Horocyclic Circles and Tubes around Complex Hypersurfaces

in a Complex Hyperbolic Space

YUSEI AOKITOSHIAKI ADACHI

Division of Mathematics and Mathematical Science & Department of Mathematics,

Nagoya Institute of Technology,

Nagoya 4668555,

JAPAN

email:yusei11291@outlook.jp & adachi@nitech.ac.jp

Abstract: We show that every horocyclic circle of nonzero complex torsion on a complex hyperbolic space is

expressed by a trajectory for a Sasakian magnetic field on some tube around totally geodesic complex hypersurface

and that such an expression is unique up to the action of isometries on the complex hyperbolic space.

KeyWords: Circles; horocycle; Sasakian magnetic fields; real hypersurfaces; complex hyperbolic space;

extrinsic shapes; congruent.

Received: March 16, 2023. Revised: November 2, 2023. Accepted: November 17, 2023. Published: December 7, 2023.

1 Introduction

For each circle of radius rin a Euclidean 3space R3,

if we take a suitable sphere S2of radius r, it can be

seen as a geodesic on this sphere. We are interested

in whether such a property holds for other symmet

ric spaces. For a complex projective space, if we re

strict ourselves to circles whose velocity and acceler

ation vectors form a complex line, they can be seen

as geodesics on some geodesic spheres. On the other

hand, for a complex hyperbolic space, even if we re

strict ourselves to such circles, if they are unbounded

and do not lie on some horospheres, they can not be

seen as trajectories for Sasakian magnetic fields, es

pecially as geodesics, on any real hypersurfaces of

type (A), [1]. Here, trajectories for Sasakian mag

netic fields are natural generalizations of geodesics

from the viewpoint of dynamical systems which are

associated with almost contact metric structures on

these real hypersurfaces. With these results it is nat

ural to consider that unbounded circles are different

from bounded circles. In this paper, we study cir

cles lying on some horospheres and show that except

the case that they lie on totally geodesic real hyper

bolic plane they can be seen as trajectories for some

Sasakian magnetic fields on some tubes around totally

geodesic complex hypersurfaces.

2 Circles on a complex hyperbolic

space

A smooth curve γon a Riemannian manifold f

Mpa

rameterized by its arclength is said to be a circle if

there exist a nonnegative constant kγand a field Yγof

unit tangent vectors along γsatisfying the equations

∇˙γ˙γ=kγYγ,

∇˙γYγ=−kγ˙γ, (1)

which is equivalent to the equation

∇˙γe

∇˙γ˙γ+k2

γ˙γ= 0.

The constant kγis called its geodesic curvature and

{˙γ, Yγ}its Frenet frame, [2]. When k= 0, it is a

geodesic with an arbitrary parallel unit vector field.

Hence, from the viewpoint of FreneSerret formula,

circles are simplest curves next to geodesics.

We say two smooth curves γ1, γ2on f

Mwhich

are parameterized by their arclengths to be congru

ent to each other (in strong sense) if there is an isom

etry φof f

Msatisfying φ◦γ1(t) = γ2(t)for all

t. In this paper, we study circles on a complex hy

perbolic space CHn(c)of constant holomorphic sec

tional curvature c. By use of complex structure J, we

set τγ=⟨˙γ, JYγ⟩for a circle γof positive geodesic

curvature, and call it its complex torsion. Clearly it is

constant along γbecause Jis parallel. Since CHn(c)

is a symmetric space of rank 1, we find that two cir

cles are congruent to each other if and only if either

they are geodesics or they satisfy kγ1=kγ2>0

and |τγ1|=|τγ2|, [3]. Therefore, the moduli space

C(CHn), which is the set of all congruence classes,

of circles on CHn(c)is set theoretically coincides

with the band [0,∞)×[0,1]/ ∼. Here we define

(k1, τ1)∼(k2, τ2)if and only if either k1=k2= 0

or k1=k2>0and τ1=τ2. When a circle γon

CHnhas complex torsion ±1, the equations (1) turn

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to e

∇˙γ˙γ=∓kγJ˙γ. We can interpret it as a trajectory

for a Kähler magnetic field ∓kγBJwith the Kähler

form BJ.

We here recall some properties of circles on

CHn(c). Congruency of circles guarantees that every

circle is a homogeneous curve, that is, it is an orbit of

one parameter family of isometries of CHn. Since

CHnis a typical example of Hadamard manifolds,

we can define its ideal boundary ∂CHnas the set of

asymptotic classes of geodesics, and have a compacti

fication CHn=CHnS∂CHn, [4, 5]. When a curve

γwhich is unbounded in both directions, that is, both

γ[0,∞)and γ−∞,0]are unbounded sets, we set

γ(∞) = limt→∞ γ(t), γ(∞) = limt→−∞ γ(t)if

they exist in ∂CHn. These are called points at infinity

of γ. We say a smooth curve γparameterized by its

arclength to be horocyclic if it satisfies the following

conditions, [6]:

i) Both γ(∞)and γ(−∞)exist and coincide with

each other;

ii) If it crosses with a geodesic βsatisfying β(∞) =

γ(∞), then they cross orthogonally.

We set a function ν: [0,∞)→Rby

ν(k) = 









0,if k < √|c|

(4k2+c)3/2

(3√3|c|k),if √|c|

2≤k≤p|c|,

1,if k > p|c|.

We take a horizontal lift ˆγof a circle γon CHn(c)

through a Hopf fibration ϖ:H2n+1

1(⊂Cn+1)→

CHn(c)of an antide Sitter space H2n+1

1and regard

it as a curve in Cn+1. When its geodesic curvature is

kγand complex torsion is τγ, by solving an ordinary

differential equation on ˆγ, we can get the following

feature of γ:

1) If kγ>p|c|or τγ< ν(kγ), it is bounded;

2) If it does not satisfies the condition in 1), it is

unbounded in both directions and has points at

infinity;

3) It is horocyclic if and only if p|c|/2 ≤kγ≤

p|c|and |τγ|=ν(kγ)hold;

4) When τγ=±1, it lies on a totally geodesic CH1,

and when τγ= 0, it lies on a totally geodesic

RH2.

The following figure shows the images of the moduli

spaces of unbounded and bounded circles on CHn(c).

√|c|

p|c|

unbounded

circles bounded circles

horocyclic circles

Figure 1: moduli space of circles on CHn

3 Horocyclic circles are expressed

uniquely by trajectories

In this paper, we study whether circles in CHn(c)can

be seen as extrinsic shapes of some “nice” curves on

a homogeneous real hypersurface. Let M=T(r)

be a tube of radius raround totally geodesic com

plex hypersurface CHn−1in CHn(c). On this hy

persurface we have an almost contact metric structure

(ξ, η, ϕ, ⟨,⟩)induced by the complex structure J

on CHn. By taking a unit normal vector field Nof

Min CHn(c)and the induced metric ⟨,⟩, we set

the characteristic vector field ξby ξ=−JN, the 1

form ηby η(v) = ⟨v, ξ⟩, the (1,1)tensor field ϕby

ϕ(v) = Jv −η(v)N. The shape operator AMof M

satisfies AMξ=δMξand AMv=λMvfor arbitrary

tangent vector vorthogonal to ξ. Here, the principal

curvatures are given as δM=p|c|coth p|c|rand

λM=p|c|/2tanhp|c|r/2, [7, 8], for example.

If we denote by ∇and e

∇the Riemannian connec

tions on Mand on CHn(c), respectively, we have the

formulae of Gauss and Weingarten:

∇XY=∇XY+⟨AMX, Y ⟩N,

∇XN=−AMX,

for arbitrary vector fields X, Y tangent to M. Since

the complex structure Jis parallel with respect to e

∇,

these formulas guarantee

∇XϕY=η(Y)AMX− ⟨AMX, Y ⟩ξ, (2)

∇Xξ=ϕAMX. (3)

We define a 2form Fϕon Mby Fϕ(v, w) =

⟨v, ϕ(w)⟩. By use of (3), we find that it is closed. We

therefore call its constant multiple Fκ=κFϕ(κ∈R)

aSasakian magnetic field. A smooth curve σparam

eterized by its arclength is said to be a trajectory for

Fκif it satisfies the equation ∇˙γ˙γ=κϕ ˙γ. Since we

find that trajectories for the null magnetic field F0are

geodesics and that every trajectory is determined by

its initial unit tangent vector, we may say that trajec

tories are generalizations of geodesics and are sim

plest curve next to geodesics from the viewpoint of

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dynamical systems on the unit tangent bundle of M.

For magnetic fields, [9]and[10].

Given a smooth curve γon CHn(c)parameterized

by its arclength, if we have a real hypersurface Mand

a smooth curve σon Msatisfying γ(t) = ι◦σ(t)for

all twith an isometric immersion ι:M→CHn(c),

we say (M, σ)is an expression of γ. We say two ex

pressions (M1, σ1)and (M2, σ2)of γto be congru

ent to each other if there is an isometry φof CHn(c)

which satisfies φ(M1) = M2and either it preserves

γor reverse γ. That is, either φ◦γ(t) = γ(t)for all

tor φ◦γ(t) = γ(−t)for all t. In [1], the authors

studied expressions of circles by geodesics on tubes.

Proposition 1 ([1]).Every circles on CHn(c)with

geodesic curvature not smaller than p|c|and com

plex torsion ±1is expressed by a geodesic on some

tube around totally geodesic complex hypersurface.

We are hence interested in expressions of circles

with complex torsion |τ|<1. But being different

from circles in a Euclidean 3space, if we stick on

expressions by geodesics on real hypersurfaces, our

study does not go through any more (see Lemma 2

below ). We therefore extend the family of curves

on tubes and study expressions by trajectories for

Sasakian magnetic fields.

Theorem 1. Let γbe a circle on CHn(c).

(1) When γis horocyclic and τγ= 0, it is expressed

by a trajectory for a Sasakian magnetic field on

some tube around totally geodesic complex hy

persurface. If the complex torsion τγof γsatis

fies 0<|τγ|<1, such an expression is unique

up to the congruence relation.

(2) When γis horocyclic and τγ= 0, it can not be

expressed by trajectories for Sasakian magnetic

fields on tubes around totally geodesic complex

hypersurfaces.

(3) When γis bounded, also it is expressed by a tra

jectory for a Sasakian magnetic field on some

tube around totally geodesic complex hypersur

face.

For a trajectory σfor Fκon a tube M=T(r)of ra

dius raround totally geodesic complex hypersurface,

we set ρσ=⟨˙σ, ξ⟩and call it its structure torsion. By

(3) we find that the derivative of the structure torsion

is given as

ρ′

σ=κ⟨ϕ˙σ, ξ⟩+⟨˙σ, ϕAM˙σ⟩

=⟨˙σ, ϕAM˙σ⟩=−⟨AMϕ˙σ, ˙σ⟩,

because AMis symmetric and ϕis antisymmetric.

Since it is known that they satisfy AMϕ=ϕAM

for our tube T(r), we find that ρσis constant along

σ. This is an important invariant for σ. Through a

Hopf fibration ϖ:H2n+1

1→CHn, the inverse im

age of T(r)is H2n−1

1×S1. Considering the action

of U(1, n)on Cn+1, we have the following result on

congruency of trajectories on T(r).

Lemma 1. Let σ1, σ2be trajectories for Sasakian

magnetic fields Fκ1,Fκ2respectively on a tube T(r)

around totally geodesic CHn−1in CHn(c). Then γ1

and γ2are congruent to each other if and only if one

of the following conditions holds:

i)|ρσ1|=|ρσ2|= 1,

ii)|ρσ1|=|ρσ2|<1,|κ1|=|κ2|and κ1ρσ1=

κ2ρσ2.

Since two circles are congruent to each other if and

only if they have the same geodesic curvatures and

the same absolute values of complex torsions, in or

der to show the existence of expressions of a circle of

geodesic curvature kand complex torsion τ, we are

enough to show that there exists a trajectory σfor a

Sasakian magnetic field on some tube whose extrinsic

shape is a circle of geodesic curvature kand complex

torsion τ.

By the formulae of Gauss and Weingarten, consid

ering a trajectory σfor Fκon M=T(r)as a curve

in CHn(c), we have

∇˙σ˙σ=κϕ ˙σ+λM(1 −ρ2

σ) + δMρ2

σN,

∇κϕ ˙σ+λM(1 −ρ2

σ) + δMρ2

σN

=−κ2(1 −ρ2

σ) + λM+ (δM−λM)ρ2

σ2˙σ

+λM−κρσ+ (δM−λM)ρ2

σ

×κ+ (δM−λM)ρσ(ρσ˙σ−ξσ).

We therefore get conditions that the extrinsic shape of

a trajectory on Mis a circle on CHn(c).

Lemma 2. Let σbe a trajectory for Fκon T(r)in

CHn(c). Its extrinsic shape is a circle on CHn(c)if

and only if one of the following condition holds:

i)ρσ=±1,

ii)λM−κρσ+ (δM−λM)ρ2

σ= 0,

iii)κ+ (δM−λM)ρσ= 0.

Corresponding to these cases, the geodesic curvature

kσand the complex torsion τσof the extrinsic shape

of σare as follows:

i)kσ=δM, τσ=∓1,

ii)kσ=|κ|, τσ=sgn(κ),

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iii)kσ=qκ2−2λMκρσ+λ2

τσ= (2κρ 2

σ−κ−λMρσ)/kσ.

Here, sgn(κ)denotes the signature of κ.

Since Proposition 1 corresponds to the first and the

second cases of Lemma 2, we study the third case.

In this case, as we have κ=−(δM−λM)ρσ=

−|c|ρσ/(4λM), we obtain

kσ=sλ2

M+|c|ρ2

2+c2ρ2

16λ2

,(4)

τσ=ρσ(|c| − 2|c|ρ2

σ−4λ2

4kσλM

.(5)

Since |ρσ| ≤ 1, by the equation (4) we see λM≤

kσ≤λM+|c|/(4λM)=δM. Moreover, by using

these two equalities we have

τ2

σ=(k2

σ−λ2

M)32λ2

Mk2

σ+ 4cλ 2

M−c22

|c|(8λ2

M−c)3k2

.(6)

We here study congruent expressions.

Lemma 3. Let γbe a circle in CHn(c). Suppose

that its complex torsion satisfies |τγ|<1. If we have

two expressions (M1, σ1),(M2, σ2)of γby trajec

tories for Sasakian magnetic fields Fκ1,Fκ2on tubes

M1, M2around totally geodesic complex hypersur

faces of the same radius, then they are congruent to

each other.

Proof. For the sake of simplicity, we consider M1and

M2as subsets of CHn(c)through isometric embed

dings. Since the extrinsic shapes of σ1and σ2coin

cide with γ, by the equation (4) we find |ρσ1|=|ρσ2|.

Hence, by the condition κi+(δMi−λMi)ρσi= 0, we

have |κ1|=|κ2|(= 0) and κ1ρσ1=κ2ρσ2.

Since M1, M2are of the same radius, they are iso

metric to each other. Hence we have an isometry

φof CHn(c)with φ(M1) = M2. Taking into ac

count of principal curvatures of M1and M2, we have

dφ(NM1) = NM2. As φis ±holomorphic, that is,

dφ ◦J=±J◦dφ, we have

ρφ◦σ1=⟨dφ ◦˙σ1, ξM2⟩=⟨dφ ◦˙σ1,−Jdφ(NM2⟩

=±⟨dφ ◦˙σ1, dφ(−JNM2⟩=±⟨˙σ1, ξM1⟩

=±ρσ1

and

∇dφ◦˙σ1dφ ◦˙σ1=dφ∇˙σ1˙σ1=κ1dφϕ˙σ1

=κ1dφJ˙σ1−ρσ1NM1

=κ1±Jdφ ◦˙σ1−ρσ1NM2

=±κ1ϕ(dφ ◦˙σ1).

Therefore, φ◦σ1is a trajectory for F±κ1on M2whose

structure torsion is ±ρσ1. Thus, we find that φ◦σ1and

σ2are congruent to each other. This means that there

is an isometry ψof M2satisfying σ2(t) = ψ◦φ◦σ1(t)

for all t. Since it is known that there is an isometry ˜

on CHn(c)satisfying ˜

ψM2=ψ, by putting Φ=

ψ◦φ, we have Φ(M1) = M2and Φ◦σ1(t) = σ2(t)

for all t. Hence (M1, σ1)and (M2, σ2)are congruent

to each other.

Proof of Theorem 1. In order to show the first and

the second assertions, we solve the equation τ2

σ=

ν(kσ)2under the assumption p|c|/2 ≤kσ≤p|c|.

Since this equation turns to

(8k2

σ+|c|)2(4λ2

M+c)2

×(32λ2

Mk2

σ+ 5cλ 2

M−ck 2

σ−c2) = 0

and 4λ2

M+c < 0, we find

σ=|c|5tanh2(p|c|r/2) + 4

48tanh2(p|c|r/2) + 1.(7)

This equality shows that if we fix the radius of a tube

we can express only one horocyclic circle on a com

plex hyperbolic space up to the action of isometries

by some trajectory on this tube. Since the righthand

side of the equation (7) is monotone decreasing with

respect to the radius r, and takes all values in the inter

val |c|/4,|c|, we find that for each horocyclic circle

γof complex torsion 0<|τ|<1it is expressed by a

trajectory on a tube whose radius is determined by kγ.

By Lemma 3, we get the first assertion. Also, we find

that if a horocyclic circle γhas null complex torsion,

it is not expressed by trajectories for Sasakian mag

netic fields on tubes around complex hypersurfaces.

In order to show the third assertion, we study the

behavior of τ2

σwith respect to kσ. We denote by

fk;T(r)the continuous function defined by the

righthand side of the equation (6) by putting k=kσ.

We set

a(r) = q−8c+ 2c2λ−2

T(r)/8, b(r) = δT(r).

They satisfy λT(r)< a(r)< b(r). Since we have

dk = 2λ2

M(8k2−c)(8k2−4λ2

M+c)

×(32λ2

Mk2+4cλ 2

M−c2)

×|c|(8λ2

M−c)3k3−1,

we find that the function fk;T(r)have the follow

ing properties:

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1) It is monotone increasing in the interval

a(r), b(r),

2) fa(r); T(r)= 0 and fb(r); T(r)= 1.

This means that the moduli space of circles which are

extrinsic shapes of trajectories satisfyiong the third

condition in Lemma 2 is like Figure 2.

Since we have

lim

r↓0a(r) = lim

r↓0b(r) = ∞,

lim

r→∞ a(r) = p|c|/2,lim

r→∞ b(r) = p|c|,

lim

r→∞ fk;T(r)= (4k2+c)3/27c2k2,

comparing the last one with ν(k)2, we find that the

union Sr>0f[a(r), b(r)]; T(r)covers the set

(k, τ )p|c|/2 < k ≤p|c|,0≤τ < ν(k)

∪p|c|,∞×[0,1].

This means that every bounded circle on CHn(c)is

expressed by some trajectory on some tube T(r).

q q

√|c|

p|c|

a(r)

λT

b(r)

Figure 2: The graph of fk;T(r)

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present re

search, at all stages from the formulation of the prob

lem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

The second author is partially supported by Grant

inAid for Scientific Research (C) (No. 20K03581),

Japan Society for the Promotion of Science.

Conflicts of Interest

The authors have no conflicts of interest to

declare that are relevant to the content of this

article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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