Mem. Gra. Sci. Eng. Shimane Univ.
Series B: Mathematics
50 (2017), pp. 1–10

GEOMETRY OF RULED REAL HYPERSURFACES IN A
NONFLAT COMPLEX SPACE FORM
SADAHIRO MAEDA
Communicated by Takumi Yamada and Shun Maeta(Received: June 19, 2016, Revised: August 31, 2016)

Abstract. This paper is the survey of the works with T. Adachi, H. Tanabe
and M. Kimura. Ruled real hypersurfaces are typical examples of non-Hopf
hypersurfaces in a nonflat complex space form. These examples are constructedby the same method in a complex projective space and a complex hyperbolic
space. We shall explain geometric properties of ruled real hypersurfaces which
depend on the sign of the sectional curvature of those ambient spaces.

1. Introduction
Among real hypersurfaces isometrically immersed into a nonflat complex space
fn (c)(= CP n (c) or CH n (c)) , n ≧ 2, we pay particular attention to ruled
form Mfn (c) is constructed
real hypersurfaces. Every ruled real hypersurface M in M(s)fn−1 (c) on a smooth realby attaching totally geodesic complex hypersurfaces M
fn (c) with its arclength s defined on some open interval I(⊂ R)
curve γ : I → M(s)fn−1
in such a way that the hypersurface M(c) is orthogonal to the real plane
spanned by {γ̇(s), J γ̇(s)} at each point γ(s), where J is the Kähler structure offthe ambient∪ (s)space Mn (c). Then we get a ruled real hypersurface M defined by
fn−1 (c).
M
M :=
s∈I

fn (c) is an abundant class which gives
The class of ruled real hypersurfaces in M
fn (c). For example, in the
fruitful results in the theory of real hypersurfaces in M
2n−1class of all homogeneous real hypersurfaces M
in CH n (c) (that is, they areorbits of some subgroups of the full isometry group I(CH n (c)) of CH n (c)), there
exists just one example which is ruled in CH n (c). Moreover, in this class M is
minimal if and only if M is ruled in the ambient space CH n (c) (cf. [4, 6, 5]). On
2010 Mathematics Subject Classification. 53B25, 53C40.
Key words and phrases. ruled real hypersurfaces, nonflat complex space forms, shape operator,
η-parallel, holomorphic distributions, the homogeneous ruled real hypersurface, minimal ruled real
hypersurfaces, geodesics, circles, sectional curvatures, characteristic vector fields, the vector U ,integral curves, the first curvature.
1

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S. MAEDA

the contrary, in the class of all homogeneous real hypersurfaces in CP n (c), there
exist six examples which are minimal in CP n (c). However, note that they are not
ruled in CP n (c) (cf. [19, 20]). So we can see that this fact depends on the sign of
fn (c).
the sectional curvature of the ambient space M
The main purpose of this paper is to clarify geometric properties of ruled realfn (c) from this viewpoint.
hypersurfaces in M
This paper is related to a talk of the author in 2016 (June 16 ∼ 18) International
Conference of the Honam Mathematical Society at Chonbuk National University,
Jeonju, 54896, Republic of Korea.
The author would like to express his hearty thanks to Professors Sang-Eon Han
and Jong Taek Cho for their warm hospitalities.
2. Ruled real hypersurfaces
fn (c) is said to be ruled if the holomorphic distribution
Real hypersurface M in M
0
T M := {X ∈ TM |η(X) = 0} is integrable and each of its leaves (i.e., maximal
fn−1 (c) of M
fn (c),
integral manifolds) is a totally geodesic complex hypersurface M
where η is the contact form on M . The construction of ruled real hypersurfaces inIntroduction means that in general a ruled real hypersurface has singularities. So
we must omit such points.
fn−1 (c) is totally geodesic in a given ruled real hypersurface
Remark 2.1. Every leaf M
fn (c) and M
fn−1 (c) is totally geodesic
M because M is isometrically immersed into M
fn (c).
in the ambient space M
We define two functions µ, ν : M → R by µ = ⟨Aξ, ξ⟩ and ν = ∥Aξ − µξ∥. These
functions µ and ν are important quantities which measure how far the characteristic
vector field ξ is from being a principal curvature vector. A characterization of ruled
real hypersurfaces in terms of the functions µ, ν and the shape operator A is given
as follows.
Proposition 2.2 ([17]). Let M be a real hypersurface in a nonflat complex space
fn (c), n ≧ 2. Then the following three conditions are mutually equivalent.
form M(1) M is a ruled real hypersurface.
(2) The shape operator A of M satisfies ⟨AX, Y ⟩ = 0 for any tangent vectors
X, Y ∈ Tx M orthogonal to ξx at each point x ∈ M .
(3) The set M1 = {x ∈ M | ν(x) ̸= 0} is an open dense subset of M and there
exists a unit vector field U on M1 such that it is orthogonal to ξ and satisfies
that
(2.1)

Aξ = µξ + νU,

AU = νξandAX = 0for an arbitrary tangent vector X orthogonal to both ξ and U .
The reason why we consider the set M1 is that we define the vector field U clearlyby U = (Aξ − µξ)/ν. By this definition, the vector field U cannot be extended toa smooth vector field on M in general. But in some cases, considering −U insteadof U and −ν instead of ν on some connected components of M1 , we can define a

GEOMETRY OF RULED REAL HYPERSURFACES

3smooth vector field U and a smooth function ν on M satisfying the equalities in(2.1). Hence in this paper, considering a ruled real hypersurface M , we treat M asa diﬀerentiable manifold such that M = M1 ∪ {x ∈ M |ν(x) = 0}.
3. A characterization of the homogeneous ruled real
hypersurfaces in CH n (c)We first show the following:
Proposition 3.1. In CP n (c), n ≧ 2 every ruled real hypersurface is not complete.
But, in CH n (c), n ≧ 2 there do exist complete ruled real hypersurfaces.
fn (c),
Proof. Let M be a ruled real hypersurface of a nonflat complex space form Mn ≧ 2. By direct computation (for details, see [8]) we havec
(3.1)
ϕU ν = ν 2 + .
4Here, ϕ is the structure tensor on M defined by ϕX := JX −η(X)ξ for all X ∈ TM .
Solving the diﬀerential equation (3.1), the function ν is of the following form oneach integral curve ρ with ρ̇(0) = ϕUρ(0) through the point ρ(0) (cf. [1]). Whenc > 0,( √c )( √c)(3.2)
ν(ρ(s)) = ±
tan
s+a
2
2
with some constant a and when c < 0,
 (√ )
(√
)
|c|
|c|


±
tanh
s
+
a
,

2

 ( √2 )|c|(3.3)
ν(ρ(s)) = ± 2 ,


(√ )
(√
)

|c|
|c|

±
coth
s
+
a
2
2
with
initial condition |ν(ρ(0))| is less than√ some constant√ a, according as the √|c| /2, equal to |c| /2 or greater than |c| /2. Since we have a choice of directions for the vector field U , we put double signs in (3.2) and (3.3). Moreover,
we can show that every integral curve ρ = ρ(s) of ϕU is a geodesic on each ruled
fn (c), which means that every ruled real hypersurface ofreal hypersurface M in MnCP (c) is not complete. When c < 0, we emphasize that there exist two minimal ruled real hypersurfaces which are complete. One is homogeneous and theother is not homogeneous in the ambient space CH n (c) (for details, see the next
section).
□Motivated by Proposition 3.1, we are interested in the homogeneous ruled real hypersurface M in CH n (c). This homogeneous real hypersurface M is an orbit of thesubgroup (of the full isometry group I(CH n (c)) which is the direct of I(CH n−1 (c))
2
and a one-parameter subgroupi.e.,√ {φs } whose orbit is a horocycle in RH (c/4),
2
a circle of positive curvature |c| /2 on a totally real totally geodesic RH (c/4) ofconstant sectional curvature c/4 in CH n (c). Here, I(CH n−1 (c)) is the full isometrygroup of a totally geodesic complex hypersurface CH n−1 (c) of CH n (c)). Note that

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S. MAEDA

the homogeneous ruled real hypersurface M is obtained by attaching holomorphic
hyperplanes CH n−1 (c) on the horocycle (for details, see [10, 6, 12]).
Our aim here is to characterize the homogeneous ruled real hypersurface M in
CH n (c) by observing some geodesics and all integral curves of the characteristic
vector field ξ on M . For this purpose, by observing the extrinsic shape of somegeodesics we prepare the following lemma which is a characterization of all ruled
fn (c).
real hypersurfaces in a nonflat complex space form M
fn (c),
Lemma 3.2. Let M be a real hypersurface of a nonflat complex space form Mn ≧ 2 through an isometric immersion. Then M is ruled in the ambient space
fn (c) if and only if at an arbitrary point p ∈ M there exist such orthonormalM
vectors v1 , v2 , . . . , v2n−2 (∈ Tp M ) orthogonal to the characteristic vector ξp that every
geodesic γij,p on M through the point γij,p (0) = p in the direction of vi + vj (1 ≦fn (c).
i ≦ j ≦ 2n − 2) is an extrinsic geodesic, namely γij,p is also a geodesic in M
Remark 3.3. In the statement of Lemma 3.2, when c > 0, at every fixed pointp ∈ M if we delete the condition that all geodesics through the point γij,p (0) = p
in the direction of vi + vj (1 ≦ i < j ≦ 2n − 2)
√ (, i.e., in this case the initial
vector γ̇ij,p (0) is expressed as γ̇ij,p (0) = (vi + vj )/ 2 ) are extrinsic geodesics, thislemma
√ is no longer true (see Remark 3 in [18]). Indeed, let M be a tube of radiusπ/(2 c ) around a totally geodesic CP (n−1)/2 (c), where n(≧ 3) is odd.√ Then M
has three distinct constant√ principal curvatures 0 with multiplicity 1, c /2 withmultiplicity n − 1 and − c /2 with multiplicity n − 1, where Aξ = 0. We√takeorthonormalbases e1 , e2 , . . . , en−1 and f1 , f2 , . . . , fn−1 of principal curvatures c /2√and − c/2, respectively. We here
2n−2 } of√ set an orthonormal basis {v1 , v2 , . . . , v√
0
T M defined by vi := (ei + fi )/ 2 (1 ≦ i ≦ n − 1) and vn−1+i := (ei − fi )/ 2 (1 ≦i ≦ n − 1). Then by the construction of the orthonormal basis {v1 , v2 , . . . , v2n−2 }e X Y = ∇X Y + ⟨AX, Y ⟩N and the property ofof T 0 M , the Gauss formula: ∇our real hypersurface M : ⟨(∇X A)X, X⟩ = 0 for all X ∈ TM we can see that
our homogeneous real hypersurface M of type (A2 ) satisfies that every geodesic
γii,p (0) = p in the direction of vi (i = 1, 2, . . . , 2n − 2) is an extrinsic geodesic.
When c < 0, in the statement of Lemma 3.2 we do not know the answer tothe question “Can we delete the condition that all geodesics through the point
γij,p (0) = p in the direction of vi + vj (1 ≦ i < j ≦ 2n − 2) are extrinsic geodesics?”.
At the end of Remark 3.3 we review the notion of real hypersurfaces of type (A2 )in a complex projective space. A real hypersurface
M of CP n (c), n ≧ 3 is of type√(A2 ) if M is a tube of radius r (0 < r < π/ c ) around a totally geodesic CP ℓ (c)with 1 ≦ ℓ ≦ n − 2 in CP n (c).
We are now in a position to show the following:
Theorem 3.4 ([12]). A real hypersurface M isometrically immersed into CH n (c),n ≧ 2 is locally congruent to the homogeneous ruled real hypersurface if and only
if it satisfies the following three conditions.
i) At an arbitrary point p ∈ M , there exist such orthonormal vectors v1 , v2 , . . . ,v2n−2 (∈ Tp M ) orthogonal to the characteristic vector ξp that every geodesic

GEOMETRY OF RULED REAL HYPERSURFACES

5γij,p on M through the point γij,p (0) = p in the direction of vi + vj (1 ≦i ≦ j ≦ 2n − 2) is an extrinsic geodesic, namely γij,p is also a geodesic in
CH n (c).
ii) At an arbitrary point p ∈ M , the integral curve γp of the characteristic
vector field ξ on M through γp (0) = p lies locally on a totally real totally
geodesic 2-dimensional real hyperbolic space RH 2 (c/4) of constant sectional
curvature c/4.
e γ̇p γ̇p ∥ of the curve γp in ii) does not dependiii) The curvature function κp = ∥∇e is the Riemannian connection of the ambienton the choice of γp , where ∇nspace CH (c). This means that for any curves γp , γq in ii) their curvaturefunctions κp (s) and κy (s) satisfy the following equality with some constants0 : κp (s) = κq (s + s0 ) for −∞ < ∀s < ∞, where p, q are any points of M .
Due to the works of J. Berndt and others ([4, 5, 6]), we find that the classof all homogeneous real hypersurfaces M in CH n (c) has just one example whichis minimal in this space. Furthermore, in this ambient space, a homogeneous
real hypersurface M is minimal if and only if it is ruled. This fact implies that
it is interesting to study this minimal homogeneous real hypersurface from theviewpoint of the geometry of ruled real hypersurfaces. At the end of this section wecharacterize this minimal homogeneous real hypersurface in CH n (c) in the class of
all ruled real hypersurfaces M by investing the first curvature of all integral curvesof the characteristic vector field ξ and the vector field U . Note that there exist
minimal non-homogeneous ruled real hypersyrfaces in CH n (c) (see [10, 1]).
We here show the following:
Theorem 3.5 ([15]). Let M 2n−1 be a ruled real hypersurface of CH n (c), n ≧ 2.
Then M is homogeneous in this ambient space, i.e., M is an orbit of a subgroup of
the full isometry group I(CH n (c)) of CH n (c) if and only if M satisfies the following
two conditions:(1) Every integral curve γU of the vector field U in (2.1)√ is mapped nto a curveeof positive first curvature κU := ∥∇U U ∥ with κU ≦ |c| in CH (c);(2) Every integral curve γξ of the characteristic vector field ξ in (2.1)√
is mappede ξ ξ∥ with κξ ≦ |c| /2 into a curve of positive first curvature κξ := ∥∇nCH (c).
e is the Riemannian connection of the ambient space CH n (c).
Here, ∇We remark that all curves γU = γU (s)√in Condition (1) in Theorem 3.5 aremapped to circles of the same curvature |c| on a holomorphic line CH 1 (c) of
CH n (c) (, i.e., they are mapped to horocycles in CH 1 (c)) and all curves γξ =√γξ (s)in Condition (2) in Theorem 3.5 are mapped to circles of the same curvature |c| /2
on a totally real totally geodesic surface RH 2 (c/4) of constant sectional curvature
c/4 (, i.e., they are mapped to horocycles in RH 2 (c/4)) in the ambient space
CH n (c).

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S. MAEDA

4. Classification theorems of minimal ruled real hypersurfaces
fn (c)in M
We first classify minimal ruled real hypersurfaces in CP n (c).
Theorem 4.1 ([1]). There exists the unique minimal ruled real hypersurface in
CP n (c), n ≧ 2 up to the action of the full isometry group I(CP n (c)). It is constructed by attaching holomorphic hyperplanes CP n−1 (c) on a geodesic on a totallyreal totally geodesic RP 2 (c/4) of constant sectional curvature c/4 in the ambient
space CP n (c).
We shall classify minimal ruled real hypersurfaces in CH n (c).
Theorem 4.2 ([1]). There exist three minimal ruled real hypersurfaces in CH n (c),n ≧ 2 up to the action of the full isometry group I(CH n (c)). They are constructedby attaching holomorphic hyperplanes CH n−1 (c) on a circle of curvature k(≧ 0) on
a totally real totally geodesic RH 2 (c/4) of constant sectional curvature c/4 in thenambient
hypersurfaces of parabolic type (, i.e.,√ space CH (c). Minimal ruled real √k = |c| /2) and axial type (, i.e., 0 ≦ k < |c| /2)√ are complete, but the minimal
ruled real hypersurface of elliptic type (, i.e., k > |c| /2) is not complete.
For the sake of your convenience we here give images of ruled{ real hypersurfacesnof axial, parabolic andelliptic
types
in
a
ball
model
D
=
(z1 , . . . , zn ) ∈ Cn}
|z1 |2 + · · · + |zn |2 < 1 of a complex hyperbolic space CH n . In figures, dotted lines
imply leaves of ruled real hypersurfaces and lines inside of the balls imply totallyreal circles on a totally real totally geodesic RH 2 , which are denoted by γk in theabove. If we regard these figures as models of RH 2 containing the totally realcircles, the dotted lines show integral curves of ϕU . We note that all the totallyreal circles have the same pairs of points at infinity as we can see in Fig. 1. In Fig.
2, two lines show horocyclic totally real circles having the same points at infinity.
One may easily guess that the only parabolic one is homogeneous.

Figure 1.
axial typeFigure 2.
parabolic typeFigure 3.
elliptic type

Remark 4.3. In the minimal ruled real hypersurface M of axial type, if we delete
the subset {x ∈ M |ν(x) = 0} of M , then M is not complete.

GEOMETRY OF RULED REAL HYPERSURFACES

7

fn (c)
5. Sectional curvatures of ruled real hypersurfaces in M
In classical diﬀerential geometry, it is well-known that every ruled surface in
3-dimensional Euclidean space R3 has nonpositive Gaussian curvature. Hence wehave an interest in the sectional curvature of every ruled real hypersurface M infn (c), n ≧ 2. Since every leaf M
fn−1 (c) is totallya nonflat complex space form M
geodesic in M (see Remark 2.1(2)), in the case of n ≧ 3 we find easily the fact thatfn (c) satisfies
the sectional curvature K of every ruled real hypersurface M in M
sharp inequalities |c/4| ≦ |K(X, Y )| ≦ |c| for an arbitrary pair of orthonormal
vectors X and Y orthogonal to the characteristic vector ξp at each point p of M .
Here, note that in the case of n = 2 we find also the fact that the sectional curvature
fn (c) satisfies K(X, Y ) = c for each pair
K of every ruled real hypersurface M in M
{X, Y } of orthonormal vectors X and Y that are orthogonal to ξ.
Inspired by this fact we study ruled real hypersurfaces M 2n−1 , n ≧ 2 having thesectional curvature K with |c/4| ≦ |K| ≦ |c|, that is, the sectional curvature K
satisfies sharp inequalities either c/4 ≦ K(X, Y ) ≦ c (c > 0) or c ≦ K(X, Y ) ≦c/4 (c < 0) for an arbitrary pair of orthonormal vectors X and Y that are notnecessarily tangent to the leaf at each point p of M .
First of all we give the following inequalities for the sectional curvature K of aruled real hypersurface M (cf. [21]).
fn (c), n ≧ 2. Then the secLemma 5.1. Let M be a ruled real hypersurface in M
tional curvature K of M satisfies the following :(1) If c > 0, we have (c/4) − ν(p)2 ≦ K ≦ c at every point p ∈ M ;(2) If c < 0, we have (c/4) − ν(p)2 ≦ K ≦ c/4 at every point p ∈ M with
ν(p)2 ≧ 3|c|/4 and c ≦ K ≦ c/4 at the point p ∈ M with ν(p)2 < 3|c|/4.
These estimates are sharp in the sense that at each point of M we can take a pairof orthonormal tangent vectors X and Y satisfying K(X, Y ) = k for given k with
Kmin ≦ k ≦ Kmax .
In [10], Lohnherr-Reckziegel studied ruled real hypersurfaces by parameterizing
fn−1 (c) → M
fn (c) and show properties on thesethem by maps of the form f : I × M
maps. The following lemma is due to them.
Lemma 5.2 ([10]). For every s ∈ I the function ν satisfies the following:(s)fn−1(1) When c > 0, on each leaf M(c) we have 0 ≦ ν < ∞ ;√(s)fn−1 (c) we have either 0 ≦ ν < |c| /2, ν ≡(2) √When c < √0, on each leaf M|c| /2 or |c| /2 < ν < ∞.
f. A
Here we recall the definition of Frenet curves in a Riemannian manifold M
smooth curve γ = γ(s) parametrized by its arclength s is called a Frenet curveof proper order d if there exist a field of orthonormal frames {V1 = γ̇, V2 , . . . , Vd }along γ and positive smooth functions κ1 (s), . . . , κd−1 (s) satisfying the following
system of ordinary diﬀerential equationse γ̇ Vj (s) = −κj−1 (s)Vj−1 (s) + κj (s)Vj+1 (s), j = 1, . . . , d,
(5.1)
∇

8

S. MAEDAe γ̇ denotes the covariant diﬀerentiation along γ with
where V0 ≡ Vd+1 ≡ 0 and ∇e of M
f. The functions κj (s) (j = 1, . . . , d −respect to the Riemannian connection ∇1) and a field of orthonormal frames {V1 , . . . , Vd } are called the curvatures andthe Frenet frame of γ, respectively. Roughly speaking, every Frenet curve can be
regarded as a smooth regular real curve admitting no inflection points. We call acurve a helix when all of its curvatures are constant functions. A curve γ is called ahelix of order d if it is a helix of proper order r(≦ d). For a helix of order d, which isof proper order r(≦ d), we use the convention in (5.1) that κj = 0 (r ≦ j ≦ d − 1)
and Vj = 0 (r + 1 ≦ j ≦ d). Needless to say, a helix of order 1 is nothing
but a geodesic and a helix of order 2 is a circle. For a Frenet curve γ of proper
fn (c), we define its complex torsions byorder d in a nonflat complex space form M
fn (c)
τij (s) = ⟨Vi (s), JVj (s)⟩ (1 ≦ i < j ≦ d). In the study of Frenet curves in M
their complex torsions play an important role (see [13]).
Now, we consider a ruled real hypersurface associated with a Frenet curve γ andinvestigate the value of the function ν along γ.
Lemma 5.3 ([16]). Let γ = γ(s) be a Frenet curve of proper order d with curvaturesκj (s) (1 ≦ j ≦ d − 1) and complex torsions τij (s) (1 ≦ i < j ≦ d) in a nonflat
fn (c), n ≧ 2 and M be a ruled real hypersurface associatedcomplex space form M
with the curve γ. Then the functions µ = ⟨Aξ, ξ⟩ and ν = ∥Aξ − µξ∥ satisfy thefollowing.
(5.2)µ(γ(s)) = −κ1 (s)τ12 (s),(5.3)

ν(γ(s))2 = κ1 (s)2 (1 − τ12 (s)2 ).

As a direct consequence of Lemmas 5.1, 5.2 and 5.3, we have the following.
Theorem 5.4 ([16]). Every ruled real hypersurface M in CP n (c), n ≧ 2 has thesectional curvature K with sharp inequalities −∞ < K ≦ c, so that there does notexist a ruled real hypersurface having the sectional curvature K with c/4 ≦ K ≦ cin this ambient space.
Theorem 5.5 ([16]). Let γ : I → CH n (c), n ≧ 2 be a Frenet curve of proper orderd defined on an open interval I(⊂ R) in CH n (c) and M be a ruled real hypersurface
associated with the curve γ in CH n (c). Then M has the sectional curvature K withsharp inequalities c ≦ K ≦ c/4 if and only if the first curvature κ1 and a complex
torsion τ12 of the Frenet curve γ satisfy κ1 (s)2 (1 − τ12 (s)2 ) ≦ |c|/4 for any s ∈ I.
Remark 5.6. Both of minimal ruled hypersurfaces M of axial type and parabolic
type satisfy the inequality in Theorem 5.5, so that the sectional curvatures K of
M satisfy c ≦ K ≦ c/4. However, note that the minimal ruled real hypersurfaceof elliptic type does not satisfy the inequality in Theorem 5.5. This ruled real
hypersurface has the sectional curvature K with −∞ < K ≦ c/4.
fn (c)
6. Characterization of all ruled real hypersurfaces in M
It is known that every real hypersurface in a nonflat complex space form does nothave parallel shape operator (for details, see [17]). So, weakening this parallelism

GEOMETRY OF RULED REAL HYPERSURFACES9fn (c).
of the shape operator, we shall characterize all ruled real hypersurfaces in M
Note that the following theorem is established from the viewpoint of geometricproperties (of all ruled real hypersurfaces) which do not depend on the sign of thesectional curvature of a nonflat complex space form.
Theorem 6.1 ([9]). A real hypersurface M isometrically immersed into a nonflat
fn (c), n ≧ 2 is ruled in this ambient space if and only if thecomplex space form M
holomorphic distribution T 0 M on M is integrable and the shape operator A of M
is η-parallel, i.e., ⟨(∇X A)Y, Z⟩ = 0 for all vectors X, Y and Z ∈ T 0 M holds on
M , where ⟨ , ⟩ and ∇ are the Riemannian metric and the Riemannian connectionof M , respectively.
References[1] T. Adachi, T. Bao and S. Maeda, Congruence classes of minimal ruled real hypersurfaces ina nonflat complex space form, Hokkaido Math. J. 43 (2014), 1–14.
[2] T. Adachi and S. Maeda, Global behaviours of circles in a complex hyperbolic space, Tsukuba
J. Math. 21 (1997), 29–42.
[3] T. Adachi, S. Maeda and S. Udagawa, Circles in a complex projective space, Osaka J. Math.
32 (1995), 709–719.
[4] J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space,
J. Reine Angew. Math. 395 (1989), 132–141.
[5] J. Berndt and J. C. Dı́az-Ramos, Homogeneous hypersurfaces in complex hyperbolic spaces,
Geom. Dedicata 138 (2009), 129–150.
[6] J. Berndt and H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces ofrank one, Trans. Amer. Math. Soc. 359 (2007), 3425–3438.
[7] B. Y. Chen and S. Maeda, Hopf hypersurfaces with constant principal curvatures in complex
projective or complex hyperbolic spaces, Tokyo J. Math. 24 (2001), 133–152.
[8] M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurface in P n (C),Math. Ann. 276 (1987), 487–497.
[9] M. Kimura and S. Maeda, On real hypersurfaces of a complex projective space, Math. Z. 202(1989), 299–311.
[10] M. Lohnherr and H. Reckziegel, On ruled real hypersurfaces in complex space forms, Geom.
Dedicata 79 (1999), 267–286.
[11] S. Maeda and T. Adachi, Integral curves of characteristic vector fields of real hypersurfaces
in nonflat complex space forms, Geom. Dedicata 123 (2006), 65–72.
[12] S. Maeda, T. Adachi and Y. H. Kim, A characterization of the homogeneous minimal ruled
real hypersurface in a complex hyperbolic space, J. Math. Soc. Japan 61 (2009), 315–325.
[13] S. Maeda and Y. Ohnita, Helical geodesic immersions into complex space forms, Geom.
Dedicata 30 (1989), 93–114.
[14] S. Maeda and H. Tanabe, Totally geodesic immersion of Kähler manifolds and Kähler Frenet
curves, Math. Z. 252 (2006), 787–795.
[15] S. Maeda and H. Tanabe, A characterization of the homogeneous ruled real hypersurface in acomplex hyperbolic space in terms of the first curvature of some integral curves, Arch Math.
105 (2015), 593–599.
[16] S. Maeda, H. Tanabe and Y. H. Kim, Ruled real hypersurfaces having the same sectional
curvature as that of an ambient nonflat complex space form, Kodai Math. J. 39 (2016),
119–128.
[17] R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, Tight and tautsubmanifolds (T.E. Cecil and S.S. Chern, eds.), Cambridge Univ. Press, 1998, 233–305.

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S. MAEDA[18] K. Okumura and S. Maeda, Three real hypersurfaces some of whose geodesics are mappedto circles with the same curvature in a nonflat complex space form, Geom. Dedicata 156(2012), 71–80.
[19] R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math.
10 (1973), 495-506.
[20] R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures
II, J. Math. Soc. Japan 27 (1975), 507–516.
[21] H. Tanabe and S. Maeda, Sectional curvatures of ruled real hypersurfaces in a nonflat complex
space form, Proceedings of the workshop on Diﬀerential Geometry of Submanifolds and its
related topics, Saga, August 4-6, 2012, World Scientific, 2013, 113–118.
S. Maeda: Department of Mathematics, Saga University, 1 Honzyo, Saga, 8408502, Japan
E-mail address: smaeda@ms.saga-u.ac.jp