# Lie derivatives and structure Jacobi operator on real hypersurfaces in
complex projective spaces II

Juan de Dios Pérez and David Pérez-López

###### Abstract

Let $\displaystyle M$ be a real hypersurface in complex projective space. The
almost contact metric structure on $\displaystyle M$ allows us to consider,
for any nonnull real number $\displaystyle k$, the corresponding
$\displaystyle k$-th generalized Tanaka-Webster connection on $\displaystyle
M$ and, associated to it, a differential operator of first order of Lie type.
Considering such a differential operator and Lie derivative we define, from
the structure Jacobi operator $\displaystyle R_{\xi}$ on $\displaystyle M$ a
tensor field of type (1,2), $\displaystyle R_{{\xi}_{T}}^{(k)}$. We obtain
some classifications of real hypersurfaces for which $\displaystyle
R_{{\xi}_{T}}^{(k)}$ is either symmetric or skew symmetric.

2000 Mathematics Subject Classification: 53C15, 53B25.

Keywords and phrases: $\displaystyle k$-th generalized Tanaka-Webster
connection; complex projective space; real hypersurface; Lie derivative;
structure Jacobi operator.

##  1 Introduction.

Consider a complex projective space $\displaystyle\mbox{{\bb C}}P^{m}$,
$\displaystyle m\geq 2$, endowed with the complex structure $\displaystyle J$
and the Fubini-Study metric $\displaystyle g$ of constant holomorphic
sectional curvature $\displaystyle 4$. Let $\displaystyle M$ be a connected
real hypersurface of $\displaystyle\mbox{{\bb C}}P^{m}$ without boundary,
$\displaystyle\nabla$ its Levi-Civita connection on $\displaystyle M$ and
$\displaystyle N$ a local unit normal vector field on $\displaystyle M$. Then
$\displaystyle\xi=-JN$ is a tangent vector field to $\displaystyle M$ called
the structure vector field on $\displaystyle M$ (or the Reeb vector field).
The Kaehlerian structure $\displaystyle(J,g)$ of $\displaystyle\mbox{{\bb
C}}P^{m}$ induces on $\displaystyle M$ an almost contact metric structure
$\displaystyle(\phi,\xi,\eta,g)$, see [1], where $\displaystyle\phi$ is the
tangent component of $\displaystyle J$, $\displaystyle\eta$ is an 1-form given
by $\displaystyle\eta(X)=g(X,\xi)$ for any $\displaystyle X$ tangent to
$\displaystyle M$ and $\displaystyle g$ is the metric induced on
$\displaystyle M$.

Takagi, see [11], [12], [13], classified homogeneous real hypersurfaces in
$\displaystyle\mbox{{\bb C}}P^{m}$ into 6 types. Kimura, [6], proved that
these 6 types are the unique Hopf real hypersurfaces in
$\displaystyle\mbox{{\bb C}}P^{m}$ with constant principal curvatures. A real
hypersurface $\displaystyle M$ is called Hopf if the structure vector field is
principal, that is, $\displaystyle A\xi=\alpha\xi$ for a certain function
$\displaystyle\alpha$ on $\displaystyle M$, where $\displaystyle A$ denotes
the shape operator of $\displaystyle M$ associated to $\displaystyle N$. We
will also denote by 𝔻 the maximal holomorphic distribution on $\displaystyle
M$, given by all vector fields orthogonal to $\displaystyle\xi$.

Among the real hypersurfaces appearing in Takagi’s list we mention the
following ones:

Type $\displaystyle(A_{1})$ real hypersurfaces are geodesic hyperspheres of
radius $\displaystyle r$, $\displaystyle 0<r<\frac{\pi}{2}$. They have 2
distinct constant principal curvatures, $\displaystyle 2\cot 2r$ with
eigenspace $\displaystyle\mbox{{\bb R}}[\xi]$ and $\displaystyle\cot r$ with
eigenspace 𝔻.

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

From now on we will call type $\displaystyle(A)$ real hypersurfaces to both of
either type $\displaystyle(A_{1})$ or type $\displaystyle(A_{2})$. The other
real hypersurfaces appearing in Takagi’s list have either 3 or 5 distinct
constant principal curvatures.

Tanaka, [14], and, independtly Webster, [16], studied the canonical affine
connection defined on a non-degenerate, pseudo-Hermitian CR-manifold, known as
the Tanaka-Webster connection. Tanno, [15], generalized this connection for
contact metric manifolds.

Cho, using the almost contact metric structure on a real hypersurface
$\displaystyle M$ of $\displaystyle\mbox{{\bb C}}P^{m}$, generalized Tanno’s
connection, defining, for any non-zero real number $\displaystyle k$, the
$\displaystyle k$-th generalized Tanaka-Webster connection
$\displaystyle\hat{\nabla}^{(k)}$, see [2], [4], by

$\begin{array}[]{c}\hat{\nabla}^{(k)}_{X}Y=\nabla_{X}Y+g(\phi
AX,Y)\xi-\eta(Y)\phi AX-k\eta(X)\phi Y\end{array}$ (1.1)

for any X,Y tangent to $\displaystyle M$. Then
$\displaystyle\hat{\nabla}^{(k)}\eta=0$,
$\displaystyle\hat{\nabla}^{(k)}\xi=0$, $\displaystyle\hat{\nabla}^{(k)}g=0$,
$\displaystyle\hat{\nabla}^{(k)}\phi=0$. In particular, if the shape operator
of a real hypersurface satisfies $\displaystyle\phi A+A\phi=2k\phi$,
$\displaystyle M$ is a contact manifold and the $\displaystyle k$-th
generalized Tanaka-Webster connection coincides with the Tanaka-Webster
connection. From (1.1) the $\displaystyle k$-th Cho operator on $\displaystyle
M$ associated to a tangent vector field $\displaystyle X$ is defined by
$\displaystyle F_{X}^{(k)}Y=g(\phi AX,Y)\xi-\eta(Y)\phi AX-k\eta(X)\phi Y$,
for any $\displaystyle Y$ tangent to $\displaystyle M$. The torsion of the
$\displaystyle k$-th generalized Tanaka-Webster connection is then given by
$\displaystyle T^{(k)}(X,Y)=F^{(k)}_{X}Y-F^{(k)}_{Y}X$. For any $\displaystyle
X$ tangent to $\displaystyle M$ we call $\displaystyle T_{X}^{(k)}$ to the
operator on $\displaystyle M$ given by $\displaystyle
T_{X}^{(k)}Y=T^{(k)}(X,Y)$, for any $\displaystyle Y$ tangent to
$\displaystyle M$ and call it the torsion operator associated to
$\displaystyle X$.

Jacobi fields along geodesics of a given Riemannian manifold
$\displaystyle(\tilde{M},\tilde{g})$ satisfy a very well-known differential
equation. This classical differential equation naturally inspires the so-
called Jacobi operator. That is, if $\displaystyle\tilde{R}$ is the curvature
operator of $\displaystyle\tilde{M}$, and $\displaystyle X$ is any tangent
vector field to $\displaystyle\tilde{M}$, the Jacobi operator (with respect to
$\displaystyle X$) at $\displaystyle p\in M$,
$\displaystyle\tilde{R}_{X}\in$End$\displaystyle(T_{p}\tilde{M})$, is defined
as $\displaystyle(\tilde{R}_{X}Y)(p)=(\tilde{R}(Y,X)X)(p)$ for all
$\displaystyle Y\in T_{p}\tilde{M}$, being a selfadjoint endomorphism of the
tangent bundle $\displaystyle T\tilde{M}$ of $\displaystyle\tilde{M}$.
Clearly, each tangent vector field $\displaystyle X$ to
$\displaystyle\tilde{M}$ provides a Jacobi operator with respect to
$\displaystyle X$.

If $\displaystyle M$ is a real hypersurface of $\displaystyle\mbox{{\bb
C}}P^{m}$ the Jacobi operator associated to the structure vector field
$\displaystyle\xi$ is called the structure Jacobi operator on $\displaystyle
M$ and we denote it by $\displaystyle R_{\xi}$.

Let $\displaystyle\cal L$ denote the Lie derivative on $\displaystyle M$.
Therefore $\displaystyle{\cal L}_{X}Y=\nabla_{X}Y-\nabla_{Y}X$ for any
$\displaystyle X,Y$ tangent to $\displaystyle M$. If we consider
$\displaystyle\hat{\nabla}^{(k)}$, we can define a differential operator of
first order $\displaystyle{\cal L}^{(k)}$ on $\displaystyle M$ by
$\displaystyle{\cal
L}_{X}^{(k)}Y=\hat{\nabla}_{X}^{(k)}Y-\hat{\nabla}_{Y}^{(k)}X={\cal
L}_{X}Y+T_{X}^{(k)}Y$, for any $\displaystyle X,Y$ tangent to $\displaystyle
M$.

We can associate to $\displaystyle R_{\xi}$ a tensor field of type (1,2),
$\displaystyle R_{{\xi}_{T}}^{(k)}$, given by $\displaystyle
R_{{\xi}_{T}}^{(k)}(X,Y)=[T_{X}^{(k)},R_{\xi}]Y=T_{X}^{(k)}R_{\xi}Y-R_{\xi}T_{X}^{(k)}Y$,
for any $\displaystyle X,Y$ tangent to $\displaystyle M$.

In the first paper of this series, [9], first author studied the condition
$\displaystyle{\cal L}^{(k)}R_{\xi}={\cal L}R_{\xi}$, that is,
$\displaystyle({\cal L}_{X}^{(k)}R_{\xi})Y=({\cal L}_{X}R_{\xi})Y$ for any
$\displaystyle X,Y$ tangent to $\displaystyle M$. This condition is equivalent
to the fact that $\displaystyle R_{{\xi}_{T}}^{(k)}$ vanishes, obtaining the
following

Theorem There do not exist real hypersurfaces in $\displaystyle\mbox{{\bb
C}}P^{m}$, $\displaystyle m\geq 3$, such that $\displaystyle{\cal
L}^{(k)}R_{\xi}={\cal L}R_{\xi}$.

The purpose of this paper is to study weaker conditions on $\displaystyle M$.
Indeed, we will consider real hypersurfaces $\displaystyle M$ in
$\displaystyle\mbox{{\bb C}}P^{m}$ whose structure Jacobi operator satisfies
the following Codazzi condition

$\begin{array}[]{c}(({\cal L}_{X}^{(k)}-{\cal L}_{X})R_{\xi})Y=(({\cal
L}_{Y}^{(k)}-{\cal L}_{Y})R_{\xi})X\end{array}$ (1.2)

for any $\displaystyle X,Y$ tangent to $\displaystyle M$. It is easy to see
that this condition is equivalent to $\displaystyle R_{{\xi}_{T}}^{(k)}$ being
symmetric, that is, $\displaystyle
R_{{\xi}_{T}}^{(k)}(X,Y)=R_{{\xi}_{T}}^{(k)}(Y,X)$ for any $\displaystyle X,Y$
tangent to $\displaystyle M$. We will prove the following theorems.

Theorem 1 There does not exist any Hopf real hypersurface $\displaystyle M$ in
$\displaystyle\mbox{{\bb C}}P^{m}$, $\displaystyle m\geq 3$, such that
$\displaystyle R_{\xi}$ satisfies (1.2) for some nonnull real number
$\displaystyle k$.

and

Theorem 2 There does not exist any non-Hopf real hypersurface $\displaystyle
M$ in $\displaystyle\mbox{{\bb C}}P^{m}$, $\displaystyle m\geq 3$, such that
$\displaystyle R_{\xi}$ satisfies (1.2) for any nonnull real number
$\displaystyle k$ such that $\displaystyle kg(A\xi,\xi)\neq-1$.

A different weaker condition is a Killing type condition given by

$\begin{array}[]{c}(({\cal L}_{X}^{(k)}-{\cal L}_{X})R_{\xi})Y+(({\cal
L}_{Y}^{(k)}-{\cal L}_{Y})R_{\xi})X=0\end{array}$ (1.3)

for any $\displaystyle X,Y$ tangent to $\displaystyle M$. Now this condition
is equivalent to the fact that $\displaystyle R_{{\xi}_{T}}^{(k)}$ is skew
symmetric, that is, $\displaystyle
R_{{\xi}_{T}}^{(k)}(X,Y)=-R_{{\xi}_{T}}^{(k)}(Y,X)$ for any $\displaystyle
X,Y$ tangent to $\displaystyle M$. In this case we have a quite different
situation as given by the following results.

Theorem 3 Let $\displaystyle M$ be a Hopf real hypersurface in
$\displaystyle\mbox{{\bb C}}P^{m}$, $\displaystyle m\geq 3$, and
$\displaystyle k$ a nonnull real number. Then $\displaystyle M$ satisfies
(1.3) if and only if either $\displaystyle k=1$ and $\displaystyle M$ is
locally congruent to a geodesic hypersphere of radius
$\displaystyle\frac{\pi}{4}$ or $\displaystyle k^{2}\geq 4$ and $\displaystyle
M$ is locally congruent to a tube of radius $\displaystyle r$, $\displaystyle
0<r<\frac{\pi}{2}$, around $\displaystyle\mbox{{\bb C}}P^{n}$, $\displaystyle
0<n<m-1$, whose radius satisfies either $\displaystyle cot(r)=k$ or
$\displaystyle cot(r)=-\frac{1}{k}$.

In the non-Hopf case we obtain

Theorem 4 There does not exist any non-Hopf real hypersurface $\displaystyle
M$ in $\displaystyle\mbox{{\bb C}}P^{m}$, $\displaystyle m\geq 3$, such that
$\displaystyle R_{{\xi}_{T}}^{(k)}$ is skew symmetric for any nonnull real
number $\displaystyle k$ satisfying $\displaystyle kg(A\xi,\xi)=1$.

Aknowledgements This work was supported by MINECO-FEDER Project MTM
2016-78807-C2-1-P.

##  2 Preliminaries.

Throughout this paper, all manifolds, vector fields, etc., will be considered
of class $\displaystyle C^{\infty}$ unless otherwise stated. Let
$\displaystyle M$ be a connected real hypersurface in $\displaystyle\mbox{{\bb
C}}P^{m}$, $\displaystyle m\geq 2$, without boundary. Let $\displaystyle N$ be
a locally defined unit normal vector field on $\displaystyle M$. Let
$\displaystyle\nabla$ be the Levi-Civita connection on $\displaystyle M$ and
$\displaystyle(J,g)$ the Kaehlerian structure of $\displaystyle\mbox{{\bb
C}}P^{m}$.

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

$\phi^{2}X=-X+\eta(X)\xi,\quad\eta(\xi)=1,\quad g(\phi X,\phi
Y)=g(X,Y)-\eta(X)\eta(Y)$ (2.1)

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

$\begin{array}[]{c}\phi\xi=0,\quad\eta(X)=g(X,\xi).\end{array}$ (2.2)

From the parallelism of $\displaystyle J$ we get

$(\nabla_{X}\phi)Y=\eta(Y)AX-g(AX,Y)\xi$ (2.3)

and

$\nabla_{X}\xi=\phi AX$ (2.4)

for any $\displaystyle X,Y$ tangent to $\displaystyle M$, where $\displaystyle
A$ denotes the shape operator of the immersion. As the ambient space has
holomorphic sectional curvature $\displaystyle 4$, the equations of Gauss and
Codazzi are given, respectively, by

$\begin{array}[]{rl}R(X,Y)Z=\displaystyle g(Y,Z)X-g(X,Z)Y+g(\phi Y,Z)\phi
X-g(\phi X,Z)\phi Y\\\ -2g(\phi X,Y)\phi Z+g(AY,Z)AX-g(AX,Z)AY\\\ \end{array}$
(2.5)

and

$(\nabla_{X}A)Y-(\nabla_{Y}A)X=\eta(X)\phi Y-\eta(Y)\phi X-2g(\phi X,Y)\xi$
(2.6)

for any vector fields $\displaystyle X,Y,Z$ tangent to $\displaystyle M$,
where $\displaystyle R$ is the curvature tensor of $\displaystyle M$. We will
call the maximal holomorphic distribution 𝔻 on $\displaystyle M$ to the
following one: at any $\displaystyle p\in M$, $\displaystyle\mbox{{\bb
D}}(p)=\\{X\in T_{p}M|g(X,\xi)=0\\}$. If $\displaystyle U\in\mbox{{\bb D}}$ we
define $\displaystyle{\mbox{{\bb D}}}_{U}=\\{X\in\mbox{{\bb
D}}|g(X,U)=g(X,\phi U)=0\\}$. We will say that $\displaystyle M$ is Hopf if
$\displaystyle\xi$ is principal, that is, $\displaystyle A\xi=\alpha\xi$ for a
certain function $\displaystyle\alpha$ on $\displaystyle M$.

From Gauss equation the structure Jacobi operator on $\displaystyle M$ is
given by

$R_{\xi}X=X-\eta(X)\xi+g(A\xi,\xi)AX-g(A\xi,X)A\xi$ (2.7)

for any $\displaystyle X$ tangent to $\displaystyle M$

In the sequel we need the following results:

Theorem 2.1. [7] Let $\displaystyle M$ be a Hopf real hypersurface of
$\displaystyle\mbox{{\bb C}}P^{m}$, $\displaystyle m\geq 2$, and let
$\displaystyle X\in\mbox{{\bb D}}$ such that $\displaystyle AX=\lambda X$.
Then $\displaystyle\alpha=g(A\xi,\xi)$ is constant, $\displaystyle
2\lambda-\alpha\neq 0$ and $\displaystyle\phi X$ is principal with principal
curvature $\displaystyle\frac{\alpha\lambda+2}{2\lambda-\alpha}$.

Theorem 2.2. [8] Let $\displaystyle M$ be a real hypersurface of
$\displaystyle\mbox{{\bb C}}P^{m}$, $\displaystyle m\geq 2$. Then the
following are equivalent:

  1. 1.

$\displaystyle M$ is locally congruent to a real hypersurface of type
$\displaystyle(A)$.

  2. 2.

$\displaystyle\phi A=A\phi$.

Theorem 2.3 [10] There do not exist real hypersurfaces $\displaystyle M$ in
$\displaystyle\mbox{{\bb C}}P^{m}$, $\displaystyle m\geq 3$, whose shape
operator is given by $\displaystyle A\xi=\alpha\xi+\beta U$, $\displaystyle
AU=\beta\xi+\frac{\beta^{2}-1}{\alpha}U$, $\displaystyle A\phi
U=-\frac{1}{\alpha}\phi U$, where $\displaystyle U$ is a unit vector field in
𝔻, $\displaystyle\alpha$ and $\displaystyle\beta$ are nonvanishing functions
on $\displaystyle M$, the eigenvalues of $\displaystyle A$ in
$\displaystyle{\mbox{{\bb D}}}_{U}$ are different from $\displaystyle 0$,
$\displaystyle-\frac{1}{\alpha}$ and $\displaystyle\frac{\beta^{2}-1}{\alpha}$
and if $\displaystyle Z\in{\mbox{{\bb D}}}_{U}$ satisfies $\displaystyle
AZ=\lambda Z$, then $\displaystyle A\phi Z=\lambda\phi Z$.

Theorem 2.4. [10] There do not exist real hypersurfaces $\displaystyle M$ in
$\displaystyle{\mbox{{\bb C}}}P^{m}$, $\displaystyle m\geq 3$, whose shape
operator is given by $\displaystyle A\xi=\xi+\beta U$, $\displaystyle
AU=\beta\xi+(\beta^{2}-1)U$, $\displaystyle A\phi U=-\phi U$, for a unit
$\displaystyle U\in{\mbox{{\bb D}}}$, being $\displaystyle\beta$ a
nonvanishing function, and there exists $\displaystyle Z\in{\mbox{{\bb
D}}}_{U}$ such that $\displaystyle AZ=-Z$, $\displaystyle A\phi Z=-\phi Z$.

##  3 Proofs of Theorems 1 and 2.

Let $\displaystyle M$ be a real hypersurface in $\displaystyle\mbox{{\bb
C}}P^{m}$ satisfying (1.2). Then $\displaystyle
F_{X}^{(k)}R_{\xi}Y-F_{R_{\xi}Y}^{(k)}X-R_{\xi}F_{X}^{(k)}Y+R_{\xi}F_{Y}^{(k)}X=F_{Y}^{(k)}R_{\xi}X-F_{R_{\xi}X}^{(k)}Y-R_{\xi}F_{Y}^{(k)}X+R_{\xi}F_{X}^{(k)}Y$
for any $\displaystyle X,Y$ tangent to $\displaystyle M$. This yields

$\begin{array}[]{c}g(\phi AX,R_{\xi}Y)\xi-k\eta(X)\phi R_{\xi}Y-g(\phi
AR_{\xi}Y,X)\xi+\eta(X)\phi AR_{\xi}Y+\eta(Y)R_{\xi}\phi AX\\\
+k\eta(X)R_{\xi}\phi Y-\eta(X)R_{\xi}\phi AY-k\eta(Y)R_{\xi}\phi X=g(\phi
AY,R_{\xi}X)\xi-k\eta(Y)\phi R_{\xi}X\\\ -g(\phi AR_{\xi}X,Y)\xi+\eta(Y)\phi
AR_{\xi}X+\eta(X)R_{\xi}\phi AY+k\eta(Y)R_{\xi}\phi X-\eta(Y)R_{\xi}\phi AX\\\
-k\eta(X)R_{\xi}\phi Y\\\ \end{array}$ (3.1)

for any $\displaystyle X,Y$ tangent to $\displaystyle M$.

If in (3.1) we take $\displaystyle X,Y\in\mbox{{\bb D}}$ we obtain

$\begin{array}[]{c}g(\phi AX,R_{\xi}Y)\xi-g(\phi AR_{\xi}Y,X)\xi=g(\phi
AY,R_{\xi}X)\xi-g(\phi AR_{\xi}X,Y)\xi\end{array}$ (3.2)

for any $\displaystyle X,Y\in\mbox{{\bb D}}$. And if we take $\displaystyle
X=\xi$, $\displaystyle Y\in\mbox{{\bb D}}$ in (3.1) it follows

$\begin{array}[]{c}g(\phi A\xi,R_{\xi}Y)\xi-k\phi R_{\xi}Y+\phi
AR_{\xi}Y+2kR_{\xi}\phi Y-2R_{\xi}\phi AY=0\end{array}$ (3.3)

for any $\displaystyle Y\in\mbox{{\bb D}}$.

Let us suppose that $\displaystyle M$ is Hopf with $\displaystyle
A\xi=\alpha\xi$. Then (3.3) gives

$\begin{array}[]{c}-k\phi R_{\xi}Y+\phi AR_{\xi}Y+2kR_{\xi}\phi Y-2R_{\xi}\phi
AY=0\end{array}$ (3.4)

for any $\displaystyle Y\in\mbox{{\bb D}}$. If $\displaystyle X\in\mbox{{\bb
D}}$ then $\displaystyle R_{\xi}X=X+\alpha AX$. Therefore, (3.2) yields
$\displaystyle g(\phi AX,Y+\alpha AY)-g(\phi AY+\alpha\phi A^{2}Y,X)=g(\phi
AY,X+\alpha AX)-g(\phi AX+\alpha\phi A^{2}X,Y$ for any $\displaystyle
X,Y\in\mbox{{\bb D}}$. Then $\displaystyle 2g(\phi AX,Y)+2\alpha g(\phi
AX,AY)-2g(\phi AY,X)-\alpha g(\phi A^{2}Y,X)+\alpha g(\phi A^{2}X,Y)=0$, for
any $\displaystyle X,Y\in\mbox{{\bb D}}$. As $\displaystyle M$ is Hopf this
implies

$\begin{array}[]{c}2\phi AX+2\alpha A\phi AX+2A\phi X+\alpha A^{2}\phi
X+\alpha\phi A^{2}X=0\end{array}$ (3.5)

for any $\displaystyle X\in\mbox{{\bb D}}$. Let now $\displaystyle
X\in\mbox{{\bb D}}$ such that $\displaystyle AX=\lambda X$. Then
$\displaystyle A\phi X=\mu\phi X$,
$\displaystyle\mu=\frac{\alpha\lambda+2}{2\lambda-\alpha}$. Then (3.5) yields
$\displaystyle
2\lambda+2\alpha\lambda\mu+2\mu+\alpha\mu^{2}+\alpha\lambda^{2}=0$. That is,
$\displaystyle 2(\lambda+\mu)+\alpha(\lambda+\mu)^{2}=0$. Thus if we suppose
$\displaystyle\lambda+\mu=0$,
$\displaystyle\frac{2\lambda^{2}+2}{2\lambda-\alpha}=0$, which is impossible.
Therefore,

$\begin{array}[]{c}\alpha\neq 0\end{array}$ (3.6)

and $\displaystyle\alpha(\lambda+\mu)=-2$. That is,
$\displaystyle\frac{2\lambda^{2}+2}{2\lambda-\alpha}=-\frac{2}{\alpha}$. This
implies $\displaystyle(\lambda^{2}+1)\alpha=\alpha-2\lambda$, that is,
$\displaystyle\lambda^{2}\alpha=-2\lambda$. Therefore, either
$\displaystyle\lambda=0$ or $\displaystyle\lambda=-\frac{2}{\alpha}$. If
$\displaystyle X$ satisfies $\displaystyle AX=0$, then $\displaystyle A\phi
X=-\frac{2}{\alpha}\phi X$. This implies $\displaystyle R_{\xi}X=X$ and
$\displaystyle R_{\xi}\phi X=\phi X+\alpha A\phi X=-\phi X$. From (3.4) we get
$\displaystyle-k\phi X-2k\phi X=0$, that is, $\displaystyle 3k\phi X=0$, which
is impossible. We obtain a similar situation if we suppose that $\displaystyle
AX=-\frac{2}{\alpha}X$ and Theorem 1 is proved.

Suppose now that $\displaystyle M$ is non-Hopf. Then, at least on a
neighbourhood of a certain point of $\displaystyle M$ we can find a unit
vector field $\displaystyle U\in\mbox{{\bb D}}$ and functions
$\displaystyle\alpha$ and $\displaystyle\beta$, where $\displaystyle\beta$
does not vanish, such that we can write $\displaystyle A\xi=\alpha\xi+\beta
U$. From now on we consider all the calculations on such a neighborhood. Then
(3.3) gives

$\begin{array}[]{c}\beta g(\phi U,R_{\xi}Y)\xi-k\phi R_{\xi}Y+\phi
AR_{\xi}Y+2kR_{\xi}\phi Y-2R_{\xi}\phi AY=0\end{array}$ (3.7)

for any $\displaystyle Y\in\mbox{{\bb D}}$. Its scalar product with
$\displaystyle\xi$ yields $\displaystyle\beta g(R_{\xi}\phi U,Y)=0$ for any
$\displaystyle Y\in\mbox{{\bb D}}$. As $\displaystyle R_{\xi}\xi=0$ we obtain
$\displaystyle R_{\xi}\phi U=0$ and then

$\begin{array}[]{c}A\phi U=-\frac{1}{\alpha}\phi U.\end{array}$ (3.8)

If in (3.7) we take $\displaystyle Y=\phi U$ we get
$\displaystyle-2kR_{\xi}U+\frac{2}{\alpha}R_{\xi}\phi^{2}U=0$. Thus
$\displaystyle-(2k+\frac{2}{\alpha})R_{\xi}U=0$ and, as we suppose that
$\displaystyle\alpha\neq-\frac{1}{k}$ we obtain $\displaystyle R_{\xi}U=0$ or

$\begin{array}[]{c}AU=\beta\xi+\frac{\beta^{2}-1}{\alpha}U.\end{array}$ (3.9)

Therefore, $\displaystyle\mbox{{\bb D}}_{U}$ is $\displaystyle A$-invariant.
From (3.2) we know that $\displaystyle g((\phi A+A\phi)X,R_{\xi}Y)=g((\phi
A+A\phi)Y,R_{\xi}X)$ for any $\displaystyle X,Y\in\mbox{{\bb D}}_{U}$.
Therefore,

$\begin{array}[]{c}-(\phi A+A\phi)R_{\xi}Y=R_{\xi}(\phi A+A\phi)Y\end{array}$
(3.10)

for any $\displaystyle Y\in\mbox{{\bb D}}_{U}$. If we take $\displaystyle
Y\in\mbox{{\bb D}}_{U}$ unit and such that $\displaystyle AY=\lambda Y$ in
(3.7) we have $\displaystyle-k\phi R_{\xi}Y+\phi
AR_{\xi}Y+2(k-\lambda)R_{\xi}\phi Y=0$. Then,
$\displaystyle-(1+\alpha\lambda)k\phi Y+(1+\alpha\lambda)\phi
AY+2(k-\lambda)(\phi Y+\alpha A\phi Y)=0$. That is,
$\displaystyle((1+\alpha\lambda)(\lambda-k)+2(k-\lambda))\phi
Y+2\alpha(k-\lambda)A\phi Y=0$. Thus $\displaystyle 2\alpha(k-\lambda)A\phi
Y=(\alpha\lambda-1)(k-\lambda)\phi Y$. Therefore, either $\displaystyle
k=\lambda$ or if $\displaystyle\lambda\neq k$,

$\begin{array}[]{c}A\phi Y=\frac{\alpha\lambda-1}{2\alpha}\phi Y.\end{array}$
(3.11)

We will call $\displaystyle\mu=\frac{\alpha\lambda-1}{2\alpha}$. If in (3.7)
we take $\displaystyle\phi Y$ instead of $\displaystyle Y$ we have
$\displaystyle-k\phi R_{\xi}\phi Y+\phi AR_{\xi}\phi Y-2kR_{\xi}Y-2R_{\xi}\phi
A\phi Y=0$. This implies
$\displaystyle-k(1+\alpha\mu)\phi^{2}Y+(1+\alpha\mu)\phi A\phi
Y-2k(1+\alpha\lambda)Y+2\mu(1+\alpha\lambda)Y=0$. Then
$\displaystyle(-1+\alpha\mu-2\alpha\lambda)(k-\mu)=0$. This gives, bearing in
mind the value of $\displaystyle\mu$,
$\displaystyle-3(1+\alpha\lambda)(k-\mu)=0$. Therefore, we have two
possibilities: either $\displaystyle\lambda=-\frac{1}{\alpha}$ and then
$\displaystyle\mu=-\frac{1}{\alpha}$, or $\displaystyle\mu=k$ and, as we
suppose $\displaystyle\lambda\neq k$, in this case
$\displaystyle\alpha\lambda-1=2k\alpha$ and
$\displaystyle\lambda=\frac{2\alpha k+1}{\alpha}$.

As $\displaystyle-(\phi A+A\phi)R_{\xi}Y=R_{\xi}(\phi A+A\phi)Y$, we have
$\displaystyle-(1+\alpha\lambda)(\phi A+A\phi)Y=(\lambda+\mu)R_{\xi}\phi Y$.
Then
$\displaystyle-(1+\alpha\lambda)(\lambda+\mu)=(1+\alpha\mu)(\lambda+\mu)$. If
$\displaystyle\lambda=\mu=-\frac{1}{\alpha}$, we obtain
$\displaystyle\lambda+\mu\neq 0$.

If $\displaystyle\mu=k$ and we suppose $\displaystyle\lambda+\mu\neq 0$, it
follows $\displaystyle-1-\alpha\lambda=1+\alpha\mu$ and
$\displaystyle\lambda+\mu=-\frac{2}{\alpha}$. Then $\displaystyle\frac{2\alpha
k+1}{\alpha}+k=\frac{3\alpha k+1}{\alpha}=-\frac{2}{\alpha}$ yields
$\displaystyle 3\alpha k=-3$ and this contradicts the fact of
$\displaystyle\alpha k\neq-1$. Therefore, $\displaystyle\lambda+\mu=0$ and
$\displaystyle 3\alpha k+1=0$ yields $\displaystyle\alpha=-\frac{1}{3k}$. In
this case, $\displaystyle\alpha$, $\displaystyle\lambda$ and
$\displaystyle\mu$ are constant.

If the unique principal curvature on $\displaystyle\mbox{{\bb D}}_{U}$ is
$\displaystyle k$, as $\displaystyle-(1+\alpha k)(\phi A+A\phi)Y=2k(1+\alpha
k)\phi Y$, we should have $\displaystyle 4k(1+\alpha k)=0$ and, as
$\displaystyle k\neq 0$, $\displaystyle\alpha k=-1$, which is impossible.
Therefore, on $\displaystyle\mbox{{\bb D}}_{U}$ there must be, at least, a
principal curvature $\displaystyle\lambda$ different from $\displaystyle k$.

Let us suppose that on $\displaystyle\mbox{{\bb D}}_{U}$ there is a principal
curvature $\displaystyle\lambda=-\frac{1}{\alpha}$ and let $\displaystyle Y$
be a unit eigenvector corresponding to $\displaystyle\lambda$. As we have seen
$\displaystyle A\phi Y=-\frac{1}{\alpha}\phi Y$ and the equation of Codazzi
implies $\displaystyle(\nabla_{Y}A)\phi Y-(\nabla_{\phi Y}A)Y=-2\xi$. This
gives $\displaystyle-Y(\frac{1}{\alpha})\phi Y-\frac{1}{\alpha}\nabla_{Y}\phi
Y-A\nabla_{Y}\phi Y+(\phi Y)(\frac{1}{\alpha})Y+\frac{1}{\alpha}\nabla_{\phi
Y}Y+A\nabla_{\phi Y}Y=-2\xi$. If we take its scalar product with
$\displaystyle\xi$ we obtain $\displaystyle\frac{1}{\alpha}g(\phi Y,\phi
AY)-g(\nabla_{Y}\phi Y,\alpha\xi+\beta U)-\frac{1}{\alpha}g(Y,\phi A\phi
Y)+g(\nabla_{\phi Y}Y,\alpha\xi+\beta U)=-2$. This implies

$\begin{array}[]{c}\beta g([\phi Y,Y],U)=\frac{2}{\alpha^{2}}.\end{array}$
(3.12)

And its scalar product with $\displaystyle U$ yields
$\displaystyle-\frac{1}{\alpha}g(\nabla_{Y}\phi
Y,U)+\frac{1}{\alpha}g(\nabla_{\phi Y}Y,U)-g(\nabla_{Y}\phi
Y,\beta\xi+\frac{\beta^{2}-1}{\alpha}U)+g(\nabla_{\phi
Y}Y,\beta\xi+\frac{\beta^{2}-1}{\alpha}U)=0$. This gives
$\displaystyle\frac{\beta^{2}}{\alpha}g([\phi
Y,Y],U)-\frac{2\beta}{\alpha}=0$. Thus

$\begin{array}[]{c}\beta g([\phi Y,Y],U)=2.\end{array}$ (3.13)

Fron (3.12) and (3.13) we have $\displaystyle\alpha^{2}=1$ and, taking
$\displaystyle-\xi$ instead of $\displaystyle\xi$ if necessary, we can suppose
$\displaystyle\alpha=1$. From Theorem 2.4 this kind of real hypersurfaces does
not exist.

Now the unique principal curvatures on $\displaystyle\mbox{{\bb D}}_{U}$ are
$\displaystyle k$ and $\displaystyle-k$, with $\displaystyle
A\xi=-\frac{1}{3k}\xi+\beta U$, $\displaystyle AU=\beta\xi+3k(1-\beta^{2})U$,
$\displaystyle A\phi U=3k\phi U$. Let a unit $\displaystyle X\in\mbox{{\bb
D}}_{U}$ such that $\displaystyle AX=\lambda X$, where we know that
$\displaystyle\lambda$ is constant. Then we have
$\displaystyle(\nabla_{X}A)\xi-(\nabla_{\xi}A)X=-\phi X$, that is,
$\displaystyle\nabla_{X}(-\frac{1}{3k}\xi+\beta U)-A\phi
AX-\nabla_{\xi}(\lambda X)+A\nabla_{\xi}X=-\phi X$. Its scalar product with
$\displaystyle\xi$ implies $\displaystyle
g(\nabla_{\xi}X,-\frac{1}{3k}\xi+\beta U)=0$. As we suppose
$\displaystyle\beta\neq 0$, we obtain $\displaystyle g(\nabla_{\xi}X,U)=0$.
But its scalar product with $\displaystyle U$ gives $\displaystyle
X(\beta)-\lambda
g(\nabla_{\xi}X,U)+g(\nabla_{\xi}X,\beta\xi+3k(\beta^{2}-1)U)=0$. Then,
$\displaystyle X(\beta)+(3k(1-\beta^{2})-\lambda)g(\nabla_{\xi}X,U)=0$. this
yields

$\begin{array}[]{c}X(\beta)=0\end{array}$ (3.14)

for any $\displaystyle X\in\mbox{{\bb D}}_{U}$.

On the other hand $\displaystyle(\nabla_{U}A)\xi-(\nabla_{\xi}A)U=-\phi U$
gives $\displaystyle-\frac{1}{3k}\phi AU+U(\beta)U+\beta\nabla_{U}U-A\phi
AU-\xi(\beta)\xi-\beta\phi
A\xi-3k\xi(1-\beta^{2})U-3k(1-\beta^{2})\nabla_{\xi}U+A\nabla_{\xi}U=\phi U$.
Its scalar product with $\displaystyle\xi$ implies
$\displaystyle-\xi(\beta)+g(\nabla_{\xi}U,-\frac{1}{3k}\xi+\beta U)=0$.
Therefore,

$\begin{array}[]{c}\xi(\beta)=0.\end{array}$ (3.15)

From the scalar product of the above expression and $\displaystyle U$ we get
$\displaystyle
U(\beta)-3k\xi(1-\beta^{2})+g(\nabla_{\xi}U,\beta\xi+3k(1-\beta^{2})U)=0$.
From (3.15) we have

$\begin{array}[]{c}U(\beta)=0.\end{array}$ (3.16)

Once more, Codazzi equation implies $\displaystyle(\nabla_{\phi
U}A)\xi-(\nabla_{\xi}A)\phi U=U$, that is, $\displaystyle-\frac{1}{3k}\phi
A\phi U+(\phi U)(\beta)U+\beta\nabla_{\phi U}U+3kAU-3k\nabla_{\xi}\phi
U+A\nabla_{\xi}\phi U=U$. Its scalar product with $\displaystyle\xi$ gives
$\displaystyle-\beta g(U,\phi A\phi U)+3k\beta+3kg(\phi U,\phi
A\xi)+g(\nabla_{\xi}\phi U,-\frac{1}{3k}\xi+\beta U)=0$. Thus $\displaystyle
9k\beta+\frac{1}{3k}g(\phi U,\phi A\xi)+\beta g(\nabla_{\xi}\phi U,U)=0$. As
$\displaystyle\beta\neq 0$ we obtain

$\begin{array}[]{c}g(\nabla_{\xi}U,\phi U)=9k+\frac{1}{3k}.\end{array}$ (3.17)

Its scalar product with $\displaystyle U$ yields
$\displaystyle\frac{1}{3k}g(A\phi U,\phi U)+(\phi
U)(\beta)+9k^{2}(1-\beta^{2})-3kg(\nabla_{\xi}\phi U,U)+g(\nabla_{\xi}\phi
U,\beta\xi+3k(1-\beta^{2})U)=1$. From (3.17) we obtain $\displaystyle(\phi
U)(\beta)+9k^{2}-9k^{2}\beta^{2}+27k^{2}+1-\beta^{2}+3k(\beta^{2}-1)(9k+\frac{1}{3k})=0$
and this implies

$\begin{array}[]{c}(\phi U)(\beta)=-9k^{2}(1+2\beta^{2}).\end{array}$ (3.18)

From (3.14), (3.15), (3.16) and (3.18) we obtain

$\begin{array}[]{c}grad(\beta)=\omega\phi U\end{array}$ (3.19)

where $\displaystyle\omega=-9k^{2}(1+2\beta^{2})$.

As $\displaystyle g(\nabla_{X}grad(\beta),Y)=g(\nabla_{Y}grad(\beta),X)$, for
any $\displaystyle X,Y$ tangent to $\displaystyle M$, we obtain $\displaystyle
X(\omega)g(\phi U,Y)+\omega g(\nabla_{X}\phi U,Y)=Y(\omega)g(\phi U,X)+\omega
g(\nabla_{Y}\phi U,X)$, for any $\displaystyle X,Y$ tangent to $\displaystyle
M$. Taking $\displaystyle Y=\xi$ we get $\displaystyle\omega g(\nabla_{X}\phi
U,\xi)=\omega g(\nabla_{\xi}\phi U,X)$, that is, $\displaystyle-\omega
g(U,AX)=\omega g(\nabla_{\xi}\phi U,X)$ for any $\displaystyle X$ tangent to
$\displaystyle M$. If $\displaystyle X=U$ we have $\displaystyle-\omega
g(AU,U)=\omega g(\nabla_{\xi}\phi U,U)=-\omega g(\nabla_{\xi}U,\phi U)$.
Therefore, either $\displaystyle\omega=0$, which is impossible, or
$\displaystyle g(AU,U)=g(\nabla_{\xi}U,\phi U)$. This yields $\displaystyle
3k(1-\beta^{2})=9k+\frac{1}{3k}$, that is, $\displaystyle
1-\beta^{2}=3+\frac{1}{9k^{2}}$. Equivalently,
$\displaystyle\beta^{2}=-2-\frac{1}{9k^{2}}$, which is impossible. This
finishes the proof of Theorem 2.

##  4 Proofs of Theorems 3 and 4.

If $\displaystyle M$ is a real hypersurface satisfying (1.3) we obtain

$\begin{array}[]{c}g(\phi AX,R_{\xi}Y)\xi-k\eta(X)\phi R_{\xi}Y-g(\phi
AR_{\xi}Y,X)\xi+\eta(X)\phi AR_{\xi}Y\\\ +\eta(Y)R_{\xi}\phi
AX+k\eta(X)R_{\xi}\phi Y-\eta(X)R_{\xi}\phi AY-k\eta(Y)R_{\xi}\phi X+g(\phi
AY,R_{\xi}X)\xi\\\ -k\eta(Y)\phi R_{\xi}X-g(\phi AR_{\xi}X,Y)\xi+\eta(Y)\phi
AR_{\xi}X+\eta(X)R_{\xi}\phi AY\\\ +k\eta(Y)R_{\xi}\phi X-\eta(Y)R_{\xi}\phi
AX-k\eta(X)R_{\xi}\phi Y=0\\\ \end{array}$ (4.1)

for any $\displaystyle X,Y$ tangent to $\displaystyle M$. If in (4.1) we take
$\displaystyle X,Y\in\mbox{{\bb D}}$ we have

$\begin{array}[]{c}g(\phi AX,R_{\xi}Y)\xi-g(\phi AR_{\xi}Y,X)\xi+g(\phi
AY,R_{\xi}X)\xi-g(\phi AR_{\xi}X,Y)\xi=0\end{array}$ (4.2)

for any $\displaystyle X,Y\in\mbox{{\bb D}}$. Taking $\displaystyle X=\xi$,
$\displaystyle Y\in\mbox{{\bb D}}$ in (4.1) we get $\displaystyle g(\phi
A\xi,R_{\xi}Y)\xi-k\phi R_{\xi}Y+\phi AR_{\xi}Y+kR_{\xi}\phi Y-R_{\xi}\phi
AY+R_{\xi}\phi AY-kR_{\xi}\phi Y=0$. This gives

$\begin{array}[]{c}g(\phi A\xi,R_{\xi}Y)\xi-k\phi R_{\xi}Y+\phi
AR_{\xi}Y=0\end{array}$ (4.3)

for any $\displaystyle Y\in\mbox{{\bb D}}$.

Let us suppose that $\displaystyle M$ is Hopf and $\displaystyle
A\xi=\alpha\xi$. From (4.2) we get $\displaystyle-A\phi R_{\xi}Y-\phi
AR_{\xi}Y+R_{\xi}\phi AY+R_{\xi}A\phi Y=0$ for any $\displaystyle
Y\in\mbox{{\bb D}}$. Take a unit $\displaystyle Y\in\mbox{{\bb D}}$ such that
$\displaystyle AY=\lambda Y$. Then $\displaystyle A\phi Y=\mu\phi Y$,
$\displaystyle\mu=\frac{\alpha\lambda+2}{2\lambda-\alpha}$. In this case,
$\displaystyle R_{\xi}Y=(1+\alpha\lambda)Y$ and $\displaystyle R_{\xi}\phi
Y=(1+\alpha\mu)\phi Y$. From (4.3) we have
$\displaystyle-k(1+\alpha\lambda)\phi Y+\lambda(1+\alpha\lambda)\phi Y=0$.
Thus $\displaystyle(\lambda-k)(1+\alpha\lambda)=0$. Therefore, if
$\displaystyle\alpha=0$, $\displaystyle\lambda=k$ and if
$\displaystyle\alpha\neq 0$, either $\displaystyle\lambda=k$ or
$\displaystyle\lambda=-\frac{1}{\alpha}$. But as $\displaystyle-A\phi
R_{\xi}Y-\phi AR_{\xi}Y+R_{\xi}\phi AY+R_{\xi}A\phi Y=0$ we get
$\displaystyle-(1+\alpha\lambda)\mu\phi Y-(1+\alpha\lambda)\lambda\phi
Y-\lambda(1+\alpha\mu)\phi Y+\mu(1+\alpha\mu)\phi Y=0$. That is,
$\displaystyle(\lambda+\mu)(\mu-\lambda)\alpha=0$. As
$\displaystyle\lambda+\mu=\lambda+\frac{\alpha\lambda+2}{2\lambda-\alpha}=\frac{2\lambda^{2}+2}{2\lambda-\alpha}\neq
0$, it follows $\displaystyle\alpha(\mu-\lambda)=0$.

If $\displaystyle\alpha=0$ the unique principal curvature on 𝔻 is
$\displaystyle k$. From [3] $\displaystyle M$ must be locally congruent to a
geodesic hypersphere of radius $\displaystyle\frac{\pi}{4}$. Therefore
$\displaystyle k=cot(\frac{\pi}{4})=1$.

If $\displaystyle\alpha\neq 0$, $\displaystyle\lambda=\mu$, that is
$\displaystyle\phi A=A\phi$ and from Theorem 2.2, $\displaystyle M$ must be
locally congruent to a real hypersurface of type $\displaystyle(A)$. Its
principal curvatures are $\displaystyle\alpha$,
$\displaystyle-\frac{1}{\alpha}$ and $\displaystyle k$. In the case of a
geodesic hypersphere we should have $\displaystyle\alpha=-\frac{1}{k}$. Thus
$\displaystyle 2cot(2r)=-\frac{1}{k}$ and $\displaystyle cot(r)=k$. Thus
$\displaystyle-\frac{1}{k}=2cot(2r)=cot(r)-tan(r)=k-\frac{1}{k}$ yields
$\displaystyle k=0$, which is impossible.

Therefore, $\displaystyle M$ is locally congruent to a tube of radius
$\displaystyle r$, $\displaystyle 0<r<\frac{\pi}{4}$, around
$\displaystyle\mbox{{\bb C}}P^{n}$, $\displaystyle 0<n<m-1$. Their principal
curvatures are $\displaystyle\alpha=2cot(2r)=cot(r)-tan(r)$, $\displaystyle
cot(r)$ and $\displaystyle-tan(r)$. We thus have two situations: either
$\displaystyle cot(r)=k$ and $\displaystyle tan(r)=\frac{1}{\alpha}$,
$\displaystyle\alpha=k-\frac{1}{\alpha}$, or $\displaystyle
cot(r)=-\frac{1}{\alpha}$, $\displaystyle tan(r)=-k$ and again
$\displaystyle\alpha=-\frac{1}{\alpha}+k$. Anyway,
$\displaystyle\alpha^{2}-k\alpha+1=0$, Thus
$\displaystyle\alpha=\frac{k\pm\sqrt{k^{2}-4}}{2}$. Therefore, $\displaystyle
k^{2}\geq 4$ and either $\displaystyle cot(r)=k$ or $\displaystyle
cot(r)=-\frac{1}{k}$, finishing the proof of Theorem 3.

Let us suppose now that $\displaystyle M$ is non-Hopf and write, as in
previous section, $\displaystyle A\xi=\alpha\xi+\beta U$. Then (4.3) gives

$\begin{array}[]{c}\beta g(\phi U,R_{\xi}Y)\xi-k\phi R_{\xi}Y+\phi
AR_{\xi}Y=0\end{array}$ (4.4)

for any $\displaystyle Y\in\mbox{{\bb D}}$. The scalar product of (4.4) and
$\displaystyle\xi$ yields $\displaystyle\beta g(R_{\xi}\phi U,Y)=0$ for any
$\displaystyle Y\in\mbox{{\bb D}}$ and, being $\displaystyle\beta\neq 0$ and
$\displaystyle R_{\xi}\xi=0$, we obtain $\displaystyle R_{\xi}\phi U=0$, that
is, $\displaystyle\phi U+\alpha A\phi U=0$. Then

$\begin{array}[]{c}\alpha\neq 0\end{array}$ (4.5)

and

$\begin{array}[]{c}A\phi U=-\frac{1}{\alpha}\phi U.\end{array}$ (4.6)

Also (4.4) becomes

$\begin{array}[]{c}\phi AR_{\xi}Y=k\phi R_{\xi}Y\end{array}$ (4.7)

for any $\displaystyle Y\in\mbox{{\bb D}}$. If we take (4.7) into (4.2) we
obtain $\displaystyle g(\phi AX,R_{\xi}Y)-kg(\phi R_{\xi}Y,X)+g(\phi
AY,R_{\xi}X)-kg(\phi R_{\xi}X,Y)=0$ for any $\displaystyle X,Y\in\mbox{{\bb
D}}$. As $\displaystyle g(R_{\xi}\phi AX+kR_{\xi}\phi X-A\phi R_{\xi}X-k\phi
R_{\xi}X,\xi)=0$ because $\displaystyle R_{\xi}\phi U=0$, we get

$\begin{array}[]{c}R_{\xi}\phi AX+kR_{\xi}\phi X-A\phi R_{\xi}X-k\phi
R_{\xi}X=0\end{array}$ (4.8)

for any $\displaystyle X\in\mbox{{\bb D}}$. Taking $\displaystyle X=\phi U$ in
(4.8) we have $\displaystyle R_{\xi}\phi A\phi U-kR_{\xi}U=0$. From (4.6) it
follows $\displaystyle(\frac{1}{\alpha}-k)R_{\xi}U=0$. As we suppose
$\displaystyle k\alpha\neq 1$ we obtain $\displaystyle R_{\xi}U=0$ and then

$\begin{array}[]{c}AU=\beta\xi+\frac{\beta^{2}-1}{\alpha}U.\end{array}$ (4.9)

Now $\displaystyle\mbox{{\bb D}}_{U}$ is A-invariant. Take a unit
$\displaystyle Y\in\mbox{{\bb D}}_{U}$ such that $\displaystyle AY=\lambda Y$.
Then $\displaystyle R_{\xi}Y=(1+\alpha\lambda)Y$ and applying
$\displaystyle\phi$ to (4.7) we have $\displaystyle-
AR_{\xi}Y+g(AR_{\xi}Y,\xi)\xi=-kR_{\xi}Y+kg(R_{\xi}Y,\xi)\xi$. As
$\displaystyle R_{\xi}U=0$ this yields $\displaystyle AR_{\xi}Y=kR_{\xi}Y$.
Therefore $\displaystyle\lambda(1+\alpha\lambda)Y=k(1+\alpha\lambda)Y$. Thus
either $\displaystyle\lambda=-\frac{1}{\alpha}$ or $\displaystyle\lambda=k$.

Moreover, from (4.8), for such a $\displaystyle Y$ we get
$\displaystyle\lambda R_{\xi}\phi Y+kR_{\xi}\phi Y-(1+\alpha\lambda)A\phi
Y-(1+\alpha\lambda)k\phi Y=0$. Now if $\displaystyle\lambda=-\frac{1}{\alpha}$
we arrive to $\displaystyle(k-\frac{1}{\alpha})R_{\xi}\phi Y=0$ and, being
$\displaystyle k\alpha\neq 1$, we obtain $\displaystyle R_{\xi}\phi Y=0=\phi
Y+\alpha A\phi Y$, which implies $\displaystyle A\phi Y=-\frac{1}{\alpha}\phi
Y$.

If $\displaystyle\lambda=k$, from (4.8) we have $\displaystyle 2kR_{\xi}\phi
Y-(1+k\alpha)A\phi Y-k(1+k\alpha)\phi Y=0$, that is, $\displaystyle 2k(\phi
Y+\alpha A\phi Y)-(1+k\alpha)A\phi Y-k(1+k\alpha)\phi Y=0$. Thus
$\displaystyle(k\alpha-1)A\phi Y=k(k\alpha-1)\phi Y$ and, as $\displaystyle
k\alpha\neq 1$ we obtain $\displaystyle A\phi Y=k\phi Y$. We have obtained
that any eigenspace in $\displaystyle\mbox{{\bb D}}_{U}$ is
$\displaystyle\phi$-invariant.

From Theorem 2.4 the case of $\displaystyle\lambda=-\frac{1}{\alpha}$ does not
occur and then the unique principal curvature on $\displaystyle\mbox{{\bb
D}}_{U}$ is $\displaystyle k$. Then, for any $\displaystyle X\in\mbox{{\bb
D}}_{U}$, Codazzi equation gives $\displaystyle(\nabla_{X}A)\phi
X-(\nabla_{\phi X}A)X=-2\xi$. This yields $\displaystyle k\nabla_{X}\phi
X-A\nabla_{X}\phi X-k\nabla_{\phi X}X+A\nabla_{\phi X}X=-2\xi$. Its scalar
product with $\displaystyle\xi$ implies $\displaystyle-kg(\phi X,\phi
AX)-g(\nabla_{X}\phi X,\alpha\xi+\beta U)+kg(X,\phi A\phi X)+g(\nabla_{\phi
X}X,\alpha\xi+\beta U)=-2$, that is

$\begin{array}[]{c}\beta g([\phi X,X],U)=2k^{2}-2k\alpha-2.\end{array}$ (4.10)

Its scalar product with $\displaystyle U$ gives $\displaystyle-kg([\phi
X,X],U)-g(\nabla_{X}\phi
X,\beta\xi+\frac{\beta^{2}-1}{\alpha}U)+g(\nabla_{\phi
X}X,\beta\xi+\frac{\beta^{2}-1}{\alpha}U)=0$. This yields

$\begin{array}[]{c}(\frac{\beta^{2}-1}{\alpha}-k)g([\phi
X,X],U)=-2k\beta.\end{array}$ (4.11)

If $\displaystyle k=\frac{\beta^{2}-1}{\alpha}$, $\displaystyle 2\beta k=0$,
which is impossible. Then $\displaystyle k\neq 0$ and $\displaystyle
k\neq\frac{\beta^{2}-1}{\alpha}$. Moreover, from (4.10) and (4.11) we obtain
$\displaystyle 2(k^{2}-k\alpha-1)(\frac{\beta^{2}-1}{\alpha}-k)=-2k\beta^{2}$.
That is,
$\displaystyle(k^{2}-(k\alpha+1))(\beta^{2}-(k\alpha+1))=-k\alpha\beta^{2}$.
If $\displaystyle k=-\frac{1}{\alpha}$, we should have $\displaystyle
k^{2}\beta^{2}=\beta^{2}$ and therefore, $\displaystyle k^{2}=1$ and
$\displaystyle\alpha^{2}=1$. This hypersurfaces do not exist from Theorem 2.4.

Thus the unique principal curvature on $\displaystyle\mbox{{\bb D}}_{U}$ is
$\displaystyle k\neq 0,-\frac{1}{\alpha},\frac{\beta^{2}-1}{\alpha}$ and such
a real hypersurface does not exist from Theore, 2.3. This finishes our proof.

$\displaystyle\Box$

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* Juan de Dios Perez<EMAIL_ADDRESS>
Departamento de Geometria y Topologia and IEMATH
Universidad de Granada
18071 Granada
Spain

* David Perez-Lopez<EMAIL_ADDRESS>
Fundación I+D del Software Libre - FIDESOL
Avda. de la Innovación, 1 - Ed. BIC CEEI (PTS)
18016 Armilla (Granada)
Spain