CURVATURE TENSORS IN KAEHLER MANIFOLDS(l) · 2018. 11. 16. · 342 MALLADI SITARAMAYYA all...

TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 183, September 1973

CURVATURE TENSORS IN KAEHLER MANIFOLDS(l)

BY

MALLADI SITARAMAYYA

ABSTRACT. Curvature tensors of Kaehler type (or type K) are defined on a

hermitian vector space and it has been proved that the real vector space £„( V)

of curvature tensors of type fï on Fis isomorphic with the vector space of sym-

metric endomorphisms of the symmetric product of V , where V = V © V

(Theorem 3-6). Then it is shown that £„( V) admits a natural orthogonal decom-

position (Theorem 5.1) and hence every L e £ ( V) is expressed as L = L. + ¿„,

+ Ly. These components are explicitly determined and then it is observed that

Ly, is a certain formal tensor introduced by Bochner. We call Ly, the Bochner-

Weyl part of L and the space of all these £-„, is called the Weyl subspace of

Introduction. Singer and Thorpe [6] established a natural decomposition of

curvature tensors on an «-dimensional vector space with an inner product and then

studied the curvature tensor of a 4-dimensional Einstein space and its canonical

form. Nomizu [5] using this decomposition discussed generalised curvature tensor

fields satisfying the second Bianchi identity (which are called proper) on a

Riemannian manifold.

The purpose of this paper is to give a similar decomposition of curvature

tensors of Kaehler type on a 2r2-dimensional hermitian vector space and then study

curvature tensors on a Kaehler manifold. In this paper the study is purely local in

nature. The contents are as follows:

In §1 some basic facts about hermitian vector spaces (V, /, (,)) are given.

Kaehler metric on a complex manifold is defined and the properties of its curvature

tensor are stated in §2. In §3 a curvature tensor of type K on a hermitian vector

space (V, /,(,)) is defined and then proved that the real vector space ^K(V) of

Received by the editors August 1, 1972.

AMS(MOS) subject classifications (1970). Primary 53B35, 53C55; Secondary 32C10.Key words and phrases. Hermitian and Kaehler metrics, curvature tensor, complex mani-

fold, hermitian vector space, fundamental 2-form, Bochner-Weyl curvature tensor, Chern

class, Einstein manifold, tangent bundle.

( i) This formed part of the author's doctoral dissertation submitted to the University of

Notre Dame under the direction of Professor Yozo Matsushima. The author is grateful to

his adviser for the help and encouragement in this wotk.

Copyright © 1973, American Mathematical Society

341

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342 MALLADI SITARAMAYYA

all curvature tensors of type K on V is isomorphic to the real vector space

s (V • V ) of all symmetric endomorphisms of the symmetric product of V

(Theorem 3.6). §4 deals with Ricci curvature K(L) and holomorphic sectional

curvature k, of L in JcK(V). In §5 we have proved that ^K(^) is an orthogonal

direct sum of three subspaces (Theorem 5.1) and hence every L £ ¿lk(V) can be

uniquely written as L = Lj + Lw + L2, these components being determined

explicitly (Theorem 5.4); L™, is observed to be a certain formal tensor introduced

by Bochner [7] and hence we call L„, the Bochner-Weyl part of L. The subspace

of these Lw's is denoted by ¿^(V) and called the Weyl subspace. In a forth-

coming paper we study curvature tensors in i.„(V).

1. Algebraic preliminaries. Let V be a 2n-dimensional real vector space

with a complex structure /. Let ( , ) be an inner product of hermitian type on V.

Denote by V the complexification of V and ~~ denotes the conjugation in V

with respect to (for short, w.r.t.) V. Further, let L be a real endomorphism of V.

Then we can extend L to a complex linear map of V to itself; in particular, /C • 2

extends to V . Since / = - identity, / is a nonsingular semisimple endomor-

phism of V with eigenvalues + i and - i, where i = \J- 1 (i is -\J— 1 always

unless i is a subscript or superscript). Hence V = V © V where

V+ = \u £ VC | Ju = z'zzi and V~ = \u £ VC\ Ju = -iu\.

Moreover, V = V and hence complex dimension of V = n = complex dimension

of V".

The hermitian inner product ( , ) on V can be extended to V as a symmetric

complex bilinear form (also denoted by ( , )) and satisfies (V , V ) = 0. If we

define iu, v) = (u, v) for u, v £ V , then ( , ) is a positive definite hermitian form

on V+.

2. Curvature tensor of a Kaehler metric. Let M be a differentiable (i.e. c°°-)

manifold of dimension n. T AM) denotes the tangent space of M at p which is an

72-dimensional vector space. If M is a complex manifold of complex dimension n

then the tangent space TziM) at Z of M as a c°°-manifold is a real vector space

of real dimension 2n with a complex structure /. A Riemannian metric g on M is

called a hermitian metric if the inner product gz on TziM) is of hermitian type for

each Z £ M.

Let g be a hermitian metric on M. Define Q(u,v) = g (ju, v) for u, v £ TZ(M).

Then O is a real differential 2-form on M of type (1,1) and is of maximal rank.

We call ß the fundamental 2-form of the hermitian manifold (M, g). A hermitian

manifold (M, g) is Kaehlerian if the fundamental 2-form 0 is closed, i.e. dû = 0.


CURVATURE TENSORS IN KAEHLER MANIFOLDS 343

We give two important examples of Kaehler manifolds.

(a) The complex number space C" with the usual inner product is a Kaehler

manifold and Í2 = \/- 1 2z/Z; A d7J where (Z1) are the canonical coordinates

of C".

(b) The complex projective space P"(C) with the Fubini-Study metric is a

Kaehler manifold and so is every complex submanifold of P"(C) (cf. [l, p. 170]).

Let V denote the connection defined by the metric g. Then the curvature

tensor R of V is a (1, 3) tensor. That is, R(X, Y, Z) is a vector field. If we

write R (X, Y, *) - R (X, Y)* then R (X, Y), for fixed X, Y, is a (1, 1) tensor.

Moreover for fixed X, Y, R (X, Y) = V^ Vy - Vy Vx _ Vr^y] is a (1, 1) tensor

(cf. [2, p. 133]).

The following gives the properties of the curvature tensor R of the Kaehler

metric g on a manifold M.

Proposition 2.1 ([3], [4]). Let (M, g) be a Kaehler manifold and R be the

curvature tensor of g. Then (1) R (X, Y) ] = JR (X, Y) and (2) R (JX, JY) =

R (X, Y) for any two vector fields X, Y on M.

3. Generalized curvature tensors. In this section, we want to study curva-

ture tensors having properties given in Proposition 2.1 in a more general alge-

braic setup.

Let V be a 2rz-dimensional real vector space with an inner product (, ).

Denote by o(V) the vector space of skew symmetric endomorphisms of V. We

define a curvature tensor L on V as an o(V)-valued 2-form on V. Then for each

x, y £ V, L (x, y) eEnd(V) such that

(1) L (x, y) is bilinear in x and y;

(2) L(x, y) = - L(y, x);

(3) (L(x, y)z, w) +{z, L{x, y)u/) = 0.

Now assume V admits a complex structure / and a hermitian inner product (, ).

A curvature tensor L on V is said to be of type K on V if L satisfies the

following:

(1) L ijx, Jy) = L ix, y) for all x, y £ V;

(2) L(x,y)] = JL(x, y);

(3) L (x, y)z + L (y, z)x + L (z, x)y = 0 for all x, y, z £ V (Bianchi identity).

Denote by ^K(V/) the real vector space of all curvature tensors of type K on

V. Our goal is to understand iK(V).

Let L £ X K(V). Extend L to V . Then L is a skew symmetric bilinear

map on Ve with values in End(Vc). Extend / and ( , ) to Ve as in §1. Then

(1) L(]x, jy) , L(x, y) for all x, y £ V is equivalent to L(v\ V*) = 0.

(2) L(x, y)j = ]L(x, y) implies L (x, y) leaves V invariant.



(3) Bianchi identity holds in the complex case.

Since L(V , V ) = 0, the Bianchi identity gives

(a) L (x, y)z = L (x, z)y and hence L (x, y)z is symmetric in y and z, and

(a ) L (x, w)y = Liy, w)x and hence L (x, w)y is symmetric in x and y.

Note that since (a ) is obtained from (a) by conjugation a curvature tensor L

satisfies Bianchi identity if and only if L satisfies (a). Now define the map F, :

V+x V+x V+ x V+-> C by

(3.1) FLiy,z,x,w) = Liy,z,x,w) = (Lix, w)z, y) for all x, y, z, w £ V+.

Then clearly this map has the properties

(a) F. is bilinear in x and y and anti-bilinear in w and z;

(b) F, is symmetric in x and y and also in z and w.

Since given any F: V x V x V x V —* C satisfying (a) and (b) there exists a

unique L £ Jl„(V) such that F = FL (as given in (3.1)); we call such an F also

a curvature tensor of type K on V. Note that FLiy, z, x, w) = FLix, w, y, z)

(from (b)).

Define hermitian inner product in V by (x, y) =(x, y). Next we extend ( , )

on V to V ® V by defining (x ® y, w ® z) = (x, zzv)(y, z).

Now we have

Proposition 3.2. // f\w, z) is an anti-bilinear function on V then there

exists u £ V ® V such that ¡{w, z) = iu, w® z).

Proof. Obvious.

Since, for every fixed x, y £ V , FL(y, *, x, *) is anti-bilinear on V , by

Proposition 3.2, there exists u (x, y) £ V ® V such that FL(y, z, x, w) =

iuix, y), w ®z). Since FL is bilinear and symmetric in x and y the map u: V

xV —> V ® V sending (x, y) to u (x, y) is bilinear and symmetric. Hence

there exists a linear endomorphism U : V ®V —> V ®V such that U (x ® y)

= z/(x, y), and hence we have

(3.3) iU'ix®y),w®z)=FLiy,z,x,w).

Moreover, using (3.3) and (3.1) we see easily that U is a symmetric endomorphism

of V ® V w.r.t. ( , ). Now we carry this Í7 one more step forward.

Consider the symmetric product V • V of V . V • V is, by definition,

the quotient of V ® V. by the subspace A consisting of all skew symmetric

tensors. Denote by 27 the canonical projection of V ® V onto V ■ V . Then

t £ V ® V implies t = s + a where s is a symmetric tensor and a £ A. Define

an inner product in V • V by (ztO), 77(/')) = (s, s') where s and s' are respec-

tively the symmetric parts of / and / and is, s ) is the inner product of s



and s' in V ® V . Since symmetric and skew symmetric tensors are perpen-

dicular, the inner product in V ■ V satisfies the condition (t, t ) = (n(t), n(t ))

+ (a, a ). Moreover, from our definition it follows that (x ■ y, w ■ z) =

Vl\(x, w)(y, z) + (x, z)(y, w)\ where we have put x ■ y = n(x ® y) for all x, y £ V .

Now we claim that there exists a symmetric linear map U: V ■ V —> V • V

such that FL(y, z, x, w) = (U (x ■ y), w ■ z). In fact, since U : V ® V —> V

®V is defined by U (x ® y) = u(x, y) and, since u(x, y) is symmetric in x and

y, U maps kernel 27 into zero and hence U induces a unique linear map U: V •

V —> V ■ V such that U (x, y) = 77.(7 (x ® y)). It remains to show

(3.4) (U(x . y),w . z)= iu'ix ®y),w ®z).

But we have ill ix ® y), w ® z) = ((7 (x • y), w ■ z) + (a, a ) where a and a

denote the skew symmetric parts of U (x®y) and w ® z respectively. Since

F j\\y, v, x, u) = (U (x ® y), u ® v) and since FL is symmetric in u and v,

U (x ® y) is orthogonal to the subspace A of skew symmetric tensors and hence

U (x ® y) is a symmetric tensor and hence a = 0. This completes the proof of

our claim. Moreover, we can prove that this U is a symmetric endomorphism of

V+ • V+.

Denote by s(V ■ V ) the vector space of all symmetric endomorphisms of

V ■ V . In summary, starting with a curvature tensor L £ JL„(V) we obtained a

UL £s(V+ ■ V+) such that

(3.5) (UL(x • y)> w • z) = FL(y, z, x, w) = (L(y, z)w, x) .

Denote this map from ^K(V) into s (V ■ V ) by <p. Now we have the following:

Theorem 3.6. The map rp defined above (by (3.5)) is an isomorphism of

£K(V) onto s(V+ ■ V+).

Before proceeding with the proof of this theorem we give some facts about

bases of V and V • V .

,+Let e,, e?»,..., c be an orthonormal basis of V , an 27-dimensional hermi-

1 ¿ 72 '

tian vector space, w.r.t. ( , ). Then ¡c,, e2, ■ ■ ■ , e , <?,, e2, ■ ■ ■ , <? ! forms a

basis of V and we have

(1) {<?•(?., z < /', z, ;' = 1, 2, • ■ . , 27 ! is a basis of V ■ V .

(2) \e. ■ e¿, \J2 e. ■ e i < j, j = 1, 2, • ■ • , n\ is an orthonormal basis of

V+ ■ V+ w.r.t. ( , ).

(3) Putting f. = (e. + cf)/v/2> //, = V~ U,- - ?)/V2. 1 = 1, 2,..., «,

•/¿' //•! 1S a basis for V as a 2/2-dimensional real vector space.

(A) L(e.,~e~)=L(e., ?.).



Proof of Theorem. That c6 is an injective real linear map is clear from (3.5). We

have to show cp is onto. That is, given (7 £ s (V • V ), we have to define for

each x, y £ V a real endomorphism L (x, y) of V such that L £X-KiV) and L

when extended to a complex linear map of V should satisfy (3.5).

Taking x = e,, y = e. ik < /), and » = c , z = e in (3.5) we get

p = 772,

<3-7) L/2T*-

ÍÍJ .. ifmm,«z

U ,, if p ¿ m ik< l, m < p).

Because of symmetry in ¿ and / and in p and 722 (3.7) defines L for all suffixes.

Now we want to define the corresponding real endomorphism L of V so that

L e£K(V). Let {/, //, z = 1, 2, • • • , 72} be real basis of V such that /fe = ek +

ë, and //, = 2 (e, - ë, ). Then we compute L in terms of components of Í7 (by

(3.7)) as in

Using (3-7) and the fact that U is hermitian we can show the right-hand side

of (1) is real.

(2) <L(/,, If )/„, /,) = ¿¡Lr r - L_+ L r -~ L - J which is alsov ' \ x>k' Jlm''p' 11' pTmlt Ip km pTkm Ipmïï

teal.

(3) (L(/,, / )//„, /.)= i\L r _ - L_+ L ,_ r - L _ 1. This is also"•■" x Vfe« Im'JIp' IV pfkm Ipkm pTmK lp mit

teal from (2).

(4) <L(/,, //)//, /.) = -¡L + L,_ -L^-I. r _}. This is'* m P '«' pTfcm /pmF /p fem pT m%

also real.

From (2) and (3) we get

(5) <L(4. //,)/„./,>-= <L (/V/J //,./,>•Now using these relations (1)—(5) we define L on the basis elements

i/t. //J of V as follows: Define L(fk, fj by (l)and (3); L (/fc, //J by (2)

and (4); L (//fe, //m) = L (/fe, /J and L (/fc. /J//p = /L (fk> fjp. Then this Lis defined as a real endomorphism of V in terms of the components of given (7

and L has the properties

(a) L(//,//) = -L(/;.,/;),

(b) L (//., //.) = L (/., /.), and

(c) L (//, f! )] = JL (//, //), where {/,', f[ ,•••,/„', /„' + 1, • • • , /2'„! denotes

[/., //z!, ¿=1,2,...,«.

Hence L belongs to J~kiV) provided L satisfies the Bianchi identity, i.e.

(3.8) L(x, y)z + Liy, z)x + Liz, x)y = 0.

We verify this on basis elements of V. Let 2L denote the left-hand side of (3.8).



Case (a). Let x = fk, y = fm, z = / and w = fr Then from relations (1)-

(5) we get

(IL, ft) = Lin_ + LpTm- - EpTk_ - Llp_mç

lkmp~ kl pm klmf Ikpm

Irnpk ml kp~ ml pk ¡mkp~

which can be seen to be zero using (3.7) and U is hermitian.

Similarly in the other cases

(b) x = fk, y = fm, z = fp and w = //,,

(c) x = fk, y = ]fm, z = fp and w = f¡ or //,,

<d> * = /*. y = JL> z = Jfk'

M * = Jfk. y = //„,. * = ]fp,etc., we can show that C^L, w) = 0 using L (Jx, Jy) = L (x, y) and L (x, y)j =

]L(x, y).

Thus L as defined above belongs to ,L.,(V) and in view of (3.5) we have

</j(L) = U, hence <p is surjective.

This completes the proof of the theorem.

Remarks. (1) Because of this isomorphism </> between ^-^^^ anc^

s(V ■ V ) we call elements of s (V ■ V ) also curvature tensors of type K

and we identify £K(V) and s (V ■ V ). Sometimes for L e£K(V) we denote

its image under (p simply by L instead of U, or fp(L).

(2) If J^R(V) denotes the vector space of curvature tensors on V satisfying

the Bianchi identity (here V is a real inner product space) then £R(^) is

isomorphic with a proper subspace of s (V A V) (cf. [6]). But in the Kaehler

case we have ¿^(V) is isomorphic with s (V • V ).

4. Ricci and holomorphic sectional curvatures of L. We define a linear

map K: £K(V) —' End(V ), called the Ricci map, as follows.

For every L ei_K(V), K(L) is an endomorphism of V such that K (L)x =

¿s_j L(x, ës)es where x £V and {ej (i = 1, 2,- • ■ , n) is an orthonormal

basis of V w.r.t. ( , ). We call K(L) the Ricci tensor of L. It is easily seen

that K(L) is a symmetric endomorphism of V w.r.t. ( , ). If we define K(x, y)

= (K(L)x, y) then K (x, y) is a symmetric bilinear form on V , called the

Ricci form of L in £K(V).

Let P(V ) denote the complex projective space of complex lines in V .

For any L ei^fV) we define a function kL: P(V ) —' C as follows.

Let v be any vector in a complex line P of V such that ||f || = 1 w.r.t.

( , ). Then



<x.L(P)= <L(i7, v)v, v) = -iL'iv • v), v . v).

Then &L(P) is well defined for any P in P ÍV ).

Remark. kL(P) is, in fact, a real number because L is hermitian. Moreover,

if we write v £ V as v = (X - i]X)/\j2, X, JX £ V (the corresponding real 2-

plane in V is spanned by X/y/2, ]X/yj2) then (L (v, v)v, v) = (L(X, JX)]X, X)

£ R. So our definition of kL is the usual definition of holomorphic sectional

curvature function.

Definition of symmetric (wedge) product of two endomorphisms. Let V be a

vector space and A, B £ End(V). Let V ■ V = V ® V/A ' (resp. V A V = V®

V/N) where A is the subspace of V ® V generated by x ® y - ® x tot all x,

y £ V (resp. N is the subspace of V ® V generated by x ® x for all x £ V).

Then A ®B £ End(V ® V) is defined by (A ® B)(x ® y) = Ax ® By and A ® B +

B ® A leaves A (resp. N) invariant and hence induces an endomorphism of

V • V (resp. V A V). We denote this endomorphism by A • B (resp. A A B).

Then we have (A . B)(x ■ y) = (Ax . By) + (Ay . Bx).

Now let A, B £ s (V ). Then we can show that A ■ B £ s (V • V ). In fact

((A • B)(x • y), w • z) = (Ax • By, w • z) + (Ay • Bx, w • z)

= (x • y, Aw ■ Bz) + (y • x, Aw • Bz) = (x • y, (A • B)(w • z)).

Therefore, by Theorem 3.6, A . B corresponds to a curvature tensor of type K on

V and we denote this by LA B £ j^k(V).

Proposition 4.1. Let L v L2 £ £K(V) such that

(LjizT, v)v, v) = (L2(v, v)v, v) for all v £ V .

Then L. = L-..

This is the same as Proposition 7.1 in [3, p. 166].

Proposition 4.2. Let L £ £„(V). Then kL is constant (say c) if and only

if L = - c LA „ with A = B = 1/\J2 £ s (V ), where I denotes the identity endo-

morphism of V .

Proof. (=») Let P eP(V+). Then kL(P) = - (L'(v ■ v),v ■ v) = c

On the other hand, / • 7/2 £ s (V+ ■ V+) and

(4.3) (L¡%¡/2(v,v)U, v) = -((!• l/2)(v • v), v • v) =-1 for all v.

Hence (L (v, v)v, v) — c = — c(L • , /, (v, v)v, v) for all v £ V and hence L =



Conversely, given L = - cL,,y2 we can see easily from (4.3) that kLiP) = c

for all P eP(V+).

Let \e., e2, • ■ • , e } be an orthonormal basis of V w.r.t. ( , ) KA ß denotes

the Ricci tensor of LA ß. Then

s = I

Then

(K, J„~{KA Re.,e.) = -y HA. B)ie..e ),e . e.)Zl'fi ¡j A • a i ' *^ j s s z

= -lÂA.. Tr ß + B.. Ti A +(Aß)..+ (ßA).J.'1 17 zj z;

So we get

(4.4) -2K. _ = (Tr A)ß + (Tr ß)A + AB + BA

and hence

(4.5) - Tr KA _ B = Tr A Tr ß + Tr Aß.

Recalling L is the image of L under </> in Theorem 3.6, we can prove that

Í4.6) Tr KiL) = 2 Tr L' = 2 Tr (7L-

We have inner product in X.AV) defined from that of V as

z,;',/t,/

The inner product ( , ) of V also induces an inner product ( , ) on s (V ■ V )

(observe that, in general, if W is a complex vector space with an inner product

and s (W) is the real vector space of all symmetric endomorphisms of W then we

can define an inner product ( , ) in s (W) by (5, T) = Tr iST)). Then we have

(4.7) iLvL2)= Z (LiWL2WTr(L'i'LV-i,j,k,l

Let £¿(V) = IL e lKiV)\ L' = XI . /, A e R¡. Then £¿(V) is a real 1-dimen-

sional subspace of £K(V). Let Orth(£K(V)) denote the orthogonal complement of

^(V) in ■»'-/((VO w.r.t. inner product defined above. Then in view of (4.6) and

(4.7) we have

Orth (£¿(10) = ! L e lK(v) \ Tr K{L) = 0},

Hence we have the decomposition



£Kiv) = £lKiv)(BOnhi£1Kiv)).

5. Decomposition of the space £K(V). Recall £¿(l0 = (L e£K(V)| L' =

A7 • 7, A eR!. Let ££(V) = {L e£K(V)| K(L) = 0| where K is the Ricci map.

Then clearly £K(tO C Orth (£K(V0). Now let £K(V) be the orthogonal complement

of ¿~K(y) in Orth (£K(V)). Therefore we have the natural decomposition of £i(-(V/)

as £K(V) = £¿(V) ©££(V) ®£2KiV). Now we have the following:

Theorem 5.1. Ler V be a real 2n-dimensional hermitian vector space. Let

JL„(V) be the real vector space of all curvature tensors of type K on V. Then

£„(V) has the decomposition

£ Jv) = £!L(lO ® S.WAV) © £1(V) (orthogonal)K K K K

where

(1) ¿lk(V) = \L e£„(V)| L has constant holomorphic sectional curvature];

(2) ££(V) = \L e£K(V)| L has Ricci curvature zero];

(3) 2%(V) © £2 (v) = \L £ £K(V)| TtKiL) = 0);

(4) £¿(10 © £*(V) = ¡L £ £K(V)| K (L) = XI, X £ R!.

Proof. All except (4) are proved above.

Suppose L e£¿(V)©££(V). Then L = Lt + L„, and hence K(L)=K(Lj)

because K is linear. But since L, = 7 • 7 and using (4.4) we get K(Lj) =

- in + 1)1 = KiD. Conversely, let K(L) = À7, A e R. Let L, be the curvature ten-

sor corresponding to L j = p7 • / £ s (V • V ) with p = - X/in + 1). Then L, e

£JC(V) and K(L - L,) = 0. Hence L - Ll £ ££(V). So L = Lj + L„,.

Recall that a Kaehler metric g on a manifold M is called Einstein if R, the

Ricci tensor of g, is a scalar multiple of g at each point p of M.

Corollary 5.2. Suppose (M, g) z's a Kaehler-Einstein manifold. Then the cur-

vature tensor L (o/ g) belongs to

£}¿Tp(M)) © £WK(Tp(M)), Vp 6 M (V rzzearzs "¡or every")

This is immediate from (A) of Theorem 5.1.

Corollary 5.3. If M ¿s a connected Kaehler-Einstein manifold of complex

dimension > 2 (i.e. K(L) = A(p)7 /or eyery p e M) ¿èer? A z's a constant function.

Proof. By (5.2), Lp 6 £¿(^0*0) © £.*& p(M)) and hence Rp = K(L)p = X(p) I,

i.e. R.-r- = A(p)g.,- (in local coordinates). Then the first Chern class of M is

represented by <A = -(l/2/r)A(p)Q where Í2 = ig^dz1 A aV is the fundamental

2-form of g and if/ = (1/2ni) R •■>- dz' A az"' is the Ricci form of g. Since if/ is a



closed 2-form we have 0 = dxft = - il/2n)dX A 0. Then dX = 0 because wedge

product with ii is an isomorphism of complex vector spaces for complex dimension

of M > 2. Hence A is a constant function on M.

Now we give the components of L £ x.AV) in the decomposition.

Theorem 5.4. Let L £X.KiV) and K(L) be the Ricci tensor of L. Further let

X = TtK(L) and L., Lw, L2 be the components of L in the decomposition of

Theorem 5.1. Then

(5.5) Lj=-A(/. /)/«(«+ 1),

(5.6) L'2 = -2(K • l)/in + 2) + 2X(I . 0/72(72 + 2),

(5.7) L'w = L' + 2iK - ¡)/in + 2) - A(/ • /)/(„ + l)in + 2).

(Here K = K(L) and I = identity endomorphism of V .)

Proof. Let R,, R2. Pw denote respectively the right-hand sides of (5.5),

(5.6) and (5.7). We show that Rj e£¿(lV), Rw e£^(K) and R2 e££(V). Clearly

R1 e£¿(V).

rí(Rw) = K(L) + 2K(LK< >/(« + 2) - \K{L¡m¡)/{n + Din + 2)

which is seen to be zero using (4.4). Hence R„, £ A.„(vO.

It remains to show R2 £x.KiV).

First we observe that TrK(R2) = 0 (use (4.5)) hence R-, e ££(V) © ££(V).

It suffices to show that R2 is perpendicular to ¡tAV). That is, Tr (RIr') = 0 for

all R e£.*(V).

We can choose an orthonormal basis ie .,•••, e ) of V such that K e. =

A.e., 2 = 1, 2,. •• ,72 (since K £s(V+)). Then (K • 00?. • e.) = (A. + A.)e. • e.,z z' ' * z ; z / z ;

i, j = 1, 2, • • • , n, and (/ • /)(«. • e.) = 2e. • e. and hence

(5.8) R,(e. • e.)= (-2/(rz +2))|(A. +A.)-2A/rz}e . e Vz,/.2 1 ; 1 ] 1 j ''

For R £££(V),

¡(5.9) V R . , = 0 for all /, /

(since /((R) = 0). In particular

(sao) y R . . = 0.s,7

Then (R2. R) = Tr(R'2R') = 1..^)..^.. (l < 7, s < /). Since R^, = 0

unless (f, /) = (s, /) we have

(5.11) Tt(R'R')=y\iX. + X)-2X/n\R'.... (/</).z.;

Now using (5.9), (5.10) and skew-symmetry in i and /, we can show that each



term in (5.H) is separately zero, and hence R2 e £K(V). Since the decomposition

is unique we have R j = L ,, R2-L2, Rw = L^,.

Remark. In the Kaehler case Lw is given by

L'w = L' + 2(K ■ /)/(« + 2) - (Tr K)(l • l)/(n + l)(n + 2).

Then if we compute the components (Ew)ag g w.r.t.an orthogonal coordinate (ea)

of V we see that this is the formal tensor introduced by Bochner (which he

denotes by K) up to sign (cf. [7, p. 161]). We call any Lw £^K(V) a generalized

Boehner-Weyl tensor and £K(V) is called the Weyl subspace of £K(V).

Now we can restate our Theorems 5.1 and 5.4 in terms of s (V ).

Theorem 5.12. Let ÍV, ],(,)) be a real 2n-dimensional hermitian vector

space. Let £K(V) = £^(V) © ££(V) © ££(V) be the decomposition of Theorem

5.1. Then

(a) £¿(V) © £¿(10 = \L £ £K(V)| L' = A . 1, A e s (V+)!.

(b) The Ricci map K: £K(V) — s(V+) z's onto, ket K = ££(V) and K is

bijective from £¿(V) © ££(V0 onto siV+). Furthermore, 7<(£¿(V)) = {A es(V+)|

A =A7, A eR¡ aTzz/ K(££(V)) = \A £siV+)\ Tr A = 0|.

Proof of (a). Let L = L^., 6 £K(V) with A e s (V+ ). Then L'= A . I. Let

L = L. + L2 + Lw. To show L^, = 0 but finding K = KÍL ) and A = Tr K from

L = A . 7, and substituting for 7< and A in Lw we see easily L w is zero. Con-

versely, if L = L, + L2 then L^ = 0 and hence L' = A . I with A = A//(t2 +1)(t2 + 2)

-2K/(t2+2) £s(V + ).

Proof of (b). Let B es(V+). Write B uniquely as B = Bn + p7 with BQ £

s(V+) and TrBQ = 0. Then, from (4.4), K ÍL,_2B g/)/(n+2)) = ß0. Also we have

K(L y. ,) = ¿^(72 + 1)7 for any real number £; in particular for Ç =— p/U + 1) we

have K(L_/_./(fi+1)M = /xi.

Set A = - ((2B0 • /)/(« + 2) + (p/ . /)/(« + 1)). Then K(LA) = BQ + pi = B.

Hence /< is onto and the other statements are all obvious.

Corollary 5.13. The real dimension of £„(10 = 22 - 1 where n = complex

dimension of V.

This is immediate from the bijectivity of K: £jt.(V)©£^(V) —> s (V ).

Corollary 5.14. Let V be a 2n-dimensional real vector space with a complex

structure ] and an hermitian inner product ( , ). Then the real dimension of £K(V)

= 722(n - l)(n + 3)/A where n = dimCV.



Proof. Since dimRs (V+ • V+ ) = in in + 1 )/2)2, dimR£K(V) = «2(r2 + l)2/4.

Hence by Corollary 5.13, dimR££(V) = t22(t2 + l)2/4 - 722 = 722(t2 - 1 )in + 3)/4.

Note that the Weyl subspace ¿LK(V) is of real dimension 5 when V is of complex

dimension 2. In general to give some characterisation of the Weyl subspace

i-K(V) seems quite difficult.

REFERENCES

1. R. C. Gunning and H. Rossi, Analytic functions of several complex variables,

Prentice-Hall, Englewood Cliffs, N. J., 1965. MR 31 #4927.2. S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, Inter-

science, New York, 1963. MR 27 #2945.

3. -, Foundations of differential geometry. Vol. II, Interscience Tracts in

Pure and Appl. Math., no. 15, Interscience, New York, 1969, MR 38 #6501.

4. A. Lascoux and M. Berger, Varietes Kahleriennes compactes, Lecture Notes in

Math., vol. 154, Springer-Verlag, Berlin and New York, 1970. MR 43 #3979.

5. K. Nomizu, On the decomposition of generalized curvature tensor fields (to

appear).

6. I. M. Singer and J. A. Thorpe, The curvature of 4-dimensional Einstein spaces,

Global Analysis (papers in Honor of K. Kodaira) (edited by D. C. Spencer and S. Iyanaga),

Univ. of Tokyo Press, Tokyo, 1969, pp. 355-365. MR 41 #959.7. K. Yano and S. Bochner, Curvature and Belti numbers, Ann. of Maths. Studies,

no. 32, Princeton Univ. Press, Princeton, N. J., 1953. MR 15, 989.

DEPARTMENT OF MATHEMATICS, MADURAI UNIVERSITY, MADURAI-21, INDIA.


CURVATURE TENSORS IN KAEHLER MANIFOLDS(l) · 2018. 11. 16. · 342 MALLADI SITARAMAYYA all...

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