Derivatives of the Operator $\phi A-A\phi$ on a Real Hypersurface in Non-flat Complex Space Forms

Abstract

On a real hypersurface M in a non-flat complex space form, two types of connections can be defined: the Levi-Civita connection, which is torsion free and, for any nonnull constant k, the kth generalized Tanaka-Webster connection, which is an affine connection with torsion. So the associated covariant derivatives can be considered. Moreover, the Lie derivative and a derivative of Lie type associated with the kth generalized Tanaka-Webster connection can be defined. In this paper, real hypersurfaces for which both covariant derivatives or Lie derivative and the derivative of Lie type associated with the kth generalized Tanaka-Webster connection coincide when they act on the operator $\phi A-A\phi$ , where $\phi$ denotes the structure operator and A the shape operator of M, either in the direction of the structure vector field $\xi$ or in any direction of the maximal holomorphic distribution of M are classified.

1 Introduction

A non-flat complex space form is a Kähler manifold of constant holomorphic sectional curvature c, which is analytically isometric either to complex projective space ${\mathbb {C}}P^{n}$, if $c>0$ or to complex hyperbolic space ${\mathbb {C}}H^{n}$, if $c<0$. We will suppose that in the case of ${\mathbb {C}}P^{n}$ the holomorphic sectional curvature is equal to 4 and in the case of ${\mathbb {C}}H^{n}$ the holomorphic sectional curvature is equal to $-$ 4. The symbol $M_{n}(c), n\ge 2,$ is used to denote them when it is not necessary to distinguish them.

Let $M_{n}(c)$, $n\ge 2$, be a non-flat complex space form endowed with the Kählerian structure (J, G), where J is the complex structure and G is the metric of constant holomorphic sectional curvature c. Suppose M to be a connected real hypersurface of $M_{n}(c)$ without boundary and consider a locally defined unit normal vector field N on M. The shape operatorA of the real hypersurface M in $M_{n}(c)$ with respect to N is given by

$$\begin{aligned} {\overline{\nabla }}_{X}N=-AX, \end{aligned}$$

where ${\overline{\nabla }}$ is the Levi-Civita connection of $M_{n}(c)$. Furthermore, the Kählerian structure of $M_{n}(c)$ induces on M an almost contact metric structure $(\phi , \xi , \eta , g)$, where $\phi X$ is the tangent component of JX, $\eta$ is an one-form given by $\eta (X)=g(X,\xi )$ for any X tangent to M, $\xi =-JN$ is the structure vector field and g is the induced Riemannian metric. The maximal holomorphic distribution on M is defined by ${\mathbb {D}}=Ker(\eta )$.

The eigenvectors of the shape operator A are called principal vectors and the corresponding eigenvalues are called principal curvatures. A real hypersurface in $M_{n}(c)$ is a Hopf hypersurface, when the structure vector field $\xi$ is a principal vector of the shape operator, i.e. $A\xi =g(A\xi ,\xi )\xi$. The corresponding eigenvalue $g(A\xi ,\xi )=\alpha$ is called Hopf principal curvature and is constant.

Takagi initiated the study of real hypersurfaces in ${\mathbb {C}}P^n,n\ge 2$. He classified the homogeneous real hypersurfaces in ${\mathbb {C}}P^n,n\ge 2$ (see [24,25,26]). In [10], Kimura proved that these are the unique Hopf hypersurfaces with constant principal curvatures in ${\mathbb {C}}P^n$. Takagi’s list contains 6 types of real hypersurfaces:

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

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

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

• Type (C) are 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 $m \ge 5$. They have 5 distinct constant principal curvatures, $2{\hbox {cot}}(2r)$ with eigenspace ${\mathbb {R}}[\xi ]$, ${\hbox {cot}}(r-\frac{\pi }{4})$ with multiplicity 2, ${\hbox {cot}}(r-\frac{\pi }{2})=-{\hbox {tan}}(r)$ with multiplicity $m-3$, ${\hbox {cot}}(r-\frac{3\pi }{4})$, with multiplicity 2 and ${\hbox {cot}}(r-\pi )={\hbox {cot}}(r)$ with multiplicity $m-3$. Moreover $\phi T_{{\hbox {cot}}(r-\frac{\pi }{4})}=T_{{\hbox {cot}}(r-\frac{3\pi }{4})}$ and $T_{-{\hbox {tan}}(r)}$ and $T_{{\hbox {cot}}(r)}$ are $\phi$-invariant.

• Type (D) are tubes of radius r, $0< r < \frac{\pi }{4}$, over the Plucker embedding of the complex Grassmannian manifold G(2, 5) in ${\mathbb {C}}P^9$. They have the same principal curvatures as type (C) real hypersurfaces, $2{\hbox {cot}}(2r)$ 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) are 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) real hypersurfaces, $2 {\hbox {cot}}(2r)$ with eigenspace ${\mathbb {R}}[\xi ]$, ${\hbox {cot}}(r-\frac{\pi }{4})$ and ${\hbox {cot}}(r-\frac{3\pi }{4})$ have multiplicities equal to 6 and $-{\hbox {tan}}(r)$ and ${\hbox {cot}}(r)$ have multiplicities equal to 8. Their corresponding eigenspaces have the same behaviour with respect to $\phi$ as in type (C).

From now on, the notion of real hypersurfaces of type (A) refers to both type $(A_1)$ and type $(A_2)$.

In the case of ${\mathbb {C}}H^{n},n\ge 2$, Montiel classified real hypersurfaces with at most two constant principal curvatures (see [15]). Later, in [1] Berndt classified Hopf hypersurfaces in ${\mathbb {C}}H^{n},n\ge 2$, with constant principal curvatures. Berndt’s list contains 4 types of real hypersurfaces:

• Type ($A_{0}$) are horospheres.

• Type ($A_{1}$) are geodesic hyperspheres or tubes over a totally geodesic complex hyperbolic hyperplane ${\mathbb {C}}H^{n - 1}$.

• Type $(A_{2})$ are tubes over a totally geodesic ${\mathbb {C}}H^{k}$, $(1\le k\le n-2)$.

• Type (B) are tubes over totally geodesic real hyperbolic space ${\mathbb {R}}H^{n}$.

From now on, the notion of real hypersurfaces of type (A) refers to types ($A_{0}$), ($A_{1}$) and ($A_{2}$). All the above Hopf hypersurfaces are homogeneous but in contrast to ${\mathbb {C}}P^{n}$ there are homogeneous real hypersurfaces which are not Hopf (see [2]).

Apart from Hopf hypersurfaces, there are ruled real hypersurfaces in $M_{n}(c)$. A ruled real hypersurface can be described as follows: consider a regular curve $\gamma$ in $M_{n}(c)$ with tangent vector field X. At each point of $\gamma$, there is a unique hyperplane of $M_{n}(c)$ cutting $\gamma$ in a way to be orthogonal to both X and JX. The union of all these hyperplanes is called ruled hypersurface. It will be an embedded hypersurface locally, although globally it will in general have self-intersections and singularities. Equivalently, a ruled real hypersurface satisfies that the maximal holomorphic distribution ${\mathbb {D}}$ of M at any point p is integrable and it has an integrable manifold $M_{n-1}(c)$, i.e. $g(A{\mathbb {D}}, {\mathbb {D}})=0$. For examples of ruled real hypersurfaces see [11, 13].

The Tanaka-Webster connection, [27, 29], is the canonical affine connection defined on a non-degenerate, pseudo-Hermitian CR-manifold. As a generalization of this connection, Tanno in [28], defined the generalized Tanaka-Webster connection for contact metric manifolds by

$$\begin{aligned} {\hat{\nabla }}_XY=\nabla _XY+(\nabla _X \eta )(Y)\xi -\eta (Y)\nabla _X \xi -\eta (X)\phi Y. \end{aligned}$$

(1.1)

Using the naturally extended affine connection of Tanno’s generalized Tanaka-Webster connection, in [5, 6] Cho defined the k-th generalized Tanaka-Webster connection${\hat{\nabla }}^{(k)}$ for a real hypersurface M in $M_{n}(c)$ given by

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

(1.2)

for any X, Y tangent to M where k is a nonnull real number and A is the shape operator of M. Then relations ${\hat{\nabla }}^{(k)}\eta =0$, ${\hat{\nabla }}^{(k)}\xi =0$, ${\hat{\nabla }}^{(k)}g=0$, ${\hat{\nabla }}^{(k)}\phi =0$ hold. In particular, if the shape operator of a real hypersurface satisfies $\phi A+A \phi =2k\phi$, the kth generalized Tanaka-Webster connection coincides with the Tanaka-Webster connection.

The tensor field of type (1,2) given by the difference of both connections $F^{(k)}(X,Y)=g(\phi AX,Y)\xi -\eta (Y)\phi AX-k\eta (X)\phi Y$, for any X, Y tangent to M is called k-th Cho tensor on M (see [12] Proposition 7.10, pp. 234–235). Associated with it, for any X tangent to M and any nonnull real number k the tensor field of type (1,1) $F_X^{(k)}$, is called k-th Cho operator corresponding toX and is given by $F_X^{(k)}Y=F^{(k)}(X,Y)$ for any $Y \in TM$. If $X \in {\mathbb {D}}$, the corresponding Cho operator does not depend on k and is simply written by $F_X$. Finally, the torsion of the connection ${\hat{\nabla }}^{(k)}$ is given by ${\hat{T}}^{(k)}(X,Y)=F_X^{(k)}Y-F_Y^{(k)}X$ for any X, Y tangent to M.

Let K be a symmetric operator on M. Then relation $\nabla _XK={\hat{\nabla }}_X^{(k)}K$ for a vector field X tangent to M due to (1.2) is equivalent to the fact that $KF_X^{(k)}=F_X^{(k)}K$, which implies that every eigenspace of K is preserved by the kth Cho operator $F_X^{(k)}$. Since, at each point $P \, \in \, M$ the tangent space is decomposed as $T_{P}M={\hbox {Span}} \{ \xi \} \oplus {\mathbb {D}}$ the above problem either for any $X \in {\mathbb {D}}$ or for $X=\xi$ will be studied.

In the case of ${\mathbb {C}}P^{n},n\ge 3,$ in [23] we consider the problem when $K=A$, obtaining ruled real hypersurfaces for any $X \in {\mathbb {D}}$ and type (A) real hypersurfaces when $X=\xi$. As a consequence, we proved non-existence of real hypersurfaces in ${\mathbb {C}}P^n,n\ge 3,$ such that $\nabla A={\hat{\nabla }}^{(k)}A$ for any nonnull k.

A similar study in the case of the structure Jacobi operator $R_{\xi }$ of M was made in [20], also obtaining ruled real hypersurfaces for any $X \in {\mathbb {D}}$ and either real hypersurfaces of type (A) or tubes of radius $\frac{\pi }{4}$ over a complex submanifold of ${\mathbb {C}}P^n,n\ge 3,$ if $X=\xi$. This second type of real hypersurfaces are those such that $A\xi =0$ (see [4]). As above, there do not exist real hypersurfaces M in ${\mathbb {C}}P^n,n\ge 3,$ such that $\nabla R_{\xi }={\hat{\nabla }}^{(k)}R_{\xi }$, for any nonnull k. In [19], the previous question was answered for real hypersurfaces in ${\mathbb {C}}H^{n},n\ge 2$, and for three-dimensional real hypersurfaces in $M_{2}(c)$. More precisely, in both cases in the case of $X \, \in \, {\mathbb {D}}$ the real hypersurface is a ruled one and in the case of $X=\xi$ the real hypersurfaces is locally congruent to a real hypersurface of type A or to a real hypersurface with $A\xi =0$ (for the construction of real hypersurfaces with $A\xi =0$ in ${\mathbb {C}}H^{2}$ see [7]).

In this paper, the analogous problem when $K=\phi A-A\phi$ will be considered. Relation $\phi A-A\phi =0$ implies that the shape operator commutes with structure tensor $\phi$. Real hypersurfaces satisfying the previous relation were studied by Okumura in the case of ${\mathbb {C}}P^{n}$, $n\ge 2$, in [18] and by Montiel and Romero in the case of ${\mathbb {C}}H^{n}$, $n\ge 2$ in [16]. The following Theorem provides the above classification of real hypersurfaces in $M_{n}(c)$, $n\ge 2$.

Theorem 1.1

Let M be a real hypersurface of $M_{n}(c)$, $n\ge 2$. Then $\phi A=A\phi$, if and only if M is locally congruent to a homogeneous real hypersurface of type (A). More precisely:

In the case of ${\mathbb {C}}P^{n}$

$(A_{1})$:

a geodesic hypersphere of radius r , where $0<r<\frac{\pi }{2}$,

$(A_{2})$:

a tube of radius r over a totally geodesic ${\mathbb {C}}P^{k}$,$(1\le k\le n-2)$, where $0<r<\frac{\pi }{2}$.

In the case of ${\mathbb {C}}H^{n}$

$(A_{0})$:

a horosphere in ${\mathbb {C}}H^{n}$,

$(A_{1})$:

a geodesic hypersphere or a tube over a totally geodesic complex hyperbolic hyperplane ${\mathbb {C}}H^{n - 1}$,

$(A_{2})$:

a tube over a totally geodesic ${\mathbb {C}}H^{k}\, (1\le k\le n-2)$.

The question if there are real hypersurfaces in $M_{n}(c),n\ge 2,$ whose eigenspaces of the symmetric tensor $\phi A-A\phi$ are preserved by the kth Cho operator will be answered. More precisely, the following Theorems are proved

Theorem 1.2

Let M be a real hypersurface in $M_{n}(c)$, $n\ge 2$. Let k be a nonnull constant. Then $\nabla _X(\phi A-A\phi )={\hat{\nabla }}_X^{(k)}(\phi A-A\phi )$ for any $X \in {\mathbb {D}}$ if and only if M is locally congruent either to a ruled real hypersurface or to a real hypersurface of type (A).

Theorem 1.3

Let M be a real hypersurface in $M_{n}(c)$, $n \ge 2$. Let k be a nonnull constant. Then $\nabla _{\xi }(\phi A-A\phi )={\hat{\nabla }}_{\xi }^{(k)}(\phi A-A\phi )$ if and only if M is locally congruent to a real hypersurface of type (A).

As a consequence of both Theorems, the following is obtained

Corollary 1.1

Let M be a real hypersurface in $M_{n}(c)$, $n \ge 2$, and a nonnull constant k. Then $\nabla (\phi A-A\phi )={\hat{\nabla }}^{(k)}(\phi A-A\phi )$ if and only if M is locally congruent to a real hypersurface of type (A).

Let ${\mathcal {L}}$ denote the Lie derivative on M. Therefore, $\mathcal{L}_XY=\nabla _XY-\nabla _YX$ for any X, Y tangent to M. Associated with the kth generalized Tanaka-Webster connection a differential operator of first order can be defined and will be called the derivative of Lie type associated with such a connection and is given by

$$\begin{aligned} {{\mathcal {L}}}^{(k)}_XY={\hat{\nabla }}^{(k)}_XY-{\hat{\nabla }}^{(k)}_YX=\mathcal{L}_XY+{\hat{T}}^{(k)}_XY, \end{aligned}$$

for any X, Y tangent to the real hypersurface M. Given X tangent to M, the k-th torsion operator associated with X, ${\hat{T}}_X^{(k)}$, is considered as the operator such that ${\hat{T}}_X^{(k)}Y={\hat{T}}^{(k)}(X,Y)$ for any Y tangent to M. Then, given a symmetric operator K on M, ${{\mathcal {L}}}_XK=\mathcal{L}_X^{(k)}K$ for a tangent vector field X is equivalent to $K{\hat{T}}_X^{(k)}={\hat{T}}_X^{(k)}K$, which implies that any eigenspace of K is preserved by ${\hat{T}}_X^{(k)}$.

In [21], this problem for $K=A$ has been studied in the case of ${\mathbb {C}}P^{n},n\ge 3$, and it has been proved that ruled real hypersurfaces for any $X \in {\mathbb {D}}$ satisfy the above condition and real hypersurfaces of type (A) when $X=\xi$. Thus, the non-existence of real hypersurfaces in ${\mathbb {C}}P^m$ such that $\mathcal{L}A = {{\mathcal {L}}}^{(k)}A$, for any nonnull k is assured. In [22] the same problem is solved when $K=R_{\xi }$, obtaining a non-existence result when $X \in {\mathbb {D}}$ and either real hypersurfaces of type (A) or tubes of radius $\frac{\pi }{4}$ over a complex submanifold in ${\mathbb {C}}P^m$ when $X=\xi$.

In this paper, the case $K=\phi A-A\phi$ is examined and the following Theorems are proved

Theorem 1.4

Let M be a real hypersurface in $M_{n}(c)$, $n\ge 2$, and k a nonnull constant. Then ${{\mathcal {L}}}_{\xi }(\phi A-A\phi )=\mathcal{L}_{\xi }^{(k)}(\phi A-A\phi )$ if and only if M is locally congruent to either

• 1. a real hypersurface of type (A), or

• 2. in the case of ${\mathbb {C}}P^{n}$, a real hypersurface of type (B), (C), (D) or (E) whose radius r, $0< r < \frac{\pi }{4}$, satisfies $\mathrm{tan}(2r)=-k$ and in the case of ${\mathbb {C}}H^{n}$, a real hypersurface of type (B), whose radius r satisfies $\mathrm{tanh}(2r)=\frac{1}{k}$.

Theorem 1.5

Let M be a real hypersurface in $M_{n}(c),n\ge 2$. Let k be a nonnull constant. Then ${{\mathcal {L}}}_X(\phi A-A\phi )=\mathcal{L}_X^{(k)}(\phi A-A\phi )$ for any $X \in {\mathbb {D}}$ if and only if M is locally congruent to a real hypersurface of type (A).

As a direct consequence of these Theorems we have

Corollary 2

Let M be a real hypersurface in $M_{n}(c),n\ge 2$, and a nonnull constant k. Then ${{\mathcal {L}}}(\phi A-A\phi )={{\mathcal {L}}}^{(k)}(\phi A-A\phi )$ if and only if M is locally congruent to a real hypersurface of type (A)

2 Preliminaries

Throughout this paper, all manifolds, vector fields, etc., will be considered of class $C^{\infty }$ unless otherwise stated. Let M be a connected real hypersurface in $M_{n}(c),n\ge 2$, without boundary. Let N be a locally defined unit normal vector field on M. Let $\nabla$ be the Levi-Civita connection on M and (J, G) the Kälerian structure of $M_{n}(c)$.

For any vector field X tangent to M we write $JX=\phi X+\eta (X)N$, and $-JN=\xi$. Then $(\phi ,\xi ,\eta ,g)$ is an almost contact metric structure on M (see [3]). That is, we have

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

(2.1)

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

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

(2.2)

From the parallelism of J, we get

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

(2.3)

for any X, Y tangent to M, where A denotes the shape operator of the immersion. As the ambient space has holomorphic sectional curvature c, the equations of Gauss and Codazzi are given, respectively, by

$$\begin{aligned} \begin{aligned} R(X,Y)Z&= \displaystyle \frac{c}{4}[ g(Y,Z)X - g(X,Z)Y + g(\phi Y,Z)\phi X - g(\phi X,Z)\phi Y \\&\quad - 2g(\phi X,Y)\phi Z] + g(AY,Z)AX - g(AX,Z)AY, \end{aligned} \end{aligned}$$

(2.4)

and

$$\begin{aligned} (\nabla _XA)Y-(\nabla _YA)X=\frac{c}{4}[\eta (X)\phi Y-\eta (Y)\phi X-2g(\phi X,Y)\xi ] \end{aligned}$$

(2.5)

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

In the sequel the following result is needed. In the case of complex projective space ${\mathbb {C}}P^{n}$ is owed to Maeda [14] and in the case of complex hyperbolic space ${\mathbb {C}}H^{n}$ is owed to Ki and Suh [9] (see also Corollary 2.3 in [17]).

Theorem 2.1

Let M be a Hopf hypersurface in $M_{n}(c)$, $n\ge 2$. Then

• (i) $\alpha =g(A\xi ,\xi )$ is constant.

• (ii) If W is a vector field which belongs to ${\mathbb {D}}$ such that $AW=\lambda W$, then

  $$\begin{aligned} \left( \lambda -\frac{\alpha }{2}\right) A\phi W=\left( \frac{\lambda \alpha }{2}+\frac{c}{4}\right) \phi W. \end{aligned}$$

• (iii) If the vector field W satisfies $AW=\lambda W$ and $A\phi W=\mu \phi W$ then

  $$\begin{aligned} \lambda \mu =\frac{\alpha }{2}(\lambda +\mu )+\frac{c}{4}. \end{aligned}$$

  (2.6)

3 Proof of Theorem 1.2

Let M a real hypersurface satisfying $(\nabla _X(\phi A-A\phi ))Y=({\hat{\nabla }}_X^{(k)}(\phi A-A\phi ))Y$ for any $X \in {\mathbb {D}}$, Y tangent to M, which is equivalent to relation $F_X(\phi A-A\phi )Y=(\phi A-A\phi )F_XY$. Therefore,

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

(3.1)

for any $X \in {\mathbb {D}}$, Y tangent to M. Suppose that M is a non-Hopf real hypersurface. Thus, locally we can write $A\xi =\alpha \xi +\beta U$, where U is a unit vector field in ${\mathbb {D}}$, $\alpha$ and $\beta$ are functions on M and $\beta \ne 0$. Moreover, ${{\mathbb {D}}}_U$ is defined to be the orthogonal complementary distribution in ${\mathbb {D}}$ to the one spanned by U and $\phi U$.

Relation (3.1) for $Y=\xi$ implies $g(\phi AX,\phi A\xi )\xi =-(\phi A-A\phi )\phi AX$, for any $X \in {\mathbb {D}}$, that is, $\beta g(AX,U)\xi =-\phi A\phi AX+A\phi ^2AX$, for any $X \in {\mathbb {D}}$. The scalar product of the latter relation with $\xi$ yields $\beta g(AX,U)=g(\phi ^2AX,A\xi )=\beta g(\phi ^2AX,U)=-\beta g(AX,U)$, for any $X \in {\mathbb {D}}$. Therefore, $g(AX,U)=0$, for any $X \in {\mathbb {D}}$,which implies

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

(3.2)

Take now $Y=U$ in (3.1). We obtain $-g(\phi AX,A\phi U)\xi =g(\phi AX,U)(\phi A-A\phi )\xi =\beta g(\phi AX,U)\phi U$, for any $X \in {\mathbb {D}}$. Its scalar product with $\phi U$ gives $g(\phi AX,U)=0$ for any $X \in {\mathbb {D}}$ and then

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

(3.3)

Take $Y=\phi U$ in (3.1). Then $\eta (AU)\phi AX=0$ for any $X \in {\mathbb {D}}$. As $\eta (AU)=\beta \ne 0$, this yields $\phi AX=0$ for any $X \in {{\mathbb {D}}}_U$. Therefore

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

(3.4)

for any $X \in {{\mathbb {D}}}_U$. From (3.2), (3.3) and (3.4) we conclude that M is ruled. Conversely, if M is ruled $F_X=0$ for any $X \in {\mathbb {D}}$ and (3.1) is satisfied.

Next the case of M being a Hopf hypersurface is examined. In this case, $A\xi =\alpha \xi$. Let $W \, \in \, {\mathbb {D}}$ be a unit principal vector field, i.e. $AW=\lambda W$, where $\lambda$ is the corresponding principal curvature. There are two cases:

CASE I$\alpha ^{2}+c\ne 0$.

In this case, $\lambda \ne \frac{\alpha }{2}$ and Theorem 2.1 holds. So, $\phi W$ is also a principal vector of the shape operator with corresponding eigenvalue $\mu$. Relation (3.1) for $Y=\xi$ implies $(\phi A-A\phi )\phi AX=0$ for any $X \in {\mathbb {D}}$. The last relation for $X=W$ and $X=\phi W$ yields

$$\begin{aligned} \lambda (\lambda -\mu )=0\;\;\text{ and }\;\;\mu (\lambda -\mu )=0. \end{aligned}$$

Combination of the last two relations results in $\lambda =\mu$, which implies $\phi A=A\phi$. Thus, because of Theorem 1.1, M is locally congruent to a real hypersurface of type (A).

CASE II$\alpha ^{2}+c=0$.

In this case due to the above relation, $c=-4$ and $\alpha \ne 0$. First the case, $\lambda \ne \frac{\alpha }{2}$ is examined. In this case Theorem 2.1 holds and let $A\phi W=\mu \phi W$. Then relation (2.6) due to $\alpha ^{2}+c=0$ implies $\mu =\frac{\alpha }{2}$. Relation (3.1) for $X=\phi W$ and $Y=\xi$ implies $\alpha =0$, which is impossible.

The remaining case is relation $AW=\frac{\alpha }{2}W$ holds, for all $W \, \in \, {\mathbb {D}}$. In this case $\phi A=A\phi$ and M is locally congruent to a horosphere. Relation (3.1) is satisfied for any X$\in \, {\mathbb {D}}$ and any $Y \, \in \, TM$ and this completes the proof of Theorem 1.2.

4 Proof of Theorem 1.3

Let M be a real hypersurface in $M_{n}(c),n\ge 2$ satisfying relation $F_{\xi }^{(k)}(\phi A-A\phi )Y=(\phi A-A\phi )F_{\xi }^{(k)}Y$ for any Y tangent to M. This yields

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

(4.1)

for any $Y \in TM$.

Let M be a non-Hopf real hypersurface satisfying relation (4.1). Then, relation $A\xi =\alpha \xi +\beta U$ for a unit $U \in {\mathbb {D}}$ holds and $\alpha$, $\beta$ are functions on M, with $\beta \ne 0$. So, (4.1) becomes

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

(4.2)

for any $Y \in TM$. Taking $Y=\xi$ in (4.2) implies $\beta g(\phi U,\phi A\xi )\xi -k\phi ^2A\xi =-\beta (\phi A-A\phi )\phi U$ and its scalar product with $\xi$ results in $\beta =0$, which is impossible.

Thus, M is a Hopf hypersurface and $A\xi =\alpha \xi$. Relation (4.1) becomes, bearing in mind that $k \ne 0$

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

(4.3)

for any Y tangent to M.

Take a unit vector field $W \in {\mathbb {D}}$ such that $AW=\lambda W$. There are two cases:

CASE I$\alpha ^{2}+c\ne 0$.

In this case, $\lambda \ne \frac{\alpha }{2}$ and Theorem 2.1 holds. So, $\phi W$ is also a principal vector of the shape operator with corresponding eigenvalue $\mu$. Relation (4.3) for $Y=W$, implies $\lambda =\mu$. The latter results in $\phi A=A\phi$ and because of Theorem 1.1, M is locally congruent to a real hypersurface of type (A).

CASE II$\alpha ^{2}+c=0$.

First suppose that $\lambda \ne \frac{\alpha }{2}$. So, Theorem 2.1 holds. Suppose that $A\phi W=\mu \phi W$, then relation (2.6) because of $\alpha ^{2}+c=0$ implies $\mu =\frac{\alpha }{2}$. Relation (4.3) for $Y=W$ implies $\lambda =\frac{\alpha }{2}$, which is a contradiction.

So, relation $AW=\frac{\alpha }{2}W$ holds, for all $W \, \in \, {\mathbb {D}}$. In this case, $\phi A=A\phi$ and M is locally congruent to a horosphere. Relation (4.1) is satisfied for any $Y \, \in \, TM$ and this completes the proof of Theorem 1.3.

5 Proof of Theorem 1.4

Let M be a real hypersurface in $M_{n}(c)$, $n\ge 2$, whose symmetric operator $\phi A-A\phi$ satisfies relation $(\mathcal{L}_{\xi }^{(k)}(\phi A-A\phi ))X=({{\mathcal {L}}}_{\xi }(\phi A-A\phi ))X$ for any vector field X tangent to M. This yields

$$\begin{aligned} F_{\xi }^{(k)}(\phi A-A\phi )X-(\phi A-A\phi )F_{\xi }^{(k)}X=\phi A^2\phi X-A\phi ^2AX \end{aligned}$$

(5.1)

for any vector field X tangent to M.

Suppose that M is a non-Hopf real hypersurface and, as in the previous section we consider $A\xi =\alpha \xi +\beta U$, with $\beta \ne 0$. Taking $X=\xi$ in (5.1) implies $\beta F_{\xi }^{(k)}\phi U-(\phi A-A\phi )(-\phi A\xi )=-A\phi ^2A\xi =-A\phi (\beta \phi U)=\beta AU$. Since, $\beta \ne 0$, the last one gives $g(\phi A\xi ,\phi U)\xi -k\phi ^2U+\phi A\phi U-A\phi ^2U=AU$, that is,

$$\begin{aligned} \beta \xi +kU+\phi A\phi U=0. \end{aligned}$$

(5.2)

The scalar product of (5.2) with $\xi$ yields $\beta =0$, which is impossible.

Therefore, M must be a Hopf hypersurface and $A\xi =\alpha \xi$. Let $W \, \in \, {\mathbb {D}}$ be a unit principal vector field such that $AW=\lambda W$. There are two cases:

CASE I$\alpha ^{2}+c\ne 0$. In this case $\lambda \ne \frac{\alpha }{2}$ and Theorem 2.1 holds. So, $\phi W$ is also a principal vector of the shape operator with $A\phi W=\mu \phi W$. Relation (5.1) for $X=W$ implies $(\lambda -\mu )F_{\xi }^{(k)}\phi X+k(\phi A-A\phi )\phi X=-\mu ^2X+\lambda ^2X$ and this results in

$$\begin{aligned} 2k(\lambda -\mu )=(\lambda +\mu )(\lambda -\mu ). \end{aligned}$$

Thus we have two possibilities

• 1. $\lambda = \mu$.

• 2. $\lambda \ne \mu$. In this case $\lambda +\mu =2k$.

If for any eigenvalue $\lambda$, we have $\lambda =\mu$, $\phi A=A\phi$ and because of Theorem 1.1M is locally congruent to a real hypersurface of type (A).

In ${\mathbb {C}}P^n$, if for any eigenvalue $\lambda$, $\lambda \ne \mu$, $\lambda +\mu =2k$ yields $\lambda +\frac{\alpha \lambda +2}{2\lambda -\alpha } =2k$. This implies $\lambda ^{2}-2k\lambda +1+k\alpha =0$. Thus $\lambda =k \pm \sqrt{k^2-k\alpha -1}$. This means that M has, at most, three distinct constant principal curvatures.

As $\lambda \ne \mu$, M must be locally congruent to a real hypersurface of type (B). In such a case we may suppose $\alpha =2{\hbox {cot}}(2r)$, $\lambda ={\hbox {cot}}(r-\frac{\pi }{4})=k+\sqrt{k^2-k\alpha -1}$ and $\mu =-{\hbox {tan}}(r-\frac{\pi }{4})=k-\sqrt{k^2-k\alpha -1}$. Then $-1=(k+\sqrt{k^2-k\alpha -1})(k-\sqrt{k^2-k\alpha -1})=k^2-(k^2-k\alpha -1)=k\alpha +1$. That is, $\alpha =-\frac{2}{k}$, that means ${\hbox {cot}}(2r)=-\frac{1}{k}$.

Suppose now there exist $X \in {\mathbb {D}}$ such that $AX=\lambda X$, $A\phi X=\lambda \phi X$ and there exists $Y \in {\mathbb {D}}$ such that $AY=\omega Y$, $A\phi Y=\epsilon \phi Y$, $\epsilon \ne \omega$ which yields $\epsilon +\omega =2k$. Bearing in mind the cases we have discussed above, M has five distinct constant principal curvatures. Therefore it is locally congruent to a real hypersurface of type either (C) or (D) or (E). From the introduction in any of these cases $\alpha =2{\hbox {cot}}(2r)$, $\omega ={\hbox {cot}}(r-\frac{\pi }{4})$, $\epsilon ={\hbox {cot}}(r-\frac{3\pi }{4})$, $\lambda =-{\hbox {tan}}(r)$, $\mu ={\hbox {cot}}(r)$.

As ${\hbox {cot}}(r-\frac{\pi }{4})+{\hbox {cot}}(r-\frac{3\pi }{4})=2k$, we have $k=\frac{2{\hbox {sin}}(r){\hbox {cos}}(r)}{{\hbox {sin}}^2(r)-{\hbox {cos}}^2(r)}=-{\hbox {tan}}(2r)$.

In ${\mathbb {C}}H^{n}$ as $\lambda \ne \mu$ and $\lambda +\mu =2k$, as above $\lambda$ and $\mu$ must be constant and M must be locally congruent to a real hypersurface of type (B). In this case $\alpha =2{\hbox {tanh}}(2r)$, $\lambda ={\hbox {tanh}}(r)$ and $\mu ={\hbox {coth}}(r)$. Substitution in $\lambda +\mu =2k$ yields ${\hbox {tanh}}(2r)=\frac{1}{k}$.

CASE II$\alpha ^{2}+c=0$.

First suppose that $\lambda \ne \frac{\alpha }{2}$. So, Theorem 2.1 holds. Suppose that $A\phi W=\mu \phi W$, then relation (2.6) because of $\alpha ^{2}+c=0$ implies $\mu =\frac{\alpha }{2}$. Relation (5.1) for $X=W$ implies that $\lambda =2k-\frac{\alpha }{2}$. Therefore, M is a Hopf real hypersurface with three constant principal curvatures and is locally congruent to a real hypersurface of type (B). Substitution of the eigenvalues in relation $\mu =\frac{\alpha }{2}$ leads to a contradiction.

Therefore, relation $AW=\frac{\alpha }{2}W$ holds, for all $W \, \in \, {\mathbb {D}}$. In this case $\phi A=A\phi$ and M is locally congruent to a horosphere. Relation (5.1) is satisfied for any $Y \, \in \, TM$ and this completes the proof of Theorem 1.4.

6 Proof of Theorem 1.5

Suppose M satisfies $({{\mathcal {L}}}_X^{(k)}(\phi A-A\phi ))Y=(\mathcal{L}_X(\phi A-A\phi ))Y$ for any $X \in {\mathbb {D}}$ and any Y tangent to M. If $n=2$, the proof is obtained in [8]. Therefore we suppose $n \ge 3$. Then we have

$$\begin{aligned} F_X(\phi A-A\phi )Y-F_{(\phi A-A\phi )Y}^{(k)}X -(\phi A-A\phi )F_XY+(\phi A-A\phi )F_Y^{(k)}X=0,\nonumber \\ \end{aligned}$$

(6.1)

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

Let M be a non-Hopf real hypersurface and write $A\xi$ as in previous Theorems. Taking $Y=\xi$ in (6.1) we get $\beta F_X\phi U-\beta F_{\phi U}X-(\phi A-A\phi )F_X\xi +(\phi A-A\phi )F_{\xi }^{(k)}X=0$. This implies

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

(6.2)

for any $X \in {\mathbb {D}}$. The scalar product of (6.2) with $\xi$ gives $\beta (g(AU,X)+g(A\phi U,\phi X))+\beta g(AX,U)-k\beta g(U,X)=0$. As $\beta \ne 0$ we obtain

$$\begin{aligned} 2g(AU,X)+g(A\phi U,\phi X)-kg(U,X)=0 \end{aligned}$$

(6.3)

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

If in (6.3) we take $X=\phi U$ we get $2g(AU,\phi U)-g(A\phi U,U)=0$. Therefore $g(AU,\phi U)=0$ and we can write $AU=\beta \xi +\gamma U+\omega Z$, $A\phi U=\mu \phi U+\epsilon W$, for certain functions $\gamma$, $\omega$, $\mu$ and $\epsilon$, where Z and W are unit vector fields in ${{\mathbb {D}}}_U$. Taking $X=Z$ in (6.3) we get $2g(AU,Z)+g(A\phi U,Z)=0$. That is

$$\begin{aligned} 2\omega +\epsilon g(W,\phi Z)=0 \end{aligned}$$

(6.4)

and taking $X=\phi W$ in (6.3) we have $2g(AU,\phi W)-g(A\phi U,W)=0$, that becomes

$$\begin{aligned} 2\omega g(Z,\phi W)- \epsilon =0. \end{aligned}$$

(6.5)

If $\epsilon =0$, (6.4) yields $\omega =0$. If $\epsilon \ne 0$, from (6.4) $g(Z,\phi W)=\frac{2\omega }{\epsilon }$ and from (6.5) $4\omega ^2-\epsilon ^2=0$. That is,

$$\begin{aligned} \epsilon ^2 =4\omega ^2. \end{aligned}$$

(6.6)

Taking $Y=U$ in (6.1) we obtain $F_X(\phi A-A\phi )U-F_{(\phi A-A\phi )U}^{(k)}X-(\phi A-A\phi )F_XU+(\phi A-A\phi )F_UX=0$ for any $X \in {\mathbb {D}}$. As $\phi AU-A\phi U=(\gamma -\mu )\phi U+\omega \phi Z-\epsilon W \in {\mathbb {D}}$, $F_X(\phi A-A\phi )U-F_{(\phi A-A\phi )U}X$ is proportional to $\xi$. Therefore $-g(F_XU,(\phi A-A\phi )\phi U)+g(F_UX,(\phi A-A\phi )\phi U)=0$. Now $(\phi A-A\phi )\phi U=\beta \xi +(\gamma -\mu )U+\epsilon \phi W+\omega Z$. Therefore $-g(\phi AX,U)g(\xi ,\beta \xi +(\gamma -\mu )U+\epsilon \phi W+\omega Z)+g(\phi AU,X)g(\xi ,\beta \xi +(\gamma -\mu )U+\epsilon \phi W+\omega Z)=0$. As $\beta \ne 0$, this yields

$$\begin{aligned} -g(\phi AX,U)+g(\phi AU,X)=0 \end{aligned}$$

(6.7)

for any $X \in {\mathbb {D}}$. Taking $X=\phi Z$ in (6.7) we have $g(A\phi U,\phi Z)+g(AU,Z)=0$. This implies

$$\begin{aligned} \omega +\epsilon g(W,\phi Z)=0. \end{aligned}$$

(6.8)

From (6.4) and (6.8), we conclude $\omega =0$ and from (6.6), $\epsilon =0$. Thus we have proved

$$\begin{aligned} AU=\beta \xi +\gamma U\;\;\text{ and }\;\;A\phi U=\mu \phi U. \end{aligned}$$

Moreover, from (6.3)

$$\begin{aligned} 2\gamma +\mu =k. \end{aligned}$$

(6.9)

Taking $Y=\phi U$ in (6.1), as $(\phi A-A\phi )\phi U=\beta \xi +(\gamma -\mu )U$ we get $\beta F_X\xi +(\gamma - \mu )F_XU-\beta F_{\xi }^{(k)}X-(\gamma -\mu )F_UX-(\phi A-A\phi )(g(AX,U)\xi +g(A\phi X,\phi U)\xi )=0$, for any $X \in {\mathbb {D}}$. Developing it we have

$$\begin{aligned}&-\beta \phi AX-\mu (\gamma -\mu )g(\phi U,X)\xi -\beta ^2g(\phi U,X)\xi +\beta k\phi X \nonumber \\&\quad -(\gamma -\mu )g(\phi AU,X)\xi -\beta g(AU,X)\phi U-\beta g(A\phi U,\phi X)\phi U =0\nonumber \\ \end{aligned}$$

(6.10)

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

If $X \in {{\mathbb {D}}}_U$, from (6.10) we get $-\beta \phi AX+\beta k\phi X=0$. As $\beta \ne 0$, if we apply $\phi$ we obtain

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

(6.11)

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

The scalar product of (6.10) with U yields $\beta g(A\phi U,X)+\beta kg(\phi X,U)=0$ for any $X \in {\mathbb {D}}$. If $X=\phi U$, bearing in mind that $\beta \ne 0$ we get

$$\begin{aligned} \mu =k \end{aligned}$$

(6.12)

and from (6.9) it follows

$$\begin{aligned} \gamma =0. \end{aligned}$$

(6.13)

The scalar product of (6.10) with $\xi$ gives $-\mu (\gamma -\mu )g(\phi U,X)-\beta ^2g(\phi U,X)-(\gamma -\mu )g(\phi AU,X)=0$ for any $X \in {\mathbb {D}}$. Taking $X=\phi U$ we have $\mu ^2 -\beta ^2=0$. From (6.12)

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

(6.14)

and therefore $\beta$ is constant.

By Codazzi equation, for any $X \in {{\mathbb {D}}}_U$ we get $(\nabla _XA)\phi X-(\nabla _{\phi X}A)X=-\frac{c}{2}\xi$. This yields

$$\begin{aligned} k\nabla _X\phi X-A\nabla _X\phi X-k\nabla _{\phi X}X+A\nabla _{\phi X}X=-\frac{c}{2}\xi . \end{aligned}$$

(6.15)

The scalar product of (6.15) with $\xi$ and with U implies respectively

$$\begin{aligned} \beta g([ \phi X,X ] ,U)=-\frac{c}{2}-2\alpha k+2k^2\;\;\text{ and }\;\;g([ \phi X,X ] ,U)=2\beta . \end{aligned}$$

(6.16)

From (6.16), bearing in mind (6.14), we get

$$\begin{aligned} \alpha =-\frac{c}{4k} . \end{aligned}$$

(6.17)

Now $(\nabla _UA)\xi -(\nabla _{\xi }A)U=-\frac{c}{4}\phi U$. If we develop it and bear in mind that $\phi AU=0$ we have $\beta \nabla _UU-\beta \phi A\xi +A\nabla _{\xi }U=-\frac{c}{4}\phi U$. Its scalar product with $\phi U$ yields

$$\begin{aligned} \beta g(\nabla _UU,\phi U)+kg(\nabla _{\xi }U,\phi U)=\beta ^2-\frac{c}{4}. \end{aligned}$$

(6.18)

Developing $(\nabla _{\phi U}A)U-(\nabla _UA)\phi U=\frac{c}{2}\xi$ we obtain

$$\begin{aligned} -\beta kU-A\nabla _{\phi U}U-k\nabla _U\phi U+A\nabla _U\phi U=\frac{c}{2}\xi . \end{aligned}$$

(6.19)

Its scalar product with $\xi$ yields $\alpha g(U,\phi A\phi U)+\beta g(\nabla _U\phi U,U)=\frac{c}{2}$. From (6.17), we have

$$\begin{aligned} \beta g(\nabla _UU, \phi U)=-\frac{c}{4}. \end{aligned}$$

(6.20)

The scalar product of (6.19) and U, bearing in mind (6.20), gives

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

(6.21)

From (6.20) and (6.21) $2\beta ^2=-\frac{c}{4}$, which is impossible if the ambient space is ${\mathbb {C}}P^{n}$. In the case of ${\mathbb {C}}H^{n}$ we have $c=-4$. Therefore $2\beta ^2=1$ and from (6.17) $k\alpha =1$. From (6.18) and (6.21) we also have $kg(\nabla _{\xi }U,\phi U)=1-\beta ^2$.

The scalar product of the Codazzi equation for $X=Z \in {{\mathbb {D}}}_U$ and $Y=\xi$ with $\xi$ yields $g(\nabla _{\xi }U,Z)=0$ for any $Z \in {{\mathbb {D}}}_U$. Then

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

(6.22)

Scalar product of the Codazzi equation for $X=Z \in {{\mathbb {D}}}_U$ and $Y=U$ with U implies $g(\nabla _UU,Z)=0$ for any $Z \in {{\mathbb {D}}}_U$. Thus

$$\begin{aligned} \nabla _UU=2\beta \phi U. \end{aligned}$$

(6.23)

Taking $X=\phi U$ and $Y=\xi$ in Codazzi equation and its scalar product with $Z \in {{\mathbb {D}}}_U$ gives $g(\nabla _{\phi U}U,Z)=0$, for any $Z \in {{\mathbb {D}}}_U$. A similar calculation taking the scalar product with $\phi U$ yields $g(\nabla _{\phi U}U,\phi U)=0$. Therefore

$$\begin{aligned} \nabla _{\phi U}U=k\xi . \end{aligned}$$

(6.24)

From Gauss equation, bearing in mind $c=-4$ we have $R(U,\phi U)U=4\phi U$. On the other hand $R(U,\phi U)U=\nabla _U\nabla _{\phi U}\phi U-\nabla _{\phi U}\nabla _UU-\nabla _{\nabla _U\phi U-\nabla _{\phi U}U}U=k\phi AU-2\beta \nabla _{\phi U}\phi U-\nabla _{\nabla _U\phi U}U+k\nabla _{\xi }U$. Its scalar product with $\phi U$ implies $4=-g(\nabla _{\nabla _U\phi U}U,\phi U)+kg(\nabla _{\xi }U,\phi U)$.

From (6.19) $g(\nabla _U\phi U,Z)=0$ for any $Z \in {{\mathbb {D}}}_U$. Then we get $\nabla _U\phi U=-2\beta U$. As $kg(\nabla _{\xi }U,\phi U)=1-\beta ^2$, we finally obtain $4=3\beta ^2+1$. Thus $\beta ^2=1=\frac{1}{2}$, which is impossible.

Therefore, M must be a Hopf hypersurface and $A\xi =\alpha \xi$. Let $W\, \in \, {\mathbb {D}}$ be unit principal vector field such that $AW=\lambda W$. There are two cases:

CASE I$\alpha ^{2}+c\ne 0$.

In this case $\lambda \ne \frac{\alpha }{2}$ and Theorem 2.1 holds. So, $\phi W$ is also a principal vector of the shape operator with $A\phi W=\mu \phi W$. Relation (6.1) for $Y=\xi$ implies

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

(6.25)

for any $X \in {\mathbb {D}}$. Suppose $X=W$ then

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

If $\lambda \ne \mu$ then $\lambda =k$. Relation (6.25) for $X=\phi W$ implies $\mu =k$. So $\lambda =\mu$, which is a contradiction. So, on M relation $\lambda =\mu$ holds and, M is locally congruent to a real hypersurface of type (A).

CASE II$\alpha ^{2}+c=0$. First suppose that $\lambda \ne \frac{\alpha }{2}$. So, Theorem 2.1 holds. Suppose that $A\phi W=\mu \phi W$, then relation (2.6) because of $\alpha ^{2}+c=0$ implies $\mu =\frac{\alpha }{2}$. Relation (6.1) for $X=W$ and $Y=\xi$ implies $\lambda =k$ and for $X=\phi W$ and $Y=\xi$ implies $\mu =k$, which is impossible.

Therefore, relation $AW=\frac{\alpha }{2}W$ holds, for all $W \, \in \, {\mathbb {D}}$. In this case $\phi A=A\phi$ and M is locally congruent to a horosphere. Relation (6.1) is satisfied for any $X \, \in \, {\mathbb {D}}$ and any $Y \, \in \, TM$ and this completes the proof of Theorem 1.5.