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. 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

includes the values of the constant principal curvatures corresponding to the real hypersurfaces above (see [1,2]).

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. 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

includes the values of the constant principal curvatures corresponding to above real hypersurfaces for ${\displaystyle n=2}$ (see [6]).

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. 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

all the existing results and also provides open problems are summarized.

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. 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 (${\displaystyle A}$)

, we summarize the obtained results.