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PureMathematics
n
Ø
ê
Æ
,2022,12(6),971-980
PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.126106
S
m
+
p
¥
f
6
/
þ
˜
‡
Ø
ª
êêê
ôôô
777
H
“
‰
Œ
Æ
ê
ÆÆ
§
H
&
²
Â
v
F
Ï
µ
2022
c
5
11
F
¶
¹
^
F
Ï
µ
2022
c
6
16
F
¶
u
Ù
F
Ï
µ
2022
c
6
23
F
Á
‡
x
:
M
m
→
S
m
+
p
(
m
≥
2
,p
≥
2)
´
m
+
p
‘
ü
¥
S
m
+
p
¥
˜
‡
m
‘Ã
ß
:
f
6
/
§
M
m
þ
M¨obius
1
Ä
/
ª
B
´
M
m
3
S
m
+
p
¥
M¨obius
C
†
+
e
ØC
þ
§
©
Ø
ª
P
α,β
tr
[(
B
α
)
2
(
B
β
)
2
]
≤
(
m
−
1)(3
m
2
−
9
m
+8)
2
m
3
§
y
²
Ò
¤
á
^
‡
"
'
…
c
M¨obius
A
Û
§
M¨obius
ØC
þ
§
Ø
ª
AnInequalityontheSubmanifoldsof
S
m
+
p
JiangtaoMa
DepartmentofMathematics,YunnanNormalUniversity,KunmingYunnan
Received:May11
th
,2022;accepted:Jun.16
th
,2022;published:Jun.23
rd
,2022
Abstract
Let
x
:
M
m
→
S
m
+
p
(
m
≥
2
,p
≥
2)
bean
m
-dimensionalnoumbilicalsubmaniflodsin
m
+
p
-dimensionalunitsphere
S
m
+
p
.TheM¨obiussecondbasicfrom
B
of
M
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)(3
m
2
−
9
m
+8)
2
m
3
. Theconditionsfortheequalitysignareproved.
©
Ù
Ú
^
:
ê
ô
7
.
S
m
+
p
¥
f
6
/
þ
˜
‡
Ø
ª
[J].
n
Ø
ê
Æ
,2022,12(6):971-980.
DOI:10.12677/pm.2022.126106
ê
ô
7
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.
Ú
ó
x
:
M
m
→
S
m
+
p
(
m>
3)
´
m
+
p
‘
ü
¥
S
m
+
p
¥
˜
‡
m
‘Ã
ß
:
f
6
/
,
d
©
z
[1]
•
,
{
e
i
}
´
Ý
þ
I
=
dx
·
dx
Û
ÜI
O
ƒ
I
e
|
,
Ù
é
ó
I
e
|
•
{
θ
i
}
,
1
Ä
/
ª
II
=
P
i,j,α
h
α
ij
θ
i
θ
j
e
α
,
²
þ
-
Ç
•
H
.
½
Â
ρ
2
=
m
m
−
1
|
II
−
1
m
tr
(
II
)
I
|
2
,
@
o
½
2-
/
ª
g
=
ρ
2
I
´
S
m
+
p
¥
M¨obius
C
†
+
e
ØC
þ
,
¡
•
x
M¨obius
Ý
þ
.
d
©
z
[2]
Ú
©
z
[3]
x
n
‡
Ä
M¨obius
ØC
þ
,
©
O
•
M¨obius
/
ª
Φ=
P
i,α
C
α
i
θ
i
e
α
,Blaschke
Ü
þ
A
=
ρ
2
P
i,j
A
ij
θ
i
θ
j
Ú
M¨obius
1
Ä
/
ª
B
=
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
(log
ρ
)
!
,
(1.1)
A
ij
=
−
ρ
−
2
Hess
ij
(log
ρ
)
−
e
i
(log
ρ
)
e
j
(log
ρ
)
−
X
α
H
α
h
α
ij
!
−
1
2
ρ
−
2
k∇
log
ρ
k
2
−
1+
k
H
k
2
δ
ij
,
(1.2)
B
α
ij
=
ρ
−
1
h
α
ij
−
H
α
δ
ij
,
(1.3)
Ù
¥
Hess
ij
Ú
∇
•
'
u
I
=
dx
·
dx
3
Ä
.
{
e
i
}
e
Hess
Ý
Ú
F
Ý
Ž
f
.
¡
B
A
Š
•
x
M¨obius
Ì
-
Ç
.
R
m
+
p
+2
´
m
+
p
+2
‘
î
ª
•
þ
˜
m
,
½
Â
S
È
h·
,
·i
X
e
:
h
X,ξ
i
=
−
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
)
,
¡
ä
k
S
È
h·
,
·i
•
þ
˜
m
R
m
+
p
+2
•
m
+
p
+2
‘
Lortentz
˜
m
,
P
•
R
m
+
p
+2
1
.
DOI:10.12677/pm.2022.126106972
n
Ø
ê
Æ
ê
ô
7
½
Â
C
m
+
p
+1
+
:=
{
X
∈
R
m
+
p
+2
1
|h
X,X
i
= 0
,x
0
>
0
}
,
(1
.
5)
•
R
m
+
p
+2
1
1
I
,
½
Â
Q
m
+
p
:=
{
[
ξ
]
∈
RP
m
+
p
+1
|h
ξ,ξ
i
= 0
}
,
(1
.
6)
•
RP
m
+
p
+1
¥
g
-
¡
.
2.
©
Ì
‡
(
J
©
3
M¨obius
A
Û
þ
X
e
½
n
:
½
n
A
x
:
M
m
→
S
m
+
p
(
m
≥
2
,p
≥
2)
´
Ã
ß
E
\
f
6
/
,
B
•
M¨obius
1
Ä
/
ª
,
K
k
e
Ø
ª
X
α,β
tr
[(
B
α
)
2
(
B
β
)
2
]
≤
(
m
−
1)(3
m
2
−
9
m
+8)
2
m
3
,
¤
á
,
Ù
¥
Ò
¤
á
…
=
M
m
´
/
²
"
…
B
α
ƒ
q
u
e
Ý
ƒ
˜
0
u
0
···
0
u
00
···
0
000
···
0
.
.
.
.
.
.
.
.
.
.
.
.
000
···
0
n
×
n
,
u
00
···
0
0
−
u
0
···
0
000
···
0
.
.
.
.
.
.
.
.
.
.
.
.
000
···
0
n
×
n
3.
S
m
+
p
f
6
/
M¨obius
ØC
þ
•
²
ï
á
¥
¥
f
6
/
1
I
.
Ú
ØC
þ
X
Ú
,
©
¦
^†
Ù
ƒ
Ó
P
ÒÚ
ú
ª
.
x
:
M
m
→
S
m
+
p
⊂
R
m
+
p
+1
´
Ã
ß
E
\
.
x
M¨obius
˜
•
þ
½
Â
•
ξ
=
ρ
(1
,x
) :
M
m
→
R
m
+
p
+2
1
,ρ
2
=
m
m
−
1
k
II
−
1
m
tr
(
II
)
I
k
2
>
0
.
(3
.
1)
K
k
X
e
½
n
:
½
n
3.1
([4])
ü
‡
f
6
/
x,
e
x
:
M
m
→
S
m
+
p
´
Moebius
d
…
=
•
3
R
m
+
p
+2
1
¥
Lorentz
C
†
T
∈
O
(
m
+
p
+1
,
1) ,
¦
ξ
=
e
ξT.
Ù
¥
O
(
m
+
p
+1
,
1)
´
R
m
+
p
+2
1
¥
±
S
È
h
,
i
ØC
Lorentz
+
,
@
o
O
(
m
+
p
+1)
´
˜
‡
3
Q
m
+
p
¥
C
†
+
½
Â
•
T
([
ξ
]) := [
ξT
]
,X
∈
C
m
+
p
+1
+
,ξ
∈
O
(
m
+
p
+1
,
1)
,
(3
.
2)
DOI:10.12677/pm.2022.126106973
n
Ø
ê
Æ
ê
ô
7
Ï
d
g
=
h
dξ,dξ
i
=
ρ
2
dx
·
dx,
(3
.
3)
´
M¨obius
ØC
þ
,
¡
•
M¨obius
Ý
þ
½
d
x
p
M¨obius
1
˜
Ä
/
ª
.
M
•
(
M,g
)
Laplace
Ž
f
.
h4
ξ,
4
ξ
i
= 1+
m
2
κ,
(3
.
4)
Ù
¥
κ
L
«
x
{
z
M¨obius
ê
þ
-
Ç
.
{
E
1
,E
2
,
···
,E
m
}
´
(
M,g
)
˜
‡
Û
ÜI
O
Ä
,
{
ω
1
,ω
2
,
···
,ω
m
}
•
Ù
é
ó
Ä
.
…
E
i
(
ξ
) =
ξ
i
,
@
o
.
h
ξ
i
,ξ
j
i
=
δ
ij
,
1
≤
i,j
≤
m,
(3
.
5)
½
Â
N
=
−
1
m
4
ξ
−
1
2
m
2
h4
ξ,
4
ξ
i
ξ,
(3
.
6)
@
o
h
ξ,ξ
i
=
h
N,N
i
= 0
,
h
ξ,N
i
= 1
,
h
ξ
i
,ξ
i
=
h
ξ
i
,N
i
= 0
,
(1
≤
i,j
≤
m
)
.
(3
.
7)
…
h
ξ,dξ
i
= 0
,
h4
ξ,ξ
i
=
−
m,
h4
ξ,ξ
k
i
= 0
,
1
≤
k
≤
m.
(3
.
8)
Ï
d
span
{
N,ξ
}⊥
span
{
ξ
1
,ξ
2
,
···
,ξ
m
}
,
(3
.
9)
½
Â
V
=
{
span
{
N,ξ
}⊕
span
{
ξ
1
,
···
,ξ
m
}}
⊥
,
(3
.
10)
V
´
f
˜
m
Span
{
ξ,N,ξ
1
,ξ
2
,
···
,ξ
m
}
3
R
m
+
p
+2
1
¥
Ö
˜
m
,
@
o
·
‚
k
X
e
©
)
.
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
}
´
{
m
V
÷
M
m
˜
‡
Û
Ü
Ä
,
@
o
{
ξ,N,ξ
1
,
···
,ξ
m
,E
m
+1
,
···
,E
m
+
p
}
´
3
R
m
+
p
+2
÷
M
m
¹
Ä
I
e
.
Ï
L
¦
^
•
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.2022.126106974
n
Ø
ê
Æ
ê
ô
7
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¨obius
Ý
þ
g
é
ä
/
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¨obius
ØC
þ
,
¡
A
•
x
Blaschke
Ü
þ
,
B
•
x
M¨obius
1
Ä
/
ª
, Φ
•
x
M¨obius
/
ª
.
½
Â
C
α
i
,A
ij
,B
α
ij
˜
C
ê
X
e
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)
(
•
§
Œ
È
^
‡
•
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
2
m
(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
,
B
Ú
Φ
'
u
g
p
é
ä
C
ê
3
I
O
Ä
e
©
þ
.
(3.24)
ª
¥
-
i
=
j
¦
Ú
−
X
B
α
ij,i
=
−
(
m
−
1)
C
α
j
,
(3
.
27)
DOI:10.12677/pm.2022.126106975
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ê
Æ
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ô
7
(3.25)
ª
¥
-
i
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k
¦
Ú
R
jl
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−
X
k
B
α
jk
B
α
kl
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tr
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A
)
δ
jl
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m
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2)
A
jl
,
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.
28)
d
(3.26)
Ú
(3.27)
Œ
X
ij
(
B
α
ij
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2
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m
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1
m
,
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.
29)
4.
½
n
y
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y
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n
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k
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y
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e
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n
.
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n
4.1
x
:
M
m
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S
m
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p
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m
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3)
´
m
‘Ã
ß
E
\
f
6
/
,
B
•
M¨obius
1
Ä
/
ª
,
K
k
Ø
ª
X
α,β
{
2[
tr
(
B
α
B
β
)]
2
+
k
B
α
B
β
−
B
β
B
α
k
2
−
2
m
m
−
2
tr
[(
B
α
)
2
(
B
β
)
2
]
}
+
2(
m
−
1)
m
2
(
m
−
2)
≥
0
,
¤
á
,
Ù
¥
Ò
¤
á
…
=
M
m
´
/
²
"
.
y
²
d
Weyl
-
Ç
Ü
þ
½
Â
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)
Ï
•
W
ijkl
•Ã
,
Ü
þ
,
Ï
d
|
W
ijkl
|
2
=
W
ijkl
R
ijkl
,
(4
.
3)
Ù
¥
S
ij
•
Shouten
Ü
þ
,
½
Â
•
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
2
mtrA
=1+
m
2
κ,
(4
.
6)
¤
±
DOI:10.12677/pm.2022.126106976
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7
mκ
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trA
−
1
m
,
(4
.
7)
u
´
S
ij
=
−
B
α
ik
B
α
kj
+
1
2
m
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)
ª
Œ
|
W
|
2
=
W
ijkl
(
B
β
ik
B
β
jl
−
B
β
il
B
β
jk
)
,
(4
.
10)
Ï
d
|
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
) =2
tr
(
B
α
B
β
)
·
tr
(
B
α
B
β
)
−
2
tr
(
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)
ª
Œ
|
W
|
2
= 2[
tr
(
B
α
B
β
)]
2
−
2
tr
(
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.2022.126106977
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7
2
d
k
B
α
B
β
−
B
β
B
α
k
2
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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
)
=2
tr
(
B
α
B
α
B
β
B
β
)
−
2
tr
(
B
α
B
β
B
α
B
β
)
,
(4.18)
Œ
−
2
tr
(
B
α
B
β
B
α
B
β
) =
k
B
α
B
β
−
B
β
B
α
k−
2
tr
[(
B
α
)
2
(
B
β
)
2
]
,
(4
.
19)
u
´
k
W
|
2
=
X
α,β
{
2[
tr
(
B
α
B
β
)]
2
+
k
B
α
B
β
−
B
β
B
α
k
2
−
2
m
m
−
2
tr
[(
B
α
)
2
(
B
β
)
2
]
}
+
2(
m
−
1)
m
2
(
m
−
2)
≥
0
,
(4.20)
Ú
n
4.1
y
.
e
5
y
²
½
n
A.
d
Ú
n
4.1
k
k
W
|
2
=
X
α,β
{
2[
tr
(
B
α
B
β
)]
2
+
k
B
α
B
β
−
B
β
B
α
k
2
−
2
m
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
+
k
B
α
B
β
−
B
β
B
α
k
2
−
2
m
m
−
2
tr
[(
B
α
)
2
(
B
β
)
2
]
}
+
2(
m
−
1)
m
2
(
m
−
2)
≥
0
,
(4
.
22)
d
DDVV
Ø
ª
(
©
z
[5])
k
X
α,β
k
B
α
B
β
−
B
β
B
α
k
2
≤
(
X
α
k
B
α
k
2
)
2
,
(4
.
23)
d
Cauchy-Schwarz
Ø
ª
k
X
α,β
[
tr
(
B
α
B
β
)]
2
≤
X
α
k
B
α
k
2
·
X
β
k
B
β
k
2
,
(4
.
24)
DOI:10.12677/pm.2022.126106978
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Ø
ê
Æ
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7
u
´
k
X
α,β
{
2[
tr
(
B
α
B
β
)]
2
+
k
B
α
B
β
−
B
β
B
α
k
2
−
2
m
m
−
2
tr
[(
B
α
)
2
(
B
β
)
2
]
}
+
2(
m
−
1)
m
2
(
m
−
2)
≤
(
X
α
k
B
α
k
2
)
2
+2
X
α
k
B
α
k
2
·
X
β
k
B
β
k
2
−
2
m
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
k
B
α
k
2
)
2
+2
X
α
k
B
α
k
2
·
X
β
k
B
β
k
2
−
2
m
m
−
2
X
α,β
tr
[(
B
α
)
2
(
B
β
)
2
]+
2(
m
−
1)
m
2
(
m
−
2)
≥
0
,
(4
.
26)
Ï
d
2
m
m
−
2
X
α,β
tr
[(
B
α
)
2
(
B
β
)
2
]
≤
(
X
α
k
B
α
k
2
)
2
+2
X
α
k
B
α
k
2
·
X
β
k
B
β
k
2
+
2(
m
−
1)
m
2
(
m
−
2)
,
(4
.
27)
…
X
α
k
B
α
k
2
=
X
β
k
B
β
k
2
=
m
−
1
m
,
(4
.
28)
u
´
k
X
α,β
tr
[(
B
α
)
2
(
B
β
)
2
]
≤
(
m
−
1)(3
m
2
−
9
m
+8)
2
m
3
,
(4
.
29)
Ù
¥
Ò
¤
á
…
=
M
m
´
/
²
"
…
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α
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q
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e
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.
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n
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n
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y
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.
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m
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Veroness
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S
2
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√
3) =
{
(
x
1
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2
,x
3
)
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3
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x
2
1
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x
2
2
+
x
2
3
= 3
}
,
½
Â
N
x
:
S
2
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√
3)
→
S
4
(1)
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e
y
1
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x
2
x
3
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2
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3
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3
x
1
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3
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1
x
2
y
4
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1
2
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3
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x
2
1
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x
2
2
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,y
5
=
1
6
(
x
2
1
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x
2
2
−
2
x
2
3
)
,
K
S
2
(
√
3)
´
S
4
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4
f
6
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,
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.
DOI:10.12677/pm.2022.126106979
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3
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2
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8
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4
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2
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>
P
α,β
tr
[(
B
α
)
2
(
B
β
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2
] =
1
8
,m
= 2
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m
2
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9
m
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©
z
[1]Li,H.Z.,Liu,H.L.,Wang,C.P.andZhao,G.S.(2002)MoebiusIsoparametricHypersurfaces
in
S
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)SubmanifoldswithConstantMoebiusScalarCurvaturein
S
n
.
ManuscriptaMathematica
,
111
,287-302.https://doi.org/10.1007/s00229-003-0368-2
[4]Wang,C.P.(1998)MoebiusGeometryofSubmanifoldsin
S
n
.
ManuscriptaMathematica
,
96
,
517-534.https://doi.org/10.1007/s002290050080
[5]Ge,J.G.andTang,Z.Z.(2008)AProofoftheDDVVConjectureandItsEqualityCase.
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,
237
,87-95.https://doi.org/10.2140/pjm.2008.237.87
DOI:10.12677/pm.2022.126106980
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