A Characterization Of Closed Hypersurfaces in 6-Dimensional Sphere

©BEYKENT UNIVERSITY

A CHARACTERIZATION OF CLOSED

HYPERSURFACES IN 6-DIMENSIONAL

SPHERE

GÜLŞEN YILMAZ

Beykent University, Faculty of Science and Letters, Department of Mathematics &Computing,

Ayazağa, Şişli, Istanbul, TURKEY. E-mail: yilmazg@beykent.edu.tr

Received: 6 November 2007, Accepted: 16 January 2008

ABSTRACT

In this work we obtain a characterization of the closed hypersurfaces with constant mean curvature and having constant length of second fundamental form in 6- dimensional sphere.

Mathematics Subject Classifications (2000): Primary 53C40; Secondary

53C15.

Key Words: Hypersurfaces; mean curvature; six-dimensional sphere.

6 - BOYUTLU KÜREN

İ

N KAPALI

H

İ

PERYÜZEYLER

İ

N

İ

N B

İ

R

KARAKTER

İ

ZASYONU

ÖZET

Bu makalede, 6-boyutlu kürenin sabit ortalama eğriliğe ve sabit uzunluklu ikinci temel forma sahip kapalı hiperyüzeylerinin bir karakteristiği elde edildi .

Anahtar Kelimeler : Hiperyüzeyler, ortalama eğrilik, 6-boyutlu küre.

1. INTRODUCTION

Let M be a closed hypersurface of constant mean curvature with second fundamental form h in S 6( 1 ) . The eigenvalues Ai ,1 < i < 5, of the second

fundamental form h are the principal curvature functions over M. Hypersurfaces of the sphere have been studied by many authors, see [1-12]. Denote by H the length of the mean curvature vector field and by Q the length of the second fundamental form of M. On the other hand, it is important to characterize those hypersurfaces immersed as the spheres in

fundamental theorem of hypersurfaces, [9].

When M is minimal, J. Simon ([10]) obtained a pinching constant 5 of Q and Chern-do Carmo-Kobayashi ([3]) showed that it is sharp and characterized all the hypersurfaces in the sphere S 6( 1 ) with Q = 5. Recently the sharp results

in this matter were established by H. Alencar-M.do Carmo ([1]) and by H.W. Xu ([12]) and by Z.H.Hou ([4]).

In the present paper, we study closed hypersurfaces of constant mean curvature in S 6( 1 ) and give a piching constant. In fact, we will prove the following

theorem, see([4]).

Theorem . Let M be a hypersurface with constant mean curvature in S 6( 1 )

with the constant length of the second fundamental form. Then,

i) If Q < 4 , M is locally a piece of small hypersphere S 5(r ) of radius

r =

V

5

5 +

Q

ii) If Q = 4 , M is locally a piece of either S5 (r ') or S !(rj)x S 4(r2)

, , Vs V3 V6

where r = , r = , r = .

3

1

3

2

3

2. PRELIMINARIES

Let M be a hypersurface isometrically immersed in S 6( 1 ) . We denote by V

(resp.

V

)

the covariant differentiation on M (resp.

S

6

(1)).

Then the second fundamental form (the shape operator) h of the immersion is given by

and it satisfies h ( X, Y ) = h(Y, X ) . Let us choose a local orthonormal frame

e

1

,e

2

,...,e

6 in

S

6

(1)

in such a way that restricted to M ,

e

15

...,e

5 are tangent to M (and consequently e6 is normal to M ). Let B be the set of all

such frames in S 6 (1) . With respect to the frame field of S6 (1) chosen above,

let G)1, G)2,..., G)6 be the field of dual frames. Then the structural equations of

S 6( 1 ) are given by A-1 6 = 7 C0AC A C0CB A C0B (2.1) C =1

where C0AB 's are the connection forms on S 6( 1 ) and C0AB + C0BA — 0 . The

Ricci tensor and the scalar curvature of S 6 (1) are given respectively by

6

RICAB = RICBA = X RACBC = 6$AB , ( 2 2)

C=1

S

= Z

RICAA = £ RACAC =

30,

(2.3)

A-1 A,C -1

where R is the Riemannian curvature tensor on S6(1) and its entries are

given by

R

MCD

— S

AC

d

BD

—

¿>

AD

S

BC . If we restrict those formulas to M, we have the structure equations of M as follows;

0

-

d® 6 = £

= > ® 6 A®i i-1

and from Cartan's lemma we have

5

®i 6 = X hv®j , (2.4)

j=1

where

h

j

.

= hji . From now on we assume that

1

<

i, j,k,...

<

5

and write

= = - ®jl, (2.5)

d

%

-

1

Z

R

and

Rj k i = J + hikhjl - huhjk

1 1 J J (2.7)

= 8*8ji ~ S J I 4 + hikhn - hahjk ,

where R is the Riemannian curvature tensor on the hypersurface M. Then, the second fundamental form h can be written as

h( X, Y)

= X

h^ (X )®J. (Y )e

6

.

J-i

The covariant derivative

V

h

of h , with components

h

ijk , is given by

V

h

= ^

h

^MWM

ijk ' •'ijk^i^j^k i, J ,k and Z ^ = d hj + Z hrJ®ri + Z hir®rJ . (2.8) k r r

Then we have

h

ijk = hikJ for any i, j and k=1,2,3,4,5, because

S

6

(1)

is of constant curvature 1.

Indeed, by exterior differentiating (2.4), we get

d

o>.

_{i 6 i}6

= V

dh.

A®. + V

_{j J im}

h.®

_{im mj —'} mi

Ao,.

jm

or

AW,—

1

V

R

i 6 / j ^i m m 6 ^ / j i

We also have from (2.4) and (2.6)

> iml 6 m l d® r 6 = " Z h mj mi j W A W . Therefore, JU - V * . „ A „ ^h r^r j . (2.9) Z d hj A = " Z hrj®ri A - Z hir A J Jr Jr So, we get Zhi j k®k A®J = o, kJ

therefore, hijk's are symmetric in all indices. Exterior differentiating the equation (2.8) and defining hijk l by

Z hVkl®l = dhijk + Z Kk^r i + Z hirk®rj + Z hjr®rk (2.10)

l r r r we have

E

( j -

T

E

hrRr f i l - 1

E

KjRnkl H A = 0 (2.11) kl 2 r 2 r

and from this we obtain

hijkl ~ hijlk ~ E hirRrjkl + E hrjRrikl . (2.12) r r

The square of the second fundamental form and the mean curvature of M are given by

Q = E hi , H = 5 E hi . (2.13)

i , j 5 i=1

Now, it is easily seen that for the hypersurface M in S 6 (1)

Q = H2 - S + 2 0 , (2.14)

where S is the scalar curvature and H is the mean curvature of M. Now, let us define the laplacian Ah of the second fundamental form h by

(Ah)

y

.

= A

hij

= £

h

_{ijkk • (2.15)}

k From (2.12) and (2.15) we obtain

E hijkk = E hkijk

k k

a n d s o Ahy = E hkijk .

k Then, from (2.11) we find

A hj

E hkikj + E ( E hriRrkjk + ^ hkrRrijk ) k k r r

=

( 5

-

Q ) h j

+

H(X

h

n

hj

}

-S

j).

r From (2.13) and (2.14), we have

2

A

Q

=

£ h jAh j + X j

=

( 5 - Q ) Q + HF -H

2

+ £

j

2 ij ijk ijk

where F - trace(h

i

).

Thus, it follows that

^

hij

A

hij

=

5Q

+

5HF

-

25H

2

-

Q

2

.

i,j

3. PROOF OF THE THEOREM

(2.16)

(2.17)

M. Okumura ([8]) established the following lemma.

Lemma 3.1. Let {t1, t2, t3, t4, t5} be a set of real numbers satisfying

±

t

t

=

0,

£

t

t 2 2

,

i—1 i—1

where 0 . Then we have

^ t i < — ^ a3 (3.1)

2 ^

" i f i

"

2 ^

( )

and equalities hold if and only if at least four of the ti's are equal to each others.

Suppose that \1, \2, \3, \4, \5 are the principal curvatures of M. Then we

have

5H

= ¿

Xi,

Q

= ¿ \

2

, F

= £

X i3. (3.2)

i—1 i—1 i— 1 Denote

Q

=

Q-5H

2

, F

=

F -3HQ + 10H3

,

\ = \

i

- H , (1

<

i

<

5).

Then the equations in (3.2) changes into

£ x i

=

0, Q

= £

X i2,

F

= £

\

i3. (3.3)

i=1 i=1 i=1

By applying Lemma 3.1. to F in (3.3) , we have

3

A ¡T _ „ „ „

3

F

>

Q

-J

Q

&

F

>

3 H Q

-

1 0 H

3

2

V

5 2

V

5

Putting this into (2.18) , we obtain

£

hj

A

hj

>

Q

|

5

-

(Q

-

5H

2

)

-

3 h

J

^ Q

J

.

Now, lets consider the quadratic form q(r\, = v\ — 32 Tp — ^2

By applying the following transformation

we get q(r\, ^) = 5 ( x2 — y2), where x2 + y2 = 2 = Q .

Now taking ^ = ^Q) and T| = A/5H in q(r\, ^) , we have

3

2'

q(

r

\, ^) = 5H

2

- - Q

Substituting this into (3.4) we find that

£ h.jAhj > Q j5 - 5 Q + 5 x2 } > Q(5 - 4 Q ) . (3.5)

Therefore we have

2

A

Q

= E

j

+ E

h

A

h

>

Q

(

5 - 5 (3.6)

2 i, j,k i, j 4

Since Q is constant, the left hand side of (3.6) is zero. Now assume that

Q < 4 . Then we have

Q

_{5 - 4 Q}

= 0, hjk = 0

for

1 < i, j,k < 5.

(3.7)

For Q < 4 , we have Q = 0 that means M is totally umbilical and so it is locally a piece of hypersphere S5 ( J ) , where j =

V

5

5 +

Q

Now suppose that Q = 4 . Then all the inequalities from (3.4) to (3.6) become equalities. Therefore from the Lemma 3.1. at least four of \i 's are equal to

each others. If \j = \2 = \3 = \4 = \5 , then M is totally umbilical and so

it is locally a piece of hypersphere S 5( J ) , where j = . When M is not

totally umbilical, there are exactly four of \i 's that are equal to one another.

By the similar arguments as those developed by Chern-do Carmo-Kobayashi ([3]) show that M is locally a piece of S!(Ji)x S4(J2) in

S

6

(1).

To determine the radius we use the same arguments in ([7]) and find

/ \ / \ / \2 / \2

H = -

!

5

We may calculate easily that

/ \ ( \ ( \ 2 ( \

22 _{H —}

4

_{, Q}

₌

+ 4

, J1

5

, J2

, Q

=

rı

and equality holds if and only if

+ 4 - i > 4

= 2 . Therefore we find that

S 4

6

—

, r

2

= .

3 3

Thus, we proved the theorem.

If Q=const. and M is a minimal hypersurface, then we obtain from (2.16) and (2.17) that

X J HV

h|2 = Q(Q-5).

ijk (3.8)

In the case Q = 5 , h will be covariantly constant on M.

REFERENCES

[1] H. Alencar and M.P.do Carmo, Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc., 120(1994), 1223-1229.

[2] Chang, S., On closed hypersurfaces of constant scalar curvatures and mean curvatures in Sn + 1, Pacific Jour.of Math., Vol 165(1994), 67-76.

[3] Chern, S.S., Carmo, M. Do and Kobayashi, S., Minimal submanifolds of the sphere with second fundamental form of constant length, Func. Anal. And Related Fields, Ed. F. Browder, NewYork, Springer (1970), 59-75.

[4] Z.H. Hou, Hypersurfaces in a sphere with constant mean curvature, Proc. Amer. Math. Soc., 125(1997), 1193-1196.

[5] Itoh, T. and Nakagawa, H., On certain hypersurfaces in a real space form, Tohoku Math. Jour., 25(1973), 445-450.

[6] Lawson, H. B., Local rigidity theorems for minimal hypersurfaces, Ann. of Math., Vol. 89(1969), 187-191.

[7] K. Nomizu and B. Smyth, A formula of Simon's type and hypersurfaces of constant mean curvature, J. Diff. Geom. 3(1969), 367-378.

[8] M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96(1974), 207-213.

[9] Sasaki, S., A proof of the fundamental theorem of hypersurfaces in a space form, Tensor, N.S., 24(1972), 363-373.

[10] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88(1968), 62-105.

[11] Tanno, S. and Takahashi, T., Some hypersurfaces of a sphere, Tohoku Math. J., 22(1970), 212-219.