SOME CHARACTE Rl ZATIONS OF A REAL HYPERSURFACE OF TYPE A

Kyungpook Mathemat.ical Jo^u.rnal

Vohune 31, Number 1, June, 1991

SOME CHARACTE Rl ZATIONS OF A REAL HYPERSURFACE OF TYPE A

U-Hang Ki'

,

Soon-Ja Kim'" and Sung-Bail

‘

^Lee"

Introduction

A Kaehlerian manifold of constant holomorphic sectional curvature c is called a complex space form

,

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

,

a complex Euclidean space Cn

,

or a complex hyperbolic space HnC according as c> O

,

C = 0 or c

<

O.

The induced almost contact metric structure of a real hypersurface M of M_{n (} c) will be denoted by (4), g,

C

TJ)

Typical examples of real hypersurfaces in PnC are homogeneous ones.

Takagi ([16]) classified homogeneous real hypersurfaces of a complex pro- jective space Pn C as the following six types.

Theorem A. Lel M be a homogeneous real hyperspace oJ PnC. Then M is locally congl-uent 10 one oJ the Jollowings:

(Ad a geodesic 1때ersphere (that is

,

a lube over a hyperplane Pn-1C)

,

(A2 ) a 1μbe over a lotaIly geodesic P_kC(l .::; k .::; n - 2)

,

(B) a lube ove,' a complex quadric Qn-l'

(C) a lube over P1C X Pn-^{1 / 2}C and n(?, 5) is odd

,

(D) a tube over a complex Grassman G2,5C a뼈 n = 9

,

(E) a lube over a Hermilian symmetric space 50(10)/U(5) and η = 15.

Received September 10, 1990

• Supported by the basic science research institute program, the Minstry of Educa- tion, 90-125

.*

Supported by TGRC-KOSEF 73

74 U-Hang Ki. Soon-Ja Kim and Sung-Baik

Lee

Trus r.않ult is generalized by many authors ([1]

,

[3],[6],[8],[9] and [17] etc.).

One of them

,

Kimura ([8]) asserts that M has constant principal curva- tures and the structure vector

ç

is principal if and only if M is locally congruent to a homogeneous real hypersurface.

On the other hand

,

real hypersurface of HnG have been also investi- gated by many authors ([2]

,

[5]

,

[13] and [14] etc.) from different points of view. 1n particular

,

Berndt ([2]) proved the follwings :

Theorem B. Let M be real hypersurface of HnG. Then M has constant principal curvatures and

ç

is principal if and 0πly if M^’ is locally congruent to one of the following:

(Ao) a seι tube， that is

,

a horosphel

"e,

(A1 ) a geodesic h νpersphere or a tμ be over a hyperplane Hn_₁G, (A2 ) a tube over a totally geodesic HkG(1 ::; k ::; n - 2),

(B) a tube over a totally real hyperbolic space H’‘R.

Let

M

be real hypersurface of type Al or type A2 in a complex projec- tive space PnG or that of type Ao

,

A₁ or A₂ in a cornplex hyperbolic space HnG. Then M is said to be of type A for sirnplicity. By a thoerern due to Okurnura ([15]) and to Montiel and Rornero ([14]) we have (see also I<i

[4])

Theorem C. If the spape operator A and the structure tensor 1> commute to each other

,

then a real hypersurface of a complex space form M^,.(c)

,

c 츄

o

^{is locally} coπgl-uent to be ⁰ f type A.

The shape operator of re때 hypersurface of M’‘

,

(C) is denoted by A

,

which is related by g(AX

,

Y)

=

^H(X

,

Y) for any vector fields X and Y

,

where H is the second fundamental form of M in Mn(c).

The main purpose of the present paper is to give another charateriza- tions of type A of Mn(c)

,

c

#

^{O. Narnely}

^,

we will prove

Theorem 1. Let M be a real hypersurface of Mn(c)

,

c

#

^{O. Then L{A = 0}

holds on M if and only if M is of tνpe A

,

ψhere L

‘

^{denotes the Lie}

derivaiive with respect to the st7'ucture vector

ç

Theorem 2. Let M be a 1연al hypersurface of Mn(c)

,

c 폼 O. Then L{H = 0 holds on M if and onlν if M is of type A provided that g(A

ç,

Ç) 폼 0

1. Preliminaries

Let Mn(c) be a real2n-dimensional complex space form equipped with

Some Characterizations of a Real Hypersurface of Type A 75

parallel almost complex structure J and a Riemannian metric tensor G which is J-Hermitian

,

and covered by a system of coordinate neighbor- hoods {W;x^A}

Let M be a real (2n-1 )-dimensional hypersu따ce of Mn(c) covered by a system of coordinate neighborhoods {V; 해 and irnrnersed isorr빠rically in Mn(c) by the immersion i M • Mn(c). Throughout the present paper the f，이lowing convention on the range of indices are used, unless otherwise stated:

A

,

B

, " . =

¹

,

²

, ".,

2n : i

,

j

, ". =

¹

,

²

,… ,

2n - 1

The summation convention wiU be used with respect to those system of indices. When the argument is local, M need not be distinguished from i(M). Thus

,

for simplicity

,

a point p in M may be identified with the point i(p) and a tangent vector X at p may also be identified with the tangent vector μ (X) at i(p) via the differential i. of i. We represent the immersion i locally by x A xA(yh) and Bj (B/) are also (2n- 1 )-linearly indenpendent local tangent vectors of M

,

^where B/ 이 x^A and Oj δ/δνJ A unit normal C to M may then be chosen. The induced Riemannian metric 9 with components gji on M is given by gji =

G BAB/ B/ because the immersion is isometric

For the unit normal C to M

,

the following representations are obtained in each coordinate neighborhoods:

JB‘ ⁼ φ‘hBh

+

ιC， JC =-~’ B‘’ ^{( 1 l i}

n ”

where we have put

<Þ

ji = G(JBj

,

B‘) and

ç

‘ = G(JBi

,

C)

,t,

h being com ponents of a vector field

t,

associated with

t,

i and

<þ,

i ⁼

<Þ

/9ri.

By the properties of the almost Hermitian structure J, it is c1ear that

<Þ

ji is skew-symmetric. A tensor field of type (1

,

^{1) with} components

<Þ

^{ih will}

be denoted by

<þ.

By the properties of the almost complex structure J

,

the following relations are then given:

￠‘，

<Þ,

h

=

-Ói h

+ t,

i

t,

h

, t;' <Þ

r h

=

⁰

, t,,<þ;' =

⁰

,

ιç'

=

¹

,

that is

,

the aggregate

(<þ,

g

, 0

defines an almost contact metric structure.

Denoting by \l ^j the operator of van der Waerden- Bortolotti covariant differentiation with respect to the induced Riemannian metric

,

equations of the Gauss and Weingarten for M are respectively obtained:

\ljBi

=

AjiC, \ljC

=

-A/ Br’ ( ^{l 끽}

76 U-Hang Ki. Soon-Ja Kim and Sung-Baik Lee

where H = (A ji ) is a second fundamentaJ form and A = (A얀)， which is reJated by Aji = A/9'i is the shape operator derived from C. By means of (1.1) and (1.2) the covariant derivatives of the structure tensors are yielded :

?j@,h

=

^{-Aj,th + Ajhe}

^{‘’ ?찌i} =

^{-Aj,<p;'. (1. 3)}

Since the ambient space is compJex space form

,

equations of the Gauss and Codazzi for M are respectively given by

Rkjlh

= i (9k싸

^{9jh9ki +}

째ji

^{- <Pjh<Pki -} 2<Pkj<Pih) (1.4) +AkhAji - AjhAki

’

\1kAji - \1jA k

‘

⁼

; (따1‘

^{Çj<Pki -}

2뼈kj) ，

^(1.5)

where Rkjih are components of the Riemannian curvature tensor R of

M

Applying <p^ki to (1.5)

,

we obtain

(와Ajl ) ￠‘

^{= -§(n -}

니

^{( 1 i}

^페

1n what follows

,

to write our formulas in convention forms

,

we denote by Ai,

=

AJTAJ, h2

=

AJIAjl, h

=

gjiAj1, a

=

A1,<jr and

a ⁼

^{/lμçJ Ç'}

If we put U_j = ç'\1_rçj

,

then U is orthogonal to the structure vector ç Because of the properties of the almost contact metric structure and the second equation of (1.

3) ,

we can get

<pjrU'

=

^{AjrC - açj}

’

^{( 1 • 끼 /}

which shows that 9( U

,

U) =

ß -

a².

From (1.3)

,

we have

\1 k \1jÇi = (Aj,çr)Aki - Ajk 2ç‘ - (\1 kAjr) <Pι with which together (1.6) implies that

?i?je‘ = hAA^r -

A파

+ 5(n

니

Since we have divU ^{= (\1}jÇi)(\1i_{çi)+çj\1;'ï1je}

,

the above equation implies

11앙i+ 재 11

²

=

^2divU

+

2{h2 - ah -

~(n

^{- 1)}}

,

Some Characterizations of a Real Hypersurface of Type A 77

where

IIXl1

² = g(X

,

X) for any vector field X on M. Thus we have

divU

= ~IIAr/>

^{- r/>}

AW -

^h2

+

ah

+ ~(n

^{- 1). (1.8)}

By the definition of U and the se∞nd equation of (1.3)

,

we esaily see that U'\1j çr ⁼ Aj/Ç' - aAjrçr (1.9)

On the other h따an때d ， d이iff.없eren따lαt“la따때t“m매l멍g( 1.7끼7끼) covaria’J따r

ing use of (1.3)

,

we find

ÇjAkrUr

+

^r/>j, \1 kur ^{= çr}\1 kAj, - AjrAbr/>" - akçj

+

aAkrr/>/, (1. 1 이

which shows that

(\1 kAji)e

ç ’

= 2Akr ur

+

ak. ( 1

씨

2. Proof of Theorem 1

This section is concerned with the proof of Theorem 1. By definition

,

the Lie derivative of the shape operator A with respect to the structure vector

ç

^{is given by}

L

‘

^{A/ = Ç'\1,A/}

⁺

^{(\1jç')A,h -}

(\1，상 )A/ ，

or using the second equation of (1.3)

,

L

‘

^A/

⁼

^{ç'\1rA/ - Aj,A,h}r/>"

+

Ajr²<J>h'. (2.1) Suppose that L

‘

^{A = O. Then (2.1)} is reduced to

Ç'\1^{j A ir}

= 함ji + Ajr ^A 펴

^-

^AJr2 ￠，

^r (2.2) because of (1.5). So we have

(\1jA,,_)(ç

’

^{= Aj}

,

Ur

hence AjrU'

+

^{aj = 0 by means of (1.11).} Substituting this and (2

찌

mto (1.10), we find

￠

^jr?kUr

= ;찌 ^k

^{- Akr2}

^{찌， +}

^aAkrr/>/

78 U-Hang Ki, Soon-Ja Kim and Sung-Baik Lee

Multiplying rþ'^k to the last equation and summing for j and k

,

we obtain

divU

+ g(ι

U)

=

a(n - 1) - h2

+

ah

+ a

^a2

and thus divU

=

^Hn-1)-h2+ah. Therefore (1.8) tells us that A4>

=

4>A.

By Theorem C

,

M is of type A

Conversely

,

if Arþ

=

4>A holds on M

,

then we have ([4])

와A1， = :(생ik +

é,;Øjk)

80 we obtain

çr

^{\7 rAji =} O. Thus the right hand side of (2.1) vanishes identically because of the fact that Arþ = rþA. This completes the proof

Remark 1. Jt was proved in [4]

,

[1 이 and [11] that a real hypersurface of Mn(c)

,

^{c 츄}

o

^is of type A if and only if L

,

^g

⁼

⁰

^,

^L

,

^\7

⁼

^{0 and L}

,

^rþ

⁼

⁰

respectively

,

where \7 denotes the R.iemannian connection induced on M.

According to Theorem 1 and Remark 1

,

we have

Theorem 3. Let M be a real hypersurface of Mπ (c) ， c

i

O. The following statements are equivalent to each other

(1) L

,

^g

⁼

⁰

^,

^{(2) L{A}

⁼

⁰

^,

^{(3) L{\7}

⁼

⁰

^,

^{(4) L{rþ}

⁼

^O.

Remark 2. By above (1) and (2)

,

it is clear that LeA

=

0 implies LeH

=

O.

3.

Proof of Theorem

2

Let M be a real hypersurface of Mn (c)

,

c 츄 O. By definition

,

the Lie derivative of the second fundamental form H with respect to

ç

is given by

LeAji =

çr

\7 rAji

+

(\7jçr)_Air

+

(\7içr)Aj^r>

with which together the second equation of (1.3) and (1.5) implies that LeAji = Çr\7jAir +

삼J'

Now

,

suppose that L

‘

^H

⁼

⁰

^,

^{Then we have}

çr\7 j A김 -%,

(3.1 )

(3.2)

Some Characterizations of a Real Hypersurface of Type A 79

Therefore (1.11) means

2Ajr^U'

+

^{Oij = 0}

Because of (3.2) and (3.3)^,‘ the equation (1.10) turns out to be tþjr ur =

-감kj

^{- AjrAk,tþT3}

+

é,jAhUr

+

^OiAkrtþ/

Transforrning by tþ‘J and taking account of (1.9)

,

we find

"h ú뇌 V' ké，r)(악;) +

^OiAki - é,;Akr ²

çr + ~

(9k; -

싫; ) ,

which shows that

çr

^{V' rUi}

=

^{U'V' r}é,i

+

OiA;rC - 야‘

(3.3)

(3.4) By the way

,

from (3.2) we haνre (V' j A，，)é，τ， = O. Differentiating this covariantly and using (1.3) and (3.2)

,

we get

gTes?k?jAsr = 5(Akj

+ ζAkrC) ，

from which

,

taking account of the Ricci formula for Aji,

R셔i， (Ar'C)é，;

⁼

~(é，jAkrC 화Ajré，r )

If we transvect

é.'

to this and make uae of (1.4)

,

then we obtain

OiAj/C = ßAjré,'. (3.5) By differentiating

ß

⁼ Aj/

é.'

é,

‘

covariantly and using (1.3) and (3.2)

,

we find

ßk

=-한 - ^2Aj썼kr

^tþ

^‘

^r

^,

with which together (3.5) yields

QaJ aQJ ⁼ 5aI/ (3.6)

So we get OißrC = 0 because of (3.3). From the last equation it is easily seen that

ßkOij -

01써

⁼

5{QJ아

^OikUj

+

^Oi(V'Pk - V'kUj)}

80 U-Hang Ki, Soon-Ja Kim and Sung-Baik Lee

Consequently we obtain

O (

'IljUk - 'IlkUj)é,' = 0 because Oj ，O뎌 and Uj are orthogonal to

ç.

^{Therefore we get} of:'IlrUj ⁼

(0 2 -

ß)

,

Ajrf: with the aid of (1.9) and (3.5). 50 it leads to oUr 'll，ζ = -ß(Aj

,f: - Oçj)

and hence oUrA:

q,

j

’ =

ß

q,

jrur

,

where we have used (1.3)

,

(1.7) and (3씨 By transforming

q,

^k'

,

we get

oAjrUr

=

ßUj. (3.7)

Thus

,

(3.3) and (3.6) are reduced respectively to

OOj + 2ß이 =

⁰

^, 0쩌 = ^{(:a2 -} 2a2)띠

(3.8) To prove Theorem 2

,

we prepare the following

Lemma 4. Let M be a real hypersurface of Mn(c)

,

c 츄 O. If the stucture vector

ç

^{salisfies L}

‘

^{H = 0}

^,

^then

^ç

^{생 principal}

Proof Let Ma be a set of points of M at which the funtion β _ 0 2

dose not vanish. Then we have 0

=1

0 on Ma because of (1.7) and (3.7). 50 we can put .\ = ~. Then (3.5)

,

(3.7) and (3.8) become on Ma respectively as follows:

Aj/ Ç'

=

.\Ajrçr

,

AjrUr ₌ .\Uj

,

@j

+

2AU1

= o

^,

a ⁼ (5 - ^2A)이

(3.9) (3.10) (3.11) If we differentiate (3 찌 covariantly along Ma and make use of (1.3) and (3.2)

,

then we find

(와Aj써rF

-

aAjr야r _

Aj/ Aksq\" (3.12)

=

써rÇ'

-

~ '\q,

kj -

셰A ks ^{1>TS ，}

from which, tansvecting

çj

and using (3.2) and (3찌，

a사 = 5Uk

^{(3 13)}

By taking the skew-symmetric part of (3.12) with respect to indices k and j

,

and using (3.13)

,

we obtain

~{2(.\ -

O)

q,

kj - Aj

,q,

kr

+

Akr

q,/ + μ U k

^{- ÇkUj}}

=

Ajr ² Ak

,q," -

_Akr ² Aj

,q," -

2.\A jr A k

,q,"

+￡{야AjrÇ'

^{- UjAkrÇ'}}

Some Characterizations of a Real Hypersurface of Type A 81

Transforming the above equation by U1 and taking account of (1.7)

,

(3.9) and (3.10), we have (2ß - 3(²)(A;

rçr -

^açj)

=

0 and hence 2ß

=

^{3a2, i.e.,}

a

=

^{2). on M". Thus,} it is not hard to see, using (3.11), that 9a² -6a+c =

o

^{on M"}

^,

which means a is constant. 80 ). is constant on M". Therefore (3.13) tells us that U 0

,

a contradiction. Thus M" is empty. Trus completes the proof of the lemma.

By Lemma 4

,

we see that ç is principal

,

namely Aç

=

aç. Hence a is constant on M (See [7] and [12]). Accordingly

(1.1

0) is reduced to

çr\7 kAjr - AjrAk'tþ"

+

aAkrtþ/

=

0

,

with wruch together (3.2) yields a(Atþ - tþA) ⁼ O. By Theorem C

,

M is of type A provided that a ￥

O .

The converse assertion is immediately from Theorem 3. This completes the proof of Theorem 2.

Remark 3. Maeda and Udagawa [11] proved Theorem 2 under the addi- tional condition that ç is principal

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KYUNGPOOK NATIONAL UNIVERSITY, TAEGU 702-701, KOREA

KYUNGPOOK NATIONAL UNIVERSITY, TAEGU 702-701, KOREA

GHOSUN UNIVERSITY, K、"ANGJU 501-759, KOREA