_X^{(k)}Y={\hat{\nabla }}_X^{(k)}Y-{\hat{\nabla }}_Y^{(k)}X={{\mathcal {L}}}_XY+T_X^{(k)}Y \end{aligned}$$

for any X, Y tangent to M.

Best known real hypersurfaces in $\mathbb {C}P^m$ are called Hopf and satisfy that the Reeb vector field $\xi$ is an eigenvector of the shape operator, that is, $A\xi =\alpha \xi$, for a certain function $\alpha$ on M, called the Reeb curvature of M. The distribution on M given by $\mathbb {D}=Ker(\eta )$ is $\phi$-invariant and called the maximal holomorphic distribution on M.

Takagi classified homogeneous real hypersurfaces in complex projective space (see [11,12,13]). Kimura [4], proved that Takagi’s real hypersurfaces are the unique ones that are Hopf and have constant principal curvatures for A in $\mathbb {C}P^m$. Takagi’s list contains the following 6 types of real hypersurfaces

• Type $(A_1)$, geodesic hyperspheres of radius r, $0< r < \frac{\pi }{2}$, with 2 distinct constant principal curvatures, $2\cot (2r)$ with eigenspace $\mathbb {R}[\xi ]$ and $\cot (r)$ with eigenspace $\mathbb {D}$.

• Type $(A_2)$, tubes of radius r, $0< r < \frac{\pi }{2}$, over totally geodesic complex projective spaces $\mathbb {C}P^n$, $0< n < m-1$, with 3 distinct constant principal curvatures, $2\cot (2r)$ with eigenspace $\mathbb {R}[\xi ]$, $\cot (r)$ and $-\tan (r)$. The corresponding eigenspaces of $\cot (r)$ and $-\tan (r)$ are complementary and $\phi$-invariant distributions in $\mathbb {D}$.

• Type (B), tubes of radius r, $0< r < \frac{\pi }{4}$, over the complex quadric $Q^{m-1}$, with 3 distinct constant principal curvatures, $2\cot (2r)$ with eigenspace $\mathbb {R}[\xi ]$, $\cot (r-\frac{\pi }{4})$ and $-\tan (r-\frac{\pi }{4})$ whose corresponding eigenspaces are complementary and equal dimensional distributions in $\mathbb {D}$ such that $\phi V_{\cot (r-\frac{\pi }{4})}=V_{-\tan (r-\frac{\pi }{4})}$.

• Type (C), tubes of radius r, $0< r < \frac{\pi }{4}$, over the Segre embedding of $\mathbb {C}P^1 \times \mathbb {C}P^n$, where $2n+1=m$ and $m \ge 5$, with 5 distinct constant principal curvatures, $2\cot (2r)$ with eigenspace $\mathbb {R}[\xi ]$, $\cot (r-\frac{\pi }{4})$ with multiplicity 2, $\cot (r-\frac{\pi }{2})=-\tan (r)$ with multiplicity $m-3$, $\cot (r-\frac{3\pi }{4})$, with multiplicity 2 and $\cot (r-\pi )=\cot (r)$ with multiplicity $m-3$. Moreover $\phi V_{\cot (r-\frac{\pi }{4})}=V_{\cot (r-\frac{3\pi }{4})}$ and $V_{-\tan (r)}$ and $V_{\cot (r)}$ are $\phi$-invariant.

• Type (D), tubes of radius r, $0< r < \frac{\pi }{4}$, over the Plucker embedding of the complex Grassmannian manifold G(2, 5) in $\mathbb {C}P^9$, with the same principal curvatures as type (C), $2\cot (2r)$ with eigenspace $\mathbb {R}[\xi ]$, and the other 4 principal curvatures have the same multiplicity 4 and their eigenspaces have the same behaviour with respect to $\phi$ as in type (C).

• Type (E), tubes of radius r, $0< r < \frac{\pi }{4}$, over the canonical embedding of the Hermitian symmetric space SO(10)/U(5) in $\mathbb {C}P^{15}$. They also have the same principal curvatures as type (C), $2 \cot (2r)$ with eigenspace $\mathbb {R}[\xi ]$, $\cot (r-\frac{\pi }{4})$ and $\cot (r-\frac{3\pi }{4})$ have multiplicities equal to 6 and $-\tan (r)$ and $\cot (r)$ have multiplicities equal to 8. Their corresponding eigenspaces have the same behaviour with respect to $\phi$ as in type (C).

We will call type (A) real hypersurfaces to both types $(A_1)$ or $(A_2)$.

Ruled real hypersurfaces in $\mathbb {C}P^m$ were introduced by Kimura [5]. The maximal holomorphic distribution $\mathbb {D}$ of such real hypersurfaces is integrable with integral manifolds $\mathbb {C}P^{m-1}$. Equivalently, $g(A\mathbb {D},\mathbb {D})=0$. Kimura gave some minimal examples of this kind of real hypersurfaces.

Let B be a symmetric operator on M. Then we can define on M a couple of tensor fields of type (1,2), for any nonnull real number k, $B_F^{(k)}$ and $B_T^{(k)}$, given, respectively, by

$$\begin{aligned} B_F^{(k)}(X,Y)=(({\hat{\nabla }}_X^{(k)}-\nabla _X)B)Y=F_X^{(k)}BY-BF_X^{(k)}Y=[F_X^{(k)},B ]Y \end{aligned}$$

(1.1)

and

$$\begin{aligned} B_T^{(k)}(X,Y)=(({{\mathcal {L}}}_X^{(k)}-{{\mathcal {L}}}_X)B)Y=T_X^{(k)}BY-BT_X^{(k)}Y=[T_X^{(k)},B ]Y \end{aligned}$$

(1.2)

for any X, Y tangent to M.

In [10] we considered the case $A=B$ in (1.1), and proved non-existence of real hypersurfaces in $\mathbb {C}P^m$, $m \ge 3$, such that $A_F^{(k)}=0$, for any nonnull real number k. A similar result for $A=B$ in (1.2) was obtained in [9]. Such conditions imply commutativity of A and either $F_X^{(k)}$ or $T_X^{(k)}$, for any X tangent to M, respectively.

In this paper we want to generalize such results. Then, we will consider the conditions $g(A_F^{(k)}(X,Y),\xi )=0$, (respectively, $g(A_F^{(k)}(X,Y),Z)=0$) for any X, Y tangent to M (respectively, for any X, Y tangent to M, $Z \in \mathbb {D}$), obtaining the following

Theorem 1.1

Let M be a real hypersurface in $\mathbb {C}P^{m}$, $m \ge 3$, and k a nonnull real number. Then $g(A_F^{(k)}(X,Y),\xi )=0$ for any X, Y tangent to M if and only if M is locally congruent to a ruled real hypersurface such that $g(A\xi ,\xi )=-k$.

And

Theorem 1.2

Let M be a real hypersurface in $\mathbb {C}P^m$, $m \ge 3$, and k a nonnull constant. Then $g(A_F^{(k)}(X,Y),Z)=0$ for any X, Y tangent to M, $Z \in \mathbb {D}$, if and only if M is locally congruent to a real hypersurface of type (A)

Similar conditions for $A_T^{(k)}$ give us the following results

Theorem 1.3

There does not exist any real hypersurface M in $\mathbb {C}P^m$, $m \ge 3$, such that $g(A_T^{(k)}(X,Y),\xi )=0$, for any X, Y tangent to M and any nonnull real number k.

And

Theorem 1.4

Let M be a real hypersurface in $\mathbb {C}P^m,m \ge 3$ and k a nonnull real number. Then $g(A_T^{(k)}(X,Y),Z)=0$, for any X, Y tangent to M, $Z \in \mathbb {D}$, if and only if M is locally congruent to a real hypersurface of type (A).

On the other hand, we will say that A is $({\hat{\nabla }}^{(k)},\nabla )$-recurrent if $(({\hat{\nabla }}_X^{(k)}-\nabla _X)A)Y=\omega (X)AY$, for any X, Y tangent to M, where $\omega$ is a nonnull 1-form on M. This is equivalent to have $A_F^{(k)}(X,Y)=\omega (X)AY$.

Similarly, we will say that A is $({{\mathcal {L}}}^{(k)},\mathcal L)$-recurrent if $(({{\mathcal {L}}}_X^{(k)}-{{\mathcal {L}}}_X)A)Y=\delta (X)AY$, for any X, Y tangent to M and a nonnull 1-form $\delta$ on M. This is equivalent to have $A_T^{(k)}(X,Y)=\delta (X)AY$.

If we consider ${\mathbb {D}}-({\hat{\nabla }}^{(k)},\nabla )$-recurrency or ${\mathbb {D}}-({{\mathcal {L}}}^{(k)},{\mathcal {L}})$-recurrency (the same conditions as above for $X,Y \in \mathbb {D}$) we obtain

Theorem 1.5

Let M be a real hypersurface in $\mathbb {C}P^m$, $m \ge 3$, and k a nonnull real number. Then $A_F^{(k)}(X,Y)=\omega (X)AY$, for any $X,Y \in \mathbb {D}$ and a nonnull 1-form $\omega$ on M if and only if M is locally congruent either to a real hypersurface of type (A) or to a ruled real hypersurface.

and

Theorem 1.6

There does not exist any real hypersurface M in $\mathbb {C}P^m$, $m \ge 3$, such that $A_T^{(k)}(X,Y)=\delta (X)AY$, for any $X,Y \in \mathbb {D}$, and a nonnull 1-form $\delta$ on M, k being a nonnull real number.

2 Preliminaries

Any mathematical object in the sequel will be considered of class $C^{\infty }$ unless otherwise stated. Let M be a connected real hypersurface without boundary in $\mathbb {C}P^m, m \ge 2$, and N a locally defined normal unit vector field on M. Let$\nabla$ be the Levi-Civita connection on M and (J, g) the Kaehlerian structure of $\mathbb {C}P^m$.

For any vector field X tangent to M, we write $JX=\phi X+\eta (X)N$, where $\phi X$ denotes the tangential component of JX, and $-JN=\xi$. Then $(\phi ,\xi ,\eta ,g)$ is an almost contact metric structure on M (see [1]). Therefore,

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

(2.1)

for any tangent vectors X, Y to M. From (2.1) we get

$$\begin{aligned} \phi \xi =0, \quad \eta (X)=g(X,\xi ). \end{aligned}$$

From the parallelism of J we obtain

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

for any X, Y tangent to M, where A denotes the shape operator of the immersion. As $\mathbb {C}P^m$ has holomorphic sectional curvature 4, the equations of Gauss and Codazzi are given, respectively, by

$$\begin{aligned} R(X,Y)Z= & {} g(Y,Z)X - g(X,Z)Y + g(\phi Y,Z)\phi X - g(\phi X,Z)\phi Y \\{} & {} - 2g(\phi X,Y)\phi Z + g(AY,Z)AX - g(AX,Z)AY, \end{aligned}$$

and

$$\begin{aligned} (\nabla _XA)Y-(\nabla _YA)X=\eta (X)\phi Y-\eta (Y)\phi X-2g(\phi X,Y)\xi \end{aligned}$$

for any tangent vectors X, Y, Z to M, where R is the curvature tensor of M.

In the sequel the following result owed to Maeda [7], is needed.

Theorem 2.1

Let M be a Hopf real hypersurface in $\mathbb {C}P^m$, $m\ge 2$. Then $\alpha =g(A\xi ,\xi )$ is constant and if W is a vector field which belongs to $\mathbb {D}$ such that $AW=\lambda W$, then $2\lambda -\alpha \ne 0$ and $A\phi W=\mu \phi W$, where $\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }$.

We will also need the following theorem proved by Okumura [8]

Theorem 2.2

Let M be a real hypersurface in $\mathbb {C}P^m$, $m \ge 2$. Then $\phi A=A\phi$ if and only if M is locally congruent to a real hypersurface of type (A).

3 Proofs of Theorems 1.1 and 1.2

Let us suppose that $g(A_F^{(k)}(X,Y),\xi )=0$ for any X, Y tangent to M. This yields $g(g(\phi AX,$ $AY)\xi -\eta (AY)\phi AX-k\eta (X)\phi AY-g(\phi AX,Y)A\xi +\eta (Y)A\phi AX+k\eta (X)A\phi Y,\xi )=0$, for any X, Y tangent to M. Therefore

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

(3.1)

for any X, Y tangent to M.

Let us suppose that M is Hopf, that is, $A\xi =\alpha \xi$. Then (3.1) gives $g(A\phi AX,Y)-\alpha g(\phi AX,Y)=0$, for any X, Y tangent to M. Thus $A\phi AX=\alpha \phi AX$, for any X tangent to M. If we choose $X \in \mathbb {D}$ such that $AX=\lambda X$, from Theorem 2.1 we should have $A\phi X=\mu \phi X$, $\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }$. Then $\lambda \mu =\lambda \alpha$ and either $\lambda =0$ or $\mu =\alpha$.

If we suppose that in $\mathbb {D}$ there exists a principal curvature $\lambda \ne 0$, $\mu =\alpha$ yields $\alpha \lambda +2=2\alpha \lambda -\alpha ^2$. That is, $\alpha \lambda =\alpha ^2+2$. This implies that $\alpha \ne 0$ and then $\lambda =\frac{\alpha ^2+2}{\alpha }$. As $\mu =\alpha$, we also have $\lambda \ne \mu$ and all the principal curvatures are constant. Therefore $A\phi \ne \phi A$ and M cannot be of type (A). If there is not a vector field $Y \in \mathbb {D}$ such that $AY=0$, the unique principal curvatures on $\mathbb {D}$ are $\alpha$ and $\frac{\alpha ^2+2}{\alpha }$. Looking at Takagi’s list, this is impossible.

Therefore, the unique principal curvature in $\mathbb {D}$ is $\lambda =0$. But then, $\mu =-\frac{2}{\alpha }$ must be equal to 0 too, which is also impossible.

Then we must suppose that M is non Hopf. So we can write $A\xi =\alpha \xi +\beta U$, where U is a unit vector field in $\mathbb {D}$ and $\beta$ is a function on M that does not vanish at least on a neighborhood of a point $p \in M$. We will make all the calculations on such a neighborhood.

If we take $Y=\xi$ in (3.1) we get $2g(A\phi A\xi ,X)=0$ for any X tangent to M. That is, $\beta g(A\phi U,X)=0$ for any X tangent to M, which yields

$$\begin{aligned} A\phi U=0. \end{aligned}$$

(3.2)

Taking $X=\xi$ in (3.1) we obtain $g(A\phi A\xi ,Y)-\alpha g(\phi A\xi ,Y)+kg(A\phi Y,\xi )=0$, for any Y tangent to M. Then, from (3.2), $-\alpha \beta g(\phi U,Y)-k\beta g(\phi U,Y)=0$. As $\beta \ne 0$, if we take $Y=\phi U$ we have

$$\begin{aligned} \alpha =-k. \end{aligned}$$

(3.3)

If now we take $Y=\phi U$ in (3.1) it follows $-\alpha g(AX,U)-k\eta (X)g(AU,\xi )=0$, for any X tangent to M. That is, $-\alpha g(AU,X)-k\beta \eta (X)=0$. From (3.3) we get $kg(AU,X)-k\beta \eta (X)=0$, for any X tangent to M. Thus

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

(3.4)

From (3.2) and (3.4) we have that ${\mathbb {D}}_U=\{ X \in \mathbb {D}| g(X,U)=g(X,\phi U)=0 \}$ is A-invariant. Take $X,Y \in {\mathbb {D}}_U$ in (3.1). Then $g(A\phi AX,Y)-\alpha g(\phi AX,Y)=0$. From (3.3) this yields

$$\begin{aligned} A\phi AX+k\phi AX=0 \end{aligned}$$

(3.5)

for any $X \in {\mathbb {D}}_U$. We can also write the equation above (3.5) as $-g(A\phi AY,X)+\alpha g(A\phi Y,X)=0$, for any $X,Y \in {\mathbb {D}}_U$. From (3.3) we obtain

$$\begin{aligned} -A\phi AX-kA\phi X=0 \end{aligned}$$

(3.6)

for any $X \in {\mathbb {D}}_U$. Adding (3.5) and (3.6) we have $k(\phi A-A\phi )X=0$ for any $X \in {\mathbb {D}}_U$ and, as $k \ne 0$, we get

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

for any $X \in {\mathbb {D}}_U$. Therefore, if $X \in {\mathbb {D}}_U$ satisfies $AX=\lambda X$, we obtain $A\phi X=\lambda \phi X$. Moreover, from (3.5) it follows $\lambda ^2+k\lambda =0$ and either $\lambda =0$ or $\lambda =-k$.

Let us suppose that there exists $Y \in {\mathbb {D}}_U$ such that $AY=-kY$ and $A\phi Y=-k\phi Y$. The Codazzi equation yields $(\nabla _YA)\phi Y-(\nabla _{\phi Y}A)Y=-2\xi$. Therefore, $-k\nabla _Y\phi Y-A\nabla _Y\phi Y+k\nabla _{\phi Y}Y+A\nabla _{\phi Y}Y=-2\xi$. Its scalar product with $\xi$ gives $kg(\phi Y,\phi AY)-g(\nabla _Y\phi Y,-k\xi +\beta U)-kg(Y,\phi A\phi Y)+g(\nabla _{\phi Y}Y,-k\xi +\beta U)=-2$. This yields $\beta g([\phi Y,Y ],U)-k^2-kg(\phi Y,\phi AY)$ $+kg(A\phi Y,\phi Y)+kg(Y,\phi A\phi Y)=-2$. Thus

$$\begin{aligned} g([\phi Y,Y ],U)=-\frac{2}{\beta }. \end{aligned}$$

(3.7)

Its scalar product with U implies $-kg(\nabla _Y\phi Y,U)-g(\nabla _Y\phi Y,\beta \xi )+kg(\nabla _{\phi Y}Y,U)+g(\nabla _{\phi Y}Y,$ $\beta \xi )=0$. That is, $kg([\phi Y,Y ],U)+\beta g(\phi Y,\phi AY)-\beta g(Y,\phi A\phi Y)=0$. Then

$$\begin{aligned} g([\phi Y,Y ],U)=2\beta . \end{aligned}$$

(3.8)

From (3.7) and (3.8) $\beta =-\frac{1}{\beta }$ would give $\beta ^2=-1$, which is impossible.

We conclude that the unique principal curvature in ${\mathbb {D}}_U$ is 0 and M is ruled. The converse is straightforward and we finish the proof of Theorem 1.1.

In order to prove Theorem 1.2 let us suppose that $g(A_F^{(k)}(X,Y),Z)=0$ for any X, Y tangent to M, $Z \in \mathbb {D}$. This implies

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

(3.9)

for any X, Y tangent to M, $Z \in {\mathbb {D}}$.

Let us suppose that M is Hopf and $A\xi =\alpha \xi$. Taking $X=\xi$ in (3.9) we get $-kg(\phi AY,Z)+kg(A\phi Y, Z)=0$ for any Y tangent to M, $Z \in {\mathbb {D}}$. As $k \ne 0$, this means that $(A\phi -\phi A)X=0$ for any $X \in \mathbb {D}$. From Theorem 2.2, M must be locally congruent to a real hypersurface of type (A).

If M is non Hopf we will write $A\xi =\alpha \xi +\beta U$ with the same conditions as in the proof of Theorem 1.1. Taking $X=Y=\xi$ in (3.9) we have $-\alpha \beta g(\phi U,Z)-k\beta g(\phi U,Z)+\beta g(A\phi U,Z)=0$ for any $Z \in \mathbb {D}$. This gives, bearing in mind that $\beta \ne 0$,

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

(3.10)

If in (3.9) we put $Y=\xi$ we get $-\alpha g(\phi AX,Z)-k\beta \eta (X)g(\phi U,Z)+g(A\phi AX,Z)=0$, for any X tangent to M, $Z \in \mathbb {D}$. If $Z=\phi U$, we obtain $-\alpha g(AU,X)-k\beta \eta (X)+(\alpha +k)g(AU,X)=0$, for any X tangent to M. This implies $kAU=k\beta \xi$. As $k \ne 0$ we obtain

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

(3.11)

If we take $Y=\xi$, $X=\phi U$ in (3.9) we have $\alpha (\alpha +k)g(U,Z)-(\alpha +k)g(AU,Z)=0$, for any $Z \in \mathbb {D}$. From (3.11) we get $\alpha (\alpha +k)g(U,Z)=0$ for any $Z \in \mathbb {D}$. Taking $Z=U$ we obtain $\alpha (\alpha +k)=0$.

Let us suppose that $\alpha =-k$. Then (3.10) and (3.11) imply $A\xi =-k\xi +\beta U$, $AU=\beta \xi$, $A\phi U=0$. If we introduce $X,Y \in {\mathbb {D}}_U$ in (3.9) we have $-g(\phi AX,Y)g(A\xi ,Z)=0$, for any $Z \in \mathbb {D}$. If $Z=U$ we get $g(\phi AX,Y)=0$ for any $X,Y \in {\mathbb {D}}_U$. Now, if we take $\phi Y$ instead of Y it follows $g(AX,Y)=0$ for any $X,Y \in {\mathbb {D}}_U$ and

$$\begin{aligned} AX=0 \end{aligned}$$

(3.12)

for any $X \in {\mathbb {D}}_U$. From (3.10), (3.11), (3.12) and the fact that $\alpha =-k$, M should be ruled. But taking $X=\xi$, $Y=U$ in (3.10) we have $k\beta g(\phi U,Z)=0$ for any $Z \in \mathbb {D}$, which is impossible.

Suppose then that $\alpha =0$. Therefore, $A\xi =\beta U$, $AU=\beta \xi$ and $A\phi U=k\phi U$. Take $X=\xi$, $Y \in {\mathbb {D}}_U$ in (3.9). Then $-kg(\phi AY,Z)+kg(A\phi Y,Z)=0$, for any $Y \in {\mathbb {D}}_U$, $Z \in \mathbb {D}$. This yields $A\phi Y=\phi AY$ for any $Y \in {\mathbb {D}}_U$. As ${\mathbb {D}}_U$ is A-invariant, if $Y \in {\mathbb {D}}_U$ satisfies $AY=\lambda Y$, $A\phi Y=\lambda \phi Y$. If we take $Y=\xi$, $X \in {\mathbb {D}}_U$ in (3.9) we obtain $g(A\phi AX,Z)=0$ for any $X \in {\mathbb {D}}_U$, $Z \in \mathbb {D}$. Therefore, $A\phi AX=0$ for any $X \in {\mathbb {D}}_U$. That is, if $Y \in {\mathbb {D}}_U$ satisfies $AY=\lambda Y$ we obtain $\lambda =0$. Therefore $AZ=0$ for any $Z \in {\mathbb {D}}_U$. For such a Z Codazzi equation gives $(\nabla _ZA)\xi -(\nabla _{\xi }A)Z=-\phi Z$. Then $\nabla _Z(\beta U)-A\phi AZ+A\nabla _{\xi }Z=-\phi Z$. This implies $Z(\beta )U+\beta \nabla _ZU+A\nabla _{\xi }Z=-\phi Z$ and its scalar product with U implies $Z(\beta )-\beta g(Z,\phi A\xi )=0$. We have proved that

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

(3.13)

for any $Z \in {\mathbb {D}}_U$.

On the other hand, $(\nabla _UA)\xi -(\nabla _{\xi }A)U=-\phi U$ implies $U(\beta )U+\beta \nabla _UU-\xi (\beta )\xi -\beta \phi A\xi +A\nabla _{\xi }U=-\phi U$. Its scalar product with $\xi$ gives $-\beta g(U,\phi AU)-\xi (\beta )+\beta g(\nabla _ {\xi }U,U)=0$. That is,

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

(3.14)

and its scalar product with U yields $U(\beta )-\beta ^2g(U,\phi U)=0$. Thus

$$\begin{aligned} U(\beta )=0. \end{aligned}$$

(3.15)

Also $(\nabla _{\phi U}A)\xi -(\nabla _{\xi }A)\phi U=U$ yields $(\phi U)(\beta )U+\beta \nabla _{\phi U}U+kAU-k\nabla _{\xi }\phi U+A\nabla _{\xi }\phi U=U$. Its scalar product with $\xi$ implies $3k\beta +\beta g(\nabla _{\xi }\phi U,U)=0$. Then

$$\begin{aligned} g(\nabla _{\xi }\phi U,U)=-3k \end{aligned}$$

(3.16)

and its scalar product with U gives $(\phi U)(\beta )-kg(\nabla _{\xi }\phi U,U)-\beta g(A\xi ,U)=1$. Therefore, from (3.16),

$$\begin{aligned} (\phi U)(\beta )=-3k^2+\beta ^2+1. \end{aligned}$$

(3.17)

From (3.13), (3.14), (3.15) and (3.17) we obtain $grad(\beta )=\gamma \phi U$, where $\gamma =-3k^2+\beta ^2+1$. As $g(\nabla _Xgrad(\beta ),Y)=g(\nabla _Ygrad(\beta ),X)$, for any X, Y tangent to M, we get $X(\gamma )g(\phi U,Y)+\gamma g(\nabla _X\phi U,Y)=Y(\gamma )g(\phi U,X)+\gamma g(\nabla _Y\phi U,X)$. If $X=\xi$ we obtain $\gamma g(\nabla _{\xi }\phi U,Y)=\gamma g(\nabla _Y\phi U,\xi )=-\gamma g(U,AY)$ for any Y tangent to M. If now $Y=U$ it follows $\gamma g(\nabla _{\xi }\phi U,U)=0$. From (3.16) we get $-3k\gamma =0$. Thus $\gamma =0$ and $\beta$ is constant.

Then $(\nabla _{\phi U}A)U-(\nabla _UA)\phi U=2\xi$ yields $\beta \phi A\phi U-A\nabla _{\phi U}U-k\nabla _U\phi U+A\nabla _U\phi U=2\xi$. Its scalar product with $\xi$ gives $kg(U,AU)+\beta g(\nabla _U\phi U,U)=2$. Therefore,

$$\begin{aligned} \beta g(\nabla _U\phi U,U)=2 \end{aligned}$$

(3.18)

and its scalar product with U implies $-\beta k+\beta g(U,\phi A\phi U)-kg(\nabla _U\phi U,U)=0$. That is, $-2\beta k=kg(\nabla _U\phi U,U)$. Then

$$\begin{aligned} g(\nabla _U\phi U,U)=-2\beta . \end{aligned}$$

(3.19)

From (3.18) and (3.19) we have $-\beta ^2=1$, which is impossible and this finishes the proof of Theorem 1.2.

4 Proofs of Theorems 1.3 and 1.4

If we suppose that $g(A_T^{(k)}(X,Y),\xi )=0$ for any X, Y tangent to M we obtain

$$\begin{aligned}{} & {} g(\phi AX,AY)-g(\phi A^2Y,X)-g(A\xi ,\xi )g(\phi AX,Y)+\eta (Y)g(\phi AX,A\xi ) \nonumber \\{} & {} \quad +k\eta (X)g(\phi Y,A\xi ) +g(A\xi ,\xi )g(\phi AY,X)-\eta (X)g(\phi AY,A\xi )\nonumber \\{} & {} \quad -k\eta (Y)g(\phi X,A\xi )=0 \end{aligned}$$

(4.1)

for any X, Y tangent to M.

Let us suppose that M is Hopf, $A\xi =\alpha \xi$, and take $X,Y \in \mathbb {D}$ in (4.1). Then $g(\phi AX,AY)-g(\phi A^2Y,X)-\alpha g(\phi AX,Y)+\alpha g(\phi AY,X)=0$. Therefore, we obtain $A\phi AX+A^2\phi X-\alpha \phi AX-\alpha A\phi X=0$ for any $X \in \mathbb {D}$. If $X \in \mathbb {D}$ satisfies $AX=\lambda X$, from Theorem 2.1 we know that $A\phi X=\mu \phi X$. Thus $\lambda \mu +\mu ^2-\alpha \lambda -\alpha \mu =0$. That is, $(\lambda +\mu )\mu -(\lambda +\mu )\alpha =0$, or $(\lambda +\mu )(\mu -\alpha )=0$. If $\lambda +\mu =0$, as $\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }$, we obtain $2\lambda ^2+2=0$, which is impossible. Therefore $\mu =\alpha$ and then, as in the proof of Theorem 1.1, this case is not possible.

Therefore M must be non Hopf. We continue writing $A\xi =\alpha \xi +\beta U$ as in Sect. 3.

Taking $X=\xi$ in (4.1) we obtain $\beta g(\phi U,AY)-\alpha \beta g(\phi U,Y)+kg(\phi Y,A\xi )-g(\phi AY,A\xi )=0$, for any Y tangent to M. This gives $\beta A\phi U-\alpha \beta \phi U-k\beta \phi U+\beta A\phi U=0$. Then $2A\phi U-(\alpha +k)\phi U=0$ and

$$\begin{aligned} A\phi U=\left( \frac{\alpha +k}{2}\right) \phi U. \end{aligned}$$

(4.2)

If now we take $Y=\xi$ in (4.1) it follows $2\,g(\phi AX,A\xi )-g(\phi A^2\xi ,X)+\alpha g(\phi A\xi ,X)-kg(\phi X,A\xi )=0$, for any X tangent to M. Then $-2\beta g(A\phi U,X)-g(\phi A(\alpha \xi +\beta U),X)+\alpha \beta g(\phi U,X)+k\beta g(\phi U,X)=0$, for any X tangent to M. From (4.2) we get $-\alpha g(\phi A\xi ,X)-\beta g(\phi AU,X)=0$, for any X tangent to M. Therefore, $-\alpha \beta \phi U-\beta \phi AU=0$, or $\phi AU=-\alpha \phi U$. Applying $\phi$ we obtain

$$\begin{aligned} AU=\beta \xi -\alpha U. \end{aligned}$$

(4.3)

Take $X=\phi U$ in (4.1). Then $g(\phi A\phi U,AY)-g(A^2Y,U)-\alpha g(\phi A\phi U,Y)+\eta (Y)g(\phi A\phi U,A\xi )$ $+\alpha g(AY,U)+k\eta (Y)g(U,A\xi )=0$, for any Y tangent to M. From (4.2) we get $-(\frac{\alpha +k}{2})$ $g(AU,Y)-g(A^2U,Y)+\alpha (\frac{\alpha +k}{2})g(U,Y)-\beta (\frac{\alpha +k}{2})\eta (Y)+\alpha g(AU,Y)+k\beta \eta (Y)=0$, for any Y tangent to M. Therefore, $(\alpha -(\frac{\alpha +k}{2}))AU-A^2U+\alpha (\frac{\alpha +k}{2})U+\beta (k-(\frac{\alpha +k}{2}))\xi =0$. This and (4.3) yield

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

(4.4)

If now we take $Y=\phi U$ in (4.1) we have $g(\phi AX,A\phi U)-g(\phi A^2\phi U,X)-\alpha g(AX,U)-k\eta (X)g(U,A\xi )+\alpha g(\phi A\phi U,X)-\beta g(\phi A\phi U,U)\eta (X)=0$, for any X tangent to M. This yields $((\frac{\alpha +k}{2})-\alpha )AU+(\frac{\alpha +k}{2})((\frac{\alpha +k}{2})-\alpha )U+\beta ((\frac{\alpha +k}{2})-\alpha )\xi =0$, that is, $(\frac{k-\alpha }{2})AU+(\frac{k+\alpha }{2})(\frac{k-\alpha }{2})U-\beta (\frac{k-\alpha }{2})\xi =0$. If $\alpha =k$, from (4.4), $\beta =0$, which is impossible. Therefore, $k \ne \alpha$ and we get

$$\begin{aligned} AU=\beta \xi -\left( \frac{\alpha +k}{2}\right) U. \end{aligned}$$

(4.5)

From (4.3) and (4.5), $\alpha =\frac{\alpha +k}{2}$, and then, $\alpha =k$, that we have seen that is impossible, finishing the proof of Theorem 1.3.

Suppose now that $g(A_T^{(k)}(X,Y),Z)=0$ for any X, Y tangent to M, $Z \in \mathbb {D}$. This implies

$$\begin{aligned}{} & {} -\eta (AY)g(\phi AX,Z)-k\eta (X)g(\phi AY,Z)+\eta (X)g(\phi A^2Y,Z)+k\eta (AY)g(\phi X,Z) \nonumber \\{} & {} \quad -g(\phi AX,Y)g(A\xi ,Z)+\eta (Y)g(\phi AX,AZ)+k\eta (X)g(\phi Y,AZ)+g(\phi AY,X)g(A\xi ,Z) \nonumber \\{} & {} \quad -\eta (X)g(\phi AY,AZ)-k\eta (Y)g(\phi X,AZ)=0 \end{aligned}$$

(4.6)

for any X, Y tangent to M, $Z \in \mathbb {D}$.

Let us suppose that M is Hopf with $A\xi =\alpha \xi$. Take $X=\xi$, $Y \in \mathbb {D}$ in (4.6). Then we get $-kg(\phi AY,Z)+g(\phi A^2Y,Z)-kg(\phi Y,AZ)-g(\phi AY,AZ)=0$, for any $Y,Z \in \mathbb {D}$. Therefore,

$$\begin{aligned} -k\phi AY+\phi A^2Y+kA\phi Y-A\phi AY=0 \end{aligned}$$

(4.7)

for any $Y \in \mathbb {D}$. If we interchange Y and Z we also obtain

$$\begin{aligned} kA\phi Y-A^2\phi Y-k\phi AY+A\phi AY=0 \end{aligned}$$

(4.8)

for any $Y \in \mathbb {D}$. If such a Y satisfies $AY=\lambda Y$, from (4.7) and Theorem 2.1 we obtain

$$\begin{aligned} (\lambda -\mu )(\lambda -k)=0 \end{aligned}$$

(4.9)

where $\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }$. From (4.8) we also get

$$\begin{aligned} (k-\mu )(\mu -\lambda )=0. \end{aligned}$$

(4.10)

From (4.9) and (4.10) either $\lambda =\mu$ for any principal curvature in $\mathbb {D}$, and in this case, from Theorem 2.2, M is locally congruent to a real hypersurface of type (A) or there exists $\lambda$ such that $\mu \ne \lambda$. Then $\lambda =\mu =k$, which is impossible.

Suppose now that M is non Hopf and write $A\xi$ as before. Take $X=Y=\xi$ in (4.6). Then $-\alpha g(\phi A\xi ,Z)-kg(\phi A\xi ,Z)+g(\phi A^2\xi ,Z)+g(\phi A\xi ,AZ)-g(\phi A\xi ,AZ)=0$. Therefore, $-(\alpha +k)\beta g(\phi U,Z)+g(\phi A(\alpha \xi +\beta U),Z)=0$, for any $Z \in \mathbb {D}$. This yields $-k\beta g(\phi U,Z)+\beta g(\phi AU,Z)=0$, for any $Z \in \mathbb {D}$. Then $\phi AU=k\phi U$, and applying $\phi$ we get

$$\begin{aligned} AU=\beta \xi +kU. \end{aligned}$$

(4.11)

Take now $X=\xi$, $Y=\phi U$ in (4.6). We obtain $-kg(\phi A\phi U,Z)+g(\phi A^2\phi U,Z)-g(A\xi ,U)g(A\xi ,Z)-kg(AU,Z)-g(\phi A\phi U,AZ)=0$, for any $Z \in \mathbb {D}$. If we take $Z=U$ we get $kg(A\phi U,\phi U)-g(A^2\phi U,\phi U)-\beta ^2-k^2+g(A\phi U,\phi AU)=0$. That is,

$$\begin{aligned} 2kg(A\phi U,\phi U)=g(A\phi U,A\phi U)+\beta ^2+k^2. \end{aligned}$$

(4.12)

If we take $X=U$, $Y=\phi U$ in (4.6) we have $-g(\phi AU,\phi U)g(A\xi ,Z)+g(\phi A\phi U,U)g(A\xi ,Z)=0$, for any $Z \in \mathbb {D}$. From (4.11) it follows $-kg(A\xi ,Z)-g(A\phi U,\phi U)g(A\xi ,Z)=0$, for any $Z \in \mathbb {D}$. If $Z=U$ we get $g(A\phi U,\phi U)=-k$, and from (4.12) $g(A\phi U,A\phi U)+\beta ^2+3k^2=0$, which is impossible, finishing the proof of Theorem 1.4.

5 Proofs of Theorems 1.5 and 1.6

If we suppose that $A_F^{(k)}(X,Y)=\omega (X)AY$ for any $X,Y \in \mathbb {D}$ we get

$$\begin{aligned} g(\phi AX,AY)\xi -\eta (AY)\phi AX-g(\phi AX,Y)A\xi =\omega (X)AY \end{aligned}$$

(5.1)

for any $X,Y \in \mathbb {D}$.

Let us suppose that M is Hopf and that $A\xi =\alpha \xi$. Then (5.1) becomes

$$\begin{aligned} g(\phi AX,AY)\xi -\alpha g(\phi AX,Y)\xi =\omega (X)AY \end{aligned}$$

(5.2)

for any $X,Y \in \mathbb {D}$. The scalar product of (5.2) and $\xi$ gives $g(\phi AX,AY)-\alpha g(\phi AX,Y)=0$, for any $X,Y \in \mathbb {D}$. Therefore, we have

$$\begin{aligned} A\phi AX-\alpha \phi AX=0 \end{aligned}$$

(5.3)

for any $X \in \mathbb {D}$, and interchanging X and Y we also get

$$\begin{aligned} -A\phi AX+\alpha A\phi X=0 \end{aligned}$$

(5.4)

for any $X \in \mathbb {D}$. From (5.3) and (5.4) it follows $\alpha (\phi A-A\phi )X=0$ for any $X \in \mathbb {D}$. Let us suppose that $\alpha =0$. Then, from (5.3) we obtain $A\phi AX=0$ for any $X \in \mathbb {D}$ and if we suppose that $AX=\lambda X$, from Theorem 2.1, $\lambda (\frac{2}{2\lambda })=0$, which is impossible. Therefore, $\phi A-A\phi =0$, and from Theorem 2.2, M must be locally congruent to a real hypersurface of type (A). In this case (5.3) gives $A^2\phi X-\alpha A\phi X=0$ for any $X \in \mathbb {D}$ and also $\phi A^2X-\alpha \phi AX=0$. Thus $\mu (\mu -\alpha )=\lambda (\lambda -\alpha )=0$. We have now that either $\mu =0$ or $\mu =\alpha$ and, at the same time, either $\lambda =0$ or $\lambda =\alpha$. These four possibilities give contradictions and M must be non Hopf.

As in previous sections we write $A\xi =\alpha \xi +\beta U$. Then (5.1) looks like

$$\begin{aligned} g(\phi AX,AY)\xi -\beta g(U,Y)\phi AX-g(\phi AX,Y)A\xi =\omega (X)AY \end{aligned}$$

(5.5)

for any $X,Y \in \mathbb {D}$. Taking $Y=U$ in (5.5) we get $g(\phi AX,AU)\xi -\beta \phi AX-g(\phi AX,U)A\xi =\omega (X)AU$. Its scalar product with U yields $-2\beta g(\phi AX,U)=\omega (X)g(AU,U)$ for any $X \in \mathbb {D}$. If, in particular, $X=U$ we obtain

$$\begin{aligned} \omega (U)g(AU,U)-2\beta g(AU,\phi U)=0. \end{aligned}$$

(5.6)

Taking the scalar product of (5.5) and $\phi U$ we have $-\beta g(U,Y)g(AX,U)=\omega (X)g(AY,\phi U)$, for any $X,Y \in \mathbb {D}$. If $X=Y=U$ it follows

$$\begin{aligned} \beta g(AU,U)+\omega (U)g(AU,\phi U)=0. \end{aligned}$$

(5.7)

The linear system given by (5.6) and (5.7) satisfies $(\omega (U))^2+2\beta ^2 \ne 0$, and therefore

$$\begin{aligned} g(AU,U)=g(AU,\phi U)=0. \end{aligned}$$

(5.8)

Taking $X \in {\mathbb {D}}_U$ in (5.5) and its scalar product with $\phi U$ we obtain $-\beta g(U,Y)g(AU,X)=\omega (X)g(AY,\phi U)$ for any $X \in {\mathbb {D}}_U$, $Y \in \mathbb {D}$. Bearing in mind (5.8), if $Y=U$ and $X \in {\mathbb {D}}_U$ we get $-\beta g(AU,X)=0$, for any $X \in {\mathbb {D}}_U$. As $\beta \ne 0$, it follows

$$\begin{aligned} g(AU,X)=0 \end{aligned}$$

(5.9)

for any $X \in {\mathbb {D}}_U$. Now (5.8) and (5.9) yield

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

(5.10)

The scalar product of (5.5) and U gives $\beta g(U,Y)g(A\phi U,X)-\beta g(\phi AX,Y)=\omega (X)g(AY,U)$ $=0$, for any $X,Y \in \mathbb {D}$. Taking $Y=U$ we have $2\beta g(A\phi U,X)=0$, for any $X \in \mathbb {D}$. Thus

$$\begin{aligned} A\phi U=0. \end{aligned}$$

(5.11)

Take now $X,Y \in {\mathbb {D}}_U$ in (5.5). Then, $g(\phi AX,AY)\xi -g(\phi AX,Y)A\xi =\omega (X)AY$, and its scalar product with U yields $\beta g(AX,\phi Y)=0$, for any $X,Y \in {\mathbb {D}}_U$. Therefore, $AX=0$, for any $X \in {\mathbb {D}}_U$. This, (5.10) and (5.11) imply that M is locally congruent to a ruled real hypersurface, finishing the proof of Theorem 1.5.

If now $A_T^{(k)}(X,Y)=\delta (X)AY$, for any $X,Y \in \mathbb {D}$, we obtain

$$\begin{aligned}{} & {} g(\phi AX,AY)\xi -\eta (AY)\phi AX-g(\phi A^2Y,X)\xi +k\eta (AY)\phi X \nonumber \\{} & {} \quad -g(\phi AX,AY)A\xi +g(\phi AY,AX)A\xi =\delta (X)AY \end{aligned}$$

(5.12)

for any $X,Y \in \mathbb {D}$.

If we suppose that M is Hopf, $A\xi =\alpha \xi$, and take the scalar product of (5.12) and $\xi$, we get

$$\begin{aligned} g(\phi AX,AY)-g(\phi A^2Y,X)-\alpha g(\phi AX,Y)+\alpha g(\phi AY,X)=0 \end{aligned}$$

(5.13)

for any $X,Y \in \mathbb {D}$. Then (5.13) yields

$$\begin{aligned} A\phi AX+A^2\phi X-\alpha \phi AX-\alpha A\phi X=0 \end{aligned}$$

(5.14)

for any $X \in \mathbb {D}$ and, interchanging X and Y,

$$\begin{aligned} -A\phi AX-\phi A^2X+\alpha A\phi X+\alpha \phi AX=0 \end{aligned}$$

(5.15)

for any $X \in \mathbb {D}$. From (5.14) and (5.15) we have $A^2\phi X-\phi A^2X=0$, for any $X \in \mathbb {D}$. If we suppose that $AX=\lambda X$, from Theorem 2.1, $A\phi X=\mu \phi X$ and $\mu ^2=\lambda ^2$. If $-\lambda =\mu =\frac{\alpha \lambda +2}{2\lambda -\alpha }$, it yields $\alpha \lambda -2\lambda ^2=\alpha \lambda +2$. Therefore, $\lambda ^2+1=0$, which is impossible. Therefore $\lambda =\mu$ and $\phi A=A\phi$. In this case (5.14) becomes $2A^2\phi X-2\alpha A\phi X=0$ and then $\mu (\mu -\alpha )=0$. In the same way, (5.15) implies $-2\phi A^2X+2\alpha \phi AX=0$ and $\lambda (\alpha -\lambda )=0$. The four possibilities that we obtain imply contradictions and M must be non Hopf. Write as usual $A\xi =\alpha \xi +\beta U$.

The scalar product of (5.12) and $\phi U$ gives

$$\begin{aligned} -\eta (AY)g(AX,U)+k\eta (AY)g(X,U)=\delta (X)g(AY,\phi U) \end{aligned}$$

(5.16)

for any $X,Y \in \mathbb {D}$. Taking $X=\phi U$ in (5.16) and $Y=U$ we get $-\beta g(A\phi U,U)=\delta (\phi U)g(AU,\phi U)$. Thus

$$\begin{aligned} (\delta (\phi U)+\beta )g(AU,\phi U)=0. \end{aligned}$$

(5.17)

If we put $Y=\phi U$ in (5.16) we obtain

$$\begin{aligned} \delta (X)g(A\phi U,\phi U)=0 \end{aligned}$$

(5.18)

for any $X \in \mathbb {D}$.

Take the scalar product of (5.12) and U. Then it follows

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

(5.19)

for any $X,Y \in \mathbb {D}$. Taking $Y=\phi U$ in (5.19) we obtain $-\beta g(AX,U)+\beta g(\phi A\phi U,X)=\delta (X)g(A\phi U,U)$. If $X=\phi U$ it follows $-\beta g(A\phi U,U)+\beta g(A\phi U,U)=\delta (\phi U)g(A\phi U,U)$. That is,

$$\begin{aligned} \delta (\phi U)g(A\phi U,U)=0. \end{aligned}$$

(5.20)

Suppose that $\delta (\phi U)=-\beta$. Then, from (5.18), $g(A\phi U,\phi U)=0$ and from (5.20), $g(A\phi U,U)=0$. If $\delta (\phi U) \ne -\beta$, from (5.17), $g(AU,\phi U)=0$. Thus we have proved that always

$$\begin{aligned} g(AU,\phi U)=0. \end{aligned}$$

(5.21)

If we take $X=Y=U$ in (5.19), bearing in mind (5.21), we obtain

$$\begin{aligned} \delta (U)g(AU,U)=0. \end{aligned}$$

(5.22)

If now we take $Y=U$ in (5.16) we get $-\beta g(AU,X)+k\beta g(U,X)=\delta (X)g(AU,\phi U)=0$, for any $X \in \mathbb {D}$. This yields

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

(5.23)

and from (5.22) we also have $\delta (U)=0$.

If we take $X=U$ in (5.19) we obtain $-\beta g(\phi AU,Y)+\beta g(\phi AY,U)=0$ for any $Y \in \mathbb {D}$. This yields $A\phi U=-\phi AU$, and bearing in mind (5.23), we arrive at

$$\begin{aligned} A\phi U=-k\phi U. \end{aligned}$$

(5.24)

From (5.23) and (5.24) we know that ${\mathbb {D}}_U$ is A-invariant. Taking $Y=U$, $X \in {\mathbb {D}}_U$ in (5.12) we have $-\beta \phi AX+k\beta \phi X=\delta (X)AU$, for any $X \in {\mathbb {D}}_U$. If we take its scalar product with $\xi$ we get $\beta \delta (X)=0$, for any $X \in {\mathbb {D}}_U$. Thus $\delta (X)=0$ for such an X, and the above equation implies $\phi AX=k\phi X$ for any $X \in {\mathbb {D}}_U$. If we apply $\phi$ we obtain $AX=kX$ for any $X \in {\mathbb {D}}_U$. For such a vector field $AX=kX$, $A\phi X=k\phi X$. Codazzi equation implies $(\nabla _XA)\phi X-(\nabla _{\phi X}A)X=-2\xi$. Therefore, $k\nabla _X\phi X-A\nabla _X\phi X-k\nabla _{\phi X}X+A\nabla _{\phi X}X=-2\xi$. Its scalar product with U gives $-kg([\phi X,X ],U)-g(\nabla _X\phi X,\beta \xi +kU)+g(\nabla _{\phi X}X,\beta \xi +kU)=0$. This yields $\beta g(\phi X,\phi AX)-\beta g(X,\phi A\phi X)=0$. Thus $2k\beta =0$, which is impossible and finishes the proof of Theorem 1.6.

Suppose finally that M satisfies $A_F^{(k)}(X,Y)=\omega (X)AY$ for any X, Y tangent to M, From Theorem 1.5M must be locally congruent to either a real hypersurface of type (A) or to a ruled real hypersurface. Moreover, M must satisfy

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

(5.25)

for any X, Y tangent to M. Suppose that M is a real hypersurface of type (A) and take $X=\xi$ in (5.25). We get

$$\begin{aligned} -k\phi AY+kA\phi Y=\omega (\xi )AY \end{aligned}$$

(5.26)

for any Y tangent to M. As our real hypersurface satisfies $A\phi =\phi A$, from (5.26) we have $\omega (\xi )AY=0$ for any Y tangent to M. If $\omega (\xi ) \ne 0$ we should have $AY=0$ for any Y tangent to M. That is, M is totally geodesic, which is impossible. Therefore $\omega (\xi )=0$.

Take then $Y=\xi$ in (5.25). We obtain

$$\begin{aligned} -\alpha \phi AX+A\phi AX=\alpha \omega (X)\xi \end{aligned}$$

(5.27)

for any X tangent to M. Consider $X \in \mathbb {D}$ and take the scalar product of (5.27) and $Z \in \mathbb {D}$. This gives $-\alpha g(\phi AX,Z)+g(A\phi AX,Z)=0$, for any $X,Z \in \mathbb {D}$. Therefore $-\alpha \phi AX+A\phi AX=0$. As $A\phi =\phi A$ we have $-\alpha \phi AX+\phi A^2X=0$ for any $X \in \mathbb {D}$ Suppose that $AX=\lambda X$. Then $-\alpha \lambda +\lambda ^2=0$, and the unique principal curvatures in $\mathbb {D}$ are $\alpha$ and 0. Thus M has, exactly, two distinct constant principal curvatures and looking at Takagi’s list M must be locally congruent to a geodesic hypersphere. But a geodesic hypersphere has not such principal curvatures.

If M is ruled and we take $Y=\phi U$ in (5.25) it follows $-g(\phi AX,\phi U)A\xi -k\eta (X)AU=\omega (X)A\phi U=0$. Then $-g(AX,U)A\xi -k\eta (X)AU=0$, for any X tangent to M. Its scalar product with U implies $-\beta g(AX,U)=0$ for any X tangent to M. If $X=\xi$ we get $\beta ^2=0$, which is impossible. Thus we have obtained the following

Corollary 5.1

There does not exist any real hypersurface M in $\mathbb {C}P^m$, $m \ge 3$, such that $A_F^{(k)}(X,Y)=\omega (X)AY$, for a certain nonnull 1-form $\omega$ on M, any X, Y tangent to M and a nonnull real number k.

Similarly, we have

Corollary 5.2

There does not exist any real hypersurface M in $\mathbb {C}P^m$, $m \ge 3$, such that $A_T^{(k)}(X,Y)=\delta (X)AY$, for a certain nonnull 1-form $\delta$ on M, any X, Y tangent to M and a nonnull real number k.