Mem. Gra. Sci. Eng. Shimane Univ.
Series B: Mathematics
52 (2019), pp. 15–19

SOME CHARACTERIZATIONS OF THE HOMOGENEOUS
RULED REAL HYPERSURFACE IN A COMPLEX HYPERBOLIC
SPACE
SADAHIRO MAEDA
Communicated by Daishi Kuroiwa(Received: April 2, 2018)
Abstract. Motivated by the notion of ruled surfaces in Euclidean 3-space, we
consider ruled real hypersurfaces isometrically immersed into an n-dimensional
complex hyperbolic space of constant holomorphic sectional curvature c(< 0).
The purpose of this expository paper is to give some geometric characterizationsof the homogeneous ruled real hypersurface, that is, this ruled real hypersurface
is an orbit of some subgroup of the full isometry group of the ambient space (see[2, 6, 7, 9, 10]).

1. Ruled real hypersurfaces
We study a real hypersurface M 2n−1 (with Riemannian metric g) in a complex
n-dimensional complex hyperbolic space CH n (c), n ≧ 2 of constant holomorphic
sectional curvature c(< 0) through an isometric immersion. We first recall thedefinition of ruled real hypersurfaces M 2n−1 in CH n (c). A real hypersurface M is
ruled if the holomorphic distribution T 0 M = {X ∈ TM |g(X, ξ) = 0} is integrable
and all of whose leaves (i.e., maximal integral manifolds) are locally congruent tototally geodesic holomorphic hypersurfaces CH n−1 (c) in the ambient space CH n (c),where ξ is the characteristic vector field with respect to the almost contact metricstructure (ϕ, ξ, η, g) on M induced from the Kähler structure (g, J) of CH n (c).
We recall the construction of ruled real hypersurfaces in CH n (c). For a realsmooth curve γ = γ(s), s ∈ I parametrized by its arclength s, where I is on
some open interval on R, we take the totally geodesic holomorphic hypersurface
CHsn−1 (c) through the point γ(s) in CH n (c) in such a way that the holomorphic
line spanned by γ̇(s) is perpendicular to the tangent space Tγ(s) CHsn−1 (c). Then∪we get a ruled real hypersurface M =CHsn−1 (c).
s∈I1991 Mathematics Subject Classification. 53B25, 53C40.
Key words and phrases. Ruled real hypersurfaces, the homogeneous ruled real hypersurface,complex hyperbolic space, characteristic vector filed, integral curves, circles, the first curvature.
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S. MAEDA

By the definition of ruled real hypersurfaces we find that a real hypersurface M
e X Y ∈ T 0 M for each X, Y ∈ T 0 M , where ∇eis ruled in CH n (c) if and only if ∇ndenotes the Riemannian connection on CH (c). Then by simple calculation weknow that a real hypersurface M is ruled in CH n (c) if and only if g(AX, Y ) = 0
for each X, Y ∈ T 0 M , where A is the shape operator of M in CH n (c).
We consider a subset M∗ = {x ∈ M |ξx is not principal} of a ruled real hypersurface M . Note that M∗ is open dense in M (see [8]). We define two functions µ = g(Aξ, ξ) and ν = ∥Aξ − µξ∥ on M . Then M is decomposed as:
M = M∗ ∪ M0 ∪ M∞ , where M0 = {x ∈ M |ν(x) = 0}, M∞ = {x ∈ M |ν(x) = ∞}and M∗ = {x ∈ M |ν(x) > 0}. Note that both of M0 and M∞ are subsets ofmeasure zero on M (cf. [8]). In general, M has singular points, that is, M is not
smooth on M∞ . So we omit M∞ and consider a ruled real hypersurface M as adiﬀerentiable manifold M = M∗ ∪ M0 . By easy computation we see that the shapeoperator A of a ruled real hypersurface M in CH n (c) is expressed on the set M∗
as: Aξ = µξ + νU, AU = νξ and AX = 0 for each X(∈ T 0 M ) perpendicular to U ,where U is a unit vector defined by U = (Aξ − µξ)/ν.
This paper is related to a talk of the author in 5Th International Conference
on Recent Advances in Pure and Applied Mathematics (ICRAPAM 2018) July 2327 2018 at Karadeniz Technical University Prof. Dr. Osman Turan Convention
Center in Trabzon, TURKEY.
2. The homogeneous ruled real hypersurface in CH n (c)We recall the classification of all homogeneous real hypersurfaces M 2n−1 in CH n (c),n ≧ 2 due to Bendt and Tamaru ([5]). Here, “homogeneous” means that these real
hypersurfaces are orbits of closed subgroups of the full isometry group I(CH n (c))(=U (1, n)) of the ambient space CH n (c). By virtue of their works ([5, 4]) we can seethat in the class of all homogeneous real hypersurfaces M of CH n (c) M is minimal
if and only if M is ruled in the ambient space. This fact implies that the homogeneous ruled real hypersurface is a nice example in the class of all homogeneous
real hypersurfaces of CH n (c).
Our aim here is to review the construction of the homogeneous ruled real hypersurface M in CH n (c). For this purpose we first review the definition of circles in
Riemannian geometry. Let γ = γ(s) be a smooth real curve parametrized by its
arclength s on a Riemannian manifold N with Riemannian metric g. If the curve
γ satisfies the following ordinary diﬀerential equations with some nonnegative constant k:∇γ̇ γ̇ = kYs and ∇γ̇ Ys = −k γ̇,where ∇γ̇ is the covariant diﬀerentiation along γ with respect to ∇ of N and Ys is
the so-called the unit principal normal vector of γ, we call γ a circle of curvature k
on N . We regard a geodesic as a circle of null curvature. By virtue of the existence
and the uniqueness of solutions to ordinary diﬀerential equations we can see that foreach point p ∈ N , an arbitrary positive constant k and every pair of orthonormal
vectors X and Y of Tp N , there exists locally the unique circle γ = γ(s) on N
satisfying the initial condition that γ(0) = p, γ̇(0) = X and Y0 = Y .

THE HOMOGENEOUS RULED REAL HYPERSURFACE

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The construction of the homogeneous √ruled real hypersurface M is very simple.
We take a circle γ = γ(s) of curvature |c| /2 on a totally real totally geodesic
surface RH 2 (c/4) of constant sectional curvature c/4 in the ambient space CH n (c).
The circle γ is said to be horocyclic on RH 2 (c/4) (for details, see [3]). Then theruled real hypersurface M associated to the circle γ is homogeneous in CH n (c).
This construction shows that the full isometry group I(M ) of our ruled real hyn−1
persurface M is a direct product of I(CH√(c)) and a one-parameter subgroup{φs } whose orbit is a circle of curvature |c| /2 on RH 2 (c/4). We remark thatin the above decomposition M = M∗ ∪ M0 ∪ M∞ of√the homogeneous ruled real
hypersurface M we see that M∗ = {x ∈ M |ν(x) = |c| /2} and both of M0 and
M∞ are empty sets (cf. [1]).
At the end of this section we recall the following which gives various characterizations of the homogeneous ruled real hypersurface in the class of all ruled real
hypersurfaces in CH n (c).
Proposition 1 ([1, 5, 8]). For a ruled real hypersurface M in CH n (c), n ≧ 2, thefollowing four conditions are mutually equivalent:(1) M is homogeneous in CH n (c);(2) All principal curvatures of M are constant on√
M;(3) The functions µ and ν satisfy µ ≡ 0 and ν ≡ |c| /2;(4) There exists an integral curve γξ of the characteristic √vector field ξ on M
such that the curve γξ is a circle of positive curvature |c| /2 on a totallyreal totally geodesic plane RH 2 (c/4) of constant sectional curvature c/4 inthe ambient space CH n (c).
3. Statements of results
We first pay attention to integral curves of the characteristic vector field ξ on thehomogeneous ruled real hypersurface M . All of them are horocyclic on a totallyreal totally geodesic RH 2 (c/4) in CH n (c), so that they are congruent with eachother up to I(CH n (c)). Motivated by this fact we establish the following
Theorem 1 ([2]). Let M be a ruled real hypersurface in CH n (c), n ≧ 2. Then M ishomogeneous in the ambient space CH n (c) if and only if M satisfies the following
two conditions:(1) Every integral curve γξ of the characteristic vector filed ξ on M lies on a totally real totally geodesic surface RH 2 (c/4) of constant sectional curvature c/4 in
CH n (c).
e ξ ξ∥(2) Every curve γξ in Condition (1) has the first curvature function κξ = ∥∇e is the Riemannian connecwhich does not depend on the choice of γξ , where ∇ntion of the ambient space CH (c). This means that for any integral curves γξ1 andγξ2 of ξ the first curvature functions κ1ξ and κ2ξ (of γξ1 and γξ2 , respectively) satisfy
κ1ξ (s) = κ2ξ (s + s0 ) for each s and some s0 .
We next characterize the homogeneous ruled real hypersurface by studying thefirst curvatures of integral curves of the vector fields U and ξ.

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S. MAEDA

Theorem 2 ([9]). Let M 2n−1 be a ruled real hypersurface of CH n (c), n ≧ 2. Then
M is homogeneous in this ambient space if and only if M satisfies the following
two conditions:(1) Every integral curve
positive first curvature κU :=√ γU of the vector field U hasne∥∇U U ∥ with κU ≦ |c| along the curve γU in CH (c);(2) Every integral curve γξ of thepositive first√ characteristic vector field ξ hasnecurvature κξ := ∥∇ξ ξ∥ with κξ ≦ |c| /2 along the curve γξ in CH (c).
e is the Riemannian connection of the ambient space CH n (c).
Here, ∇We remark that all(1) in Theorem 2 have√ curves γU = γU (s) in Condition1
the same curvature |c| on a holomorphic line CH (c) of CHn (c) and√ all curvesγξ = γξ (s) in Condition (2) in Theorem 2 have the same curvature |c| /2 on a
totally real totally geodesic surface RH 2 (c/4) of constant sectional curvature c/4
in the ambient space CH n (c).
Paying particular attention to the first curvature and the first holomorphic torsion of integral curves of the characteristic vector field ξ on the homogeneous ruled
real hypersurface, we get the following
Proposition 2 ([6]). Let M be a ruled real hypersurface in CH n (c), n ≧ 2. Then
M is homogeneous in this ambient space if and only if every integral curve γξ ofthe characteristic vector field ξ on M , considered as a curve in CH n (c), satisfies
the following two conditions:e ξ ξ∥ of the curve γξ is a constant function which is(1) The first curvature κξ := ∥∇e is the Riemannian connection of CH n (c);independent of the choice of γξ , where ∇(2) The first holomorphic torsion τ12 := g(ξ, JYγ ) is a constant function which is
independent of the choice of γξ , where J is the complex structure of CH n (c) and Yγe ξ ξ = κξ Yγ .
is the unit principal normal vector of the curve γξ , so that Yγ satisfies ∇The following theorem gives a necessary and suﬃcient condition for a ruled real
hypersurface having constant mean curvature to be homogeneous in CHn (c).
Theorem 3 ([7]). Let M be a ruled real hypersurface of CH n (c), n ≧ 2. Then M
is homogeneous in CH n (c) if and only if M satisfies the following two conditions:(1) M has constant mean curvature in CH n (c), i.e., Trace A is constant on M ,where A is the shape operator of M in CH n (c);(2) Every integral curve γU of the vector field U has the first curvature functione U U ∥ which does not depend on the choice of γU , where ∇e is the RiemannκU = ∥∇nian connection of the ambient space CH (c). This means that for any integral
curves γU1 and γU2 of U the first curvature functions κ1U and κ2U (of γU1 and γU2 ,respectively) satisfy κ1U (s) = κ2U (s + s0 ) for each s and some s0 .
Observing integral curves of the characteristic vector field ξ on a ruled real
hypersurface M , we obtain the following characterization of the homogeneous ruled
real hypersurface in CH n (c):Theorem 4 ([10]). For a ruled real hypersurface M in CH n (c), n ≧ 2 the following
two conditions are equivalent:THE HOMOGENEOUS RULED REAL HYPERSURFACE

19(1) M is homogeneous in the ambient space CH n (c);(2) Every integral curve γξ of the characteristic vector field ξ on M is mapped toa circle of the same positive curvature which is independent of the choice of γξ inthe ambient space CH n (c).
Note that in Theorem 4(2) a circle is not supposed to be lying locally on a totallyreal totally geodesic surface RH 2 (c/4) in CH n (c), n ≧ 2.
The following result is a characterization of the homogeneous ruled real hypersurface in the class of all real hypersurfaces in CH 2 (c).
Theorem 5 ([6]). Let M be a real hypersurface in a complex hyperbolic plane
CH 2 (c). Then M is locally congruent to the homogeneous ruled real hypersurface
in this space if and only if M satisfies the following two conditions:(1) M has constant mean curvature, i.e., Trace A is constant on M , where A isthe shape operator of M in CH 2 (c);(2) Every integral curve γξ of the characteristic vector field ξ on M is lying on
RH 2 (c/4) as a circle of the same curvature k which is independent of the choice ofγξ in CH 2 (c).
We finally pose the following open problem:
Problem 1. If we delete Condition (1) in Theorem 5, does this theorem hold true?
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S. Maeda: Department of Mathematics, Saga University, Saga 840-8502, Japan
E-mail address: sadahiromaeda0801@gmail.com