ON RULED REAL HYPERSURFACES IN A COMPLEX SPACE FORM 11

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ON RULED REAL HYPERSURFACES IN A COMPLEX SPACE FORM 11

Seong-Soo Ahn

,

Sung-BaiJ

‘

^{Lee* and}Young-Jin Suht

Dedicaled 10 Professor Younki Chae on his sixlielh birthday

30. Introduction

A complex n( 즈2)-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called a complex space form

,

which is denoted by Mn(c). A complete and simply connectcd complex space form is a complex projective space PnC

,

a complex Euclidean space cn or a complex hyperbolic space HnC

,

according as c> 0

,

c

=

0 or c

<

O. The induced almost contact metric structure of real hypersu따.ces of Mn(c) is

denoted by

(rþ

, E" T/, g)

Now

,

there exist many studies about real hypersurfaces of A1n(c). One of the first researches is the c1assifìcation of homogen∞ us real hypersurfaces of a complex projective space PnC by Takagi [12]

,

who showed that

these hypersurfaces of PnC could be divided into six types which are said to be of type A1

,

A2

,

B

,

C

,

D

,

and E

,

and in [2] Cecil-Ryan and [4] Kimura proved that they were realized as the tubes of constant radius over com pact Hermitian symmetric spaιes of rank 1 or rank 2 if the structure vector fìeld E, is principal. Also Berndt

[1]

showed recently that all real hypersurfaces with constant principal curvatures of a complex hyperbolic space HnC are realized as the tubes of constant radius over certain submanifolds if E, is principal

Received March 23, 1992

*

The present studies were supported by the basic science research institute program, Korea rninistry of Education, 1991-114

t Partially supported by TGRιKOSEF

301

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302 Seong-Soo Ahn, Sung-Baik Lee and Young-Jin Suh

On the other hand

,

for some characterizations of real hypersurfaces of type A j or _{A 2}in PnC or

Ao,

Aj

,

A2 in HnC Okumura[10]

,

Maeda[8]

,

Montiel and Romero [9] gave the condition that the second fundamental tensor A of M satisfies

(1) (Y'xA)Y=

힘 (X)4>Y

^- g( 4>X , Y)Ç},

(IJ) (A4> - 4>A)X = 0 for any tangent vector fields X and Y on M

Obviously the fact that ~ is principal can be implied by the condition (1) or (11). But until now for real hypersurfaces with the structure vec tor field ~ is not princi pal it has not been well known to us. From this point of view a ruled real hypersurface M of Pn C is constructed and some properties are investigated by Kimura[5], Kimura and Maeda[6]. Also the present authors[12] constucted an example of rninimal ruled real hyper surfaces of HnC and gave a characterization of ruled real hypersurfaces of Mn(c)

,

c:;60

Let us deβne a distribution To ^byTo(x) = {XETx(M)IX~~(x)} for any point x in M. Then it is seen in [5] that the properties (1) and (11) hold along the distribution To .

The purpose of this paper is also to give some sufficient conditions for real hypersurfaces of Mn(c)

,

c:;60 to be ruled real hypersurfaces as follows

Theorem A. Let M be a connecled real hypersurfaces of Mn(c)

,

c￥ 0， aηd n으 3. [f it satisfies

(1IJ) (Y'xA)Y =

숨(4)X，

^Y)ç

for anν νector fields X and Y in To

,

then M is /ocallν congruent to a ruled real hypersurface.

Theorem B. Let M be a connecled rea/ hypersurface of Mn(c)，c￥o and n2':3. [f it satisfies

(1V) (A4> -

4>A

)X = 0

for an ν veclor 껴eld X in T o and if the structure vector ^βeld

ç

^엉^not^pηη

cipal

,

then M is local/y congruent to a ru/ed real hypersurface

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'0ie will recall some fundamental properties and structure equations for real hypersurfaces in Mn(c) ，혜o in 31. In 32 we will introduce the notion of ruled real hypersurfaces in Mn (c) ，c￥ o which are constucted in Kimura[5], Kimura and Maeda[6] for c

>

0

,

and the present authors[12]

for c

<

0 and give the complete proof of Theorem A. Finally the proof of Therem B w^il\be treated in 33

~ 1. Preliminaries

We begin with recalling basic properties of real hypersurfaces of a complex space form. Let M be a real hypersurface of n(;::::2)-dimensional complex space form Mn(c) of constant holomorphic sectional curvature

c( 카이 and let

C

be a unit normal field on a neighborhood of a point x in M. We denote by J an almost complex structure of Mn(c). For a local vector field X on a neighborhood of x in M

,

the transformation of X and C under J can be represented as

J X = <þX

+

⁷⁾(X)C

,

JC =-Ç,

where <þ defines a skew-symrnetric transformation on the tangent bundle T M of M

,

while ⁷⁾ and ( denote a l-form and a vector field on a neighborhood of x in M, respectively. Moreover, it is seen that g((, X)

7)(X)

,

where 9 denotes the induced Riemannian metric on M. By properties of the almost complex structure J, the set (<þ,(,7),g) of tensors satisfies

<þ²

=

-1

+

η0( ， <þ(

=

0, η( <þX)

=

0, 7)(()

=

1,

where 1 denotes the identity transformation. Accordingly

,

the set is so called an almost contact metric structure. Furthermore the covariant derivative of the structure tensors are given by

(\7 x<þ)Y = 7)(Y)AX - g(AX

,

Y)ζ \7x( =

<ÞA

X

,

where ^\7 is the Riemannian connection of 9 and A denotes the shape operator with respect to the unit normal C on M.

Since the ambient space is of constant holomorphic sectional curvature c

,

the equation of Gauss and Coda.zzi are respectively given as follows

R(X

,

Y)Z (1.3)

:i

^{g(Y^,^{Z)X -} ^g(X^,^Z)Y

⁺

^g(^<þ^Y^,^Z)^<þ^X^{- g(}^<þ^X^,^Z)^<Þ^Y

- 2g(<þX, Y)<þZ}

+

g(AY, Z)AX - g(AX, Z)AY,

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(1.4) (VxA)Y - (VyA)X

= 쉰η (X) <j>Y

^-

η ( Y)<j>X

^- ^2g⁽^<j>^X

,

Y)ç}, where R denotes the Riemannian curvature tensor of M and V x A is the covariant derivative of the shape operator A with respect to X.

32. The proof of Theorem A

Before to prove Theorem A let us recall a characterization of a real hypersurface of type A of a complex space form Mn(c) ，c￥O. It is seen by a theorem due to Maeda[8] and a theorem due to Chen

,

Ludden and Montiel[3] that the covariant derivative of the shape operator A satisfies the condition (1)

,

then a real hypersurface M is locally congruent to a tube of radius r(O

<

r

<

~) over PkC (1::;k::;n - 1) or a horosphere

,

a tube over HkC (O::;k::;n - 1), which are said to be of type A. Restrict- ing this condition to the orthogonal distrihution To to

ç ,

we shall give a characterization of ruled real hypersurfaces in Mn(c)，c￥o

Remark 2.1. Now

,

a ruled real hypersurface M of Mn(c)，혜 o can be constructed as follows: we shall construct M in which the orthogonal distribution To defined by To(x) = {XE 끄(M)IX μ} for xEM is in따te-

-

g망rab비)“le and i“t싫s i n tegr al maJl씨]끼if，μo이l떠di샌5 a totally geodesic complex hypersurfac야 e Mn^샤시^_^1“(μ^써^c이^)，CfO. Let 에(“에1 t) (따tκe타J) be a뻐n arh에it뻐ary띠y씨(re멍gu뼈l

Then [，ι。아r any t( E

I)

there exists a totally geodesic compJex hypersu따ce

Mn_l(c) of Mn(c) which is orthogonal to the holomorphic plane spanned by 7(1) and Fy(t). Here we denote by

M낀 I

(c) such a totally geodesic compJex hypersurface Mn_l(c). Set M

{XEM낀l(c)ltEl}

. Then the construction of M asserts that M is a reaJ hypersur[ace o[ j\1η (c) ， which is called a ruled real hypersurface. Moreover the construction o[ M tells us that there are many ruled real hypersurfaces of Mn(c).

Then it is seen in [5] and [12] that the shape operator A of a ruled real hypersurface M of Mn(c),cfO satisfies

(2.1) Aç = aç

+

ßU，(ß￥ 0)， AU = 야 ， AX =0

[or any vector X orthogonal to

ç

and U

,

where U is a unit vector orthog- o^Jlal to

C

and a and

ß

are smooth fuctions on M.

Kimura[5] constructed an exampJe of minimal ruled real hypersurface in PnC and the present authors[12] aJso constructed an example of minimal ruled real hypersurface in HnC. Moreover it is known that these examples are ^Jlot complete.

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The second fundamental form is said to be 1/-parallel if the shape operator A satisfies g( (\7 x A) Y

,

Z) 0 for any vector βelds X

,

Y and Z orthogonal to

ç.

^It^is^see^{n in}^{[6] that}^thesecond fundamental form A is

1/-parallel by (1.2) and (2.1). Thus the properties (1) and (II) hold along the distribution To for a ruled real hypersurface of PnC. Namely

,

it satisfies the conditions (III) and (IV). By the similar argument to those in PnC we can also 잃sert that a ruled real hypersurface of HnC satisfies the conditions (III) and (IV)

Now for a characterization of ruled real hypersurface in Mn(c) ，c카 o let us introduce the following theorem.

Theorem C. Let M be a real hypersurJace oJ Mn(c)

,

c#O. Then the sec- ond Jundamenlal J01γn oJ M is ^1/-parallel and the holomorphic dislribution T_o(= {XETx(M)IX上Ç} Jor xEM) is integrable iJ and only iJ M is locally

congruent to a ηded real hypersurJace.

Remark 2.2. Though the result in Theorem C is proved by Kimura and Maeda [6] for a case where M is a real hypersurface of PnC, by using similar argument to those in PnC we can also obtain the same result for a real hypersurface in HnC. Thus we omit the proof of Theorem C.

Proof of Theorem A: Since the condition (III) implies that the second fundamental tensor A is 1}-parallel

,

it remains to show that the dislribution To is integrable.

Differentiating the condition (I1I) yields

(\7x^\7y A)Z - (\7vxy A)Z

=-삼(<ÞY，

^Z)^<þ^AX

for any vector fields X

,

Y and Z in To

,

w here (\7 x \7y A) Z is defined by (\7x^\7y A)Z = \7x {(^\7y A)Z} - (\7y A)(\7xZ)

From this

,

taking skew-syrnmetric part

,

it follows that

(2.1) (R(X

,

Y)A)Z = -.;c {g(<þY

,

Z)<þAX - g(<þX

,

Z)iþAY}, 4

where the curvature tensor R(X

,

Y) is defined by R(X

,

Y)Z ⁼ \7 x \7y Z-

\7y\7xZ - \7lx,Y1Z for any vector field Z in M.

The Ricci-identity (R(X

,

Y)A)Z

=

R(X

,

Y)AZ - A(R(X

,

Y)Z) and the first Bianchi- identity implies that

S(R(X

,

Y)A)Z = SR(X

,

Y)AZ

,

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306 Seong-Soo Ahn. Sung-Baik Lee and Young-Jin Suh

where S denotes the cyclic sum with respect to X

,

Y and Z in T_o. From this

,

if we take a cyclic sum of (2.1) to the left side

,

we get

SR(X, Y)AZ

=

^{S -}

감g (4)Y， Z )4>AX

^{- g(}⁴⁾^X^,^Z)^4>^AY^}

From this together with

(1.

3) it follows (2.2)

g((A4>+ 4>A)Y, Z)4>X

+

g((A4>+ 4>A)Z,X)4>Y

+

g((A4>+ q,A)X, Y)4>Z = 0 for any vector fìelds X

,

Y and Z in T_o

Since it is well known that the rank of the almost contact structure tensor 4> is 2n - 2,4> is regular on To• Thus (2.2) implies that for any X, Y and Z in To

(2.3) g((A4>H A)Y, Z)X +g((A4>HA)Z, X)Y +g((A4>H A)X, Y)Z = O. Since we have assumed n:::::3

,

we can always take linearly independent vector fields X, Y and Z in To • Thus (2.3) implies g((A4>

+

4>A)X, Y) = 0 for any X, Y in T_o, that is, To is integrable. From this fact together with Theorem C M is locally congruent to a ruled real hypersurface.

~3. The proof of Theorem B

This section is concerned with the proof of Theorem B. Let M be an

n(n 즈 3)-dimensional real hypersurface of Mn(c)，생 o satisfies the condition (IV). Then taking covariant derivative to (IV) yields

(\7_{x A)}4>Y

+

A(\7_x4>)Y - (\7_x4>)AY - 4>(\7_{x A)Y}

+

(A4> - 4>A)\7_{x Y}= 0,

for any vector fìelds X and Y in To. Hereafter unless otherwise stated

,

let us promise that vector fields X

,

Y and Z lie in To

From this

,

substituting

(1.

2) and using the equation of Codazzi

(1.

4)

,

we get

(\7_{x A)}4>Y - (\7y A)4>X- 깨 (AY)AX

+

^1](AX)AY

(3.1) +(A4> - 4>A)(\7_{x Y -} \7yX) = 0

Now we can put \7 x Y - \7y X = (\7 x Y - \7 y X)O + μC where (\7 x Y

\7y X)O and μ denote the To-comp이lent of \7 x Y - \7y X and an arbitary smooth Íunction defìned on M respectively. Then using the condition (IV) in Theorem B yields the following

(A4> - 4>A)(\7 x Y - \7y X) = μ( A4> - 4>A){ = -μ4>A{.

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Substituting this into (3.1) and taking inner product with a vector field Z in To

,

we get

(3.2)

g((\7 x A)cþY - (\7y A)cþX

,

Z) - T/(AY)g(AX

,

Z)

+

T/(AX)g(AY

,

Z) = 0

by virtue of g(cþA~， Z)

=

-g(~，AcþZ)

=

-g(~，cþAZ)

=

0

From (3찌， replacing X and Y by cþX and cþY respectively and using (IV) and the equation of Codazzi

,

we get

(3.3)

g( -(\7φx A)Y

+

^(\7^~^Y^A)X

,

Z)

=

^g(^{-(\7y A)cþ}^X

+

^(\7^x^A)cþY

,

Z)

= o.

Thus from this and (3.2) it follows that

T/(AX)g(AY

,

Z) - T/(AY)g(AX

,

Z) = 0

This fact implies T/(AX)AY -T/ (AY) AX 때， where 떠 means a 1-dimensional vector subspace of Tx(M) for any xEM spanned by the structure vector field ~. Thus we can put

’

1(AX)AY - T/ (AY)AX = vf,.

From this

,

if we take inner product with the structure vector field ~， then we get

v= η (AX) T/ (AY) - T/ (AY)η (AX) = 0

This means T/(AX)AY - T/(AY)AX = 0 for any X

,

Y in To• Thus (3.1) reduces to

(3.5) (\7x A)cþY - (\7yA)CÞX

+

(Acþ- cþA)(\7x Y - \7yX) =

o.

On the other hand

,

we can also put

\7xY - \7yX

=

(\7 x Y - \7yX)"

+

^{η ( \7}x^Y- \7yX)

ç,

where T/(\7x Y - \7yX) is given by

T/(\7 x Y - \7y X) = g(\7 x Y - \7y X,f,) = -g((Acþ

+

^cþ^A)X^,^Y^).

Accordingly, (3.5) reduces to

(3.6) (\7 x A)cþY - (\7yA)cþX - g((Acþ

+

cþA)X

,

Y)cþA~ = 0

Since we have assumed that f, is not principal

,

we can put Af,

=

af, +ßZ for some Z ETo ^orth^{ogonal to}^f,^an^dsome arbitary smooth functions a and

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ß =l

O on M. Then "i>A

ç

⁼

ß

4>Z. Taking account of this

,

if we take inner

product (3.6) with ßφZ and use (3.3)

,

then we get

(3.7) g((A 4>

+

^4>A)X

,

Y) = O.

That is

,

the distribution To is integrable. To complete the proof of The orem B it remains to show that the second fundamental form A is η

pamllel

From (3.7) together with the condition (IV) it follows g(ArþY

,

Z) = 0

for any Y and Z in To • From this

,

replacing Y by 4>Y

,

we get

(3.8) g(AY

, Z)

O.

Taking covariant derivative (3.8) along X

,

then we get

(3.9) g(( V' x A)Y, Z)

+

g(AV' x Y, Z)+ g(AY, V' xZ) = O.

On the other hand

,

V' x Z can be decomposed as following

V'xZ = (V'xzt

+

η( V' xZ)ç

,

where (V' x Z)O denotes the To-component of V' x Z. Thus η ( V'xZ) =

-g(Z,4>AX) = 0 implies that the second and the third terms of (3.9) vanish. From this it follows that the second fundamental form A is η

pamllel. Thus by Theorem C we complete the proof of Theorem B

References

[1] J. Berndt, Real hypers.rfaces with constant principal curvature in complex hyper- bolic space, J.reine angew, Math.395(1989),132-141

[2] T.E. Cecil and P.J. Ryan, Focal sets and real hypersurfaces in complex projective space, 까ans .Amer. M ath ^.Soc. 269( 1982),481-499.

[3] B.Y. Chen, G.D. Ludden 뻐 d S. Montiel, Real s.bmanifold of a K aehler manifold, Algebra,Groups and Geometries 1(1984),176-212

[4] M. Kimura, Real hypersurfaces and complex s.bmanifolds in complex projective space, Trans.Amer .Math .Soc.296( 1986) ,137-149

I히 M. Kimura, Sectional curvatures 01 holomorphic p{anes on a real hypersurface π

PnC, Math. Ann. 276(1987),487-497

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[6] M. Kimura and S. Maeda, On πal hypersurfaces

01

a complex projective space,

Math. Z. 202(1989),299-311

[7] M. Kimura and S. Maeda, On πaJ hypersurjaces 01 a complex projedive space,

Tsukuba J. Math. 15(1991),546-561

[8] Y. Maeda, On real hypersurfaces of a complex projeclive space, J .Math. Soc. J apan 28(1976),529-540^‘

[9] S. Montiel and A. Romero, On some πal hypersurfaces of a complex hyperbolic space, Geometriae Dedicata 20(1986),245-261

[10] M. Okumura, On some real hypersurfaces of a complex projectve space, Trans Amer. Math. Soc. 212(1975) ，35상 364‘

[11] Y.J. Suh, On type number of πa/ hypersurfaces i1l PnC, Tsukuba J. Math 15(1991),99-104

[12] S.S. Ahn, S.8. Lee and Y.J. Suh, 0" ruled real hypersurface in a complex space form, to appear in Tsukuba J. Math.

[13] R. Takagi, On homogeneous real hypersuψaces in a complex projective space, Osaka J. Math. 10(1973),493-506

[14] K. Yano and M. Kon, CR-submanifolds of Kaeh/erian and Sasakian manifold,

Birkhäuser, Boslon Basel Strutgart,1983

DEPARTMENT OF MATHEMATICS, CHOSUN UNIVERSITY, KWANGJU 501-759, Ko- REA

DEPARTMENT OF MATHEMATICS, ANDONG NATIONAL UNIVERSITY, ANDONG 760- 749, KOREA