球空间中具有拟平行第二基本形式的超曲面 Hypersurfaces with Quasi-Parallel SecondBasic Form in Spherical Space

doi:10.12677/PM.2023.134112

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PureMathematicsnØêÆ,2023,13(4),1062-1072

PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.134112

¥˜m¥äk[²11Ä/ª‡-¡

€€€¸¸¸

H“‰ŒÆêÆÆ§H&²

ÂvFÏµ2023c320F¶¹^FÏµ2023c421F¶uÙFÏµ2023c428F

Á‡

x:M

→S

n+1

•iù6/ü ¥˜m¥åE\,gÚB©O•x#'¿dÝþÚ#'¿d

1Ä/ª,R•gp -ÇÜþ.©ïÄ÷v^‡RB=0‡- ¡,¼A‡Ð Ú(

'…c

AÛ§‡-¡§[²1§1Ä/ª§M¨obius/ª

HypersurfaceswithQuasi-ParallelSecond

BasicForminSphericalSpace

FengSu

SchoolofMathematics,YunnanNormalUniversity,KunmingYunnan

Received:Mar.20

,2023;accepted:Apr.21

,2023;published:Apr.28

,2023

Abstract

Letx:M

→S

n+1

beisometriimmersionofRiemannianmanifoldintounitsphere

space,andgandBbeMobiusmetricandMobiussecondfundamentalformofx

©ÙÚ^:€¸.¥˜m¥äk[²11Ä/ª‡-¡[J].nØêÆ,2023,13(4):1062-1072.

DOI:10.12677/pm.2023.134112

€¸

respectivelyRisacurvaturetensorinducedbyg.Inthispaper,westudythehyper-

surfaceoperatorsatisfyingtheconditionRB= 0,andobtainsomepreliminaryresults.

Keywords

M¨obiusGeometry,Hypersurface,Quasi-Parallel,M¨obiusSecondFundamentalForm,

M¨obiusForm

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó9Ì‡(J

M¨obiusAÛ´¥¡þ3M¨obiusC†+ef6/5AÛ,éM¨obiusØCþ\þØÓ•

›^‡,Ò¬ØÓk¿ÂïÄ••.•C,LÇkJÑ¿ïÄ[²1f6/¿¼

k(J,~XB

ij,kl

ij,lk

i,j

j,i

.ÉÙéu,©´3M¨obius1Ä/ªB

[²1Ä:þ‰ïÄ.k{ü0˜e[²1:∇´dM¨obiusÝþg¤péä,-Ç

Žf´R.X,Y,Z•f6/þ1w•þ|.Kk R(X,Y)=−∇

∇

+ ∇

∇

+ ∇

[X,Y]

JR(X,Y)B=0,K¡M¨obius1Ä/ªB´[²1.©?˜ÚïÄ[²1‡-¡˜

AÛ5Ÿ,¼eA‡ÐÚ(J.

½n1.1¥S

n+1

¥[²1‡-¡M

,XJM¨obius/ªΦ²1,K‡-¡M

ÛÜM¨obius

due‡-¡ƒ˜:

(i)‚¡S

(a)×S

n−k



√

1−a



,Ù¥1 ≤k≤n−1;

(ii)R

n+1

¥IOÎ¡S

(a)×R

n−k

3Nσe”,Ù¥1 ≤k≤n−1;

(iii)H

n+1

¥IOÎ¡S

(a)×H

n−k



√

1+a



3Nτe”,Ù¥1 ≤k≤n−2;

(iv)‡-¡CSS(p,q,a).

Ù¥˜m!N!CSS½Â•„ý•£.

½n1.2S

n+1

¥;‡-¡M

,eM¨obius1Ä/ªB´[²1,Kkeª¤á:

n(n−1)

|Φ|

dM=

|∇B|

dM.

½n1.3S

n+1

¥;‡-¡M

,eA=λg,λ∈C

∞

(M),…M¨obius1Ä/ªB[²1,K

DOI:10.12677/pm.2023.1341121063nØêÆ

€¸

k±eØª¤á.

(4λ)

dM≥



λ−



|∇λ|

dM.

©(Sü

©©•o‡Ü©§1˜Ü©•ÚóÚÌ‡½n§ùÜ©Ì‡08cM¨obiusAÛïÄ9

:§Óž‰Ñ©ïÄ8Ú(J;1Ü©•ý•£§ùÜ©½ÂM¨obius1Ä/ª

[²1§¿˜‡'…·K.1nÜ©Ì‡0[²1V gÚÚ^½n.1oÜ©•Ì‡

½ny²§ùÜ©ÏLé®kü‡(J(B

ij,kl

= B

ij,lk

i,j

= C

j,i

)\±|^§¿?˜Ú

(J.

2.ý•£

•²31998cïá¥˜m¥f6/1I.Ú#'¿dØCþXÚ,©÷^ Ù¥

úª†PÒ,[!•„©z[1].

·‚Äk½ÂS

m+p

¥M¨obiusØCþ¿…‰Ñ(•§.

R

m+p+2

´Lorentz˜m,KÙLorentzSÈ½Â•µ

hx,ξi= −x

+···+x

m+p+1

Ù¥x= (x

,...,x

m+p+1

);ξ= (ξ

,ξ

,...,ξ

m+p+1

x:M

→S

m+p

⊂R

m+p+1

´S

m+p

¥ÃßE\f6/.òxM¨obius ˜•þX:M

→

m+2

Xeµ

X= ρ(1,x) : M

→R

m+2

,ρ

m−1



kIIk−mH



>0.

½n2.1eü‡f6/x,˜x: M

→S

m+p

´M¨obiusd,…=•3R

m+p+1

¥LorentzC

†T∈O(m+p+1,1),¦X=

XT.

Ù¥O(m+p+1,1)´R

m+p+2

¥±SÈh·,·iØCLorentz+,Ï•S

m+p

¥M¨obius

+åuO(m+p+1,1)±1If+O

(m+p+1,1),Œ

g= hdX,dXi= ρ

dx·dx,(2.1)

´M¨obiusØCþ,·‚òg¡•M¨obiusÝþ½öM¨obius1˜Ä/ª.4•(M,g)

LaplaceŽf,k

h4X,4Xi= 1+m

Ù¥k•ÝþgXþ-Ç.

DOI:10.12677/pm.2023.1341121064nØêÆ

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{E

,...,E

}´(M,g)˜‡ÛÜIOÄ,{ω

,ω

,...,ω

}•ÙéóÄ.¿…E

(X) =

,@okµ

i= δ

,1 ≤i,j≤m,

½Â

N= −

4X−

h4X,4XiX,(2.2)

@ok

hX,Xi= hN,Ni= 0,hX,Ni= 1,hX

,Xi= 0,(1 ≤i,j≤m).(2.3)

…

hX,dXi= 0,h4X,Xi= −m,h4X,X

i= 0,1 ≤k≤m.(2.4)

Ïd

span{N,X}⊥span{X

,...,X

½Â

V= {span{N,X}⊕span{X

,...,X

}}

⊥

,(2.5)

-V´f˜mspan{X,N,X

,...,X

}3R

m+p+2

¥Ö˜m,Œ±e¡©).

m+p+2

= span{X,N}⊕span{X

,...,X

}⊕V,(2.6)

·‚é©•IkXe5½:1≤i,j,k,···≤m;m+1≤α≤m+p,·‚„UìOÏd"½%

@-E•IL«3ˆg‰ŒS¦Ú.¡V´x:M

→S

m+p

M¨obius{m.{mV÷M

˜‡

ÛÜIOÄ•{E

m+1

,...,E

m+p

@o{X,N,X

,...,X

m+1

,...,E

m+p

}¤R

m+p+2

÷M

¹ÄIe.Ù(•§Xe:

dX=

,(2.7)

dN=

i,j

i,α

,(2.8)

= −

X−ω

N+

i,α

,(2.9)

DOI:10.12677/pm.2023.1341121065nØêÆ

€¸

= −

X−

i,j

αβ

,(2.10)

Ù¥{ω

}´M¨obiusÝþgéä/ª,{ω

αβ

}´M

þ{éä,…kA

= A

= B

.?

i,j

⊗ω

,(2.11)

i,j,α

⊗ω

,(2.12)

Φ =

i,α

,(2.13)

Ñ´M¨obiusØCþ¶©O¡A•xBlaschkeÜþ,B•xM¨obius1Ä/ª,Φ•xM¨obius

/ª.

©O½ÂC

˜CêXe

i,j

= dC

βα

,(2.14)

ij,k

= dA

,(2.15)

ij,k

= dB

βα

,(2.16)

…

dω

−

∧ω

= −

ijkl

∧ω

ijkl

= −R

ijlk

,(2.17)

@oŒ(•§ŒÈ^‡•

ij,k

−A

ik,j

= B

−B

,(2.18)

i,j

−C

j,i

= B

−B

,(2.19)

ij,k

−B

ik,j

= δ

−δ

,(2.20)

ijkl



−B



+(δ

+δ

−δ

),(2.21)

tr(A) =



m−1



= 0.(2.22)

DOI:10.12677/pm.2023.1341121066nØêÆ

€¸

Ù¥´{A

ij,k



ij,k





i,j



´A,BÚΦ'ugpéäCê2IOÄe©þ.

i= j¦Ú

−

ij,i

= (m−1)C

,(2.23)

i= k¦Ú





m−1

,(2.24)

½ÂA

ÚB

Cê

ij,kl

= dA

ij,k

lj,k

il,k

ij,l

,(2.25)

ij,kl

= dB

ij,k

lj,k

il,k

ij,l

ij,k

βα

,(2.26)

3.[²1Vg

½Â3.1(M

,g)´˜‡iù6/,∇ÚR©O•Ýþg¤pé äÚ-ÇÜþ.XJ˜‡Ü

þT÷vRT= 0,@oK¡ÜþT´[²1.

X,Y,Z´M

þ1w•þ|,Ù¥

R(X,Y)Z= −



∇

−∇

∇

−∇

[X,Y]



Z,(3.1)

T•CÜþ|,KòRT½Â•

(R(X,Y)T)(Z,W) := R(X,Y)(T(Z,W))−T(R(X,Y)Z,W)−T(Z,R(X,Y)W).(3.2)

·K3.2CÜþT[²1…=

T(R(X,Y)Z,W)+T(Z,R(X,Y)W) = 0.

yé∀f∈C

∞

(M),k∇

f= X(f).K

(∇

∇

)f= ∇

(∇

f) = ∇

(Y(f)) = X(Y(f)),(3.3)

Óžd(3.1),k

R(X,Y)(f) = −



∇

−∇

∇

−∇

[X,Y]



(f)

= −(X◦Y−Y◦X−[X,Y])(f)

= 0.(3.4)

DOI:10.12677/pm.2023.1341121067nØêÆ

€¸

ddy²T•CÜþž,duT(Z,W)•1w¼ê,k

R(X,Y)(T(Z,W)) = 0,(3.5)

¤±d½Â3.1•,CÜþT[²1…=

0 = (R(X,Y)T)(Z,W) = −T(R(X,Y)Z,W)−T(Z,R(X,Y)W).(3.6)

=T(R(X,Y)Z,W)+T(Z,R(X,Y)W) = 0,·K¤á.

{e

,...,e

}´(M

,g)˜‡ÛÜIOÄ,{ω

,ω

,...,ω

}•ÙéóÄ.K

R(e

= R

ijkl

,(3.7)

·K3.2Œ¤CÜþT[²1…=

ijkm

ijlm

= 0.(3.8)

½Â3.3x:M

→S

n+1

•åE\,gÚB©O•xM¨obiusÝþÚM¨obius1Ä/ª.

XJB'ugpéä´[²1,K¡x•M¨obius[²1‡-¡,½M-[²1‡-¡.

32004c,LÚo°¥3©z[3]¥kXe©a½nµ

½n3.4x:M

→S

n+1

(n≥2)´äk²1M¨obius1Ä/ªÃßE\‡-¡.@

ÛÜM¨obiusde‡-¡ƒ˜µ

(i)‚¡S

(a)×S

n−k



√

1−a



,Ù¥1 ≤k≤n−1;

(ii)R

n+1

¥IOÎ¡S

(a)×R

n−k

3Nσe”,Ù¥1 ≤k≤n−1;

(iii)H

n+1

¥IOÎ¡S

(a)×H

n−k



√

1+a



3Nτe”,Ù¥1 ≤k≤n−2;

(iv)d~1‰Ñ‡-¡CSS(p,q,a).

Ù¥H

n+1

´n+1‘V-˜m,½Â•

n+1



) ∈R

×R

n+1

|−y

·y

= −1



σ: R

n+1

→S

n+1

´¥4ÝK_,½Â•

σ(u) =

1−|u|

1+|u|

,u∈R

n+1

τ: H

n+1

→S

n+1

´5N,½Â•

τ(y

) =





,(y

) ∈H

n+1

~1CSS(p,q,a)

DOI:10.12677/pm.2023.1341121068nØêÆ

€¸

é‰½g,êp,q,p+ q<n,9¢êa∈(0,1)Úb=

√

1−a

,•ÄÛ-¦Èi\‡-

¡u: S

(a)×S

(b)×R

n−p−q−1

→R

n+1

u= (tu

,tu

000

),u

∈S

(a),u

∈S

(b),t∈R

000

∈R

n−p−q−1

x= σ◦u: S

(a)×S

(b)×R

×R

n−p−q−1

→S

n+1

½Â

CSS(p,q,a) = x



u: S

(a)×S

(b)×R

×R

n−p−q−1



§´S

n+1

¥‡-¡.

4.Ì‡½ny²

½n1.1yd©z[1]•

|B|

n−1

?•Ä4|B|



0 =

4|B|

i,j

)

i,j

i,j,k

ij,k

)

i,j,k

ij,kk

.(4.1)

òB

ij,kk

•

ij,kk

= (B

ij,kk

−B

ik,jk

)+(B

ki,jk

−B

ki,kj

)+(B

ki,kj

−B

kk,ij

)+B

kk,ij

.(4.2)

éŒÈ^‡B

ij,k

−B

ik,j

= δ

−δ

¦Œ

ij,kk

−B

ik,jk

= δ

k,k

−δ

j,k

,(4.3)

DOI:10.12677/pm.2023.1341121069nØêÆ

€¸

d‡-¡M

[²1,Ly²B

ij,kl

= B

ij,lk

,2òƒ†(3.3)˜Ó“\(3.2)Œ

ij,kk

= (δ

k,k

−δ

j,k

)+0+(δ

k,j

−δ

i,j

)+B

kk,ij

,(4.4)

2ò(3.4)ék¦Ú

ij,kk

= δ

k,k

−C

j,i

i,j

−nC

i,j

(4.5)

d‡-¡M

[²1,Ly²C

i,j

= C

j,i

,(3.5)Œz•

ij,kk

= δ

k,k

−nC

i,j

.(4.6)

ŒOŽ

i,j,k

ij,kk

i,j,k

(δ

k,k

−nC

i,j

)

= −

i,j

.(4.7)

•ò|∇B|

i,j,k

ij,k

)

9(3.7)“\(3.1)k

|∇B|

= −

i,j,k

ij,kk

i,j

.(4.8)

duf6/M

M¨obius/ªΦ²1(∇Φ=0),=C

i,j

= 0,¤±d(3.8)•ªŒ±y∇B= 0,=B

²1.KŠâ½n2.4Œ•,½n1.1¤á.

½n1.2y²

y?˜Ú,e‡-¡M

´;,KŒé(3.8)3‡-¡M

þÈ©,k

|∇B|

dM= n

i,j

dM,(4.9)

|^©ÜÈ©,Œ

|∇B|

dM= n

i,j

·C

)

−B

ij,j

dM,(4.10)

dÑÝ½nŒ

|∇B|

dM= −n

i,j

ij,j

dM,(4.11)

DOI:10.12677/pm.2023.1341121070nØêÆ

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ò(2.1.23)ª“\(3.0.30)

|∇B|

dM= n(n−1)

= n(n−1)

|Φ|

dM.(4.12)

ddy²½n1.2¤á.

½n1.3y²

4|∇λ|

i,j



i,j

+λ

i,jj



i,j



i,j

+λ

j,ij



(4.13)

dRicciðªŒ

4|∇λ|

i,j

i,j,m

(λ

j,ji

+λ

mjij

)

i,j

(4λ)

i,m

,(4.14)

d©z[1]•,R

= −

+tr(A)δ

+(n−2)A

,òƒ“\(3.14)

4|∇λ|

i,j

(4λ)

i,m

−

+tr(A)δ

+(n−2)A

i,j

(4λ)

i,m

−

+nλδ

+(n−2)λδ

i,j

(4λ)

−

i,m

+2(n−1)|∇λ|

λ,(4.15)

i,j

≥

i,i

,2d…Ü)–]Øªk

i,i

≥

(

i,i

)

,(3.15)Œ˜ •

4|∇λ|

≥

i,i

(4λ)

−

i,m

+2(n−1)|∇λ|

(4λ)

−

+2(n−1)|∇λ|

λ,(4.16)

DOI:10.12677/pm.2023.1341121071nØêÆ

€¸

)

(λ

)

≥

(

)

,KŒUY˜

4|∇λ|

≥

(4λ)

+2(n−1)λ|∇λ|

−

)

(λ

)

(4λ)

+2(n−1)λ|∇λ|

−

n−1

|∇λ|

,(4.17)

df6/M

;,Ké(3.17)ªü>3M

þÈ©k

0 ≥

(4λ)

dM+

(4λ)

dM+

2(n−1)



λ−



|∇λ|

(4λ)

dM+

(λ

·4λ)

dM−

i,i

·4λdM+

2(n−1)



λ−



|∇λ|



−1



(4λ)

dM+

2(n−1)



λ−



|∇λ|

(4.18)

nk

(4λ)

dM≥



λ−



|∇λ|

dM(4.19)

½n1.3y.

ë•©z

[1]Wang,C.P.(1998)MoebiusGeometryofSubmanifoldsins

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517-534.https://doi.org/10.1007/s002290050080

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