LEVI-UMBILICAL REAL HYPERSURFACES IN A COMPLEX SPACE FORM

Abstract

We give a classification of Levi-umbilical real hypersurfaces in a complex space form $\widetilde{M}_{n}(c)$, $n\geqslant 3$, whose Levi form is proportional to the induced metric by a nonzero constant. In a complex projective plane $\mathbb{C}\mathbb{P}^{2}$, we give a local construction of such hypersurfaces and moreover, we give new examples of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$.

1 Introduction

Let $M$ be a $(2n-1)$ -dimensional manifold and $TM$ be its tangent bundle. A CR-structure on $M$ is a complex rank $(n-1)$ subbundle ${\mathcal{H}}\subset \mathbb{C}TM=TM\otimes \mathbb{C}$ satisfying

$$\begin{eqnarray}\begin{array}{@{}ll@{}} & \text{(i)}\qquad {\mathcal{H}}\cap \bar{{\mathcal{H}}}=\{0\},\\ & \text{(ii)}\qquad [{\mathcal{H}},{\mathcal{H}}]\subset {\mathcal{H}}~(integrability),\end{array}\end{eqnarray}$$

where $\bar{{\mathcal{H}}}$ denotes the complex conjugation of ${\mathcal{H}}.$

Then there exists a unique subbundle $D=Re\{{\mathcal{H}}\oplus \bar{{\mathcal{H}}}\}$ , called the Levi subbundle (maximally holomorphic subbundle) of $(M,{\mathcal{H}})$ , and a unique bundle map $J$ such that $J^{2}=-I$ and ${\mathcal{H}}=\{X-iJX|X\in D\}$ . We call $(D,J)$ the real representation of ${\mathcal{H}}$ . Let $E\subset T^{\ast }M$ be the conormal bundle of $D$ . If $M$ is an oriented CR-manifold then $E$ is a trivial bundle, hence admits globally defined a nowhere zero section $\unicode[STIX]{x1D702}$ , that is, a real one-form on $M$ such that Ker $(\unicode[STIX]{x1D702})=D$ . For $(D,J)$ we define the Levi form by

$$\begin{eqnarray}L:D\times D\rightarrow {\mathcal{F}}(M),\qquad L(X,Y)=d\unicode[STIX]{x1D702}(X,JY)\end{eqnarray}$$

where ${\mathcal{F}}(M)$ denotes the algebra of differential functions on $M$ . If the Levi form is nondegenerate (positive or negative definite, resp.), then the CR-structure is called a nondegenerate (strongly pseudo-convex, resp.) pseudo-Hermitian CR-structure.

Now, let $\widetilde{M}^{n}$ be an $n$ -dimensional Kähler manifold and let $M^{2n-1}$ be a real hypersurface in $\widetilde{M}$ . Then $M$ is called Levi-flat if the Levi form vanishes. In the present paper, we introduce the so-called Levi-umbilicity. If the Levi form $L$ is proportional to the induced metric $g$ by a nonzero constant $k$ , then $M$ is said to be Levi-umbilical.

A complex $n$ -dimensional complete and simply connected Kähler manifold of constant holomorphic sectional curvature $c$ is called a complex space form, which is denoted by $\widetilde{M}_{n}(c)$ . A complex space form consists of a complex projective space $\mathbb{C}\mathbb{P}^{n}$ , a complex Euclidean space $\mathbb{CE}^{n}$ or a complex hyperbolic space $\mathbb{CH}^{n}$ , according as $c>0$ , $c=0$ or $c<0$ . Recently, Siu [14] proved the nonexistence of compact smooth Levi-flat hypersurfaces in $\mathbb{C}\mathbb{P}^{n}$ of dimensions ${\geqslant}3$ . When $n=2$ , Ohsawa [13] proved the nonexistence of compact real analytic Levi-flat hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$ . Here, it is remarkable that the assumption of compactness has a crucial role. Indeed, there are noncomplete examples which are realized as ruled hypersurfaces and Levi-flat in $\mathbb{C}\mathbb{P}^{n}$ (see Section 3). We also find that there does not exist a Levi-flat Hopf hypersurface in $\mathbb{C}\mathbb{P}^{n}$ or $\mathbb{CH}^{n}$ (cf. [6]). In the present paper, we give noncompact examples of Levi-flat real hypersurfaces which are not ruled hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$ (see Section 5).

On the other hand, Takagi [16], [17] classified the homogeneous real hypersurfaces in $\mathbb{C}\mathbb{P}^{n}$ into six types. Cecil and Ryan [4] extensively studied a real hypersurface whose structure vector $\unicode[STIX]{x1D709}$ is a principal curvature vector, which is realized as tubes over certain submanifolds in $\mathbb{C}\mathbb{P}^{n}$ , by using its focal map. A real hypersurface of a complex space form is said to be a Hopf hypersurface if its structure vector is a principal curvature vector. By making use of those results and the mentioned work of Takagi, Makoto Kimura [8] proved the classification theorem for Hopf hypersurfaces of $\mathbb{C}\mathbb{P}^{n}$ whose all principal curvatures are constant. For the case $\mathbb{CH}^{n}$ , Berndt [2] proved the classification theorem for Hopf hypersurfaces whose all principal curvatures are constant.

The main purpose of the present paper is to give a classification of Levi-umbilical real hypersurfaces in a complex space form.

Theorem 1. If a real hypersurface $M$ of a complex space form $\widetilde{M}_{n}(c)$ is Levi-umbilical, then $n=2$ or $M$ is a Hopf hypersurface. Moreover, in case that $M$ is connected, complete and $n\geqslant 3$ , we have the following.

1. (I) If $\widetilde{M}_{n}(c)=\mathbb{C}\mathbb{P}^{n}$ , then $M$ is congruent to one of the following:

   1. (1) a geodesic hypersphere, that is, a tube of radius $r$ over $\mathbb{C}\mathbb{P}^{n-1}$ , where $0<r<\frac{\unicode[STIX]{x1D70B}}{2}$ ,

   2. (2) a tube of radius $r$ over a complex quadric $\mathbb{CQ}^{n-1}$ , where $0<r<\frac{\unicode[STIX]{x1D70B}}{4}$ .

2. (II) If $\widetilde{M}_{n}(c)=\mathbb{CH}^{n}$ , then $M$ is congruent to one of the following:

   1. (1) a horosphere in $\mathbb{CH}^{n}$ ,

   2. (2) a geodesic hypersphere or a tube of radius $r\in \mathbb{R}_{+}$ over a totally geodesic $\mathbb{CH}^{n-1}$ ,

   3. (3) a tube of radius $r\in \mathbb{R}_{+}$ over a totally real hyperbolic space $\mathbb{RH}^{n}$ .

3. (III) If $\widetilde{M}_{n}(c)=\mathbb{CE}^{n}$ , then $M$ is locally congruent to one of the following:

   1. (1) a sphere $S^{2n-1}(r)$ of radius $r\in \mathbb{R}_{+}$ ,

   2. (2) a generalized cylinder $S^{n-1}(r)\times \mathbb{E}^{n}$ of radius $r\in \mathbb{R}_{+}$ .

In Section 5, we give a construction of Levi-umbilical non-Hopf hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$ .

2 Almost contact metric structures and the associated CR-structures

In this paper, all manifolds are assumed to be connected and of class $C^{\infty }$ . First, we give a brief review of several fundamental concepts and formulas which we need later on. An odd-dimensional differentiable manifold $M$ has an almost contact structure if it admits a (1,1)-tensor field $\unicode[STIX]{x1D719}$ , a vector field $\unicode[STIX]{x1D709}$ and a 1-form $\unicode[STIX]{x1D702}$ satisfying

(1) $$\begin{eqnarray}\unicode[STIX]{x1D719}^{2}=-I+\unicode[STIX]{x1D702}\otimes \unicode[STIX]{x1D709},~\unicode[STIX]{x1D702}(\unicode[STIX]{x1D709})=1.\end{eqnarray}$$

Then we can find always a compatible Riemannian metric, namely which satisfies

(2) $$\begin{eqnarray}g(\unicode[STIX]{x1D719}X,\unicode[STIX]{x1D719}Y)=g(X,Y)-\unicode[STIX]{x1D702}(X)\unicode[STIX]{x1D702}(Y)\end{eqnarray}$$

for all vector fields on $M$ . We call $(\unicode[STIX]{x1D702},\unicode[STIX]{x1D719},\unicode[STIX]{x1D709},g)$ an almost contact metric structure of $M$ and $M=(M;\unicode[STIX]{x1D702},\unicode[STIX]{x1D719},\unicode[STIX]{x1D709},g)$ an almost contact metric manifold. The fundamental 2-form $\unicode[STIX]{x1D6F7}$ is defined by $\unicode[STIX]{x1D6F7}(X,Y)=g(\unicode[STIX]{x1D719}X,Y)$ . If $M$ satisfies in addition $d\unicode[STIX]{x1D702}=\unicode[STIX]{x1D6F7}$ , then $M$ is called a contact metric manifold, where $d$ is the exterior differential operator. From (1) and (2) we easily get

(3) $$\begin{eqnarray}\unicode[STIX]{x1D719}\unicode[STIX]{x1D709}=0,\qquad \unicode[STIX]{x1D702}\circ \unicode[STIX]{x1D719}=0,\qquad \unicode[STIX]{x1D702}(X)=g(X,\unicode[STIX]{x1D709}).\end{eqnarray}$$

The tangent space $T_{p}M$ of $M$ at each point $p\in M$ is decomposed as $T_{p}M=D_{p}\oplus \{\unicode[STIX]{x1D709}\}_{p}$ (direct sum), where we denote $D_{p}=\{v\in T_{p}M|\unicode[STIX]{x1D702}(v)=0\}$ . Then $D:p\rightarrow D_{p}$ defines a distribution orthogonal to $\unicode[STIX]{x1D709}$ . For an almost contact metric manifold $M$ , one may define naturally an almost complex structure on the product manifold $M\times \mathbb{R}$ , where $\mathbb{R}$ denotes the real line. If the almost complex structure is integrable, $M$ is said to be normal. The integrability condition for the almost complex structure is the vanishing of the tensor $[\unicode[STIX]{x1D719},\unicode[STIX]{x1D719}]+2d\unicode[STIX]{x1D702}\otimes \unicode[STIX]{x1D709}$ , where $[\unicode[STIX]{x1D719},\unicode[STIX]{x1D719}]$ denotes the Nijenhuis torsion of $\unicode[STIX]{x1D719}$ . For more details about the general theory of almost contact metric manifolds, we refer to [3].

On the other hand, for an almost contact metric manifold $M$ , the restriction $J=\unicode[STIX]{x1D719}|D$ of $\unicode[STIX]{x1D719}$ to $D$ defines an almost complex structure in $D$ . As soon as $M$ satisfies

(4) $$\begin{eqnarray}[JX,JY]-[X,Y]\in D~(or~[JX,Y]+[X,JY]\in D)\end{eqnarray}$$

and

(5) $$\begin{eqnarray}[J,J](X,Y)=0\end{eqnarray}$$

for all $X,Y\in D$ , where $[J,J]$ is the Nijenhuis torsion of $J$ , then the pair $(\unicode[STIX]{x1D702},J)$ is called an (integrable) CR-structure associated with the almost contact metric structure $(\unicode[STIX]{x1D702},\unicode[STIX]{x1D719},\unicode[STIX]{x1D709},g)$ . For example, a normal almost contact metric manifold has an integrable CR-structure [7]. In addition, the associated Levi form $L$ defined by $L(X,Y)=d\unicode[STIX]{x1D702}(X,JY)$ , $X,Y\in D$ , is nondegenerate (positive or negative definite, resp.), then $(\unicode[STIX]{x1D702},J)$ is called a nondegenerate (strongly pseudo-convex, resp.) pseudo-Hermitian CR-structure. We may refer to [5], [7], [18] about CR-structures associated with (almost) contact metric structures.

3 Real hypersurfaces in a complex space form

Let $M$ be an immersed real hypersurface of a Kähler manifold $\widetilde{M}=(\widetilde{M};\tilde{J},\tilde{g})$ and $N$ a local unit normal vector in a neighborhood of each point. By $\tilde{\unicode[STIX]{x1D6FB}}$ , $\unicode[STIX]{x1D70E}$ we denote the Levi-Civita connection in $\widetilde{M}$ and the second fundamental form associated with the shape operator $A$ with respect to $N$ , respectively. Then the Gauss and Weingarten formulas are given respectively by

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D6FB}}_{X}Y=\unicode[STIX]{x1D6FB}_{X}Y+\unicode[STIX]{x1D70E}(X,Y)N,\qquad \tilde{\unicode[STIX]{x1D6FB}}_{X}N=-AX\end{eqnarray}$$

for any vector fields $X$ and $Y$ tangent to $M$ . Here, we note that $\unicode[STIX]{x1D70E}(X,Y)=g(AX,Y)$ , where $g$ denotes the Riemannian metric of $M$ induced from $\tilde{g}$ . An eigenvector (resp. eigenvalue) of the shape operator $A$ is called a principal curvature vector (resp. principal curvature). For any vector field $X$ tangent to $M$ , we put

(6) $$\begin{eqnarray}\tilde{J}X=\unicode[STIX]{x1D719}X+\unicode[STIX]{x1D702}(X)N,\qquad \tilde{J}N=-\unicode[STIX]{x1D709}.\end{eqnarray}$$

We easily see that the structure $(\unicode[STIX]{x1D702},\unicode[STIX]{x1D719},\unicode[STIX]{x1D709},g)$ is an almost contact metric structure on $M$ , that is, satisfies (1) and (2). From the condition $\tilde{\unicode[STIX]{x1D6FB}}\tilde{J}=0$ , the relations (6) and by making use of the Gauss and Weingarten formulas, we have

(7) $$\begin{eqnarray}\displaystyle & (\unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D719})Y=\unicode[STIX]{x1D702}(Y)AX-g(AX,Y)\unicode[STIX]{x1D709}, & \displaystyle\end{eqnarray}$$

(8) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D709}=\unicode[STIX]{x1D719}AX. & \displaystyle\end{eqnarray}$$

From now, let $\widetilde{M}_{n}(c)$ be a complex space form of constant holomorphic sectional curvature $c$ . Then, from the Codazzi equation, we have

(9) $$\begin{eqnarray}(\unicode[STIX]{x1D6FB}_{X}A)Y-(\unicode[STIX]{x1D6FB}_{Y}A)X=\frac{c}{4}\{\unicode[STIX]{x1D702}(X)\unicode[STIX]{x1D719}Y-\unicode[STIX]{x1D702}(Y)\unicode[STIX]{x1D719}X-2g(\unicode[STIX]{x1D719}X,Y)\unicode[STIX]{x1D709}\}.\end{eqnarray}$$

By using (7) and (8), we see that a real hypersurface in a Kähler manifold always satisfies (4) and (5), the integrability condition of the associated CR-structure. From (8) we find that $M$ is Levi-flat if and only if

(10) $$\begin{eqnarray}g((\unicode[STIX]{x1D719}A+A\unicode[STIX]{x1D719})X,Y)=0,\quad X,Y\bot \unicode[STIX]{x1D709},\end{eqnarray}$$

and $M$ is Levi-umbilical if and only if there exists nonzero constant $k\in \mathbb{R}$ such that

(11) $$\begin{eqnarray}g((\unicode[STIX]{x1D719}A+A\unicode[STIX]{x1D719})X,Y)=kg(\unicode[STIX]{x1D719}X,Y),\quad X,Y\bot \unicode[STIX]{x1D709}.\end{eqnarray}$$

Here we recall ruled real hypersurfaces in $\mathbb{C}\mathbb{P}^{n}$ or $\mathbb{CH}^{n}$ . Such a space is a foliated real hypersurface whose leaves are complex hyperplanes $\mathbb{C}\mathbb{P}^{n-1}$ or $\mathbb{CH}^{n-1}$ , respectively in $\mathbb{C}\mathbb{P}^{n}$ or $\mathbb{CH}^{n}$ . That is, let $\unicode[STIX]{x1D6FE}:I\rightarrow \widetilde{M}_{n}(c)$ be a regular curve in $\widetilde{M}_{n}(c)$ $(\mathbb{C}\mathbb{P}^{n}or\mathbb{CH}^{n})$ . Then for each $t\in I$ , let $M_{n-1}^{(t)}(c)$ be a totally geodesic complex hypersurface which is orthogonal to holomorphic plane Span $\{\dot{\unicode[STIX]{x1D6FE}},J\dot{\unicode[STIX]{x1D6FE}}\}$ . We have a ruled real hypersurface $M=\bigcup _{t\in I}M_{n-1}^{(t)}(c)$ . A ruled real hypersurface is non-Hopf and particularly it is noncomplete real hypersurface in $\mathbb{C}\mathbb{P}^{n}$ (see, [10] for the case $\mathbb{C}\mathbb{P}^{n}$ and see [1] for the case $\mathbb{CH}^{n}$ , respectively). The shape operator $A$ is written by the following form:

(12) $$\begin{eqnarray}\begin{array}{@{}rcl@{}}A\unicode[STIX]{x1D709}\ & =\ & \unicode[STIX]{x1D707}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D708}V~(\unicode[STIX]{x1D708}\neq 0),\\ AV\ & =\ & \unicode[STIX]{x1D708}\unicode[STIX]{x1D709},\\ AX\ & =\ & 0\quad \text{for any}~X\bot \unicode[STIX]{x1D709},V,\end{array}\end{eqnarray}$$

where $V$ is a unit vector orthogonal to $\unicode[STIX]{x1D709}$ , and $\unicode[STIX]{x1D707}$ , $\unicode[STIX]{x1D708}$ are differentiable functions on $M$ . Then, we easily see that ruled real hypersurfaces in $\mathbb{C}\mathbb{P}^{n}$ or in $\mathbb{CH}^{n}$ are Levi-flat.

4 Proof of Theorem 1

In this section, we prove Theorem 1. Let $M$ be a Levi-umbilical real hypersurface in a complex space form $\widetilde{M}_{n}(c)$ . If we differentiate (11) covariantly, then we have

(13) $$\begin{eqnarray}\displaystyle & & \displaystyle g(\,(\unicode[STIX]{x1D6FB}_{X}A)\unicode[STIX]{x1D719}Y+A(\unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D719})Y+A\unicode[STIX]{x1D719}\unicode[STIX]{x1D6FB}_{X}Y+(\unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D719})AY\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D719}(\unicode[STIX]{x1D6FB}_{X}A)Y+\unicode[STIX]{x1D719}A\unicode[STIX]{x1D6FB}_{X}Y,Z)\!\,+\,g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)Y,\unicode[STIX]{x1D6FB}_{X}Z)\nonumber\\ \displaystyle & & \displaystyle \quad =k(g((\unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D719})Y,Z)+g(\unicode[STIX]{x1D719}\unicode[STIX]{x1D6FB}_{X}Y,Z)+g(\unicode[STIX]{x1D719}Y,\unicode[STIX]{x1D6FB}_{X}Z)),\end{eqnarray}$$

for any vector fields $X,Y,Z\bot \unicode[STIX]{x1D709}$ . Use (7) to get

(14) $$\begin{eqnarray}\displaystyle & & \displaystyle g((\unicode[STIX]{x1D6FB}_{X}A)\unicode[STIX]{x1D719}Y+\unicode[STIX]{x1D719}(\unicode[STIX]{x1D6FB}_{X}A)Y,Z)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,g(\unicode[STIX]{x1D702}(Y)A^{2}X-g(AX,Y)A\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702}(AY)AX-g(AX,AY)\unicode[STIX]{x1D709},Z)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)\unicode[STIX]{x1D6FB}_{X}Y,Z)-g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)\unicode[STIX]{x1D6FB}_{X}Z,Y)\nonumber\\ \displaystyle & & \displaystyle \quad =k(g(\unicode[STIX]{x1D719}\unicode[STIX]{x1D6FB}_{X}Y,Z)-g(\unicode[STIX]{x1D719}\unicode[STIX]{x1D6FB}_{X}Z,Y)).\end{eqnarray}$$

We decompose $\unicode[STIX]{x1D6FB}_{X}Y=\unicode[STIX]{x1D6FB}_{X}Y^{\bot }+\unicode[STIX]{x1D702}(\unicode[STIX]{x1D6FB}_{X}Y)\unicode[STIX]{x1D709}$ , where $\unicode[STIX]{x1D6FB}_{X}Y^{\bot }$ denotes the part of $\unicode[STIX]{x1D6FB}_{X}Y$ orthogonal to $\unicode[STIX]{x1D709}$ . Using (8) and (11), (14) becomes

(15) $$\begin{eqnarray}\displaystyle & & \displaystyle g((\unicode[STIX]{x1D6FB}_{X}A)\unicode[STIX]{x1D719}Y+\unicode[STIX]{x1D719}(\unicode[STIX]{x1D6FB}_{X}A)Y,Z)\nonumber\\ \displaystyle & & \displaystyle \quad +\,g(\unicode[STIX]{x1D702}(Y)A^{2}X-g(AX,Y)A\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702}(AY)AX-g(AX,AY)\unicode[STIX]{x1D709},Z)\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D702}(\unicode[STIX]{x1D6FB}_{X}Y)g(U,Z)-\unicode[STIX]{x1D702}(\unicode[STIX]{x1D6FB}_{X}Z)g(U,Y)=0,\end{eqnarray}$$

where we have put $U=\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D709}$ . Use (8) to obtain

(16) $$\begin{eqnarray}\displaystyle & & \displaystyle g((\unicode[STIX]{x1D6FB}_{X}A)Z,\unicode[STIX]{x1D719}Y)-g((\unicode[STIX]{x1D6FB}_{X}A)Y,\unicode[STIX]{x1D719}Z)\nonumber\\ \displaystyle & & \displaystyle \quad =g(\unicode[STIX]{x1D719}AX,Y)g(U,Z)-g(\unicode[STIX]{x1D719}AX,Z)g(U,Y)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D702}(Z)g(A^{2}X,Y)-\unicode[STIX]{x1D702}(Y)g(A^{2}X,Z)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D702}(AZ)g(AX,Y)-\unicode[STIX]{x1D702}(AY)g(AX,Z).\end{eqnarray}$$

Taking the cyclic sum of (16) for $X,Y,Z$ , using (9) we have

(17) $$\begin{eqnarray}\displaystyle & & \displaystyle g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)X,Y)g(U,Z)+g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)Y,Z)g(U,X)\nonumber\\ \displaystyle & & \displaystyle \quad +\,g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)Z,X)g(U,Y)=0.\end{eqnarray}$$

Using (11) in (17) again, we have

(18) $$\begin{eqnarray}k(g(\unicode[STIX]{x1D719}X,Y)g(U,Z)+g(\unicode[STIX]{x1D719}Y,Z)g(U,X)+g(\unicode[STIX]{x1D719}Z,X)g(U,Y))=0.\end{eqnarray}$$

If we put $Z=U$ in (18), then we have

(19) $$\begin{eqnarray}k(g(\unicode[STIX]{x1D719}X,Y)\Vert U\Vert ^{2}+g(\unicode[STIX]{x1D719}Y,U)g(U,X)+g(\unicode[STIX]{x1D719}U,X)g(U,Y))=0.\end{eqnarray}$$

Replace $Y$ by $\unicode[STIX]{x1D719}X$ in (19), then it turns to

(20) $$\begin{eqnarray}k(g(X,X)\Vert U\Vert ^{2}-g(X,U)g(U,X)+g(\unicode[STIX]{x1D719}U,X)g(U,\unicode[STIX]{x1D719}X))=0.\end{eqnarray}$$

For an adapted orthonormal basis $\{e_{i},\unicode[STIX]{x1D709}\}$ , $i=1,\cdots \,,2n-2$ , we put $X=e_{i}$ and taking the sum for $i=1,\cdots \,,2n-2$ , then since $k\neq 0$ we have

$$\begin{eqnarray}(n-2)\Vert U\Vert ^{2}=0.\end{eqnarray}$$

From this, we find that $n=2$ or $M$ is a Hopf hypersurface, that is, $A\unicode[STIX]{x1D709}=\unicode[STIX]{x1D707}\unicode[STIX]{x1D709}$ , where we have used (8). Now, we assume that $n\geqslant 3$ . Then Levi-umbilicity condition (11) yields that $\unicode[STIX]{x1D719}A+A\unicode[STIX]{x1D719}=k\unicode[STIX]{x1D719}$ , $k\neq 0$ . Due to results of [11] (in case of $\mathbb{C}\mathbb{P}^{n}$ ), [19], [15] (in case of $\mathbb{CH}^{n}$ ), and [12] (in case of $\mathbb{CE}^{n}$ ) we find the following.

1. (I) If $\widetilde{M}_{n}(c)=\mathbb{C}\mathbb{P}^{n}$ , then $M$ is locally congruent to one of the following:

   1. (1) a geodesic hypersphere, that is, a tube of radius $r$ over $P_{n-1}\mathbb{C}$ , where $0<r<\frac{\unicode[STIX]{x1D70B}}{2}$ ,

   2. (2) a tube of radius $r$ over a complex quadric $\mathbb{CQ}^{n-1}$ , where $0<r<\frac{\unicode[STIX]{x1D70B}}{4}$ .

2. (II) If $\widetilde{M}_{n}(c)=\mathbb{CH}^{n}$ , then $M$ is locally congruent to one of the following:

   1. (1) a horosphere in $\mathbb{CH}^{n}$ ,

   2. (2) a geodesic hypersphere or a tube of radius $r\in \mathbb{R}_{+}$ over a totally geodesic $\mathbb{CH}^{n-1}$ ,

   3. (3) a tube of radius $r\in \mathbb{R}_{+}$ over a totally real hyperbolic space $\mathbb{RH}^{n}$ .

3. (III) If $\widetilde{M}_{n}(c)=\mathbb{CE}^{n}$ , then $M$ is locally congruent to one of the following:

   1. (1) a sphere $S^{2n-1}(r)$ of radius $r\in \mathbb{R}_{+}$ ,

   2. (2) a generalized cylinder $S^{n-1}(r)\times \mathbb{E}^{n}$ of radius $r\in \mathbb{R}_{+}$ .

Then, we have Theorem 1. ◻

5 Three-dimensional Levi-umbilical hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$

In this section, we give a construction of $3$ -dimensional Levi-flat or Levi-umbilical real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$ . First, we prepare

Lemma 2. Let $M^{2n-1}$ $(n\geqslant 2)$ be a Levi-flat hypersurface in a Kähler manifold $\widetilde{M}^{n}$ . Then $\operatorname{trace}A=\unicode[STIX]{x1D702}(A\unicode[STIX]{x1D709})$ on $M$ . The converse holds when $n=2$ .

Lemma 3. Let $M^{2n-1}$ $(n\geqslant 2)$ be a Levi-umbilical hypersurface in a Kähler manifold $\widetilde{M}^{n}$ . Then $\operatorname{trace}A-\unicode[STIX]{x1D702}(A\unicode[STIX]{x1D709})$ is a nonzero constant on $M$ . The converse holds when $n=2$ .

Now, according to [9], we construct Levi-flat or Levi-umbilical hypersurfaces respectively in $\mathbb{C}\mathbb{P}^{2}$ . We denote $S^{n}$ as the unit sphere of which the center is the origin in $\mathbb{R}^{n+1}$ . We consider the following submanifolds of $\mathbb{C}^{3}$ :

$$\begin{eqnarray}\displaystyle \mathbb{C}^{3} & \supset & \displaystyle S^{5}\nonumber\\ \displaystyle & \supset & \displaystyle \sin rS^{3}\times \cos rS^{1}\nonumber\\ \displaystyle & \supset & \displaystyle \sin r(\sin \unicode[STIX]{x1D703}S^{1}\times \cos \unicode[STIX]{x1D703}S^{1})\times \cos rS^{1},\nonumber\end{eqnarray}$$

where $0<r,\unicode[STIX]{x1D703}<\unicode[STIX]{x1D70B}/2$ . Let $\unicode[STIX]{x1D6FE}:I\rightarrow (0,\unicode[STIX]{x1D70B}/2)\times (0,\unicode[STIX]{x1D70B}/2)$ , $\unicode[STIX]{x1D6FE}(s)=(r(s),\unicode[STIX]{x1D703}(s))$ be a (nonconstant) curve defined on an interval $I$ . We put

(21) $$\begin{eqnarray}\begin{array}{@{}l@{}}\widetilde{M}_{\unicode[STIX]{x1D6FE}}:=\mathop{\bigcup }_{s\in I}\sin r(s)(\sin \unicode[STIX]{x1D703}(s)S^{1}\times \cos \unicode[STIX]{x1D703}(s)S^{1})\times \cos r(s)S^{1}\\ \qquad \quad \text{and}\qquad M_{\unicode[STIX]{x1D6FE}}:=\unicode[STIX]{x1D70B}(\widetilde{M}_{\unicode[STIX]{x1D6FE}}),\end{array}\end{eqnarray}$$

where $\unicode[STIX]{x1D70B}:S^{5}\rightarrow \mathbb{C}\mathbb{P}^{2}$ is the Hopf fibration. Then $\widetilde{M}_{\unicode[STIX]{x1D6FE}}$ is a hypersurface of $S^{5}$ , and since $\widetilde{M}_{\unicode[STIX]{x1D6FE}}$ is invariant under the $S^{1}$ -action, $M_{\unicode[STIX]{x1D6FE}}$ is a real hypersurface of $\mathbb{C}\mathbb{P}^{2}$ . Note that $M_{\unicode[STIX]{x1D6FE}}$ is foliated by flat Lagrangian torus $T^{2}$ in $\mathbb{C}\mathbb{P}^{2}$ .

Let $x,y,z\in S^{1}\subset \mathbb{C}$ and denote

$$\begin{eqnarray}\tilde{x}=\sin r\sin \unicode[STIX]{x1D703}x,\qquad {\tilde{y}}=\sin r\cos \unicode[STIX]{x1D703}y,\qquad \tilde{z}=\cos rz,\end{eqnarray}$$

where $0<r,\unicode[STIX]{x1D703}<\unicode[STIX]{x1D70B}/2$ . Then the position vector $\unicode[STIX]{x1D6F9}$ of $\widetilde{M}_{\unicode[STIX]{x1D6FE}}$ is given by

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}=\unicode[STIX]{x1D6F9}(r(s),\unicode[STIX]{x1D703}(s))=(\tilde{x},{\tilde{y}},\tilde{z})=(\sin r\sin \unicode[STIX]{x1D703}x,\sin r\cos \unicode[STIX]{x1D703}y,\cos rz)\end{eqnarray}$$

and unit normal vectors $N_{1}$ and $N_{2}$ of $3$ -dimensional submanifold

$$\begin{eqnarray}\sin r(\sin \unicode[STIX]{x1D703}S^{1}\times \cos \unicode[STIX]{x1D703}S^{1})\times \cos rS^{1}\end{eqnarray}$$

in $S^{5}$ at $\unicode[STIX]{x1D6F9}$ are given as

$$\begin{eqnarray}N_{1}:=\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}r}=(\cot r\tilde{x},\cot r{\tilde{y}},-\!\tan r\tilde{z})=(\cos r\sin \unicode[STIX]{x1D703}x,\cos r\cos \unicode[STIX]{x1D703}y,-\!\sin rz)\end{eqnarray}$$

and

$$\begin{eqnarray}N_{2}:=\frac{1}{\sin r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}=\left(\frac{\cot \unicode[STIX]{x1D703}}{\sin r}\tilde{x},-\frac{\tan \unicode[STIX]{x1D703}}{\sin r}{\tilde{y}},0\right)=(\cos \unicode[STIX]{x1D703}x,-\!\sin \unicode[STIX]{x1D703}y,0).\end{eqnarray}$$

Put $\dot{\unicode[STIX]{x1D6F9}}=\frac{d}{ds}\unicode[STIX]{x1D6F9}(r(s),\unicode[STIX]{x1D703}(s))$ . Then we have

$$\begin{eqnarray}\dot{\unicode[STIX]{x1D6F9}}={\dot{r}}N_{1}+\dot{\unicode[STIX]{x1D703}}\sin rN_{2}\quad \left({\dot{r}}=\frac{dr}{ds},\dot{\unicode[STIX]{x1D703}}=\frac{d\unicode[STIX]{x1D703}}{ds}\right).\end{eqnarray}$$

By taking an arc-length parameterization, we may put $({\dot{r}})^{2}+(\dot{\unicode[STIX]{x1D703}})^{2}\sin ^{2}r=1$ and

(22) $$\begin{eqnarray}{\dot{r}}=\cos \unicode[STIX]{x1D6FC},\qquad \dot{\unicode[STIX]{x1D703}}=\frac{\sin \unicode[STIX]{x1D6FC}}{\sin r}.\end{eqnarray}$$

Hence $\dot{\unicode[STIX]{x1D6F9}}=\cos \unicode[STIX]{x1D6FC}N_{1}+\sin \unicode[STIX]{x1D6FC}N_{2}$ . Let

$$\begin{eqnarray}\widetilde{N}=-\!\sin \unicode[STIX]{x1D6FC}N_{1}+\cos \unicode[STIX]{x1D6FC}N_{2}.\end{eqnarray}$$

Then $\widetilde{N}$ is a unit normal vector field of $\widetilde{M}_{\unicode[STIX]{x1D6FE}}$ in $S^{5}$ . Since $\widetilde{N}$ is $S^{1}$ -invariant, $N:=\unicode[STIX]{x1D70B}_{\ast }(\widetilde{N})$ is a unit normal vector field of $M_{\unicode[STIX]{x1D6FE}}$ in $\mathbb{C}\mathbb{P}^{2}$ . We have

$$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D6F9}} & = & \displaystyle \left(\left(\cos \unicode[STIX]{x1D6FC}\cot r+\sin \unicode[STIX]{x1D6FC}\frac{\cot \unicode[STIX]{x1D703}}{\sin r}\right)\tilde{x},\right.\nonumber\\ \displaystyle & & \displaystyle \left.\left(\cos \unicode[STIX]{x1D6FC}\cot r-\sin \unicode[STIX]{x1D6FC}\frac{\tan \unicode[STIX]{x1D703}}{\sin r}\right){\tilde{y}},-\cos \unicode[STIX]{x1D6FC}\tan r\tilde{z}\right)\nonumber\\ \displaystyle & = & \displaystyle \left(\left(\cos \unicode[STIX]{x1D6FC}\cos r\sin \unicode[STIX]{x1D703}+\sin \unicode[STIX]{x1D6FC}\cos \unicode[STIX]{x1D703}\right)x,\right.\nonumber\\ \displaystyle & & \displaystyle \left.\left(\cos \unicode[STIX]{x1D6FC}\cos r\cos \unicode[STIX]{x1D703}-\sin \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D703}\right)y,-\cos \unicode[STIX]{x1D6FC}\sin rz\right),\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle \widetilde{N} & = & \displaystyle \left(\left(-\!\sin \unicode[STIX]{x1D6FC}\cot r+\cos \unicode[STIX]{x1D6FC}\frac{\cot \unicode[STIX]{x1D703}}{\sin r}\right)\tilde{x},\right.\nonumber\\ \displaystyle & & \displaystyle \left.\left(-\!\sin \unicode[STIX]{x1D6FC}\cot r-\cos \unicode[STIX]{x1D6FC}\frac{\tan \unicode[STIX]{x1D703}}{\sin r}\right){\tilde{y}},\sin \unicode[STIX]{x1D6FC}\tan r\tilde{z}\right)\nonumber\\ \displaystyle & = & \displaystyle \left(\left(-\!\sin \unicode[STIX]{x1D6FC}\cos r\sin \unicode[STIX]{x1D703}+\cos \unicode[STIX]{x1D6FC}\cos \unicode[STIX]{x1D703}\right)x,\right.\nonumber\\ \displaystyle & & \displaystyle \left.(-\!\sin \unicode[STIX]{x1D6FC}\cos r\cos \unicode[STIX]{x1D703}-\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D703})y,\sin \unicode[STIX]{x1D6FC}\sin rz\right).\nonumber\end{eqnarray}$$

The tangent space of $\widetilde{M}_{\unicode[STIX]{x1D6FE}}$ at $\unicode[STIX]{x1D6F9}$ is spanned by the following orthonormal vectors:

$$\begin{eqnarray}i\unicode[STIX]{x1D6F9},\qquad i\widetilde{N},\qquad i\dot{\unicode[STIX]{x1D6F9}}\qquad \text{and}\qquad \dot{\unicode[STIX]{x1D6F9}}.\end{eqnarray}$$

Here $i\unicode[STIX]{x1D6F9}$ is a unit vertical vector of the Hopf fibration $\unicode[STIX]{x1D70B}:S^{5}\rightarrow \mathbb{C}\mathbb{P}^{2}$ and the others are horizontal.

Let $D$ and $\widetilde{A}$ be the flat connection of $\mathbb{C}^{3}$ and the shape operator of the hypersurface $\widetilde{M}_{\unicode[STIX]{x1D6FE}}$ in $S^{5}$ , respectively. Then by the Weingarten formula, we have

$$\begin{eqnarray}\widetilde{A}W=-D_{W}\widetilde{N}\quad \text{for }W\in T_{\unicode[STIX]{x1D6F9}}(\widetilde{M}_{\unicode[STIX]{x1D6FE}}).\end{eqnarray}$$

Covariant differentiation of $\widetilde{N}$ for $\dot{\unicode[STIX]{x1D6F9}}$ is given by

$$\begin{eqnarray}\displaystyle & & \displaystyle D_{\dot{\unicode[STIX]{x1D6F9}}}\widetilde{N}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}s}\widetilde{N}=-\dot{\unicode[STIX]{x1D6FC}}\dot{\unicode[STIX]{x1D6F9}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\Big((-\!\sin \unicode[STIX]{x1D6FC}(-{\dot{r}}\sin r\sin \unicode[STIX]{x1D703}+\dot{\unicode[STIX]{x1D703}}\cos r\cos \unicode[STIX]{x1D703})-\dot{\unicode[STIX]{x1D703}}\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D703})x,\nonumber\\ \displaystyle & & \displaystyle \quad (-\!\sin \unicode[STIX]{x1D6FC}(-{\dot{r}}\sin r\cos \unicode[STIX]{x1D703}-\dot{\unicode[STIX]{x1D703}}\cos r\sin \unicode[STIX]{x1D703})-\dot{\unicode[STIX]{x1D703}}\cos \unicode[STIX]{x1D6FC}\cos \unicode[STIX]{x1D703})y,{\dot{r}}\sin \unicode[STIX]{x1D6FC}\cos rz\Big)\nonumber\\ \displaystyle & & \displaystyle ~=-\dot{\unicode[STIX]{x1D6FC}}\dot{\unicode[STIX]{x1D6F9}}+\!\left((\sin \unicode[STIX]{x1D6FC}\cos \unicode[STIX]{x1D6FC}\sin r\sin \unicode[STIX]{x1D703}-\frac{\sin \unicode[STIX]{x1D6FC}}{\sin r}(\sin \unicode[STIX]{x1D6FC}\cos r\cos \unicode[STIX]{x1D703}\,+\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D703}))x,\right.\nonumber\\ \displaystyle & & \displaystyle \quad (\sin \unicode[STIX]{x1D6FC}\cos \unicode[STIX]{x1D6FC}\sin r\cos \unicode[STIX]{x1D703}+\frac{\sin \unicode[STIX]{x1D6FC}}{\sin r}(\sin \unicode[STIX]{x1D6FC}\cos r\sin \unicode[STIX]{x1D703}-\cos \unicode[STIX]{x1D6FC}\cos \unicode[STIX]{x1D703}))y,\nonumber\\ \displaystyle & & \displaystyle \qquad \cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}\cos rz\Big)\nonumber\\ \displaystyle & & \displaystyle ~=-(\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\cot r)\dot{\unicode[STIX]{x1D6F9}}.\nonumber\end{eqnarray}$$

Hence we obtain

(23) $$\begin{eqnarray}\widetilde{A}\dot{\unicode[STIX]{x1D6F9}}=(\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\cot r)\dot{\unicode[STIX]{x1D6F9}}.\end{eqnarray}$$

Also we have

$$\begin{eqnarray}\displaystyle & \widetilde{A}(i\unicode[STIX]{x1D6F9})=-i\widetilde{N}, & \displaystyle \nonumber\\ \displaystyle & \widetilde{A}(i\widetilde{N})=-i\unicode[STIX]{x1D6F9}+\unicode[STIX]{x1D707}i\widetilde{N}+\unicode[STIX]{x1D708}i\dot{\unicode[STIX]{x1D6F9}}, & \displaystyle \nonumber\\ \displaystyle & \widetilde{A}(i\dot{\unicode[STIX]{x1D6F9}})=\unicode[STIX]{x1D708}i\widetilde{N}+\unicode[STIX]{x1D706}i\dot{\unicode[STIX]{x1D6F9}}, & \displaystyle \nonumber\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}:=-\langle D_{i\widetilde{N}}\widetilde{N},i\widetilde{N}\rangle ,\qquad \unicode[STIX]{x1D708}:=-\langle D_{i\widetilde{N}}\widetilde{N},i\dot{\unicode[STIX]{x1D6F9}}\rangle ,\qquad \unicode[STIX]{x1D706}:=-\langle D_{i\dot{\unicode[STIX]{x1D6F9}}}\widetilde{N},i\dot{\unicode[STIX]{x1D6F9}}\rangle . & & \displaystyle \nonumber\end{eqnarray}$$

Computations (2.8) of [9] yield:

(24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707} & = & \displaystyle \sin ^{3}\unicode[STIX]{x1D6FC}(\cot r-\tan r)+3\sin \unicode[STIX]{x1D6FC}\cos ^{2}\unicode[STIX]{x1D6FC}\cot r-\frac{\cos ^{3}\unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703}),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(25) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D708} & = & \displaystyle \cos \unicode[STIX]{x1D6FC}\!\left(\!\sin ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})-\cos ^{2}\unicode[STIX]{x1D6FC}\cot r\!\right)\!,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(26) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706} & = & \displaystyle \sin \unicode[STIX]{x1D6FC}\!\left(\!\sin ^{2}\unicode[STIX]{x1D6FC}\cot r-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})-\cos ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)\!\right)\!.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Let $U=-\unicode[STIX]{x1D70B}_{\ast }(i\dot{\unicode[STIX]{x1D6F9}})$ . Then $\unicode[STIX]{x1D719}U=\unicode[STIX]{x1D70B}_{\ast }(\dot{\unicode[STIX]{x1D6F9}})$ . Also we have $\unicode[STIX]{x1D709}=-JN=-\unicode[STIX]{x1D70B}_{\ast }(i\widetilde{N})$ . Then the shape operator $A$ of $M_{\unicode[STIX]{x1D6FE}}$ in $\mathbb{C}\mathbb{P}^{2}$ with respect to $N$ is given by

(27) $$\begin{eqnarray}A\unicode[STIX]{x1D709}=\unicode[STIX]{x1D707}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D708}U,\qquad AU=\unicode[STIX]{x1D708}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D706}U,\qquad A\unicode[STIX]{x1D719}U=(\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\cot r)\unicode[STIX]{x1D719}U.\end{eqnarray}$$

Hence with respect to $M_{\unicode[STIX]{x1D6FE}}$ in $\mathbb{C}\mathbb{P}^{2}$ , we have

(28) $$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{trace}A-\unicode[STIX]{x1D702}(A\unicode[STIX]{x1D709})=\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\cot r+\unicode[STIX]{x1D706}\nonumber\\ \displaystyle & & \displaystyle \quad =\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\bigg((1+\sin ^{2}\unicode[STIX]{x1D6FC})\cot r-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \qquad -\cos ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)\bigg).\end{eqnarray}$$

Proposition 4. Let $(r(s),\unicode[STIX]{x1D703}(s),\unicode[STIX]{x1D6FC}(s))$ be a solution of the system of nonlinear ODE,

(29) $$\begin{eqnarray}\displaystyle {\dot{r}} & = & \displaystyle \cos \unicode[STIX]{x1D6FC},\qquad \dot{\unicode[STIX]{x1D703}}=\frac{\sin \unicode[STIX]{x1D6FC}}{\sin r},\nonumber\\ \displaystyle & & \displaystyle \dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\bigg((1+\sin ^{2}\unicode[STIX]{x1D6FC})\cot r-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle -\cos ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)\bigg)=0,\end{eqnarray}$$

such that the initial condition satisfying $0<r(0),\unicode[STIX]{x1D703}(0)<\unicode[STIX]{x1D70B}/2$ . Then the real hypersurface $M_{\unicode[STIX]{x1D6FE}}$ in $\mathbb{C}\mathbb{P}^{2}$ , defined by (21) is Levi-flat.

A special solution of (29) is given by

$$\begin{eqnarray}\unicode[STIX]{x1D703}=\text{constant},\qquad \unicode[STIX]{x1D6FC}\equiv 0\hspace{0.6em}{\rm mod}\hspace{0.2em}\unicode[STIX]{x1D70B}.\end{eqnarray}$$

In this case, we have $\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\cot r=\unicode[STIX]{x1D706}=0$ and $M_{\unicode[STIX]{x1D6FE}}$ is a ruled real hypersurface.

Proposition 5. Let $k$ be a nonzero constant and let $(r(s),\unicode[STIX]{x1D703}(s),\unicode[STIX]{x1D6FC}(s))$ be a solution of the system of nonlinear ODE,

(30) $$\begin{eqnarray}\displaystyle {\dot{r}} & = & \displaystyle \cos \unicode[STIX]{x1D6FC},\qquad \dot{\unicode[STIX]{x1D703}}=\frac{\sin \unicode[STIX]{x1D6FC}}{\sin r},\nonumber\\ \displaystyle & & \displaystyle \dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\bigg((1+\sin ^{2}\unicode[STIX]{x1D6FC})\cot r-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle -\,\cos ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)\bigg)=k,\end{eqnarray}$$

such that the initial condition satisfying $0<r(0),\unicode[STIX]{x1D703}(0)<\unicode[STIX]{x1D70B}/2$ . Then the real hypersurface $M_{\unicode[STIX]{x1D6FE}}$ in $\mathbb{C}\mathbb{P}^{2}$ , defined by (21) is Levi-umbilical.

A special solution of (30) is given by

$$\begin{eqnarray}r=\text{constant},\qquad \unicode[STIX]{x1D6FC}\equiv \unicode[STIX]{x1D70B}/2\hspace{0.6em}{\rm mod}\hspace{0.2em}\unicode[STIX]{x1D70B}.\end{eqnarray}$$

In the case $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70B}/2$ , we have $\unicode[STIX]{x1D707}=2\cot 2r$ , $\unicode[STIX]{x1D708}=0$ and $\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\cot r=\unicode[STIX]{x1D706}=\cot r$ . Hence $M_{\unicode[STIX]{x1D6FE}}$ is a geodesic sphere of radius $r$ $(0<r<\unicode[STIX]{x1D70B}/2)$ with $k=2\cot r$ .