einstein-like manifolds which are not einstein

ALFRED GRAY

E I N S T E I N - L I K E M A N I F O L D S W H I C H

A R E N O T E I N S T E I N

1. INTRODUCTION

Two important classes of Riemannian manifolds are the Einstein manifolds and the Riemannian manifolds with constant scalar curvature. We denote these classes by e* and c~; also, let ~ be the class of manifolds with parallel Ricci tensor. Then we have

The main point of this paper is that there are two interesting classes of Riemannian manifolds, to be denoted by ~ ' and ~ , which lie between and ~. The defining relations are:

~ : V~pj.k + V~p~s + Vjpk~ = 0,

~ : V~ps~ - Vjp~ = 0,

where p,j denotes the Ricci tensor (see Section 2 for the exact definitions). The classes d and ~ do not seem to have been studied extensively.

Although these classes are simple enough to define, I have been unable to find a discussion of them in any of the classical books on differential geometry. However, recently there has been some investigation of the class d in [21], [22] and ~ in [2], [3], [16], [18], [20], [21]. In [22] Sumitomo proves that if the algebra of differential operators of a compact homogeneous space M is commutative, then M ~ d . For the class ~ one approach is to investigate 'Codazzi-tensors', that is, the general tensors which satisfy the identity g . According to [2], [3] this is the same as saying that the curvature operator is harmonic, viewed as a vector valued 2-form.

There are many compact homogeneous spaces in the class ~'. We give examples of such spaces, including homogeneous metrics on S 3, in Sections 4 and 7.

In Part I of this paper-- that is, Sections 2 through 7--we show more precisely how the classes d and g fit between ~ and ~. In Section 2 we show that the following inclusions exist between the various classes:

(1.1) g c ~ = d n U ~ c ~ c

where ~ denotes the class of all Riemannian manifolds. All of the inclusions in (1.1) are strict, as we demonstrate in Sections 4 and 5. Section 3 is devoted

Geometriae Dedieata 7 (1978) 259-280. All Rights Reserved Copyright © 1978 by D. Reidel Publishing Company, Dordrecht, Holland

260 A L F R E D G R A Y

to an interpretation of the classes d and ~ based on the representations of the orthogonal group. We prove in Sections 3 and 5 that from this point of view d and M are the only classes between ~ and c~. The inclusions (1.1) are discussed for K/ihler manifolds in Section 6 and nearly Khhler manifolds in Section 7.

In Part II of the paper (Sections 8 through 11) we prove some global theorems for the classes d and ~. Similar theorems have been proved by a different method by Berger and Ebin [2], Ryan [18], Simon [20], [21] and Wegner [24], [25]. For example, we prove

THEOREM 1.1. Let M be a Riemannian manifold whose Ricci curvature satisfies V~p~ = V@~, and suppose that M has positive sectional curvature. Then the Rieci curvature has no local maximum or minimum on the unit sphere bundle of M. In particular, i f M is compact, then M must be an Einstein manifold.

THEOREM 1.2. Let M be a Riemannian manifold whose Rieci tensor satisfies V~pj~ + Vkp~j + Vjpk~ = O, and suppose that M has negative sectional curvature. Then the Rieci curvature has no local maximum or minimum on the unit sphere bundle of M. In particular, i f M is compact, then M must be an Einstein manifold.

We also give generalizations of Theorems 1.1 and 1.2 in which we assume that M has nonnegative or nonpositive sectional curvature. It should be pointed out that Theorems 1.1 and 1.2 and our generalizations of these theorems follow from results in the papers previously mentioned, together with Hopf's maximum principle. Our point in proving these theorems here is to make use of a class of differential operators on the unit sphere bundle of a Riemannian manifold. These differential operators are also useful in other contexts (e.g. [12]).

I wish to thank M. Berger, O. Kowalski, A. Lichnerowicz, and P. Ryan for useful advice concerning the subject matter of this paper.

PART I

THE I N T E R M E D I A R Y CLASSES d AND

2. DEFINITIONS AND NOTATION

Let M be an n-dimensional C °%Riemannian manifold, and let f ( M ) be the Lie algebra of C ~ vector fields on M. Denote by ( , ~ the metric tensor of M, V the Riemannian connection on M, and Rxy = Vfx,y~- [Vx, V~,]

E I N S T E I N - L I K E M A N I F O L D S W H I C H ARE NOT E I N S T E I N 261

(X, Ye ~ (M)) the symmetric bilinear form p on ~ ( M ) given by

p(X, Y) = ~ <Rx~, Y, E,>, i=X

for X, YeS , (M) , Further, the scalar

~" = ~ p(E,, E,) = ~ <RE,E,E,, Ej>. i=1 i = l

curvature operator of M. The Ricci tensor of M is the

where {El . . . . . E,} is an arbitrary local flame field. curvature r of M is the real-valued function given by

For m e M we denote by Mm the tangent space to M at m. Then we write Rwx~ for the value of the curvature tensor on tangent vectors w, x, y, z • Mm ; similarly, we use pwx to denote the value of the Ricci tensor on w, x e Mm. Also, we shall sometimes use such expressions as Rx,uB (instead of

RxE~(m)vEa(m))" For simplicity we consider only positive definite metrics in this paper.

However, all of the definitions that we give make sense for indefinite metrics, in particular, Lorentz metrics. Some, but not all, of our results remain valid. We shall not go into the details.

By definition, a Riemannian manifold is in the class 6 o if and only if there exists a real-valued function h on M such that

p(X, Y) = a<X, Y>

for all X, Y e ~ (M) . If dim M/> 3, then ~ must be a constant (see for example [13, vol. 1, p. 293]). If dim M = 2, we require that ~ be constant in order that M • d ~.

Denote by ~ the class of all Riemannian manifolds with the property that

Vx(p)(Y, Z) = 0

for all X, Y, Z e Y'(M). There is not a great deal of difference between the classes ~ and ~, because of the following (well-known) result:

T H E O R E M 2.1. Let M • ~. Then M is locally the Riemannian product of Einstein manifolds. (This follows from the fact that the Ricci curvature remains invariant under the action by the holonomy groups of M, and the de Rham decomposition theorem.)

Next we define the classes d and ~ :

(2.1) M • d if and only if Vx(p)(X, X) = 0 for all X • Y'(M);

(2.2) M • ~ if and only if Vx(p)( Y, Z) = Vr(p)(X, Z) for all X, Y, Z • ~r(M).

2 6 2 A L F R E D G R A Y

It is easy to prove, using polarization, that (2.1) is equivalent to

(2.3) Vx(p)(Y,Z) + Vz(p)(X, Y) + Vr(p)(Z, X) = 0

for all X, Y, Z e Y'(M). Furthermore, using the second Bianchi identity it can be shown that for any Riemannian manifold

(2.4) ~ <VE~(R)E,xY, Z} = Vr(p)(X, Z) - Vz(p)(X, Y) i=1

for X, Y, Z e Y'(M). From (2.4) it follows that (2.2) is equivalent to

(2.5) ~ <VE,(R)E,xY, Z> = 0

for X, Y, Z ~ 5f(M). Some authors describe Equation (2.5) by saying that the divergence of the curvature tensor is zero.

The basic inclusion relations between the various classes are given by the following theorem. (We consider here only the existence of the various inclusions; the strictness of the inclusions will be taken care of in Theorem 5.3.)

THEOREM 2.2. We have

(where ~ denotes the class of all Riemannian manifolds). Proof. Everything is immediate except the equality ~ = d c~ ~ and the

inclusion d u ~ ~ ft. It is easy to check that if the Ricci tensor of M satisfies (2.2) and (2.3),

then it is parallel. Hence d c~ ~ _c ~ ; the reverse inclusion is obvious. Next assume M e ~¢. Then from (2.3) we have at each point m e M that

(2.6) Xz = ~ Vx(p)(E,, E,) = - 2 ~ VE,(p)(X, EO, I=I i=l

where X e W(M) and {El . . . . . E~} is a local frame field. On the other hand, it follows from the second Bianchi identity that

(2.7) X~- = 2 ~. VE,(p)(X, E,)

for X e 5f(M). From (2.6) and (2.7) we have that X~ = 0; thus d c c#. Finally, assume M ~ ~'. Then from (2.2) we have

(2.8) X~--= ~ V~,(p)(X, E,) /=1

for X e ~(M). But (2.7) is valid for any Riemannian manifold. Then (2.7) and (2.8) imply that Xr = 0. Hence ~ _ cg, and so ~¢ u ~' c_ c~.

E I N S T E I N - L I K E M A N I F O L D S W H I C H A R E N O T E I N S T E I N 263

3. T H E I N T E R M E D I A R Y C L A S S E S F R O M T H E P O I N T OF V I E W

OF G R O U P R E P R E S E N T A T I O N S

In this section we motivate the study of the classes d and ~ from the stand- point of representations of the orthogonal group. Consider the usual representation of O(n) on an n-dimensional vector space V with inner product ( , ). Let V* be the dual space of V, and consider the space V* ® V* ® V*. This space is naturally isomorphic to the space of all trilinear covariant tensors on V. Let W be the subspace of V* ® V* ® V*, defined by

W = ~¢ E V* ® V* ® V* [ ¢(x, y, z) = ¢(x, z, y) and x

2 ¢(x, ei, eO--2 ~ ¢(ei, ei, x) forx, y , z ~ V}. t - 1 i = l

(Here {el,. •., en} denotes an arbitrary orthonormal basis of V.) Thus W is the space of tensors which satisfy all the identities of the covariant derivative of the Ricci tensor, and no others. There is a naturally defined inner product on W, given by

(¢' ¢) = 2 ¢(e,, ej, ek)¢(e, ej, ek). t , . ~ , / c = 1

We define three subspaces of W as follows:

A = { ¢ e wI ¢ (x , y , z ) + ¢ ( z , x , y ) + ¢ ( y , z , x ) = o for all x, y, zE V}

B = {¢ e W I ¢(x, y, z) = ¢(y, x, z) for all x e V}

C = ( ¢ e W [ t = l ~ ¢(x, e, e , ) = 0 for all x e / I } .

Consider now the induced representation of O(n) on W. This representation is reducible. The next theorem describes the decomposition of W into subspaces on which the induced representation of O(n) is irreducible.

THEOREM 3.1 Wehave

(3.1) C = A @ B

(3.2) W = A @ B @ C ±.

These direct sums are orthogonal and are preserved under the indueed representation of O(n) on A, B, and C a are irreducible.

Proof It is easy to check that A (~ B = {0}. Furthermore for ¢ E C we define Ca and ~bB by

eB(x, y, z) = / { ¢ ( x , y, z) + ¢(z, x, y) + ¢(y, z, x)},

CA(x, y, z) = 1{2¢(x, y, z) - ¢(z, x, y) - ¢(y, z, x)}.

Then CA e A, ¢B e B, and ¢ = ¢~ + eB.


Hence (3.1) and (3.2) hold. Furthermore, it is easy to check that the direct sums are orthogonal and are preserved under the induced representation of O(n).

To each component of the induced representation of O(n) on W we assign an O(n)-invariant symmetric bilinear form which vanishes precisely on that component. Thus the number of components of the induced representation of O(n) on W is ~< equal to the dimension of the space of quadratic invariants.

We now show that this dimension is equal to 3. Define ~ 115 ~, and fl by

~¢ll z = ~ ¢(e,e,,ek) 2 i , J , k = l

a(~) = ~ ¢(e,, e~, eu)@(ek, e,, ej) i , ] , Ic = 1

/3(~) = ~(e,, ej, ej) . t = 1

It is clear that 1] 112, ~, and/3 are quadratic invariants of the induced representation of O(n) on W. That they are linearly independent can be proved directly, or it follows from Theorem 5.3.

To prove that [[ ~2, a, and/3 span the space of quadratic invariants, we must use Weyl's theorem on invariants of the orthogonal group. It is most convenient to use it in the form of [1, p. 76]. According to this theorem every quadratic invariant of W is a linear combination of invariants P, of the form

P,(~b) = ~ cr(~ ® $)(e,, e,, e s, ei, eu, ek), / . , ] , k = 1

where ~ is a permutation of degree 6, and a(~ Q ~) denotes the natural action of cr on 4, ® ~. It is then easy to check that up to a scalar multiple the only possibilities for Po are ]1 112, a, and/3.

Thus the induced representation of O(n) on W has precisely the three components, namely A, B, and C ±. Hence Theorem 3.1 follows.

For a Riemannian manifold there is a representation of O(n) on each tangent space M,~. Let

= X~b ~ M* ® M* ® M* [ 4,(x, y, z) = ~b(x, z, y) and Wm \

~(X, e,, e 0 = 2 ~1 ~(e,,e,,x) for x, y, z~ Mm). t = 1 '=

Then the induced representation of O(n) on Wm has the three components Am, Bin, and C~, as described above. Thus we have the following alternate descriptions for the classes d and M:

M ~ ~ ' if and only if for each m ~ M, (Vp)m lies in the subspace Am. M ~ M if and only if for each rn ~ 34, (Vp)m lies in the subspace B,~.


Remark. One can also consider the class cg± of manifolds corresponding to C ' . The defining condition for this class is

1 Vx(p)(Y, Z) = (n + 2)(n - 1){n(Xr)( Y, Z )

+ ½(n - 2)((Yr)(Z, Z) + (Zr)(X, Y))}

for X, Y, Z ~ d (M) . This condition is satisfied by any two-dimensional Riemannian manifold,

but I do not know if there are other interesting manifolds in c8±. Similarly it is possible to consider the classes d Q ( ~ and M @ (£~

corresponding to A @ C s and B @ C ±, respectively. The defining relations are as follows:

~' • c~: ~xrz (Vx(p ) (y , z ) - n +2 ~ (X'O<Y'Z>) = 0

® ~ ' : vx(p)(Y, z ) - vy(p)(x, z )

1 - n - 1 { ( X z ) ( Y, Z ) - ( Y ' O ( X , Z ) } .

4. H O M O G E N E O U S M A N I F O L D S A N D THE S T R I C T N E S S

OF THE INCLUSION P c d

The following theorem is obvious:

THEOR EM 4.1. I f M is a homogeneous Riemannian manifold, then M ~ cg. More particularly, we shall show that there are many homogeneous spaces

in the class d . Let M be a compact homogeneous space. Then M = G/K, where G is a

compact Lie group and K is a compact subgroup of G. Put a bi-invariant metric on G. Then we choose the metric on M to be a projection of the bi-invariant metric of G. Let ® and ~ be the Lie algebras of G and K respectively. Denote by ~r: G --> M the projec, tion, and put p = ¢r(e). If m = t±, then the tangent map ~ , is an isometry between m and Mp. We have

(4.1) [~, m] _~

(4.2) ([/1, B], C) = (A, [B, C])

for A, B, C ~ ~. (Property (4.2) is a consequence of the bi-invariance of the metric of G.)

Furthermore, the Riemannian connection and curvature operator of M are given by Nomizu's formula (see [9], [15]):

(4.3) v x r = ½IX, Ylm,

(4.4) (RxrX, Y) = ll[x, Y]~I? + ¼ll[x, Y],.H'


for X, Y ~ m. Furthermore from [9]

(4.5) (Vx(R)xyX, Y) = - ( [ X , [X, r]], [X, r l t ) .

LEMMA 4.2. Let X ~ ~7(M). Then

(4.6) Vx(p)(X, X) = ~ ([X, [X, E~]], [X, E.]t), 01=1

where {El . . . . . E.} is an orthogonal basis o f re. Proof. This follows immediately from (4.5).

THEOREM 4.3. Let M = 0(4)/0(2) where 0(2) is imbedded as a subgroup o f 0(4) as the set o f matrices of the form

(i b°i)0°a 0°1 where a s + b 2 = 1. Let M have a bi-invar&nt metric. Then M ~ d but M ~ ~.

Proof. Using (4.6) it can be verified that M e d . Then using (4.4) one checks that M $ g. Since M is not reducible, M ~ ~.

The details of the calculation are as follows. Let S,j be the 4 x 4 matrix with + 1 in the (i, j ) place and - 1 in the (j, i) place. Then the tangent space Mp can be identified with the space m of skew-symmetric matrices spanned by {Sla, $14, $23, $24, $34}. Then for x ~ m we have

v~(p)(x, x ) = i<J

(~j)~(1,2)

(~,1)~(1,2)

(IX, [x, sCj]], [x, s~j]~)

([X, [X, Sij]], $12)([X, S~j], &2)

: ~, <[Jr, s,j], [x, s12]Xx, [s12, s~j]) (LD¢(1,2)

= ([x, ~ <x, [&~, S~jl)S,j], [x, sly]) t<1

(t,D 4, (1,2)

= 0 .

That M is not reducible can be checked by computing the brackets of the S~[s. Furthermore, from (4.4) the sectional curvatures can be computed. Let A = IIS~jll 2. Then all sectional curvatures are equal to ¼A, except for Ks~3s~3 = Ks~4s~4 = A and Ks~3s2~ = Ks~4s~3 = 0. Moreover, the basis {Sla, $14, S2a,

E I N S T E I N - L I K E M A N I F O L D S W H I C H ARE N O T E I N S T E I N 267

$24, $34} diagonalizes the Ricci curvature. From this fact and the computa- tion of the sectional curvatures it follows that the Ricci curvature has precisely two distinct eigenvalues. These are p(Sla, S l a ) = p($14, S~4)= p($23, S2a) = p(S2~, $2~) = ~-A arid p(S3~, $34) = ~.

Next we show that there are many homogeneous metrics on the three- dimensional sphere which are in ~/ but not in ~.. Let N denote the unit outward normal to the unit sphere S a in ~4. Regarding ~4 as the quaternions we obtain vector fields IN, JN, K N tangent to S 3. Let ~bi, q~, q~K be the 1-forms on S 3 given by ~z(X) = (X , IN} , etc. The metrics that we shall consider are those of the form

( , ) = ~ ,~ + ~ + ~,~.

The isometry group of ( , ) always contains Sp(1). The curvature of the metric ( , ) can be computed using the Cartarl structure equations together with the relations d4,i = 24,~ A ~br, dq~z~ = 2 ~ A 4'7, dq~ = 2~x A 4)~. The Ricci curvature ( , ) is given by

2 /32 /32 p(IU, I N ) = ~ (a 2 - + y2)(a2 + _ y2)

2 /32 /32 (4.8) p(JU, JU) = - ~ ( - a 2 + + y2)(az + _ y~)

2 /3 = /3 ~ o(KN, K N ) = - ~ ( -¢z 2 + + y2)(a= _ + y2).

Furthermore, the connection D of ( , ) is given by

I D , u I N = DsNJN = DKNKN = 0

(4.9) t D m d N = ~ ( - a ~ + /3~ +

l DzxKU = -~ (a2 - /3 ~ - y2)JU, etc.

Let Sa(a,/3, V) denote S a with the metric ( , ) = a~b 2 + /3 2 ~ + y~4'~.

T H E O R E M 4.4. We have

(i) Sa(a, /3, Y) ~ ~ i f and only i f a 2 =/32 = y2;

(ii) S*(a, /3, 7') ~ ~ i f and only i f Sa(a, /3, Y) e ~ ;

(iii) SZ(a, fl, y) c d i f and only i f two o f a 2, /32, y2 are equal.

Proof. From (4.8) and (4.9) it follows that

(4.10) DzN(p)(IN, I N ) = Dm(p)(IX, SN) = 0, etc.

268 A L F R E D GRAY

and

4 (4.11) D,u(p ) ( JN , K N ) = a ~ (~2 _ f12 _ ),2)~(fl2 _ ),2).

From (4.10) and (4.11) we get (i) immediately. Furthermore, (ii) follows with some calculation. Moreover, from (4.11) we have that

- 1 6 (4.12) ~DIN(p ) ( JN , K N ) = ~ (o~ 2 - fi2)(fi2 _ ~,2)(~2 _ a2).

Now (iii) is immediate from (4.12). We shall show in Section 6 that there is a large class of manifolds, namely

3-symmetric spaces with a bi-invariant metric, which are in d but not in ~.

5. C O N F O R M A L L Y FLAT R I E M A N N I A N M A N I F O L D S AND THE

S T R I C T N E S S OF THE I N C L U S I O N .~ c , ~

We shall need the following (known) result (for example, see [4], [5], [6], [7], [8], [181, [23]).

THEOREM 5.1. L e t M be a conformal ly f l a t R iemannian mani fo ld with

dim M >1 3, and suppose M ~ ~. Then M E ~ . Proof . First suppose dim M/> 4. Since M is conformally flat, the curvature

operator of M is given by

1 (5.1) ( R w x Y , Z ) = n - 2 (p (W' Y ) ( X , Z ) - p ( W , Z ) ( X , Y )

+ p(X, Z)<W, Y> - p(X, Y)<W, Z>} T

- ( n - 1 ) ( n - 2)

× ((w, r ) < x , z ) - < w , z ) ( x , r ) }

for IV, X, Y, Z E ~(M) . We suppose that M ~ ~'; that is, the scalar curvature ~- of M is constant. Taking the covariant derivative of (5.1) we obtain

1 (5.2) ( V v ( R ) w x Y, Z ) = n - 2 {Vv(p ) (W, Y)KX, Z )

- V v ( p ) ( W , Z ) K X , Y ) + V v ( p ) ( X , Z ) ( W , Y )

- Vv(p)(x, r ) ( w , z ) )

for V, W, X, Y, Z ~ f ( M ) . Contracting (5.2) and using (2.7) together with the assumption that M E (g, we get

(5.3) ~ (V~,(R)E,~ Y, Z) 1

= n -----5-7 (vy(e)(x, z ) - vz(o)(x, r)}. n = l


In view of (2.4), Equation (5.3) reduces to

(5.4) (n - 3){Vr(p)(X, Z) - Vx(p) (Y, Z)} = 0.

Since n > 3, the result follows. Next assume n = 3. The argument above does not work, because Equation

(5.4) yields no new information. However, if M is a conformally flat three- dimensional manifold, then

(5.5) V~(p)(X, Z) - Vz(p)(X, Y) = ¼{(YO(X, Z ) - (Z~) (X , r ) }

(see [19, p. 306] or [26, p. 16]). Thus, if M e ~, then M e ~ in this case also. Using Theorem 5.1 we show that in all dimensions larger than 2, there is

a manifold M e ~' such that M 6 ~.

THEOR EM 5.2. Le t M = {(p~ . . . . , p~) e R ~ [ p~ > 0} where n >1 3. Define

a metric ds 2 on M by

2 " d x ~ ) ds ~ = x~(dxl + " +

where a = 4/(n - 2). Then M e ~ , but M ¢ ~.

P r o o f The curvature of (M, d s0 can be easily computed using the Cartan structure equations. It can be verified that the scalar curvature of (M, ds 2) is identically zero; in particular, it is constant. (M, ds 2) is obviously conformally flat and so by Theorem 5.1 we have (M, ds 2) sM. On the other hand, it is easy to check by computing the Ricci curvature that (M, ds z) ~ ~. Since M is irreducible, (M, ds ~) ¢ ~.

We now have the necessary ingredients to establish the strictness of the inclusions.

T H E O R E M 5.3. Al l o f the inclusions in (1.1) are strict.

P r o o f By Theorems 4.3 and 5.2 we have the strict inclusions ~ = d and = ~'. It remains to demonstrate the strictness of the inclusions 6 ~ = ~,

~ = d w ~ , ~ c d u ~ , d w ~ = ~ , a n d ~ f c ~ . ~ ~ : Take two Einstein manifolds 341 and M2 such that the Ricci

curvatures of M1 and 3/2 are different multiples of the respective metric tensors. Then M~ x 342 ~ ~ but M1 × M2 6 &

d ~ d w ~ and M = d w ~ : These strict inclusions follow from

~ ¢ w ~ = rg: Let M ~ z ¢ - ~ and 3 / 2 ~ - ~ ¢ . Then M~ × M2~ butM~ x M ~ 6 ~ C W ~ .

c~ c ~ : Let M be any surface of nonconstant Gaussian curvature. Then M¢~.

6. THE CLASSES ~ AND ,~ FOR K~,HLER MANIFOLDS

Let M be a K/ihler manifold. This means that M has an almost complex structure J such that (JX, J Y ) = ( X , Y ) and Vx(J) Y = 0 for X, Y~ ~(M) .

270 ALFRED GRAY

Let gff be the class of all K/ihler manifolds. We show that for K~thler manifolds, the inclusion relations of (1.1) simplify considerably.

T H E O R E M 6.1. We have ~ n d ~ n ~ = Jt ~ n ~. Furthermore, we have the following strict inclusions:

(6.1) ~ n ~ c ~(¢" n ~ c ~Y( n cg c o~.

Proof. To show ~ n ~ -- ~ n ~ it suffices to prove that ~Y" n ~ _ n ~. To do this, we use the first Chern form ~'1, which is given by

2~ryl(X, Y) - p(JX, Y)

for X, Y E Y'(M). I f M e 3((, then p(gX, Y) = - p ( X , JY) and Vx(g)Y = 0. Hence for X, II, Z e W(M) we have

(6.2) 2ZtVx(y~)( Y, Z) = Vx(p)(J Y, Z ) = - Vx(p)( Y, JZ).

I f in addition M ~ &, it follows from (6.2) that

(6.3) Vx(el)(Y, Z) = Vy(71)(X, Z).

On the other hand, since ~'1 is a 2-form, we have

(6.4) Vx(Vl)(Y, Z) = - Vx(70(Z, Y).

Thus (W, Y, Z) --> Vx(Tz)( Y, Z ) is symmetric in the first two variables, and skew-symmetric in the last two. It follows that 71 is parallel. Then p must also be parallel, and so M ~ S n ~.

Next we show that ~ ' n ~ _ ~ ~ of'. I f M e ~¢ n ~ then

(6.5) ¢b Vx(7~)(JY , z ) = O. X Y,~

Taking Y = Z in (6.5) we obtain Vy(Tt)(Y, JX) = 0 for all X, YE ~¢(M). Thus Vx(7~)(Y, Z) is skew-symmetric in all 3 variables. But we always have d~,l = 0 and so

0 = dT~(X, Y, Z) = 3Vx0'I)(Y, Z).

Since yl is parallel, so is p. Thus M ~ JY" n ~. The only inclusion in (6.1) which is not obvious is ~¢" n ~ c oU n cg. To

establish this strict inclusion we use a result of Matsushima [14]: Let M be a compact homogeneous K~hler manifold. Then up to homothety there is a unique K~,hler metric on M which is Einstein.

Thus if M is a compact irreducible homogeneous Kahler manifold with reducible isotropy representation (e.g. G/T where T is a maximal torus of G), we can choose a metric ds 2 on M which is not Einstein but is homogeneous. Then

(M, ds2) ¢ ~ n ~, M E Jg" n c~.

Thus, either ~ n ~ :~ ~ n ~¢ or ~ n zJ -~ ~ n (g.


7. T H E C L A S S E S d~ ¢' A N D , ~ F OR N E A R L Y K A H L E R M A N I F O L D S

A nearly K/ihler manifold is an almost Hermitian manifold M which satisfies the condition

(7.1) Vx(J)X = 0

for X~ W(M) [10]. Condition (7.1) is weaker than the defining property for K~ihler manifolds. The most well-known example of a nearly K/ihler manifold is the sphere S 6. Nearly K~hler manifolds enjoy many of the properties of K/ihler manifolds, because the curvature tensor of a nearly K/ihler manifold is only slightly more complicated than that of a K~ihler manifold.

In this section we shall also consider 3-symmetric spaces [11 ]. By definition, a 3-symmetric space is a Riemannian manifold M such that at each point p ~ M there is an isometry 0p of M such that 0 F = I and p is an isolated fixed point of 0p. Such a manifold must be a homogeneous almost complex manifold. We require one further condition, that each 0p be a holomorphic isometry.

The sphere S 6 is both nearly K~ihlerian and 3-symmetric. More generally every 3-symmetric space has a metric (namely a bi-invariant metric) which is nearly K/ihlerian.

THEOREM 7.1. Let M be a nearly K6hlerian 3-symmetric space. Then M e d .

Proof. In [11] it is shown that the following curvature identity is satisfied for M:

(7.2) (Vv(R)wx Y, Z} + (Vv(R)swsxJY, JZ~ = 0,

for V, W, X, Y, Z E Y'(M). From (7.2) we obtain

(7.3) vv(p)(w, r ) + vv(p)(Jw, J r ) = 0.

On the other hand, we also have the identity (see [10])

(7.4) (Rwx Y, Z} = (RlwsxJY, JZ).

From (7.4) it follows that

(7.5) p(W, X) = p(JW, JX)

and from (7.5) we obtain

(7.6) Vv(p)(W, X) - Vv(p)(SW, JX) = p(Vv(J) W, JX) - p(jrv, vv(s)x).

In particular, taking V = W = X in (7.6), we obtain

(7.7) Vx(p)(X, X) - Vx(p)(JX-, JX) = 2p(Vx(J)X, X).

Then (7.3) and (7.7) imply

(7.8) Vx(p)(X, X) = p(Vx(J )X , X).

Thus, from (7.8) we see that M e ~o:U implies that M E d .


The simplest example of a nearly K/ihler 3-symmetric space not in ~ is perhaps

U(4) U(2) x U(1) x U(1)"

(In [I I] it was erroneously stated that a 3-symmetric space must have parallel Ricci tensor.) Let .A/'~F denote the class of nearly Kfihler manifolds. We have

T H E O R E M 7.2. The following inclusions hold:

n .ACY{" c ~ n .A/ 'X" & n .A/',.~g/-fld u gg) n ,/V'~" c_ ~ n .As'..~l c ,,4,".~

Proof. From Theorem 7.1 we have the strict inclusion ~ n .AS'~ c d n

JV)/", and hence also the strict inclusion ~ ' n . , ,e~ = (d u ~) n ~ . All the other inclusions are obvious.

P A R T I I

S O M E G L O B A L T H E O R E M S

8. P A R T I A L D E R I V A T I V E S ON T H E U N I T S P H E R E B U N D L E

OF A R I E M A N N I A N M A N I F O L D

I f M is any Riemannian manifold, we define

S(M) = {(m, x) I m e M, x e Mm, IIxL[ = 1},

= {x I Ilxll = l } ;

then S(M) is the unit sphere bundle of M. We denote by ~r: S(M) --+ M the projection.

It will be necessary to perform some operations with normal coordinates on a fiber Sin. I f dim M = l, then Sm is isometric to a standard sphere SZ-l(1). For x ~ Sm we take an orthonormal base {el . . . . , et} of Mm such that x = e~. Denote by (Y2 . . . . . Y3 the corresponding system of normal coordinates defined on a neighborhood of x in Sin. We require that y=(x) = 0 f o r = = 2 . . . . . I.

L E M M A 8.1. Let F: Sm-+ R be a C~ function. Then

(8 1) ~y--l~"=-- ? Oy~-- ~ (m, x) =

where r 2 --- ~=2 u~.

Ou~l . . . Oug~

× F( ( c ° s r ) x + ( ~ 7 ~ ) y~=2 u~e~)(O)

E I N S T E I N - L I K E M A N I F O L D S W H I C H ARE NOT E I N S T E I N 273

Proof. Let exp~: (Sm)~ --~ S,, be the exponential map. Then, by definition of the normal coordinates {Y2, • •., Y~}, we have

- ~ ~ F-exp~ u~ey . (8.2) ~ y ~ - " ~ y ~ ~u2~ ' "~u~ \

Now the exponential map for a sphere can be explicitly computed (compare [I]):

(8.3) exp~ u~e v = (cos r ) x + uye~.

Hence (8.1) follows from (8.2) and (8.3).

9. D I F F E R E N T I A L O P E R A T O R S ON THE U N I T S P H E R E B U N D L E

OF A R I E M A N N I A N M A N I F O L D

D E F I N I T I O N . Let M be a Riemannian manifold and let m E M, x ~ Mm. We take an orthonormal basis { e l , . . . , e,} of Mm such that x = el. Then we lift this frame to an orthonormal basis { f ~ , . . . , f , , .g2 . . . . , g,} of the tangent space S(M)<m,~>. Here we require that f l , . . . , f~ are horizontal and g2 . . . . . g , are vertical. Denote by (xl . . . . . x , , y 2 , . . . , y,) the corresponding normal coordinate system on a neighborhood of (m, x) in S ( M ) . We define a second- order linear differential operator L(A, ~) by

(9.1) L(A,/~)(m.x) = ~ + A ~.a =2 h~a ~Y,~--~ + v ~=2 q~ (r~,x)

where h~a(m, x) = R~B~, q~(m, x) = P~x, and A,/~ are constants to be chosen later. This definition is independent of the choice of normal coordinates at (m, x). Hence L(A,/~)o~.~> is well defined.

We now view the Ricci curvature as a real-valued C o~ function on S ( M ) . We show that if M is in d or ~', there is a differential operatorL(A, ~) of the type described above such that L(A, t*)(p) vanishes on S ( M ) .

L E M M A 9.1. Suppose M ~ ~ . Then L(½, - ½)(p) = O. Proof. We compute ~2=1 (V~p)(m, x). Using the fact that M ~ ~ and

the Ricci identity we find that

= ~ V~p~

= ~ {vLp~-R~(p)~A

(vLp)(m, x) =


The notation Rax(p),~x indicates that the curvature tensor, considered as a derivation, is operating on the Ricci tensor. We continue the calculation by writing out the definition of R,~=(p),~,:

(9.2) ~ (V~p)(m, x) = ~ (R~x~p~ + R~=~poA 6 = 1 ct ,B= l

B = l (z,/~ = 1

It is easy to see that (82p/ax~)(m, x) = (V~,p)(m, x). We must also compute (82p/~y,~ ~y~)(m, x). By Lemma 8.1 we have

82p (9.3) 8y'~ 8y~ (m, x)

Similarly

(9.4)

((cosr,x + (si+r) = ~ (cos r)2p~ + 7 (cos r)(sin r) ~ u,o~,

7 > 1

1 (sinr) 2 ~ . 2 + -~ u~o~ y > l

2 (sin r) 2 ~ u,uop,o}(O) +~ 1 <?<d~

- 8u~ Ou B P~x + 2 ~ uyp~ V>I

+ ~.. (-p~x + p~)u~ + 2 ~ pT~u~uo}(O) 7>1 1 < } ' < 6

= 2(-p~=3~ + P~e).

~P (re, x )= ~---~p((cosr)x+ (~f-)7_/lu~e~)(O) 8y~ = 2p=,~.

From (9.2), (9.3), and (9.4) we find

( (9.5) ~ (m, x) = p~= - p~a~B a = i ~ = 1 ,~, 1 R c ~ x B x +

1 02p 2 8y~ 8y B

= p~x - ~ R~x~x (m, x)

1 ~ Ozp = --2 R~x~x (m, x) ,~,B= I ay~ 8y~

+ ~O~x-~(m,x).

Thus from (9.5) we see that L(½, -½)(p) = 0.

- - (m, x))


L E M M A 9.2. Suppose M ~ d . Then L ( - 1, 1)(p) = 0. Proof The method of proof of Lemma 9.2 is the same as that of Lemma

9.1, but the calculations are slightly different. We find

(Ve=ap)(m, x)= ~ V~p~x ~=i ~=i

= -2 ~ {V~p~x}

= - 2 ~ {v~p~x - R.~(p)..}

= 2 ~ Rox(p)~ o:=1

= - 2 2 {R.x~ePBx + R~x~BP.B} o:,B = 1

B=I a,fl=l

~ a2P = - 2 O~x + R~xBx ~ (m, x) B=2 g,B=I

~2p n ap = R~xzx ~ (m, x) - B~=2 PB,¢ 8y---~ (m, x).

~,B=l

Hence L ( - 1, 1)(p) = 0 in this case. We can now give the proofs of Theorems 1.1 and 1.2 which were stated

in the introduction. Proof of Theorem 1.1. Let M e ~ ' and assume that M has positive sec-

tional curvature. Then L(5, - 5)(0) = 0 and L(5, - 5) is elliptic. By E. Hopf ' s maximum principle (see, for example, [27]) the function 0 has no local maximum or minimum on the unit sphere bundle of M.

Proof of Theorem 1.2. Let M e ~ ' and assume that M has negative sectional curvature. Then L ( - 1 , 1)(p) = 0 and L ( - 1 , 1) is elliptic. By E. Hopf ' s maximum principle, R has no local maximum or minimum on the unit sphere bundle of M.

We remark that the techniques of this section are used in [12] to prove theorems about K~ihler manifolds with constant scalar curvature. In [12] it is shown that L(5, 0)(H) = 0, where H denotes the holomorphic sectional curvature. This fact is used to prove the following theorem: A compact Kiihler manifoM with constant scalar curvature and nonnegative sectional curvature is locally symmetric.


10. D I F F E R E N T I A L O P E R A T O R S D E F I N E D BY

S Y M M E T R I C T E N SO R F I E L D S

Let ¢ be a symmetric tensor field of type (1, 1) on an n-dimensional Rieman- nian manifold. In this section we define a second-order differential operator L~ whose symbol is ¢. The operator L~ turns out to be self-adjoint.

We regard ~b as a function ~b: 5f(M) --~ ~ ( M ) which is linear with respect to functions. A vector field div¢ is defined by the formula

(10.1) (div ~b, X) = ~ (V~.(~b)E., X)

for X ~ ~(M), where {El, • . . , E~} is an arbitrary local frame field. There is a similar formula for the divergence of a vector field (see [11, vol. 1]):

(10.2) div X = ~ <V~X, E~)

for Xe:T(M). From (10.1) and (10.2) it is easy to check that for a C ° real-valued function f we have the formulas

(10.3) div (fX) = f d i v X + X f

(10.4) div(f~b) = f d i v ¢ + ¢ gradfi

where g rad f i s the vector field defined by (gradfi X) = Xfl Next we define tr(~b o VX) for X e ~ ( M ) by

(10.5) tr(¢ o VX) = ~ (¢VE~X, E~).

From (10.1), (10.2), and (10.5) it follows that for X e £r(M),

(10.6) tr(¢ o VX) = div(~bX) - (div ~b, X).

The point of defining all of these operations is to give a simple description of the differential operator L, . We define

(10.7) Lof = tr(¢ o V grad f ) + (div ¢, grad f ) = div(¢ grad f ) ,

where f is a C ~ real-valued function on M. Thus ~b is the symbol of La. If M is a compact oriented Riemannian manifold, there is an inner

product ( , ) on the space of functions given by

(f, g) = fMfgo~,

where ¢o is the volume element of M. Although this inner product is defined for all L: functions, we shall use it only on C ~ functions.

The following theorem generalizes a result of Cheng and Yau [5]. In this

EINSTEIN-LIKE MANIFOLDS WHICH ARE NOT EINSTEIN 277

theorem, and in the rest of the paper, 'self-adjoint' means self-adjoint with respect to CO° functions.

THEOR EM 10.1. Suppose M is a compact oriented Riemannian manifold. Then L , is self-adjoint with respect to ( , ).

Proof Using (10.3), (10.4), (10.6), and (10.7) we find

(10.8) g L f f - f L , g = div(¢(g g r a d f - f g r a d g)).

According to Green's theorem [11, vol. 1], we have

(10.9) f u (div X)oJ = 0

for XE f ( M ) . From (10.8) and (10.9) we obtain

(L, f , g) - ( f , L ,g ) = fM ( g L , f -- f L , g ) w = O.

Thus L e, is self-adjoint.

11. THE SELF-ADJOINTNESS OF CERTAIN DIFFERENTIAL OPERATORS ON THE UNIT SPHERE BUNDLES

We apply the results of Section 10 to the differential operators on S ( M ) defined by Equation (9.1).

LEMMA 11.1. The differential operator L(A, k~) defined by (9.1) is self-adjoint provided that A = -t*.

Proof We define a symmetric tensor field ¢ of type (1, 1) on S ( M ) as follows:

(11.1) (¢u, v)(m, x) = Rx~x~

for (m, x) E S ( M ) and u, v ~ Mm with (u, x) = (v, x) = 0. (Here we identify the tangent space (Sm)x to the fiber Sm with the subspace of M~ which is orthogonal to x. This identification is just parallel translation.)

The divergence of ¢ is computed in [12]; we have

(11.2) (div 4J, v>(m, x) = - px~

for (m, x) ~ S ( M ) and v ~ M~ with (x, v> = 0. Thus from (9.1), (11.1), and (11.2) it follows that

(11.3) L(A, g ) f = Ahf + h tr(¢ o V grad f ) - ~<div ¢, grad f>,

where f is a C °o real-valued function on S(M) . Here A~ is the horizontal Laplacian.

Now it is easy to check that A ~ is self-adjoint. Suppose h = - ~ . Then L(A,/x) = A h + AL,. It follows from Theorem 10.1 that L(A,/~) is self- adjoint.


If f is a C °~ real-valued function on S ( M ) we denote by g r a & f and grad~fthe vertical and horizontal components of grad f

LEMMA 11.2. Let p be the Ricci curvature of M viewed as a C o~ real-valued function on S(M) . Then

L(~t, - a)(p=)(m, x) = {2pL(,L - ,~)(p) + 2]]grad h pllZ}(m, x)

+ 2hKx(grao~o~(x~.

Proof We have

L(A, - h)(p2)(m, x) = (A~(p 2) + hL,(p2)}(m, x)

= (2pAh(p) + 2 IIgrad ~ pH2)(rn, x)

+ h{2pL~(p)(m, x)

= {2pL(a, - A)(p) + 2 II g rad~ P II~}(m, x)

+ 2AKx(~rago)(x~.

LEMMA 11.3. We have

(11.4) ~ {pL(A, -A)(p)(m, x) + I[grad n p[[2(m, x) J s (M)

+ hKx(gradVo)(x))dV = O.

Proof By Lemma 11.1 the operator L(A, - h ) is self-adjoint. Thus, if F is any C ® real-valued function on M, it follows that

(11.5) f L(h, - h ) ( F ) d V = 0 -- (L(h, -h)(F) , 1) as (M)

= ( F , L ( a , - ~ ) 1 ) = 0 .

In particular, we take F = p 2 in (11.5) and use Lemma 11.2. The result is (11.4).

We now prove theorems analogous to Theorems 1.1 and 1.2 for compact manifolds. The point here is that the sectional curvature is allowed to be nonnegative instead of positive, or noupositive instead of negative.

THEOREM 11.4. Let M be a Riemannian manifold with nonnegative sectional curvature and suppose M ~ ~. Then M E ~.

Proof If M ~ ,~ we have L(½, - ½)(p) = 0. Thus (11.4) reduces to

(11.6) ( (Hgradh pll2(m, x) + dV 0. J s (M)

EINSTEIN-LIKE MANIFOLDS WHICH ARE NOT EINSTEIN 279

All the terms on the left-hand side o f (11.6) are nonnegative. It follows that each must vanish identically. In particular, grad n p is identically zero. This is equivalent to saying that M has parallel Ricci tensor.

T H E O R E M 11.5. Let M be a Riemannian manifold with nonpositive sectional

curvature, and suppose M ~ ~ . Then M ~ ~.

Proo f I f M e d , we have L ( - 1, 1)(p) = 0. Thus (11.4) reduces to

(11.7) fs (]]grad ~ pll~(m, x) - Kx(gradvo)(x)} d V = O. (M)

All the terms on the r ight-hand side o f (11.7) are nonnegative. It follows that each must vanish identically. In particular, grad~ p is identically zero. This is equivalent to saying that M has parallel Ricci tensor.

Theorem 11.4 was originally proved by Ryan [18], Simon [20], and Wegner [25] and Theorem 11.5 was originally proved by Simon [21]. Our point here was to give a method of p r o o f that works for both the classes d and ~ .

B I B L I O G R A P H Y

1. Berger, M., Gauduchon, P. and Mazet, E., 'Le spectre d'une vari6t6 riemannienne', Lecture Notes in Mathematics, Vol. 194, Springer Verlag, Berlin and New York, 1971.

2. Berger, M. and Ehin, D.J., 'Some Decompositions on the Space of Riemannian Manifolds', J. Diff. Geom. 3, 379-392 (1969).

3. Bourguignon, J.P., 'On Harmonic Forms of Curvature Type', (to appear). 4. Chaki, M. C. and Gupta, B., 'On Conformally Symmetric Spaces', lndian J. Math.

5, 113-122 (1963). 5. Cheng, S.Y. and Yau, S.T., 'Hypersurfaces with Constant Scalar Curvature',

Math. Ann. 225, 195-204 (1977). 6. Glodek, E., 'Some Remarks on Conformally Symmetric Riemannian Spaces',

Coll. Math. 23, 121-123 (1971). 7. Goldberg, S.I., 'On Conformally Flat Spaces with Definite Ricci Curvature',

Kodai Math. Sem. Rep. 21, 226-232 (1969). 8. Goldberg, S.I. and Okumura, M., 'Conformally Flat Manifolds and a Pinching

Problem on the Ricci Tensor', Proe. Am. Math. Soc. 58, 234-236 (1976). 9. Gray, A., 'Pseudo-Riemannian Almost Product Manifolds and Submersions', J.

Math. Mech. 16, 715-737 (1967). 10. Gray, A., 'Nearly K~hler Manifolds', J. Diff. Geom. 4, 283-310 (1970). 11. Gray, A., 'Riemannian Manifolds with Geodesic Symmetries of Order 3', J. Diff.

Geom. 7, 343-369 (1972). 12. Gray, A., 'Compact Ktihler Manifolds with Nonnegative Sectional Curvature',

Invent. Math. 41, 33-43 (1977). 13. Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, John Wiley,

New York, 1961. 14. Matsushima, Y., 'Remarks on Kt~hler-Einstein Manifolds', Nagoya Math. J. 46,

161-173 (1972). 15. Nomizu, K., 'Invariant Affine Connections on Homogeneous Spaces', Am. d.

Math. 76, 33-65 (1954).


16. Nomizu, K., 'On the Decomposition of Generalized Curvature Tensor Fields', Differential Geometry in Honor of K. Yano, Kinokunuja, Tokyo, pp. 335-345 (1972).

17. Nomizu, K. and Smythe, B., 'A Formula of Simons' Type and Hypersurfaces with Constant Mean Curvature', J. Diff. Geom. 3, 367-377 (1969).

18. Ryan, P., 'A Note on Conformally Flat Spaces with Constant Scalar Curvature', Proc. 13th Biennial Seminar of the Canadian Math. Congress.

19. Schouten, J.A., 'Ricci Calculus, Der mathematischen Wissenschaften' in Ein- zeldarstellungen, Band X, Springer Verlag, Berlin, 1954.

20. Simon, U., 'Compact Conformally Symmetric Riemannian Spaces', Math. Z. 132, 173-177 (1973).

21. Simon, U., 'On Differential Operators of Second Order on Riemannian Manifolds with Nonpositive Curvature', Coll. Math. 31, 223-229 (1974).

22. Sumitomo, T., 'On a Certain Class of Riemannian Homogeneous Spaces', Coll. Math. 26, 129-133 (1972).

23. Tani, M., 'On a conformally Flat Riemannian Space with Positive Ricci Curvature', Tdhoku Math. J. 19, 227-231 (1967).

24. Wegner, B., 'Codazzi-Tensoren und Kennzeichnungen sph/irischer Immersionen', J. Diff. Geom. 9, 61-70 (1974).

25. Wegner, B., 'Kennzeichnungen yon R/iumen konstanter Krtimmung unter local konformeuklidischen Riemannschen Mannigfaltigkeiten', Geom. Dedicata (to appear).

26. Yano, K., Integral Formulas in Riemannian Geometry, Marcel Dekker, Inc., New York, 1970.

27. Yano, K. and Bochner, S., 'Curvature and Betti Numbers', Ann. Math. Studies 32, Princeton University Press, Princeton, New Jersey, 1953.

Author's address: Alfred Gray , Dept . o f Mathemat ics , Univers i ty o f Mary land , College Park, M d 20742

U.S.A.

(Received November 4, 1976; in revised form February 14, 1977)

einstein-like manifolds which are not einstein

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