Comparison of Differential Operators with Lie Derivative of Three-Dimensional Real Hypersurfaces in Non-Flat Complex Space Forms

Abstract

In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field ${\displaystyle P=\varphi A-A\varphi }$ is given and three-dimensional real hypersurfaces in non-flat complex space forms whose tensor field ${\displaystyle P}$ satisfies geometric conditions are classified.

1. Introduction

A real hypersurface is a submanifold of a Riemannian manifold with a real co-dimensional one. Among the Riemannian manifolds, it is of great interest in the area of Differential Geometry to study real hypersurfaces in complex space forms. A complex space form is a Kähler manifold of dimension ${\displaystyle n}$ and constant holomorphic sectional curvature c. In addition, complete and simply connected complex space forms are analytically isometric to complex projective space ${\displaystyle \mathbb{C}{P}^n}$ if ${\displaystyle c>0}$, to complex Euclidean space ${\displaystyle {\mathbb{C}}^n}$ if ${\displaystyle c=0}$, or to complex hyperbolic space ${\displaystyle \mathbb{C}{H}^n}$ if ${\displaystyle c<0}$. The notion of non-flat complex space form refers to complex projective and complex hyperbolic space when it is not necessary to distinguish between them and is denoted by ${\displaystyle M_n\left(c\right),n\ge 2}$.

Let ${\displaystyle J}$ be the Kähler structure and ${\displaystyle \tilde{\nabla }}$ the Levi–Civita connection of the non-flat complex space form ${\displaystyle M_n\left(c\right),n\ge 2}$. Consider ${\displaystyle M}$ a connected real hypersurface of ${\displaystyle M_n\left(c\right)}$ and ${\displaystyle N}$ a locally defined unit normal vector field on ${\displaystyle M}$. The Kähler structure induces on ${\displaystyle M}$ an almost contact metric structure ${\displaystyle (\varphi ,\xi ,\eta ,g)}$. The latter consists of a tensor field of type (1, 1) ${\displaystyle \varphi }$ called structure tensor field, a one-form ${\displaystyle \eta }$, a vector field ${\displaystyle \xi }$ given by ${\displaystyle \xi =-JN}$ known as the structure vector field of ${\displaystyle M}$ and ${\displaystyle g}$, which is the induced Riemannian metric on ${\displaystyle M}$ by ${\displaystyle G}$. Among real hypersurfaces in non-flat complex space forms, the class of Hopf hypersurfaces is the most important. A Hopf hypersurface is a real hypersurface whose structure vector field ${\displaystyle \xi }$ is an eigenvector of the shape operator ${\displaystyle A}$ of ${\displaystyle M}$ .

Takagi initiated the study of real hypersurfaces in non-flat complex space forms. He provided the classification of homogeneous real hypersurfaces in complex projective space ${\displaystyle \mathbb{C}{P}^n}$ and divided them into five classes (${\displaystyle A}$), (${\displaystyle B}$), (${\displaystyle C}$), (${\displaystyle D}$) and (${\displaystyle E}$) (see [1,2,3]). Later, Kimura proved that homogeneous real hypersurfaces in complex projective space are the unique Hopf hypersurfaces with constant principal curvatures, i.e., the eigenvalues of the shape operator ${\displaystyle A}$ are constant (see [4]). Among the above real hypersurfaces, the three-dimensional real hypersurfaces in ${\displaystyle \mathbb{C}{P}^2}$ are geodesic hyperspheres of radius ${\displaystyle r}$, ${\displaystyle 0<r<\frac{\pi }{2}}$, called real hypersurfaces of type (${\displaystyle A}$) and tubes of radius ${\displaystyle r}$, ${\displaystyle 0<r<\frac{\pi }{4}}$, over the complex quadric called real hypersurfaces of type (${\displaystyle B}$). Table 1 includes the values of the constant principal curvatures corresponding to the real hypersurfaces above (see [1,2]).

Table 1. Principal curvatures of real hypersurfaces in ${\displaystyle \mathbb{C}{P}^2.}$

Type	${\displaystyle \mathit{\alpha }}$	${\displaystyle {\mathit{\lambda }}_1}$	${\displaystyle \mathit{\nu }}$	${\displaystyle {\mathit{m}}_{\mathit{\alpha }}}$	${\displaystyle {\mathit{m}}_{{\mathit{\lambda }}_1}}$	${\displaystyle {\mathit{m}}_{\mathit{\nu }}}$
(${\displaystyle A}$)	${\displaystyle 2cot\left(2r\right)}$	${\displaystyle cot\left(r\right)}$	-	1	2	-
(${\displaystyle B}$)	2${\displaystyle cot\left(2r\right)}$	${\displaystyle cot(r-\frac{\pi }{4})}$	${\displaystyle -tan(r-\frac{\pi }{4})}$	1	1	1

The study of Hopf hypersurfaces with constant principal curvatures in complex hyperbolic space ${\displaystyle \mathbb{C}{H}^n,n\ge 2}$, was initiated by Montiel in [5] and completed by Berndt in [6]. They are divided into two types: type (${\displaystyle A}$), which are open subsets of horospheres (${\displaystyle A_0}$), geodesic hyperspheres (${\displaystyle A_{1,0}}$), or tubes over totally geodesic complex hyperbolic hyperplane ${\displaystyle \mathbb{C}{H}^{n-1}}$ (${\displaystyle A_{1,1}}$) and type (${\displaystyle B}$), which are open subsets of tubes over totally geodesic real hyperbolic space ${\displaystyle \mathbb{R}{H}^n}$. Table 2 includes the values of the constant principal curvatures corresponding to above real hypersurfaces for ${\displaystyle n=2}$ (see [6]).

Table 2. Principal curvatures of real hypersurfaces in ${\displaystyle \mathbb{C}{H}^2.}$

Type	${\displaystyle \mathit{\alpha }}$	${\displaystyle \mathit{\lambda }}$	${\displaystyle \mathit{\nu }}$	${\displaystyle {\mathit{m}}_{\mathit{\alpha }}}$	${\displaystyle {\mathit{m}}_{\mathit{\lambda }}}$	${\displaystyle {\mathit{m}}_{\mathit{\nu }}}$
(${\displaystyle A_0}$)	2	1	-	1	2	-
(${\displaystyle A_{1,1}}$)	2${\displaystyle coth\left(2r\right)}$	${\displaystyle coth\left(r\right)}$	-	1	2	-
(${\displaystyle A_{1,2}}$)	2${\displaystyle coth\left(2r\right)}$	${\displaystyle tanh\left(r\right)}$	-	1	2	-
(${\displaystyle B}$)	2${\displaystyle tanh\left(2r\right)}$	${\displaystyle tanh\left(r\right)}$	${\displaystyle coth\left(r\right)}$	1	1	1

The Levi–Civita connection ${\displaystyle \tilde{\nabla }}$ of the non-flat complex space form ${\displaystyle M_n\left(c\right),n\ge 2}$ induces on ${\displaystyle M}$ a Levi–Civita connection ${\displaystyle \nabla }$. Apart from the last one, Cho in [7,8] introduces the notion of the k-th generalized Tanaka–Webster connection ${\displaystyle {\widehat{\nabla }}^{\left(k\right)}}$ on a real hypersurface in non-flat complex space form given by

$$\begin{array}{c}{\widehat{\nabla }}_X^{\left(k\right)}Y={\nabla }_{X}Y+g(\varphi AX,Y)\xi -\eta \left(Y\right)\varphi AX-k\eta \left(X\right)\varphi Y,\end{array}$$

(1)

for all ${\displaystyle X}$, ${\displaystyle Y}$ tangent to ${\displaystyle M}$, where ${\displaystyle k}$ is a nonnull real number. The latter is an extension of the definition of generalized Tanaka–Webster connection for contact metric manifolds given by Tanno in [9] and satisfying the relation

$$\begin{array}{c}{\widehat{\nabla }}_{X}Y={\nabla }_{X}Y+\left({\nabla }_X\eta \right)\left(Y\right)\xi -\eta \left(Y\right){\nabla }_X\xi -\eta \left(X\right)\varphi Y.\end{array}$$

The following relations hold:

$${\widehat{\nabla }}^{\left(k\right)}\eta =0,{\widehat{\nabla }}^{\left(k\right)}\xi =0,{\widehat{\nabla }}^{\left(k\right)}g=0,{\widehat{\nabla }}^{\left(k\right)}\varphi =0.$$

In particular, if the shape operator of a real hypersurface satisfies ${\displaystyle \varphi A+A\varphi =2k\varphi }$, the generalized Tanaka–Webster connection coincides with the Tanaka–Webster connection.

The k-th Cho operator on ${\displaystyle M}$ associated with the vector field ${\displaystyle X}$ is denoted by ${\displaystyle {\widehat{F}}_X^{\left(k\right)}}$ and given by

$$\begin{array}{c}\hfill {\widehat{F}}_X^{\left(k\right)}Y=g(\varphi AX,Y)\xi -\eta \left(Y\right)\varphi AX-k\eta \left(X\right)\varphi Y,\end{array}$$

(2)

for any ${\displaystyle Y}$ tangent to ${\displaystyle M}$. Then, the torsion of the k-th generalized Tanaka–Webster connection ${\displaystyle {\widehat{\nabla }}^{\left(k\right)}}$ is given by

$$T^{\left(k\right)}(X,Y)={\widehat{F}}_X^{\left(k\right)}Y-{\widehat{F}}_Y^{\left(k\right)}X,$$

for any ${\displaystyle X}$, ${\displaystyle Y}$ tangent to ${\displaystyle M}$. Associated with the vector field ${\displaystyle X}$, the k-th torsion operator ${\displaystyle T_X^{\left(k\right)}}$ is defined and given by

$$T_X^{\left(k\right)}Y=T^{\left(k\right)}(X,Y),$$

for any ${\displaystyle Y}$ tangent to ${\displaystyle M}$.

The existence of Levi–Civita and k-th generalized Tanaka–Webster connections on a real hypersurface implies that the covariant derivative can be expressed with respect to both connections. Let ${\displaystyle K}$ be a tensor field of type (1, 1); then, the symbols ${\displaystyle \nabla K}$ and ${\displaystyle {\widehat{\nabla }}^{\left(k\right)}K}$ are used to denote the covariant derivatives of ${\displaystyle K}$ with respect to the Levi–Civita and the k-th generalized Tanaka–Webster connection, respectively. Furthermore, the Lie derivative of a tensor field ${\displaystyle K}$ of type (1, 1) with respect to Levi–Civita connection ${\displaystyle \mathcal{L}K}$ is given by

$$\begin{array}{c}\hfill \left({\mathcal{L}}_{X}K\right)Y={\nabla }_X\left(KY\right)-{\nabla }_{KY}X-K{\nabla }_{X}Y+K{\nabla }_{Y}X,\end{array}$$

(3)

for all ${\displaystyle X,Y}$ tangent to ${\displaystyle M}$ . Another first order differential operator of a tensor field ${\displaystyle K}$ of type (1, 1) with respect to the k-th generalized Tanaka–Webster connection ${\displaystyle {\widehat{\mathcal{L}}}^{(k)}K}$ is defined and it is given by

$$\begin{array}{c}\hfill ({\widehat{\mathcal{L}}}_X^{\left(k\right)}K)Y={\widehat{\nabla }}_X^{\left(k\right)}\left(KY\right)-{\widehat{\nabla }}_{KY}^{\left(k\right)}X-K({\widehat{\nabla }}_X^{\left(k\right)}Y)+K({\widehat{\nabla }}_Y^{\left(k\right)}X),\end{array}$$

(4)

for all ${\displaystyle X,Y}$ tangent to ${\displaystyle M}$ .

Due to the existence of the above differential operators and derivatives, the following questions come up

1.

Are there real hypersurfaces in non-flat complex space forms whose derivatives with respect to different connections coincide?

2.

Are there real hypersurfaces in non-flat complex space forms whose differential operator ${\displaystyle {\widehat{\mathcal{L}}}^{\left(k\right)}}$ coincides with derivatives with respect to different connections?

The first answer is obtained in [10], where the classification of real hypersurfaces in complex projective space ${\displaystyle \mathbb{C}{P}^n}$, ${\displaystyle n\ge 3}$, whose covariant derivative of the shape operator with respect to the Levi–Civita connection coincides with the covariant derivative of it with respect to the k-th generalized Tanaka–Webster connection is provided, i.e., ${\displaystyle {\nabla }_{X}A={\widehat{\nabla }}_X^{\left(k\right)}A}$, where ${\displaystyle X}$ is any vector field on ${\displaystyle M}$. Next, in [11], real hypersurfaces in complex projective space ${\displaystyle \mathbb{C}{P}^n}$, ${\displaystyle n\ge 3}$, whose Lie derivative of the shape operator coincides with the operator ${\displaystyle {\widehat{\mathcal{L}}}^{\left(k\right)}}$ are studied, i.e., ${\displaystyle {\mathcal{L}}_{X}A={\widehat{\mathcal{L}}}_X^{\left(k\right)}A}$, where ${\displaystyle X}$ is any vector field on ${\displaystyle M}$. Finally, in [12], the problem of classifying three-dimensional real hypersurfaces in non-flat complex space forms ${\displaystyle M_2\left(c\right)}$, for which the operator ${\displaystyle {\widehat{\mathcal{L}}}^{\left(k\right)}}$ applied to the shape operator coincides with the covariant derivative of it, has been studied, i.e., ${\displaystyle {\widehat{\mathcal{L}}}_X^{\left(k\right)}A={\nabla }_{X}A}$, for any vector field ${\displaystyle X}$ tangent to ${\displaystyle M}$.

In this paper, the condition ${\displaystyle {\mathcal{L}}_{X}A={\widehat{\mathcal{L}}}_X^{\left(k\right)}A}$, where ${\displaystyle X}$ is any vector field on ${\displaystyle M}$ is studied in the case of three-dimensional real hypersurfaces in ${\displaystyle M_2\left(c\right)}$.

The aim of the present paper is to complete the work of [11] in the case of three-dimensional real hypersurfaces in non-flat complex space forms ${\displaystyle M_2\left(c\right)}$. The equality ${\displaystyle {\mathcal{L}}_{X}A={\widehat{\mathcal{L}}}_X^{\left(k\right)}A}$ is equivalent to the fact that ${\displaystyle T_X^{\left(k\right)}A=A{T}_X^{\left(k\right)}}$. Thus, the eigenspaces of ${\displaystyle A}$ are preserved by the k-th torsion operator ${\displaystyle T_X^{\left(k\right)}}$, for any ${\displaystyle X}$ tangent to ${\displaystyle M}$. First, three-dimensional real hypersurfaces in ${\displaystyle M_2\left(c\right)}$ whose shape operator ${\displaystyle A}$ satisfies the following relation:

$$\begin{array}{c}\hfill {\widehat{\mathcal{L}}}_X^{\left(k\right)}A={\mathcal{L}}_{X}A,\end{array}$$

(5)

for any ${\displaystyle X}$ orthogonal to ${\displaystyle \xi }$ are studied and the following Theorem is proved:

Theorem 1.

There do not exist real hypersurfaces in ${\displaystyle M_2\left(c\right)}$ whose shape operator satisfies relation (5).

Next, three-dimensional real hypersurfaces in ${\displaystyle M_2\left(c\right)}$ whose shape operator satisfies the following relation are studied:

$$\begin{array}{c}\hfill {\widehat{\mathcal{L}}}_{\xi }^{\left(k\right)}A={\mathcal{L}}_{\xi }A,\end{array}$$

(6)

and the following Theorem is provided:

Theorem 2.

Every real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose shape operator satisfies relation (6) is locally congruent to a real hypersurface of type (${\displaystyle A}$).

As an immediate consequence of the above theorems, it is obtained that

Corollary 1.

There do not exist real hypersurfaces in ${\displaystyle M_2\left(c\right)}$ such that ${\displaystyle {\widehat{\mathcal{L}}}_X^{\left(k\right)}A={\mathcal{L}}_{X}A}$, for all ${\displaystyle X}$ ${\displaystyle \in }$ ${\displaystyle TM}$.

Next, the following tensor field ${\displaystyle P}$ of type (1, 1) is introduced:

$$PX=\varphi AX-A\varphi X,$$

for any vector field ${\displaystyle X}$ tangent to ${\displaystyle M}$. The relation ${\displaystyle P=0}$ implies that the shape operator commutes with the structure tensor ${\displaystyle \varphi }$. Real hypersurfaces whose shape operator ${\displaystyle A}$ commutes with the structure tensor ${\displaystyle \varphi }$ have been studied by Okumura in the case of ${\displaystyle \mathbb{C}{P}^n}$, ${\displaystyle n\ge 2}$, (see [13]) and by Montiel and Romero in the case of ${\displaystyle \mathbb{C}{H}^n}$, ${\displaystyle n\ge 2}$ (see [14]). The following Theorem provides the above classification of real hypersurfaces in ${\displaystyle M_n\left(c\right)}$, ${\displaystyle n\ge 2}$.

Theorem 3.

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

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

(A₁)

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

(A₂)

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

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

(A₀)

a horosphere in ${\displaystyle \mathbb{C}{H}^n}$, i.e., a Montiel tube,

(A₁)

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

(A₂)

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

Remark 1.

In the case of three-dimensional real hypersurfaces in ${\displaystyle M_2\left(c\right)}$, real hypersurfaces of type (${\displaystyle A_2}$) do not exist.

It is interesting to study real hypersurfaces in non-flat complex spaces forms, whose tensor field ${\displaystyle P}$ satisfies certain geometric conditions. We begin by studying three-dimensional real hypersurfaces in ${\displaystyle M_2\left(c\right)}$ whose tensor field ${\displaystyle P}$ satisfies the relation

$$\begin{array}{c}\hfill ({\widehat{\mathcal{L}}}_X^{\left(k\right)}P)Y=\left({\mathcal{L}}_{X}P\right)Y,\end{array}$$

(7)

for any vector fields ${\displaystyle X}$, ${\displaystyle Y}$ tangent to ${\displaystyle M}$.

First, the following Theorem is proved:

Theorem 4.

Every real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose tensor field P satisfies relation (8) for any ${\displaystyle X}$ orthogonal to ${\displaystyle \xi }$ and ${\displaystyle Y}$ ${\displaystyle \in }$ ${\displaystyle TM}$ is locally congruent to a real hypersurface of type (A).

Next, we study three-dimensional real hypersurfaces in ${\displaystyle M_2\left(c\right)}$ whose tensor field ${\displaystyle P}$ satisfies relation (7) for ${\displaystyle X=\xi }$, i.e.,

$$\begin{array}{c}\hfill (\widehat{{\mathcal{L}}_{\xi }^{\left(k\right)}}P)Y=\left({\mathcal{L}}_{\xi }P\right)Y,\end{array}$$

(8)

for any vector field ${\displaystyle Y}$ tangent to ${\displaystyle M}$. Then, the following Theorem is proved:

Theorem 5.

Every real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose tensor field P satisfies relation (8) is a Hopf hypersurface. In the case of ${\displaystyle \mathbb{C}{P}^2,}$ M is locally congruent to a real hypersurface of type (A) or to a real hypersurface of type (B) with ${\displaystyle \alpha =-2k}$ and in the case of ${\displaystyle \mathbb{C}{H}^2}$ M is a locally congruent either to a real hypersurface of type (A) or to a real hypersurface of type (B) with ${\displaystyle \alpha =\frac{4}{k}}$.

This paper is organized as follows: in Section 2, basic relations and theorems concerning real hypersurfaces in non-flat complex space forms are presented. In Section 3, analytic proofs of Theorems 1 and 2 are provided. Finally, in Section 4, proofs of Theorems 4 and 5 are given.

2. Preliminaries

Throughout this paper, all manifolds, vector fields, etc. are considered of class ${\displaystyle C^{\infty }}$ and all manifolds are assumed to be connected.

The non-flat complex space form ${\displaystyle M_n\left(c\right)}$, ${\displaystyle n\ge 2}$ is equipped with a Kähler structure ${\displaystyle J}$ and ${\displaystyle G}$ is the Kählerian metric. The constant holomorphic sectional curvature ${\displaystyle c}$ in the case of complex projective space ${\displaystyle \mathbb{C}{P}^n}$ is ${\displaystyle c=4}$ and in the case of complex hyperbolic space ${\displaystyle \mathbb{C}{H}^n}$ is ${\displaystyle c=-4}$. The Levi–Civita connection of the non-flat complex space form is denoted by ${\displaystyle \overline{\nabla }}$.

Let ${\displaystyle M}$ be a connected real hypersurface immersed in ${\displaystyle M_n\left(c\right)}$, ${\displaystyle n\ge 2}$, without boundary and ${\displaystyle N}$ be a locally defined unit normal vector field on ${\displaystyle M}$. The shape operator ${\displaystyle A}$ of the real hypersurface ${\displaystyle M}$ with respect to the vector field ${\displaystyle N}$ is given by

$${\overline{\nabla }}_{X}N=-AX.$$

The Levi–Civita connection ${\displaystyle \nabla }$ of the real hypersurface ${\displaystyle M}$ satisfies the relation

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+g(AX,Y)N.$$

The Kähler structure of the ambient space induces on ${\displaystyle M}$ an almost contact metric structure ${\displaystyle (\varphi ,\xi ,\eta ,g)}$ in the following way: any vector field ${\displaystyle X}$ tangent to ${\displaystyle M}$ satisfies the relation

$$JX=\varphi X+\eta \left(X\right)N.$$

The tangential component of the above relation defines on ${\displaystyle M}$ a skew-symmetric tensor field of type (1, 1) denoted by ${\displaystyle \varphi }$ known as the structure tensor. The structure vector field ${\displaystyle \xi }$ is defined by ${\displaystyle \xi =-JN}$ and the 1-form ${\displaystyle \eta }$ is given by ${\displaystyle \eta \left(X\right)=g(X,\xi )}$ for any vector field ${\displaystyle X}$ tangent to ${\displaystyle M}$. The elements of the almost contact structure satisfy the following relation:

$${\varphi }^{2}X=-X+\eta \left(X\right)\xi ,\eta \left(\xi \right)=1,g(\varphi X,\varphi Y)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right)$$

(9)

for all tangent vectors ${\displaystyle X,Y}$ to ${\displaystyle M}$. Relation (9) implies

$$\begin{array}{c}\hfill \varphi \xi =0,\eta \left(X\right)=g(X,\xi ).\end{array}$$

Because of ${\displaystyle \overline{\nabla }J=0}$, it is obtained

$$\begin{array}{c}\hfill \left({\nabla }_X\varphi \right)Y=\eta \left(Y\right)AX-g(AX,Y)\xi and{\nabla }_X\xi =\varphi AX\end{array}$$

for all ${\displaystyle X,Y}$ tangent to ${\displaystyle M}$. Moreover, the Gauss and Codazzi equations of the real hypersurface are respectively given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill R(X,Y)Z=\frac{c}{4}[g(Y,Z)X-g(X,Z)Y+g(\varphi Y,Z)\varphi X-g(\varphi X,Z)\varphi Y\\ \hfill -2g(\varphi X,Y)\varphi Z]+g(AY,Z)AX-g(AX,Z)AY,\end{array}\end{array}$$

(10)

and

$$\begin{array}{c}\hfill \left({\nabla }_{X}A\right)Y-\left({\nabla }_{Y}A\right)X=\frac{c}{4}[\eta \left(X\right)\varphi Y-\eta \left(Y\right)\varphi X-2g(\varphi X,Y)\xi ],\end{array}$$

(11)

for all vectors ${\displaystyle X,Y,Z}$ tangent to ${\displaystyle M}$, where ${\displaystyle R}$ is the curvature tensor of ${\displaystyle M}$.

The tangent space ${\displaystyle T_{p}M}$ at every point ${\displaystyle p}$ ${\displaystyle \in }$${\displaystyle M}$ is decomposed as

$$\begin{array}{c}\hfill T_{p}M=span\left\{\xi \right\}\oplus \mathbb{D},\end{array}$$

(12)

where ${\displaystyle \mathbb{D}=ker\eta =\{X\in T_{p}M:\eta \left(X\right)=0\}}$ and is called (maximal) holomorphic distribution (if ${\displaystyle n\ge 3}$).

Next, the following results concern any non-Hopf real hypersurface ${\displaystyle M}$ in ${\displaystyle M_2\left(c\right)}$ with local orthonormal basis ${\displaystyle \{U,\varphi U,\xi \}}$ at a point ${\displaystyle p}$ of ${\displaystyle M}$.

Lemma 1.

Let M be a non-Hopf real hypersurface in ${\displaystyle M_2\left(c\right)}$. The following relations hold on M:

$$\begin{array}{c}AU=\gamma U+\delta \varphi U+\beta \xi ,A\varphi U=\delta U+\mu \varphi U,A\xi =\alpha \xi +\beta U,\hfill \\ {\nabla }_U\xi =-\delta U+\gamma \varphi U,{\nabla }_{\varphi U}\xi =-\mu U+\delta \varphi U,{\nabla }_{\xi }\xi =\beta \varphi U,\hfill \\ {\nabla }_{U}U={\kappa }_1\varphi U+\delta \xi ,{\nabla }_{\varphi U}U={\kappa }_2\varphi U+\mu \xi ,{\nabla }_{\xi }U={\kappa }_3\varphi U,\hfill \\ {\nabla }_U\varphi U=-{\kappa }_{1}U-\gamma \xi ,{\nabla }_{\varphi U}\varphi U=-{\kappa }_{2}U-\delta \xi ,{\nabla }_{\xi }\varphi U=-{\kappa }_{3}U-\beta \xi ,\hfill \end{array}$$

(13)

where ${\displaystyle \alpha ,\beta ,\gamma ,\delta ,\mu ,{\kappa }_1,{\kappa }_2,{\kappa }_3}$ are smooth functions on M and ${\displaystyle \beta \ne 0}$.

Remark 2.

The proof of Lemma 1 is included in [15].

The Codazzi equation for ${\displaystyle X}$ ${\displaystyle \in }$ ${\displaystyle \{U,\varphi U\}}$ and ${\displaystyle Y=\xi }$ implies, because of Lemma 1, the following relations:

$$\begin{array}{ccc}\hfill \xi \delta & =& \alpha \gamma +\beta {\kappa }_1+{\delta }^2+\mu {\kappa }_3+\frac{c}{4}-\gamma \mu -\gamma {\kappa }_3-{\beta }^2,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \xi \mu & =& \alpha \delta +\beta {\kappa }_2-2\delta {\kappa }_3,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill \left(\varphi U\right)\alpha & =& \alpha \beta +\beta {\kappa }_3-3\beta \mu ,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill \left(\varphi U\right)\beta & =& \alpha \gamma +\beta {\kappa }_1+2{\delta }^2+\frac{c}{2}-2\gamma \mu +\alpha \mu ,\hfill \end{array}$$

(17)

and for ${\displaystyle X=U}$ and ${\displaystyle Y=\varphi U}$

$$\begin{array}{ccc}\hfill U\delta -\left(\varphi U\right)\gamma & =& \mu {\kappa }_1-{\kappa }_1\gamma -\beta \gamma -2\delta {\kappa }_2-2\beta \mu .\hfill \end{array}$$

(18)

The following Theorem refers to Hopf hypersurfaces. In the case of complex projective space ${\displaystyle \mathbb{C}{P}^n,}$ it is given by Maeda [16], and, in the case of complex hyperbolic space ${\displaystyle \mathbb{C}{H}^n,}$ it is given by Ki and Suh [17] (see also Corollary 2.3 in [18]).

Theorem 6.

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

(i)

${\displaystyle \alpha =g(A\xi ,\xi )}$ is constant.

(ii)

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

$$\begin{array}{c}\hfill (\lambda -\frac{\alpha }{2})A\varphi W=(\frac{\lambda \alpha }{2}+\frac{c}{4})\varphi W.\end{array}$$

(iii)

If the vector field ${\displaystyle W}$ satisfies ${\displaystyle AW=\lambda W}$ and ${\displaystyle A\varphi W=\nu \varphi W,}$ then

$$\begin{array}{c}\hfill \lambda \nu =\frac{\alpha }{2}(\lambda +\nu )+\frac{c}{4}.\end{array}$$

(19)

Remark 3.

Let M be a three-dimensional Hopf hypersurface in ${\displaystyle M_2\left(c\right)}$. Since M is a Hopf hypersurface relation ${\displaystyle A\xi =\alpha \xi }$, it holds when ${\displaystyle \alpha =constant}$. At any point ${\displaystyle p}$${\displaystyle \in }$${\displaystyle M}$, we consider a unit vector field ${\displaystyle W}$ ${\displaystyle \in }$${\displaystyle \mathbb{D}}$ such that ${\displaystyle AW=\lambda W}$. Then, the unit vector field ${\displaystyle \varphi W}$ is orthogonal to ${\displaystyle W}$ and ${\displaystyle \xi }$ and relation ${\displaystyle A\varphi W=\nu \varphi W}$ holds. Therefore, at any point ${\displaystyle p}$ ${\displaystyle \in }$ ${\displaystyle M}$, we can consider the local orthonormal frame ${\displaystyle \{W,\varphi W,\xi \}}$ and the shape operator satisfies the above relations.

3. Proofs of Theorems 1 and 2

Suppose that ${\displaystyle M}$ is a real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose shape operator satisfies relation (5), which because of the relation of k-th generalized Tanaka-Webster connection (1) becomes

$$\begin{array}{c}g((A\varphi A+A^2\varphi )X,Y)\xi -g((A\varphi +\varphi A)X,Y)A\xi +k\eta \left(AY\right)\varphi X+\eta \left(Y\right)A\varphi AX\hfill \\ -\eta \left(AY\right)\varphi AX-k\eta \left(Y\right)A\varphi X=0,\hfill \end{array}$$

(20)

for any ${\displaystyle X}$ ${\displaystyle \in }$ ${\displaystyle \mathbb{D}}$ and for all ${\displaystyle Y}$ ${\displaystyle \in }$ ${\displaystyle TM}$.

Let ${\displaystyle \mathcal{N}}$ be the open subset of ${\displaystyle M}$ such that

$$\mathcal{N}=\{p\in M:\beta \ne 0,\mathrm{in}\mathrm{a}\mathrm{neighborhood}\mathrm{of}p\}.$$

The inner product of relation (20) for ${\displaystyle Y=\xi }$ with ${\displaystyle \xi }$ due to relation (13) implies ${\displaystyle \delta =0}$ and the shape operator on the local orthonormal basis ${\displaystyle \{U,\varphi U,\xi \}}$ becomes

$$\begin{array}{c}\hfill A\xi =\alpha \xi +\beta U,AU=\gamma U+\beta \xi andA\varphi U=\mu \varphi U.\end{array}$$

(21)

Relation (20) for ${\displaystyle X=Y=U}$ and ${\displaystyle X=\varphi U}$ and ${\displaystyle Y=\xi }$ due to (21) yields, respectively,

$$\begin{array}{c}\hfill \gamma =kand\mu =0.\end{array}$$

(22)

Differentiation of ${\displaystyle \gamma =k}$ with respect to ${\displaystyle \varphi U}$ taking into account that ${\displaystyle k}$ is a nonzero real number implies ${\displaystyle \left(\varphi U\right)\gamma =0}$. Thus, relation (18) results, because of ${\displaystyle \delta =\mu =0}$, in ${\displaystyle {\kappa }_1=-\beta }$. Furthermore, relations (14)–(17) due to ${\displaystyle \delta =0}$ and relation (22) become

$$\begin{array}{ccc}\hfill \alpha k+\frac{c}{4}& =& 2{\beta }^2+k{\kappa }_3,\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\kappa }_2& =& 0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill \left(\varphi U\right)\alpha & =& \beta (\alpha +{\kappa }_3),\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill \left(\varphi U\right)\beta & =& \alpha k-{\beta }^2+\frac{c}{2}.\hfill \end{array}$$

(26)

The inner product of Codazzi equation (11) for ${\displaystyle X=U}$ and ${\displaystyle Y=\xi }$ with ${\displaystyle U}$ and ${\displaystyle \xi }$ implies because of ${\displaystyle \delta =0}$ and relation (21),

$$\begin{array}{c}\hfill U\alpha =U\beta =\xi \beta =\xi \gamma =0.\end{array}$$

(27)

The Lie bracket of ${\displaystyle U}$ and ${\displaystyle \xi }$ satisfies the following two relations:

$$\begin{array}{c}[U,\xi ]\beta =U\left(\xi \beta \right)-\xi \left(U\beta \right),\hfill \\ [U,\xi ]\beta =({\nabla }_U\xi -{\nabla }_{\xi }U)\beta .\hfill \end{array}$$

A combination of the two relations above taking into account relations of Lemma 1 and (27) yields

$$(k-{\kappa }_3)\left[\left(\varphi U\right)\beta \right]=0.$$

Suppose that ${\displaystyle k\ne {\kappa }_3}$, then ${\displaystyle \left(\varphi U\right)\beta =0}$ and relation (26) implies ${\displaystyle \alpha k+\frac{c}{2}={\beta }^2}$. Differentiation of the last one with respect to ${\displaystyle \varphi U}$ results, taking into account relation (25), in ${\displaystyle {\kappa }_3=-\alpha }$. The Riemannian curvature satisfies the relation

$$R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z,$$

for any ${\displaystyle X}$, ${\displaystyle Y}$, ${\displaystyle Z}$ tangent to ${\displaystyle M}$. Combination of the last relation with Gaussian Equation (10) for ${\displaystyle X=U}$, ${\displaystyle Y=\varphi U}$ and ${\displaystyle Z=U}$ due to relation (22) and relation (24), ${\displaystyle {\kappa }_1=-\beta }$, ${\displaystyle {\kappa }_3=-\alpha }$ and ${\displaystyle \left(\varphi U\right)\beta =0}$ implies ${\displaystyle c=0}$, which is a contradiction.

Therefore, on ${\displaystyle M}$, relation ${\displaystyle k={\kappa }_3}$ holds. A combination of ${\displaystyle R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z}$ with Gauss Equation (10) for ${\displaystyle X=U}$, ${\displaystyle Y=\varphi U}$ and ${\displaystyle Z=U}$ because of relations (22) and (26) and ${\displaystyle {\kappa }_1=-\beta }$ yields

$$k^2=-\alpha k-\frac{3c}{2}.$$

A combination of the latter with relation (23) implies

$${\beta }^2+k^2=-\frac{5c}{8}.$$

Differentiation of the above relation with respect to ${\displaystyle \varphi U}$ gives, due to relation (26) and ${\displaystyle k^2=-\alpha k-\frac{3c}{2}}$,

$${\beta }^2+k^2=-\frac{c}{2}.$$

If the ambient space is the complex projective space ${\displaystyle \mathbb{C}{P}^2}$ with ${\displaystyle c=4}$, then the above relation leads to a contradiction. If the ambient space is the complex hyperbolic space ${\displaystyle \mathbb{C}{H}^2}$ with ${\displaystyle c=-4}$, combination of the latter relation with ${\displaystyle {\beta }^2+k^2=-\frac{5c}{8}}$ yields ${\displaystyle c=0}$, which is a contradiction.

Thus, ${\displaystyle \mathcal{N}}$ is empty and the following proposition is proved:

Proposition 1.

Every real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose shape operator satisfies relation (5) is a Hopf hypersurface.

Since ${\displaystyle M}$ is a Hopf hypersurface, Theorem 6 and remark 3 hold. Relation (20) for ${\displaystyle X=W}$ and for ${\displaystyle X=\varphi W}$ implies, respectively,

$$\begin{array}{c}\hfill (\lambda -k)(\nu -\alpha )=0and(\nu -k)(\lambda -\alpha )=0.\end{array}$$

(28)

Combination of the above relations results in

$$(\nu -\lambda )(\alpha -k)=0.$$

If ${\displaystyle \lambda \ne \nu }$, then ${\displaystyle \alpha =k}$ and relation ${\displaystyle (\lambda -k)(\nu -\alpha )=0}$ becomes

$$(\lambda -\alpha )(\nu -\alpha )=0.$$

If ${\displaystyle \nu \ne \alpha }$, then ${\displaystyle \lambda =\alpha }$ and relation (19) implies that ${\displaystyle \nu }$ is also constant. Therefore, the real hypersurface is locally congruent to a real hypersurface of type (${\displaystyle B}$). Substitution of the values of eigenvalues in relation ${\displaystyle \lambda =\alpha }$ leads to a contradiction. Thus, on ${\displaystyle M}$, relation ${\displaystyle \nu =\alpha }$ holds. Following similar steps to the previous case, we are led to a contradiction.

Therefore, on ${\displaystyle M}$, we have ${\displaystyle \lambda =\nu }$ and the first of relations (28) becomes

$$(\lambda -k)(\lambda -\alpha )=0.$$

Supposing that ${\displaystyle \lambda \ne k}$, then ${\displaystyle \lambda =\nu =\alpha }$. Thus, the real hypersurface is totally umbilical, which is impossible since there do not exist totally umbilical real hypersurfaces in non-flat complex space forms [18].

Thus, on ${\displaystyle M}$ relation ${\displaystyle \lambda =k}$ holds. Relation (20) for ${\displaystyle X=W}$ and ${\displaystyle Y=\varphi W}$ implies, because of ${\displaystyle \lambda =\nu =k}$, ${\displaystyle \lambda =\alpha }$. Thus, ${\displaystyle \lambda =\nu =\alpha }$ and the real hypersurface is totally umbilical, which is a contradiction and this completes the proof of Theorem 1.

Next, suppose that ${\displaystyle M}$ is a real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose shape operator satisfies relation (6), which, because of the relation of the k-th generalized Tanaka-Webster connection (1), becomes

$$\begin{array}{c}(A\varphi -\varphi A)AX-g(\varphi A\xi ,AX)\xi +\eta \left(AX\right)\varphi A\xi +k\varphi AX+g(\varphi A\xi ,X)A\xi \hfill \\ -\eta \left(X\right)A\varphi A\xi -kA\varphi X=0,\hfill \end{array}$$

(29)

for any ${\displaystyle X}$ ${\displaystyle \in }$ ${\displaystyle TM}$.

Let ${\displaystyle \mathcal{N}}$ be the open subset of ${\displaystyle M}$ such that

$$\mathcal{N}=\{p\in M:\beta \ne 0,inaneighborhoodofp\}.$$

The inner product of relation (29) for ${\displaystyle X=U}$ with ${\displaystyle \xi }$ implies, due to relation (13), ${\displaystyle \delta =0}$ and the shape operator on the local orthonormal basis ${\displaystyle \{U,\varphi U,\xi \}}$ becomes

$$\begin{array}{c}\hfill A\xi =\alpha \xi +\beta U,AU=\gamma U+\beta \xi andA\varphi U=\mu \varphi U.\end{array}$$

(30)

Relation (29) for ${\displaystyle X=\xi }$ yields, taking into account relation (30), ${\displaystyle \gamma =k}$. Finally, relation (29) for ${\displaystyle X=\varphi U}$ implies, due to relation (30) and the last relation,

$$({\mu }^2-2k\mu +k^2)+{\beta }^2=0.$$

The above relation results in ${\displaystyle \beta =0}$, which implies that ${\displaystyle \mathcal{N}}$ is empty. Thus, the following proposition is proved:

Proposition 2.

Every real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose shape operator satisfies relation (6) is a Hopf hypersurface.

Due to the above Proposition, Theorem 6 and Remark 3 hold. Relation (29) for ${\displaystyle X=W}$ and for ${\displaystyle X=\varphi W}$ implies, respectively,

$$(\lambda -k)(\lambda -\nu )=0and(\nu -k)(\lambda -\nu )=0.$$

Suppose that ${\displaystyle \lambda \ne \nu }$. Then, the above relations imply ${\displaystyle \lambda =\nu =k}$, which is a contradiction.

Thus, on ${\displaystyle M}$, relation ${\displaystyle \lambda =\nu }$ holds and this results in the structure tensor ${\displaystyle \varphi }$ commuting with the shape operator ${\displaystyle A}$, i.e., ${\displaystyle A\varphi =\varphi A}$ and, because of Theorem 3 ${\displaystyle M}$ , is locally congruent to a real hypersurface of type (${\displaystyle A}$), and this completes the proof of Theorem 2.

4. Proof of Theorems 4 and 5

Suppose that ${\displaystyle M}$ is a real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose tensor field ${\displaystyle P}$ satisfies relation (7) for any ${\displaystyle X}$${\displaystyle \in }$ ${\displaystyle \mathbb{D}}$ and for all ${\displaystyle Y}$ ${\displaystyle \in }$ ${\displaystyle TM}$. Then, the latter relation becomes, because of the relation of the k-th generalized Tanaka-Webster connection (1) and relations (3) and (4),

$$\begin{array}{c}g(\varphi AX,PY)\xi -\eta \left(PY\right)\varphi AX-g(\varphi APY,X)\xi +k\eta \left(PY\right)\varphi X-g(\varphi AX,Y)P\xi \hfill \\ +\eta \left(Y\right)P\varphi AX+g(\varphi AY,X)P\xi -k\eta \left(Y\right)P\varphi X=0,\hfill \end{array}$$

(31)

for any ${\displaystyle X}$ ${\displaystyle \in }$ ${\displaystyle \mathbb{D}}$ and for all ${\displaystyle Y}$ ${\displaystyle \in }$ ${\displaystyle TM}$.

Let ${\displaystyle \mathcal{N}}$ be the open subset of ${\displaystyle M}$ such that

$$\mathcal{N}=\{p\in M:\beta \ne 0,inaneighborhoodofp\}.$$

Relation (31) for ${\displaystyle Y=\xi }$ implies, taking into account relation (13),

$$\begin{array}{c}\hfill \beta \{g(AX,U)+g(A\varphi U,\varphi X)\}\xi +P\varphi AX+{\beta }^{2}g(\varphi U,X)\varphi U-kP\varphi X=0,\end{array}$$

(32)

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

The inner product of relation (32) for ${\displaystyle X=\varphi U}$ with ${\displaystyle \xi }$ due to relation (13) yields ${\displaystyle \delta =0}$. Moreover, the inner product of relation (32) for ${\displaystyle X=\varphi U}$ with ${\displaystyle \varphi U}$, taking into account relation (13) and ${\displaystyle \delta =0}$, results in

$$\begin{array}{c}\hfill {\beta }^2+k(\gamma -\mu )=\mu (\gamma -\mu ).\end{array}$$

(33)

The inner product of relation (32) for ${\displaystyle X=U}$ with ${\displaystyle U}$ gives, because of relation (13) and ${\displaystyle \delta =0}$,

$$(\gamma -k)(\gamma -\mu )=0.$$

Suppose that ${\displaystyle \gamma \ne k}$, then the above relation implies ${\displaystyle \gamma =\mu }$ and relation (33) implies ${\displaystyle \beta =0}$, which is impossible.

Thus, relation ${\displaystyle \gamma =k}$ holds and relation (33) results in

$${\beta }^2+{(\gamma -\mu )}^2=0.$$

The latter implies ${\displaystyle \beta =0}$, which is impossible.

Thus, ${\displaystyle \mathcal{N}}$ is empty and the following proposition has been proved:

Proposition 3.

Every real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose tensor field P satisfies relation (7) is a Hopf hypersurface.

As a result of the proposition above, Theorem 6 and remark 3 hold. Thus, relation (31) for ${\displaystyle X=W}$ and ${\displaystyle Y=\xi }$ and for ${\displaystyle X=\varphi W}$ and ${\displaystyle Y=\xi }$ yields, respectively,

$$(\lambda -k)(\lambda -\nu )=0and(\nu -k)(\lambda -\nu )=0.$$

Supposing that ${\displaystyle \lambda \ne \nu }$, the above relations imply ${\displaystyle \lambda =\nu =k}$, which is a contradiction.

Therefore, relation ${\displaystyle \lambda =\nu }$ holds and this implies that ${\displaystyle A\varphi =\varphi A}$. Thus, because of Theorem 3, ${\displaystyle M}$ is locally congruent to a real hypersurface of type (${\displaystyle A}$) and this completes the proof of Theorem 4.

Next, we study three-dimensional real hypersurfaces in ${\displaystyle M_2\left(c\right)}$ whose tensor field ${\displaystyle P}$ satisfies relation (8). The last relation becomes, due to relation (2),

$$\begin{array}{c}\hfill F_{\xi }^{\left(k\right)}PY-P{F}_{\xi }^{\left(k\right)}Y+\varphi APY-P\varphi AY=0,\end{array}$$

(34)

for any ${\displaystyle Y}$ tangent to ${\displaystyle M}$.

Let ${\displaystyle \mathcal{N}}$ be the open subset of ${\displaystyle M}$ such that

$$\mathcal{N}=\{p\in M:\beta \ne 0,inaneighborhoodofp\}.$$

The inner product of relation (34) for ${\displaystyle Y=\xi }$ implies, taking into account relation (13), ${\displaystyle \beta =0}$, which is impossible. Thus, ${\displaystyle \mathcal{N}}$ is empty and the following proposition has been proved

Proposition 4.

Every real hypersurface in ${\displaystyle M_2\left(c\right)}$ whose tensor field P satisfies relation (8) is a Hopf hypersurface.

Since ${\displaystyle M}$ is a Hopf hypersurface, Theorems 6 and 3 hold. Relation (34) for ${\displaystyle Y=W}$ implies, due to ${\displaystyle AW=\lambda W}$ and ${\displaystyle A\varphi W=\nu \varphi W}$,

$$(\lambda -\nu )(\nu +\lambda -2k)=0.$$

We have two cases:

Case I: Supposing that ${\displaystyle \lambda \ne \nu }$, then the above relation implies ${\displaystyle \nu +\lambda =2k}$. Relation (19) implies, due to the last one, that ${\displaystyle \lambda }$, ${\displaystyle \nu }$ are constant. Thus, ${\displaystyle M}$ is locally congruent to a real hypersurface with three distinct principal curvatures. Therefore, it is locally congruent to a real hypersurface of type (${\displaystyle B}$).

Thus, in the case of ${\displaystyle \mathbb{C}{P}^2}$, substitution of the eigenvalues of real hypersurface of type (${\displaystyle B}$) in ${\displaystyle \nu +\lambda =2k}$ implies ${\displaystyle \alpha =-2k}$. In the case of ${\displaystyle \mathbb{C}{H}^2}$, substitution of the eigenvalues of real hypersurface of type (${\displaystyle B}$) in ${\displaystyle \nu +\lambda =2k}$ yields ${\displaystyle \alpha =\frac{4}{k}}$.

Case II: Supposing that ${\displaystyle \lambda =\nu }$, then the structure tensor ${\displaystyle \varphi }$ commutes with the shape operator ${\displaystyle A}$, i.e., ${\displaystyle A\varphi =\varphi A}$ and, because of Theorem 3, ${\displaystyle M}$ is locally congruent to a real hypersurface of type (${\displaystyle A}$) and this completes the proof of Theorem 5.

As a consequence of Theorems 4 and 5, the following Corollary is obtained:

Corollary 2.

A real hypersurface M in ${\displaystyle M_2\left(c\right)}$ whose tensor field P satisfies relation (7) is locally congruent to a real hypersurface of type (A).

5. Conclusions

In this paper, we answer the question if there are three-dimensional real hypersurfaces in non-flat complex space forms whose differential operator ${\displaystyle {\mathcal{L}}^{\left(k\right)}}$ of a tensor field of type (1, 1) coincides with the Lie derivative of it. First, we study the case of the tensor field being the shape operator ${\displaystyle A}$ of the real hypersurface. The obtained results complete the work that has been done in the case of real hypersurfaces of dimensions greater than three in complex projective space (see [11]). In Table 3 all the existing results and also provides open problems are summarized.

Table 3. Results on condition ${\displaystyle {\widehat{\mathcal{L}}}_X^{\left(k\right)}A={\mathcal{L}}_{X}A.}$

Condition	${\displaystyle {\mathit{M}}_2\left(\mathit{c}\right)}$	${\displaystyle \mathbb{C}{\mathit{P}}^{\mathit{n}},\mathit{n}\ge 3}$	${\displaystyle \mathbb{C}{\mathit{H}}^{\mathit{n}},\mathit{n}\ge 3}$
${\displaystyle {\widehat{\mathcal{L}}}_X^{\left(k\right)}A={\mathcal{L}}_{X}A,X\in \mathbb{D}}$	does not exist	does not exist	open
${\displaystyle {\widehat{\mathcal{L}}}_{\xi }^{\left(k\right)}A={\mathcal{L}}_{\xi }A}$	type (${\displaystyle A}$)	type (${\displaystyle A}$)	open
${\displaystyle {\widehat{\mathcal{L}}}_X^{\left(k\right)}A={\mathcal{L}}_{X}A,X\in TM}$	does not exist	does not exist	open

Next, we study the above geometric condition in the case of the tensor field being ${\displaystyle P=A\varphi -\varphi A}$, which is introduced here. In Table 4, we summarize the obtained results.

Table 4. Results on condition ${\displaystyle {\widehat{\mathcal{L}}}_X^{\left(k\right)}P={\mathcal{L}}_{X}P.}$

Condition	${\displaystyle \mathbb{C}{\mathit{P}}^2}$	${\displaystyle \mathbb{C}{\mathit{H}}^2}$
${\displaystyle {\widehat{\mathcal{L}}}_X^{\left(k\right)}P={\mathcal{L}}_{X}P,X\in \mathbb{D}}$	type (${\displaystyle A}$)	type (${\displaystyle A}$)
${\displaystyle {\widehat{\mathcal{L}}}_{\xi }^{\left(k\right)}P={\mathcal{L}}_{\xi }P}$	type (${\displaystyle A}$) and	type (${\displaystyle A}$) and
	type (${\displaystyle B}$) with ${\displaystyle \alpha =-2k}$	type (${\displaystyle B}$) with ${\displaystyle \alpha =\frac{4}{k}}$
${\displaystyle {\widehat{\mathcal{L}}}_X^{\left(k\right)}P={\mathcal{L}}_{X}P,X\in TM}$	type (${\displaystyle A}$)	type (${\displaystyle A}$)