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 [Reference Siu14] proved the nonexistence of compact smooth Levi-flat hypersurfaces in
$\mathbb{C}\mathbb{P}^{n}$
of dimensions
${\geqslant}3$
. When
$n=2$
, Ohsawa [Reference Ohsawa13] 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. [Reference Cho6]). 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 [Reference Takagi16], [Reference Takagi17] classified the homogeneous real hypersurfaces in
$\mathbb{C}\mathbb{P}^{n}$
into six types. Cecil and Ryan [Reference Cecil and Ryan4] 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 [Reference Kimura8] 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 [Reference Berndt2] 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.
-
(I) If
$\widetilde{M}_{n}(c)=\mathbb{C}\mathbb{P}^{n}$
, then
$M$
is congruent to one of the following:-
(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) a tube of radius
$r$
over a complex quadric
$\mathbb{CQ}^{n-1}$
, where
$0<r<\frac{\unicode[STIX]{x1D70B}}{4}$
.
-
-
(II) If
$\widetilde{M}_{n}(c)=\mathbb{CH}^{n}$
, then
$M$
is congruent to one of the following:-
(1) a horosphere in
$\mathbb{CH}^{n}$
, -
(2) a geodesic hypersphere or a tube of radius
$r\in \mathbb{R}_{+}$
over a totally geodesic
$\mathbb{CH}^{n-1}$
, -
(3) a tube of radius
$r\in \mathbb{R}_{+}$
over a totally real hyperbolic space
$\mathbb{RH}^{n}$
.
-
-
(III) If
$\widetilde{M}_{n}(c)=\mathbb{CE}^{n}$
, then
$M$
is locally congruent to one of the following:-
(1) a sphere
$S^{2n-1}(r)$
of radius
$r\in \mathbb{R}_{+}$
, -
(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
$$\begin{eqnarray}\unicode[STIX]{x1D719}^{2}=-I+\unicode[STIX]{x1D702}\otimes \unicode[STIX]{x1D709},~\unicode[STIX]{x1D702}(\unicode[STIX]{x1D709})=1.\end{eqnarray}$$
Then we can find always a compatible Riemannian metric, namely which satisfies
$$\begin{eqnarray}g(\unicode[STIX]{x1D719}X,\unicode[STIX]{x1D719}Y)=g(X,Y)-\unicode[STIX]{x1D702}(X)\unicode[STIX]{x1D702}(Y)\end{eqnarray}$$
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
$$\begin{eqnarray}\unicode[STIX]{x1D719}\unicode[STIX]{x1D709}=0,\qquad \unicode[STIX]{x1D702}\circ \unicode[STIX]{x1D719}=0,\qquad \unicode[STIX]{x1D702}(X)=g(X,\unicode[STIX]{x1D709}).\end{eqnarray}$$
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 [Reference Blair3].
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
$$\begin{eqnarray}[JX,JY]-[X,Y]\in D~(or~[JX,Y]+[X,JY]\in D)\end{eqnarray}$$
and
$$\begin{eqnarray}[J,J](X,Y)=0\end{eqnarray}$$
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 [Reference Ianus7]. 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 [Reference Cho5], [Reference Ianus7], [Reference Tanno18] 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
$$\begin{eqnarray}\tilde{J}X=\unicode[STIX]{x1D719}X+\unicode[STIX]{x1D702}(X)N,\qquad \tilde{J}N=-\unicode[STIX]{x1D709}.\end{eqnarray}$$
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
$$\begin{eqnarray}\displaystyle & (\unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D719})Y=\unicode[STIX]{x1D702}(Y)AX-g(AX,Y)\unicode[STIX]{x1D709}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D709}=\unicode[STIX]{x1D719}AX. & \displaystyle\end{eqnarray}$$
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
$$\begin{eqnarray}(\unicode[STIX]{x1D6FB}_{X}A)Y-(\unicode[STIX]{x1D6FB}_{Y}A)X=\frac{c}{4}\{\unicode[STIX]{x1D702}(X)\unicode[STIX]{x1D719}Y-\unicode[STIX]{x1D702}(Y)\unicode[STIX]{x1D719}X-2g(\unicode[STIX]{x1D719}X,Y)\unicode[STIX]{x1D709}\}.\end{eqnarray}$$
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
$$\begin{eqnarray}g((\unicode[STIX]{x1D719}A+A\unicode[STIX]{x1D719})X,Y)=0,\quad X,Y\bot \unicode[STIX]{x1D709},\end{eqnarray}$$
and
$M$
is Levi-umbilical if and only if there exists nonzero constant
$k\in \mathbb{R}$
such that
$$\begin{eqnarray}g((\unicode[STIX]{x1D719}A+A\unicode[STIX]{x1D719})X,Y)=kg(\unicode[STIX]{x1D719}X,Y),\quad X,Y\bot \unicode[STIX]{x1D709}.\end{eqnarray}$$
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, [Reference Kimura and Maeda10] for the case
$\mathbb{C}\mathbb{P}^{n}$
and see [Reference Ahn, Lee and Suh1] for the case
$\mathbb{CH}^{n}$
, respectively). The shape operator
$A$
is written by the following form:
$$\begin{eqnarray}\begin{array}{@{}rcl@{}}A\unicode[STIX]{x1D709}\ & =\ & \unicode[STIX]{x1D707}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D708}V~(\unicode[STIX]{x1D708}\neq 0),\\ AV\ & =\ & \unicode[STIX]{x1D708}\unicode[STIX]{x1D709},\\ AX\ & =\ & 0\quad \text{for any}~X\bot \unicode[STIX]{x1D709},V,\end{array}\end{eqnarray}$$
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
$$\begin{eqnarray}\displaystyle & & \displaystyle g(\,(\unicode[STIX]{x1D6FB}_{X}A)\unicode[STIX]{x1D719}Y+A(\unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D719})Y+A\unicode[STIX]{x1D719}\unicode[STIX]{x1D6FB}_{X}Y+(\unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D719})AY\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D719}(\unicode[STIX]{x1D6FB}_{X}A)Y+\unicode[STIX]{x1D719}A\unicode[STIX]{x1D6FB}_{X}Y,Z)\!\,+\,g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)Y,\unicode[STIX]{x1D6FB}_{X}Z)\nonumber\\ \displaystyle & & \displaystyle \quad =k(g((\unicode[STIX]{x1D6FB}_{X}\unicode[STIX]{x1D719})Y,Z)+g(\unicode[STIX]{x1D719}\unicode[STIX]{x1D6FB}_{X}Y,Z)+g(\unicode[STIX]{x1D719}Y,\unicode[STIX]{x1D6FB}_{X}Z)),\end{eqnarray}$$
for any vector fields
$X,Y,Z\bot \unicode[STIX]{x1D709}$
. Use (7) to get
$$\begin{eqnarray}\displaystyle & & \displaystyle g((\unicode[STIX]{x1D6FB}_{X}A)\unicode[STIX]{x1D719}Y+\unicode[STIX]{x1D719}(\unicode[STIX]{x1D6FB}_{X}A)Y,Z)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,g(\unicode[STIX]{x1D702}(Y)A^{2}X-g(AX,Y)A\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702}(AY)AX-g(AX,AY)\unicode[STIX]{x1D709},Z)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)\unicode[STIX]{x1D6FB}_{X}Y,Z)-g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)\unicode[STIX]{x1D6FB}_{X}Z,Y)\nonumber\\ \displaystyle & & \displaystyle \quad =k(g(\unicode[STIX]{x1D719}\unicode[STIX]{x1D6FB}_{X}Y,Z)-g(\unicode[STIX]{x1D719}\unicode[STIX]{x1D6FB}_{X}Z,Y)).\end{eqnarray}$$
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
$$\begin{eqnarray}\displaystyle & & \displaystyle g((\unicode[STIX]{x1D6FB}_{X}A)\unicode[STIX]{x1D719}Y+\unicode[STIX]{x1D719}(\unicode[STIX]{x1D6FB}_{X}A)Y,Z)\nonumber\\ \displaystyle & & \displaystyle \quad +\,g(\unicode[STIX]{x1D702}(Y)A^{2}X-g(AX,Y)A\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702}(AY)AX-g(AX,AY)\unicode[STIX]{x1D709},Z)\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D702}(\unicode[STIX]{x1D6FB}_{X}Y)g(U,Z)-\unicode[STIX]{x1D702}(\unicode[STIX]{x1D6FB}_{X}Z)g(U,Y)=0,\end{eqnarray}$$
where we have put
$U=\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D709}$
. Use (8) to obtain
$$\begin{eqnarray}\displaystyle & & \displaystyle g((\unicode[STIX]{x1D6FB}_{X}A)Z,\unicode[STIX]{x1D719}Y)-g((\unicode[STIX]{x1D6FB}_{X}A)Y,\unicode[STIX]{x1D719}Z)\nonumber\\ \displaystyle & & \displaystyle \quad =g(\unicode[STIX]{x1D719}AX,Y)g(U,Z)-g(\unicode[STIX]{x1D719}AX,Z)g(U,Y)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D702}(Z)g(A^{2}X,Y)-\unicode[STIX]{x1D702}(Y)g(A^{2}X,Z)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\unicode[STIX]{x1D702}(AZ)g(AX,Y)-\unicode[STIX]{x1D702}(AY)g(AX,Z).\end{eqnarray}$$
Taking the cyclic sum of (16) for
$X,Y,Z$
, using (9) we have
$$\begin{eqnarray}\displaystyle & & \displaystyle g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)X,Y)g(U,Z)+g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)Y,Z)g(U,X)\nonumber\\ \displaystyle & & \displaystyle \quad +\,g((A\unicode[STIX]{x1D719}+\unicode[STIX]{x1D719}A)Z,X)g(U,Y)=0.\end{eqnarray}$$
Using (11) in (17) again, we have
$$\begin{eqnarray}k(g(\unicode[STIX]{x1D719}X,Y)g(U,Z)+g(\unicode[STIX]{x1D719}Y,Z)g(U,X)+g(\unicode[STIX]{x1D719}Z,X)g(U,Y))=0.\end{eqnarray}$$
If we put
$Z=U$
in (18), then we have
$$\begin{eqnarray}k(g(\unicode[STIX]{x1D719}X,Y)\Vert U\Vert ^{2}+g(\unicode[STIX]{x1D719}Y,U)g(U,X)+g(\unicode[STIX]{x1D719}U,X)g(U,Y))=0.\end{eqnarray}$$
Replace
$Y$
by
$\unicode[STIX]{x1D719}X$
in (19), then it turns to
$$\begin{eqnarray}k(g(X,X)\Vert U\Vert ^{2}-g(X,U)g(U,X)+g(\unicode[STIX]{x1D719}U,X)g(U,\unicode[STIX]{x1D719}X))=0.\end{eqnarray}$$
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 [Reference Kon11] (in case of
$\mathbb{C}\mathbb{P}^{n}$
), [Reference Vernon19], [Reference Suh15] (in case of
$\mathbb{CH}^{n}$
), and [Reference Okumura12] (in case of
$\mathbb{CE}^{n}$
) we find the following.
-
(I) If
$\widetilde{M}_{n}(c)=\mathbb{C}\mathbb{P}^{n}$
, then
$M$
is locally congruent to one of the following:-
(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) a tube of radius
$r$
over a complex quadric
$\mathbb{CQ}^{n-1}$
, where
$0<r<\frac{\unicode[STIX]{x1D70B}}{4}$
.
-
-
(II) If
$\widetilde{M}_{n}(c)=\mathbb{CH}^{n}$
, then
$M$
is locally congruent to one of the following:-
(1) a horosphere in
$\mathbb{CH}^{n}$
, -
(2) a geodesic hypersphere or a tube of radius
$r\in \mathbb{R}_{+}$
over a totally geodesic
$\mathbb{CH}^{n-1}$
, -
(3) a tube of radius
$r\in \mathbb{R}_{+}$
over a totally real hyperbolic space
$\mathbb{RH}^{n}$
.
-
-
(III) If
$\widetilde{M}_{n}(c)=\mathbb{CE}^{n}$
, then
$M$
is locally congruent to one of the following:-
(1) a sphere
$S^{2n-1}(r)$
of radius
$r\in \mathbb{R}_{+}$
, -
(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 [Reference Kimura9], 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
$$\begin{eqnarray}\begin{array}{@{}l@{}}\widetilde{M}_{\unicode[STIX]{x1D6FE}}:=\mathop{\bigcup }_{s\in I}\sin r(s)(\sin \unicode[STIX]{x1D703}(s)S^{1}\times \cos \unicode[STIX]{x1D703}(s)S^{1})\times \cos r(s)S^{1}\\ \qquad \quad \text{and}\qquad M_{\unicode[STIX]{x1D6FE}}:=\unicode[STIX]{x1D70B}(\widetilde{M}_{\unicode[STIX]{x1D6FE}}),\end{array}\end{eqnarray}$$
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
$$\begin{eqnarray}{\dot{r}}=\cos \unicode[STIX]{x1D6FC},\qquad \dot{\unicode[STIX]{x1D703}}=\frac{\sin \unicode[STIX]{x1D6FC}}{\sin r}.\end{eqnarray}$$
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
$$\begin{eqnarray}\widetilde{A}\dot{\unicode[STIX]{x1D6F9}}=(\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\cot r)\dot{\unicode[STIX]{x1D6F9}}.\end{eqnarray}$$
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 [Reference Kimura9] yield:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707} & = & \displaystyle \sin ^{3}\unicode[STIX]{x1D6FC}(\cot r-\tan r)+3\sin \unicode[STIX]{x1D6FC}\cos ^{2}\unicode[STIX]{x1D6FC}\cot r-\frac{\cos ^{3}\unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703}),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D708} & = & \displaystyle \cos \unicode[STIX]{x1D6FC}\!\left(\!\sin ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})-\cos ^{2}\unicode[STIX]{x1D6FC}\cot r\!\right)\!,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706} & = & \displaystyle \sin \unicode[STIX]{x1D6FC}\!\left(\!\sin ^{2}\unicode[STIX]{x1D6FC}\cot r-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})-\cos ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)\!\right)\!.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
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
$$\begin{eqnarray}A\unicode[STIX]{x1D709}=\unicode[STIX]{x1D707}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D708}U,\qquad AU=\unicode[STIX]{x1D708}\unicode[STIX]{x1D709}+\unicode[STIX]{x1D706}U,\qquad A\unicode[STIX]{x1D719}U=(\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\cot r)\unicode[STIX]{x1D719}U.\end{eqnarray}$$
Hence with respect to
$M_{\unicode[STIX]{x1D6FE}}$
in
$\mathbb{C}\mathbb{P}^{2}$
, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \operatorname{trace}A-\unicode[STIX]{x1D702}(A\unicode[STIX]{x1D709})=\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\cot r+\unicode[STIX]{x1D706}\nonumber\\ \displaystyle & & \displaystyle \quad =\dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\bigg((1+\sin ^{2}\unicode[STIX]{x1D6FC})\cot r-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \qquad -\cos ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)\bigg).\end{eqnarray}$$
Proposition 4. Let
$(r(s),\unicode[STIX]{x1D703}(s),\unicode[STIX]{x1D6FC}(s))$
be a solution of the system of nonlinear ODE,
$$\begin{eqnarray}\displaystyle {\dot{r}} & = & \displaystyle \cos \unicode[STIX]{x1D6FC},\qquad \dot{\unicode[STIX]{x1D703}}=\frac{\sin \unicode[STIX]{x1D6FC}}{\sin r},\nonumber\\ \displaystyle & & \displaystyle \dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\bigg((1+\sin ^{2}\unicode[STIX]{x1D6FC})\cot r-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle -\cos ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)\bigg)=0,\end{eqnarray}$$
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,
$$\begin{eqnarray}\displaystyle {\dot{r}} & = & \displaystyle \cos \unicode[STIX]{x1D6FC},\qquad \dot{\unicode[STIX]{x1D703}}=\frac{\sin \unicode[STIX]{x1D6FC}}{\sin r},\nonumber\\ \displaystyle & & \displaystyle \dot{\unicode[STIX]{x1D6FC}}+\sin \unicode[STIX]{x1D6FC}\bigg((1+\sin ^{2}\unicode[STIX]{x1D6FC})\cot r-\frac{\cos \unicode[STIX]{x1D6FC}\sin \unicode[STIX]{x1D6FC}}{\sin r}(\cot \unicode[STIX]{x1D703}-\tan \unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle -\,\cos ^{2}\unicode[STIX]{x1D6FC}(\cot r+\tan r)\bigg)=k,\end{eqnarray}$$
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
$k=2\cot r$
.
