on 80me alm08t hermitian manifold8 찌tith con8tant

Kyungpook Mathematical Jownal

Volume 29, Number 1, June 1989

ON 80ME ALM08T HERMITIAN MANIFOLD8 찌TITH CON8TANT

HOLOMORPHIC 8ECTIONAL CURVATU RE

Takuji Sat。

1. Introduction

1n the present paper , we shall study some almost Hermi tian [lla.rüfo!ds with constant holomorpbic sectional curvature. For Käbler manifolds (or more generally, nea싸 Käbler manifolds ) witb constant holomorphic sec t ional curvature, it is well kn。、.vn that the lliemannian curvature tcns。야 r

IS g밍i ven by the fo。이nl1l

mu비11a잃s to a따.lmos야t He앉rml“띠tian manif.“Olc띠ds (πThe。αrem 4.2잉). Es윈렌pec미la떠lly이’ we glve an expression for the curvature tensor in quasi- Kählcr manifolds with constant holomorphic sectional cur、'ature under some curvature condi­tions. 1n the last section , we shall deal with so called the SCIWT싱 theo­rem. 1n particular, we shall give an another proof of t he theorem due t。Gray and Vanhecke [4].

The author wishes to express his hear ty thanks to P rof. K. Sekiga、，va

for his valuable suggest ions

2. Preliminaries

1n this section, we prepare some elementary formulas in an almost Hermit ian manifold. Let M = (M , J, <, >) be an n( = 2m)- dimensional almost Hermitiall manifold. We denote by \l and R the Riemannian connection and the curvature tensor of /vf , respect ively. \Ve assume that the curvature tensor R is defined by

R (X , Y)Z = \l [X ,l' JZ - [\l x , \ll' ]Z ,

and R(X , Y, Z , W ) = (R(X, Y )Z, W ),

11

12 Takuji Sat。

for X , Y, Z , W E X(M). '0le denote by r and ,.- the Ricci tensor and the Ricci * tensor of Af, respectively. The Ricci * tensor r- is defined by

r*(x , ν) = trace of (z •• R(Jz,x) Jy ),

for x , y, z E TpM. The tensor fields r and r* satisfy the foll。、.ving equali ties ’

r (X , Y) ~ ,'(Y, X) , r* (JX,JY) = ,'*(Y,X) ,

(2. 1 )

(2.2)

for X , Y E X (M ). We denote by s and s* the scalar curvature and the * scalar curvature of M , respectively.

The sectional curvature and the holomorphic sect ional curvature are defined respectively by

R(x , y, x , y) C(x ， ν) = ----­

싸11 2 11ν11 2 ’

for x ,y E TpM(p E M ) wi th x f O,y f O,(x ,y) = 0, and

H(x)=K(x ,Jx ),

for x E 깐M(p E M) with x f O. We define

G(X ,Y,Z ,W ) = R(X ,Y,Z ,W )- R(X ,Y,JZ,JW)

= ((Vx Vy J)Z,JW) - ((Vl' Vx J )Z,JW)

-((VrX,y)J)Z , JW)

It seems that the tensor field G is important in the study of 떠most Hermitian manifolds (cf. Gray [lJ , [3], Vanhecke [9J , Tl;cceri- Vanhecke [8J , ... ). For X , Y, Z , W E X(M ), G satisfies the following relations ;

G(X, Y, Z , W) = -G(Y, X , Z, W) = -G(X, Y, W, Z ), (2. 3)

G(X ,Y,JZ,JW) = - G(X ,Y,Z , W) , (2 .4)

G(X, Y, JZ, W) = G(X , Y, Z , JW ), (2.5)

G(X,Y,Z ,JZ ) =O, (2.6)

G(X , Y, Z , W) + G(JX , JY,JZ, JW ) (2 .7)

= G(Z, W,X ,Y ) + G(JZ, JW, JX, JY ),

On Some Almost Hermitian Manifolds 13

G(X ,JY,Z ,JW )+ G(JX ,Y , JZ ,W) (2.8)

= G(Z , JW, X , JY) + G(JZ , H끼 JX ,Y ),

ε G(X ,E ,‘ ’ Y ,E‘) = r(X , Y) - ,."(X , Y ), (2 .9)

where {B‘} is an orthonormal frame of M We shall recall the definitions of special kinds of a.lmost Hermi ti떠1

manifolds. An almost Hermi tian manifold M is called Kähleriaπ if

t7xJ = 0,

for all X E ,Y (M ), M is c떠led H ermitian if

(\l x J )Y - (\l Jx J )( JY ) = 0,

for all X , Y E X (M ), M is called nearly Kählerian if

(\lx J )Y + (\l y J )X = 0,

for 떠1 X ,Y ε X (M ), M is called almost Kählerian if

((\l x J )Y , Z) + ((\ld )Z, X ) + ((\l zJ)X, Y) = 0,

for all X , Y , Z E X (M ), and M is called quas i-Kählerian if

(\lxJ )Y + (\l JxJ )( JY ) = 0,

for 외1 X ,Y E X (M )

Gray has obtained the following

Lemma 2.1 ([3]) Let M be a Hermitian manifold. Then

G(X, Y , Z , W ) + G(JX, JY,JZ , JW ) (2. 10 )

-G( JX , Y , JZ , W ) - G(X , JY, Z , JW ) = O.

Lemma 2 .2 ([3]) Let M be a quasi-Kähler manifold. Th en

G(X , Y , Z , W ) + G(JX, JY, JZ , JW )

+G(JX ,Y , JZ ,W) + G(X, JY,Z , JW ) (2.11)

= 2(( \l('VxJ) l,J)Z , W) + 2((\l('V yJ)x J )W , Z) .

14 Takuji Sat。

Le mma 2.3 ([3]) Let M be an almost Kãhler manifold. Then

G(X, Y, Z , W) + G( J X , JY, J Z, JW)

+G(JX, Y, JZ, W ) + G(X , JY, Z , JW) (2.12)

= - 2((VxJ)Y - (<;h J)X, (VzJ)W - (V IVJ )Z)

L e mma 2.4 ([1]) Let M be a nearly K ãhler manifold. Then

G(X,Y,Z , W) = ((VxJ)Y,(Vz J)W) ‘ (2.13)

3. Some conditions on G

We shall consider the following conditions in an almost Hermitian manifold M

(a) G = 0, (b ) G(JX,Y,JZ, W) = G(X,Y,Z , W) forX, Y,Z , W E X (M) , (c) G(X, Y,Z , W) = G(Z, W,X , Y) forX, Y,Z , W E X (AI).

It iseasilyshown tha t (a) =} (b ) =} (c). Weremark that thecondition (a) is equivalent to

R(X, Y) 0 J = J 0 R(X, Y) , n / n ‘ u

(

for X , Y E X (M ). An almost Hermi tian manifold M sat isfying (3. 1) (and so, the condi tion (a)) is called a para-Kãhler manifold or an F-space ([5 ], [9]). The condition (b) is equivalent to

R(X, Y, Z , W) (3.2)

= R(JX ,JY,Z , W) + R(JX,Y,J Z, W) + R(J X ,Y,Z ,JW) ,

for X , Y, Z , W E X(M). The condition (c) is equivalent to

R(JX ,JY,JZ ,J W) = R(X,Y,Z, W ), (3.3)

for X , Y, Z , VV E X(M ). An almost Hermitian manifold M satisfying (3.3) (and so, the condi tion (c)) is called an RK- manifold ([9]).

The identi ties (3.1), (3.2) and (3.3) in almost Hermitian manifolds are studied by Gray, Tricerri, Vanhecke, etc.. It is well kn。、.vn t 'mt

On Some Almost Hermitian Manifolds 15

Kähler manifolds sat isfy the condition (a). Nearly Kähler manifolds and fuemannian locally 3-symmetric spaces satisfy the condition(b) ([2]). It is proved in [3], some quasi- Kähler manifolds satisfy the c∞。ond버l“띠tion (μ에c이) . In He앙rml따i삐tμla때nma뻐n띠1니ifolc닝ds ，

in [7J끼J a con찌l띠d며l“tμion for an RK - manifold to be of constant holomorphic sectional curvature

In an RK-manifold M , we get easily the following

G(JX,JY,JZ ,JW ) = G(X,Y ,Z , W ),

,.( J X , JY ) = ’ (X , Y ), ?“(X, Y) = ,'"(Y, X) ,

,."(J X , JY) = ,."(X, Y ), for X , Y, Z, W E X (M ).

(3.4)

(3.5)

(3.6 )

(3.7)

4. Curvature tensors on almost Hermitian manifolds with constant holomorphic sectional curvature

In an almost Hermitian manifold jμ ， we put

.\(x , ν ) = G(x ,y ,x , ν ) ，

and Q(x) = R (x , J x ,x , J x) = H(x) 1I씨 1 4 ’

for x , y E TpM. Vanhecke proved the foll。、，Y 1Ilg

Propositio l1 4.1 ([1이 ) Let M be an almost Hermitian manifold and x ,y E 감JvI . Then

R (x ,y ,x ,y) = 깊 {3Q(x + Jy) + 3Q(x - Jν) - Q(x + ν) -Q(x-y)-4Q(x)-4Q(y)} (4.1 )

+l {13A(z, y) - 3A(J z , Jν)+ .\ (x ， Jν ) + .\(Jx , y)} 16

Now, we assume that the holomorphic sect ional curvature H (x) is con­stant c(p) for all x E 과M at each point p of M. For Kähler manifolds M of pointwise constant holomorphic sectional curvature c(p) , it is well known that the curvature tensor of M at p is given by

c(p) R (x , y,z , ω)= 그:.L Ro(x ， y ， z ， w) ， (4.2)

16 Takuji Sat。

where

Ro(x ,y,z , ω)=(x ， z)(y ， ψ) - (x , ω)(y ， z)

+(x , J z)( y, Jω) - (:r;, Jw)(y , J z) + 2(:r;, Jy)(z , J ω) ,

and c is a constant function on M , if 1.1 is connected. For 외most Hermitian manifolds, we have the following

Theorem 4 .2 Let JvJ be an almost H ermitian manifold of point'W ise constant holomorphic sectional curvature c(p) . Th en

c(p) R (x ,y,z , ω) 그:.L Ro(x ， y, z , w) + P(x , Y, z, ω) ， (4.3)

'Where

P (:t , y , z ， ψ) = ^lJ26{G(x ,y ,z, ψ) + G(z ,w,x ,y)} 96 -6{G( Jx ， Jν ， Jz ， Jψ )+G(Jz ， Jw ， Jx ， Jν)}

+13{G(x ， z ， ν ， ω) + G(ν ， ω ， x , z)

-G(x , ω ， y ， z) - G(y,z ,x ,w)}

-3{G(Jx ， Jz ， Jν ， Jψ ) + G(Jy , Jω ， Jx ， Jz)

-G(Jx ， Jψ ， Jy , J z) - G(Jy , J z , J x, Jψ)} +4{G(x ， Jν ， z ， Jω) + G( J x , ν ， Jz ， ψ) }

+2{G(x ,J z , ν ， Jω)+G( Jx， z ， Jν， ω)

-G(x ， Jω ， y ， Jz)-G(Jx ， ψ ， Jy ， z)}]

Proof Since Q(x) = c(p) lI xll<' we have fl'Om (4.1)

c(p) (11 __ 1I 2IL. 1I 2 , __ .\2 R(x ， ν ， x ， y ) = 그:.L{lI xll 2 l1 yll2 _ (x , y)2 + 3(x , Jy )2) (4 .4)

+]」 {13A(I ， y) - 3A(JI , Jy) + A(x, Jy) + A(Jt , y)} 16

By linearlizing (4씨， 、ve get

R(x , ν ， z , ω) + R(z , ν ， x ， ψ)

c(p) = 그:.L {2(x , z)(y ， ψ) - (x ， ν )( z ， ω) - (x씨 (z ， ν)

+3(x , Jy)( z , Jω)+3(x ， Jψ)(z ， Jy)} (4.5)

On Some Almost Hermitia n Manifolds 17

+끓[13{G(x ， y ， z ， ψ) + G(써 x ， ω) + G(x ， ψ ， z ， y) + G(z ， ω ， x ， ν)} 3{G( Jx， Jy ， Jz ， Jω)+G(Jz ， Jν， Jx ， Jψ)

+G(Jx ， Jψ ， Jz ， Jy ) + G(Jz, Jw , Jx , Jν)} +2{G(x ， Jy ， z ， Jψ)+G(z ， Jν ， x ， Jψ)

+G(Jx, ν ， Jz ， ψ)+G(Jz ， ν ， Jx , ω)}]

lnterchanging x and y in (4.5) and subtracting the resulting equation from (4.5) , we have finally (4.3)

From this theorem and Lemma 2.1 , we have easily the following

COl'ollary 4.3 Let M be a Hermitian mainlold 01 pointψ!se constant holomorphic sectional curvature. Theπ the curvature tensor 01 NI is given by (4. 3), where

P(x , y , z , ψ) = l{14{G(x,y , z, ψ) + G(z ， ψ ， x ， y)} 48 -2{G( Jx， Jν ， Jz ， Jω ) +G(Jz ， Jω ， Jx ， Jν)}

+7{G(x ,z ,y, ψ) + G(y , ψ ， x ， z)-G(x ， ω ， y ， z )

-G(ν ， Z ， X ， ψ)}

-{G(Jx ,Jz, Jy , Jω) + G( Jν ， Jω ， Jx ， Jz)

-G(Jx ， Jω ， Jν ， Jz)-G(Jν，Jz ， Jx , Jw )}]

Cor olla ry 4 .4 Let M be an RK .manilold 01 pointwise constant holomorphic sectional curvature. Then the curvature tensor 01 M is given by (4.3) , where

P (x ,y , z , ψ) = ]」 { 1OG(z ， ν ， z , ψ) + 5G(x , z ， ν ， ψ) - 5G(x , ψ ， y ， z) 24 +2G(x ， Jy ， z ， Jψ)+G(x ， Jz ， y ， Jψ) - G(x , Jw , ν ， Jz)}

Cor olla r y 4 . 5(에 ) Let M be an almost Hermitian manilold 01 pointwise constant holomorphic sectional curvature. 11 M satisfies the condition (b) , then the curvature tensor 01 M is given by (4 .3) , ψhere

P(찌찌) = i{2G(Ly， z ， ψ) + G(x ，씨 ψ) - G(x ， ψ ， ν ， z) }

18 Takuji Sat。

Corollary 4.6([5]) Let M be a para-Kμler mani센ψfμo이ld of poin떠ttuψOIS constant ho이lomorψphic sectiona띠1 cuπ띠Tπva띠a띠t1ωuπTπe. Then the cu‘π떠Tπrva따a띠t1ωu‘πre tensor of Mi생s given b야y (μ4.‘.2낀 )

By (4.3), (2.9), (3.4) , (3 .5) , (3 .6) and (3.7) , we have

Corollary 4.7 Let lv1 be an n-dimensional almost Hermitian manifold of pointwise constant holomorphic sectional curvature c(p) . Then

,'(x , y) + "(h , Jy) + 3(r ‘ (x , y) + r*(ν ， x ))

= 2(n + 2)c(p)(x , ν) ,

s + 3s ‘ = n(n + 2)c(p).

Iπ particular, if M is an RK -manifold, then

( 4.6)

(4.7)

r(x , ν) + 31'‘(x , ν) = (n + 2)c(p)(x , y) (4.8)

Fω n Lemma 2.2 and Corollary 4.5, we have

Theorem 4.8 Let M be a quasi-K ãhler manifold of pointwise constant holomol따ic sectional curvatμre c(p). 1f M satisfies the condition (b) , theη the cμrvatμre tensor of M is given by

R(X,Y ,Z , W)

、I

，

J

찌 찌 찌

K K

7ι

써 써 n씨

씨 씨 씨

r」

?

, ,

”끼 까 」‘커

%

… M

””

니 「

여

잉 ?

γA‘

?

「시 ?

) ( ‘ (

C4

1훤애 싸

-+

( 4.9)

From Lemma 2.3 a.nd Corollary 4.5, wc have

Theorem 4.9 Let M be an almost Kãhler manifold of pointwise constant holomorphic sectional curvatμre c(p). 1f M satisβes the condition (b) , then th e c'urvature tensor of M is given by

R(X, Y , Z , W) = 눔(X， Y , Z , W) (4 .10)

- $힐늙떼{μ띠(“(\7잉xJ끼)Zι낀-커(\7z링얘Z성써J끼)Xχ’(\7안얘Y너J)Wι-카(\7，‘W꾀V -(“('vx J끼)W - (\7 w J )X, (\7yJ )Z - (\7zJ)Y)

+2((\7 xJ)Y - (\7yJ )X, (\7zJ )W - (\7wJ)Z)}

On Some Almost Hermitian Manifolds 19

From Lemma 2.4 and Corollary 4.5, we have

Theorem 4.10 ([이) Let M be a nearly K ãhler manifold of pointu떠e

con !Jtant holomorphic !J ectional curvature c(p) . Then the curvature ten !J or of M i !J given by

R(X,Y,Z , W) - 늄(X， Y， Z ， W) (4.11)

+i{((?xJ)Z, (?vJ)W) - (“(\7x헤헤X써싸J끼)YV，찌V +2끽(“(\7 x’ J끼)Y， (\7 z J끼)W씨)}샤 } . .

5. On the theorem of Schur

In this sect ion, we sh따1 consider the fo11。、.ving problem "Let M be an almo !J t Hermitian manifold of pointwi!J e con!J tant holo­

morphic !J ectional cur'Uature c(p) . When is c a con !J tant function ?" As for this problem, Gray and Vanhecke [4] have proved an interesting

theorem (see Theorem 4.7 below). Moreover , they have shown that this problem does not hold for the class of Henni tian manifolds, that is, there exists a Hermitian manifold of pointwise constant holomorphic sectional curvature which is not globally constant. We shall make a slightly different a.pproach to this problem

Let M be a para-I<하hler manifold of point、.vise constant holomorphic sectional curvature. Then , by Corollary 4.6, we have

R(X, Y, Z , W) = 챔(X， Y， Z ， W ) (5 .1 )

Contracting (5 .1), we have

n+2 T(X,Y) = -x-c(X,Y),

which shows that M is an Einstein manifold and c is constant. Thus we have

Theorem 5.1 Let M be a connected para-Kähler manifold of pointwi !J e holomorphic !J ectional cμrvature c(p) with dim M = n 즈 4 . Then c i!J a con !J tant function on JvI

20 Takuji Sat。

The Bochner curvature tensor B on an n = 2m-dimensional almost Hermitian manifold M is defined by

B=R - --L-S+ Rn 2(m+2)- '4(m+1)(m+2f-u

where 5 is given by

5(X , Y , Z , W) = ,'(X , Z)(Y, W) - r(Y, Z)(X , W)

+(X, Z)r (Y , W) - (Y , Z)r(X , W)

+,'(JX , Z)( JY, W) - r ( JY, Z) (JX , W )

+(JX,Z)r(JY, W) - (JY,Z )7-( JX , W)

+2r(J X , Y) (JZ , W) + 2(JX, Y )r( J Z , W )

Suppose that B = 0 and ,'( JX , JY ) = r(X , Y) for all X 'y E X (M ), then we have directly G = 0, that is , M is para- Kählerian. Thus we have from Theorem 5.1 Theorem 5.2 Let 1.1 be a connected RK .manifold of pointwise constant holomorphic s ectional curvature c(p) with dim M 즈 4. If the Bochner cμrvatμre tensor vanishes , then c is a constant function

Let {E;} be an orthonormal frame of an almost Hermitian manifold M. We prepare the following

Lemma 5.3 Let M be a quasi.Kähler manifold satisfying the coπdition (c). The n,

2ε(v타 r*)(X ， Ej ) - Vxs* = 2ε R(X， E; , J Ej , (V E, J )E ‘) , (5.2)

forX ε X(M)

Proof. By definition, we have

,'*(X , Y) = ε R(X,E ‘’ JY, JE;) ,

and s · = ε r*(Ej ， Ej ) = εR(Ej ， E‘， JEj , JE; )

By the standard calculation,

(Vvr*)(X, Y ) = ε{(VvR)(X， E; , JY, J E ‘) (5.3)

+ .1(X,E ,‘ , (Vv J )Y, J E;) + R (X , E ‘ , J}' (\7 vJ)E )}

On Some Almost Hermitian Manifolds 21

and malöng use of Bianchi identity,

\1x S ‘ = ε{ ( \1xR)(Ej ， Ei , J Ej , J Ei)

+2R(E j ,E‘ , J E;, (\1 xJ)E‘)} (5.4)

=2 ε{ (\1s，R)(X， Ei, J Ej , J Ei ) + R(Ej , Ei , J E;,(\1 xJ)E;)}

From (5.3) and (5.4), we have

2ε( \1ι r*)(X， Ej) - \1xs ‘ = 2ε{R(X， E;，(\1타 J )Ej ， .] E ‘ ) (5.5)

十R(X， E;, .] Ej , (\1 SJ)Ei) - R(Ej , Ei , J Ej , (\1xJ )Ei ) }

It is clear that in quasi- Kähler manifolds,

ε(\1sJ)Ej = 0,

therefore the first term of the right hand side of (5.5) is eqr때 to zero. By making use of (3 .3),

ε R(Ej , Ei , JEj ,(\1x J)E;) = 'L, R(JEj , JE‘ ’ Ej ,( \1x J )JEi )

ε R(Ej , Ei , J Ej , (\1 xJ)Ei) ,

so the last term is also zero. Thus, we obtain (5 .2)

Lemma 5.4 Let ÑI be a q'uas i- K ähler manifold satisfying the condition (c) . Then

2 ε R(X,E‘ ’ JEj ,(\1sJ)E‘) ‘ J

(5.6)

= ε{ ( \1 s;R)(JX， JE，‘’ Ej ,E‘) - (\1s,R)(X, Ei ,JEj ,JE‘ )},

for X E X(M)

Proof. By the condition (c),

R(JX,JY,Z , W ) = R(X,Y, JZ, JW) ,

Differentiating this

(\1vR )( JX,JY, Z , W) + R(( \1vJ)X, JY, Z , W ) + R(JX, (\1 vJ )Y, Z , W )

= (\1vR )(X , Y, JZ, JW) + R(X, Y, (\1vJ )Z, JW) + R(X , Y, JZ, (\1vJ)W).

22 Takuji Sat。

Putting Y = vV = E; , Z = V = Ej and summing up with respect to i ,j ,

ε{ ( \7 s， R)(JX， JE‘’ Ej , E;) - (\7s,R)(X, E; , J Ej , J E;)} ’J

= ε{R(X ， E; , (\7 s,J )Ej, J E;) + R(X, E; , J Ej , (\7s,J)Ei) (5.7) ‘J

-R((\7s,J)X, JEi ,Ej ,E;) - R(JX,(\7s,J)E; ,Ej ,E‘)}.

By the same reason as in the proof of Lemma 5.3, 、ve sce that t hc first and thi1'd term of the right hand side of (5.7) vanish. Taking account of

εR( JX, l \7 sJ)E;, Ej , Ei) = ε R(JX ， E‘’ EJ ' (\7 S ,J )Ei)

‘，~ '( u btaill (ι6).

= εR(X‘ JE‘’ JEj ， J( \7 s，J )E;) -εR(X， JE; , J EJ , (\7s,J)JE;)

-ε R(X, E; , JEj , (\7 s,J)E,‘),

~ow ， we prove the followiug Sch 1Lr ’s theorem fo1' qUi.1.Si- Kähler mani folds under some condi tions

Theorem 5.5 Let M be a connected q1Lasi-Kähler manifolrl of pointwise constant holomorphic sectional C1Lrvat1Lre c(p) with dim A1 > 4. 1f M satisfies the condition (c) and

(\7vR )(X , Y, Z , W) + (\7vR)(JX, JY, JZ, JW ) = 0, (5.8)

for X , Y , Z , W E X(M) , then c is a constant f l!nction. 1n particular, we have

Corollary 5.6 Let M be a connecterl locally symmetric quasi-K ähler manifold of pointwise constant holomorphic sectional C1Lrvat1Lre c(p) 떼th

dim M ~ 4. 1f M satisβes the condition (c) , then c is a constant jl!nc­tion

Proof of Th eorem 5.5. By the condition (5.8) and Bia.nchi identity, we have

ε{(\7 s,R)(J X , J E; , Ej , E;) - (\7s,R)(X, E‘ , JEj , JE‘)}

= -2 ε(\7작R)(X ， E; , J Ej , J E;)

- ε(\7xR)(Ej ， E‘ ， JEj ， JE;) .

On Some Almost Hermitian Manifolds 23

By using (5 .8) again,

('V xR)(Ej, E. , J Ej , J E ;) -(VxR )(J Ej , J E; , Ej ,E.‘) -(VxR)(Ej , E; , J Ej , J E;) ,

from which we see that the right hand side of (5 .6) vanishes. From Lemma 5.3 and Lemma 5.4, 、'Ie have

2ε(V타r*)(X， Ej ) = Vx s' ‘ (5.9)

On the other hand , it is well known that

2ε(Vζ?‘ )(X， Ej) = V xs. ( 5.10)

Differentiating (4.7) with respect to arbitrary E;, we have

VE, S + 3 VE, s' = n(n + 2) VE, C. (5.11 )

On one hand, by making use of (5 .9), (5.10) and (4.8),

VE,S + 3 VE, s'

=2 ε((VEJI')(E; ， Ej) + 3(VE/'*)(E‘’ Ej)} (5 .12) 1

= 2(n + 2) VE, C.

By (5 .11) anc\ (5 .12),

(n - 2)(n + 2) VE, C = 0, from which it foll。、.vs that C is a constant function

The following theorem is due to Gray and Vanhecke.

Theorem 5.7([4]) Let M be a connected quasi.Kä hler manifold of pOl떠'wise constant holomorphic sectional curvature c(p) with dim M = n 즈 4 . 1f M satisβes the coπdition (b), then c is a constant function .

Proof Observing the proof of Theorem 5.5, it is sufficient to prove (5.9). By Corollary 4.5, the curvature tensor of M is given by

R(X ,Y,Z ,W) = 첼(X， Y, Z , W) (5 .13)

+ ~{2G(X ， Y, Z , W ) + G(X , Z , Y, W ) - G(X, W, Y, Z)}

24 Takuji Sat。

It is easy to check

εR，， (X ， Ei ， JEj ， (VE，J)Ei ) = 0

By using the condition (b) ,

ε G(X , Ei , J Ej , (VE,J)Ei ) εG(X， JEi 、 J Ej , J (VE,J)E;)

-ε G(X,JEi ,JEj’ (VE,J)J Ei)

-ε G(X , E i , J Ej , (VE,J)Ei ),

from which we have

ε G(X , E i, J Ej , (VE,J)Ei ) = 0

Similarly by using the condit ion (b ) and the definition of quasi-Kähler manifold, we get

ε G(χ， J Ej , E;,CVE,J)E‘) = 0 、

and

L, G(X,(V E,J)E‘’ Ei ,JEj ) = 0

Hence we have from (5.13) ,

ε R(X， E，‘ ’ JEj ， (VE，J)Ei ) = O.

This proves (5.9) by virtue of Lemma 5.3. Thus wc obtain the theorem

Refere nces [1] A. Gray, Nearly Kähler manifold5, J . Differential Geometry 4(1970) , 283-309

(2] A. Gray, Riemannian manifolds with geodesic symm e1ries 0/ order 9, J . Different ial Geometry 7(1972), 343-369

(3] A. Gray, CU ï/Jature iden. tities /0 1' f{e,mitian and almosi Hermitian manifolds, Tôhoku Math. J. 28(1976) , 601-612

[4] A. Gray and L. Vanhecl‘e, Almost JJermitian ma띠ifolds with co7l stant holomorpkic seciional curvatuπ ， ëasopis pro pèstováni mat . 104(1 979) , 170-179

[5] S. Sawaki and K. Sekigawa, Almo5t llennitian manifold5 with con51ant holomor­phic sectional curvatu떠 J. Differential Geometry 9(1974) , 123-134

On Some Almost Hermitian Manifolds 25

[이 S. Sawaki, Y. Watanabe and T . Sato, Not es on a f{ -spa ce o[ co"slant holomorphic sectional curvatuπ， Kodai Math . Sem 바p . 26(1975) , 438-445

[끼 S. Tanno, Constancy 01 holomorph-ic seclional curvatu re ;n almost Hennitian man­i[olds, Kodai Math . Sem. Rep. 25(1973) , 190-201

[8] F 까icerri and L. Vanhecke, Cuπalure lensors on al11'/.05t Henllitian manifolds, 자ans. Amer. Math . Soc. 267(1981) , 365, 398

[9] L. Vanhecke, Almosl JIermitian ma’‘’folds wilh J-‘’l,va7'ianl Riemann curvaluπ tenso r, Rend . Sem. Mat. Univ . e Politec. 'lb rino 34 (1975-76) , 4 8ι498

[10 ] L. Vanhecke, 50m e almosl Heπnilian ma71ψIds wilh consla71l holom orphíc sec. tional curvature , J . Differential Geomctry 12( 197ï), 461-471

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