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PureMathematics
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,2023,13(6),1728-1743
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.136177
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ConstructionofHolomorphicEquivalence
RelationsonRiemannianSpheresandTheir
Applications
YulanLv
1
,
2
,LiningGan
1
,ZhimingHuang
1
,QiuhuaYang
1
,
3
,WeijunLu
1
1
CollegeofMathematicsandPhysics,GuangxiMinzuUniversity,NanningGuangxi
2
JiangmenPeiyingSeniorHighSchool,JiangmenGuangdong
3
CollegeofMathematicsandElectronicInformationEngineering,GuangxiNormalUniversityfor
Nationalities,ChongzuoGuangxi
Received:May21
st
,2023;accepted:Jun.22
nd
,2023;published:Jun.29
th
,2023
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2023,13(6):1728-1743.DOI:10.12677/pm.2023.136177
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Abstract
Inthispaper,westudysomespecialRiemannsurfacesoncomplexone-dimensional
connectedcomplexanalyticmanifolds,includingcomplexone-dimensionalprojection
space
C
P
1
,extendedcomplexplane
C
∞
andcomplexsphere
S
2
.Inthesenseofholo-
morphicmappingandbiholomorphicmapping,thesethreetypicalRiemannsurfaces
areholomorphicequivalent.Furthermore,underHopfmapping,theholomorphice-
quivalencebetween
S
3
and
C
P
1
isderived.BasedonFrankle’sconjecture,theproblem
ofholomorphicmappingofenergyminimizationoncomplexone-dimensionalprojec-
tivespaces
C
P
1
tocompactK¨ahlermanifoldsisdiscussed.
Keywords
RiemannSurfaces,HolomorphicMapping,HolomorphicEquivalence,HopfMapping,
HolomorphicMinimizationMapping
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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∂u
∂w
=
∂
(
1
w
)
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.
u
=
g
21
(
w
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1
w
´
'
u
w
X
¼
ê
.
Ï
d
,
C
∞
´
˜
‡
i
ù
-
¡
.
·
K
2.1.4
E
˜
‘
Ý
K
˜
m
C
P
1
=
C
2
−{
0
}
/
∼
þ
½
Â
û
N
π
:
C
2
−{
0
}→
C
P
1
(
z
0
,z
1
)
7→
π
(
z
0
,z
1
) = (
z
0
,z
1
)
/
∼
= [(
z
0
,z
1
)]
.
´
÷
.
½
Â
û
ÿ
À
µ
τ
C
2
−{
0
}
=
τ
E
C
2
∩
(
C
2
−{
0
}
)
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C
P
1
=
π
(
U
)
|∀
U
∈
τ
C
2
−{
0
}
.
…
C
P
1
´
˜
‡
i
ù
-
¡
.
y
²
e
U
⊂
C
2
−{
0
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,
K
½
Â
π
(
U
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{
π
(
z
0
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1
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z
0
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1
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|∀
(
z
0
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1
)
∈
U
}
•
C
P
1
m
8
.
-
V
1
=
[(
z
0
,z
1
)]
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C
P
1
|
(
z
0
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1
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∈
C
2
−{
0
}
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0
6
= 0
,
(2.9)
V
2
=
[(
z
0
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1
)]
∈
C
P
1
|
(
z
0
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1
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2
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0
}
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1
6
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,
(2.10)
V
1
∪
V
2
=
C
P
1
,V
1
∩
V
2
6
=
∅
.
V
1
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2
•
C
P
1
¥
m
8
.
•
Ä
N
:
ψ
1
:
V
1
→
C
.
[(
z
0
,z
1
)]
7→
w
=
ψ
1
(
z
0
,z
1
) =
ψ
1
[(
z
0
,z
1
)] =
z
1
z
0
,
(2.11)
ψ
2
:
V
2
→
C
.
[(
z
0
,z
1
)]
7→
u
=
ψ
2
(
z
0
,z
1
) =
ψ
2
[(
z
0
,z
1
)] =
z
0
z
1
.
(2.12)
´
y
ψ
1
,ψ
2
´
V
,
ψ
1
,ψ
2
,ψ
−
1
1
,ψ
−
1
2
´
ë
Y
,
ψ
1
,ψ
2
´
Ó
.
ψ
1
,ψ
2
ƒ
m
=
£
N
:
ψ
2
◦
ψ
-1
1
:
ψ
1
(
V
1
∩
V
2
)
→
ψ
2
(
V
1
∩
V
2
)
,w
7→
u
=
ψ
2
◦
ψ
-1
1
(
w
)
,
DOI:10.12677/pm.2023.1361771732
n
Ø
ê
Æ
½
Œ
=
l
u
=
ψ
2
◦
ψ
-1
1
(
w
) =
ψ
2
◦
ψ
-1
1
(
z
1
z
0
) =
ψ
2
([(
z
0
,z
1
)]) =
z
0
z
1
=
1
z
1
z
0
=
1
w
.
(2.13)
d
£
2.13
¤
Œ
∂u
∂w
=
∂
(
1
w
)
∂w
= 0.
u
=
1
w
´
'
u
w
X
¼
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.
Ï
d
C
P
1
•
i
ù
-
¡
.
±
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‡
~
f
š
~
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q
,
¢
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n
‡
i
ù
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X
d
,
ä
N
y
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3
e
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Ñ
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2.2.
X
N
†
X
d
X
Ã
A
O
(
²
,
b
½
i
ù
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ë
Ï
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…
=
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o
´
X
.
•
d
Ú
\
i
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m
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d
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g
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½
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2.2.1
M
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ü
‡
i
ù
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¡
,
©
O
±
{
(U
α
,
ϕ
α
)
}
α
∈
A
,
{
(V
β
,
ψ
β
)
}
β
∈
B
•
Ù
Û
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‹
I
k
,
f
:
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N
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ë
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N
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X
J
é
z
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é
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k
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f
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α
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1
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V
β
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U
6
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…
E
Ü
N
ψ
β
◦
f
◦
ϕ
−
1
α
:
ϕ
α
(
U
α
∩
V
−
1
β
)
→
ψ
β
(
V
β
)
´
X
N
,
K
¡
f
:
M
→
N
•
X
N
.
Ù
¥
,
ψ
β
◦
f
◦
ϕ
−
1
α
¡
•
f
Û
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«
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½
Â
2.2.2
M
!
N
´
ü
‡
i
ù
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¡
,
X
J
f
:
M
→
N,g
:
N
→
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þ
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X
N
,
…
g
◦
f
=
Id
M
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◦
g
=
Id
N
.
K
¡
f
½
g
•
V
X
N
,
¡
M
†
N
•
X
d
.
2.3.Hopf
n
‘
z
½
Â
2.3.1
‘
k
n
‘
S
1
∼
=
U
(1)
∼
=
SO
2
N
P
:
S
2
n
+1
→
C
P
n
.
l
π
1
(
S
1
)
∼
=
Z
,
π
i
(
S
1
)=0,
i>
1,
·
‚
Œ
±
l
n
‘
z
Ü
S
¥
±
e
Ó
'
X
µ
π
i
(
S
2
n
+1
)
∼
=
π
i
(
C
P
n
)
,i
6
= 2
,
π
2
n
+1
(
S
2
n
+1
)
∼
=
π
2
n
+1
(
C
P
n
)
∼
=
Z
,
π
2
(
C
P
2
)
∼
=
Z
∼
=
π
1
(
S
1
)
.
½
Â
2.3.2
¡
N
f
:
X
→
Y
´
˜
‡
n
‘
z
,
X
J
é
u
.
˜
m
Y
?
¿
N
ϕ
=
ϕ
0
?
Û
Ó
Ô
Φ =
{
ϕ
t
}
:
K
×
I
→
Y
Ñ
•
3
,
‡
Ó
Ô
e
Φ =
{
e
ϕ
t
}
:
K
×
I
→
X
CX
Φ.
Ù
¥
,
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m
Y
¡
•
.
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m
,
X
¡
•
n
‘
z
˜
m
.
”
F
y
=
f
−
1
(
y
)
¡
•
n
‘
,
N
f
K
¡
•
K
.
DOI:10.12677/pm.2023.1361771733
n
Ø
ê
Æ
½
Œ
=
3.
X
d
'
X
E
3.1.
i
ù
¥
¡
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m
X
d
3ù
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!
¥
,
·
‚
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E
Ñ
Î
Ü
ý
Ï
Ž
{
X
N
,
•
Œ
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K
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)û
ü
‡
i
ù
¡
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I
ã
k
ƒ
m
‹
I
=
†
.
3
X
N
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V
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¿Â
e
,
y
²
ü
‡
i
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¡
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X
d
'
X
.
·
K
3.1
‘
¥
¡
S
2
=
{
(
x,y,z
)
∈
R
3
|
x
2
+
y
2
+
z
2
= 1
}
†
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¡
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∞
=
C
∪{∞}
X
d
.
y
²
‘
¥
¡
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2
E
)
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(
•
{
(
U
1
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1
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,
(
U
2
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2
)
}
,
d
·
K
2.1.2
¥
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2.2
¤
±
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2.3
¤
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Ñ
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¿
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∞
=
C
∪{∞}
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•
n
(
f
U
1
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1
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(
f
U
2
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2
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o
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d
·
K
2.1.3
¥
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¤
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2.6
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±
9
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2.7
¤
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Ñ
.
E
N
f
:
S
2
→
C
∞
.
(
x,y,z
)
7→
w
=
f
(
x,y,z
) =
x
+
iy
1
−
z
,
(
x,y,z
)
6
= (0,0,1)
.
∞
,
(
x,y,z
) = (0,0,1)
.
(3.1)
K
´
•
f
´
V
,
Ù
_
N
•
g
=
f
−
1
:
C
∞
→
S
2
.
w
7→
(
x,y,z
)=
g
(
w
) =
(
2Re
w
|
w
|
2
+1
,
2Im
w
|
w
|
2
+1
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|
w
|
2
−
1
|
w
|
2
+1
)
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6
=
∞
.
(0,0,1)
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=
∞
.
(3.2)
1
¤
é
uN
f
,
©
ü
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¹
5
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1
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‹
I
•
Ä
.
e
(
x,y,z
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K
f
(0,0,1) =
∞
.
©
O
:
(0,0,1)
∈
U
2
,
3
S
2
Û
Ü
)
Û
ã
k
(
U
2
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2
)
Ú
∞
3
C
∞
Û
Ü
)
Û
ã
k
(
f
U
2
,φ
2
),
ù
ž
f
Û
ÜL
«
µ
f
f
22
=
φ
2
◦
f
◦
ϕ
−
1
2
:
ϕ
2
(
U
2
∩
f
−
1
(
f
U
2
))
⊂
C
→
φ
2
(
f
(
U
2
)
∩
f
U
2
)
⊂
C
.u
7→
w
=
f
f
22
(
u
)
.
(
Ü
£
2.3
¤
!
£
2.7
¤
!
£
3.1
¤
,
w
=
f
f
22
(
u
) =
φ
2
◦
f
◦
ϕ
−
1
2
(
u
) =
φ
2
◦
f
◦
ϕ
−
1
2
(
x
−
iy
1+
z
) =
φ
2
◦
f
(
x,y,z
)
=
φ
2
(
x
+
iy
1
−
z
)
,
(
x,y,z
)
6
= (0,0,1)
φ
2
(
∞
)
,
(
x,y,z
) = (0,0,1)
=
1
−
z
x
+
iy
0
=
u
0
.
=
w
=
f
f
22
(
u
) =
u
.
Œ
„
∂w
∂u
=
∂
(
u
)
∂
(
u
)
= 0
,
w
=
f
f
22
(
w
)
'
u
u
∈
C
´
X
¼
ê
.
e
(
x,y,z
)
6
= (0,0,1),
K
f
(
x,y,z
) =
x
+
iy
1
−
z
6
=
∞
,
:
(
x,y,z
)
3
S
2
Û
Ü
‹
I
ã
k
(
U
1
,ϕ
1
)
Ú
E
DOI:10.12677/pm.2023.1361771734
n
Ø
ê
Æ
½
Œ
=
ê
x
+
iy
1
−
z
3
C
∞
Û
Ü
+
•
‹
I
ã
k
(
f
U
1
,φ
1
),
K
f
Û
ÜL
«
•
:
f
f
11
=
φ
1
◦
f
◦
ϕ
−
1
1
:
ϕ
1
(
U
1
∩
f
−
1
(
f
U
1
))
⊂
C
→
φ
1
(
f
(
U
1
)
∩
f
U
1
)
⊂
C
.
w
=
x
+
iy
1
−
z
7→
u
=
f
f
11
(
w
)
.
(
Ü
£
2.2
¤
!
£
2.6
¤
!
£
3.1
¤
,
u
=
f
f
11
(
w
) =
φ
1
◦
f
◦
ϕ
−
1
1
(
w
) =
φ
1
◦
f
◦
ϕ
−
1
1
(
x
+
iy
1
−
z
) =
φ
1
◦
f
(
x,y,z
)
=
φ
1
(
x
+
iy
1
−
z
) =
id
(
x
+
iy
1
−
z
) =
x
+
iy
1
−
z
=
w.
=
u
=
f
f
11
(
w
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w.
u
´
∂u
∂w
=
∂
(
w
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∂
(
w
)
= 0,
u
=
f
f
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(
w
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w
∈
C
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¼
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2
¤
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uN
g
,
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©
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¹
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µ
e
w
=
∞
,
K
g
(
w
)=(0,0,1).
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ž
∞
3
C
∞
Û
Ü
)
Û
ã
k
(
f
U
2
,φ
2
)
Ú
g
(
∞
)=(0,0,1)
3
S
2
Û
Ü
)
Û
ã
k
(
U
2
,ϕ
2
),
g
Û
Ü
‹
IL
«
•
:
f
g
22
=
ϕ
2
◦
g
◦
φ
−
1
2
:
φ
2
(
f
U
2
∩
f
−
1
(
U
2
))
→
ϕ
2
(
f
(
f
U
2
)
∩
U
2
)
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7→
u
=
f
g
22
(
w
)
.
(
Ü
£
2.3
¤
!
£
2.7
¤
!
£
3.2
¤
,
u
=
f
g
22
(
w
) =
ϕ
2
◦
g
◦
φ
−
1
2
(
w
) =
ϕ
2
◦
g
◦
φ
−
1
2
(
x
−
iy
1+
z
)
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6
= 0
ϕ
2
◦
g
◦
φ
−
1
2
(0)
,w
= 0
=
ϕ
2
◦
g
(
x
+
iy
1
−
z
)
ϕ
2
◦
g
(
∞
)
=
ϕ
2
(
x,y,z
)
ϕ
2
(0,0,1)
=
w
0
.
Ù
¥
,
w
=
x
−
iy
1+
z
,x
2
+
y
2
+
z
2
= 1
,g
(
x
+
iy
1
−
z
)=(
x,y,z
).
K
k
1
w
=
1+
z
x
−
iy
=
x
+
iy
1
−
z
.
Œ
„
,
u
=
f
g
22
(
w
) =
w,
ù
´
'
u
w
X
¼
ê
.
w
3
C
∞
Û
Ü
‹
I
ã
k
(
f
U
1
,φ
1
)
Ú
g
(
w
)= (
x,y,z
)
3
S
2
Û
Ü
‹
I
ã
k
(
U
1
,ϕ
1
),
g
Û
Ü
‹
IL
«
•
:
f
g
11
=
ϕ
1
◦
g
◦
φ
−
1
1
:
φ
1
(
f
U
1
∩
g
−
1
(
U
1
))
→
ϕ
1
(
g
(
f
U
1
)
∩
U
1
)
.w
7→
u
=
f
g
11
(
w
)
.
(
Ü
£
2.2
¤
!
£
2.6
¤
!
£
3.2
¤
,
u
=
f
g
11
(
w
) =
ϕ
1
◦
g
◦
φ
−
1
1
(
x
+
iy
1
−
z
) =
ϕ
1
◦
g
(
x
+
iy
1
−
z
) =
ϕ
1
(
x,y,z
) =
x
+
iy
1
−
z
=
w.
DOI:10.12677/pm.2023.1361771735
n
Ø
ê
Æ
½
Œ
=
Œ
„
,
f
g
11
´
X
N
.
n
þ
¤
ã
,
•
g
=
f
−
1
•
´
X
N
,
l
f
´
˜
‡
V
X
N
.
Ï
d
,
S
2
†
C
∞
´
X
d
.
·
K
3.2
‘
¥
¡
S
2
=
{
(
x,y,z
)
∈
R
3
|
x
2
+
y
2
+
z
2
= 1
}
†
E
˜
‘
Ý
K
˜
m
C
P
1
=
C
2
−{
0
}
/
∼
X
d
.
y
²
‘
¥
¡
S
2
E
)
Û
(
•
{
(
U
1
,ϕ
1
)
,
(
U
2
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2
)
}
,
d
·
K
2.1.2
¥
ª
£
2.1
¤
!
£
2.2
¤
±
9
£
2.3
¤
‰
Ñ
.
E
˜
‘
Ý
K
˜
m
C
P
1
g
,
Ý
K
π
:
C
2
−{
0
}→
C
P
1
.
(
z
0
,z
1
)
7→
π
(
z
0
,z
1
) = (
z
0
,z
1
)
/
∼
= [(
z
0
,z
1
)]
.
û
ÿ
À
∀
A
∈
τ
E
C
2
−{
0
}
=
τ
E
C
2
∩
(
C
2
−{
0
}
)
,
½
Â
π
(
A
) =
{
π
(
z
0
,z
1
)
∈
C
P
1
|∀
(
z
0
,z
1
)
∈
A
}
•
C
P
1
m
8
.
=
τ
=
π
(
A
)
|∀
A
∈
τ
C
2
−{
0
}
.
E
˜
‘
Ý
K
˜
m
C
P
1
˜
‡E
)
Û
(
•
{
(
V
1
,ψ
1
)
,
(
V
2
,ψ
2
)
}
.
d
·
K
2.1.4
¥
ª
£
2.9
¤
-
£
2.12
¤
‰
Ñ
.
E
N
F
:
C
P
1
→
S
2
,
[(
z
0
,z
1
)]
7→
F
([(
z
0
,z
1
)]) = (
x,y,z
) =
2Re
z
1
z
0
z
1
z
0
2
+1
,
2Im
z
1
z
0
z
1
z
0
2
+1
,
z
1
z
0
2
−
1
z
1
z
0
2
+1
,z
0
6
= 0
.
(0,0,1)
,z
0
= 0
.
(3.3)
K
F
´
V
.
Ù
_
N
•
G
=
F
−
1
:
S
2
→
C
P
1
,
(
x,y,z
)
7→
G
(
x,y,z
) = [(
z
0
,z
1
)] =
[(1
−
z,x
+
iy
)]
,
(
x,y,z
)
6
= (0
,
0
,
1)
.
[(0,
z
1
)]
,
(
x,y,z
) = (0
,
0
,
1)
.
(3.4)
1
¤
é
uN
F
,
©
ü
«
œ
¹
5
?
1
Û
Ü
‹
I
•
Ä
.
e
z
0
=0
ž
,
K
F
([(0,
z
1
)])=(0,0,1)
∈
S
2
.
du
[(0,
z
1
)]
á
u
V
2
,
Û
Ü
‹
I
ã
k
(
V
2
,ψ
2
)
Ú
:
(0,0,1)
3
S
2
Û
Ü
‹
I
ã
k
(
U
2
,ϕ
2
),
ù
ž
F
Û
ÜL
«
µ
g
F
22
=
ϕ
2
◦
F
◦
ψ
−
1
2
:
ψ
2
(
V
2
∩
F
−
1
(
U
2
))
→
ϕ
2
(
F
(
V
2
)
∩
U
2
)
.w
7→
u
=
g
F
22
(
w
)
.
DOI:10.12677/pm.2023.1361771736
n
Ø
ê
Æ
½
Œ
=
(
Ü
£
2.3
¤
!
£
2.12
¤
!
£
3.3
¤
,
u
=
g
F
22
(
w
) =
ϕ
2
◦
F
◦
ψ
−
1
2
(
w
)=
ϕ
2
◦
F
◦
ψ
−
1
2
(
z
0
z
1
)
,z
0
6
= 0
ϕ
2
◦
F
◦
ψ
−
1
2
(
z
0
z
1
)
,z
0
= 0
=
ϕ
2
◦
F
([(
z
0
,z
1
)])
ϕ
2
◦
F
([(0,
z
1
)])
=
ϕ
2
2Re
z
1
z
0
z
1
z
0
2
+1
,
2Im
z
1
z
0
z
1
z
0
2
+1
,
z
1
z
0
2
−
1
z
1
z
0
2
+1
ϕ
2
(0,0,1)
=
z
0
z
1
0
=
w
0
.
u
´
∂u
∂w
=
∂
(
w
)
∂
(
w
)
= 0,
g
F
22
´
X
N
.
e
z
0
6
= 0
ž
,
F
([
z
0
,z
1
])
6
= (0,0,1).
[(
z
0
,z
1
)]
3
C
P
1
Û
Ü
‹
I
ã
k
(
V
1
,ψ
1
)
Ú
S
2
Û
Ü
‹
I
ã
k
(
U
1
,ϕ
1
),
ù
ž
F
Û
ÜL
«
µ
g
F
11
=
ϕ
1
◦
F
◦
ψ
−
1
1
:
ψ
1
(
V
1
∩
F
−
1
(
U
1
))
→
ϕ
1
(
F
(
V
1
)
∩
U
1
)
.u
7→
w
=
g
F
11
(
u
)
.
(
Ü
£
2.2
¤
!
£
2.11
¤
!
£
3.3
¤
,
w
=
g
F
11
(
u
) =
ϕ
1
◦
F
◦
ψ
−
1
1
(
z
1
z
0
) =
ϕ
1
◦
F
([
z
0
,z
1
])
=
ϕ
1
2Re
z
1
z
0
z
1
z
0
2
+1
,
2Im
z
1
z
0
z
1
z
0
2
+1
,
z
1
z
0
2
−
1
z
1
z
0
2
+1
=
x
+
iy
1
−
z
=
z
1
z
0
.
d
w
=
g
F
11
(
u
) =
u
•
g
F
11
´
X
N
.
n
þ
ü
«
œ
/
,
B
•
F
:
C
P
1
→
S
2
´
X
N
.
2
¤
é
uN
G
,
Ó
©
ü
«
œ
¹
?
Ø
µ
e
(
x,y,z
) = (0,0,1)
ž
,
K
G
(0,0,1) = [(0,
z
1
)].
ù
ž
(0,0,1)
3
S
2
Û
Ü
‹
I
ã
k
(
U
2
,ϕ
2
)
Ú
C
P
1
Û
Ü
‹
I
ã
k
(
V
2
,ψ
2
),
G
Û
ÜL
«
µ
g
G
22
=
ψ
2
◦
G
◦
ϕ
−
1
2
:
ϕ
2
(
U
2
∩
G
−
1
(
V
2
))
→
ψ
2
(
G
(
U
2
)
∩
V
2
)
.u
7→
w
=
g
G
22
(
u
)
.
(
Ü
£
2.3
¤
!
£
2.12
¤
!
£
3.4
¤
,
w
=
g
G
22
(
u
) =
ψ
2
◦
G
◦
ϕ
−
1
2
(
u
) =
ψ
2
◦
G
◦
ϕ
−
1
2
(
x
−
iy
1+
z
)
,u
6
= 0
ψ
2
◦
G
◦
ϕ
−
1
2
(0)
,u
= 0
=
ψ
2
◦
G
(
x,
−
y,
−
z
)
ψ
2
◦
G
(0,0,1)
=
ψ
2
([(1+
z,x
−
iy
)])
ψ
2
([(0
,z
1
)])
=
1
u
0
.
l
d
w
=
g
G
22
(
u
) =
1
u
•
g
G
22
´
X
N
.
e
(
x,y,z
)
6
= (0,0,1)
ž
,
K
G
(
x,y,z
) = [(
z
0
,z
1
)] = [(1
−
z,x
+
iy
)].
S
2
Û
Ü
‹
I
ã
k
(
U
1
,ϕ
1
)
DOI:10.12677/pm.2023.1361771737
n
Ø
ê
Æ
½
Œ
=
Ú
C
P
1
Û
Ü
‹
I
ã
k
(
V
1
,ψ
1
),
G
Û
ÜL
«
µ
g
G
11
=
ψ
1
◦
G
◦
ϕ
−
1
1
:
ϕ
1
(
U
1
∩
G
−
1
(
V
1
))
→
ψ
1
(
G
(
U
1
)
∩
V
1
)
.w
7→
u
=
g
G
11
(
w
)
.
(
Ü
£
2.2
¤
!
£
2.11
¤
!
£
3.4
¤
,
u
=
g
G
11
(
w
) =
ψ
1
◦
G
◦
ϕ
−
1
1
(
w
) =
ψ
1
◦
G
◦
ϕ
−
1
1
(
x
+
iy
1
−
z
)
=
ψ
1
◦
G
(
x,y,z
) =
ψ
1
([(
z
0
,z
1
)]) =
x
+
iy
1
−
z
=
w.
l
d
w
=
g
G
11
(
u
) =
u
•
g
G
12
´
X
N
.
•
,
d
1
¤
,2
¤
•
F
:
C
P
1
→
S
2
•
V
X
N
,
C
P
1
†
S
2
´
X
d
.
±
þ
ü
‡
·
K
´
i
ù
¡
ƒ
m
X
d
Ä
¯¢
,
Ó
n
Œ
C
P
1
†
C
∞
•
´
X
d
.
3
i
ù
-
¡
Ø
¥
,
X
d
i
ù
-
¡
,
Ó
•
˜
‡
i
ù
-
¡
,
Ú
¡
i
ù
¥
¡
.
3.2.Hopf
n
‘
z
e
X
d
3
!
¥
,
‰
Ñ
Hopf
n
‘
z
e
N
Ú
N
,
d
i
ù
¡
ƒ
m
X
d
'
X
,
í
Ñ
n
‘
¥
¡
S
3
3
Hopf
n
‘
z
†
E
˜
‘
K
˜
m
C
P
1
ƒ
m
X
d
.
·
K
3.3
[11]
3
Ó
Ô
n
Ø
¥-
‡
Hopf
N
h
:
S
3
→
S
2
´
N
Ú
N
.
y
²
S
3
=
{
(
z
1
,z
2
)
∈
C
2
||
z
1
|
2
+
|
z
2
|
2
= 1
}
,
é
∀
(
z
1
,z
2
)
∈
S
2
,
½
Â
h
(
z
1
,z
2
) = (2Rez
1
z
2
,
2Im
z
1
z
2
,
|
z
1
|
2
−|
z
2
|
2
)
.
(3.5)
K
|
h
(
z
1
,z
2
)
|
2
=
|
2
z
1
z
2
|
2
+
|
z
1
|
2
−|
z
2
|
2
2
= (2
z
1
z
2
)
·
(2
z
1
z
2
)+
|
z
1
|
4
−
2
|
z
1
|
2
|
z
2
|
2
+
|
z
2
|
4
=
|
z
1
|
4
+2
|
z
1
|
2
|
z
2
|
2
+
|
z
2
|
4
=
|
z
1
|
2
+
|
z
2
|
2
2
= 1.
h
(
z
1
,z
2
)
ü
‡
©
þ
¼
ê
h
1
(
z
1
,z
2
)=2
z
1
z
2
= 2(
x
1
+
iy
1
)(
x
2
−
iy
2
)=2(
x
1
x
2
+
y
1
y
2
)+
i
(
−
x
1
y
2
+
x
2
y
1
)
,
h
2
(
z
1
,z
2
)=
|
z
1
|
2
−|
z
2
|
2
=
x
1
2
+
y
1
2
−
x
2
2
−
y
2
2
.
Ñ
´
C
2
∼
=
R
4
þ
2
g
à
g
N
Ú¼
ê
.
l
¯¢
d
n
+1
‡
k
g
à
g
N
Ú
õ
‘
ª
f
:
R
n
→
S
n
→
R
n
+1
,
K
f
3
¥
¡
S
n
þ
•
›
f
|
S
n
=
S
m
→
S
n
´
N
Ú
N
•
,
h
½
Â
S
3
S
2
þ
N
Ú
N
.
DOI:10.12677/pm.2023.1361771738
n
Ø
ê
Æ
½
Œ
=
Š
Hopf
n
‘
z
N
,
π
:
S
3
→
C
P
1
,
é
∀
(
z
0
1
,z
0
2
)
∈
S
3
,
=
z
0
1
2
+
z
0
2
2
= 1.
e
∃
λ
∈
C
,
|
λ
|
=1,
¦
(
z
0
1
,z
0
2
)=
λ
(
z
1
,z
2
).
K
¡
(
z
0
1
,z
0
2
)
d
u
(
z
1
,z
2
).
S
3
'
u
ù
‡
d
'
X
û
˜
m
•
C
P
1
=
C
2
−{
0
}
/
∼
=
{
[(
z
1
,
z
2
)] = (
z
1
,
z
2
)
/
∼|∀
(
z
1
,
z
2
)
∈
C
2
−{
0
}}
.
½
Â
X
N
h
1
:
C
P
1
→
C
∞
=
C
∪{∞}
X
e
:
∀
[(
z
1
,
z
2
)]
∈
C
P
1
,
k
h
1
[(
z
1
,z
2
)] =
z
1
z
2
,z
2
6
= 0,
∞
,z
2
=0.
(3.6)
d
·
K
3.1,
½
Â
X
N
g
:
C
∞
→
S
2
X
e
µ
é
∀
z
∈
C
∞
,
k
g
(
z
) =
(
2Re
z
|
z
|
2
+1
,
2Im
z
|
z
|
2
+1
,
|
z
|
2
−
1
|
z
|
2
+1
)
,z
6
=
∞
(0
,
0
,
1)
,z
=
∞
(3.7)
Œ
±
w
Ñ
,Hopf
N
´
d
˜
‡
n
‘
z
N
†
ü
‡
X
N
E
Ü
(
J
,
=
h
=
g
◦
h
1
◦
π
.
d
d
Œ
,
n
‘
¥
¡
S
3
3
Hopf
n
‘
z
†
C
P
1
X
d
.
4.
E
˜
‘
K
˜
m
C
P
1
X
N
¯
K
Frankel
3
[12]
¥
Q
Š
X
e
ß
Ž
:
z
‡
;
K¨ahler
6
/
,
X
J
§
X
V
-
Ç
•
,
@
o
§
X
d
u
E
K
˜
m
C
P
n
[13].
é
u
E
˜
‘
œ
/
,
!
3
?
¿
U
þ
•
z
þ
½
Â
E
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‘
K
˜
m
C
P
1
K¨ahler
6
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X
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,
•
d
‰
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O
ó
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C
P
1
´
ä
k
½
/
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ω
E
˜
‘
K
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m
,
M
´
ä
k
K¨ahler
Ý
þ
h
;
K¨ahler
6
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3
(
C
P
1
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)
Ú
(
M,h
)
X
Û
Ü
‹
I
þ
,
©
O
k
µ
ω
=
λ
2
dω
⊗
d
¯
ω,h
=
√
−
1
h
ij
dz
i
∧
d
¯
z
j
.
Ù
¥
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[1]
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[2]Jiang, Y.F.(2004)RealizabilityofSomeClassesofAbstractBranchDataoverRiemannSphere.
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