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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