# Hypersurfaces with Two Distinct Para-Blaschke Eigenvalues inS

Authors: Junfeng Chen; Shichang Shu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

---

## Abstract

Letx
:
M
↦
S
n
+
1
(
1
) be an n
(
n
≥
3
)-dimensional immersed hypersurface without umbilical points and with vanishing Möbius form in a unit sphere S
n
+
1
(
1
), and let A and B be the Blaschke tensor and the Möbius second fundamental form of x, respectively. We define a symmetric (
0,2
) tensor D
=
A
+
λ
B which is called the para-Blaschke tensor of x, where λ is a constant. An eigenvalue of the para-Blaschke tensor is called a para-Blaschke eigenvalue of x. The aim of this paper is to classify the oriented hypersurfaces in S
n
+
1
(
1
) with two distinct para-Blaschke eigenvalues under some rigidity conditions.


---

## Body

## 1. Introduction

In the Möbius geometry of hypersurfaces, Wang [1] studied invariants of hypersurfaces in a unit sphere S
n
+
1
(
1
) under the Möbius transformation group. Let x
:
M
↦
S
n
+
1
(
1
) be an n-dimensional immersed hypersurface without umbilical points in S
n
+
1
(
1
). We choose a local orthonormal basis {
e
i
} for the induced metric I
=
d
x
·
d
x with dual basis {
θ
i
}. Let I
I
=
∑
i
,
j
‍
h
i
j
θ
i
⊗
θ
j be the second fundamental form of the immersion x and H
=
(
1
/
n
)
∑
i
‍
h
i
i the mean curvature of the immersion x. By putting ρ
2
=
(
n
/
(
n
-
1
)
)
{
∑
i
,
j
‍
h
i
j
2
-
n
H
2
}, theMöbius metric of the immersion x is defined by g
=
ρ
2
d
x
·
d
x which is a Möbius invariant. Φ
=
ρ
∑
i
‍
C
i
θ
i, A
=
ρ
2
∑
i
,
j
‍
A
i
j
θ
i
⊗
θ
j, and B
=
ρ
2
∑
i
,
j
‍
B
i
j
θ
i
⊗
θ
j are called the Möbius form, the Blaschke tensor, and the Möbius second fundamental formof the immersion x, respectively (see [1]), where
(1)
A
i
j
=
-
ρ
-
2
{
Hes
s
i
j
(
log
⁡
⁡
ρ
)
-
e
i
(
log
⁡
⁡
ρ
)
e
j
(
log
⁡
⁡
ρ
)
-
H
h
i
j
}
-
1
2
ρ
-
2
(
|
∇
(
log
⁡
⁡
ρ
)
|
2
-
1
+
|
H
→
|
2
)
δ
i
j
,
(2)
B
i
j
=
ρ
-
1
(
h
i
j
-
H
δ
i
j
)
,
(3)
C
i
=
-
ρ
-
2
{
H
,
i
+
∑
j
‍
(
h
i
j
-
H
δ
i
j
)
e
j
(
log
⁡
⁡
ρ
)
}
,
and Hes
s
i
j and ∇ are the Hessian matrix and the gradient with respect to the induced metric d
x
·
d
x. It was proved by [1] that Φ, A, and B are Möbius invariants.In the study of the Möbius geometry of hypersurfaces, one of the important aims is to characterize hypersurfaces in terms of Möbius invariants. Concerning this topic, there are many important results; one can see [2–9]. We should notice that [5] classified all umbilic-free hypersurfaces with parallel Möbius second fundamental form. Recently, by making use of the two important Möbius invariants, the Blaschke tensor A and the Möbius second fundamental form B of x, Zhong and Sun [10] defined a symmetric (
0,2
) tensor D
=
A
+
λ
B which is called the para-Blaschke tensor of x, where λ is a constant. An eigenvalue of the para-Blaschke tensor is called a para-Blaschke eigenvalue of x. In [7], Li and Wang investigated and completely classified hypersurfaces x
:
M
↦
S
n
+
1
(
1
) without umbilical points and with vanishing Möbius form Φ in S
n
+
1
(
1
), which satisfy A
+
λ
B
+
μ
g
=
0. It should be noted that the condition A
+
λ
B
+
μ
g
=
0 implies that the para-Blaschke eigenvalues of x are all equal. If x has two distinct constant para-Blaschke eigenvalues, Zhong and Sun [10] obtained the following.Theorem 1 (see [10]).
Letx
:
M
↦
S
n
+
1
(
1
) be an n-dimensional immersed hypersurface without umbilical points. If x is of two distinct constant para-Blaschke eigenvalues and of vanishing Möbius form Φ, then x is locally Möbius equivalent to(1)
the Riemannian productS
k
(
r
)
×
S
n
-
k
(
1
-
r
2
) in S
n
+
1
(
1
), or
(2)
the image ofσ of the standard cylinder S
k
(
r
)
×
R
n
-
k in R
n
+
1, or
(3)
the image ofτ of the Riemannian product S
k
(
r
)
×
H
n
-
k
(
1
+
r
2
) in H
n
+
1, or
(4)
the hypersurface as indicated in Example11, or
(5)
the hypersurface as indicated in Example12.Remark 2.
In Theorem1 and the following Theorems, the conformal diffeomorphisms σ
:
R
n
+
1
↦
S
n
+
1
(
1
)
∖
{
(
-
1,0
)
} and τ
:
H
n
+
1
↦
S
+
n
+
1
(
1
) are defined by
(4)
σ
(
u
)
=
(
1
-
|
u
|
2
1
+
|
u
|
2
,
2
u
1
+
|
u
|
2
)
,
u
∈
R
n
+
1
,
τ
(
y
0
,
y
1
)
=
(
1
y
0
,
y
1
y
0
)
,
(
y
0
,
y
1
)
∈
H
n
+
1
,
respectively, where S
+
n
+
1
(
1
) is the open hemisphere in S
n
+
1
(
1
) whose first coordinate is positive (see [1]).In this paper, we study hypersurfacesx
:
M
↦
S
n
+
1
(
1
) without umbilical points and with vanishing Möbius form. If x is of two distinct para-Blaschke eigenvalues, we obtain some classification Theorems in terms of some rigidity conditions.Theorem 3.
Letx
:
M
↦
S
n
+
1
(
1
) be an n  (
n
≥
3
)-dimensional immersed hypersurface without umbilical points and with vanishing Möbius form Φ and constant Möbius scalar curvature n
(
n
-
1
)
R in S
n
+
1
(
1
). If x is of two distinct para-Blaschke eigenvalues and the multiplicities of these two distinct para-Blaschke eigenvalues are greater than 1, then x is locally Möbius equivalent to(1)
the Riemannian productS
k
(
r
)
×
S
n
-
k
(
1
-
r
2
) in S
n
+
1
(
1
), or
(2)
the image ofσ of the standard cylinder S
k
(
r
)
×
R
n
-
k in R
n
+
1, or
(3)
the image ofτ of the Riemannian product S
k
(
r
)
×
H
n
-
k
(
1
+
r
2
) in H
n
+
1, or
(4)
the hypersurface as indicated in Example11, or
(5)
the hypersurface as indicated in Example12, where 1
<
k
<
n
-
1.Theorem 4.
Letx
:
M
↦
S
n
+
1
(
1
) be an n  (
n
≥
3
)-dimensional immersed complete hypersurface without umbilical points and with vanishing Möbius form Φ and constant Möbius scalar curvature n
(
n
-
1
)
R in S
n
+
1
(
1
). If x is of two distinct para-Blaschke eigenvalues one of which is simple, then we have the following.(1)
If the Möbius sectional curvature ofx is nonnegative and R
>
(
(
n
-
2
)
/
n
2
)
(
1
+
n
|
λ
|
), then x is locally Möbius equivalent to
(i)
the Riemannian productS
n
-
1
(
r
)
×
S
1
(
1
-
r
2
) in S
n
+
1
(
1
), or
(ii)
the image ofσ of the standard cylinder S
n
-
1
(
r
)
×
R in R
n
+
1.
(2)
If(5)
|
D
~
|
≤
n
-
1
n
{
n
n
-
2
R
-
1
+
n
|
λ
|
n
}
,
thenx is locally Möbius equivalent to:
(i)
the Riemannian productS
n
-
1
(
r
)
×
S
1
(
1
-
r
2
) in S
n
+
1
(
1
), or
(ii)
the image ofσ of the standard cylinder S
n
-
1
(
r
)
×
R in R
n
+
1, in this case λ
≥
0, or
(iii)
the image ofτ of the Riemannian product S
n
-
1
(
r
)
×
H
1
(
1
+
r
2
) in H
n
+
1, in this case λ
≤
0, where D
~
i
j
=
D
i
j
-
(
1
/
n
)
∑
k
‍
D
k
k
δ
i
j.Ifλ
=
0, then the para-Blaschke tensor is the Blaschke tensor of x. From Theorems 3 and 4, we have the following.Corollary 5.
Letx
:
M
↦
S
n
+
1
(
1
) be an n  (
n
≥
3
)-dimensional immersed hypersurface without umbilical points and with vanishing Möbius form Φ and constant Möbius scalar curvature n
(
n
-
1
)
R in S
n
+
1
(
1
). If x is of two distinct Blaschke eigenvalues and the multiplicities of these two distinct Blaschke eigenvalues are greater than 1, then x is locally Möbius equivalent to(1)
the Riemannian productS
k
(
r
)
×
S
n
-
k
(
1
-
r
2
) in S
n
+
1
(
1
), or
(2)
the image ofσ of the standard cylinder S
k
(
r
)
×
R
n
-
k in R
n
+
1, or
(3)
the image ofτ of the Riemannian product S
k
(
r
)
×
H
n
-
k
(
1
+
r
2
) in H
n
+
1, or
(4)
the minimal hypersurface as indicated in Example13, or
(5)
the nonminimal hypersurface as indicated in Example14, where 1
<
k
<
n
-
1.Corollary 6.
Letx
:
M
↦
S
n
+
1
(
1
) be an n  (
n
≥
3
)-dimensional immersed complete hypersurface without umbilical points and with vanishing Möbius form Φ and constant Möbius scalar curvature n
(
n
-
1
)
R in S
n
+
1
(
1
). If x is of two distinct Blaschke eigenvalues one of which is simple, then we have the following.(1)
If the Möbius sectional curvature ofx is nonnegative and R
>
(
n
-
2
)
/
n
2, then x is locally Möbius equivalent to:
(i)
the Riemannian productS
n
-
1
(
r
)
×
S
1
(
1
-
r
2
) in S
n
+
1
(
1
), or
(ii)
the image ofσ of the standard cylinder S
n
-
1
(
r
)
×
R in R
n
+
1.
(2)
If(6)
|
A
~
|
≤
n
-
1
n
{
n
n
-
2
R
-
1
n
}
,
thenx is locally Möbius equivalent to
(i)
the Riemannian productS
n
-
1
(
r
)
×
S
1
(
1
-
r
2
) in S
n
+
1
(
1
), or
(ii)
the image ofσ of the standard cylinder S
n
-
1
(
r
)
×
R in R
n
+
1, or
(iii)
the image ofτ of the Riemannian product S
n
-
1
(
r
)
×
H
1
(
1
+
r
2
) in H
n
+
1, where A
~
i
j
=
A
i
j
-
(
1
/
n
)
∑
k
‍
A
k
k
δ
i
j.


## 2. Preliminaries

In this section, we review the definitions of Möbius invariants and the fundamental formulas on Möbius geometry of hypersurfaces inS
n
+
1
(
1
); for more details see [1].Letx
:
M
↦
S
n
+
1
(
1
) be an n-dimensional hypersurface of S
n
+
1
(
1
) without umbilical points. We use the following range of indices throughout this paper:
(7)
1
≤
i
,
j
,
k
,
l
,
m
≤
n
.
The structure equations on M with respect to the Möbius metric g can be written as follows:
(8)
d
Y
=
∑
i
‍
ω
i
Y
i
,
d
N
=
∑
i
‍
ψ
i
Y
i
+
ϕ
E
n
+
1
,
d
Y
i
=
-
ψ
i
Y
-
ω
i
N
+
∑
j
‍
ω
i
j
Y
j
+
ω
i
n
+
1
E
n
+
1
,
d
E
n
+
1
=
-
ϕ
Y
-
∑
i
‍
ω
i
n
+
1
Y
i
,
where {
ψ
i
,
ω
i
j
,
ω
i
n
+
1
,
ϕ
} are 1-forms on M with
(9)
ω
i
j
+
ω
j
i
=
0
.
By exterior differentiation of these equations, we get
(10)
∑
i
‍
ω
i
∧
ψ
i
=
0
,
∑
i
‍
ω
i
n
+
1
∧
ω
i
=
0
,
(11)
d
ω
i
=
∑
j
‍
ω
i
j
∧
ω
j
,
(12)
d
ψ
i
=
∑
j
‍
ω
i
j
∧
ψ
j
+
ω
i
n
+
1
∧
ϕ
,
(13)
d
ϕ
=
-
∑
i
‍
ω
i
n
+
1
∧
ψ
i
,
(14)
d
ω
i
j
=
∑
k
‍
ω
i
k
∧
ω
k
j
-
ω
i
n
+
1
∧
ω
j
n
+
1
-
ω
i
∧
ψ
j
-
ψ
i
∧
ω
j
,
(15)
d
ω
i
n
+
1
=
∑
j
‍
ω
i
j
∧
ω
j
n
+
1
-
ω
i
∧
ϕ
,
where
(16)
ψ
i
=
∑
j
‍
A
i
j
ω
j
,
A
i
j
=
A
j
i
,
ω
i
n
+
1
=
∑
j
‍
B
i
j
ω
j
,
B
i
j
=
B
j
i
,
ϕ
=
∑
i
‍
C
i
ω
i
,
and A
i
j, B
i
j, and C
i are locally defined functions and satisfy (1), (2), and (3). We have
(17)
d
ω
i
j
=
∑
k
‍
ω
i
k
∧
ω
k
j
-
1
2
∑
k
,
l
‍
R
i
j
k
l
ω
k
∧
ω
l
,
R
i
j
k
l
=
-
R
j
i
k
l
,
(18)
∑
i
‍
B
i
i
=
0
,
∑
i
,
j
‍
B
i
j
2
=
n
-
1
n
,
tr
⁡
A
=
1
2
n
(
1
+
n
2
R
)
.
Let C
i
,
j, A
i
j
,
k, B
i
j
,
k be the covariant derivative of C
i, A
i
j, B
i
j. From the structure equations (8), we infer
(19)
A
i
j
,
k
-
A
i
k
,
j
=
B
i
k
C
j
-
B
i
j
C
k
,
(20)
C
i
,
j
-
C
j
,
i
=
∑
k
‍
(
B
i
k
A
k
j
-
B
k
j
A
k
i
)
,
(21)
B
i
j
,
k
-
B
i
k
,
j
=
δ
i
j
C
k
-
δ
i
k
C
j
,
(22)
R
i
j
k
l
=
B
i
k
B
j
l
-
B
i
l
B
j
k
+
δ
i
k
A
j
l
+
δ
j
l
A
i
k
-
δ
i
l
A
j
k
-
δ
j
k
A
i
l
,
where R
i
j
k
l denotes the curvature tensor with respect to the Möbius metric g on M and n
(
n
-
1
)
R
=
∑
i
,
j
R
i
j
i
j is the Möbius scalar curvature of the immersion x
:
M
→
S
n
+
1
(
1
).Since the Möbius formΦ
=
∑
i
C
i
ω
i
E
n
+
1
≡
0, by (19)–(21), we have for all indices i, j, k that
(23)
A
i
j
,
k
=
A
i
k
,
j
,
B
i
j
,
k
=
B
i
k
,
j
,
∑
k
‍
B
i
k
A
k
j
=
∑
k
‍
B
k
j
A
k
i
.Denote byD
=
∑
i
,
j
‍
D
i
j
ω
i
⊗
ω
j the (
0,2
) para-Blaschke tensor; then,
(24)
D
i
j
=
A
i
j
+
λ
B
i
j
,
1
≤
i
,
j
≤
n
,
where λ is constant. The covariant derivative of D
i
j is defined by
(25)
∑
k
‍
D
i
j
,
k
ω
k
=
d
D
i
j
+
∑
k
‍
D
i
k
ω
k
j
+
∑
k
‍
D
k
j
ω
k
i
.
From (24), we have
(26)
D
i
j
,
k
=
A
i
j
,
k
+
λ
B
i
j
,
k
,
D
i
j
,
k
-
D
i
k
,
j
=
A
i
j
,
k
-
A
i
k
,
j
+
λ
(
B
i
j
,
k
-
B
i
k
,
j
)
.
From (23), we have for all indices i, j, k that
(27)
D
i
j
,
k
=
D
i
k
,
j
.


## 3. Lemma and Examples

In this section, we make the following convention on the ranges of indices:(28)
1
≤
i
,
j
≤
n
,
1
≤
a
,
b
,
c
≤
k
,
k
+
1
≤
s
,
t
≤
n
.
The following Lemma 7 due to Li et al. [6] will be used.Lemma 7 (see [6]).
Letx
:
M
↦
S
n
+
1
(
1
) be an n-dimensional hypersurface with two distinct principal curvatures with multiplicities k and n
-
k, respectively. Then, the principal curvatures of the Möbius second fundamental form B of x are constant, which are given by
(29)
μ
1
=
1
n
(
n
-
1
)
(
n
-
k
)
k
,
μ
2
=
-
1
n
(
n
-
1
)
k
n
-
k
.In [4], Cheng and Shu studied the Möbius invariants on some typical examples.Example 8 (see [4]).
Letx
1
:
S
k
(
1
)
↦
R
k
+
1 and x
2
:
S
n
-
k
(
1
)
↦
R
n
-
k
+
1 be the standard embedding of the unit spheres. Then, the Riemannian product x
:
S
k
(
r
)
×
S
n
-
k
(
1
-
r
2
)
↦
S
n
+
1
(
1
) defined by x
=
(
r
x
1
,
1
-
r
2
x
2
), for any 1
≤
k
≤
n
-
1 and any 0
<
r
<
1, is an immersed hypersurface without umbilical points and with vanishing Möbius form in S
n
+
1
(
1
). By (3.4) in [4], we know that the Möbius sectional curvature of x is nonnegative. From (3.9) to (3.11) in [4], Lemma 7, and (24) and by a simple and direct calculation, we see that
(30)
D
a
b
=
{
(
n
-
1
)
(
n
-
k
)
k
n
-
1
2
k
(
n
-
k
)
n
2
[
k
(
2
n
-
k
)
-
n
2
r
2
]
+
λ
1
n
(
n
-
1
)
(
n
-
k
)
k
}
δ
a
b
,
D
s
t
=
{
n
-
1
2
k
(
n
-
k
)
n
2
(
n
2
r
2
-
k
2
)
-
λ
1
n
(
n
-
1
)
k
n
-
k
}
δ
s
t
,
mmm
m
m
,
m
m
m
m
m
m
m
m
m
m
n
m
m
ii
D
a
s
=
0
.
Thus, x has exactly two distinct constant para-Blaschke eigenvalues. Defining the trace-free tensor D
~
i
j
=
D
i
j
-
(
1
/
n
)
tr
⁡
D
δ
i
j, we see that
(31)
(
n
-
2
k
)
2
|
D
~
|
2
=
{
k
(
n
-
k
)
n
(
n
R
-
n
-
2
n
)
-
(
n
-
2
k
)
λ
n
-
1
n
k
(
n
-
k
)
n
}
2
,
or
(32)
(
n
-
2
k
)
2
|
D
~
|
2
=
{
k
(
n
-
k
)
n
(
n
R
-
n
-
2
n
)
+
(
n
-
2
k
)
λ
n
-
1
n
k
(
n
-
k
)
n
}
2
.
Thus, we know that |
D
~
|
=
(
n
-
1
)
/
n
(
(
n
/
(
n
-
2
)
)
R
-
(
(
1
+
n
λ
)
/
n
)
), if and only if k
=
1 or k
=
n
-
1, and |
D
~
|
=
(
n
-
1
)
/
n
(
(
n
/
(
n
-
2
)
)
R
-
(
(
1
-
n
λ
)
/
n
)
), if and only if k
=
1 or k
=
n
-
1.Example 9 (see [4]).
Letx
^
:
S
k
(
1
)
×
R
n
-
k
↦
R
n
+
1 be the standard cylinder. Then, the hypersurface x
=
σ
∘
x
^
:
S
k
(
1
)
×
R
n
-
k
↦
S
n
+
1
(
1
) is an immersed hypersurface without umbilical points and with vanishing Möbius form in S
n
+
1
(
1
). By (3.25) in [4], we know that the Möbius sectional curvature of x is nonnegative. From (3.31)–(3.33) in [4], Lemma 7 and (24), we can calculate that
(33)
D
a
b
=
{
-
(
n
-
1
)
(
k
-
2
n
)
2
(
n
-
k
)
n
2
+
λ
1
n
(
n
-
1
)
(
n
-
k
)
k
}
δ
a
b
,
D
s
t
=
{
-
(
n
-
1
)
k
2
(
n
-
k
)
n
2
-
λ
1
n
(
n
-
1
)
k
n
-
k
}
δ
s
t
,
D
a
s
=
0
.
Thus x has exactly two distinct constant para-Blaschke eigenvalues. By the similar calculation in Example 8, we have
(34)
(
n
-
2
k
)
2
|
D
~
|
2
=
{
k
(
n
-
k
)
n
(
n
R
-
n
-
2
n
)
+
(
n
-
2
k
)
λ
n
-
1
n
k
(
n
-
k
)
n
}
2
.
We know that |
D
~
|
=
(
n
-
1
)
/
n
(
(
n
/
(
n
-
2
)
)
R
-
(
(
1
+
n
λ
)
/
n
)
) if and only if k
=
n
-
1. If k
=
1, from (3.23) in [4], we have R
=
0.Example 10 (see [4]).
Letx
¯
:
S
k
(
r
)
×
H
n
-
k
(
1
+
r
2
)
↦
H
n
+
1 be the standard embedding. Then the hypersurface x
=
τ
∘
x
¯
:
S
k
(
r
)
×
H
n
-
k
(
1
+
r
2
)
↦
S
n
+
1
(
1
) is an immersed hypersurface without umbilical points and with vanishing Möbius form in S
n
+
1
(
1
). By (3.41) in [4], we know that the Möbius sectional curvature of x is not nonnegative. From (3.45)–(3.47) in [4], Lemma 7 and (24), we can calculate that
(35)
D
a
b
=
{
(
n
-
1
)
(
n
-
k
)
k
n
-
1
2
k
(
n
-
k
)
n
2
[
k
(
2
n
-
k
)
+
n
2
r
2
]
+
λ
1
n
(
n
-
1
)
(
n
-
k
)
k
}
δ
a
b
,
(36)
D
s
t
=
-
{
(
n
-
1
)
k
n
-
k
n
-
1
2
k
(
n
-
k
)
n
2
[
k
2
+
n
2
r
2
]
+
λ
1
n
(
n
-
1
)
k
n
-
k
}
δ
s
t
,
D
a
s
=
0
.
Thus, x has exactly two distinct constant para-Blaschke eigenvalues. By the similar calculation in Example 8, we have
(37)
(
2
k
-
n
)
2
|
D
~
|
2
=
{
k
(
n
-
k
)
n
(
n
R
-
n
-
2
n
)
h
h
h
h
h
h
+
(
2
k
-
n
)
λ
n
-
1
n
k
(
n
-
k
)
n
}
2
.
We know that |
D
~
|
=
(
(
n
-
1
)
/
n
)
(
(
n
/
(
n
-
2
)
)
R
-
(
(
1
-
n
λ
)
/
n
)
) if and only if k
=
n
-
1. If k
=
1, from (3.39) in [4], we have R
<
0.Recently, Zhong and Sun [10] studied some important examples, which can be stated as follows.Example 11 (see [10]).
Hypersurfacex
~
=
(
x
~
1
,
x
~
2
)
:
M
~
1
k
×
H
n
-
k
(
r
)
↦
S
n
+
1, where x
~
1
=
y
1
/
y
0,  x
~
2
=
y
2
/
y
0,  y
0
∈
R
+, y
1
∈
R
k
+
2, y
2
∈
R
n
-
k, 2
≤
k
≤
n
-
1,  r
>
0. Here, y
1
:
M
~
1
k
↦
S
k
+
1
(
r
)
↪
R
k
+
2 is an immersed hypersurface in (
k
+
1
)-dimensional sphere S
k
+
1
(
r
), which is at least of two nonzero principal curvatures and with constant mean curvature H
1 and constant scalar curvature R
1:
(38)
H
1
=
-
n
k
λ
,
R
1
=
{
n
k
(
k
-
1
)
-
(
n
-
1
)
r
2
}
n
r
2
+
n
(
n
-
1
)
λ
2
,
and (
y
0
,
y
2
)
:
H
n
-
k
(
r
)
↦
R
1
n
-
k
+
1 is the canonical embedding of (
n
-
k
)-dimensional hyperbolic space H
n
-
k
(
r
) to (
n
-
k
+
1
)-dimensional Lorentzian space R
1
n
-
k
+
1 and y
0
>
0,  -
y
0
2
+
y
2
·
y
2
=
-
r
2. By [10], we know that the Möbius sectional curvature of x
~ is not nonnegative. From (3.12) to (3.21) in [10], by a simple and direct calculation, we see that the para-Blaschke tensor of x
~ is
(39)
D
a
b
=
(
1
2
r
2
+
λ
2
2
)
δ
a
b
,
D
a
s
=
0
,
D
s
t
=
(
-
1
2
r
2
+
λ
2
2
)
δ
s
t
.
Clearly, the Möbius form Φ
≡
0 and x
~ has exactly two distinct constant para-Blaschke eigenvalues. We can easily check that |
D
~
|
≠
(
n
-
1
)
/
n
(
(
n
/
(
n
-
2
)
)
R
-
(
(
1
+
n
|
λ
|
)
/
n
)
) if k
=
1 or k
=
n
-
1.Example 12 (see [10]).
Hypersurfacex
~
=
(
x
~
1
,
x
~
2
)
:
M
~
1
k
×
S
n
-
k
(
r
)
↦
S
n
+
1, where x
~
1
=
y
1
/
y
0,  x
~
2
=
y
2
/
y
0,  y
0
∈
R
+, y
1
∈
R
k
+
1,  y
2
∈
R
n
-
k
+
1,  2
≤
k
≤
n
-
1,  r
>
0. Here, (
y
0
,
y
1
)
:
M
~
1
k
↦
H
k
+
1
(
r
)
↪
R
1
k
+
2 is an immersed hypersurface in (
k
+
1
)-dimensional hyperbolic space H
k
+
1
(
r
), which is at least of two nonzero principal curvatures and with constant mean curvature H
1 and constant scalar curvature R
1:
(40)
H
1
=
-
n
k
λ
,
R
1
=
{
-
n
k
(
k
-
1
)
-
(
n
-
1
)
r
2
}
n
r
2
+
n
(
n
-
1
)
λ
2
,
and y
0
>
0, -
y
0
2
+
y
1
·
y
1
=
-
r
2 and y
2
:
S
n
-
k
(
r
)
↦
R
n
-
k
+
1 is the canonical embedding of (
n
-
k
)-dimensional sphere S
n
-
k
(
r
) to (
n
-
k
+
1
)-dimensional Euclidean space R
n
-
k
+
1. By [10], we know that the Möbius sectional curvature of x
~ is not nonnegative. From (3.39) to (3.48) in [10], by a simple and direct calculation, we see that the para-Blaschke tensor of x
~ is
(41)
D
a
b
=
(
-
1
2
r
2
+
λ
2
2
)
δ
a
b
,
D
a
s
=
0
,
D
s
t
=
(
1
2
r
2
+
λ
2
2
)
δ
s
t
.
Clearly, the Möbius form Φ
≡
0 and x
~ has exactly two distinct constant para-Blaschke eigenvalues. We can easily check that |
D
~
|
≠
(
n
-
1
)
/
n
(
(
n
/
(
n
-
2
)
)
R
-
(
(
1
+
n
|
λ
|
)
/
n
)
) if k
=
1 or k
=
n
-
1.Ifλ
=
0, Examples 11 and 12 decrease to the following Examples (see [8]).Example 13.
Minimal hypersurfacex
~
=
(
x
~
1
,
x
~
2
)
:
M
~
1
k
×
H
n
-
k
(
r
)
↦
S
n
+
1, where x
~
1
=
y
1
/
y
0, x
~
2
=
y
2
/
y
0, y
0
∈
R
+, y
1
∈
R
k
+
2, y
2
∈
R
n
-
k, 2
≤
k
≤
n
-
1, r
>
0. Here, y
1
:
M
~
1
k
↦
S
k
+
1
(
r
)
↪
R
k
+
2 is an immersed minimal hypersurface without umbilical points in (
k
+
1
)-dimensional sphere S
k
+
1
(
r
) and with constant scalar curvature R
1:
(42)
R
1
=
{
n
k
(
k
-
1
)
-
(
n
-
1
)
r
2
}
n
r
2
,
and (
y
0
,
y
2
)
:
H
n
-
k
(
r
)
↦
R
1
n
-
k
+
1 is the canonical embedding of (
n
-
k
)-dimensional hyperbolic space H
n
-
k
(
r
) to (
n
-
k
+
1
)-dimensional Lorentzian space R
1
n
-
k
+
1.Example 14.
Nonminimal hypersurfacex
~
=
(
x
~
1
,
x
~
2
)
:
M
~
1
k
×
S
n
-
k
(
r
)
↦
S
n
+
1, where x
~
1
=
y
1
/
y
0, x
~
2
=
y
2
/
y
0, y
0
∈
R
+, y
1
∈
R
k
+
1, y
2
∈
R
n
-
k
+
1, 2
≤
k
≤
n
-
1, r
>
0. Here, (
y
0
,
y
1
)
:
M
~
1
k
↦
H
k
+
1
(
r
)
↪
R
1
k
+
2 is an immersed minimal hypersurface without umbilical points in (
k
+
1
)-dimensional hyperbolic space H
k
+
1
(
r
) and with constant scalar curvature R
1:
(43)
R
1
=
{
-
n
k
(
k
-
1
)
-
(
n
-
1
)
r
2
}
n
r
2
,
and y
2
:
S
n
-
k
(
r
)
↦
R
n
-
k
+
1 is the canonical embedding of (
n
-
k
)-dimensional sphere S
n
-
k
(
r
) to (
n
-
k
+
1
)-dimensional Euclidean space R
n
-
k
+
1.


## 4. Proof of Theorems

LetA, B, and D denote the n
×
n-symmetric matrices (
A
i
j
), (
B
i
j
), and (
D
i
j
), respectively, where A
i
j, B
i
j, and D
i
j are defined by (1), (2), and (24). Since from (23), we know that B
A
=
A
B, D
A
=
A
D, and B
D
=
D
B hold, we can choose a local orthonormal basis {
E
1
,
E
2
,
…
,
E
n
} such that A
i
j
=
A
i
δ
i
j, B
i
j
=
B
i
δ
i
j, and D
i
j
=
D
i
δ
i
j, where A
i, B
i, and D
i are the Blaschke eigenvalues, the Möbius principal curvatures, and the para-Blaschke eigenvalues of the immersion x.Since the Möbius formΦ is zero, from (27), we know that the para-Blaschke tensor is Codazzi tensor. By the similar proof of Proposition  3.1 in [11], we have the following.Proposition 15.
Letx
:
M
↦
S
n
+
1
(
1
) be an n-dimensional hypersurface with vanishing Möbius form in a unit sphere S
n
+
1
(
1
). If the multiplicity of a para-Blaschke eigenvalue is constant and greater than 1, then this para-Blaschke eigenvalue is constant along its leaf.Proof of Theorem3.
Letx
:
M
↦
S
n
+
1
(
1
) be an immersed hypersurface without umbilical points and with vanishing Möbius form Φ and constant Möbius scalar curvature n
(
n
-
1
)
R in S
n
+
1
(
1
). Denote by μ
,
ν the para-Blaschke eigenvalues of multiplicities k and n
-
k, respectively, where 1
<
k
<
n
-
1. We have from (18) and (24) that
(44)
k
μ
+
(
n
-
k
)
ν
=
tr
⁡
D
=
tr
⁡
A
=
1
2
n
(
1
+
n
2
R
)
.
Denote by 𝒟
μ and 𝒟
ν the leaf of the para-Blaschke eigenvalues μ and ν, respectively. From Proposition 15, we know that μ is constant on 𝒟
μ. Since the Möbius scalar curvature n
(
n
-
1
)
R is constant, (44) implies that ν is constant on 𝒟
μ. By Proposition 15, again, ν is constant on 𝒟
ν. Therefore, we know that ν is constant on M. By the same assertion, we know that μ is constant on M. Therefore x
:
M
↦
S
n
+
1
(
1
) has two distinct constant para-Blaschke eigenvalues. By Theorem 1, we know that Theorem 3 is proved.Next, we consider the case thatx is of two distinct para-Blaschke eigenvalues, one of which is simple. We assume that x is of vanishing Möbius form and constant Möbius scalar curvature n
(
n
-
1
)
R. Without loss of generality, we may assume
(45)
D
1
=
D
2
=
⋯
=
D
n
-
1
=
μ
,
D
n
=
ν
,
where D
i for i
=
1,2
,
…
,
n are the para-Blaschke eigenvalues of x. Therefore, we know that
(46)
(
n
-
1
)
μ
+
ν
=
tr
⁡
D
=
tr
⁡
A
=
1
2
n
(
1
+
n
2
R
)
,
(47)
|
D
|
2
=
(
n
-
1
)
μ
2
+
ν
2
.
Since μ
-
ν
=
(
(
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
)
/
2
)
≠
0, we have 2
n
μ
-
(
(
1
/
n
)
+
n
R
)
≠
0.We choose a local orthonormal frame{
E
1
,
…
,
E
n
} such that E
i is a unit principal vector with respect to D
i. From (25), we have
(48)
D
i
j
,
k
=
E
k
(
D
i
)
δ
i
j
+
Γ
i
k
j
(
D
i
-
D
j
)
,
where Γ
i
k
j is the Levi-Civita connection for the Möbius metric g given by
(49)
ω
i
j
=
∑
k
‍
Γ
i
k
j
ω
k
,
Γ
i
k
j
=
-
Γ
j
k
i
.From (27) and (48), we easily see that
(50)
D
a
b
,
k
=
0
,
∀
a
≠
b
,
k
,
D
a
a
,
a
=
D
n
n
,
a
=
0
,
E
a
(
μ
)
=
E
a
(
ν
)
=
0
,
D
a
a
,
n
=
E
n
(
μ
)
,
D
n
n
,
n
=
E
n
(
ν
)
,
where 1
≤
a, b
≤
n
-
1. From (50) and the formula
(51)
∑
k
‍
D
a
n
,
k
ω
k
=
d
D
a
n
+
∑
k
‍
D
k
n
ω
k
a
+
∑
k
‍
D
a
k
ω
k
n
=
(
μ
-
ν
)
ω
a
n
,
we have
(52)
ω
a
n
=
E
n
(
μ
)
μ
-
ν
ω
a
=
2
E
n
(
μ
)
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
ω
a
,
1
≤
a
≤
n
-
1
.
Therefore, from (11) we have
(53)
d
ω
n
=
∑
k
=
1
n
-
1
‍
ω
k
∧
ω
k
n
+
ω
n
n
∧
ω
n
=
0
.
We may consider locally that μ is a function of the arc length s of the integral curve of the principal vector field E
n corresponding to the principal curvature ν. Therefore, we may put ω
n
=
d
s. Thus, for μ
=
μ
(
s
), we get
(54)
d
μ
=
E
n
(
μ
)
d
s
,
E
n
(
μ
)
=
d
μ
d
s
,
and for 1
≤
a
≤
n
-
1
(55)
ω
a
n
=
2
(
d
μ
/
d
s
)
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
ω
a
=
d
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
1
/
n
}
d
s
ω
a
.
From (17), (22), and (55), we have for 1
≤
a
≤
n
-
1 that
(56)
d
ω
a
n
=
∑
b
=
1
n
-
1
‍
ω
a
b
∧
ω
b
n
-
(
B
a
B
n
+
A
a
+
A
n
)
ω
a
∧
d
s
=
d
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
}
1
/
n
d
s
×
∑
b
=
1
n
-
1
ω
a
b
∧
ω
b
‍
-
(
B
a
B
n
+
μ
+
ν
-
λ
B
a
-
λ
B
n
)
ω
a
∧
d
s
.
From (55), we have for 1
≤
a
≤
n
-
1 that
(57)
d
ω
a
n
=
d
2
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
1
/
n
}
d
s
2
d
s
∧
ω
a
+
d
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
1
/
n
}
d
s
d
ω
a
=
{
[
d
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
1
/
n
}
d
s
]
2
-
d
2
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
1
/
n
}
d
s
2
+
[
d
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
1
/
n
}
d
s
]
2
}
ω
a
∧
d
s
+
d
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
1
/
n
}
d
s
∑
b
=
1
n
-
1
‍
ω
a
b
∧
ω
b
.
From the above two equalities, we have for 1
≤
a
≤
n
-
1
(58)
d
2
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
1
/
n
}
d
s
2
-
{
d
{
log
⁡
⁡
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
1
/
n
}
d
s
}
2
-
(
B
a
B
n
+
μ
+
ν
-
λ
B
a
-
λ
B
n
)
=
0
.
We define ϖ
=
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
-
1
/
n and obtain from the above equation that
(59)
d
2
ϖ
d
s
2
+
ϖ
(
B
a
B
n
+
μ
+
ν
-
λ
B
a
-
λ
B
n
)
=
0
,
1
≤
a
≤
n
-
1
.On the other hand, from (55), we have ∇
E
n
E
n
=
∑
i
=
1
n
‍
ω
n
i
(
E
n
)
E
i
=
0. By the definition of geodesic, we know that any integral curve of the principal vector field corresponding to the principal curvature μ is a geodesic. Thus, we see that ϖ
(
s
) is a function defined in (
-
∞
,
+
∞
) since x is complete and any integral curve of the principal vector field corresponding to μ is a geodesic.We can prove the following.Proposition 16.
IfR
>
(
(
n
-
2
)
/
n
2
)
+
(
(
n
-
2
)
/
n
)
|
λ
|, then the positive function ϖ is bounded.Proof.
By (59) and taking summation over a, 1
≤
a
≤
n
-
1, we have
(60)
(
n
-
1
)
d
2
ϖ
d
s
2
+
ϖ
{
∑
a
=
1
n
-
1
‍
B
a
B
n
+
(
n
-
1
)
μ
+
(
n
-
1
)
ν
-
λ
∑
a
=
1
n
-
1
‍
B
a
-
(
n
-
1
)
λ
B
n
}
=
0
;
that is, from (18)
(61)
d
2
ϖ
d
s
2
+
ϖ
{
-
1
n
-
1
B
n
2
-
n
-
2
n
-
1
λ
B
n
+
μ
+
ν
}
=
0
.
From the definition of ϖ and (46), we see that
(62)
d
2
ϖ
d
s
2
+
ϖ
{
-
1
n
-
1
B
n
2
-
n
-
2
n
-
1
λ
B
n
+
2
-
n
2
n
ϖ
-
n
+
1
n
2
+
R
}
=
0
,
for 2
n
μ
-
(
(
1
/
n
)
+
n
R
)
>
0, or
(63)
d
2
ϖ
d
s
2
+
ϖ
{
-
1
n
-
1
B
n
2
-
n
-
2
n
-
1
λ
B
n
-
2
-
n
2
n
ϖ
-
n
+
1
n
2
+
R
}
=
0
,
for 2
n
μ
-
(
(
1
/
n
)
+
n
R
)
<
0.
From (18), we have B
n
2
=
(
∑
i
=
1
n
-
1
B
i
)
2
≤
(
n
-
1
)
∑
i
=
1
n
-
1
B
i
2
=
(
n
-
1
)
(
(
(
n
-
1
)
/
n
)
-
B
n
2
). Thus, we have B
n
2
≤
(
(
n
-
1
)
/
n
)
2. We infer that
(64)
-
1
n
-
1
B
n
2
-
n
-
2
n
-
1
λ
B
n
≥
-
n
-
1
n
2
-
n
-
2
n
|
λ
|
.
From (63) and (64), we have
(65)
d
2
ϖ
d
s
2
+
ϖ
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
+
2
-
n
2
n
ϖ
1
-
n
≤
0
,
for 2
n
μ
-
(
(
1
/
n
)
+
n
R
)
>
0, or
(66)
d
2
ϖ
d
s
2
+
ϖ
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
-
2
-
n
2
n
ϖ
1
-
n
≤
0
,
for 2
n
μ
-
(
(
1
/
n
)
+
n
R
)
<
0.
Sinced
ϖ
/
d
s is existing, we know that for some fixed s
0
∈
(
-
∞
,
+
∞
), ϖ
(
s
0
) and d
ϖ
(
s
0
)
/
d
s are two constants. Multiplying (65) and (66) by 2
d
ϖ
/
d
s, we have two cases.(i)
Ifd
ϖ
/
d
s
≥
0, for any s
∈
(
-
∞
,
+
∞
), integrating from s
0 to s, then
(67)
∫
s
0
s
{
2
d
ϖ
d
s
d
2
ϖ
d
s
2
+
2
d
ϖ
d
s
ϖ
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
‍
+
2
d
ϖ
d
s
2
-
n
2
n
ϖ
1
-
n
d
ϖ
d
s
d
2
ϖ
d
s
2
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
}
d
s
≤
0
,
for2
n
μ
-
(
(
1
/
n
)
+
n
R
)
>
0, or
(68)
∫
s
0
s
{
2
d
ϖ
d
s
d
2
ϖ
d
s
2
+
2
d
ϖ
d
s
ϖ
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
‍
-
2
d
ϖ
d
s
2
-
n
2
n
ϖ
1
-
n
d
ϖ
d
s
d
2
ϖ
d
s
2
}
d
s
≤
0
,
for2
n
μ
-
(
(
1
/
n
)
+
n
R
)
<
0.
(ii)
Ifd
ϖ
/
d
s
≤
0, for any s
∈
(
-
∞
,
+
∞
), integrating from s to s
0, then
(69)
∫
s
s
0
‍
{
2
d
ϖ
d
s
d
2
ϖ
d
s
2
+
2
d
ϖ
d
s
ϖ
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
+
2
d
ϖ
d
s
2
-
n
2
n
ϖ
1
-
n
d
ϖ
d
s
d
2
ϖ
d
s
2
}
d
s
≥
0
,
for2
n
μ
-
(
(
1
/
n
)
+
n
R
)
>
0, or
(70)
∫
s
s
0
‍
{
2
d
ϖ
d
s
d
2
ϖ
d
s
2
+
2
d
ϖ
d
s
ϖ
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
-
2
d
ϖ
d
s
2
-
n
2
n
ϖ
1
-
n
d
ϖ
d
s
d
2
ϖ
d
s
2
}
d
s
≥
0
,
for2
n
μ
-
(
(
1
/
n
)
+
n
R
)
<
0.
For the two cases above, we all have(71)
(
d
ϖ
(
s
)
d
s
)
2
+
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
ϖ
2
(
s
)
+
1
n
ϖ
2
-
n
(
s
)
≤
(
d
ϖ
(
s
0
)
d
s
)
2
+
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
ϖ
2
(
s
0
)
+
1
n
ϖ
2
-
n
(
s
0
)
,
for 2
n
μ
-
(
(
1
/
n
)
+
n
R
)
>
0, or
(72)
(
d
ϖ
(
s
)
d
s
)
2
+
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
ϖ
2
(
s
)
-
1
n
ϖ
2
-
n
(
s
)
≤
(
d
ϖ
(
s
0
)
d
s
)
2
+
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
ϖ
2
(
s
0
)
-
1
n
ϖ
2
-
n
(
s
0
)
,
for 2
n
μ
-
(
(
1
/
n
)
+
n
R
)
<
0.
Therefore, we have(73)
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
ϖ
2
(
s
)
+
1
n
ϖ
2
-
n
(
s
)
≤
(
d
ϖ
(
s
0
)
d
s
)
2
+
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
ϖ
2
(
s
0
)
+
1
n
ϖ
2
-
n
(
s
0
)
,
for 2
n
μ
-
(
(
1
/
n
)
+
n
R
)
>
0, or
(74)
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
ϖ
2
(
s
)
-
1
n
ϖ
2
-
n
(
s
)
≤
(
d
ϖ
(
s
0
)
d
s
)
2
+
(
R
-
n
-
2
n
2
-
n
-
2
n
|
λ
|
)
ϖ
2
(
s
0
)
-
1
n
ϖ
2
-
n
(
s
0
)
,
for 2
n
μ
-
(
(
1
/
n
)
+
n
R
)
<
0.
SinceR
>
(
(
n
-
2
)
/
n
2
)
+
(
(
n
-
2
)
/
n
)
|
λ
|, from (73) and (74), we see that the positive function ϖ is bounded from above. Proposition 16 is proved.Proof of Theorem4.
(1) Since the Möbius sectional curvature ofx is nonnegative, from (22), we have R
i
j
i
j
=
B
i
B
j
+
A
i
+
A
j
=
B
i
B
j
+
D
i
-
λ
B
i
+
D
j
-
λ
B
j
≥
0 for i
≠
j. Therefore, we know that B
a
B
n
+
μ
+
ν
-
λ
B
a
-
λ
B
n
≥
0, 1
≤
a
≤
n
-
1. From (59), we have d
2
ϖ
/
d
s
2
≤
0. Thus, d
ϖ
/
d
s is a monotonic function of s
∈
(
-
∞
,
+
∞
). Therefore, as observed by Wei [12], ϖ
(
s
) must be monotonic when s tends to infinity. Since we assume that R
>
(
(
n
-
2
)
/
n
2
)
+
(
(
n
-
2
)
/
n
)
|
λ
|, from Proposition 16, we know that the positive function ϖ
(
s
) is bounded. Since ϖ
(
s
) is bounded and monotonic when s tends to infinity, we know that both lim
⁡
s
→
-
∞
ϖ
(
s
) and lim
⁡
s
→
+
∞
ϖ
(
s
) exist and then we get
(75)
lim
⁡
s
→
-
∞
⁡
d
ϖ
(
s
)
d
s
=
lim
⁡
s
→
+
∞
⁡
d
ϖ
(
s
)
d
s
=
0
.
By the monotonicity of d
ϖ
(
s
)
/
d
s, we have d
ϖ
(
s
)
/
d
s
≡
0 and ϖ
(
s
)
=
c
o
n
s
t
a
n
t. From ϖ
=
|
2
n
μ
-
(
(
1
/
n
)
+
n
R
)
|
-
1
/
n, we know that μ is constant, and therefore ν is also constant from (46). We obtain that x
:
M
↦
S
n
+
1
(
1
) has two distinct constant para-Blaschke eigenvalues. From Theorem 1 and Examples 8–12, we know that the result (1) of Theorem 4 is proved.
(2) From (46) and (47), we have
(76)
n
(
n
-
1
)
μ
2
-
n
-
1
n
(
1
+
n
2
R
)
μ
+
1
4
n
2
(
1
+
n
2
R
)
2
-
|
D
|
2
=
0
.
Putting D
~
i
j
=
D
i
j
-
(
1
/
n
)
tr
⁡
D
δ
i
j, we have |
D
|
2
=
|
D
~
|
2
+
(
1
/
n
)
(
tr
⁡
D
)
2
=
|
D
~
|
2
+
(
1
/
4
n
3
)
(
1
+
n
2
R
)
2. Therefore, (76) can be rewritten as
(77)
n
(
n
-
1
)
μ
2
-
n
-
1
n
(
1
+
n
2
R
)
μ
+
n
-
1
4
n
3
(
1
+
n
2
R
)
2
-
|
D
~
|
2
=
0
.
We have from (77)
(78)
μ
=
1
2
n
2
(
1
+
n
2
R
)
+
|
D
~
|
n
(
n
-
1
)
or
,
μ
=
1
2
n
2
(
1
+
n
2
R
)
-
|
D
~
|
n
(
n
-
1
)
.
From (46), we obtain
(79)
μ
+
ν
=
1
n
2
(
1
+
n
2
R
)
-
(
n
-
2
)
|
D
~
|
n
(
n
-
1
)
,
or
(80)
μ
+
ν
=
1
n
2
(
1
+
n
2
R
)
+
(
n
-
2
)
|
D
~
|
2
n
(
n
-
1
)
.
If there exists a point p on M such that (79) and (80) hold at p; that is, we have |
D
~
|
=
0 at p. From |
D
~
|
=
∑
i
(
D
~
i
i
)
2
=
∑
i
(
D
i
i
-
(
1
/
n
)
tr
⁡
D
)
2
=
0, we have D
i
i
=
(
1
/
n
)
tr
⁡
D at p; this is in contradiction to the assumption that M has two distinct para-Blaschke eigenvalues. Therefore, we only consider two cases.
Case (i). If (79) holds on M, by (64) and (5), we have
(81)
-
1
n
-
1
B
n
2
-
n
-
2
n
-
1
λ
B
n
+
μ
+
ν
≥
-
n
-
1
n
2
-
n
-
2
n
|
λ
|
+
1
n
2
(
1
+
n
2
R
)
-
(
n
-
2
)
|
D
~
|
n
(
n
-
1
)
=
1
n
(
n
R
-
n
-
2
n
-
(
n
-
2
)
|
λ
|
)
-
(
n
-
2
)
|
D
~
|
n
(
n
-
1
)
=
n
-
2
n
(
n
-
1
)
{
n
-
1
n
(
n
n
-
2
R
-
1
+
n
|
λ
|
n
)
-
|
D
~
|
}
≥
0
.
Case (ii). If (80) holds on M, by (64) and (5), we have
(82)
-
1
n
-
1
B
n
2
-
n
-
2
n
-
1
λ
B
n
+
μ
+
ν
≥
-
n
-
1
n
2
-
n
-
2
n
|
λ
|
+
1
n
2
(
1
+
n
2
R
)
+
(
n
-
2
)
|
D
~
|
n
(
n
-
1
)
=
1
n
(
n
R
-
n
-
2
n
-
(
n
-
2
)
|
λ
|
)
+
(
n
-
2
)
|
D
~
|
n
(
n
-
1
)
≥
1
n
n
n
-
1
(
n
-
2
)
|
D
~
|
+
(
n
-
2
)
|
D
~
|
n
(
n
-
1
)
=
2
(
n
-
2
)
|
D
~
|
n
(
n
-
1
)
≥
0
.
From (61), (81), and (82), we all have d
2
ϖ
/
d
s
2
≤
0. Thus, d
ϖ
/
d
s is a monotonic function of s
∈
(
-
∞
,
+
∞
). Therefore, we have the following: ϖ
(
s
) must be monotonic when s tends to infinity. Since |
D
~
|
≠
0 and |
D
~
|
≤
(
n
-
1
)
/
n
{
(
n
/
(
n
-
2
)
)
R
-
(
(
1
+
n
|
λ
|
)
/
n
)
}, we have R
>
(
(
n
-
2
)
/
n
2
)
+
(
(
n
-
2
)
/
n
)
|
λ
|. From Proposition 16, we know that the positive function ϖ
(
s
) is bounded. Therefore, by the similar assertion in the proof of (1), we obtain that x
:
M
↦
S
n
+
1
(
1
) has two distinct constant para-Blaschke eigenvalues. From Theorem 1 and Examples 8–12, we know that Theorem 4 is proved.



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*Source: 398746-2014-04-14.xml*