Rend. Sem. Mat. Univ. Poi. Torino Voi. 50, 2 (1992)

K. Sekigawa - L. Vanhecke

ALMOST HERMITIAN MANIFOLDS WITH VANISHING FIRST CHERN CLASSES OR CHERN NUMBERS

A b s t r a c t . One derives some properties for nearly Kàhler, almost Kàhler and

Hermitian manifolds with vanishing first Chern classes or vanishing first Chern

numbers.

1. Introduction

The relations between the properties of the almost complex structure and the curvature ofan almost Hermitian manifold have been studied by many people and there are a lot of papers devoted to this broad subject. (See [3] for some examples.) In this paper we consider three special classes of almost Hermitian manifolds : nearly Kàhler, almost Kàhler and Hermitian manifolds. In parti cular we study such manifolds with vanishing first Ghern classes or vanishing first Chern numbers. (Related problems have been treated in [8], [11], [12], [15], [22].)

2. Preliminaries

Let (M,g,.J) he a 2n-dimensional almost Hermitian manifold and let fì denote the Kàhler forai defìned by Ù(X, Y) = g(X, JY) where X, Y G X(M). (X(M) denotes the Lie algebra of smooth vector fields on M.) We assume that M- is oriented by the volume form

dm = ^-/-nn. n!

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Further, let V be the Riemannian connection and R its curvature tensor given by

(2.1) R(X,Y) = [VX,VY]-V\X,YÌ

for I , F G X(M). We denote by p the associated Ricci tensor and by r the corresponding scalar curvature. Moreover, let p*, respectively r*, denote the Ricci *-tensor, respectively the *-scalar curvature, defìned by

p*(x, y) = g(Q*x, y) = trace \z H+ R(x, Jz)Jy),

(2.2)

r* = trace Q*

for ali x,y,z E TpM,p E M. Hence, by using the first Bianchi identity, we have

1 v-^ (2.3) p*(x,y) = --2^R{x,Jy,ei,Jei)

. *

for an arbitrary orthonormal basis {e2, i = l , . . . , 2n} . (See for example [21]). Here R{x, y, z, w) = g(R(x, y)z, w) for ali x, y,z,w E TpM. From (2.3) we get easily

(2.4) p\x,y) = p*(Jy,Jx),

x,yETpM.

The Ricci *-tensor and the *-scalar curvature will play an important role in the rest of this paper. For a Kàhler manifold p* coincides with p but this does not necessarily hold on a general almost Hermitian manifold. Further, (M,g,J) is said to be a weakly ^-Einstein manifold if /?* = ^g holds. In particular, (M,g, J) is called a ^-Einstein spa.ce if, in addition, r* is Constant. Note that, in contrast to what holds for the scalar curvature of an Einstein space, r* is not automatically Constant on weakly *-Einstein manifolds. (We refer to the remark after Corollary 5.2.)

Next, we defìne the two-forms <p and t/> on (M,g,J) by

(2.5) (f(x;y) = trace (z H-» J ( V X J){S/yJ)z)

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and

(2.6) i>(xi y) = trace {z >-• R(x, y)Jz)

for ali x,y,z e TpM and ali p£ M. Then (2.3) and (2.6) yield

(2.7) t/>(xìy)=-2p*(xìJy).

Further, the first Chern form 7 on (M,g,J) is given by

(2.8) 8TT7 = -<p + 2ip

(see, for example, [9], [16]). The two-form 7 represents the first Chern class c\(M) of (M,g,J) (in the de Rham cohomology group) [10].

Now, for a one-form a on (M,g,J), let Ja be the one-form defined by (Ja)(X) = —ot(JX),X E X(M). Further, for an orthonormal basis {ej,i = l , . . . ,2n} of the tangent space TpM at p G M , w e denote by {el,i = l,...,2ra} its dual basis. Then, for an orthonormal basis {e,-}-of the form {ei,e2 = Je i , . . . , e 2 w - i , e2n = Je2n-i}, we get that {e1, J e 1 . , . . . , e 2 n _ 1 , Jc2"""1} is its dual basis. As a consequence, the Kàhler form fì is expressed by

(2.9) Q = - ( e 1 A J e 1 + e3 A J e 3 + ... + e271'1 A Je 2 7 1" 1) .

Finally, we mention that we shall adopt the following notational convention :

Rhijk = g(R(eh,ei)ej,ek), Rhijk =-9{R(J^h,ei)ejìe0y

Rhrfk = 9(R(JehJei)Jej,Jekyi

Pij ~ P\eiiej)i'"" iPìj = P\Jeii^ ej)i

Pij = P*(eiiej)i'" ,PÌJ = P*(Jei, Jej),

Jij = 9{Jei,ej),ViJjk = 9((VeiJ)ej,.ek).

and so on. Here'.{e,-,i = 1, ...,2?i} is an orthonormal basis of TpM at each point p E M.

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3. Nearly Kàhler manifolds

Now we consider the first special class of almost Hermitian manifolds. (M, #, J) is said to be a nearly Kàhler manifold if

( V x J)Y + (Vy J)X = 0

for ali X,Y £ X(M). Hence, a Kàhler manifold (VJ = 0) is automatically nearly Kàhlerian. A non-Kàhler nearly Kàhler manifold is said to be strictly nearly Kàhlerian. We refer to [8] for more details and references. Many examples are given in [7] and [23].

For a nearly Kàhler manifold we have the following identities (see [8] for more details):

(3.1) (VJXJ)JY + (VXJ)Y = 0

(i.e., (M,g,J) is a quasi-Kàhler manifold);

(3.2) p(x, y) = p(Jx, Jy), p*(x, y) = p*(Jx, Jy)

(and p* is symmetric),

(3-3) Yl ViP*i = 2XT' S Vip*i = 2XT*' i i

(3.4) p(x,y) - p\x,y) = Y^iVzJijWyJij)

for x, y e TpM,p e M, and

(3.5) T-T* = ||VJ||2 = Constant.

Finally, using (3.2), (3.4), we get that the first Chern forni 7 takes the following form [22]:

(3.6). 8x~fij = pfj - òp*j.

The geometry of nearly Kàhler manifolds has been studied extensively. It is known'that any six-dimensional strictly nearly Kàhler manifold is an

199

Einstein space with positive scalar curvature and its first Chern forni vanishes [14]. A. Gray proved that any eight-dimensional strictly nearly Kàhler space is reducible and is locally a product of a two-dimensionai Kàhler manifold and a six-dimensional strictly nearly Kàhler space. He also studied the structure of ten-dimensional strictly nearly Kàhler manifolds. (See [8] for more details.) Further, Y. Watanabe and K. Takamatsu proved in [22] that an irreducible strictly nearly Kàhler manifold with vanishing first Chern form is an Einstein space and M. Matsumoto proved that, conversely, the first Chern form of an irreducible strictly nearly Kàhler Einstein space vanishes. These observations lead to the following problem: Is a compact irreducible strictly nearly Kàhler manifold with vanishing first Chern class necessarily an Einstein manifold ?

T. Koda considered this question and gave an affirmative answer for compact irreducible strictly nearly Kàhler Riemannian 3-symmetric spaces [11], [12]. Here, we shall prove the following generalization of this result :

THEO REM 3.1. A compact irreducible strictly nearly Kàhler manifold with vanishing first Chern class and with Constant scalar curvature is an Einstein space.

Proof. Since r is Constant, (3.5) yields that r* is also Constant.

Next, since c\(M) = 0, the first Chern form 7 is exact and hence, there exists a one-form rj on M sudi that

(3.7) pfj - 5p*j = Virjj - VJTJÌ.

Transvecting (3.7) with pf- — 5/9?-= and using (3.2), (3.5) and the hypothesis, we get

(3.8) \\p - 5 / 1 | 2 - 2 £ Jjt(plt - 5pS)V,-f7>

= 2div ( ~ 2JT(V;J j t)(p i t - 5^)7/7

-2^J|<(V l-p.rt-.5Vj/>S)»/j

= 2div ( - Yl JJi(Vtr - ^tr*)Vj

=-2divC

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at each point p E M. Here ( is the vector field defined by

C = ^(52jjt(Pit-ZPit)Vj)ei i j,t

at p. Hence, from (3.8) and Green's theorem we obtain at once

(3.9) / \\p - bp*\\2dM = 0 JM

and so, p = bp*. The required result follows then from Corollary 3.5 in [22].

4. Almost Kàhler manifolds

The second special class of almost Hermitian manifolds is that formed by the almost Kàhler manifolds. (M,g, J) is said to be almost Kàhlerian if its Kàhler form Q, is closed or, equivalently, if

®X,Y,Z9{{VXJ)Y, Z) = 0

for ali X, Y, Z £ X(M), where (5 denotes the cyclic sum. Hence, any Kàhler manifold is almost Kàhlerian. A non-Kàhler almost Kàhler manifold is said to be strictly almost Kàhlerian. The first and important example of a compact strictly almost Kàhler manifold is Thurston's four-dimensional example given in [20]. See also [1] for an explicit description. Higher-dimensional compact examples are given in [4]. We note that the tangent bundles of these examples of strictly almost Kàhler manifolds are trivial.

For an almost Kàhler manifold we have [19], [24]

(4.1) (VJXJ)JY + (VXJ)Y = 0

for X,Y G X(M), i.e. (M, g,J) is a quasi-Kàhler manifold, and further

(4.2) r _ T * = _ ì | | V J | | » .

First, we prove

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THEOREM 4.1. A compact almost Kahler manifold {M,g,J) with vanishing first Chern class and non-positive ^-scalar curvature is a Kahler manifold with zero scalar curvature.

Proof. Using (2.5) and (4.1) we see that the two-form <p takes the form

(4.3) y>(*,y) = - Y^(V*Jij)(Vj,Jij)

at each point p G M. Further, (4.1) and (4.3) yield that (p is J-invariant. Now, we define the tensor fìeld <j> of type (0,2) by

<j>(X, Y) = <p(X, JY.),

X, Y G X(M). It follows that <f> is J-invariant and symmetric. Therefore we can choose an orthonormal basis {e^i = l , . . . ,2n} = {ej,e2 " = • Je i , ...,e2n-i?e2n =" Jc2n-i.} °f TpM, at each point p G. M, such that

/*?

(4.4) (^c,-,C>)) =

A? \

A2

A2

A 3

V

0

A2

A2 2ìi- J

for some real numbers Ai, A3,..., A2n-i- Then, <p may be expressed in the form

(4.5) ; \p = Aje1 A Je 1 + ... + A ^ e 2 " " 1 A Je2n~l

at each p € M. Further, (4.3), (4.4) and (4.5) yield

n (4.6) |VJ||2 = 2^AL_ 1 .

a = l

Next, let [fì] G #2(M;IR) denote the de Rham cohomology class represented by the Kahler form fì. Since c\(M) • = 0, we get at once c i ( M ) - [ft]w' = 0andso

(4.7) / 7 A i r - 1 = 0.

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Using (2.7) - (2.9) and (4.5) - (4.7), we then get

(4.8) / (4r* - | |VJ||2)dM = 0. JM

This, tpgether with r* < 0, implies VJ = 0 and r* = 0. The required result now follows since r = r* for a Kàhler manifold.

REMARK. The generalizations of Thurston's example, given in [4], are ali parallelizable and hence their first Chern classes vanish. So, by the theorem above, the *-scalar curvature of any almost Hermitian structure (#, J) tamed to a iìxed symplectic structure on these generalizations must have positive value at some point of M. (Note that it is incorrectly stated in [4] that r* - 0.)

Secondly, we prove

THEOREM 4.2. Let (M,g,J) be a compact 2n-dimensionai almost Kahler manifold with vanishing first Chern number and let (M, g, J) be weakly *-Einsteinian. Then the ^-scalar curvature r* iias to take a non-negative value at some point ofM.

Proof. For a weakly *-Einstein manifold, (2.7) - (2.9) and (4.5) yield

2r* (4.9) 8TT7 =-(p Q,

n

C l = l

at each p £ M. Hence, we obtain

• • . " ^ < - \ • • •

(4.10) (87r)w7n = 77,! T T ( — ~ A L - i ) ( e l A Je 1 A • • • A e271"1 A Je2"'1) TI

fi ci %

. I l

a—l

For r* < 0, (4.10) implies c\(M)n ^ 0 and this proves the required result.

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REMARKS. a) It is worthwile to compare Theorem 4.1 with Goldberg's conjecture stating that a compact almost Kàhler Einstein space is a Kàhler manifold [5]. (See also [16].)

b) Since in Theorem 4.2 we essentially used only (4.1), the result also holds for quasi-Kàhler manifolds.

•e

5. Hermit ian manifolds

Finally, we consider the class of Hermitian manifolds, i.e. almost Hermitian manifolds (M,g,J) with integrable almost complex structure J or, equivalente,

(5.1) (VJXJ)JY ~ (VXJ)Y = 0

for ali X,Y G X(M) [6]. From this condition it follows that the two-form ip may be expressed as

(5.2) <t>(x,y)=J2(^xJ)ab(^JyJ)ab a,b

for ali x,y G TpM,p G M. So, <p (respectively 0) is a J-invariant two-form (respectively a J-invariant symmetric tensor fìeld of type (0,2)) on M. We have

THEOREM 5.1. Let (M, J ) be a compact almost complex manifold with vanishing first Chern number. IfM admits a Riemannian metric g such that J is orthogonal with respect to g and the almost Hermitian manifold (M,g,J) is weakly *-Einsteinian with positive ^-scalar curvature, then J cannot be integrable.

Proof. Let J be integrable. Then (5.2) yields that we may consider an orthonormal basis {ei,i = l,...,2n} =. {ei,C2 = Je i , • • • j - ^ n - i j ^ n = J^2n-i} ofTpM, such that

n

i=l

(5.3)

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for some real numbers Ai, A3, • • •, \2n-i- Then, by (2.7) - (2.9) and (5.3) we obtain

(5.4) 8TT7 = - — n-cp

n

= D ? + A'-i)e2a_1 A Je2a~1

a = l

at p G M . This yields at once TI

(5.5) (8x7)" = n! f [ ( — + >&>-i)dM

at each p £ M. So, if r* > 0 011 M, the first Chern number C\(M)n must be positive and hence, the required result follows.

From this result we get a short proof for a result of C. Lebrun [13] :

COROLLARY 5.2. S6 has no integrable orthogonal almost complex structure.

Proof. Let ( 5 6 , #) be a six-dimensional unit sphere with the canonical Riemannian metric g of Constant sectional curvature 1. Further, let J be any orthogonal almost complex structure with respect to g on (S6,g). Then we get easily

P = %, p* = g

and hence, (6'6,</, J ) is a ^-Einstein space with Constant *-scalar curvature r* = 6. Since 7/2(56 , IR) = 0 ,c i (M)-= 0 and hence, cv(Mf vanishes. So, by Theorem 5.1, S6 cannot admit an integrable orthogonal almost complex structure.

R E M A R K . AS is now well-known, L. Bérard Bergery constructed an Einstein metric g with positive scalar curvature on M — C P 2 # C P . (See [18], for example.) T. Koda showed in an unpublished note [17] that (M,g) with a suitable almost complex structure J is a weakly *-Einstein Hermitian surface with non-constant positive *-scalar curvature. (Note that the first Chern number c i (M) 3 of Bérard Bergery's example equals 8.)

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Now, let (M,g,J) be again a compact Hermitian manifold. First we have

(5.6) ^ìjJab ~ V2jiJab = ~22RijaiJìb ~ Z^RijbtJat-t t

From this we get

i t

where

i

Now, we transvect (5.7) with Jja to get

(5-8) . £ JjaVÌjJai + Y^Jjoy^a = T* - T. i,j,a a,j

Put

si = / j JjgvjJgii Qj — / ; Jjawa-j,a a

Th.en (5.8) becomes

E v^« + E v>6 - E(v«y«»)(vi*«) - H I 2 - r*-r-i i i j , *

Using (5.1) and VjJja = — VjJia we see that the tliird term vanishes. Then, integration and Green's theorem yield

(5.9) f \\w\\2dM = f (T-T*)CIM = K-K* hi hi

where

(5.10) K= f rdM, K* = f r*dM. JM JM

So, we get from (5.9)

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THEOREM 5.3. Let (M,g,J) he a compact Hermitian manifold. Then K* < K with the equality sign if and only if (M,g,J) is a semi-Kàhler manifold, i.e. 60, = 0.

REMARK. Examples of non-compact homogeneous Hermitian manifolds of Constant sectional curvature —1 are given in [2]. For a general 2/z-dimensional almost Hermitian manifold of Constant sectional curvature e we have

r = 2n(2n — l)c, r* = 2nc.

Hence, by Theorem 5.3, we see that there do not exist 2n-dimensionai compact Hermitian manifolds of Constant negative sectional curvature.

The examples in [2], mentioned above, are group manifolds and their underlying Lie group G is a semi-direct product of the multiplicative group IR+ of positive real numbers and the additive group IR2w_1 defined by the follòwing :

(xUX2,' • ' ,X2n)(yi,y2,- ' ,y2n) = (#l2/ l ,« l2/2 + X2,--,X1y2n + X2n)

for («i,a?2,---,«2n),(2/1,2/2,••*,2/2n) G G= IR+ix IR2n_1.

We finish with

THEOREM 5.4. Let (M,g,J) he a In-dimensionai compact Hermitian semi-Kahler manifold with n > 3 and with vanishing first Chern class. If K > 0, then M is a Kahler manifold with K = 0.

Proof. Since (ilf,#, J ) is semi-Kàhler, we have

(5.11) w = 0.

Moreover, c\(M) = 0 implies that there exists a one-form t] on (M,g, J) such that 7 = drj. Then, using (5.11) we get

(5.12) r, Adii A Sln~2 = ^ ^ ] T « f c W r t + V ^ 6 a + ^lhb)dM = 0

a,b

at each p E M. Hence, we have

(5.13) / 7 A fìn_1 = / di] A nn~l = -{n -1)1 77 A dSl A Sln~2 = 0. hi hi JM

207

Now, taking account of Theorem 5.3, we get

(5.14) / T A ft"-1 = ( n " 1 ) ! / (4r* + \\VJ\\2)dM JM IGTT JM

= {J^JM(4T+ì]VJÌÌ2)dM'

So, (5.13) and (5.14) yield

/ (4r + ||VJ||2)</À/ = 4 A ' + / | |VJ | | 2 dM = 0 JM JM

and hence, the hypothesis implies K = 0 and V J = 0. This proves the required result.

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Kouei SEKIGAWA

Department of Mathematics, Niigata University

Niigata, 950-21, Japan.

Lieven VANHECKE

Department of Mathematics, Katholieke Universiteit Leuven

Celestijjienlaan 200B, B-3001 Leuven, Belgium.

Lavoro pervenuto in redazione il 20.12.1991.