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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
S
m+p
¥f6/þ˜‡Øª
êêêôôô777
H“‰ŒÆêÆÆ§H&²
ÂvFϵ2022c511F¶¹^Fϵ2022c616F¶uÙFϵ2022c623F
Á‡
x:M
m
→S
m+p
(m≥2,p≥2)´m+p‘ü ¥S
m+p
¥˜‡m‘Ãß:f6/§M
m
þ
M¨obius 1Ä/ ªB´M
m
3S
m+p
¥M¨obius C†+eØCþ§©Øª
P
α,β
tr[(B
α
)
2
(B
β
)
2
] ≤
(m−1)(3m
2
−9m+8)
2m
3
§y²Ò¤á^‡"
'…c
M¨obius AÛ§M¨obius ØCþ§Øª
AnInequalityontheSubmanifoldsofS
m+p
JiangtaoMa
DepartmentofMathematics,YunnanNormalUniversity,KunmingYunnan
Received:May11
th
,2022;accepted:Jun.16
th
,2022;published:Jun.23
rd
,2022
Abstract
Letx:M
m
→S
m+p
(m≥2,p≥2)beanm-dimensionalnoumbilicalsubmaniflodsin
m+ p-dimensionalunitsphereS
m+p
.TheM¨obiussecondbasicfromBofM
m
isthe
invariant ofunder thegroup ofM¨obius transformationsin S
m+p
.We obtaininequality
P
α,β
tr[(B
α
)
2
(B
β
)
2
] ≤
(m−1)(3m
2
−9m+8)
2m
3
. 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
Copyright
c
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
m
→S
m+p
(m>3)´m+p‘ü ¥S
m+p
¥˜‡m‘Ãß:f6/, d©z[1]•,
{e
i
}´ÝþI=dx·dxÛÜIOƒIe|,ÙéóIe|•{θ
i
},1Ä/ª
II=
P
i,j,α
h
α
ij
θ
i
θ
j
e
α
, ²þ-Ç•H. ½Âρ
2
=
m
m−1
|II−
1
m
tr(II)I|
2
,@o½2-/ªg=ρ
2
I
´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=
P
i,α
C
α
i
θ
i
e
α
,Blaschke ÜþA=ρ
2
P
i,j
A
ij
θ
i
θ
j
Ú
M¨obius1Ä/ªB=
P
i,j,α
B
α
i,j
ω
i
ω
j
(ρ
−1
e
α
), d©z[4]‰Ñ,
C
α
i
= −ρ
−2
H
α
,i
+
X
j

h
α
ij
−H
α
δ
ij

e
j
(logρ)
!
,(1.1)
A
ij
=−ρ
−2
Hess
ij
(logρ)−e
i
(logρ)e
j
(logρ)−
X
α
H
α
h
α
ij
!
−
1
2
ρ
−2

k∇logρk
2
−1+kHk
2

δ
ij
,(1.2)
B
α
ij
= ρ
−1

h
α
ij
−H
α
δ
ij

,(1.3)
Ù¥Hess
ij
Ú∇•'uI= dx·dx3Ä.{e
i
}eHessÝÚFÝŽf.¡BAŠ•x
M¨obiusÌ-Ç.R
m+p+2
´m+p+2‘þ˜m,½ÂSÈh·,·iXe:
hX,ξi= −x
0
ξ
0
+x
1
ξ
1
+x
2
ξ
2
+x
3
ξ
3
+...+x
m+p+1
ξ
m+p+1
,(1.4)
Ù¥
X= (x
0
,x
1
,x
2
···x
m+p+1
);ξ= (ξ
0
,ξ
1
,ξ
2
···ξ
m+p+1
),
¡äkSÈh·,·i•þ˜mR
m+p+2
•m+p+2‘Lortentz˜m, P•R
m+p+2
1
.
DOI:10.12677/pm.2022.126106972nØêÆ
êô7
½Â
C
m+p+1
+
:= {X∈R
m+p+2
1
|hX,Xi= 0,x
0
>0},(1.5)
•R
m+p+2
1
1I,½Â
Q
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
m
→S
m+p
(m≥2,p≥2) ´ÃßE\f6/, B•M¨obius1Ä/ª,K
keØª
X
α,β
tr[(B
α
)
2
(B
β
)
2
] ≤
(m−1)(3m
2
−9m+8)
2m
3
,
¤á,Ù¥Ò¤á…=M
m
´/²"…B
α
ƒqueÝƒ˜












0u0···0
u00···0
000···0
.
.
.
.
.
.
.
.
.
.
.
.
000···0












n×n
,












u00···0
0−u0···0
000···0
.
.
.
.
.
.
.
.
.
.
.
.
000···0












n×n
3.S
m+p
f6/M¨obiusØCþ
•²ïá¥¥f6/1I.ÚØCþXÚ,©¦^†ÙƒÓPÒÚúª.
x: M
m
→S
m+p
⊂R
m+p+1
´ÃßE\.xM¨obius ˜•þ½Â•
ξ= ρ(1,x) :M
m
→R
m+p+2
1
,ρ
2
=
m
m−1
kII−
1
m
tr(II)Ik
2
>0.(3.1)
KkXe½n:
½n3.1([4])ü‡f6/x,ex: M
m
→S
m+p
´Moebius d…=• 3R
m+p+2
1
¥
LorentzC†T∈O(m+p+1,1) ,¦ξ=
e
ξT.
Ù¥O(m+p+1,1) ´R
m+p+2
1
¥±SÈh,iØCLorentz +,@oO(m+p+1) ´˜‡
3Q
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
2
dx·dx,(3.3)
´M¨obiusØCþ,¡•M¨obiusÝþ½dxpM¨obius1˜Ä/ª.
M•(M,g) Laplace Žf.
h4ξ,4ξi= 1+m
2
κ,(3.4)
Ù¥κL«x{zM¨obiusêþ-Ç.
{E
1
,E
2
,···,E
m
}´(M,g)˜‡ÛÜIOÄ,{ω
1
,ω
2
,···,ω
m
}•ÙéóÄ.…
E
i
(ξ) = ξ
i
, @o.
hξ
i
,ξ
j
i= δ
ij
,1 ≤i,j≤m,(3.5)
½Â
N= −
1
m
4ξ−
1
2m
2
h4ξ,4ξiξ,(3.6)
@o
hξ,ξi= hN,Ni= 0,hξ,Ni= 1,hξ
i
,ξi= hξ
i
,Ni= 0,(1 ≤i,j≤m).(3.7)
…
hξ,dξi= 0,h4ξ,ξi= −m,h4ξ,ξ
k
i= 0,1 ≤k≤m.(3.8)
Ïd
span{N,ξ}⊥span{ξ
1
,ξ
2
,···,ξ
m
},(3.9)
½Â
V= {span{N,ξ}⊕span{ξ
1
,···,ξ
m
}}
⊥
,(3.10)
V´f˜mSpan{ξ,N,ξ
1
,ξ
2
,···,ξ
m
}3R
m+p+2
1
¥Ö˜m,@o·‚kXe
©).
R
m+p+2
1
= span{ξ,N}⊕span{ξ
1
,ξ
2
,···,ξ
m
}⊕V,(3.11)
¡V´x:M
m
→S
m+p
M¨obius{m.{E
m+1
,···,E
m+p
}´{mV÷M
m
˜‡ÛÜ
Ä,@o{ξ,N,ξ
1
,···,ξ
m
,E
m+1
,···,E
m+p
}´3R
m+p+2
÷M
m
¹ÄIe.ÏL¦^•I‰
Œ:
1 ≤i,j,k,l≤m;m+1 ≤α,β≤m+p;
Ù(•§•:
dξ=
X
i
ω
i
ξ
i
,(3.12)
dN=
X
ij
A
ij
ω
j
ξ
i
+
X
i,α
C
α
i
ω
i
E
α
,(3.13)
DOI:10.12677/pm.2022.126106974nØêÆ
êô7
dξ
i
= −
X
j
A
ij
ω
j
ξ−ω
i
N+
X
j
ω
ij
ξ
j
+
X
i
B
α
ij
ω
j
E
α
,(3.14)
dE
α
= −
X
i
C
α
i
ω
i
ξ−
X
i,j
B
α
ij
ω
j
ξ
i
,(3.15)
Ù¥{ω
ij
}´M¨obiusÝþgéä/ª
A
ij
= A
ji
,B
α
ij
= B
α
ji
.(3.16)
…
A=
X
i,j,
A
ij
ω
i
⊗ω
j
,B=
X
i,j,α
B
α
ij
ω
i
⊗ω
j
E
α
,Φ =
X
α
φ
α
E
α
=
X
i,α
C
α
i
ω
i
E
α
.(3.17)
Ñ´M¨obiusØCþ, ¡A•xBlaschkeÜþ,B•xM¨obius1Ä/ª, Φ•xM¨obius
/ª.
½ÂC
α
i
,A
ij
,B
α
ij
˜CêXe
X
j
C
α
i,j
ω
j
= dC
α
i
+
X
j
C
α
j
ω
ji
+
X
β
C
β
i
ω
βα
,(3.18)
X
k
A
ij,k
= dA
ij
+
X
A
ik
ω
kj
+
X
k
A
kj
ω
ki
,(3.19)
X
B
α
ij,k
ω
k
= dB
ij
+
X
k
B
α
ik
ω
kj
+
X
k
B
α
kj
ω
ki
+
X
β
B
β
ij
ω
βα
,(3.20)
…
dω
ij
−
X
k
ω
ik
∧ω
kj
= −
1
2
X
kl
R
ijkl
ω
k
∧ω
l
,R
ijkl
= −R
ijlk
,(3.21)
@o(3.12)-(3.15)(•§ŒÈ^‡•
A
ij,k
−A
ik,j
=
X
α
(B
α
ik
C
α
j
−B
α
ij
C
α
k
),(3.22)
C
α
i,j
−C
α
j,i
=
X
k
(B
α
ik
A
kj
−B
α
kj
A
ki
),(3.23)
B
α
ij,k
−B
α
ik,j
= δ
ij
C
α
k
−δ
ik
C
α
j
,(3.24)
R
ijkl
=
X
α
(B
α
ik
B
α
jl
−B
α
il
B
α
jk
)+(δ
ik
A
jl
+δ
jl
A
ik
−δ
il
A
jk
−δ
jk
A
il
),(3.25)
tr(A) =
1
2m
(1+
m
m−1
R) ,
X
i
B
α
ii
= 0.(3.26)
Ù¥{A
ij,k
}, {B
α
ij,k
}Ú{C
α
i,j
}´A,BÚΦ 'ugpéäCê3IOÄe©þ.
(3.24)ª¥-i= j¦Ú
−
X
B
α
ij,i
= −(m−1)C
α
j
,(3.27)
DOI:10.12677/pm.2022.126106975nØêÆ
êô7
(3.25)ª¥-i= k¦Ú
R
jl
= −
X
k
B
α
jk
B
α
kl
+tr(A)δ
jl
+(m−2)A
jl
,(3.28)
d(3.26)Ú(3.27)Œ
X
ij
(B
α
ij
)
2
=
m−1
m
,(3.29)
4.½ny²
y²©½n,ÄkI‡y²e¡Ún.
Ún4.1x: M
m
→S
m+p
(m≥3) ´m‘ÃßE\f6/,B•M¨obius1Ä/ª, KkØ
ª
X
α,β
{2[tr(B
α
B
β
)]
2
+kB
α
B
β
−B
β
B
α
k
2
−
2m
m−2
tr[(B
α
)
2
(B
β
)
2
]}+
2(m−1)
m
2
(m−2)
≥0,
¤á,Ù¥Ò¤á…=M
m
´/²".
y²dWeyl-ÇÜþ½Â
W
ijkl
= R
ijkl
−
1
m−2
(S
ik
δ
jl
+S
jl
δ
ik
−S
il
δ
jk
−S
jk
δ
il
),(4.1)
Œ
|W
ijkl
|
2
= W
ijkl
[R
ijkl
−
1
m−2
(S
ik
δ
jl
+S
jl
δ
ik
−S
il
δ
jk
−S
jk
δ
il
)],(4.2)
ϕW
ijkl
•Ã,Üþ, Ïd
|W
ijkl
|
2
= W
ijkl
R
ijkl
,(4.3)
Ù¥S
ij
•ShoutenÜþ,½Â•
S
ij
= R
ij
−
R
2(m−1)
δ
ij
,κ=
R
m(m−1)
,(4.4)
d(4.4)ª
S
ij
=R
ij
−
m(m−1)κ
2(m−1)
δ
ij
=R
ij
−
mκ
2
δ
ij
,(4.5)
d(3.26),(4.4) ª
2mtrA=1+m
2
κ,(4.6)
¤±
DOI:10.12677/pm.2022.126106976nØêÆ
êô7
mκ= 2trA−
1
m
,(4.7)
u´
S
ij
= −B
α
ik
B
α
kj
+
1
2m
δ
ij
+(m−2)A
ij
,(4.8)
d(3.26), (4.1)Ú(4.8)Œ
W
ijkl
=B
α
ik
B
α
jl
−B
α
il
B
α
jk
+
1
m−2
[B
α
im
B
α
mk
δ
jl
+B
α
jm
B
α
ml
δ
ik
]
−B
α
im
B
α
ml
δ
jk
−B
α
jm
B
α
mk
δ
il
]−
1
m(m−2)
(δ
ik
δ
jl
−δ
il
δ
jk
),(4.9)
d(4.3)Ú(4.9)ªŒ
|W|
2
= W
ijkl
(B
β
ik
B
β
jl
−B
β
il
B
β
jk
),(4.10)
Ïd
|W|
2
=[(B
α
ik
B
α
jl
−B
α
il
B
α
jk
+
1
m−2
(B
α
im
B
α
mk
δ
jl
+B
α
jm
B
α
ml
δ
ik
−B
α
im
B
α
ml
δ
jk
−B
α
jm
B
α
mk
δ
il
)−
1
m(m−2)
(δ
ik
δ
jl
−δ
il
δ
jk
)](B
β
ik
B
β
jl
−B
β
il
B
β
jk
)
=2(B
α
ik
B
β
ki
B
α
jl
B
β
lj
−B
α
ik
B
β
kj
B
α
jl
B
β
li
)−
4
m−2
B
α
im
B
α
ml
B
β
lk
B
β
ki
+
2(m−1)
m
2
(m−2)
,(4.11)
Ù¥
B
α
ik
B
α
jl
B
β
ik
B
β
jl
= B
α
il
B
α
jk
B
β
il
B
β
jk
,B
α
ik
B
α
jl
B
β
il
B
β
lk
= B
α
il
B
α
jk
B
β
ik
B
β
jl
,(4.12)
X
k,l,β
(B
β
kl
)
2
=
m−1
m
,(4.13)
¤±k
|W|
2
= 2(B
α
ik
B
β
ki
B
α
jl
B
β
lj
−B
α
ik
B
β
kj
B
α
jl
B
β
li
)−
4
m−2
B
α
im
B
α
ml
B
β
lk
B
β
ki
+
2(m−1)
m
2
(m−2)
,(4.14)
d
2
X
i,k,j,l,α,β
(B
α
ik
B
β
ki
B
α
jl
B
β
lj
−B
α
ik
B
β
kj
B
α
jl
B
β
li
) =2tr(B
α
B
β
)·tr(B
α
B
β
)−2tr(B
α
B
β
B
α
B
β
),(4.15)
X
i,k,l,m,α,β
B
α
im
B
α
ml
B
β
lk
B
β
ki
= tr(B
α
B
α
B
β
B
β
),(4.16)
(Ü(4.14)ªŒ
|W|
2
= 2[tr(B
α
B
β
)]
2
−2tr(B
α
B
β
B
α
B
β
)−
4
m−2
tr(B
α
B
α
B
β
B
β
)+
2(m−1)
m
2
(m−2)
(4.17)
DOI:10.12677/pm.2022.126106977nØêÆ
êô7
2d
kB
α
B
β
−B
β
B
α
k
2
=tr[(B
α
B
β
−B
β
B
α
)(B
α
B
β
−B
β
B
α
)
>
]
=tr[(B
α
B
β
−B
β
B
α
)(B
β
B
α
−B
α
B
β
)]
=tr(B
α
B
β
B
β
B
α
−B
α
B
β
B
α
B
β
−B
β
B
α
B
β
B
α
+B
β
B
α
B
α
B
β
)
=tr(B
α
B
β
B
β
B
α
)−tr(B
α
B
β
B
α
B
β
)−tr(B
β
B
α
B
β
B
α
)+tr(B
β
B
α
B
α
B
β
)
=tr(B
α
B
α
B
β
B
β
)−tr(B
α
B
β
B
α
B
β
)−tr(B
α
B
β
B
α
B
β
)+tr(B
α
B
α
B
β
B
β
)
=2tr(B
α
B
α
B
β
B
β
)−2tr(B
α
B
β
B
α
B
β
),(4.18)
Œ
−2tr(B
α
B
β
B
α
B
β
) = kB
α
B
β
−B
β
B
α
k−2tr[(B
α
)
2
(B
β
)
2
],(4.19)
u´
kW|
2
=
X
α,β
{2[tr(B
α
B
β
)]
2
+kB
α
B
β
−B
β
B
α
k
2
−
2m
m−2
tr[(B
α
)
2
(B
β
)
2
]}
+
2(m−1)
m
2
(m−2)
≥0,(4.20)
Ún4.1y.
e5y²½nA.dÚn4.1k
kW|
2
=
X
α,β
{2[tr(B
α
B
β
)]
2
+kB
α
B
β
−B
β
B
α
k
2
−
2m
m−2
tr[(B
α
)
2
(B
β
)
2
]}
+
2(m−1)
m
2
(m−2)
,(4.21)
¤±
X
α,β
{2[tr(B
α
B
β
)]
2
+kB
α
B
β
−B
β
B
α
k
2
−
2m
m−2
tr[(B
α
)
2
(B
β
)
2
]}+
2(m−1)
m
2
(m−2)
≥0,(4.22)
dDDVVØª(©z[5])k
X
α,β
kB
α
B
β
−B
β
B
α
k
2
≤(
X
α
kB
α
k
2
)
2
,(4.23)
dCauchy-SchwarzØªk
X
α,β
[tr(B
α
B
β
)]
2
≤
X
α
kB
α
k
2
·
X
β
kB
β
k
2
,(4.24)
DOI:10.12677/pm.2022.126106978nØêÆ
êô7
u´k
X
α,β
{2[tr(B
α
B
β
)]
2
+kB
α
B
β
−B
β
B
α
k
2
−
2m
m−2
tr[(B
α
)
2
(B
β
)
2
]}+
2(m−1)
m
2
(m−2)
≤(
X
α
kB
α
k
2
)
2
+2
X
α
kB
α
k
2
·
X
β
kB
β
k
2
−
2m
m−2
X
α,β
tr[(B
α
)
2
(B
β
)
2
]+
2(m−1)
m
2
(m−2)
,(4.25)
(Ü(4.22)Ú(4.25)ª
(
X
α
kB
α
k
2
)
2
+2
X
α
kB
α
k
2
·
X
β
kB
β
k
2
−
2m
m−2
X
α,β
tr[(B
α
)
2
(B
β
)
2
]+
2(m−1)
m
2
(m−2)
≥0,(4.26)
Ïd
2m
m−2
X
α,β
tr[(B
α
)
2
(B
β
)
2
] ≤(
X
α
kB
α
k
2
)
2
+2
X
α
kB
α
k
2
·
X
β
kB
β
k
2
+
2(m−1)
m
2
(m−2)
,(4.27)
…
X
α
kB
α
k
2
=
X
β
kB
β
k
2
=
m−1
m
,(4.28)
u´k
X
α,β
tr[(B
α
)
2
(B
β
)
2
] ≤
(m−1)(3m
2
−9m+8)
2m
3
,(4.29)
Ù¥Ò¤á…=M
m
´/²"…B
α
ƒqueÝƒ˜











0u0···0
u00···0
000···0
.
.
.
.
.
.
.
.
.
.
.
.
000···0











n×n
,











u00···0
0−u0···0
000···0
.
.
.
.
.
.
.
.
.
.
.
.
000···0











n×n
ùÒ¤½nAy².
5Pþãx(M
m
)M¨obiusduVeroness-¡ž,KÒ¤á.
~1S
2
(
√
3) = {(x
1
,x
2
,x
3
) ∈R
3
|x
2
1
+x
2
2
+x
2
3
= 3},½ÂNx: S
2
(
√
3) →S
4
(1)Xe
y
1
=
1
√
3
x
2
x
3
,y
2
=
1
√
3
x
3
x
1
,y
3
=
1
√
3
x
1
x
2
y
4
=
1
2
√
3
(x
2
1
−x
2
2
),y
5
=
1
6
(x
2
1
+x
2
2
−2x
2
3
),
KS
2
(
√
3)´S
4
(1)4f6/,¡•Veroness -¡.
DOI:10.12677/pm.2022.126106979nØêÆ
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ÀTIe,K1Ä/ª©O•
"
1
√
3
0
0−
1
√
3
#
,
"
0
1
√
3
1
√
3
0
#
dρ
2
=
8
3
Ïd
B
3
= ρ
−1
"
1
√
3
0
0−
1
√
3
#
=
"
1
2
√
2
0
0−
1
2
√
2
#
,B
4
= ρ
−1
"
0
1
√
3
1
√
3
0
#
=
"
0
1
2
√
2
1
2
√
2
0
#
džM
m
= S
2
(
√
3)´S
4
(1)¥Veroness-¡.
…
(B
3
)
2
=
"
1
2
√
2
0
0−
1
2
√
2
#"
1
2
√
2
0
0−
1
2
√
2
#
=
"
1
8
0
0
1
8
#
,
(B
4
)
2
=
"
0
1
2
√
2
1
2
√
2
0
#"
0
1
2
√
2
1
2
√
2
0
#
=
"
1
8
0
0
1
8
#
,
þãØª†>
P
α,β
tr[(B
α
)
2
(B
β
)
2
] =
1
8
,m= 2 “\m>
(m−1)(3m
2
−9m+8)
2m
3
=
1
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
n
.
ManuscriptaMathematica,111,287-302.https://doi.org/10.1007/s00229-003-0368-2
[4]Wang,C.P.(1998)MoebiusGeometryofSubmanifoldsinS
n
.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

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