Sm+p中子流形上的一个不等式 An Inequality on the Submanifolds of Sm+p

doi:10.12677/PM.2022.126106

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PureMathematicsnØêÆ,2022,12(6),971-980

PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.126106

m+p

¥f6/þ˜‡Øª

êêêôôô777

H“‰ŒÆêÆÆ§H&²

ÂvFÏµ2022c511F¶¹^FÏµ2022c616F¶uÙFÏµ2022c623F

Á‡

x:M

→S

m+p

(m≥2,p≥2)´m+p‘ü ¥S

m+p

¥˜‡m‘Ãß:f6/§M

M¨obius 1Ä/ ªB´M

m+p

¥M¨obius C†+eØCþ§©Øª

α,β

tr[(B

)

] ≤

(m−1)(3m

−9m+8)

§y²Ò¤á^‡"

'…c

M¨obius AÛ§M¨obius ØCþ§Øª

AnInequalityontheSubmanifoldsofS

m+p

JiangtaoMa

DepartmentofMathematics,YunnanNormalUniversity,KunmingYunnan

Received:May11

,2022;accepted:Jun.16

,2022;published:Jun.23

,2022

Abstract

Letx:M

→S

m+p

(m≥2,p≥2)beanm-dimensionalnoumbilicalsubmaniﬂodsin

m+ p-dimensionalunitsphereS

m+p

.TheM¨obiussecondbasicfromBofM

isthe

invariant ofunder thegroup ofM¨obius transformationsin S

m+p

.We obtaininequality

α,β

tr[(B

)

] ≤

(m−1)(3m

−9m+8)

. Theconditionsfortheequalitysignareproved.

©ÙÚ^:êô7.S

m+p

¥f6/þ˜‡Øª[J].nØêÆ,2022,12(6):971-980.

DOI:10.12677/pm.2022.126106

êô7

Keywords

M¨obiusGeometry,M¨obiusInvariant,Inequation

2022byauthor(s)andHansPublishersInc.

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

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

1.Úó

x: M

→S

m+p

(m>3)´m+p‘ü ¥S

m+p

¥˜‡m‘Ãß:f6/, d©z[1]•,

{e

}´ÝþI=dx·dxÛÜIOƒIe|,ÙéóIe|•{θ

},1Ä/ª

II=

i,j,α

, ²þ-Ç•H. ½Âρ

m−1

|II−

tr(II)I|

,@o½2-/ªg=ρ

´S

m+p

¥M¨obius C†+eØCþ,¡•xM¨obius Ýþ.d©z[2]Ú©z[3]xn‡Ä

M¨obius ØCþ,©O•M¨obius/ªΦ=

i,α

,Blaschke ÜþA=ρ

i,j

M¨obius1Ä/ªB=

i,j,α

i,j

(ρ

−1

), d©z[4]‰Ñ,

= −ρ

−2



−H



(logρ)

,(1.1)

=−ρ

−2

Hess

(logρ)−e

(logρ)e

(logρ)−

−

−2



k∇logρk

−1+kHk



,(1.2)

= ρ

−1



−H



,(1.3)

Ù¥Hess

Ú∇•'uI= dx·dx3Ä.{e

}eHessÝÚFÝŽf.¡BAŠ•x

M¨obiusÌ-Ç.R

m+p+2

´m+p+2‘îª•þ˜m,½ÂSÈh·,·iXe:

hX,ξi= −x

+...+x

m+p+1

,(1.4)

Ù¥

X= (x

···x

m+p+1

);ξ= (ξ

,ξ

···ξ

m+p+1

¡äkSÈh·,·i•þ˜mR

m+p+2

•m+p+2‘Lortentz˜m, P•R

m+p+2

DOI:10.12677/pm.2022.126106972nØêÆ

êô7

½Â

m+p+1

:= {X∈R

m+p+2

|hX,Xi= 0,x

>0},(1.5)

•R

m+p+2

1I,½Â

m+p

:= {[ξ] ∈RP

m+p+1

|hξ,ξi= 0},(1.6)

•RP

m+p+1

¥g-¡.

2.©Ì‡(J

©3M¨obiusAÛþXe½n:

½nAx:M

→S

m+p

(m≥2,p≥2) ´ÃßE\f6/, B•M¨obius1Ä/ª,K

keØª

α,β

tr[(B

)

] ≤

(m−1)(3m

−9m+8)

¤á,Ù¥Ò¤á…=M

´/²"…B

ƒqueÝƒ˜







0u0···0

u00···0

000···0







n×n







u00···0

0−u0···0

000···0







n×n

3.S

m+p

f6/M¨obiusØCþ

•²ïá¥¥f6/1I.ÚØCþXÚ,©¦^†ÙƒÓPÒÚúª.

x: M

→S

m+p

⊂R

m+p+1

´ÃßE\.xM¨obius ˜•þ½Â•

ξ= ρ(1,x) :M

→R

m+p+2

,ρ

m−1

kII−

tr(II)Ik

>0.(3.1)

KkXe½n:

½n3.1([4])ü‡f6/x,ex: M

→S

m+p

´Moebius d…=• 3R

m+p+2

¥

LorentzC†T∈O(m+p+1,1) ,¦ξ=

ξT.

Ù¥O(m+p+1,1) ´R

m+p+2

¥±SÈh,iØCLorentz +,@oO(m+p+1) ´˜‡

m+p

¥C†+½Â•

T([ξ]) := [ξT],X∈C

m+p+1

,ξ∈O(m+p+1,1),(3.2)

DOI:10.12677/pm.2022.126106973nØêÆ

êô7

Ïd

g= hdξ,dξi= ρ

dx·dx,(3.3)

´M¨obiusØCþ,¡•M¨obiusÝþ½dxpM¨obius1˜Ä/ª.

M•(M,g) Laplace Žf.

h4ξ,4ξi= 1+m

κ,(3.4)

Ù¥κL«x{zM¨obiusêþ-Ç.

{E

,···,E

}´(M,g)˜‡ÛÜIOÄ,{ω

,ω

,···,ω

}•ÙéóÄ.…

(ξ) = ξ

, @o.

hξ

,ξ

i= δ

,1 ≤i,j≤m,(3.5)

½Â

N= −

4ξ−

h4ξ,4ξiξ,(3.6)

hξ,ξi= hN,Ni= 0,hξ,Ni= 1,hξ

,ξi= hξ

,Ni= 0,(1 ≤i,j≤m).(3.7)

…

hξ,dξi= 0,h4ξ,ξi= −m,h4ξ,ξ

i= 0,1 ≤k≤m.(3.8)

Ïd

span{N,ξ}⊥span{ξ

,ξ

,···,ξ

},(3.9)

½Â

V= {span{N,ξ}⊕span{ξ

,···,ξ

}}

⊥

,(3.10)

V´f˜mSpan{ξ,N,ξ

,ξ

,···,ξ

}3R

m+p+2

¥Ö˜m,@o·‚kXe

©).

m+p+2

= span{ξ,N}⊕span{ξ

,ξ

,···,ξ

}⊕V,(3.11)

¡V´x:M

→S

m+p

M¨obius{m.{E

m+1

,···,E

m+p

}´{mV÷M

˜‡ÛÜ

Ä,@o{ξ,N,ξ

,···,ξ

m+1

,···,E

m+p

}´3R

m+p+2

÷M

¹ÄIe.ÏL¦^•I‰

Œ:

1 ≤i,j,k,l≤m;m+1 ≤α,β≤m+p;

Ù(•§•:

dξ=

,(3.12)

dN=

i,α

,(3.13)

DOI:10.12677/pm.2022.126106974nØêÆ

êô7

dξ

= −

ξ−ω

N+

,(3.14)

= −

ξ−

i,j

,(3.15)

Ù¥{ω

}´M¨obiusÝþgéä/ª

= A

= B

.(3.16)

…

i,j,

⊗ω

,B=

i,j,α

⊗ω

,Φ =

i,α

.(3.17)

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

/ª.

½ÂC

˜CêXe

i,j

= dC

βα

,(3.18)

ij,k

= dA

,(3.19)

ij,k

= dB

βα

,(3.20)

…

dω

−

∧ω

= −

ijkl

∧ω

ijkl

= −R

ijlk

,(3.21)

@o(3.12)-(3.15)(•§ŒÈ^‡•

ij,k

−A

ik,j

−B

),(3.22)

i,j

−C

j,i

−B

),(3.23)

ij,k

−B

ik,j

= δ

−δ

,(3.24)

ijkl

−B

)+(δ

+δ

−δ

),(3.25)

tr(A) =

(1+

m−1

R) ,

= 0.(3.26)

Ù¥{A

ij,k

}, {B

ij,k

}Ú{C

i,j

}´A,BÚΦ 'ugpéäCê3IOÄe©þ.

(3.24)ª¥-i= j¦Ú

−

ij,i

= −(m−1)C

,(3.27)

DOI:10.12677/pm.2022.126106975nØêÆ

êô7

(3.25)ª¥-i= k¦Ú

= −

+tr(A)δ

+(m−2)A

,(3.28)

d(3.26)Ú(3.27)Œ

)

m−1

,(3.29)

4.½ny²

y²©½n,ÄkI‡y²e¡Ún.

Ún4.1x: M

→S

m+p

(m≥3) ´m‘ÃßE\f6/,B•M¨obius1Ä/ª, KkØ

ª

α,β

{2[tr(B

)]

+kB

−B

−

m−2

tr[(B

)

]}+

2(m−1)

(m−2)

≥0,

¤á,Ù¥Ò¤á…=M

´/²".

y²dWeyl-ÇÜþ½Â

ijkl

= R

ijkl

−

m−2

−S

),(4.1)

Œ

ijkl

= W

ijkl

−

m−2

−S

)],(4.2)

Ï•W

ijkl

•Ã,Üþ, Ïd

ijkl

= W

ijkl

,(4.3)

Ù¥S

•ShoutenÜþ,½Â•

= R

−

2(m−1)

,κ=

m(m−1)

,(4.4)

d(4.4)ª

−

m(m−1)κ

2(m−1)

−

mκ

,(4.5)

d(3.26),(4.4) ª

2mtrA=1+m

κ,(4.6)

¤±

DOI:10.12677/pm.2022.126106976nØêÆ

êô7

mκ= 2trA−

,(4.7)

u´

= −B

+(m−2)A

,(4.8)

d(3.26), (4.1)Ú(4.8)Œ

ijkl

−B

m−2

]

−B

]−

m(m−2)

(δ

−δ

),(4.9)

d(4.3)Ú(4.9)ªŒ

|W|

= W

ijkl

−B

),(4.10)

Ïd

|W|

=[(B

−B

m−2

−B

)−

m(m−2)

(δ

−δ

)](B

−B

)

=2(B

−B

)−

m−2

2(m−1)

(m−2)

,(4.11)

Ù¥

= B

,(4.12)

k,l,β

)

m−1

,(4.13)

¤±k

|W|

= 2(B

−B

)−

m−2

2(m−1)

(m−2)

,(4.14)

i,k,j,l,α,β

−B

) =2tr(B

)·tr(B

)−2tr(B

),(4.15)

i,k,l,m,α,β

= tr(B

),(4.16)

(Ü(4.14)ªŒ

|W|

= 2[tr(B

)]

−2tr(B

)−

m−2

tr(B

2(m−1)

(m−2)

(4.17)

DOI:10.12677/pm.2022.126106977nØêÆ

êô7

−B

=tr[(B

−B

)(B

−B

)

]

=tr[(B

−B

)(B

−B

)]

=tr(B

−B

)

=tr(B

)−tr(B

)+tr(B

)

=tr(B

)−tr(B

)+tr(B

)

=2tr(B

)−2tr(B

),(4.18)

Œ

−2tr(B

) = kB

−B

k−2tr[(B

)

],(4.19)

u´

kW|

α,β

{2[tr(B

)]

+kB

−B

−

m−2

tr[(B

)

]}

2(m−1)

(m−2)

≥0,(4.20)

Ún4.1y.

e5y²½nA.dÚn4.1k

kW|

α,β

{2[tr(B

)]

+kB

−B

−

m−2

tr[(B

)

]}

2(m−1)

(m−2)

,(4.21)

¤±

α,β

{2[tr(B

)]

+kB

−B

−

m−2

tr[(B

)

]}+

2(m−1)

(m−2)

≥0,(4.22)

α,β

−B

≤(

)

,(4.23)

dCauchy-SchwarzØªk

α,β

[tr(B

)]

≤

,(4.24)

DOI:10.12677/pm.2022.126106978nØêÆ

êô7

u´k

α,β

{2[tr(B

)]

+kB

−B

−

m−2

tr[(B

)

]}+

2(m−1)

(m−2)

≤(

)

−

m−2

α,β

tr[(B

)

2(m−1)

(m−2)

,(4.25)

(Ü(4.22)Ú(4.25)ª

(

)

−

m−2

α,β

tr[(B

)

2(m−1)

(m−2)

≥0,(4.26)

Ïd

m−2

α,β

tr[(B

)

] ≤(

)

2(m−1)

(m−2)

,(4.27)

…

m−1

,(4.28)

u´k

α,β

tr[(B

)

] ≤

(m−1)(3m

−9m+8)

,(4.29)

Ù¥Ò¤á…=M

´/²"…B

ƒqueÝƒ˜







0u0···0

u00···0

000···0







n×n







u00···0

0−u0···0

000···0







n×n

ùÒ¤½nAy².

5Pþãx(M

)M¨obiusduVeroness-¡ž,KÒ¤á.

~1S

(

√

3) = {(x

) ∈R

= 3},½ÂNx: S

(

√

3) →S

(1)Xe

√

−x

),y

−2x

(

√

3)´S

(1)4f6/,¡•Veroness -¡.

DOI:10.12677/pm.2022.126106979nØêÆ

êô7

√

0−

√

dρ

Ïd

= ρ

−1

√

0−

√

0−

√

= ρ

−1

√

džM

= S

(

√

3)´S

(1)¥Veroness-¡.

…

)

√

0−

√

0−

√

)

√

þãØª†>

α,β

tr[(B

)

] =

,m= 2 “\m>

(m−1)(3m

−9m+8)

džþãØªÒ¤á.

ë•©z

[1]Li,H.Z.,Liu,H.L.,Wang,C.P.andZhao,G.S.(2002)MoebiusIsoparametricHypersurfaces

inS

n+1

withTwoDistinctPrincinalCurvatures.ActaMathematicaSinica,18,437-446.

https://doi.org/10.1007/s10114-002-0173-y

[2]Hu, Z.J. and Li, H.Z. (2004) The Classiﬁcation of Hyperfurfaces in S

n+1

with Parallel Moebius

SecondFundamentalForm.ScienceinChina,SeriesA,34,28-39.(InChinese)

[3]Hu,Z.J.andLi,H.Z.(2003)SubmanifoldswithConstantMoebiusScalarCurvatureinS

ManuscriptaMathematica,111,287-302.https://doi.org/10.1007/s00229-003-0368-2

[4]Wang,C.P.(1998)MoebiusGeometryofSubmanifoldsinS

.ManuscriptaMathematica,96,

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

[5]Ge,J.G.andTang,Z.Z.(2008)AProofoftheDDVVConjectureandItsEqualityCase.

PaciﬁcJournalofMathematics,237,87-95.https://doi.org/10.2140/pjm.2008.237.87

DOI:10.12677/pm.2022.126106980nØêÆ