Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 4(2011), Pages 15-30.
CURVATURE AND RIGIDITY THEOREMS OF SUBMANIFOLDS
IN A UNIT SPHERE
(COMMUNICATED BY UDAY CHAND DE)
SHICHANG SHU, YANYAN LI
Abstract. In this paper, we investigate n-dimensional submanifolds with
higher codimension in a unit sphere S n+p (1). We obtain some rigidity results of submanifolds in S n+p (1) with parallel mean curvature vector or with
constant scalar curvature, which generalize some related rigidity results of hypersurfaces.
1. Introduction
Let M n be an n-dimensional hypersurface in a unit sphere S n+1 (1). It is well
known that there are many rigidity results for hypersurfaces in S n+1 (1) with constant mean curvature or constant scalar curvature (see [1], [4], [10]), but few of
submanifolds with higher codimension in S n+p (1), especially, if the submanifolds
are complete.
It is well known that H. Alencar, M. do Carmo [1] and H. Li [10] obtained some
important results of compact hypersurface with constant mean curvature or constant scalar curvature in a unit sphere S n+1 (1) , respectively.
Theorem 1.1([1]). Let M n be an n-dimensional compact hypersurface in a unit
sphere S n+1 (1) with constant mean curvature. Assume that |ϕ|2 ≤ BH,n , then
(1) |ϕ|2 = 0, M n is totally umbilical; or
(2) |ϕ|2 = BH,n if and only if
√
√
(i) H = 0, M n is a Clifford torus S k ( k/n) × S n−k ( (n − k)/n), with 1 ≤ k ≤
n − 1;
√
(ii) H ̸= 0, n ≥ 3, and M n is an H(r)-torus S n−1 (r) × S 1 ( 1 − r2 ) with
r2 < (n − 1)/n;
√
(iii) H ̸= 0, n = 2, and M n is an H(r)-torus S 1 (r)×S 1 ( 1 − r2 ) with 0 < r < 1,
r2 ̸= 21 .
Theorem 1.2([10]). Let M n be an n-dimensional (n ≥ 3) compact hypersurface in
2000 Mathematics Subject Classification. 53C42, 53A10.
Key words and phrases. submanifolds; mean curvature; scalar curvature; rigidity theorems.
c
⃝2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted Jul 21, 2011. Published August 28, 2011.
15
16
SHICHANG SHU, YANYAN LI
a unit sphere S n+1 (1) with constant scalar curvature n(n − 1)R and R̄ = R − 1 ≥ 0.
If
n
nR̄ ≤ S ≤
{n(n − 1)R̄2 + 4(n − 1)R̄ + n},
(1.1)
(n − 2)(nR̄ + 2)
n
then either S = nR̄ and M n is totally umbilical, or S = (n−2)(n
{n(n − 1)R̄2 +
R̄+2)
√
√
4(n − 1)R̄ + n} and M n is a product S 1 ( 1 − r2 ) × S n−1 (r), r = n−2
nR .
Remark 1.1.
We should notice that in Theorem 1.1, |ϕ|2 = S − nH 2 is the
non-negative function on M n , S and H the squared norm of the second fundamental form and mean curvature of M n , BH,n the square of the positive real root
of
n−2
PH,n (x) = x2 + √
nHx − n(1 + H 2 ) = 0.
n(n − 1)
We should notice that W. Santos [14], Cheng [5] obtained some important results
of compact submanifolds with higher codimension and parallel mean curvature
vector or constant scalar curvature in S n+p (1), but to our knowledge, the results
of complete submanifolds in S n+p (1) are very few.
In this paper, we study n-dimensional compact or complete submanifolds with
higher codimension and parallel mean curvature vector or constant scalar curvature
in S n+p (1). In order to present our result, we define a function QR̄,p,n (x) by
1
1 n−1
QR̄,p,n (x) =n + nR̄ + [ − (2 − )
](x − nR̄)
n
p n
√
n−2
[x + n(n − 1)R̄](x − nR̄),
−
n
then we may obtain the following result:
(1.2)
Theorem 1.3. Let M n be an n-dimensional compact submanifold in a unit sphere
S n+p (1) with constant scalar curvature n(n − 1)R and R̄ = R − 1 ≥ 0. If the
normalized mean curvature vector is parallel and the squared norm S of the second
fundamental form of M n satisfies
QR̄,p,n (S) ≥ 0,
(1.3)
then
(1) S = nR̄ and M n is totally umbilical; or
(2) QR̄,p,n (S) = 0. In the latter case, either
√
√
(a) p = 1 and M n is a product S 1 ( 1 − r2 ) × S n−1 (r), r = n−2
nR ; or
(b) n = 2, p = 2 and M n is Veronese surface in S 4 .
Remark 1.2. We note that if p = 1, Theorem 1.3 reduces to Theorem 1.2. We
should notice that in [11], J.T. Li obtained some results of compact submanifold
in S n+p (1) with constant scalar curvature and parallel normalized mean curvature
vector, but his results are very different from us.
We define a polynomial PH,p,n (x) by
n−2
1
nHx − n(1 + H 2 ).
PH,p,n (x) = (2 − )x2 + √
p
n(n − 1)
(1.4)
CURVATURE AND RIGIDITY THEOREMS OF SUBMANIFOLDS
17
We easily know that PH,p,n (x) = 0 has a positive real root, and denoted by BH,p,n
the square of the positive real root.
If M n is an n-dimensional complete submanifold with higher codimension in a
unit sphere S n+p (1), we obtain the following results:
Theorem 1.4. Let M n be an n-dimensional complete submanifold in a unit sphere
S n+p (1) with parallel mean curvature vector. Assume that sup |ϕ|2 ≤ BH,p,n , then
(1) sup |ϕ|2 = 0, M n is totally umbilical; or
(2) sup |ϕ|2 = BH,p,n . If the supremum sup |ϕ|2 is attained on M n , then either
(a) p = 1 and
√
√
(i) H = 0, M n is an open piece of Clifford torus S k ( k/n)×S n−k ( (n − k)/n),
with 1 ≤ k ≤ n − 1;
√
(ii) H ̸= 0, n ≥ 3, and M n is an open piece of H(r)-torus S n−1 (r)×S 1 ( 1 − r2 )
with r2 < (n − 1)/n;
√
(iii) H ̸= 0, n = 2, and M n is an open piece of H(r)-torus S 1 (r) × S 1 ( 1 − r2 )
with 0 < r < 1, r2 ̸= 12 ; or
(b) n = 2, p = 2 and M n is an open piece of Veronese surface in S 4 .
Theorem 1.5. Let M n be an n-dimensional complete submanifold in a unit sphere
S n+p (1) with constant scalar curvature n(n − 1)R and R̄ = R − 1 > 0. If the normalized mean curvature vector is parallel and the squared norm S of the second
fundamental form of M n satisfies
QR̄,p,n (sup S) ≥ 0,
(1.5)
then
(1) sup S = nR̄ and M n is totally umbilical; or
(2) QR̄,p,n (sup S) = 0. In the latter case, if the supremum sup S is attained on
M n , then either
√
(i) p = 1 and M n is an open piece of H(r)-torus S n−1 (r) × S 1 ( 1 − r2 ) with
0 < r < 1; or
(ii) n = 2, p = 2 and M n is an open piece of Veronese surface in S 4 , where
QR̄,p,n (x) is defined by (1.2).
Remark 1.3. We note that Theorem 1.4 and Theorem 1.5 generalize the results of H. Alencar, M.do Carmo [1] and H. Li [10](Theorem 1.1 and Theorem 1.2)
to complete submanifold with higher codimension.
2. Preliminaries
n
Let M be an n-dimensional submanifold in an (n + p)-dimensional unit sphere
S n+p (1). Let {e1 , . . . , en } be a local orthonormal basis of M n with respect to the
induced metric, {ω1 , . . . , ωn } are their dual form. Let en+1 , . . . , en+p be the local
unit orthonormal normal vector field. We make the following convention on the
range of indices:
1 ≤ i, j, k, . . . ≤ n;
n + 1 ≤ α, β, γ, . . . ≤ n + p.
Then the structure equations are
∑
dωA = −
ωAB ∧ ωB , ωAB + ωBA = 0,
B
(2.1)
18
SHICHANG SHU, YANYAN LI
dωAB = −
∑
ωAC ∧ ωCB +
C
1∑
KABCD ωC ∧ ωD ,
2
(2.2)
C,D
KABCD = δAC δBD − δAD δBC .
The Gauss equations are
Rijkl = (δik δjl − δil δjk ) +
(2.3)
∑
α
α α
(hα
ik hjl − hil hjk ),
(2.4)
α
where S =
∑
i,j,α
n(n − 1)(R − 1) = n2 H 2 − S,
⃗ = ∑ H α eα , H α = 1 ∑ hα ,
H
kk
n
2
(hα
ij ) ,
α
(2.5)
⃗
H = |H|, R is the
k
normalized scalar curvature of M n .
α
The first covariant derivative {hα
ijk } and the second covariant derivative {hijkl }
of hα
are
defined
by
ij
∑
∑
∑
∑ β
α
hα
hα
hij ωβα ,
(2.6)
hα
ijk ωk = dhij −
kj ωki −
ik ωkj −
∑
k
hα
ijkl ωl
=
dhα
ijk
−
∑
k
hα
ljk ωli
−
∑
l
l
k
hα
ilk ωlj
−
l
∑
β
hα
ijl ωlk
−
∑
l
hβijk ωβα .
(2.7)
β
Then, we have the Codazzi equations and the Ricci identities
α
hα
ijkl − hijlk
α
hα
ijk = hikj ,
∑
∑
∑ β
hij Rβαkl .
=
hα
hα
mj Rmikl +
im Rmjkl +
m
m
The Ricci equations are
Rαβij =
(2.8)
(2.9)
β
∑
β
β α
(hα
ik hkj − hik hkj ).
(2.10)
k
From (2.8) and (2.9), we have
∑ β
∑
∑
∑
hki Rβαjk .
hα
hα
hα
∆hα
mi Rmkjk +
km Rmijk +
kkij +
ij =
k,m
k
(2.11)
k,β
k,m
Denote by |ϕ|2 = S − nH 2 the non-negative function |ϕ| on M n . We know that
|ϕ|2 = 0 exactly at the umbilical points of M n . Define the first, second covariant
⃗ = ∑ H α eα in the
derivatives and Laplacian of the mean curvature vector field H
α
normal bundle N (M n ) as follows
∑
∑
H β θβα ,
H,iα θi = dH α +
∑
i
α
H,ij
θj = dH,iα +
∑
j
β
H,jα θji +
j
∆⊥ H α =
∑
∑
(2.12)
H,iβ θβα ,
(2.13)
β
α
H,ii
,
Hα =
i
1∑ α
hkk .
n
(2.14)
k
Let f be a smooth function on M n . The first, second covariant derivatives fi , fi,j
and Laplacian of f are defined by
∑
∑
∑
∑
df =
fi θi ,
fi,j θj = dfi +
fj θji ,
∆f =
fi,i .
(2.15)
i
j
j
i
CURVATURE AND RIGIDITY THEOREMS OF SUBMANIFOLDS
19
For the fix index α(n + 1 ≤ α ≤ n + p), we introduce an operator α due to
Cheng-Yau [4] by
∑
α f =
(nH α δij − hα
(2.16)
ij )fi,j .
i,j
Since M n is compact, the operator α is self-adjoint (see[4]) if and only if
∫
∫
(α f )gdv =
f (α g)dv,
(2.17)
M
M
where f and g are any smooth functions on M n .
In general, for a matrix A = (aij ) we denote by N (A) the square of the norm of
A, that is,
∑
N (A) = tr(A · At ) =
(aij )2 .
i,j
Clearly, N (A) = N (T t AT ) for any orthogonal matrix T .
We need the following Lemmas due to Chern-Do Carmo-Kobayashi [7], Cheng
[5] and the author [15].
Lemma 2.1([7]). Let A and B be symmetric (n × n)-matrices. Then
N (AB − BA) ≤ 2N (A)N (B),
(2.18)
and the equality holds for nonzero matrices A and B if and only if A and B can
be transformed simultaneously by on orthogonal matrix into multiples of à and B̃
respectively, where




0 1 0 ··· 0
1 0 0 ··· 0
 1 0 0 ··· 0 
 0 −1 0 · · · 0 




 0 0 0 ··· 0 


à = 
B̃ =  0 0 0 · · · 0  .

 .. .. .. . .
 ..
.. 
..
.. . .
.. 
 . . .
 .
. . 
. . 
.
.
0
0
0 ···
0
0
0
0
···
0
Moreover, if A1 , A2 and A3 are (n × n)-symmetric matrices and if
N (Aα Aβ − Aβ Aα ) = 2N (Aα )N (Aβ ), 1 ≤ α, β ≤ 3
then at least one of the matrices Aα must be zero.
Lemma 2.2([5]). Let bi for i = 1, · · · , n be real numbers satisfying
and
n
∑
bi = 0
i=1
n
∑
i=1
b2i = B. Then
n
∑
i=1
b4i −
(n − 2)2 2
B2
≤
B .
n
n(n − 1)
(2.19)
Lemma 2.3 ([5], [15]).
Let ai and bi for i = 1, · · · , n be real numbers satn
n
∑
∑
isfying
ai = 0 and
a2i = a. Then
i=1
i=1
v
n
u
∑
n
u n
( b2i )2
∑
u∑
√
t
a.
b4i − i=1
ai b2i ≤
n
i=1
i=1
(2.20)
20
SHICHANG SHU, YANYAN LI
If ai = bi for i = 1, · · · , n, then Lemma 2.3 becomes to the well-known Lemma
of M. Okumura [12].
Lemma 2.4 ([12]). Let {ai }ni=1 be a set of real numbers satisfying
∑
0, a2i = a, where a ≥ 0. Then we have
i
∑
ai =
i
∑
n−2
n−2
−√
a3/2 ≤
a3/2 ,
a3i ≤ √
n(n − 1)
n(n
−
1)
i
(2.21)
and the equalities hold if and only if at least (n − 1) of the ai are equal.
3. Proof of Theorem 1.3
Define tensors
σ̃αβ
α
α
h̃α
ij = hij − H δij ,
∑
∑
β
β
=
h̃α
σαβ =
hα
ij h̃ij ,
ij hij .
i,j
(3.1)
(3.2)
i,j
Then the (p × p)-matrix (σ̃αβ ) is symmetric and can be assumed to be diagonized
for a suitable choice of en+1 , . . . , en+p . We set
σ̃αβ = σ̃α δαβ .
(3.3)
By a direct calculation, we have
∑
∑
α β
2
h̃α
σ̃α = S − nH 2 ,
kk = 0, σ̃αβ = σαβ − nH H , |ϕ| =
∑
i,j,k,α
k
α
hβkj hα
ij hik =
∑
α
h̃βkj h̃α
ij h̃ik + 2
∑
(3.4)
α
β
β
2
2 β
H α h̃α
ij h̃ij + H |ϕ| + nH H .
(3.5)
i,j,α
i,j,k,α
Setting f = nH α in (2.16), we have
∑
α
α (nH α ) =
(nH α δij − hα
ij )(nH )i,j
(3.6)
i,j
=
∑
∑
α
(nH α )(nH α )i,i −
hα
ij (nH )i,j .
i
i,j
We also have
1 ∑
1∑
1
∆(nH)2 = ∆
(nH α )2 =
∆(nH α )2
2
2
2
α
α
∑
∑
1∑
α 2
=
[(nH ) ]i,i =
[(nH α ),i ]2 +
(nH α )(nH α )i,i
2 α,i
α,i
α,i
∑
2
⊥⃗ 2
α
α
=n |∇ H| +
(nH )(nH )i,i .
(3.7)
α,i
Therefore, from (2.5), (3.6), (3.7), we get
∑
∑
1
α
⃗ 2−
α (nH α ) = ∆(nH)2 − n2 |∇⊥ H|
hα
ij (nH )i,j
2
α
i,j,α
∑
1
1
α
⃗ 2−
= n(n − 1)∆R + ∆S − n2 |∇⊥ H|
hα
ij (nH )i,j .
2
2
i,j,α
(3.8)
CURVATURE AND RIGIDITY THEOREMS OF SUBMANIFOLDS
21
From (2.11), we have
∑
∑
1
2
α
∆S =
(hα
hα
ijk ) +
ij ∆hij
2
i,j,α
i,j,k,α
∑ ∑
∑
α
α
α
=|∇h|2 +
hα
hα
ij (nH )i,j +
ij (hkl Rlijk + hli Rlkjk )
α i,j,k,l
i,j,α
+
∑∑
(3.9)
β
hα
ij hki Rβαjk .
α,β i,j,k
Putting (3.9) into (3.8), we have
∑
α
⃗ 2 + 1 n(n − 1)∆R
α (nH α ) =|∇h|2 − n2 |∇⊥ H|
(3.10)
2
∑ ∑
∑∑
β
α
α
+
hα
hα
ij (hkl Rlijk + hli Rlkjk ) +
ij hki Rβαjk .
α i,j,k,l
α,β i,j,k
Thus, if M n is compact, from (2.17) and Stokes formula, we have
∫
⃗ 2 }dv
{|∇h|2 − n2 |∇⊥ H|
0=
∫
Mn
+
Mn
(3.11)
∫
∑ ∑
α
α
{
hα
(h
R
+
h
R
)}dv
+
lijk
lkjk
ij
kl
li
α i,j,k,l
∑∑
M n α,β i,j,k
β
hα
ij hki Rβαjk dv.
From (2.10), we have
∑
(Rβαjk )2 =
α,β,k
∑
β α
(hβji hα
ik − hki hij )Rβαjk = −2
∑∑
β
hα
ij hki Rβαjk .
α,β i,j,k
α,β,i,j,k
Thus, we have
∑∑
β
hα
ij hki Rβαjk = −
α,β i,j,k
1 ∑
(Rβαjk )2
2
(3.12)
α,β,k
=−
1 ∑ ∑ β α ∑ α β 2
(
hjl hlk −
hjl hlk )
2
α,β,j,k
l
l
α,β,j,k
l
l
1 ∑ ∑ β α ∑ α β 2
=−
(
h̃jl h̃lk −
h̃jl h̃lk )
2
1∑
=−
N (Ãα Ãβ − Ãβ Ãα ),
2
α,β
α
α
where Ãα := (h̃α
ij ) = (hij − H δij ).
22
SHICHANG SHU, YANYAN LI
From (2.4), (2.10), (3.2), (3.4), (3.5) and (3.12), we have
∑ ∑
α
α
hα
ij (hkl Rlijk + hli Rlkjk )
α i,j,k,l
=n|ϕ|2 −
∑ ∑
β α β
hα
ij hij hlk hlk + n
α,β i,j,k,l
=n|ϕ| −
2
∑
2
σαβ
α,β
+n
∑∑
∑∑
∑
α
H β hβkj hα
ij hik +
α,β i,j,k
α
H β h̃βkj h̃α
ij h̃ik
(3.13)
+ 2n
∑∑
β
hα
ji hik Rβαjk
α,β,i,j,k
β
H H β h̃α
ij h̃ij
α
α,β i,j
α,β i,j,k
∑
∑
1∑
+n
(H β )2 |ϕ|2 + n2 H 2
(H β )2 −
N (Ãα Ãβ − Ãβ Ãα )
2
β
β
α,β
∑
∑∑
2
α
=n|ϕ|2 −
σ̃αβ
+ nH 2 |ϕ|2 + n
H β h̃βkj h̃α
ij h̃ik
α,β
1∑
N (Ãα Ãβ − Ãβ Ãα ).
−
2
α,β i,j,k
α,β
∑
∑
∑ β
∑ α
Let (h̃βii )2 = τβ . Then τβ ≤ (h̃βij )2 = σ̃β . Since
h̃ii = 0,
µi = 0 and
i,j
i
i
∑ αi 2
(µi ) = σ̃α . We have from Lemma 2.2 and Lemma 2.3 that
i
∑∑
β β
H α h̃α
ij h̃kj h̃ik =
α,β i,j,k
∑∑
β,α i,j,k
∑
α
H β h̃βij h̃α
kj h̃ik
(3.14)
∑
n−2 ∑ β
√
2
h̃βii (µα
|H |σ̃α τβ
i ) ≥ −√
n(n − 1) α,β
i
α,β
∑
∑
√
n−2
≥− √
σ̃α
|H β | σ̃β
n(n − 1) α
β
√∑
∑
n−2
σ̃β
≥− √
|ϕ|2
(H β )2
n(n − 1)
β
β
=
Hβ
n−2
|H||ϕ|3 .
=− √
n(n − 1)
From Lemma 2.1, (3.3), (3.4), we have
∑
∑
∑
∑
2
−
σ̃αβ
−
N (Ãα Ãβ − Ãβ Ãα ) = −
σ̃α2 −
N (Ãα Ãβ − Ãβ Ãα )
α,β
α
α,β
≥−
∑
α,β
σ̃α2 − 2
α
∑
σ̃α σ̃β
α̸=β
∑
∑
σ̃α2
σ̃α )2 +
= − 2(
α
α
1 ∑
≥ − 2|ϕ|4 + (
σ̃α )2
p α
1
= − (2 − )|ϕ|4 .
p
Therefore, from (3.11), (3.12)-(3.15), we have the following:
(3.15)
CURVATURE AND RIGIDITY THEOREMS OF SUBMANIFOLDS
23
Proposition 3.1. Let M n be an n-dimensional compact submanifolds in a unit
sphere S n+p (1). Then there holds the following
∫
⃗ 2 }dv
0≥
{|∇h|2 − n2 |∇⊥ H|
(3.16)
Mn
∫
n−2
1
+
|ϕ|2 {n + nH 2 − √
n|H||ϕ| − (2 − )|ϕ|2 }dv.
p
n(n − 1)
Mn
Proof of Theorem 1.3. Since R ≥ 1 and the normalized mean curvature vector is
parallel, we easily know that
⃗ 2.
|∇h|2 ≥ n2 |∇⊥ H|
In fact, from (2.5) and R ≥ 1, we have S ≤ n2 H 2 . Taking covariant derivative on
(2.5), we get
∑
α
n2 HH,k =
hα
ij hijk .
i,j,α
From Cauchy-Schwarz’s inequality, we get
∑ ∑
∑
∑
α 2
2
⃗ 2 = n4 H 2
(H,k )2 =
(
hα
(hα
n4 H 2 |∇⊥ H|
ij hijk ) ≤ S
ijk ) ,
k
k
i,j,α
(3.17)
i,j,k,α
and we conclude. Denote R̄ = R − 1, by (2.5) we have S − nH 2 = n−1
n (S − nR̄).
Thus, we obtain
n−2
1
n+nH 2 − √
n|H||ϕ| − (2 − )|ϕ|2
p
n(n − 1)
1
1 n−1
=n + nR̄ + [ − (2 − )
](S − nR̄)
n
p n
√
n−2
−
[S + n(n − 1)R̄](S − nR̄).
n
From the assumption of Theorem 1.3 and the Proposition 3.1, we have
∫
n−2
1
0≥
|ϕ|2 {n + nH 2 − √
(3.18)
n|H||ϕ| − (2 − )|ϕ|2 }dv
p
n(n − 1)
Mn
∫
n−1
1
1 n−1
=
(S − nR̄){n + nR̄ + [ − (2 − )
](S − nR̄)
n
n
p n
Mn
√
n−2
−
[S + n(n − 1)R̄](S − nR̄)}dv ≥ 0.
n
Therefore, we have
(1) S = nR̄, that is, M n is totally umbilical;
(2) or
1
1 n−1
− (2 − )
](S − nR̄)
(3.19)
n
p n
√
n−2
−
[S + n(n − 1)R̄](S − nR̄) = 0.
n
In this case, the equalities in (3.18) (3.16), (3.15), (3.14) and Lemma 2.1 hold.
Thus, we have
1
n−2
n|H||ϕ| − (2 − )|ϕ|2 = 0.
(3.20)
n + nH 2 − √
p
n(n − 1)
n+nR̄ + [
24
SHICHANG SHU, YANYAN LI
We see that M n is not totally umbilical and the equalities in (3.16), (3.15), (3.14)
and Lemma 2.1 hold. Thus, we have ∇h = 0,
∑
∑
p
σ̃α2 = (
σ̃α )2 ,
α
α
that is
σ̃n+1 = · · · = σ̃n+p ,
(3.21)
N (Ãα Ãβ − Ãβ Ãα ) = 2N (Ãα )N (Ãβ ),
∑
and
α ̸= β,
√
|H β | σ̃β = |H||ϕ|.
(3.22)
(3.23)
β
We may consider the case p = 1 and p ≥ 2 separately.
Case (i). If p = 1, from (3.19), we have n ̸= 2 and
n
S=
{n(n − 1)R̄2 + 4(n − 1)R̄ + n}.
(n − 2)(nR̄ + 2)
√
Thus, by the Theorem
1.1 of H. Li [10], we know that M n is a product S 1 ( 1 − r2 )×
√
S n−1 (r), r = n−2
nR .
Case (ii). If p ≥ 2, from (3.21) and (3.23), we have
√∑
√∑
√∑
∑
√
√
β
β
2
(H )
σ̃β = pσ̃n+1
(H β )2 .
σ̃n+1
|H | =
β
β
β
β
Since M is not totally umbilical, we have σ̃n+1 ̸= 0. Thus, we have
∑
∑
(
|H β |)2 = p
(H β )2 ,
n
β
β
that is,
|H n+1 | = · · · = |H n+p |.
(3.24)
α
From Lemma 2.1, we know that at most two of Ãα = (h̃ij ), α = n + 1, . . . , n + p, are
n
different from zero. If all of Ãα = (h̃α
ij ) are zero, which is contradiction with M is
not totally umbilical. If only one of them, say Ãα , is different from zero, which is
contradiction with (3.21). Therefore, we may assume that
Ãn+1 =λÃ,
Ãn+2 = µB̃, λ, µ ̸= 0,
Ãα = 0,
α ≥ n + 3,
where à and B̃ are defined in Lemma 2.1.
From (3.23), we have
∑
√
√
(H α )2 (2λ2 + 2µ2 ).
( 2λ|H n+1 | + 2µ|H n+2 |)2 = H 2 |ϕ|2 =
α
Thus, from (3.24), we have
(H n+1 )2 (λ + µ)2 = p(H n+1 )2 (λ2 + µ2 ),
that is,
(H n+1 )2 [(p − 1)λ2 − 2λµ + (p − 1)µ2 ] = 0.
Since λ, µ ̸= 0, we infer that H n+1 = 0. Thus, from (3.24), we have H α = 0, n+1 ≤
⃗ = 0, M n is a minimal submanifold in S n+p (1) and from (3.20),
α ≤ n+p, that is, H
n
we have S = 2−1/p on M n . From the Theorem of Chern-Do Carmo-Kobayashi [7],
CURVATURE AND RIGIDITY THEOREMS OF SUBMANIFOLDS
25
we know that n = 2, p = 2 and M n is Veronese surface in S 4 . This completes the
proof of Theorem 1.3.
4. Proof of Theorem 1.4 and 1.5
The important maximum principle of Omori [13], Yau [16] and Cheng [6] are
useful to us.
Proposition 4.1 ([13], [16]). Let M n be a complete Riemannian manifold whose
Ricci curvature is bounded from below. If f is a C 2 -function bounded from above
on M n , then for any ε > 0, there is a point x ∈ M n such that
sup f − ε < f (x), |∇f |(x) < ε, ∆f (x) < ε.
(4.1)
Proposition 4.2 ([6]). Let M n be a complete Riemannian manifold whose Ricci
curvature is bounded from below. Let f be a C 2 -function which bounded from above.
Then there exists a sequence {xk } in M n such that
lim f (xk ) = sup f, lim |∇f (xk )| = 0, lim sup Lf (xk ) ≤ 0,
m→∞
m→∞
∑
where Lf =
bj fj,j is a differential operator, and bj ≥ 0 is bounded.
(4.2)
m→∞
j
We need the following Lemma.
Lemma 4.3 ([2], [9]).
Let A = (aij ), i, j∑= 1, · · · , n be a symmetric (n × n)
matrix, n ≥ 2. Assume that A1 = trA, A2 = (aij )2 . Then
i,j
∑
1
(4.3)
(ain )2 − A1 ann ≤ 2 {n(n − 1)A2
n
i
√
√
+ (n − 2) n − 1|A1 | nA2 − (A1 )2 − 2(n − 1)(A1 )2 },
the equality holds if and only if n = 2 or n > 2, (aij ) is of the following form


a
0


..


.

,


a
0
A1 − (n − 1)a
where (na − A1 )A1 ≥ 0.
Proof of Theorem 1.4. We assume that sup |ϕ|2 ≤ BH,p,n , then 0 ≤ |ϕ| ≤
we have PH,p,n (|ϕ|) ≤ 0, that is
n + nH 2 − √
n−2
n(n − 1)
√
BH,p,n ,
1
n|H||ϕ| − (2 − )|ϕ|2 ≥ 0.
p
Since the mean curvature vector is parallel, we know that the mean curvature is
constant. From (3.9), (3.12)-(3.15), we have
n−2
1
1
∆|ϕ|2 ≥ |∇ϕ|2 + |ϕ|2 {n + nH 2 − √
n|H||ϕ| − (2 − )|ϕ|2 } ≥ 0. (4.4)
2
p
n(n − 1)
26
SHICHANG SHU, YANYAN LI
For any point and any unit vector v ∈ Tp M n , we choose a local orthonormal
frame field e1 , . . . , en such that en = v, we have from Gauss equation (2.4) that the
Ricci curvature Ric(v, v) of M n with respect to v is expressed as
∑
∑
2
Ric(v, v) = (n − 1) +
[(trHα )hα
(hα
(4.5)
nn −
in ) ],
α
i
where Hα is the (n × n)-matrix (hα
ij ). Assume that Tα = trHα , Sα =
∑
∑
we have n2 H 2 = Tα2 , S = Sα . By Lemma 4.3, we have
α
∑
i,j
2
(hα
ij ) , then
α
∑ 1
Ric(v, v) ≥(n − 1) −
{n(n − 1)Sα
(4.6)
n2
α
√
√
+ (n − 2) n − 1|Tα | nSα − Tα2 − 2(n − 1)Tα2 }
√
√
n−1
n−2 n−1∑
T2
2(n − 1) ∑ 2
=(n − 1) −
S−
|Tα | Sα − α +
Tα
n
n
n
n
n2
α
α
√
∑
∑
n−1
n−2
T2
2
(
≥
{n + 2nH − S − √
Tα2 )[ (Sα − α )]}
n
n
n(n − 1)
α
α
n−1
n−2
{n + nH 2 − √
n|H||ϕ| − |ϕ|2 }
n
n(n − 1)
n−1
n−2
1
≥
{n + nH 2 − √
n|H||ϕ| − (2 − )|ϕ|2 } ≥ 0.
n
p
n(n − 1)
=
Therefore, we know that the Ricci curvature Ric(v, v) is bounded from below.
Now we consider the following smooth function on M n defined by f = −(|ϕ|2 +
−1/2
a)
, where a(> 0) is a real number. Obviously, f is bounded, so we can apply
Proposition 4.1 to f . For any ε > 0, there is a point x ∈ M n , such that at which f
satisfies the (4.1). By a simple and direct calculation, we have
1
f ∆f = 3|df |2 − f 4 ∆|ϕ|2 .
2
(4.7)
From (4.1) and (4.7), we have
1
∆|ϕ|2 (x) = f −4 (x)[3|df |2 (x) − f (x)∆f (x)] < f −4 (x)[3ε2 − εf (x)].
2
(4.8)
Thus, for any convergent sequence {εm } with εm > 0 and limm→∞ εm = 0, there
exists a point sequence {xm } such that the sequence {f (xm )} converges to f0 (we
can take a subsequence if necessary) and satisfies (4.1), hence, limm→∞ εm [3εm −
f (xm )] = 0. From the definition of supremum and (4.1), we have limm→∞ f (xm ) =
f0 = sup f and hence the definition of f gives rise to limm→∞ |ϕ|2 (xm ) = sup |ϕ|2 .
From (4.4) and (4.8), we have
1
∆|ϕ|2 (xm )
(4.9)
2
n−2
1
≥ |ϕ|2 (xm ){n + nH 2 − √
n|H||ϕ|(xm ) − (2 − )|ϕ|2 (xm )} ≥ 0.
p
n(n − 1)
f −4 (xm )[3ε2m − εm f (xm )] >
CURVATURE AND RIGIDITY THEOREMS OF SUBMANIFOLDS
27
Putting m → ∞ in (4.9), we have
sup |ϕ|2 {n + nH 2 − √
n−2
1
n|H| sup |ϕ| − (2 − ) sup |ϕ|2 } = 0.
p
n(n − 1)
Thus, we have
(1) sup |ϕ|2 = 0 and M n is totally umbilical; or
(2)
n−2
1
n + nH 2 − √
n|H| sup |ϕ| − (2 − ) sup |ϕ|2 = 0.
p
n(n − 1)
(4.10)
From (4.4), we know that |ϕ|2 is a subharmonic function on M n . Since the supremum sup |ϕ|2 is attained at some point of M n , by the maximum principle, we have
|ϕ|2 = const. = BH,p,n . Thus, (4.10) becomes
n + nH 2 − √
n−2
1
n|H||ϕ| − (2 − )|ϕ|2 = 0.
p
n(n − 1)
(4.11)
and (4.4) becomes equality. We may consider the case p = 1 and p ≥ 2 separately.
Case (i). If p = 1, from equality in (4.4), we obtain that ∇ϕ = ∇h = 0, that
is, the second fundamental form is parallel. If H = 0, then by a classical local
n
rigidity result of Lawson (see Proposition 1 in Lawson√
[8]), we know that
√ M is an
k
n−k
open piece of a minimal Clifford torus of the form S ( k/n) × S
( (n − k)/n)
with 1 ≤ k ≤ n − 1. If H ̸= 0, then from the equality in (4.4), we also obtain
the equality in Lemma 2.4 of Okumura, which implies that M n has exactly two
constant principal curvatures, with multiplicities n−1 and 1. Then, by the classical
result on isoparametric hypersurfaces of E. Cartan
[3] we conclude that if n ≥ 3,
√
M n must be an open piece of H(r)-torus S 1 ( 1 − r2 ) × S n−1 (r), with 0 √
< r < 1,
2
n
1
1
r < (n − 1)/n; if n = 2, M is an open piece of H(r)-torus S (r) × S ( 1 − r2 )
with 0 < r < 1, r2 ̸= 12 .
Case (ii). If p ≥ 2, since (4.11) holds, M n is not totally umbilical and the
equalities in (3.16), (3.15), (3.14) and Lemma 2.1 hold. Thus, we have ∇h = 0,
and (3.21), (3.22), (3.23) hold. By the same assertion in the proof of Theorem 1.3,
n
we know that M n is a minimal submanifold in S n+p (1) and S = 2−1/p
on M n .
From the Theorem of Chern-Do Carmo-Kobayashi [7], we know that n = 2, p = 2
and (4.11) reduces to S = 34 , M n is an open piece of Veronese surface in S 4 . This
completes the proof of Theorem 1.4.
Proof of Theorem 1.5.
f =
Let
∑
α f =
α
∑∑
( (nH α δij − hα
ij ))fi,j .
i,j
α
We may prove that the operator is elliptic. In fact, for a fixed α, n+1 ≤ α ≤ n+p,
α
we can take a local orthonormal frame field {e1 , . . . , en } such that hα
ij = λi δij , then
∑∑
(4.12)
f =
( (nH α − λα
i ))fi,i .
i
α
Since R > 1, from (2.5), we have S < n2 H 2 . For a fixed
α ≤ n + p, if
∑ α, n α+21 ≤ ∑
α
α
2 2
2
there is a λα
(nH ) ≤ (λα
i such that nH − λi ≤ 0, then n H =
i ) ≤ S,
α
α
28
SHICHANG SHU, YANYAN LI
this is a contradiction. Thus, we have nH α − λα
i > 0, then
∑
α
(nH α − λα
i ) > 0 and
the operator is elliptic.
From (1.5), we have
1
1 n−1
n + nR̄ + [ −(2 − )
](sup S − nR̄)
n
p n
√
n−2
−
[sup S + n(n − 1)R̄](sup S − nR̄) ≥ 0.
n
n−2
Since n1 − (2 − p1 ) n−1
n ≤ − n ≤ 0, (4.13) implies that sup S < +∞ and
(4.13)
1 n−1
1
](S − nR̄)
(4.14)
n + nR̄ + [ −(2 − )
n
p n
√
n−2
−
[S + n(n − 1)R̄](S − nR̄) ≥ 0.
n
Thus, from (2.5) and (4.6), we get the Ricci curvature Ric(v, v) is bounded from
below.
From (3.10), (3.12)-(3.15), (4.14) and R > 1 , we have
∑
⃗ 2
(nH α ) =
α (nH α ) = |∇h|2 − n2 |∇⊥ H|
(4.15)
α
n−2
1
n|H||ϕ| − (2 − )|ϕ|2
p
n(n − 1)
n−1
1
1 n−1
≥
(S − nR̄){n + nR̄ + [ − (2 − )
](S − nR̄)
n
n
p n
√
n−2
−
[S + n(n − 1)R̄](S − nR̄)} ≥ 0.
n
∑
Putting f = nH α in (4.12), by H 2 = (H α )2 , we have |nH α | ≤ nH. From
+ |ϕ|2 {n + nH 2 − √
α
α
(2.5) and sup S < +∞, we
∑ haveα f =αnH is bounded from above. Since we know
(nH
−
λ
)
>
0, we have
that nH α − λα
>
0
and
i
i
α
∑
∑
∑
(nH α − λα
Hα
0<
(nH α − λα
i ) = n(n − 1)
i )≤
α
α
α,i
α
is bounded. We may use Proposition 4.2 to f = nH . Thus, we have
lim (nH α )(xm ) = sup(nH α ),
m→+∞
lim sup (nH α )(xm ) ≤ 0,
m→+∞
(4.16)
where {xm } is a sequence on M n . From (2.5) and (4.16), we have limm→∞ S(xm ) =
sup S.
From (4.13), (4.15) and (4.16), we have
0 ≥ lim sup (nH α )(xm )
m→+∞
n−1
1
1 n−1
≥
(sup S − nR̄){n + nR̄ + [ − (2 − )
](sup S − nR̄)
n
n
p n
√
n−2
−
[sup S + n(n − 1)R̄](sup S − nR̄)} ≥ 0.
n
Thus, we have
(i) sup S − nR̄ = 0, that is, sup S = nR̄. From (2.5), we have sup(S − nH 2 ) = 0,
thus, S = nH 2 and M n is totally umbilical; or
CURVATURE AND RIGIDITY THEOREMS OF SUBMANIFOLDS
29
(ii)
1
1 n−1
n + nR̄ + [ −(2 − )
](sup S − nR̄)
(4.17)
n
p n
√
n−2
−
[sup S + n(n − 1)R̄](sup S − nR̄) = 0.
n
(4.17) implies that (4.14) and (4.15) hold. From the assumption, we know that
sup S is attained at some point of M n . Thus, ∑
from (2.5), we have sup(nH)2 is
n
2
attained at this point of M . By sup(nH) =
sup(nH α )2 , we have sup(nH α )
α
is attained at this point of M n . Since the operator is elliptic, we have nH α is
⃗ 2 . From (2.5)
constant. Thus, the equalities in (4.15) hold and |∇h|2 = n2 |∇⊥ H|
and (3.17), we have
⃗ 2 ≤ S(|∇h|2 − n2 |∇⊥ H|
⃗ 2 ).
0 ≤ n3 (n − 1)(R − 1)|∇⊥ H|
⃗ = 0. Therefore, we know that M n is a
Since we assume that R > 1, we have ∇⊥ H
n+p
complete submanifold in S
(1) with parallel mean curvature vector.
From (4.17), we have
n−2
1
n + nH 2 − √
(4.18)
n|H| sup |ϕ| − (2 − ) sup |ϕ|2 = 0.
p
n(n − 1)
Thus, we have sup |ϕ|2 = BH,p,n . Since n2 H 2 > S, we have H > 0. By the
n
result of Theorem
√ 1.4, we have (i) p = 1 and M is an open piece ofn H(r)-torus
n−1
1
2
S
(r) × S ( 1 − r ) with 0 < r < 1; or (ii) n = 2, p = 2 and M is an open
piece of Veronese surface in S 4 . This completes the proof of the Theorem 1.5. Acknowledgments. Project supported by NSF of Shaanxi Province(SJ08A31)and
NSF of Shaanxi Educational Committee(11JK0479).
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SHICHANG SHU, YANYAN LI
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Shichang Shu
Institute of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000 Shaanxi, P.R. China
E-mail address: shushichang@126.com
Yanyan Li
Institute of Mathematics and Information Science, Xianyang Normal University, Xianyang 712000 Shaanxi, P.R. China
E-mail address: muzilyy@163.com