黎曼球面上全纯等价关系的构造及其应用 Construction of Holomorphic EquivalenceRelations on Riemannian Spheres and TheirApplications

doi:10.12677/PM.2023.136177

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PureMathematicsnØêÆ,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

,ZhimingHuang

,QiuhuaYang

1,3

,WeijunLu

CollegeofMathematicsandPhysics,GuangxiMinzuUniversity,NanningGuangxi

JiangmenPeiyingSeniorHighSchool,JiangmenGuangdong

CollegeofMathematicsandElectronicInformationEngineering,GuangxiNormalUniversityfor

Nationalities,ChongzuoGuangxi

Received:May21

,2023;accepted:Jun.22

,2023;published:Jun.29

,2023

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2023,13(6):1728-1743.DOI:10.12677/pm.2023.136177

½Œ=

Abstract

Inthispaper,westudysomespecialRiemannsurfacesoncomplexone-dimensional

connectedcomplexanalyticmanifolds,includingcomplexone-dimensionalprojection

spaceCP

,extendedcomplexplaneC

∞

andcomplexsphereS

.Inthesenseofholo-

morphicmappingandbiholomorphicmapping,thesethreetypicalRiemannsurfaces

areholomorphicequivalent.Furthermore,underHopfmapping,theholomorphice-

quivalencebetweenS

andCP

isderived.BasedonFrankle’sconjecture,theproblem

ofholomorphicmappingofenergyminimizationoncomplexone-dimensionalprojec-

tivespacesCP

tocompactK¨ahlermanifoldsisdiscussed.

Keywords

RiemannSurfaces,HolomorphicMapping,HolomorphicEquivalence,HopfMapping,

HolomorphicMinimizationMapping

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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DOI:10.12677/pm.2023.1361771730nØêÆ

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DOI:10.12677/pm.2023.1361771734nØêÆ

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

(x,y,z)

(0,0,1)







Ù¥,w=

x−iy

1+z

= 1,g(

x+iy

1−z

)=(x,y,z).Kk

1+z

x−iy

x+iy

1−z

Œ„,u=fg

(w) = w,ù´'uwX¼ê.

w3C

∞

ÛÜ‹Iãk(

,φ

)Úg(w)= (x,y,z)3S

ÛÜ‹Iãk(U

,ϕ

),gÛÜ‹

IL«•:

= ϕ

◦g◦φ

−1

: φ

(

∩g

−1

)) →ϕ

(g(

)∩U

).w7→u=fg

(w).

(Ü£2.2¤!£2.6¤!£3.2¤,

u=fg

(w) = ϕ

◦g◦φ

−1

(

x+iy

1−z

) = ϕ

◦g(

x+iy

1−z

) = ϕ

(x,y,z) =

x+iy

1−z

= w.

DOI:10.12677/pm.2023.1361771735nØêÆ

½Œ=

Œ„,fg

´XN.

nþ¤ã,•g= f

−1

•´XN,lf´˜‡VXN.

Ïd,S

†C

∞

´Xd.

·K3.2‘¥¡S

={(x,y,z) ∈R

= 1}†E˜‘ÝK˜mCP

−{0}/∼

Xd.

y²‘¥¡S

E)Û(•{(U

,ϕ

),(U

,ϕ

)},d·K2.1.2¥ª£2.1¤!£2.2¤±9

£2.3¤‰Ñ.

E˜‘ÝK˜mCP

g,ÝK

π: C

−{0}→CP

.(z

) 7→π(z

) = (z

)/∼= [(z

)].

ûÿÀ

∀A∈τ

−{0}

= τ

∩(C

−{0}),

½Âπ(A) = {π(z

) ∈CP

|∀(z

) ∈A}•CP

m8.=

τ=



π(A)|∀A∈τ

−{0}



E˜‘ÝK˜mCP

˜‡E)Û(•{(V

,ψ

),(V

,ψ

)}.d·K2.1.4¥ª£2.9¤-£2.12¤‰

Ñ.

ENF: CP

→S

[(z

)] 7→F([(z

)]) = (x,y,z) =

















2Re



2Im



−1









6= 0.

(0,0,1),z

= 0.

(3.3)

KF´V.Ù_N•G= F

−1

: S

→CP

(x,y,z) 7→G(x,y,z) = [(z

)] =







[(1−z,x+iy)],(x,y,z) 6= (0,0,1).

[(0,z

)],(x,y,z) = (0,0,1).

(3.4)

1¤éuNF,©ü«œ¹5?1ÛÜ‹I•Ä.

=0ž,KF([(0,z

)])=(0,0,1)∈S

.du[(0,z

)]áuV

,ÛÜ‹Iãk(V

,ψ

)Ú

:(0,0,1)3S

ÛÜ‹Iãk(U

,ϕ

),ùžFÛÜL«µ

= ϕ

◦F◦ψ

−1

: ψ

∩F

−1

)) →ϕ

(F(V

)∩U

).w7→u=

(w).

DOI:10.12677/pm.2023.1361771736nØêÆ

½Œ=

(Ü£2.3¤!£2.12¤!£3.3¤,

(w) = ϕ

◦F◦ψ

−1

(w)=







◦F◦ψ

−1

(

),z

6= 0

◦F◦ψ

−1

(

),z

= 0







◦F([(z

)])

◦F([(0,z

)])

















2Re



2Im



−1









(0,0,1)













u´

∂u

∂w

∂(w)

= 0,

´XN.ez

6= 0ž,F([z

]) 6= (0,0,1).[(z

)]3CP

ÛÜ

‹Iãk(V

,ψ

)ÚS

ÛÜ‹Iãk(U

,ϕ

),ùžFÛÜL«µ

= ϕ

◦F◦ψ

−1

: ψ

∩F

−1

)) →ϕ

(F(V

)∩U

).u7→w=

(u).

(Ü£2.2¤!£2.11¤!£3.3¤,

(u) = ϕ

◦F◦ψ

−1

(

) = ϕ

◦F([z

])

= ϕ







2Re



2Im



−1









x+iy

1−z

dw=

(u) = u•

´XN.

nþü«œ/,B•F: CP

→S

´XN.

2¤éuNG,Ó©ü«œ¹?Øµ

e(x,y,z) = (0,0,1)ž,KG(0,0,1) = [(0,z

)].ùž(0,0,1)3S

ÛÜ‹Iãk(U

,ϕ

)ÚCP



ÛÜ‹Iãk(V

,ψ

),GÛÜL«µ

= ψ

◦G◦ϕ

−1

: ϕ

∩G

−1

)) →ψ

(G(U

)∩V

).u7→w=

(u).

(Ü£2.3¤!£2.12¤!£3.4¤,

(u) = ψ

◦G◦ϕ

−1

(u) =







◦G◦ϕ

−1

(

x−iy

1+z

),u6= 0

◦G◦ϕ

−1

(0),u= 0







◦G(x,−y,−z)

◦G(0,0,1)







([(1+z,x−iy)])

([(0,z

)])







ldw=

(u) =

•

´XN.

e(x,y,z) 6= (0,0,1)ž,KG(x,y,z) = [(z

)] = [(1−z,x+iy)].S

ÛÜ‹Iãk(U

,ϕ

)

DOI:10.12677/pm.2023.1361771737nØêÆ

½Œ=

ÚCP

ÛÜ‹Iãk(V

,ψ

),GÛÜL«µ

= ψ

◦G◦ϕ

−1

: ϕ

∩G

−1

)) →ψ

(G(U

)∩V

).w7→u=

(w).

(Ü£2.2¤!£2.11¤!£3.4¤,

(w) = ψ

◦G◦ϕ

−1

(w) = ψ

◦G◦ϕ

−1

(

x+iy

1−z

)

= ψ

◦G(x,y,z) = ψ

([(z

)]) =

x+iy

1−z

= w.

ldw=

(u) = u•

´XN.

•,d1¤,2¤•F: CP

→S

•VXN,CP

†S

´Xd.

±þü‡·K´iù¡ƒmXdÄ¯¢,ÓnŒCP

†C

∞

•´Xd.3iù

-¡Ø¥,Xdiù-¡,Ó•˜‡iù-¡,Ú¡iù¥¡.

3.2.Hopfn‘zeXd

3!¥,‰ÑHopfn‘zeNÚN,diù¡ƒmXd'X,íÑn‘¥

¡S

3Hopfn‘z†E˜‘K˜mCP

ƒmXd.

·K3.3[11]3ÓÔnØ¥-‡HopfNh: S

→S

´NÚN.

y²S

= {(z

) ∈C

||z

+|z

= 1},é∀(z

) ∈S

,½Â

h(z

) = (2Rez

,2Imz

,|z

−|z

).(3.5)

|h(z

= |2z



−|z



= (2z

)·(2z

)+|z

−2|z

+|z

=|z

+2|z

+|z

=|z

+|z

= 1.

h(z

)ü‡©þ¼ê

)=2z

= 2(x

+iy

)(x

−iy

)=2(x

)+i(−x

)=|z

−|z

−x

−y

Ñ´C

∼

þ2gàgNÚ¼ê.

l¯¢dn+1‡kgàgNÚõ‘ªf: R

→S

→R

n+1

,Kf3¥¡S

þ•›f|

= S

→

´NÚN•,h½ÂS

S

þNÚN.

DOI:10.12677/pm.2023.1361771738nØêÆ

½Œ=

ŠHopfn‘zN,π: S

→CP

,é∀(z

) ∈S



= 1.e∃λ∈C,|λ|=1,¦

(z

)=λ(z

).K¡(z

)du(z

).S

'uù‡d'Xû˜m•

−{0}/∼= {[(z

)] = (z

)/∼|∀(z

) ∈C

−{0}}.

½ÂXNh

: CP

→C

∞

= C∪{∞}Xe:∀[(z

)] ∈CP

[(z

)] =







6= 0,

∞,z

=0.

(3.6)

d·K3.1,½ÂXNg: C

∞

→S

Xeµé∀z∈C

∞

g(z) =







(

2Rez

|z|

2Imz

|z|

−1

|z|

),z6= ∞

(0,0,1),z= ∞

(3.7)

Œ±wÑ,HopfN´d˜‡n‘zN†ü‡XNEÜ(J,=h= g◦h

◦π.

ddŒ,n‘¥¡S

3Hopfn‘z†CP

Xd.

4.E˜‘K˜mCP

XN¯K

Frankel3[12]¥QŠXeßŽ:z‡;K¨ahler6/,XJ§XV-Ç•,@o§X

duEK˜mCP

[13].éuE˜‘œ/,!3?¿Uþ•zþ½ÂE˜‘K˜mCP

K¨ahler6/þXN,•d‰˜OóŠ.

CP

´äk½/(ωE˜‘K˜m,M´äkK¨ahlerÝþh;K¨ahler6/.

3(CP

,ω)Ú(M,h)XÛÜ‹Iþ,©Okµ

ω= λ

dω⊗d¯ω,h=

√

−1h

∧d¯z

Ù¥h



∂

∂z

∂

∂¯z



é?¿1wNf: (CP

,ω) →(M,h),fUþ•¼•

E(f) =



∂f

∂ω

∂f

∂ω



√

−1dω∧d¯ω

∂f

∂ω

∂f

∂ω

∂f

∂¯ω

∂f

∂¯ω

√

−1dω∧d¯ω.(4.1)

DOI:10.12677/pm.2023.1361771739nØêÆ

½Œ=

Ù¥,

∂f

∂ω

= f

∂f

∂¯ω

= f

¯ω

∂f

∂ω

= f

∗



∂

∂ω



= f

∂

∂z

¯ω

∂

∂¯z

Ún4.1E(f)'uf´Œ‡,KkUþ1˜C©úª

(f)|

t=0

= −2



∂f

∂t

∂¯ω



∂f

∂ω



√

−1dω∧d¯ω.(4.2)

y²f(t) : CP

→M,t∈C,|t|≤ε´m˜Cþëêz1wNq,d£4.1¤k

(f) =

∂

∂t



∂f

∂ω

∂f

∂ω



√

−1dω∧d¯ω



∇

∂

∂t

∂f

∂ω

∂f

∂ω





∂f

∂ω

,∇

∂

∂f

∂ω



√

−1dω∧d¯ω



∇

∂

∂ω

∂f

∂t

∂f

∂ω





∂f

∂ω

,∇

∂

∂ω

∂f

∂



√

−1dω∧d¯ω



∇

∂

∂ω

∂f

∂t

∂f

∂ω





∇

∂

∂ω

∂f

∂t

∂f

∂ω



√

−1dω∧d¯ω



∂

∂ω



∂f

∂t

∂f

∂ω



−



∂f

∂t

,∇

∂

∂¯ω

∂f

∂ω



∂

∂ω



∂f

∂t

∂f

∂ω



−



∂f

∂t

,∇

∂

∂¯ω

∂f

∂ω



√

−1dω∧d¯ω

= −2



∂f

∂t

,∇

∂

∂¯ω

∂f

∂ω



√

−1dω∧d¯ω.

f´NÚN,=f÷v

∇

∂

∂¯ω

∂f

∂ω

= 0.(4.3)

du

∂

∂ω∂¯ω

+Γ

∂f

∂¯ω

∂f

∂ω

= 0.(4.4)

(f) = 0ž,E

(f)âk¿Â

Ún4.2[14]f: (CP

(f)|

t=0

= 2



∇

∂

∂ω

∂f

∂t



−



∂f

∂t



∂f

∂t

∂f

∂¯ω



∂f

∂ω



√

−1dω∧d¯ω.(4.5)

DOI:10.12677/pm.2023.1361771740nØêÆ

½Œ=

y²dÚn4.1

(f) = −2

∂

∂t



∂f

∂t

,∇

∂

∂¯ω

∂f

∂ω



√

−1dω∧d¯ω

= −2



∂f

∂t

,∇

∂

∂t

∇

∂

∂¯ω

∂f

∂ω



√

−1dω∧d¯ω

= −2



∂f

∂t



∂

∂t

∂

∂¯ω



∂f

∂ω





∂f

∂t

,∇

∂

∂¯ω

∇

∂

∂t

∂f

∂ω



√

−1dω∧d¯ω

= −2



∂f

∂t



∂

∂t

∂

∂¯ω



∂f

∂ω



∂

∂ω



∂f

∂t

,∇

∂

∂t

∂f

∂ω



−



∇

∂

∂ω

∂f

∂t

,∇

∂

∂t

∂f

∂ω



√

−1dω∧d¯ω

= 2



∇

∂

∂ω

∂f

∂t



−



∂f

∂t



∂f

∂t

∂f

∂¯ω



∂f

∂ω



√

−1dω∧d¯ω.

½Â4.3[15](M,h)´˜‡‘ên≥2;K¨ahler6/,é?¿š"•þ|X,Y∈T

1,0

X,Y>= 0,K

R(X,

X,Y,

Y) >0

¡•K¨ahler6/þV¡-Ç.

½n4.4(M,h)´˜‡‘ên≥2;K¨ahler6/,äkV¡-Ç,K?¿Uþ•

zNf: (CP

,ω) →(M,h)7L´X½öÝX.

y²bUþ•zNfQØ´X,•Ø´ÝX.

5¿dimH

(CP

,TCP

) = 3,Œ±CP

š"X•þv

∂

∂ω

,ü‡Mþš0£1,0¤.

•þ|



∗

∂

∂ω

)



(1,0)

= v

∂f

∂ω

∂

∂z

,Y=



∗

∂

∂ω

)



(1,0)

=¯v

∂f

∂¯ω

∂

∂z

f´NÚN,÷v£4.4¤,Kk

∇

∂

∂¯ω

X= ∇

∂

∂¯ω



∂f

∂ω

∂

∂z



= v



∇

∂

∂¯ω

∂f

∂ω



∂

∂z

= v



∂

∂ω∂¯ω

+Γ

∂f

∂¯ω

∂f

∂ω



∂

∂z

= 0.(4.6)

∇

∂

∂ω

Y= ∇

∂

∂ω



¯v

∂f

∂¯ω

∂

∂z



=¯v



∇

∂

∂ω

∂f

∂¯ω



∂

∂z

=¯v



∂

∂ω∂¯ω

+Γ

∂f

∂¯ω

∂f

∂ω



∂

∂z

= 0.(4.7)

d£4.6¤!£4.7¤k

∂

∂¯ω

hX,Yi=

∇

∂

∂¯ω

X,Y

X,∇

∂

∂ω

=0.ù¿›XhX,Yi´CP

X¼

ê.=hX,Yi3CP

þk":hX,Yi≡0.•Ò´`,(1,0).•þ|X,Yƒp,3Mþ•kk•

DOI:10.12677/pm.2023.1361771741nØêÆ

½Œ=

‡ú":.

d£4.7¤±9C©••

∂f(t)

∂t



t=0

= Y,Œ



∇

∂

∂ω

∂f

∂t





t=0

= ∇

∂

∂ω



∂f

∂t



t=0



= ∇

∂

∂ω

Y= 0.(4.8)

(f) = 2



∇

∂

∂ω

∂f

∂t



−



∂f

∂t



∂

∂t

∂

∂¯ω



∂f

∂ω



√

−1dω∧d¯ω

= −2





∂f

∂

∂f

∂ω



∂f

∂¯ω

∂f

∂



√

−1dω∧d¯ω

= −2

ijkl



¯ω

−f



¯ω

−f



√

−1dω∧d¯ω

= −2

|v|

ijkl

¯ω

√

−1dω∧d¯ω

= −2

|v|

−2

R(X,

X,Y,

√

−1dω∧d¯ω.

Ù¥|v|

−2

R(X,

X,Y,

Y)3v":??•0.

?d½Â4.3,

(f) = −2

|v|

−2

R(X,

X,Y,

√

−1dω∧d¯ω<0.

,˜•¡,duf´Uþ•zN,E

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Ä7‘8

2020437)]Ï"

ë•©z

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