complete submanifolds in euclidean spaces with parallel mean curvature vector

manuscripta math. 105, 353 – 366 (2001) © Springer-Verlag 2001

Qing-Ming Cheng· Kazuhiro Nonaka

Complete submanifolds in Euclidean spaceswith parallel mean curvature vector

Received: 28 November 2000 / Revised version: 7 May 2001

Abstract. In this paper, we prove thatn-dimensional complete and connected submanifoldswith parallel mean curvature vectorH in the(n+p)-dimensional Euclidean spaceEn+p arethe totally geodesic Euclidean spaceEn, the totally umbilical sphereSn(c)or the generalizedcylinderSn−1(c) × E1 if the second fundamental formh satisfies〈h〉2 ≤ n2|H |2/(n − 1).

1. Introduction

The purpose of this paper is to studyn-dimensional complete and connected sub-manifolds in an(n + p)-dimensional Euclidean spaceEn+p. In 1900, Liebmannproved that compact surfaces with constant Gauss curvature inE3 are the standardspheres. In 1953, Hopf proved that compact surfaces with constant mean curvatureand with genus zero inE3 are the standard spheres. Hopf’s result was extended tocomplete surfaces inE3 by Klotz–Osserman as following.

Klotz–Osserman’s Theorem (see [5]).Let M be a complete and connected sur-face with constant mean curvature inE3. If the Gauss curvature ofM is nonnegative,then M is the plane E2, the sphere S2(c) or the cylinder S1(c) × E1.

From the equation of Gauss, we know that the Gaussian curvature of a surfaceM in E3 is nonnegative if and only if〈h〉2 ≤ n2 |H |2 /(n− 1), n = 2, where〈h〉 isthe length of the second fundamental formh of M and|H | is the mean curvatureof M. It is natural to extend the result due to Klotz–Osserman to higher dimensionsand higher codimensions. In this paper, we shall prove the following result.

Main Theorem. Let M be an n-dimensional complete and connected submanifoldwith parallel mean curvature vector H in En+p, n ≥ 3. If the second fundamentalform h of M satisfies

〈h〉2 ≤ n2 |H |2n − 1

, (1.1)

Research partially Supported by the Grant-in-Aid for Scientific Research of the Ministry ofEducation, Science, Sports and Curvature, Japan.

Q.-M. Cheng: Department of Mathematics, Faculty of Science and Engineering, Saga Uni-versity, Saga 840-8502, Japan. e-mail: [email protected]

K. Nonaka: Graduate School of Science, Josai University, Sakado, Saitama 350-0295, Japan

Mathematics Subject Classification (2000): 53C42

354 Q.-M. Cheng, K. Nonaka

then M is the totaly geodesic Euclidean space En, the totally umbilical sphereSn(c) or the generalized cylinder Sn−1(c) × E1 in En+1.

Remark 1. In [9], Shen intended to prove the above theorem by making use of theresult of Motomiya [7]. But, since the result of Motomiya is wrong (see the Sect. 2),the proof of Shen about the above theorem is not valid. In the Sect. 4, we shall givea proof of the Main Theorem.

2. The maximum principles

In this section, we shall mention the maximum principles which play on an impor-tant role in the study of differential geometry on Riemannian manifolds. First ofall, we state the well known theorem which is called Hopf’s maximum principle asthe following:

Hopf’s maximum principle. Let M be a compact Riemannian manifold. If aC2-functionf onM satisfies∆f ≥ 0 or∆f ≤ 0, thenf is a constant function, where∆ denotes the Laplacian onM.

As a generalization of Hopf’s maximum principle, Omori [8] and Yau [10]proved a very important theorem, which is called the generalized maximum prin-ciple.

The generalized maximum principle. LetM be a complete and connected Rieman-nian manifold with Ricci curvature bounded from below. If aC2-functionf onM

is bounded from above, then for allε > 0, there exists a pointx ∈ M such that

supf − ε < f (x) , (2.1)

||gradf (x) || < ε , (2.2)

∆f (x) < ε . (2.3)

In [7], Motomiya intended to improve the generalized maximum principle dueto Omori and Yau as the following:

Motomiya’s wrong result. LetM be a complete and connected Riemannian mani-fold with Ricci curvature bounded from below. If aC2-functionf is bounded fromabove and has no maximum onM, then for allε > 0, there exists a pointx ∈ M

such that

supf − ε < f (x) < supf − ε/2 , (2.4)

||gradf (x) || < ε , (2.5)

∆f (x) < ε . (2.6)

But this result is wrong. In fact, Cheng and Wu gave the following counterexample in [2].

Complete submanifolds in Euclidean spaces with parallel mean curvature vector 355

Counter example of Motomiya’s result: LetM = E2, f (x, y) = − exp(cx)(c ≥ 2). Obviously,M is a complete and connected Riemannian manifold withRicci curvature zero andf is a smooth function bounded from above by zero andf has no maximum onM. So, if for ε > 0, there exists a pointq = (x, y) ∈ M

such that (2.4), (2.5) and (2.6) hold , then from (2.5) and (2.6), we would have

||gradf (q) || = c exp(cx) < ε ,

∆f (q) = −c2 exp(cx) < ε.

Hence,

−ε

c< − exp(cx) = f (q) . (2.7)

On the other hand, since supf = 0, from (2.4), we have

−ε < f (q) < −ε

2. (2.8)

Takingc ≥ 2, we find that (2.7) and (2.8) are contradictory.The Motomiya’s wrong result was used by Shen in [9] in order to prove the

Main Theorem in the Sect. 1. Hence, Shen’s proof is not valid. In the Sect. 4, wewill give a proof of the Main Theorem by making use of the generalized maximumprinciple due to Omori and Yau.

3. Definitions and fundamental equations

Let M be ann-dimensional submanifold in an(n + p)-dimensional RiemannianmanifoldN of constant curvaturek. We denote byg, g the Riemannian metric onN ,M, respectively and by∇, ∇ the Riemannian connection onN ,M, with respectto g,g, respectively. Then the formula of Gauss (3.1) and the formula of Weingarten(3.2) hold.

∇XY = ∇XY + h (X, Y ) , (3.1)

∇Xξ = −Aξ(X) + ∇⊥Xξ, (3.2)

whereX, Y are any tangent vector fields onM andξ is any normal vector field onM. We callh the second fundamental form ofM andAξ the second fundamentaltensor ofM with respect toξ and∇⊥ the normal connection ofM. A normal vectorfield ξ of M is said to be parallel in the normal bundle, or simply, parallel if wehave∇⊥ ξ = 0 identically. From the formula of Gauss (3.1) and the formula ofWeingarten (3.2), we have immediately,

g (h (X, Y ) , ξ) = g(Aξ (X) , Y

). (3.3)


Let E1, E2, · · · , En be orthonormal tangent vectors at a pointx ∈ M

andξ1, ξ2, · · · , ξp be orthonormal normal vectors atx ∈ M, then

H = 1

n

p∑α=1

(traceAα) ξα (3.4)

is a normal vector atx, which is independent of the choice of the othonormal basisE1, · · · , En, ξ1, · · · , ξp, whereAα = Aξα, traceAα = ∑n

i=1 g (Aα (Ei) , Ei).The vector fieldH is called mean curvature vector field ofM. Obviously, if themean curvature vector fieldH of M is parallel in the normal bundle, then the meancurvature|H | of M is constant.

We denote byXi = ∂/∂xi (i = 1,2, · · · , n) the natural frame of submanifoldM and byK, K the curvature tensor ofN , M, respectively. Letξ1, ξ2, · · · , ξpbe orthonormal normal vector fields ofM andhα be the corresponding secondfundamental form, that is,

h (X, Y ) = hα (X, Y ) ξα.

Now, since the ambient spaceN is of constat curvaturek, then we have

K(X, Y )Z = k{g(Z, Y )X − g(Z, X)Y }, (3.5)

for any tangent vector fieldsX, Y , Z on N . Thus, from the formula of Gauss(3.1), the formula of Weingarten (3.2) and (3.3), we have the equation of Gauss asfollowing.

K(X, Y,Z,W) = k{g(X,W)g(Y, Z) − g(X,Z)g(Y,W)}+ g (h (X,W) , h (Y, Z)) − g (h (X,Z) , h (Y,W)) ,

(3.6)

whereK(X, Y,Z,W) = g (K (X, Y )Z,W) andX, Y, Z, W are any tangent vec-tor fields onM. In local components,

Kkji% = k(gk%gji − gkigj%

)+p∑

α=1

(hk%αhji

α − hkiαhj%

α). (3.7)

Moreover, from (3.5), we see that the normal components ofK(X, Y )Z vanish,then we have,

(∇Xhα)(Y, Z)ξα + hα(Y, Z)∇⊥

Xξα = (∇Y hα)(X,Z)ξα + hα(X,Z)∇⊥

Y ξα. (3.8)

We define the covariant derivative, denoted by∇, to be

(∇Xh)(Y, Z) = (∇Xhα)(Y, Z)ξα + hα(Y, Z)∇⊥

Xξα. (3.9)

In general, for example, for a tensor fields, sayT , of type (0.3) with values in thenormal bundleT ⊥(M) of M, we define

(∇WT )(X, Y,Z) = (∇WT α)(X, Y,Z)ξα + T α(X, Y,Z)∇⊥Wξα.


Remark 2. We call the connection∇ on M the connection of van der WaerdenBortolotti.

Then We have the equation of Codazzi as following,

(∇Xh)(Y, Z) = (∇Y h)(X,Z), (3.10)

in local component,

∇khjiα = ∇j hki

α. (3.11)

We denote byKN the curvature tensor of the normal connection∇⊥ onM, that is,

KN(X, Y )ξ = ∇⊥X∇⊥

Y ξ − ∇⊥Y ∇⊥

Xξ − ∇⊥[X,Y ]ξ,

then, from the formula of Gauss (3.1), the formula of Weingarten (3.2) and (3.5),we have the equation of Ricci as following,

KN(X, Y, ξ, η ) = g([Aξ ,Aη

](X), Y

), (3.12)

where

KN(X, Y, ξ, η ) = g(KN(X, Y )ξ, η

)[Aξ ,Aη

] = AξAη − AηAξ .

If we setKjiβαξα = KN(Xj ,Xi)ξβ , then we shall give the local expression of the

equation of Ricci as following,

Kjiβα = hjt

αhitβ − hit

αhjtβ, (3.13)

wherehjkα

= gkihjiα. In this paper, we shall use the Einstein convention, that

is, repeated indices, with one upper index and one lower index, denote summationover its range.

We may define the second covariant derivative∇ ∇h of the second fundamentalform h by

(∇W∇Xh)(Y, Z) = ∇⊥W(∇Xh(Y,Z)) − (∇Xh)(∇WY,Z)

− (∇Xh)(Y,∇WZ) − (∇∇WXh)(Y, Z),(3.14)

for any tangent vector fieldsX, Y, Z, W onM. By a direct computation, we find

(∇W∇Xh)(Y, Z) − (∇X∇Wh)(Y, Z)

= h(K(X,W)Y,Z) + h(Y,K(X,W)Z) − KN(X,W)h(Y,Z),(3.15)

in local component,

∇%∇khjiα − ∇k∇%hji

α = Kk%jthti

α + Kk%ithtj

α − Kk%βαhji

β . (3.16)

If the mean curvature|H | of M is nowhere zero, then we may choose or-thonormal normal vector fieldsξ1, ξ2, · · · , ξp such thatH = |H |ξ1. Obviously,if we chooseξ1, ξ2, · · · , ξp as above, then we have traceA1 = n|H |, traceAα =


0 (α = 2, 3, · · · , p). In the following, for any tangent vector fieldsX, Y onM,we set

U(X, Y ) = {h1(X, Y ) − |H | g(X, Y )}ξ1, (3.17)

T (X, Y ) =p∑

α=2

hα(X, Y )ξα, (3.18)

in local component,

Uji = hji1 − |H | gji, (3.19)

Tjiα = hji

α (α = 2, · · · , p). (3.20)

From (3.19) and (3.20), we have immediately the following,

|U |2 = traceA21 − n |H |2 , (3.21)

|T |2 =p∑

α=2

traceA2α, (3.22)

〈h〉2 = |U |2 + |T |2 + n|H |2. (3.23)

From these formulas, we know that|U |2, |T |2 is independent of the choice of theorthonormal basisE1, · · · , En. Then, functions|U |2, |T |2 are defined globally onM if H is nowhere zero. Moreover, From (3.16) and (3.22), we have the followinglemma.

Lemma 1. If the nonzero mean curvature vector H of submanifold M is parallel inthe normal bundle, then we have

1

2∆|T |2 =

P∑α=2

(Kjthit

αhjiα − Kkjith

ktαh

jiα)

+p∑

α,β=2

Kjiβαht

jβh

itα + |∇T |2,

(3.24)

where Kji = Ktjit denotes the component of Ricci tensor of M, Kj

t = gtiKji ,hji

α = gjkgi%hk%α , ∆ denotes the Laplacian on M.

Proof. Let ξ1, ξ2, · · · , ξp be orthonormal normal vector fields onM such thatH = |H |ξ1 . Then, from (3.22), we have

1

2∆|T |2 =

p∑α=2

gk%(∇k∇%hjiα)hji

α +p∑

α=2

gk%(∇khjiα)(∇%h

jiα), (3.25)

by virtue of traceA2α = hi

jαhj

iα

.


If we set ∇⊥Xk

ξα = %kαβξβ , then we find immediately%kαβ = −%kβ

α byg(ξα, ξβ) = δαβ . SinceH = |H |ξ1 is parallel in the normal bundle, we have%kα

1 = −%k1α = 0. Hence, from the definition of∇, we obtain

1

2∆|T |2 =

p∑α=2


α +p∑

α=2


jiα)

−p∑

α,β=2

gk%∇%(hjiβ%kβ

α)hjiα −

p∑α,β=2

gk%(∇%hji

α)hjiβ%kβ

α

−p∑

α,β,γ=2

gk%hjiβhji

α%kβγ %%γ

α −p∑

α,β,γ=2

gk%hjiβhji

γ %kβα%%γ

α

−p∑

α,β=2

gk%(∇khjiβ)%%β

αhjiα −

p∑α,β=2

gk%(∇khjiα)hji

β%%βα.

(3.26)

Moreover, from%kαβ = −%kβ

α , we find

p∑α,β=2

gk%(∇khjiβ%kβ

α)hjiα +

p∑α,β=2

gk%(∇%hji

α)hjiβ%kβ

α = 0,

p∑α,β,γ=2

gk%hjiβhji

α%kβγ %%γ

α +p∑

α,β,γ=2

gk%hjiβhji

γ %kβα%%γ

α = 0,

p∑α,β=2

gk%(∇khjiβ)%%β

αhjiα +

p∑α,β=2

gk%(∇khjiα)hji

β%%βα = 0.

Thus, we infer

1

2∆|T |2 =

p∑α=2


α +p∑

α=2


jiα) . (3.27)

By using the equation of Codazzi (3.11) and (3.16), we know that formula (3.24)holds. This finished the proof of Lemma 1.

4. Proof of Main Theorem

Proof of Main Theorem. Since the mean curvature vectorH of M is parallel in thenormal bundle,|H | is constant. We now consider the case|H | = 0 and|H | �= 0separately.

Case(1): |H | = 0 on M.From the assumption of the Main Theorem, we have

〈h〉2 ≤ n2|H |2n − 1

= 0 ,


that is,M is ann-dimensional totally geodesic submanifoldEn inEn+p. Therefore,the Main Theorem holds.

Case(2): |H | �= 0 on M.Let ξ1, ξ2, · · · , ξp be an orthonormal normal vector fields onM such that

H = |H |ξ1. Since the mean curvature vectorH of M is parallel in the normalbundle, from the equation of Ricci (3.12), it is easy to prove that,

A1Aα = AαA1

for eachα. Then, substituting the equation of Gauss (3.7) and the equation of Ricci(3.13) into (3.24), we find

1

2∆|T |2 =

p∑α=2

(hk

k1hj

t1hit

αhjiα − hk

t1hj

k1hit

αhjiα

− hkt1hji

1hktαh

jiα + hki

1hjt1hkt

αhji

α

)+

p∑α,β=2

{− hktβhj

kβhit

αhjiα − (

hktβhji

β − hkiβhjt

β)hkt

αhji

α

+ (hjtαhi

tβ − hit

αhjtβ)hk

jβh

ikα

}+ |∇T |2

=p∑

α=2

{traceA1traceA1A2α − (traceA1Aα)

2}

−p∑

α,β=2

{(hk

jαhi

kβ − hk

jβhi

kα

)(ht

iαhj

tβ

− htiβhj

tα

)+ (

hjiαhi

jβhk

tαht

kβ

)}+ |∇T |2

=p∑

α=2

{traceA1traceA1A2α − (traceA1Aα)

2}

−p∑

α,β=2

{trace[(AαAβ − AβAα)t (AαAβ − AβAα)]

+ (traceAαAβ)2} + |∇T |2.

By settingN(Aα) = traceAαtAα andSαβ = traceAαAβ , we have

1

2∆|T |2 =

p∑α=2

{traceA1traceA1A2α − (traceA1Aα)}

−p∑

α,β=2

{N(AαAβ − AβAα) + S2αβ} + |∇T |2.

(4.1)

By virtue of

A1Aα = AαA1


for eachα, then, for a fixedα, A1 andAα can be simultaneously diagonalized.We denote byρi , ρα

i (i = 1, 2, · · · , n) the principal curvatures of the secondfundamental tensorsA1, Aα, respectively. Then we obtain

traceA1traceA1A2α − (traceA1Aα)

2

=n∑

i,j=1

ρiρj (ραj )

2 −n∑

i,j=1

ρiρjραi ρ

αj

= 1

2

n∑i,j=1

ρiρj (ραj )

2 + 1

2

n∑i,j=1

ρjρi(ραi )

2 −n∑

i.j=1

ρiρjραi ρ

αj

= 1

2

n∑i,j=1

ρiρj (ραi − ρα

j )2.

(4.2)

Condition (1.1) and (3.23) imply that(n∑

i=1

ρi

)2

≥ (n − 1)n∑

i=1

(ρi)2 + (n − 1)|T |2. (4.3)

The following lemma is an algebraic result.

Lemma 2 (Chen [1, p. 55]).Let a1, a2, · · · , an, b be n+1 (n > 1) real numberssatisfying the following inequality,(

n∑i=1

ai

)2

≥ (n − 1)n∑

i=1

a2i + b (resp. >).

Then we have

2aiaj ≥ b

n − 1(resp. >)

for any distinct i and j .

By applying the Lemma 2 to (4.3), we find

ρiρj ≥ |T |22

(4.4)

for any distincti andj . From this inequality and (4.2), we obtain, for anyα,


2

≥ 1

4|T |2

n∑i,j=1

(ραi − ρα

j )2

= 1

2|T |2{

n∑i,j=1

(ραi )

2 −n∑

i,j=1

ραi ρ

αj }

= n

2|T |2

n∑i=1

(ραi )

2 − 1

2|T |2

(n∑

i=1

ραi

)2

= n

2|T |2traceA2

α − 1

2|T |2(traceAα)

2.


From traceAα = 0 for α ≥ 2, we have

p∑α=2

(traceA1traceA1A

2α − (traceA1Aα)

2) ≥ n

2|T |4 (4.5)

by (3.22).On the other hand, from the following result, we have

p∑α,β=2

{N(AαAβ − AβAα) + S2αβ} ≤ 3

2|T |4. (4.6)

Lemma 3 (Li, A. M. and Li, J. M. [6]). Let A1, A2, · · · , Ap be symmetric ( n×n )-matrices. Then we have

p∑α,β=1

(N(AαAβ − AβAα) + S2

αβ

) ≤ 3

2

(p∑

α=1

N(Aα)

)2

and the equality holds if and only if one of the following conditions holds:

(i) A1 = A2 = · · · = Ap = 0,(ii) only two of A1, A2, · · · , Ap are different from zero.

Moreover, assuming A1 �= 0, A2 �= 0, then S1 = S2 and there exists an orthogonal( n × n )-matrix T such that

t T A1 T =√

S1

2

1 0 0 · · · 00 −1 0 · · · 00 0 0 · · · 0...

....... . .

...

0 0 0 · · · 0

, tT A2 T =√

S1

2

0 1 0 · · · 01 0 0 · · · 00 0 0 · · · 0.......... . .

...

0 0 0 · · · 0

,

where Sα = N(Aα).

By virtue of (4.5), (4.6), we have

1

2∆|T |2 ≥ n − 3

2|T |4. (4.7)

Condition (1.1) implies that|T | is bounded from above byn2|H |2/(n− 1) and bythe Lemma 2, we can prove that the sectional curvatures ofM are nonnegative.Hence, we can apply the generalized maximum principle due to Omori and Yau tofunction|T |2. Then, there exists a sequence{xk} ⊂ M such that

limk→∞ |T |2(xk) = sup|T |2, (4.8)

lim supk→∞

∆|T |2(xk) ≤ 0. (4.9)


From (4.7), (4.8) and (4.9), we see that

0 ≥ lim supk→∞

∆|T |2(xk) ≥ (n − 3)(

limk→∞ |T |2(xk)

)2 = (n − 3)(sup|T |2)2 ≥ 0,

that is,

(n − 3) sup|T |2 = 0. (4.10)

Hence, ifn ≥ 4, then we have|T |2 = 0 onM.Now we considern = 3. From (4.7), we see that∆|T |2 ≥ 0. From this

inequality and (4.9), we have

lim supk→∞

∆|T |2(xk) = 0. (4.11)

Hence, the above inequalities become equalities when we take the limitation. From(1.1), we know that{hji

α(xk)} is a bounded sequence for anyi, j andα. Therefore,we can assume that limk→∞ hji

α(xk) exists for anyi ,j andα, if necessary, we cantake subsequence of{xk}. Then we set

Aα = limk→∞Aα(xk) =

(limk→∞hj

iα(xk)

).

From (4.5) and (4.6), we have

p∑α=2

(traceA1traceA1A

2α − (traceA1Aα)

2)= 3

2sup|T |2

p∑α=2

traceA2α,

(4.12)

p∑α,β=2

(N(AαAβ − AβAα) + (traceAαAβ)

2) = 3

2

(p∑

α=2

N(Aα)

)2

. (4.13)

Applying the Lemma 3 to (4.13), we see that

(i) A2 = A3 = · · · = Ap = 0 or(ii) A2 �= 0, A3 �= 0, A4 = · · · = Ap = 0 and there exists an orthogonal matrix

T such that

t T A2T =√

S2

2

1 0 00 −1 00 0 0

, tT A3T =√

S2

2

0 1 01 0 00 0 0

, (4.14)

whereS2 = traceA22.


If (i) holds, we have sup|T |2 = ∑pα=2 traceA2

α = 0, that is,|T |2 = 0 onM.Now we assume that (ii) holds. For eachα, by virtue of


2 ≥ 3

2|T |2traceA2

α,

(4.12) implies that

n∑i,j=1

ρiρj (ρiα − ρj

α)2 = sup|T |22

n∑i,j=1

(ρiα − ρj

α)2, (4.15)

whereρi = limk→∞ ρi(xk) , ρiα = limk→∞ ρi

α(xk). Moreover, from (4.14), wehaveρi

2 �= ρj2 for any distincti andj . Then, from (4.4) and (4.15), we find

ρiρj = 1

2sup|T |2 (4.16)

for distincti andj . It can be easily verified that∑i �=j

ρiρj = 6|H |2 − |U |2 (4.17)

by (3.21). From (4.16) and (4.17) we have immediately

limk→∞ |U |2(xk) + 3 sup|T |2 = 6|H |2. (4.18)

On the other hand, by the Lemma 2, (4.16) implies that(n∑

i=1

ρi

)2

= 2n∑

i=1

(ρi)2 + 2 sup|T |2,

that is,

limk→∞ 〈h〉2 (xk) = 9

2|H |2. (4.19)

Then, from (3.23) and (4.19), we have

limk→∞ |U |2(xk) + sup|T |2 = 3

2|H |2. (4.20)

Solving the equations (4.18) and (4.20), we find

limk→∞ |U |2(xk) = −3

4|H |2 < 0. (4.21)

This is contradictory to that|U |2 is nonnegative. That is, (ii) does not hold. Thus,|T |2 = 0 for n ≥ 3 onM. Hence, we have traceA2

α = 0(α = 2, · · · , p). Namely,we obtain

A2 = A3 = · · · = Ap = 0.


Hence, if we denote byN1 the normal subbundle spanned byξ2, ξ3, · · · , ξp of thenormal bundleT ⊥(M) of M, thenM is totally geodesic with respect toN1. Sincethe mean curvature vectorH = |H |ξ1 of M is parallel in the normal bundle, setting

∇⊥Xξα = ωβ

α(X)ξβ

for any tangent vectorX onM, we find

∇⊥Xξα =

p∑β=2

ωβα(X)ξβ ∈ N1,

for α = 2, 3, · · · , p, that is, the normal subbundleN1 is invariant under paralleltranslation with respect to the normal connection∇⊥ of M. From the theorem 1 in[11], we conclude thatM lies in an(n+1)-dimensional totally geodesic submanifoldEn+1 of En+p. We denote by|H ′| the mean curvature ofM in En+1. SinceEn+1

is totally geodesic inEn+p, we have|H | = |H ′|, that is, the mean curvature|H ′|of M in En+1 is constant. Thus, M is a complete and connected hypersurface withconstant mean curvature|H | in En+1 and its sectional curvatures are nonnegative.From the following proposition 1, we know thatM is the hypersphereSn(c) or ageneralized cylinderSn−k(c)×Ek , (1 ≤ k ≤ n−1), inEn+1. SinceSn−k(c)×Ek

for 1 < k ≤ n − 1 satisfiesS >n2|H |2n − 1

, we infer that Main Theorem is true.

Proposition 1 (Cheng, S.Y. andYau, S.T. [3]).Let M be a complete and connectedhypersurface with nonnegative sectional curvatures in En+1. If the mean curvature|H | of M is constant, then M is the hyperplane En, the hypersphere Sn(c) or ageneralized cylinder Sn−k(c) × Ek , (1 ≤ k ≤ n − 1).

Therefore, the Main Theorem is proved completely.

References

[1] Chen, B.Y.: Geometry of submanifolds. New York: Marcel Dekker, INC., 1973[2] Cheng, Q.M. and Wu, B.Q.: The generalized maximum principle and conformally flat

spaces, Northeastern Math. J.8, 54–56 (1992)[3] Cheng, S.Y. andYau, S.T.: Differential equations on Riemannian submanifolds and their

geometric applications. Comm. Pure Appl. Math.28, 333–354 (1975)[4] Hasanis, Th.: Conformally flat spaces and a piching problem on Ricci tensor. Proc.

Amer. Math. Soc.86, 312–215 (1982)[5] Klotz, T. and Osserman, R.: On complete surfaces inE3 with constant mean curvature.

Comm. Math. Helv.41, 313–318 (1966–67)[6] Li, A.M. and Li, J.M.: An intrinsic rigidity theorem for minimal submanifolds. Arch.

Math.58, 582–594 (1992)[7] Motomiya, K.: On functions which satisfy some differential inequalities on Riemannian

manifolds. Nagoya Math. J.,81, 57–72 (1981)[8] Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Japan19,

205–214 (1967)


[9] Shen, B.Y.: Complete submanifolds inEn+p with parallel mean curvature. Chin. Ann.of Math. Ser. B6, 345–350 (1985)

[10] Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl.Math.28, 201–228 (1975)

[11] Yau, S.T.: Submanifolds with constant mean curvature I, II.Amer. J. Math.96, 346–366(1974);97, 76–100 (1975)

complete submanifolds in euclidean spaces with parallel mean curvature vector

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