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# On the structure Lie operator of a real hypersurface in the complex quadric

Juan de Dios Pérez, David Pérez-López and Young Jin Suh

###### Abstract

The almost contact metric structure that we have on a real hypersurface
$\displaystyle M\;$in the complex quadric $\displaystyle
Q^{m}=SO_{m+2}/SO_{m}SO_{2}$ allows us to define, for any nonnull real number
$\displaystyle k$, the $\displaystyle k$-th generalized Tanaka-Webster
connection on $\displaystyle M\;$, $\displaystyle\hat{\nabla}^{(k)}$.
Associated to this connection we have Cho and torsion operators,
$\displaystyle F_{X}^{(k)}$ and $\displaystyle T_{X}^{(k)}$, respectively, for
any vector field $\displaystyle X$ tangent to $\displaystyle M$. From them and
for any symmetric operator $\displaystyle B$ on $\displaystyle M$ we can
consider two tensor fields of type (1,2) on $\displaystyle M$ that we will
denote by $\displaystyle B_{F}^{(k)}$ and $\displaystyle B_{T}^{(k)}$,
respectively. We will classify real hypersurfaces $\displaystyle M$ in
$\displaystyle Q^{m}$ for which any of those tensors identically vanishes, in
the particular case of $\displaystyle B$ being the structure Lie operator
$\displaystyle L_{\xi}$ on $\displaystyle M$.

2010 Mathematics Subject Classification: 53C15, 53B25.

Keywords and phrases: Complex quadric, real hypersurface, structure Lie
operator, $\displaystyle k$-th generalized Tanaka-Webster connection, Lie
derivative.

##  1 Introduction.

Let $\displaystyle M$ be a real hypersurface of the complex quadric
$\displaystyle Q^{m}=SO_{m+2}/SO_{m}SO_{2}$ endowed with the Kählerian
structure $\displaystyle(J,G)$. Let $\displaystyle N$ be a unit local normal
vector field on $\displaystyle M$. $\displaystyle(J,G)$ induces on
$\displaystyle M$ an almost contact metric structure
$\displaystyle(\phi,\eta,g,\xi)$, see [4]. Then, if $\displaystyle\nabla$
denotes its Levi-Civita connection, for any nonnull real number $\displaystyle
k$, we can define on $\displaystyle M$ the so-called $\displaystyle k$-th
generalized Tanaka-Webster connection $\displaystyle\hat{\nabla}^{(k)}$ by

$\displaystyle\hat{\nabla}_{X}^{(k)}Y=\nabla_{X}Y+g(\phi SX,Y)\xi-\eta(Y)\phi
SX-k\eta(X)\phi Y$

for any $\displaystyle X,Y$ tangent to $\displaystyle M\;$(see [5], [6], [14],
[15] and [16]), where $\displaystyle S$ denotes the shape operator on
$\displaystyle M$ associated to $\displaystyle N$.

We will write $\displaystyle F_{X}^{(k)}Y=g(\phi SX,Y)\xi-\eta(Y)\phi
SX-k\eta(X)\phi Y$, for any $\displaystyle X,Y$ tangent to $\displaystyle
M\;$and call it the $\displaystyle k$-th Cho operator on $\displaystyle M$
associated to $\displaystyle X$. The torsion of the $\displaystyle k$-th
generalized Tanaka-Webster connection is given by $\displaystyle
T^{(k)}(X,Y)=F_{X}^{(k)}Y-F_{Y}^{(k)}X$, for any $\displaystyle X,Y$ tangent
to $\displaystyle M$. We will call $\displaystyle k$-th torsion operator
associated to the vector field $\displaystyle X$ to the operator given by
$\displaystyle T_{X}^{(k)}Y=T^{(k)}(X,Y)$ for any $\displaystyle Y$ tangent to
$\displaystyle M$.

If $\displaystyle\cal L$ denotes the Lie derivative on $\displaystyle M$, we
have that $\displaystyle{\cal L}_{X}Y=\nabla_{X}Y-\nabla_{Y}X$, for any
$\displaystyle X,Y$ tangent to $\displaystyle M$. We can also consider on
$\displaystyle M$ a differential operator of first order associated to
$\displaystyle\hat{\nabla}^{(k)}$, $\displaystyle{\cal L}^{(k)}$, given by
$\displaystyle{\cal
L}_{X}^{(k)}Y=\hat{\nabla}_{X}^{(k)}Y-\hat{\nabla}_{Y}^{(k)}X$, for any
$\displaystyle X,Y$ tangent to $\displaystyle M$. It is easy to see that
$\displaystyle{\cal L}_{X}^{(k)}Y={\cal L}_{X}Y+T_{X}^{(k)}Y$, for any
$\displaystyle X,Y$ tangent to $\displaystyle M$.

Let $\displaystyle B$ be a symmetric operator on $\displaystyle M$. Then we
can consider $\displaystyle((\hat{\nabla}_{X}^{(k)}-\nabla_{X})B)Y$ for any
$\displaystyle X$ and $\displaystyle Y$ tangent to $\displaystyle M$. It is
easy to see that
$\displaystyle((\hat{\nabla}_{X}^{(k)}-\nabla_{X})B)Y=F_{X}^{(k)}BY-
BF_{X}^{(k)}Y$. Therefore, from $\displaystyle B$, we can define on
$\displaystyle M$ the tensor field of type (1,2) $\displaystyle B_{F}^{(k)}$
that is given by

$\displaystyle B_{F}^{(k)}(X,Y)=[F_{X}^{(k)},B]Y=F_{X}^{(k)}BY-BF_{X}^{(k)}Y$

for any $\displaystyle X,Y$ tangent to $\displaystyle M$. Thus the vanishing
of such a tensor field is equivalent to the commutativity of the operators
$\displaystyle F_{X}^{(k)}$ and $\displaystyle B$ for any $\displaystyle X$
tangent to $\displaystyle M$. We have studied this problem in the cases of
$\displaystyle B=S$, the shape operator of $\displaystyle M$, in [11] and
$\displaystyle B=\bar{R}_{N}$, the normal Jacobi operator of $\displaystyle
M$, in [12].

We can also consider $\displaystyle(({\cal L}_{X}^{(k)}-{\cal L}_{X})B)Y$ for
any $\displaystyle X,Y$ tangent to $\displaystyle M$. As above, it is easy to
see that $\displaystyle(({\cal L}_{X}^{(k)}-{\cal L}_{X})B)Y=T_{X}^{(k)}BY-
BT_{X}^{(k)}Y$, and we can define another tensor field of type (1,2) on
$\displaystyle M$ associated to $\displaystyle B$, $\displaystyle
B_{T}^{(k)}$, by

$\displaystyle B_{T}^{(k)}(X,Y)=[T_{X}^{(k)},B]Y=T_{X}^{(k)}BY-BT_{X}^{(k)}Y$

for any $\displaystyle X,Y$ tangent to $\displaystyle M$. Such a tensor field
vanishes if and only if, for any $\displaystyle X$ tangent to $\displaystyle
M$, the operators $\displaystyle T_{X}^{(k)}$ and $\displaystyle B$ commute.
This problem has been solved in [11] for $\displaystyle B=S$ and in [10] for
$\displaystyle B=\bar{R}_{N}$.

Let us define the structure Lie operator on $\displaystyle M$, $\displaystyle
L_{\xi}$, by $\displaystyle g(L_{\xi}X,Y)=({\cal
L}_{\xi}g)(X,Y)=g(\nabla_{X}\xi,Y)+g(X,\nabla_{Y},\xi)=g(\phi SX,Y)+g(X,\phi
SY)=g((\phi S-S\phi)X,Y)$, for any $\displaystyle X,Y$ tangent to
$\displaystyle M$. Therefore we have

$\displaystyle L_{\xi}X=(\phi S-S\phi)X$

for any $\displaystyle X$ tangent to $\displaystyle M$. Berndt and Suh, [2],
classified real hypersurfaces in $\displaystyle Q^{m}$ with isometric Reeb
flow, proving that such a condition is equivalent to the vanishing of the
structure Lie operator. They obtained the following

###### Theorem A

Let $\displaystyle M$ be a real hypersurface in the complex quadric
$\displaystyle Q^{m}$, $\displaystyle m\geq 3$. The Reeb flow on
$\displaystyle M$ is isometric if and only if $\displaystyle m$ is even, say
$\displaystyle m=2n$, and $\displaystyle M$ is an open part of a tube around a
totally geodesic $\displaystyle\mathbb{C}P^{n}\subset Q^{m}$.

A real hypersurface $\displaystyle M$ in $\displaystyle Q^{m}$ is Hopf if the
Reeb vector field $\displaystyle\xi$ is principal, that is, $\displaystyle
S\xi=\alpha\xi$, and $\displaystyle\alpha$ is called the Reeb function of
$\displaystyle M$. The tubes appearing in Theorem 1.1 are examples of Hopf
real hypersurfaces in $\displaystyle Q^{m}$. We will denote by
$\displaystyle\mathcal{C}$ the maximal holomorphic distribution on
$\displaystyle M$, given by $\displaystyle{\mathcal{C}}=\\{X\in TM|$
$\displaystyle g(X,\xi)=0\\}$.

Recently, in [7], a family of non Hopf real hypersurfaces of $\displaystyle
Q^{m}$, called ruled real hypersurfaces, was introduced and some examples were
provided. $\displaystyle M$ is ruled in $\displaystyle Q^{m}$ if the
distribution $\displaystyle\mathcal{C}$ is integrable and $\displaystyle M$ is
foliated by totally geodesic complex hyperplanes $\displaystyle Q^{m-1}$ in
$\displaystyle Q^{m}$. In such a case, the shape operator $\displaystyle S$ of
$\displaystyle M$ in $\displaystyle Q^{n}$ is given by $\displaystyle
S\xi=\alpha\xi+\beta U$, $\displaystyle SU=\beta\xi$ and $\displaystyle SX=0$
for any vector field $\displaystyle X\perp Span\\{\xi,U\\}$, where
$\displaystyle U$ is a unit vector field in $\displaystyle\mathcal{C}$,
$\displaystyle\alpha$ and $\displaystyle\beta$ are functions on $\displaystyle
M$ and $\displaystyle\beta$ does not vanish.

This paper is devoted to study the vanishing of the tensors $\displaystyle
L_{\xi_{F}}^{(k)}$ and $\displaystyle L_{\xi_{T}}^{(k)}$. We will prove the
following

###### Theorem 1.1

Let $\displaystyle M\;$be a real hypersurface in $\displaystyle Q^{m}\;$,
$\displaystyle m\geq 3$, and $\displaystyle k$ a nonnull real number. Then
$\displaystyle L_{\xi_{F}}^{(k)}(\xi,X)=0$, for any $\displaystyle X$ tangent
to $\displaystyle M$, if and only if $\displaystyle M$ is an open part of a
tube of radius $\displaystyle r$, $\displaystyle 0<r<\frac{\pi}{2}$, around a
totally geodesic $\displaystyle\mathbb{C}P^{n}\subset Q^{m}$, $\displaystyle
m=2n$.

As $\displaystyle TM=\mathbb{R}\xi\oplus\mathcal{C}$, we can prove also the

###### Theorem 1.2

Let $\displaystyle M\;$be a real hypersurface of $\displaystyle Q^{m}\;$,
$\displaystyle m\geq 3$, and $\displaystyle k$ a nonnull real number. Then
$\displaystyle L_{\xi_{F}}^{(k)}(X,Y)=0$ for any $\displaystyle
X\in\mathcal{C}$, $\displaystyle Y$ tangent to $\displaystyle M$ if and only
if $\displaystyle M\;$is an open part of either a tube of radius
$\displaystyle r$, $\displaystyle 0<r<\frac{\pi}{2}$, around a totally
geodesic $\displaystyle\mathbb{C}P^{n}\subset Q^{m}$, $\displaystyle m=2n$, or
a ruled real hypersurface.

From both Theorems we obtain the

###### Corollary 1.1

Let $\displaystyle M$ be a real hypersurface in $\displaystyle Q^{m}\;$,
$\displaystyle m\geq 3$, and $\displaystyle k$ a nonnull real number. Then
$\displaystyle L_{\xi_{F}}^{(k)}\equiv 0$ if and only if $\displaystyle M$ is
an open part of a tube of radius $\displaystyle r$, $\displaystyle
0<r<\frac{\pi}{2}$, around a totally geodesic
$\displaystyle\mathbb{C}P^{n}\subset Q^{m}$, $\displaystyle m=2n$.

In the case of $\displaystyle L_{\xi_{T}}^{(k)}$ we have, bearing in mind
again that $\displaystyle TM=\mathbb{R}\xi\oplus\mathcal{C}$, we have

###### Theorem 1.3

Let $\displaystyle M\;$be a real hypersurface in $\displaystyle Q^{m}\;$,
$\displaystyle m\geq 3$, and $\displaystyle k$ and nonnull real number. Then
$\displaystyle L_{\xi_{T}}^{(k)}(X,Y)=$0 for any $\displaystyle
X\in\mathcal{C}$, $\displaystyle Y$ tangent to $\displaystyle M$, if and only
if $\displaystyle M$ is an open part of a tube of radius $\displaystyle r$,
$\displaystyle 0<r<\frac{\pi}{2}$, around $\displaystyle\mathbb{C}P^{n}$,
$\displaystyle m=2n$.

On the other hand

###### Theorem 1.4

There does not exist any non Hopf real hypersurface $\displaystyle M$ in
$\displaystyle Q^{m}\;$, $\displaystyle m\geq 3$, such that $\displaystyle
L_{\xi_{T}}^{(k)}(\xi,X)=0$ for any $\displaystyle X$ tangent to
$\displaystyle M\;$and any nonnull real number $\displaystyle k$.

As a consequence of these results we conclude with the

###### Corollary 1.2

Let $\displaystyle M\;$be a real hypersurface in $\displaystyle Q^{m}\;$,
$\displaystyle m\geq 3$, and $\displaystyle k$ a nonnull real number. Then
$\displaystyle L_{\xi_{T}}^{(k)}\equiv 0$ if and only if $\displaystyle M$ is
an open part of a tube of radius $\displaystyle r$, $\displaystyle
0<r<\frac{\pi}{2}$, around $\displaystyle\mathbb{C}P^{n}$, $\displaystyle
m=2n$.

##  2 Preliminaries.

For more details about this section we refer to [1], [2], [3], [8], [9] and
[13]. The complex quadric $\displaystyle Q^{m}$ is the complex hypersurface in
the complex projective space $\displaystyle\mathbb{C}P^{m+1}$ defined by the
equation $\displaystyle z_{1}^{2}+\cdots+z_{m+2}^{2}=0$, where $\displaystyle
z_{1},\ldots,z_{m+2}$ are homogeneous coordinates on
$\displaystyle\mathbb{C}P^{m+1}$. The Kähler structure on
$\displaystyle\mathbb{C}P^{m+1}$ induces canonically a Kähler structure
$\displaystyle(J,G)$ on the complex quadric, where $\displaystyle G$ is a
Riemannian metric induced on $\displaystyle Q^{m}$ by the Fubini-Study metric
of $\displaystyle\mathbb{C}P^{m+1}$.

A point $\displaystyle[z]$ in $\displaystyle\mathbb{C}P^{m+1}$ is given by
$\displaystyle[z]=\\{\lambda z|$ $\displaystyle z\in\mathbb{C}\\}$, where
$\displaystyle z$ is a nonzero vector in $\displaystyle\mathbb{C}^{m+2}$. If
$\displaystyle\bar{z}$ is the conjugated of $\displaystyle z$ , $\displaystyle
T_{[z]}Q^{m}$, for any $\displaystyle[z]\in Q^{m}$, can be identified
canonically with $\displaystyle\mathbb{C}^{m+2}\ominus([z]\oplus[\bar{z}])$,
the orthogonal complement of $\displaystyle[z]\oplus[\bar{z}]$ in
$\displaystyle\mathbb{C}^{m+2}$. $\displaystyle\bar{z}$ is a unit normal
vector of $\displaystyle Q^{m}$ in $\displaystyle\mathbb{C}P^{m+1}$ at the
point $\displaystyle[z]$. The shape operator $\displaystyle A_{\bar{z}}$ of
$\displaystyle Q^{m}$ with respect to the unit normal vector
$\displaystyle\bar{z}$ is given by

$\displaystyle A_{\bar{z}}w=\bar{w}$

for any $\displaystyle w\in T_{[z]}Q^{m}$. Then $\displaystyle A_{\bar{z}}$
defines a real structure on the complex vector space $\displaystyle
T_{[z]}Q^{m}$. The set of all shape operators $\displaystyle
A_{\lambda\bar{z}}$ defines a parallel subbundle of rank 2,
$\displaystyle\mathfrak{A}$, of $\displaystyle End(TQ^{m})$, consisting of all
the real structures on the tangent space of $\displaystyle Q^{m}$. For any
$\displaystyle A\in\mathfrak{A}$, $\displaystyle A^{2}=I$, the identity
endomorphism, and $\displaystyle AJ=-JA$.

A nonzero tangent vector $\displaystyle W\in T_{[z]}Q^{m}$ is called singular
if it is tangent to more than one maximal flat in $\displaystyle Q^{m}$. There
are two types of singular tangent vectors in $\displaystyle Q^{m}$:

  1. 1.

$\displaystyle W$ is $\displaystyle\mathfrak{A}$-principal if there exists a
real structure $\displaystyle A\in\mathfrak{A}$ such that $\displaystyle
AW=W$.

  2. 2.

$\displaystyle W$ is $\displaystyle\mathfrak{A}$-isotropic if there exist
$\displaystyle A\in\mathfrak{A}$ and two orthonormal vectors $\displaystyle
X,Y$ such that $\displaystyle AX=X$, $\displaystyle AY=Y$ and $\displaystyle
W/\|W\|=(X+JY)/\sqrt{2}$.

If $\displaystyle M$ is a real hypersurface in $\displaystyle Q^{m}\;$with
unit local normal vector field $\displaystyle N$ and Levi-Civita connection
$\displaystyle\nabla$, let $\displaystyle S$ denote the shape operator on
$\displaystyle M$ with respect to $\displaystyle N$. For any vector field
$\displaystyle X$ tangent to $\displaystyle M\;$we write $\displaystyle
JX=\phi X+\eta(X)N$. The tangential component of $\displaystyle JX$,
$\displaystyle\phi X$ defines on $\displaystyle M$ a skew-symmetric tensor
field of type (1,1) called the structure tensor. The vector field
$\displaystyle\xi=-JN$ is called the Reeb vector field (or structure vector
field) of $\displaystyle M\;$. Thus we naturally obtain that the 1-form
$\displaystyle\eta$ is given by $\displaystyle\eta(X)=g(X,\xi)$ for any vector
field $\displaystyle X$ tangent to $\displaystyle M\;$. Therefore on
$\displaystyle M$ we have an almost contact metric structure
($\displaystyle\phi,\xi,\eta,g$), where $\displaystyle g$ is the restriction
to $\displaystyle M$ of the metric $\displaystyle G$. Thus, see [4], we have
the following relations

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

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

$\phi\xi=0,\quad\nabla_{X}\xi=\phi SX$ (2.2)

for any $\displaystyle X$ tangent to $\displaystyle M$, where we have applied
that $\displaystyle J$ is parallel respect to the Levi-Civita connection on
$\displaystyle Q^{m}$

The tangent bundle $\displaystyle TM$ of $\displaystyle M$ splits orthogonally
into

$\displaystyle TM=\mathcal{C}\oplus\mathcal{F},$

where $\displaystyle\mathcal{C}=ker(\eta)=\\{X\in TM|g(X,\xi)=0\\}$ is the
maximal complex (holomorphic) subbundle of $\displaystyle TM$ and
$\displaystyle\mathcal{F}=\mathbb{R}\xi$. If we restrict $\displaystyle\phi$
to $\displaystyle\mathcal{C}$ it coincides with the complex structure
$\displaystyle J$.

For a tube M of radius r, $\displaystyle 0<r<\pi/2$, around the totally
geodesic $\displaystyle\mathbb{C}P^{n}$ in $\displaystyle Q^{m}\;$,
$\displaystyle m=2n$, its normal bundle consists of
$\displaystyle\mathfrak{A}$-isotropic singular tangent vectors of
$\displaystyle Q^{m}\;$, see [2].

In the case of a ruled real hypersurface of $\displaystyle Q^{m}$,
$\displaystyle N$ is $\displaystyle\mathfrak{A}$-principal, see [7].

##  3 Proofs of Theorems 1.1 and 1.2.

Suppose that $\displaystyle M$ satisfies $\displaystyle
L_{\xi_{F}}^{(k)}(\xi,X)=0$ for any $\displaystyle X$ tangent to
$\displaystyle M$. This means that $\displaystyle F_{\xi}^{(k)}(\phi
S-S\phi)X-(\phi S-S\phi)F_{\xi}^{(k)}X=0$ for any $\displaystyle X$ tangent to
$\displaystyle M$. Therefore,

$\begin{array}[]{c}g(\phi S\xi,(\phi S-S\phi)X)\xi-\eta((\phi S-S\phi)X)\phi
S\xi-k\phi(\phi S-S\phi)X-g(\phi S\xi,X)\phi S\xi\\\ +\eta(X)(\phi
S-S\phi)\phi S\xi+k(\phi S-S\phi)\phi X=0\end{array}$ (3.1)

for any $\displaystyle X$ tangent to $\displaystyle M$. Suppose first that
$\displaystyle M\;$is Hopf with Reeb function $\displaystyle\alpha$. In this
case (3.1) yields

$\begin{array}[]{c}-k\phi(\phi S-S\phi)X+k(\phi S-S\phi)\phi X=0\end{array}$
(3.2)

for any $\displaystyle X$ tangent to $\displaystyle M$. As $\displaystyle
k\neq 0$, from (3.2) we get $\displaystyle 2SX+2\phi S\phi X=0$ for any
$\displaystyle X\in\mathcal{C}$. If we apply $\displaystyle\phi$ to this
equation we obtain $\displaystyle S\phi X=\phi SX$ for any $\displaystyle
X\in\mathcal{C}$. This yields $\displaystyle S\phi=\phi S$ and the results
follows from Theorem A.

If $\displaystyle M$ is non Hopf, we write $\displaystyle S\xi=\alpha\xi+\beta
U$, where $\displaystyle U$ is a unit vector field in
$\displaystyle\mathcal{C}$, $\displaystyle\alpha$ and $\displaystyle\beta$ are
functions on $\displaystyle M$ and $\displaystyle\beta$ does not vanish at
least on a neighbourhood of a point $\displaystyle p\in M$. The following
calculations will be made on such a neighbourhood. From (3.1) we have

$\begin{array}[]{c}\beta g(SU,X)\xi-\beta g(S\phi U,\phi
X)\xi-2\beta^{2}g(\phi U,X)\phi U-k\phi(\phi S-S\phi)X\\\ +\beta\eta(X)(\phi
S-S\phi)\phi U+k(\phi S-S\phi)\phi X=0\end{array}$ (3.3)

for any $\displaystyle X$ tangent to $\displaystyle M$. Taking $\displaystyle
X=\xi$ in (3.3) it follows that $\displaystyle\beta^{2}\xi+k\beta U+\beta(\phi
S-S\phi)\phi U=0$. Its scalar product with $\displaystyle\xi$ yields
$\displaystyle 2\beta^{2}=0$, which is impossible and finishes the proof of
Theorem 1.1

If we now suppose that $\displaystyle L_{\xi_{F}}^{(k)}(X,Y)=0$, for any
$\displaystyle X\in\mathcal{C}$, $\displaystyle Y$ tangent to $\displaystyle
M$, we obtain

$\begin{array}[]{c}g(\phi SX,\phi SY)\xi-g(\phi SX,S\phi Y)\xi-g((\phi
S-S\phi)Y,\xi)\phi SX\\\ -g(\phi SX,Y)\phi S\xi+\eta(Y)(\phi S-S\phi)\phi
SX=0\\\ \end{array}$ (3.4)

for any $\displaystyle X\in\mathcal{C}$, $\displaystyle Y$ tangent to
$\displaystyle M$. First suppose that $\displaystyle M$ is Hopf with Reeb
function $\displaystyle\alpha$. Then (3.4) becomes

$\begin{array}[]{c}g(SX,SY)\xi-g(\phi SX,S\phi Y)\xi+\eta(Y)(\phi S-S\phi)\phi
SX=0\end{array}$ (3.5)

for any $\displaystyle X\in\mathcal{C}$, $\displaystyle Y$ tangent to
$\displaystyle M$. Take $\displaystyle Y\in\mathcal{C}$ in (3.5). Then
$\displaystyle g(S^{2}X,Y)\xi+g(\phi S\phi SX,Y)\xi=0$ for any $\displaystyle
X,Y\in\mathcal{C}$. This means that $\displaystyle S^{2}X+\phi S\phi SX=0$ for
any $\displaystyle X\in\mathcal{C}$. Apply $\displaystyle\phi$ to obtain
$\displaystyle\phi S^{2}X=S\phi SX$, for any $\displaystyle X\in\mathcal{C}$.
Suppose that $\displaystyle X\in\mathcal{C}$ satisfies $\displaystyle
SX=\lambda X$. Then we obtain $\displaystyle\lambda^{2}\phi X=\lambda S\phi
X$. Therefore, either $\displaystyle\lambda=0$ or $\displaystyle\lambda\neq 0$
and $\displaystyle S\phi X=\lambda\phi X$. This means that for any nonnull
eigenvalue of $\displaystyle S$ in $\displaystyle\mathcal{C}$ the
corresponding eigenspace is $\displaystyle\phi$-invariant. Thus also the
eigenspace corresponding to the possible eigenvalue $\displaystyle 0$ is also
$\displaystyle\phi$-invariant and $\displaystyle S\phi=\phi S$. We conclude as
in Theorem 1.1.

If we suppose now that $\displaystyle M$ is non Hopf, we write, as above,
$\displaystyle S\xi=\alpha\xi+\beta U$, with the same conditions. From (3.4)
we have

$\begin{array}[]{c}g(\phi SX,\phi SY)\xi-g(\phi SX,S\phi Y)\xi-\beta g(Y,\phi
U)S\phi X\\\ -\beta g(\phi SX,Y)\phi U+\eta(Y)(\phi S-S\phi)\phi
SX=0\end{array}$ (3.6)

for any $\displaystyle X\in\mathcal{C}$, $\displaystyle Y$ tangent to
$\displaystyle M$. If we take $\displaystyle Y=\xi$ in (3.6) we get
$\displaystyle\beta g(SU,X)\xi+(\phi S-S\phi)\phi SX=0$, for any
$\displaystyle X\in\mathcal{C}$. Its scalar product with $\displaystyle\xi$
yields $\displaystyle 2\beta g(SU,X)=0$, and as $\displaystyle\beta\neq 0$, we
obtain $\displaystyle g(SU,X)=0$, for any $\displaystyle X\in\mathcal{C}$.
That is,

$\begin{array}[]{c}SU=\beta\xi\end{array}$ (3.7)

If we take $\displaystyle Y=\phi U$ in (3.6) it follows $\displaystyle g(\phi
SX,\phi S\phi U)\xi-\beta\phi SX=0$, for any $\displaystyle X\in\mathcal{C}$.
Its scalar product with $\displaystyle U$ implies $\displaystyle\beta g(S\phi
U,X)=0$ for any $\displaystyle X\in\mathcal{C}$. $\displaystyle\beta$ being
non vanishing implies

$\begin{array}[]{c}S\phi U=0.\end{array}$ (3.8)

We will denote $\displaystyle\mathcal{C}_{U}=\\{X\in\mathcal{C}|$
$\displaystyle g(X,U)=g(X,\phi U)=0\\}$. From (3.7) and (3.8) we know that
$\displaystyle\mathcal{C}_{U}$ is $\displaystyle S$-invariant. Take
$\displaystyle X\in\mathcal{C}_{U}$, $\displaystyle Y=\phi U$ in (3.6). This
gives $\displaystyle-\beta\phi SX=0$. Therefore, $\displaystyle\phi SX=0$ for
any $\displaystyle X\in\mathcal{C}_{U}$, and applying $\displaystyle\phi$ we
get

$\begin{array}[]{c}SX=0\end{array}$ (3.9)

for any $\displaystyle X\in{\mathcal{C}}_{U}$. From (3.7), (3.8) and (3.9) we
obtain that $\displaystyle M$ is ruled. The converse is trivial and we have
finished the proof of Theorem 1.2.

##  4 Proofs of Theorems 1.3 and 1.4.

If we suppose that $\displaystyle T_{X}^{(k)}L_{\xi}Y-L_{\xi}T_{X}^{(k)}Y=0$,
for any $\displaystyle X\in\mathcal{C}$, $\displaystyle Y$ tangent to
$\displaystyle M$, we should have $\displaystyle F_{X}^{(k)}(\phi
S-S\phi)Y-F_{(\phi S-S\phi)Y}^{(k)}X-(\phi S-S\phi)F_{X}^{(k)}Y+(\phi
S-S\phi)F_{Y}^{(k)}X=0$, for any $\displaystyle X\in\mathcal{C}$,
$\displaystyle Y$ tangent to $\displaystyle M$. Therefore,

$\begin{array}[]{c}g(\phi SX,(\phi S-S\phi)Y)\xi-\eta((\phi S-S\phi)Y)\phi
SX-g(\phi S(\phi S-S\phi)Y,X)\xi\\\ +k\eta((\phi S-S\phi)Y)\phi X-g(\phi
SX,Y)\phi S\xi+\eta(Y)(\phi S-S\phi)\phi SX\\\ +g(\phi SY,X)\phi
S\xi-\eta(Y)(\phi S-S\phi)\phi X=0\\\ \end{array}$ (4.1)

for any $\displaystyle X\in\mathcal{C}$, $\displaystyle Y$ tangent to
$\displaystyle M$. Suppose first that $\displaystyle M$ is Hopf and write
$\displaystyle S\xi=\alpha\xi$. Then (4.1) gives

$\begin{array}[]{c}g(SX,SY)\xi-g(\phi SX,S\phi Y)\xi-g(\phi S(\phi
S-S\phi)Y,X)\xi\\\ +\eta(Y)(\phi S-S\phi)\phi SX-k\eta(Y)(\phi S-S\phi)\phi
X=0\end{array}$ (4.2)

for any $\displaystyle X\in\mathcal{C}$, $\displaystyle Y$ tangent to
$\displaystyle M$. If we take $\displaystyle Y=\xi$ in (4.2) we obtain
$\displaystyle(\phi S-S\phi)\phi SX-k(\phi S-S\phi)\phi X=0$, for any
$\displaystyle X\in\mathcal{C}$. That is, $\displaystyle\phi S\phi
SX+S^{2}X-k\phi S\phi X-kSX=0$, for any $\displaystyle X\in\mathcal{C}$. If we
suppose that $\displaystyle X\in\mathcal{C}$ satisfies $\displaystyle
SX=\lambda X$, we have $\displaystyle\lambda\phi S\phi X+\lambda^{2}X-k\phi
S\phi X-k\lambda X=0$. Applying $\displaystyle\phi$ we get
$\displaystyle-\lambda S\phi X+\lambda^{2}\phi X+kS\phi X-k\lambda\phi X=0$.
Then, $\displaystyle(k-\lambda)S\phi X=\lambda(k-\lambda)\phi X$. Thus either
$\displaystyle\lambda=k$ or, if $\displaystyle\lambda\neq k$, $\displaystyle
S\phi X=\lambda\phi X$. We have found that for any eigenvalue
$\displaystyle\lambda\neq k$ in $\displaystyle\mathcal{C}$, the corresponding
eigenspace is $\displaystyle\phi$-invariant. This yields that also the
eigenspace corresponding to $\displaystyle k$ is
$\displaystyle\phi$-invariant. Therefore $\displaystyle\phi S=S\phi$ and
$\displaystyle M$ is an open part of a tube of radius $\displaystyle r$,
$\displaystyle 0<r<\frac{\pi}{2}$ around a totally geodesic
$\displaystyle\mathbb{C}P^{n}\subset Q^{m}$, with $\displaystyle m=2n$.

Let us suppose now that $\displaystyle M$ is non Hopf and write again
$\displaystyle S\xi=\alpha\xi+\beta U$ as in previous section. From (4.1) we
get

$\begin{array}[]{c}g(\phi SX,\phi SY)\xi-g(\phi SX,S\phi Y)\xi-\beta g(Y,\phi
U)\phi SX+g((\phi S-S\phi)Y,S\phi X)\xi\\\ +k\beta g(Y,\phi U)\phi X-\beta
g(\phi SX,Y)\phi U+\eta(Y)(\phi S-S\phi)\phi SX+\beta g(\phi SY,X)\phi U\\\
-k\eta(Y)(\phi S-S\phi)\phi X=0\\\ \end{array}$ (4.3)

for any $\displaystyle X\in\mathcal{C}$, $\displaystyle Y$ tangent to
$\displaystyle M$. If we take $\displaystyle X=\xi$ in (4.3) it follows

$\begin{array}[]{c}\beta g(SU,X)\xi+\beta g(S\phi U,\phi X)\xi+(\phi
S-S\phi)\phi SX+\beta^{2}g(\phi U,X)\phi U\\\ -k(\phi S-S\phi)\phi
X=0\end{array}$ (4.4)

for any $\displaystyle X\in\mathcal{C}$. The scalar product of (4.4) and
$\displaystyle\xi$ gives $\displaystyle\beta g(SU,X)+\beta g(S\phi U,\phi
X)+g(\phi SX,\phi S\xi)-kg(\phi X,\phi S\xi)=0$ for any $\displaystyle
X\in\mathcal{C}$. Therefore, $\displaystyle 2g(SU,X)+g(S\phi U,\phi
X)-kg(U,X)=0$ for any $\displaystyle X\in\mathcal{C}$. For $\displaystyle
X=\phi U$ we obtain

$\begin{array}[]{c}g(SU,\phi U)=0\end{array}$ (4.5)

and for $\displaystyle X=U$ it yields

$\begin{array}[]{c}2g(SU,U)+g(S\phi U,\phi U)=k.\end{array}$ (4.6)

Taking $\displaystyle Y\in\mathcal{C}$ in (4.3) we have

$\begin{array}[]{c}g(\phi SX,\phi SY)\xi-g(\phi SX,S\phi Y)\xi-\beta g(Y,\phi
U)\phi SX+g((\phi S-S\phi)Y,S\phi X)\xi\\\ +k\beta g(Y,\phi U)\phi X-\beta
g(\phi SX,Y)\phi U+\beta g(\phi SY,X)\phi U=0\end{array}$ (4.7)

for any $\displaystyle X,Y\in\mathcal{C}$. Its scalar product with
$\displaystyle\phi U$ gives

$\begin{array}[]{c}-g(Y,\phi U)g(SX,U)+kg(Y,\phi U)g(X,U)-g(\phi SX,Y)+g(\phi
SY,X)=0\end{array}$ (4.8)

for any $\displaystyle X,Y\in\mathcal{C}$. If in (4.8) we take $\displaystyle
Y=U$ we obtain $\displaystyle-g(\phi SX,U)+g(S\phi U,X)=0$ for any
$\displaystyle X\in\mathcal{C}$. Thus

$\begin{array}[]{c}S\phi U+\phi SU=0.\end{array}$ (4.9)

Taking $\displaystyle Y=\phi U$ in (4.8) it follows
$\displaystyle-2g(SU,X)+kg(U,X)+g(\phi S\phi U,X)=0$ for any $\displaystyle
X\in\mathcal{C}$. Then $\displaystyle-2SU+kU+\phi S\phi U$ has no component in
$\displaystyle\mathcal{C}$. That means that $\displaystyle-2SU+kU+\phi S\phi
U=g(-2SU+kU+\phi S\phi U,\xi)\xi=-2\beta\xi$. On the other hand, from (4.9),
$\displaystyle-2\beta\xi=-2SU+kU-\phi^{2}SU=-SU+kU-\beta\xi$. That is,

$\begin{array}[]{c}SU=\beta\xi+kU.\end{array}$ (4.10)

From (4.10) and (4.9),

$\begin{array}[]{c}S\phi U=-k\phi U.\end{array}$ (4.11)

Therefore, $\displaystyle\mathcal{C}_{U}$ is $\displaystyle S$-invariant.
Taking $\displaystyle X\in\mathcal{C}_{U}$ in (4.7) we get

$\begin{array}[]{c}g(S^{2}Y,X)\xi+g(S\phi S\phi Y,X)\xi-g(\phi S(\phi
S-S\phi)Y,X)\xi\\\ +\beta g(S\phi Y,X)\phi U+\beta g(\phi SY,X)\phi
U=0\end{array}$ (4.12)

for any $\displaystyle X\in\mathcal{C}$, $\displaystyle Y\in\mathcal{C}_{U}$.
Its scalar product with $\displaystyle\phi U$ yields $\displaystyle g(S\phi
Y,X)+g(\phi SY,X)=0$, for any $\displaystyle Y\in\mathcal{C}_{U}$,
$\displaystyle X\in\mathcal{C}$. Then

$\begin{array}[]{c}S\phi Y+\phi SY=0\end{array}$ (4.13)

for any $\displaystyle Y\in\mathcal{C}_{U}$.

Taking $\displaystyle Y=\phi U$ in (4.3) we obtain $\displaystyle g(SX,S\phi
U)\xi+g(SU,\phi SX)\xi-\beta\phi SX+g((\phi S-S\phi)\phi U,S\phi
X)\xi+k\beta\phi X-\beta g(SX,U)\phi U+\beta g(\phi S\phi U,X)\phi U=0$, for
any $\displaystyle X\in\mathcal{C}$. If $\displaystyle X\in\mathcal{C}_{U}$
this yields $\displaystyle-\beta\phi SX+k\beta\phi X=0$. As we suppose
$\displaystyle\beta\neq 0$, $\displaystyle SX=kX$, for any $\displaystyle
X\in\mathcal{C}_{U}$, but then (4.13) yields $\displaystyle 2k=0$, which is
impossible, finishing the proof of Theorem 1.3.

If now we suppose that $\displaystyle M$ is non Hopf and $\displaystyle
L_{\xi_{T}}^{(k)}(\xi,X)=0$ for any $\displaystyle X$ tangent to
$\displaystyle M$, we obtain

$\begin{array}[]{c}\beta g(\phi U,(\phi S-S\phi)X)\xi-2\beta^{2}(X,\phi U)\phi
U-k\phi(\phi S-S\phi)X\\\ +\phi S(\phi S-S\phi)X+\beta\eta(X)(\phi
S-S\phi)\phi U+k(\phi S-S\phi)\phi X\\\ -(\phi S-S\phi)\phi SX=0\end{array}$
(4.14)

for any $\displaystyle X$ tangent to $\displaystyle M$. Taking $\displaystyle
X=\xi$ in (4.14) we get $\displaystyle\beta^{2}\xi-k\phi^{2}S\xi+\phi S\phi
S\xi+\beta(\phi S-S\phi)\phi U-\beta(\phi S-S\phi)\phi U=0$. Therefore,
$\displaystyle\beta^{2}\xi+k\beta U+\beta\phi S\phi U=0$. Its scalar product
with $\displaystyle\xi$ yields $\displaystyle\beta=0$, a contradiction that
finishes the proof of Theorem 1.4.

In order to prove Corollary 1.2, if $\displaystyle M$ satisfies $\displaystyle
L_{\xi_{T}}^{(k)}\equiv 0$, from Theorems 1.3 and 1.4, $\displaystyle M$ must
be Hopf. Now from Theorem 1.3 $\displaystyle\phi S=S\phi$, that is,
$\displaystyle L_{\xi}\equiv 0$. Therefore $\displaystyle
L_{\xi_{T}}^{(k)}(\xi,X)=0$, for any nonnull real number $\displaystyle k$ and
any $\displaystyle X$ tangent to $\displaystyle M$ and the proof finishes.

## Acknowledgements

First author is supported by Projects PID 2020-11 6126GB-I00 from MICINN and
PY20-01391 from Junta de Andalucía. Third author by grant Proj. No.
NRF-2018-R1D1A1B-05040381 from National Research Foundation of Korea.

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* Juan de Dios Pérez<EMAIL_ADDRESS>
Departamento de Geometría y Topología and IMAG (Instituto de Matemáticas)
Universidad de Granada
18071 Granada
Spain

* David Pérez-López<EMAIL_ADDRESS>
18003 Granada
Spain

* Young Jin Suh<EMAIL_ADDRESS>
Kyungpook National University
College of Natural Sciences
Department of Mathematics
and Research Institute of Real and Complex Manifolds
Daegu 41566, Republic of Korea