JOHANN DAVIDOV1, GUEO GRANTCHAROV2 and OLEG MUS̆KAROV1
1Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str.Bl. 8, 1113 Sofia, Bulgaria. e-mail: [email protected]; [email protected] of Mathematics, University of Sofia, bul. J. Bautcher 5, 1126 Sofia, Bulgaria, andDepartment of Mathematics, University of Connecticut, Storrs, CT 06269, U.S.A.e-mail: [email protected]
(Received: 15 March 2000; accepted: 10 November 2000)
Communicated by Claude LeBrun (Stony Brook)
Abstract. In this paper we study the twistor spaces of oriented Riemannian four-manifolds asa source of almost-Hermitian ∗-Einstein manifolds and show that some results in dimension fourrelated to the Riemannian Goldberg–Sachs theorem cannot be extended to higher dimensions.
It is well known that the Ricci/Chern form of a Kähler manifold is the image R(�)
of the Kähler form � under the action of the curvature operator R ∈ End(�2).For an arbitrary almost-Hermitian manifold (M, g, J ) the 2-form R(�) is neitherclosed nor of type (1, 1) but it is still closely related to the Ricci form of thecanonical Hermitian connection which represents the first Chern class of (M, J ).
The tensor ρ∗ associated to R(�) by ρ∗(X, Y ) = R(�)(X, JY ) is known inthe literature as the ∗-Ricci tensor and appears in almost-Hermitian geometry indifferent contexts. For example, it has been used by Gray [14] for studying nearly-Kähler manifolds and by Tricceri and Vanhecke [26] for describing the irreduciblecomponents of the curvature operator under the action of the unitary group. The ∗-Ricci tensor also plays an important role in the theory of harmonic almost-complexstructures, developed recently by Wood [31].
An almost-Hermitian manifold is said to be weakly ∗-Einstein if its ∗-Riccitensor is a multiple of the metric, i.e. if the Kähler form is an eigenvector of thecurvature operator. Unlike Kähler–Einstein manifolds, the multiple (usually called∗-scalar curvature) need not be a constant and when this holds, we shall simply saythat the manifold is ∗-Einstein.
As we have already mentioned, for Kähler manifolds the Einstein and weakly∗-Einstein conditions coincide, so it is natural to ask whether there is a relation
104 JOHANN DAVIDOV ET AL.
between them for more general almost-Hermitian manifolds. In real dimensionfour, the weakly ∗-Einstein condition holds if and only if the traceless Ricci tensoris J -anti-invariant and the Kähler form is an eigenvector of the self-dual Weyloperator W+. Since, for a Hermitian 4-manifold, the latter condition is equiva-lent to W+ being degenerate (cf. [2]), it follows from the Riemannian Goldberg–Sachs theorem [2, 23, 25] that any Einstein Hermitian metric is weakly ∗-Einstein.For almost-Kähler 4-manifolds, it is still an open question whether the Einsteincondition implies the weakly ∗-Einstein one, although Armstrong [4] has expli-citly described all weakly ∗-Einstein strictly almost-Kähler Einstein 4-manifolds.This, combined with a result of Sekigawa [27], shows that such manifolds cannever be compact, so the positive answer to the question above would imply thewell-known Goldberg conjecture [13] that any compact almost-Kähler Einstein4-manifold must be Kähler.
In higher dimensions, the (weakly) ∗-Einstein condition has not been so wellstudied and it seems that the main reason for that is the lack of interesting ex-amples. The purpose of this paper is to investigate the twistor spaces of orientedRiemannian 4-manifolds as a source of examples of ∗-Einstein almost-Hermitianmanifolds and to show that some four-dimensional results on ∗-Einstein conditioncan not be extended in higher dimensions.
The twistor space of an oriented Riemannian 4-manifold (M, g) is the 2-spherebundle Z on M whose fibre at a point p ∈ M consists of all complex structures onTpM compatible with the metric and the opposite orientation of M. The 6-manifoldZ admits a 1-parameter family of Riemannian metrics ht , t > 0, such that thenatural projection π : (Z, ht ) → (M, g) is a Riemannian submersion with totallygeodesic fibres (see, for example, [6, sec. 14H]). These metrics are compatible withthe almost-complex structures J1 and J2 on Z introduced, respectively, by Atiyahet al. [5] and Eells and Salamon [10].
Our main result is the following theorem:
THEOREM 1. Let M be an oriented Riemannian 4-manifold with scalar curva-ture s.
(i) The twistor space (Z, ht , J1) is ∗-Einstein if and only if M is Einstein, self-dual and t|s| = 12;
(ii) The twistor space (Z, ht , J2) is ∗-Einstein if and only if M is Einstein, self-dual and ts = 6.
Note that any compact Einstein self-dual manifold with positive scalar curvature isisometric to the sphere S4 or the complex projective space CP
2 with their standardmetrics [11, 16]. In the case of negative scalar curvature, a complete classificationis not available yet and the only known compact examples are quotients of the unitball in C
2 with the metric of constant negative curvature or the Bergman metric.In contrast, there are many local examples of self-dual Einstein metrics with aprescribed sign of the scalar curvature (cf., e.g., [9, 17, 19–21, 24, 29].
TWISTORIAL EXAMPLES OF ∗-EINSTEIN MANIFOLDS 105
The proof of Theorem 1 is based on an explicit formula for the ∗-Ricci tensorof (Z, ht , Jn), n = 1, 2, in terms of the curvature of the base manifold M (Pro-position 1). Using this formula, we show that either M is Einstein and self-dual, orM is self-dual with constant scalar curvature s �= 0 and at every point three of theeigenvalues of the Ricci operator are equal to (9s + 4|s|)/36 and the fourth one is(3s −4|s|)/12 . However, the latter case can never happen, as follows from a resultby LeBrun [18] and Apostolov [3] (cf. Lemma 1).
Remarks. (1) A Hermitian metric on a compact complex surface (M, J ) is∗-Einstein iff it is locally conformally Kähler and the traceless Ricci tensor is J -anti-invariant [2]. In higher dimensions, however, the ∗-Einstein condition doesnot imply any of these two properties, as can be seen by considering the twistorspace (Z, ht , J1) of a compact self-dual Einstein manifold M with negative scalarcurvature s and t = −12/s. By Theorem 1, the six-dimensional Hermitian mani-fold (Z, ht , J1) is ∗-Einstein but neither locally conformally Kähler [22] nor withJ1-anti-invariant traceless Ricci tensor [7].
(2) By a result of Apostolov [1], any compact ∗-Einstein Hermitian surface ofnegative ∗-scalar curvature is Kähler. The twistorial example above shows that theanalogous statement is false in higher dimensions.
(3) It is well known [12] that the twistor space (Z, ht ) is an Einstein manifoldiff the base manifold M is Einstein, self-dual and with positive scalar curvature s =6/t , or s = 12/t . Thus (Z, ht , J1), t = s/6, is an Einstein Hermitian 6-manifoldwhich is neither locally conformally Kähler [22] nor ∗-Einstein (Theorem 1). IfM = S4 or M = CP
2, then Z = CP3 or Z = SU(3)/S(U(1) × U(1) × U(1))
and (ht , J1) for t = 12/s is the standard Kähler–Einstein structure on Z. For t =6/s, (Z, ht ) is a Riemannian 3-symmetric space (in the sense of Wolf and Gray[30]) and J2 is its canonical almost-complex structure. In this case (Z, ht, J2) is anearly-Kähler manifold and, hence, ∗-Einstein by a result of Gray [14]. Note alsothat for M = S4 and t = 6/s, ht is the ‘squashed’ Einstein metric on CP
3 [6,example 9.83].
2. Preliminaries
Let M be a (connected) oriented Riemannian 4-manifold with metric g. Then g
induces a metric on the bundle �2TM of 2-vectors by the formula
The Riemannian connection of M determines a connection on the vector bundle�2TM (both denoted by ∇) and the respective curvatures are related by
R(X, Y )(Z ∧ T ) = R(X, Y )Z ∧ T + X ∧ R(Y,Z)T
for X,Y,Z, T ∈ χ(M); χ(M) stands for the Lie algebra of smooth vector fieldson M. (For the curvature tensor R we adopt the following definition R(X, Y ) =
106 JOHANN DAVIDOV ET AL.
∇[X,Y ] − [∇X,∇Y ].) The curvature operator R is the self-adjoint endomorphism of�2TM defined by
g(R(X ∧ Y ), Z ∧ T ) = g(R(X, Y )Z, T ),
for all X,Y,Z, T ∈ χ(M). The Hodge star operator defines an endomorphism ∗of �2TM with ∗2 = Id. Hence
�2TM = �2+TM ⊕ �2
−TM,
where �2±TM are the subbundles of �2TM corresponding to the (±1)-eigen-vectors of ∗. Let (E1, E2, E3, E4) be a local oriented orthonormal frame of TM.Set
Then (s1, s2, s3) (resp. (s̄1, s̄2, s̄3) is a local oriented orthonormal frame of�2−TM (resp. �2+TM). The matrix of R with respect to the frame (s̄i, si) of�2TM has the form
R =[
A BtB C
],
where the 3×3-matrices A and C are symmetric and have equal traces. Let B, W+and W− be the endomorphisms of �2TM with matrices
B =[
0 BtB 0
], W+ =
[A − s
6I 0
0 0
], W− =
[0 0
0 C − s
6I
],
where s is the scalar curvature and I is the unit 3 × 3-matrix. Then
R = s
6Id + B + W+ + W−
is the irreducible decomposition of R under the action of SO(4) found by Singerand Thorpe [28]. Note that B and W = W+ + W− represent the traceless Riccitensor and the Weyl conformal tensor, respectively. The manifold M is called self-dual (anti-self-dual) if W− = 0 (W+ = 0). It is Einstein exactly when B = 0.
The twistor space of M is the subbundle Z of �2−TM consisting of all unitvectors. The Riemannian connection ∇ of M gives rise to a splitting T Z = H ⊕Vof the tangent bundle of Z into horizontal and vertical components. More precisely,let π : �2−TM → M be the natural projection. By definition, the vertical space atσ ∈ Z is Vσ = Kerπ∗σ (TσZ is always considered as a subspace of Tσ (�
2−TM)).Note that Vσ consists of those vectors of TσZ which are tangent to the fibre Zp =π−1(p)∩Z, p = π(σ ), of Z through the point σ . Since Zp is the unit sphere in the
TWISTORIAL EXAMPLES OF ∗-EINSTEIN MANIFOLDS 107
vector space �2−TpM, Vσ is the orthogonal complement of σ in �2−TpM. Let s be alocal section of Z such that s(p) = σ . Since s has a constant length, ∇Xs ∈ Vσ forall X ∈ TpM. Given X ∈ TpM, the vector Xh
σ = s∗X − ∇Xs ∈ TσZ depends onlyon p and σ . By definition, the horizontal space at σ is Hσ = {Xh
σ : X ∈ TpM}.Note that the map X → Xh
σ is an isomorphism between TpM and Hσ with inversemap π∗ | Hσ .
Each point σ ∈ Z defines a complex structure Kσ on TpM by
g(KσX, Y ) = 2g(σ,X ∧ Y ), X, Y ∈ TpM. (2)
Note that Kσ is compatible with the metric g and the opposite orientation of M atp. The 2-vector 2σ is dual to the fundamental 2-form of Kσ .
Denote by × the usual vector product in the oriented three-dimensional vectorspace �2±TpM, p ∈ M. Then it is easily checked that
g(R(a)b, c) = −g(R(a), b × c)) (3)
for a ∈ �2TpM, b, c ∈ �2−TpM and
g(σ × V,X ∧ KσY ) = g(σ × V,KσX ∧ Y ) = −g(V,X ∧ Y ) (4)
for V ∈ Vσ , X,Y ∈ TpM.Following [5] and [10], define two almost-complex structures J1 and J2 on Z
by
JnV = (−1)nσ × V for V ∈ Vσ
JnXhσ = (KσX)hσ for X ∈ TpM,p = π(σ ).
It is well known [5] that J1 is integrable (i.e. comes from a complex structure)if and only if M is self-dual. Unlike J1, the almost-complex structure J2 is neverintegrable [10].
Let ht be the Riemannian metric on Z given by ht = π∗g + tgv , where t > 0,g is the metric of M and gv is the restriction of the metric of �2TM on the verticaldistribution V. Then π : (Z, ht) → (M, g) is a Riemannian submersion with totallygeodesic fibres and the almost-complex structures J1 and J2 are compatible withthe metrics ht .
3. Proof of Theorem 1
First we give an explicit formula for the ∗-Ricci tensor of the twistor spaces(Z, ht , Jn), n = 1, 2 in terms of the curvature of the base manifold (M, g).
Recall that given an almost-Hermitian manifold (N, g, J ) with curvature tensorR, its ∗-Ricci tensor ρ∗ and ∗-scalar curvature s∗ are defined by (cf. [26])
ρ∗(X, Y ) = Trace(Z → R(JZ,X)JY ), s∗ = Trace ρ∗.
108 JOHANN DAVIDOV ET AL.
An explicit formula for the sectional curvature of (Z, ht ) was obtained in [8].Using this formula and the well-known expression of the Riemannian curvaturetensor by means of sectional curvatures (cf. [15]), one can compute the curvaturetensor of the twistor space (Z, ht ) in terms of the curvature of the base manifold(M, g). Then one can easily obtain the following formula for the ∗-Ricci tensorρ∗t,n of (Z, ht , Jn):
PROPOSITION 1. Let E,F ∈ TσZ and
X = π∗E, Y = π∗F, A = VE, B = VF,
where V means ‘vertical component’. Then
ρ∗t,n(E, F ) = [1 + (−1)n+1]g(R(σ ),X ∧ KσY ) −
− t
2g(R(X ∧ KσY )σ,R(σ )σ ) +
+ t
4Trace(Z → g(R(X ∧ Z)σ,R(KσZ ∧ KσY )σ )) +
+ t
4(−1)n+1Trace(Vσ � C → g(R(C)X,R(σ × C)KσY )) +
+ t
2(−1)ng((∇XR)(σ ), B) + t
2g((∇KσYR)(σ ), σ × A) +
+ [1 + (−1)n+1tg(R(σ ), σ )]g(A,B) +
+ (−1)n+1 t2
4Trace(Z → g(R(σ × A)KσZ,R(B)Z)),
where Kσ is the complex structure on TpM, p = π(σ ), determined by σ via (2).
In the case when the base manifold (M, g) is Einstein and self-dual, this formulasimplifies significantly:
COROLLARY 1. Let M be an Einstein self-dual 4-manifold with scalar curvatures. Then
ρ∗t,n(E, F ) = 1
12
[(1 + (−1)n+1)s + t
24(1 + (−1)n)s2
]g(X, Y ) +
+[
1 + (−1)n+1 ts
6+ (−1)n
(ts
12
)2]g(A,B).
Proof. In this case, R = (s/6)Id + W+. Since W+ maps �2TM into �2+TM
and ∇ preserves �2+TM, one gets
g((∇XR)(Y ∧ Z), σ × A) = 0. (5)
TWISTORIAL EXAMPLES OF ∗-EINSTEIN MANIFOLDS 109
Let σ, τ ∈ Z and π(σ ) = π(τ). Then it is easy to check that
Kσ ◦ Kτ = −g(σ, τ)Id − Kσ×τ (here K0 = 0).
Using this identity, (3) and (4), one obtains that
g(R(σ × A)X,R(σ × B)Y )
=( s
12
)2 [g(X, Y )g(A,B) − 2g(X ∧ Y,A × B)] (6)
and
g(R(X ∧ Y )σ,R(Z ∧ T )σ )
=( s
12
)2 [2g(X ∧ Y,Z ∧ T ) − g(KσX,Z)g(KσY, T ) +
+ g(KσX, T )g(KσY,Z)]. (7)
Now the corollary follows from Proposition 1 and formulas (5)–(7). �Next we recall some well-known algebraic facts. Let 〈A,B〉=TracetAB be thestandard metric of End(TpM). Then we have the orthogonal decomposition
End(TpM) = R · Id ⊕ �2 ⊕ S0,
where �2 and S0 are the spaces of skew-symmetric and traceless symmetric endo-morphisms, respectively. Further, we shall identify �2 with the space of 2-vectorsσ at p via the formula
g(σ (X), Y ) = 2g(σ,X ∧ Y ), σ ∈ �2TpM, X, Y ∈ TpM
(so σ ∈ Z corresponds to the complex structure Kσ defined by (2)). Then 〈σ, τ 〉End
= 4g(σ, τ), σ, τ ∈ �2TpM. It is easily seen that if σ, τ ∈ �2±TpM, the vector2τ × σ corresponds to the commutator ∓[σ, τ ]. Moreover,
σ ◦ τ = − 14〈σ, τ 〉Id ∓ 1
2 [σ, τ ], for σ, τ ∈ �2±TpM, (8)
σ ◦ τ ∈ S0 and [σ, τ ] = 0, for σ ∈ �2−TpM, τ ∈ �2
+TpM. (9)
Now we are ready to prove Theorem 1. Suppose the twistor space (Z, ht , Jn),n = 1 or 2, is ∗-Einstein, i.e. ρ∗
t,n = cht for some constant c. First we shallshow that W− = 0. Note that for any a ∈ �2TpM and X ∈ TpM, R(a)X =1/2R(a)(X). Then, by Proposition 1, the ∗-Einstein condition on the vertical vec-tors is equivalent to the identity
4〈R(σ ), σ 〉 + t〈R(σ × A) ◦ σ,R(B)〉 = 16
t(−1)n+1(ct − 1)g(A,B)
110 JOHANN DAVIDOV ET AL.
for any σ ∈ Z and A,B ∈ Vσ . Using (8) and (9), one easily sees that the latteridentity can be rewritten as
Let λ1, λ2, λ3 be the eigenvalues of the symmetric operator W−: �2−TpM →�2−TpM and let σ1, σ2, σ3 be an orthonormal basis of �2−TpM consisting of eigen-vectors of W− corresponding to λ1, λ2, λ3 and such that σ1 × σ2 = σ3. Then itfollows from (11) that
16 (4 + ts(p))λi − tλjλk = 1
9t(144(−1)n+1(ct − 1) − 24ts(p) + t2s2(p)),
where {i, j, k} = {1, 2, 3}. Since λ1 + λ2 + λ3 = 0, one sees that either λ1 = λ2 =λ3 = 0 or two of the eigenvalues λi are equal to
−(
4
t+ s(p)
6
)
and the third one is
2
(4
t+ s(p)
6
).
Suppose W−(σ ) �= 0 for some σ . Then, W− has two distinct nonzero smootheigenvalues near the point p = π(σ ). Hence, there exists a smooth section ω of Zin a neighbourhood U of p such that, for each q ∈ U , ω(q) is an eigenvector of(W−)q corresponding to the eigenvalue
λ(q) = 2
(4
t+ s(q)
6
)�= 0.
Therefore
W−(τ ) = λ(q)(τ − 3g(τ, ω(q))ω(q), for q = π(τ) ∈ U. (12)
On the other hand, according to Proposition 1, the ∗-Einstein condition givesg((∇XR)(ω), B) = 0 for X ∈ TqM and B ∈ Vω(q). Since ∇ preserves �2±TM,
TWISTORIAL EXAMPLES OF ∗-EINSTEIN MANIFOLDS 111
one gets g((∇XW−)(ω), B) = 0 and, using (12), one obtains g(∇Xω,B) = 0.Hence, ∇Xω = 0. Thus, the almost-complex structure on U determined by ω via(2) is Kählerian. It is well known (cf. [6]) that the eigenvalues of W− on a Kählermanifold endowed with the orientation opposite to that determined by the complexstructure are equal to −s/6, −s/6, s/3. But we have seen that the eigenvalues ofW− on U are equal to
−(
4
t+ s
6
), −
(4
t+ s
6
), 2
(4
t+ s
6
),
a contradiction. Thus W− = 0 and one obtains from (11) that
144(−1)n+1(ct − 1) − 24ts(p) + t2s2(p) = 0. (13)
Now we shall consider the ∗-Einstein condition on the horizontal vectors. Letσ ∈ Z, p = π(σ ), X,Y ∈ TpM. Since W− = 0,
g(R(σ ),X ∧ KσY ) = − 12g(σ ◦ R(σ )(X), Y ) and R(σ )σ = 0. (14)
Let (E1, E2, E3, E4) be an oriented orthonormal basis of TpM and (s1, s2, s3) theorthonormal basis of �2−TpM defined by (1). Then, using (3), one gets
4∑i=1
g(R(X ∧ Ei)σ,R(KσEi ∧ KσY )σ )
=∑i,k
g(R(X ∧ Ei)σ, sk)g(R(KσEi ∧ KσY )σ, sk)
= − 14
∑i,k
g(R(σ × sk)(X),Ei)g(R(σ × sk)(KσY ),KσEi)
= − 14
3∑k=1
g(σ ◦ R(σ × sk) ◦ σ ◦ R(σ × sk)(X), Y ). (15)
Since R(a)X = (1/2)R(a)(X) for a ∈ �2TpM, X ∈ TpM, one has
g(R(A)X,R(σ × A)KσY ) − g(R(σ × A)X,R(A)KσY )
= 14g(σ ◦ [R(σ × A),R(A)](X), Y ), for A ∈ Vσ . (16)
Now, in view of (14)–(16) and Proposition 1, the ∗-Einstein condition on thehorizontal vectors is equivalent to the identity
− 12 (1 + (−1)n+1)σ ◦ R(σ ) − t
16
3∑k=1
(σ ◦ R(σ × sk))2+
+ t
16(−1)n+1σ ◦ [R(σ × A),R(A)] = c Id, (17)
112 JOHANN DAVIDOV ET AL.
where A is a g-unit vertical vector at σ . Since W− = 0 and R = (s/6)Id + B on�2−TM, using (8) and (9), one obtains easily
σ ◦ R(σ ) = − s
6Id + σ ◦ B(σ ),
3∑k=1
(σ ◦ R(σ × sk))2 =
(− s2
18+
3∑k=1
‖B(σ × sk)‖2g
)Id,
σ ◦ [R(σ × A),R(A)] = − s2
18Id + σ ◦ [B(σ × A),B(A)].
Thus (17) becomes(s
12(1 + (−1)n+1) + ts2
16.18(1 + (−1)n) − t
16
3∑k=1
‖B(σ × sk)‖2g
)Id+
+ σ ◦(
− 12(1 + (−1)n+1)B(σ ) + t
16[B(σ × A),B(A)]
)= c Id.
According to (9), the second summand in the latter formula belongs to S0 and onegets
24s(1 + (−1)n+1) + (1 + (−1)n)ts2 − 18t3∑
k=1
‖B(σ × sk)‖2g = 16.18c, (18)
4(1 + (−1)n+1)B(σ ) + tB(A) × B(σ × A) = 0. (19)
Now varying σ on the fibre Zp, one gets, from (18) and (19),
Consider first the case n = 1. If Bp = 0, one has from (20) that s(p) = 6c.This together with (13) gives ts(p) = ±12. If Bp �= 0, it follows from (20) and(21) that s(p) = 6c + 48/t which together with (13) gives ts(p) = ±36. Since M
is connected, one sees that s is constant and either ts = ±12 or ts = ±36. In thefirst case B ≡ 0 whereas in the second case, B �= 0 at each point of M and theidentities (20) and (21) become
g(B(si),B(sj )) = 64δijt2
, (22)
TWISTORIAL EXAMPLES OF ∗-EINSTEIN MANIFOLDS 113
B(si × sj ) = − t
8B(si) × B(sj ). (23)
Assume that Bp �= 0 for each p ∈ M. Let C: TpM → TpM be the Riccioperator and (E1, E2, E3, E4) an oriented orthonormal basis of TpM consistingof eigenvectors of C. Denote by λi , i = 1, . . . , 4, the corresponding eigenvalues.Let (s̄k, sk), k = 1, 2, 3, be the basis of �2TpM defined by (1). Since
This, together with (22), (23) and t|s| = 36, implies that, at each point of M, threeof the eigenvalues of C are equal to s/4 + |s|/9 and the fourth one is s/4 − |s|/3.But the latter case can never happen because of Lemma 1 below.
In the case when n = 2, it easily follows from (20) and (21) that B = 0 andt2s2 = 144ct . The latter identity together with (13) gives st = 6.
Conversely, if M is Einstein and self-dual with st = ±12 or st = 6, then thetwistor space (Z, ht , Jn), n = 1 or 2, is ∗-Einstein according to Corollary 1.
Finally, the following result of LeBrun [18] and Apostolov [3] completes theproof of Theorem 1:
LEMMA 1. There is no self-dual manifold (M, g) whose Ricci operator has con-stant eigenvalues (λ, µ,µ,µ) with λ �= 0 and λ �= µ.
Proof. Suppose that such a manifold (M, g) exists. Then its Ricci tensor ρ
locally has the form
ρ = (λ − µ)α ⊗ α + µg, (24)
where α is a unit 1-form. Denoting the dual vector field of α by A, we obtain from(24) that
Since the scalar curvature is constant, the second Bianchi identity implies
δρ = − 12ds = 0 and (dρ)− = −2δW− = 0,
(dρ)− being the restriction of dρ on TM ⊗ �2−TM. So, it follows from (25) thatδα = 0 and ∇Aα = 0. The identities (26) (for X = A) and (dρ)− = 0 imply(dα)− = 0 which together with ∇Aα = 0 gives dα = 0. Now it follows from (26)that dρ(X, Y,Z) = 0 whenever Y,Z ⊥ A and we get dρ = 0 since (dρ)− = 0.Applying again identity (26) we see that ∇α = 0. Therefore, λ = ρ(A,A) = 0which contradicts to the assumption λ �= 0. �
Remark. Lemma 1 has been first proved by LeBrun [18] in the case λ > 0.Recently, Apostolov [3] has pointed out to us that the same result holds for anyλ �= 0.
Acknowledgements
We are very grateful to C. LeBrun and V. Apostolov for sending us their proofs ofLemma 1. The authors would like also to thank the referee for his valuable remarks.J.D. and O.M. were supported in part by the NSF grant INT-9903302.
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