Curvature properties of twistor spaces
aa r X i v : . [ m a t h . DG ] F e b CURVATURE PROPERTIES OF TWISTOR SPACES
JOHANN DAVIDOV AND OLEG MUSHKAROV
Dedicated to Professor Armen Sergeev on the occasion of his 70th birthday
Abstract.
In this paper we review some results on the Riemannian and al-most Hermitian geometry of twistor spaces of oriented Riemannian 4-manifoldswith emphasis on their curvature properties.2010
Mathematics Subject Classification . Primary 53C28, Secondary 53C56.
Key words: twistor spaces, almost Hermitian structures, Riemannian andHermitian curvatures of twistor spaces . Introduction
The twistor theory has its origin in Mathematical Physics. Inspired by thePenrose program [81, 82], Atiyah, Hitchin and Singer [16] developed this theory onoriented Riemannian 4-manifolds. They defined the twistor space of such a manifold M as the 2-sphere bundle Z on M whose fiber at any point p ∈ M consists of allcomplex structures on the tangent space T p M compatible with the metric and theopposite orientation of M . The 6-manifold Z admits two natural almost complexstructures J and J introduced, respectively, by Atiyah, Hitchin, and Singer [16]and by Eells and Salamon [47]. The almost complex structure J is never integrablebut it plays an important role in the theory of harmonic maps. The almost complexstructure J is integrable if and only if the base manifold M is self-dual, i.e. itsWeyl conformal tensor W satisfies the equation ∗W = W , where ∗ is the Hodgestar-operator [19]. So, in the case when M is self-dual, Z is a complex 3-manifoldand each fibre Z p = π − ( p ) , p ∈ M , is a complex submanifold of Z biholomorphicto CP . The normal bundle of Z p is biholomorphically equivalent to H ⊕ H , where H is the positive Hopf bundle on CP . The antipodal map j → − j on each fibreinduces an anti-biholomorphic involution of Z without fixed points.The above construction depends only on the conformal class of the given metricon M and, conversely, the complex structure of the twistor space determines theself-dual conformal structure on M . More precisely, let Z be a complex 3-manifoldwith the following properties: (1) Z is fibered by projective lines whose normalbundle is isomorphic to H ⊕ H ; (2) Z possesses a free anti-holomorphic involutionwhich transforms each fibre to itself. Then Z is the twistor space of a self-dualmanifold M ([16], see also [19]). The described correspondence between self-dualmanifolds and twistor spaces is called the Penrose correspondence and it has beenused for years to study the conformal geometry of four-manifolds by means ofcomplex geometry techniques. The authors are partially supported by the National Science Fund, Ministry of Education andScience of Bulgaria under contract DN 12/2 .
The twistor space Z admits a 1-parameter family of Riemannian metrics whichare compatible with the almost complex structures J and J and the naturalprojection π : Z → M is a Riemannian submersion [19]. These natural almostHermitian structures are very interesting geometric objects in their own right whosedifferential geometric properties have been studied by many authors, to cite herejust a few [14, 42, 48, 49, 50, 51, 62, 63, 83].Motivated by open questions in non-K¨ahler geometry in a series of papers [77,36, 35, 33, 34, 39, 37, 9, 2], the authors studied the twistor spaces of orientedRiemannian 4-manifolds as a source of examples of almost Hermitian 6-manifoldswith interesting geometric properties. In the present paper we review some of theseresults with emphasis on the curvature properties of twistor spaces. Table of contents
2. Preliminaries3. Riemannian curvature of twistor spaces3.1. Riemannian sectional curvature3.2. Ricci curvature4. Twistor spaces with Hermitian Ricci tensor5. K¨ahler curvature identities on twistor spaces6. ∗ -Einstein twistor spaces7. Curvature properties of the Chern connection on twistor spaces8. Holomorphic curvatures of twistor spaces8.1. Holomorphic bisectional curvature8.2. Orthogonal bisectional curvatures8.3. Hermitian bisectional curvature Acknowledgements . The authors would like to thank the editors of this volumefor the invitation to submit a paper in honour of Professor Armen Sergeev, ourdear friend and colleague. 2.
Preliminaries
Let (
M, g ) be an oriented (connected) Riemannian manifold of dimension four.The metric g induces a metric on the bundle of two-vectors π : Λ T M → M by theformula g ( v ∧ v , v ∧ v ) = 12 det [ g ( v i , v j )] , the factor 1 / ∗ : Λ k T M → Λ − k M , k = 0 , ...,
4, be the Hodge star operator. For k = 2, it is an involution of Λ T M ,thus we have the orthogonal decompositionΛ T M = Λ − T M ⊕ Λ T M, where Λ ± T M are the subbundles of Λ T M corresponding to the eigenvalues ± ∗ . Given a local oriented orthonormal frame ( E , E , E , E ) of T M we set(1) s ± = E ∧ E ± E ∧ E , s ± = E ∧ E ± E ∧ E , s ± = E ∧ E ± E ∧ E . Then ( s ± , s ± , s ± ) is a local orthonormal frame of Λ ± T M . This frame defines an ori-entation on Λ ± T M which does not depend on the choice of the frame ( E , E , E , E )(see, for example, [32]). We call this orientation ”canonical”.For every a ∈ Λ T M , define a skew-symmetric endomorphism of T π ( a ) M by (2) g ( K a X, Y ) = 2 g ( a, X ∧ Y ) , X, Y ∈ T π ( a ) M. Denoting by G the standard metric − T race ( P Q ) on the space of skew-symmetricendomorphisms, we have G ( K a , K b ) = 2 g ( a, b ) for a, b ∈ Λ T M . The assignment a → K a is the standard isomorphism of the bundle Λ T M with the bundle A ( T M )of g -skew-symmetric endomorphism of T M . Let × be the usual vector cross producton the oriented 3-dimensional vector space Λ ± T p M , p ∈ M , endowed with themetric g . If a, b ∈ Λ ± T p M , the isomorphism Λ T M ∼ = A ( T M ) sends a × b to ± [ K a , K b ]. In the case when a ∈ Λ T p M , b ∈ Λ − T p M , the endomorphisms K a and K b of T p M commute. For a, b ∈ Λ ± T p M , we have K a ◦ K b = − g ( a, b ) Id ± K a × b . In particular, K a and K b , a, b ∈ Λ ± T p M , anti-commute if and only if a and b areorthogonal.If σ ∈ Λ T M is of unit length, then K σ is a complex structure on the vector space T π ( σ ) M compatible with the metric g , i.e., g -orthogonal. Conversely, the 2-vector σ dual to one half of the K¨ahler 2-form of such a complex structure is a unit vectorin Λ T M . Therefore the unit sphere bundle Z of Λ T M parametrizes the complexstructures on the tangent spaces of M compatible with the metric g . This bundleis called the twistor space of the Riemannian manifold ( M, g ). Since M is oriented,the manifold Z has two connected components Z ± called the positive and negativetwistor spaces of ( M, g ). These are the unit sphere subbundles of Λ ± T M . Thebundle π : Z ± → M parametrizes the complex structures on the tangent spacesof M compatible with the metric and the ± orientation via the correspondence Z ± ∋ σ → K σ .The connection ∇ on Λ T M induced by the Levi-Civita connection of M pre-serves the bundles Λ ± T M , so it induces a metric connection on each of themdenoted again by ∇ . The horizontal distribution of Λ ± T M with respect to ∇ istangent to the twistor space Z ± . Thus, we have the decomposition T Z ± = H ⊕ V of the tangent bundle of Z ± into horizontal and vertical components. The verticalspace V σ = { V ∈ T σ Z ± : π ∗ V = 0 } at a point σ ∈ Z ± is the tangent space to thefibre of Z ± through σ . If we consider T σ Z ± as a subspace of T σ (Λ ± T M ), then thespace V σ is the orthogonal complement of σ in Λ ± T π ( σ ) M .The 6-manifold Z ± admits two almost complex structures J and J introduced,respectively, by Atiyah, Hitchin, and Singer [16] and by Eells and Salamon [47]. For σ ∈ Z ± , the horizontal space H σ is isomorphic to the tangent space T π ( σ ) M via thedifferential π ∗ σ , and the structures J and J on the space H σ are both definedas the lift to H σ of the complex structure K σ on T π ( σ ) M . The vertical space V σ is tangent to the unit sphere in the 3-dimensional vector space (Λ ± T π ( σ ) M, g ), andwe denote by J σ the standard complex structure of the unit sphere restricted to V σ . It is given by J σ V = ± ( σ × V ) , V ∈ V σ , On a vertical space V σ , J is defined to be the complex structure J σ of the fibrethrough σ , while J is defined as the conjugate complex structure, i.e., J |V σ = JOHANN DAVIDOV AND OLEG MUSHKAROV −J σ . Thus, for σ ∈ Z ± ,(3) J n |H σ = ( π ∗ |H σ ) − ◦ K σ ◦ π ∗ |H σ J n V = ± ( − n +1 ( σ × V ) for V ∈ V σ , n = 1 , . It is a result of Eells and Salamon [47] that the almost complex structure J is never integrable, so it does not come from a complex structure. Nevertheless, J is very useful for constructing harmonic maps. The integrability condition for J has been found by Atiyah, Hitchin, and Singer [16]. To state their result,we first recall the well-known curvature decomposition in dimension four. Notethat for the curvature tensor R , we adopt the following definition: R ( X, Y ) = ∇ [ X,Y ] − [ ∇ X , ∇ Y ]. The curvature operator R corresponding to the curvaturetensor is the endomorphism of Λ T M defined by g ( R ( X ∧ Y ) , Z ∧ U ) = g ( R ( X, Y ) Z, U ) , X, Y, Z, U ∈ T M.
Denote by ρ the Ricci tensor of ( M, g ) and by A : T M → T M its Ricci operator, g ( A ( X ) , Y ) = ρ ( X, Y ). Then the endomorphism B : Λ T M → Λ T M correspond-ing to the traceless Ricci tensor is given by(4) B ( X ∧ Y ) = A ( X ) ∧ Y + X ∧ A ( Y ) − s X ∧ Y, where s is the scalar curvature. Note that B sends Λ ± T M into Λ ∓ T M . Let W : Λ T M → Λ T M be the endomorphism corresponding to the Weyl conformaltensor. Denote the restriction of W to Λ ± T M by W ± , so W ± sends Λ ± T M toΛ ± T M and vanishes on Λ ∓ T M .It is well known that the curvature operator decomposes as ([87], [19, Chapter1 H])(5) R = s Id + B + W + + W − Note that this differs from [19] by a factor of 1 / / T M . The Riemannian manifold (
M, g ) is Ein-stein exactly when B = 0. It is called self-dual (anti-self-dual) if W − = 0 ( W + = 0).The self-duality (anti-self-duality) condition is invariant under conformal changesof the metric since the Weyl tensor is so. Note also that changing the orientationof M interchanges the roles of Λ − T M and Λ T M (respectively, of Z − and Z + ),hence the roles of W − and W + .The famous Atiyah-Hitchin-Singer theorem [16] states that the almost complexstructure J on Z − (resp. Z + ) is integrable if and only if ( M, g ) is self-dual (resp.anti-self-dual).The twistor space Z ± of an oriented Riemannian 4-manifold ( M, g ) admits anatural 1-parameter family of Riemannian metrics h t defined by h t = π ∗ g + tg v where t > g v is the restriction of the metric of Λ T M on the vertical dis-tribution V . Then π : ( Z ± , h t ) → ( M, g ) is a Riemannian submersion with totallygeodesic fibres, and the almost-complex structures J and J are compatible withthe metrics h t . The Gray-Hervella classes [59] of the almost Hermitian structures( h t , J n ) , t > , n = 1 ,
2, have been determined in [77]. Riemannian curvature of twistor spaces
The O’Neill formulas [79], [19, Ch. 9 G] can be used to obtain coordinate-freeformulas for various curvatures of the metric h t on the twistor space in termsof the curvature of its base manifold M . This is done in [36] in the case when dim M = 4 and in [31] for the general twistor space of partially complex structures( f -structures) on a Riemannian manifold of any dimension ≥
3. We shall discusshere the most interesting case of the negative twistor space of an oriented fourdimensional Riemannian manifold. The reason to choose the negative twistor spaceis connected with the Atiyah-Hitchin-Singer theorem mentioned above. As smoothmanifolds, the positive and the negative twistor spaces of CP coincide with thecomplex flag manifold F , . The Atiyah-Hitchin-Singer almost complex structureon the negative twistor space of CP is integrable and coincides with the standardcomplex structure of F , , while it is not integrable on the positive twistor space.In what follows, ( M, g ) will denote an oriented Riemannian manifold of dimen-sion four, and Z will stand for its negative twistor space Z − .3.1. Riemannian sectional curvature.
Let (
M, g ) be an oriented Riemannian4-manifold with Levi-Civita connection ∇ and Riemannian curvature tensor R . Forany t > R t the Riemannian curvature tensor of the metric h t on thetwistor space Z of ( M, g ). Applying the O’Neill formulas [79] for the Riemann-ian submersion π : ( Z , h t ) → ( M, g ), one can compute the Riemannina sectionalcurvature of ( Z , h t ). Proposition 1. ([36]) Let
E, F ∈ T σ Z , X = π ∗ E, Y = π ∗ F, V = V E and W = V F .Then h t ( R t ( E ∧ F ) E, F ) = g ( R ( X ∧ Y ) X, Y ) − tg (( ∇ X R )( X ∧ Y ) , σ × W )+ tg (( ∇ Y R )( X ∧ Y ) , σ × V ) − tg ( R ( σ ) , X ∧ Y ) g ( σ × V, W ) − t g ( R ( σ × V ) X, R ( σ × W ) Y ) + t k R ( σ × W ) X + R ( σ × V ) Y k − t k R ( X ∧ Y ) σ k + t ( k V k k W k − g ( V, W ) ) . In the case when the base manifold (
M, g ) is self-dual and Einstein the aboveformula takes an apparently simple form.
Corollary 1.
Let ( M, g ) be a self-dual Einstein manifold with scalar curvature s .Then h t ( R t ( E ∧ F ) E, F ) = g ( R ( X ∧ Y ) X, Y ) − ts g ( σ, X ∧ Y ) g ( σ × V, W ) −
12 ( ts
12 ) g ( X, Y ) g ( V, W ) + 3( ts
12 ) g ( X ∧ Y, V ∧ W )+( ts
24 ) ( k X k k W k + k Y k k V k ) − t ( s
24 ) ( k X ∧ Y k − g ( σ, X ∧ Y ) )+ t ( k V k k W k − g ( V, W ) ) . JOHANN DAVIDOV AND OLEG MUSHKAROV
Ricci curvature.
The study of the Ricci curvature of a twistor space is basedon the following explicit formula for the Ricci tensor which is a consequence ofProposition 1.
Proposition 2. ([36]) Let ρ t be the Ricci tensor of the twistor space ( Z , h t ). If E ∈ T σ Z , X = π ∗ E , and V = V E , then ρ t ( E, E ) = ρ ( X, X ) + tg ( δ R ( X ) , σ × V ) + t ||R ( σ × V ) || + t || ı X ◦ R ( σ ) || − t || ı X ◦ R − || + || V || , where ρ is the Ricci tensor of ( M, g ), δ R is the co-differential of R , R − is therestriction of R on Λ − T M , and ı X : Λ T M → T M is the interior product.Taking the trace of ρ t , we obtain the following formula for the scalar curvature s t of the twistor space ( Z , h t ). Corollary 2. ([36]) Let s be the scalar curvature of ( M, g ). Then s t ( σ ) = s ( π ( σ )) + t ||R ( σ ) || − ||R − || π ( σ ) ) + 2 t . In the case when the base manifold of a twistor space is Einstein and self-dual,these formulas can significantly be simplified as follows.
Corollary 3. ([36]) If (
M, g ) is Einstein and self-dual, the Ricci tensor ρ t of ( Z , h t )and its scalar curvature s t are given by ρ t ( E, E ) = [ s − t ( s
12 ) ] || X || + [1 + ( ts
12 ) ] || V || , E = X hσ + V,s t = 2 t + s − t s . As an application of Proposition 2, one can prove the following result of T.Friedrich and R. Grunewald [49] about the Einstein condition on ( Z , h t ). Theorem 1. ([49, 36]) The Riemannian manifold ( Z , h t ) is Einstein if and only if( M, g ) is a self-dual Einstein manifold with scalar curvature s = 6 /t or s = 12 /t .The next property of the Ricci tensor ρ t is an easy consequence of Corollary 3. Proposition 3. ([38]) If (
M, g ) is Einstein and self-dual, the covariant derivativeof the Ricci tensor ρ t of ( Z , h t ) satisfies the identity(6) ( D E ρ t )( E, E ) = 0 , E ∈ T Z , where D is the Levi-Civita connection of h t . Moreover, ρ t is parallel if and only if st = 6, st = 12, or s = 0. Remark . Condition (6) for the Ricci tensor ρ of a Riemannian manifold ( N, h )is known as the third Ledger condition [72], [98, Sec. 6.8]. It is easy to see bypolarization that (6) is equivalent to the identity( ∇ X ρ )( Y, Z ) + ( ∇ Y ρ )( Z, X ) + ( ∇ Z ρ )( X, Y ) = 0 , X, Y, Z ∈ T N, where ∇ is the Levi-Chivita connection of ( N, h ). If this condition is satisfied, themanifold is real-analytic [90], and the scalar curvature is constant [30, Proposition2.3]. Condition (6) appears in the study of the so-called D’Atri spaces which arecharacterized by the property that the geodesic symmetries preserve the volume up to sign [30]. It is one of the Einstein-like conditions introduced and studied byA. Gray [57], and discussed also in Besse’s book [19, Sec. 16G] as an interestinggeneralization of the Einstein condition. We refer the reader to [19, 31, 57, 65, 67, 80]for examples of Riemannian manifolds satisfying condition (6). Proposition 3, whichgives twistorial examples of such manifolds, seems to be interesting in the case ofnegative scalar curvature of (
M, g ) since the complete classification of compactEinstein self-dual manifolds with negative scalar curvature is not available yet. Ithas been conjectured by A. Vitter [97] that every such a manifold is a quotient ofthe unit ball in C with the metric of negative constant sectional curvature or theBergman metric.Corollary 3 can be used to show that an isometry of the twistor space preservesvertical, and hence horizontal, spaces. This implies the following. Lemma 1. ([31]) If (
M, g ) is an Einstein and self-dual manifold with scalar cur-vature s , then every (local) isometry of the twistor space π : ( Z , h t ) → ( M, g )descends to a (local) isometry of the metric g provided ts = 6 and ts = 12. Remarks . ([31]) Suppose that the manifold (
M, g ) is Einstein and self-dual, and ts = 6 or ts = 12. Lemma 1 does not hold as there may exist an isometry of the twistor space of(
M, g ) which does not descend to an isometry of g . For example, it is well-knownthat the twistor space Z of the sphere S considered with its standard metric is thecomplex projective space CP . To describe the twistor projection π : CP → S , itis convenient to identify S with the quaternionic projective space HP . Then π isgiven in homogeneous coordinates by [ z , z , z , z ] → [ z + z j , z + z j ]. If ts = 12,the metric h t is a multiple of the Fubini-Study metric. The map Ψ : CP → CP defined by Ψ([ z , z , z , z ]) = [ √ ( z + z ) , √ ( z − z ) , z , z ] is an isometry of theFubini-Study metric which does not preserve all fibres of the twistor projection π . The scalar curvature s of M is positive and, by a result of Hitchin [62] and ofFriedrich and Kurke [50], see also [19, Theorem 13.30], ( M, g ) is isometric to thesphere S or the complex projective space CP with their standard metrics. Inparticular, the metric g is homogeneous, hence all of the metrics h t on the twistorspace are also homogeneous.The latter remark and Lemma 1 give the following result which seems to be”folklore”. Proposition 4. ([31]) Let (
M, g ) be a complete Einstein self-dual manifold. Themetric h t (with arbitrary t ) on the twistor space Z is (locally) homogeneous if andonly if the metric g on the base manifold M is (locally) homogeneous.Proposition 3 and Lemma 1 imply the following Proposition 5. ([31]) Let M be an inhomogeneous Einstein self-dual 4-manifoldwith non-zero scalar curvature s . Then, for any t > ts = 6 and ts = 12, thetwistor space ( Z , h t ) is non-homogeneous, has non-parallel Ricci tensor satisfyingthe third Ledger condition (6) and is not locally isometric to a Riemannian product.Moreover, if M is locally non-homogeneous, then so is its twistor space. Remark . ([31]) If the base manifold M is locally homogeneous, so is its twistorspace. There are a lot of examples of (non-compact) locally non-homogeneous, self-dual, Einstein manifolds with non-zero scalar curvature, to cite just a few paperswhere such examples (complete or not) can be found:[13, 26, 41, 63, 70]. JOHANN DAVIDOV AND OLEG MUSHKAROV Twistor spaces with Hermitian Ricci tensor
It is well-known that on any symplectic manifold N with symplectic form Ωthere exist a Riemannian metric h and a h -orthogonal almost complex structure J such that Ω is the K¨ahler 2-form of the almost Hermitian manifold ( N, h, J ), i.e.,Ω(
X, Y ) = h ( JX, Y ) for
X, Y ∈ T N . Recall that an almost Hermitian manifoldwith closed K¨ahler 2-form is called almost K¨ahler. A Riemannian metric h on N is said to be associated to the symplectic form Ω if there exists a h -orthogonalalmost complex structure J for which Ω( X, Y ) = h ( JX, Y ). Note that such analmost complex structure is unique. Assume that N is compact, and denote by A the set of all Riemannian metrics on N associated to Ω. If h ∈ A and J is thecorresponding almost complex structure, let s and s ∗ be the scalar curvature of themetric h and the ∗ -scalar curvature of the almost Hermitian structure ( h, J ) (werecall the definition of s ∗ in Section 6). Then we can consider the integrals Z N s vol h and Z N ( s − s ∗ ) vol h as functionals on the set A . D. Blair and S. Ianu¸s [23] have proved that the criticalpoints of both functionals are the Riemannian metrics h ∈ A whose Ricci tensor ρ is Hermitian with respect to the corresponding almost complex structure J , i.e.,(7) ρ ( JX, JY ) = ρ ( X, Y ) , X, Y ∈ T N.
The K¨ahler metrics satisfy this condition, and Blair and Ianu¸s raised the ques-tion of whether a compact almost K¨ahler manifold with Hermitian Ricci tensor isK¨ahlerian. This question motivated the following result.
Theorem 2. ([35]) Let (
M, g ) be a connected oriented real-analytic Riemannianmanifold. If the Ricci tensor of the twistor space ( Z , h t ) is J n -Hermitian, n = 1 or n = 2, then either( i ) ( M, g ) is Einstein and self-dualor( ii ) ( M, g ) is self-dual with constant scalar curvature s = 12 /t and, for each pointof M , at least three eigenvalues of its Ricci operator coincide.Conversely, if ( M, g ) is a smooth oriented Riemannian four-manifold satisfying( i ) or ( ii ), then the Ricci tensor of ( Z , h t ) is J n -Hermitian. Examples . ([35]) . Let M be an Einstein self-dual manifold with negative scalarcurvature. Then, by [77], ( J , h t ) for t = − /s is an almost K¨ahler structure onthe twistor space Z . This structure is not K¨ahlerian since, by the Eells-Salamonresult mentioned above, the almost complex structure J is not integrable. Onthe other hand, the Ricci tensor of the metric h t is J -Hermitian by Theorem 2.Thus, if M is compact, the twistor space ( Z , h t , J ) gives a negative answer tothe Blair-Ianu¸s question. Examples of compact Einstein self-dual manifolds withnegative scalar curvature can be found in [97]. Multiplying the twistor space ofsuch a manifold by K¨ahler manifolds, one can construct examples of non-K¨ahleralmost K¨ahler manifolds of arbitrary even dimension ≥ . The Riemannian product M = S × S is a non-Einstein manifold satisfying theconditions (ii) of Theorem 2. Other examples of such manifolds can be obtained aswarped-products of S and S , see, for example, [40].The twistor space construction can be used to obtain other examples of almostHermitian manifolds with Hermitian Ricci tensor. Let ( M, g, J ) be a 4-dimensionalalmost Hermitian manifold with the orientation induced by the almost complexstructure J . Then J is a section of the positive twistor bundle π : Z + → M .Taking the horizontal lift of J and the complex structure of the fibre of Z + wedefine an almost complex structure J compatible with the metrics h t , t >
0. Moreprecisely, for σ ∈ Z , X ∈ T π ( σ ) M , and V ∈ V σ we set J X hσ = ( JX ) hσ , J V = σ × V. The geometric conditions for integrability of J have been obtained in [42] and theGray-Hervella classes of the almost Hermitian structure ( h t , J ) have been deter-mined in [2]. Theorem 3. ([37]) The Ricci tensor of the almost Hermitian manifold ( Z + , h t , J )is Hermitian if and only if the base manifold ( M, g ) is Einstein and anti-self-dual.
Examples . ([37]) According to [2, Theorem 1], the almost Hermitian structure( h t , J ) is K¨ahler exactly when ( M, g, J ) is K¨ahler and Ricci flat. Thus, in orderto construct compact non-K¨ahler twistor spaces ( Z , h t , J ) with Hermitian Riccitensor we need examples of compact, Einstein, anti-self-dual, non-K¨ahler almostHermitian manifolds ( M, g, J ). We consider three cases according to the sign of thescalar curvature s of such a manifold. . Case s >
0. In this case, by the Hitchin and Friedrich-Kurke result we havementioned, (
M, g ) is isometric either to the 4-sphere S with the round metric orto the complex projective space CP with the opposite orientation and the Fubini-Study metric. As is well-known, none of these manifolds admits an almost complexstructure for topological reasons. . Case s <
0. As C. LeBrun pointed out to us, Conder and Maclachlan [29] haveconstructed a compact orientable Riemannian manifold (
M, g ) of constant negativesectional curvature with Euler characteristic χ = 16. The signature of M is zeroby the well-known integral formula τ = 112 π Z M ( ||W + || − ||W − || ) vol g since both half-Weyl tensors W ± vanish. In particular, the intersection form of M is indefinite. We also have τ + χ ≡ mod
4. Hence, by a version of Ehresmann-Wutheorem due to O. Saeki (see, for example, [74, Theorem 8 (A)], M admits an almostcomplex structure I . Then, as is well-known, M admits also an almost complexstructure J compatible with the metric of M . Indeed, let g be a Riemannianmetric on M compatible with I , for example take g ( X, Y ) = g ( X, Y ) + g ( IX, IY ).Define a symmetric positive endomorphism G of T M by g ( GX, Y ) = g ( X, Y ).Then J = G − / IG / is an almost complex structure compatible with the metric g . This almost complex structure is not integrable as the following lemma shows. Lemma 2. ([37]) Every compact anti-self-dual Hermitian surface (
M, g, J ) of non-positive scalar curvature is K¨ahler and scalar flat.
Note also that if a Riemannian manifold (
M, g ) admits a compatible almost com-plex structure J , it possesses many such structures inducing the same orientationas J . This can be seen, for example, by means of the exponential map of the twistorspace ( Z + , h t ) of ( M, g ) endowed with the orientation induced by J [32, 37]. . Case s = 0. It is a result of Hitchin [61] that every compact, Ricci flat, anti-self-dual, 4-manifold is either flat, or is a K W + of a K¨ahler surface are s/ , − s/ , − s/
12 (see, forexample, [8]) .Now, let (
M, g, J ) be a compact Ricci flat K¨ahler surface (a Calabi-Yau surface).Let J u ( p ) = exp J ( p ) [ uV ( p )] be a 1-parameter deformation of the K¨ahler structure J , where V is a non-zero compactly supported section of the pull-back bundle J ∗ V → M . Then every J u , u = 0, is non-integrable. Otherwise, by Lemma 2, J u would be K¨ahler and we would have J u = J everywhere since J u = J outside of supp V . Hence V = 0 on M , a contradiction. Thus ( M, g, J u ) is a compact, Ricciflat, anti-self-dual, strictly almost Hermitian manifold.Finally, let us note that the twistor spaces ( Z + , h t , J u ) of the almost Hermit-ian 4-manifolds ( M, g, J u ) belong to the Gray-Hervella class G [2]. Moreover, byProposition 3 and [2, Theorem 1], it follows that G is the only possible Gray-Hervella class of the twistor spaces ( Z + , h t , J ) with Hermitian Ricci tensor.5. K¨ahler curvature identities on twistor spaces
In order to generalize results in K¨ahler geometry, A. Gray [55] has introducedthree classes of almost Hermitian manifolds whose curvature tensor resembles thatof a K¨ahler manifold. On an almost Hermitian manifold (
N, h, J ), these classes aredefined by the following curvature identities: AH : R ( X, Y, Z, W ) = R ( X, Y, JZ, JW ) AH : R ( X, Y, Z, W ) = R ( JX, JY, Z, W ) + R ( JX, Y, JZ, W ) + R ( JX, Y, Z, JW ) AH : R ( X, Y, Z, W ) = R ( JX, JY, JZ, JW ) , where, as usual, R ( X, Y, Z, W ) = h ( R ( X, Y ) Z, W ) for
X, Y, Z, W ∈ T N . Theseidentities have been used in [93] for finding irreducible components of the spaceof curvature tensors on an Hermitian vector space under the action of the unitarygroup. They have also been a useful tool for characterizing the K¨ahler manifolds invarious classes of almost Hermitian manifolds, to quote just a few papers [6, 7, 10,85, 94, 95]. Note that in the last years there has been an intensive study of Hermitianmetrics which are K¨ahler-like in the sense that the curvature tensor of either theLevi-Civita, Chern, Bismut-Str¨ominger, or more generally, a Gauduchon connection[52] has the same symmetries as the curvature tensor of a K¨ahler metric(see, forexample, [4, 101, 102]).The next theorem gives geometric characterizations of the oriented Riemannianfour-manifolds (
M, g ) whose twistor spaces ( Z , h t , J n ) , n = 1 ,
2, belong to one ofthe K¨ahler curvature classes listed above.
Theorem 4. ([39]) ( i ) ( Z , h t , J n ) ∈ AH is equivalent to ( Z , h t , J n ) ∈ AH andholds if and only if ( M, g ) is Einstein and self-dual, n = 1 or 2. ( ii ) ( Z , h t , J ) ∈ AH if and only if ( M, g ) is Einstein and self-dual with scalarcurvature s = 0 or s = 12 /t .( iii ) ( Z , h t , J ) ∈ AH if and only if ( M, g ) is Einstein and self-dual with s = 0, Remarks . ([39]) . By a result of S. Goldberg [53], every compact almost K¨ahlermanifold of class AH is K¨ahler, and A. Gray [55, Theorem 5.3] has raised thequestion of whether the same is true under the weaker condition AH . Now, let( M, g ) be an Einstein self-dual four-manifold with negative scalar curvature s . For t = − /s , the twistor space ( Z , h t , J ) is an almost K¨ahler manifold of class AH by [77] and Theorem 4. This manifold is not K¨ahler, since the almost complexstructure J is never integrable. So, we have a negative answer to Gray’s question. . Let ( M, g ) be a Ricci-flat self-dual four-manifold. Then ( Z , h t , J ), t >
0, is aquasi K¨ahler manifold [77] of class AH which is not K¨ahler. Thus, the Goldbergresult cannot be extended to quasi K¨ahler manifolds. In the case when M = R the twistor space is Z = R × S and we recover an example of A. Gray [55].By a result of I. Vaisman [95], every compact Hermitian surface of class AH is K¨ahler. The twistor space ( Z , h t , J ) is a non-K¨ahler Hermitian manifold ofcomplex dimension 3 and of class AH by [50],[16] and Theorem 4. If M is compact,then Z is also compact, and we see that the Vaisman result is not true in complexdimensions greater than 2. . If ( M, g ) is an Einstein self-dual four-manifold with scalar curvature s > Z , h t , J ) , t = 12 /s, is a K¨ahler manifold [50] and hence of class AH . Infact, in this case, as we have already mentioned, either M = S or M = CP ,so ( Z , h t , J ) is either CP or the complex flag manifold F , with their standardK¨ahler structures. 6. ∗ -Einstein twistor spaces It is well known that the Ricci/Chern form of a K¨ahler manifold is the image R (Ω) of the K¨ahler form Ω under the action of the curvature operator R ∈
End (Λ ).For an arbitrary almost Hermitian manifold ( N, h, J ), the 2-form R (Ω) is neitherclosed nor of type (1 , N, J ).The tensor ρ ∗ associated to R (Ω) by ρ ∗ ( X, Y ) = R (Ω)( X, JY ) =
T race ( Z → R ( JZ, X ) JY )has been introduced by S. Tachibana [91], and is known in the literature as the ∗ -Ricci tensor. This tensor then appeared in almost Hermitian geometry in differentcontexts. For example, it has been used by A. Gray [56] for studying nearly K¨ahlermanifolds and by F. Tricceri and L. Vanhecke [93] for describing the irreduciblecomponents of the space of curvature tensors on a Hermitian vector space underthe action of the unitary group. The ∗ -Ricci tensor also plays an important role inthe theory of harmonic almost complex structures, developed recently by C. Wood[100].An almost Hermitian manifold is said to be weakly ∗ -Einstein if its ∗ -Ricci ten-sor is a multiple of the metric, i.e. if the K¨ahler form is an eigenvector of thecurvature operator. Unlike K¨ahler-Einstein manifolds, the multiple (usually calledthe ∗ -scalar curvature) need not be a constant and when this holds the manifold iscalled ∗ -Einstein. As we have already mentioned, for K¨ahler manifolds the Einstein and weakly ∗ -Einstein conditions coincide, so it is natural to ask whether there is arelation between them for more general almost Hermitian manifolds. The curvaturedecomposition (5) implies that in real dimension four the weakly ∗ -Einstein condi-tion holds if and only if the traceless Ricci tensor is J -anti-invariant and the K¨ahlerform is an eigenvector of the self-dual Weyl operator W + . Since, for a Hermitian4-manifold, the latter condition is equivalent to W + being degenerate (see [12]), itfollows from the Riemannian Goldberg-Sachs theorem [12, 78, 84] that any EinsteinHermitian metric is weakly ∗ -Einstein. For almost K¨ahler 4-manifolds, it is still anopen question whether the Einstein condition implies the weakly ∗ -Einstein one,although J. Armstrong [15] has explicitly described all weakly ∗ -Einstein strictlyalmost K¨ahler Einstein 4-manifolds. This, combined with a result of K. Sekigawa[86], shows that such manifolds can never be compact, so the positive answer tothe question above would imply the well-known Goldberg conjecture [53] that anycompact almost K¨ahler Einstein 4-manifold must be K¨ahler. In higher dimensions,the (weakly) ∗ -Einstein condition has not been so well studied and it seems thatthe main reason for that is the lack of interesting examples. Because of that in[33] the authors studied the twistor spaces of oriented Riemannian 4-manifolds asa source of 6-dimensional examples of ∗ -Einstein almost Hermitian manifolds andshowed that some four-dimensional results on the ∗ -Einstein condition cannot beextended to higher dimensions.The ∗ -Ricci tensor ρ ∗ t,n of the twistor space ( Z , h t , J n ), t > , n = 1 ,
2, can becomputed in terms of the curvature of the base manifold (
M, g ) using the formulafor the sectional curvature of ( Z , h t ) in Proposition 1 and the well-known expressionof the Riemannian curvature tensor by means of sectional curvatures. Proposition 6. ([33]) Let
E, F ∈ T σ Z and X = π ∗ E , Y = π ∗ F , A = V E , B = V F .Then ρ ∗ t,n ( E, F ) = [1 + ( − n +1 ] g ( R ( σ ) , X ∧ K σ Y ) − t g ( R ( X ∧ K σ Y ) σ, R ( σ ) σ )+ t Z → g ( R ( X ∧ Z ) σ, R ( K σ Z ∧ K σ Y ) σ ))+( − n +1 Trace( V σ ∋ C → g ( R ( C ) X, R ( σ × C ) K σ Y ))+ t − n g (( ∇ X R )( σ ) , B ) + t g (( ∇ K σ Y R )( σ ) , σ × A )+[1 + ( − n +1 tg ( R ( σ ) , σ )] g ( A, B )+( − n +1 t Z → g ( R ( σ × A ) K σ Z, R ( B ) Z )) , where K σ is the complex structure on T π ( σ ) M determined by σ .In the case when the base manifold ( M, g ) is Einstein and self-dual the formulafor ρ ∗ t,n simplifies significantly: Corollary 4.
Let ( M, g ) be an Einstein self-dual -manifold with scalar curvature s . Then ρ ∗ t,n ( E, F ) = 112 [(1 + ( − n +1 ) s + t
24 (1 + ( − n ) s ] g ( X, Y )++[1 + ( − n +1 ts − n ( ts
12 ) ] g ( A, B ) . The above formulas can be used to obtain the following geometric characteriza-tion of the ∗ -Einstein twistor spaces. Theorem 5. ([33]) Let (
M, g ) be an oriented Riemannian 4-manifold with scalarcurvature s .( i ) The twistor space ( Z , h t , J ) is ∗ -Einstein if and only if ( M, g ) is Einstein,self-dual and t | s | = 12,( ii ) The twistor space ( Z , h t , J ) is ∗ -Einstein if and only if ( M, g ) is Einstein,self-dual and ts = 6 .A crucial role in the proof of Theorem 5 is played by the following result essen-tially due to C. LeBrun and V. Apostolov (private communications, 2000) which isalso of independent interest. Lemma 3. ( [33] ) There is no self-dual manifold ( M, g ) whose Ricci operator hasconstant eigenvalues ( λ, µ, µ, µ ) with λ = 0 and λ = µ . Remarks . ([33]) . A Hermitian metric on a compact complex surface ( M, J ) is ∗ -Einstein if and only if it is locally conformally K¨ahler and the traceless Ricci tensoris J -anti-invariant [ ? ]. In higher dimensions however the ∗ -Einstein condition doesnot imply any of these two properties, as can be seen by considering the twistorspace ( Z , h t , J ) of a compact self-dual Einstein manifold ( M, g ) with negativescalar curvature s and t = − /s . By Theorem 5, the 6-dimensional Hermitianmanifold ( Z , h t , J ) is ∗ -Einstein, but is neither locally conformally K¨ahler [77],nor with J -anti-invariant traceless Ricci tensor [35]. . By a result of V. Apostolov [5], any compact ∗ -Einstein Hermitian surface ofnegative ∗ -scalar curvature is K¨ahler. The twistorial example above shows that theanalogous statement is false in higher dimensions. . Recall that the twistor space ( Z , h t ) is an Einstein manifold if and only thebase manifold M is Einstein and self-dual with positive scalar curvature s = 6 /t or s = 12 /t . Thus ( Z , h t , J ) , t = s/
6, is an Einstein Hermitian manifold ofreal dimension 6 which is neither locally conformally K¨ahler [77] nor ∗ -Einstein(Theorem 5). Recall also that if M = S or M = CP , then Z = CP or Z = F , = SU (3) /S ( U (1) × U (1) × U (1)), and ( h t , J ) for t = 12 /s is the standardK¨ahler-Einstein structure on Z . For t = 6 /s , ( Z , h t ) is a Riemannian 3-symmetricspace [99] and J is its canonical almost complex structure. In this case ( Z , h t , J )is a ∗ -Einstein nearly K¨ahler manifold by a result of Gray [56]. Note also that for M = S and t = 6 /s , h t is the ”squashed” Einstein metric on CP [19, Example9.83].7. Curvature properties of the Chern connection on twistor spaces
It is well-known [73, 52] that every almost Hermitian manifold admits a uniqueconnection for which the almost complex structure and the metric are parallel, andthe (1 , manifolds with flat Chern connection are exactly the quotients of complex Liegroups equipped with left invariant Hermitian metrics.Motivated by the works of S. Donaldson [43] and C. LeBrun [71], V. Apostolovand T. Dragichi [11] have proposed to study the problem of existence of almostK¨ahler structures of constant Hermitian scalar curvature and/or type (1 ,
1) Ricciform of its Chern connection (from now on we refer to it as the first Chern form).One of our goals in [34] was to show that the twistor space of any self-dual Einstein4-manifold of negative scalar curvature admits such an almost K¨ahler structure.Given an almost Hermitian manifold (
N, g, J ), denote by ∇ the Levi-Civitaconnection of h . Then the Chern connection ∇ c of ( N, g, J ) is defined by (see, forexample, [58, Theorem 6.1]):(8) g ( ∇ cX Y, Z ) = g ( ∇ X Y, Z ) + 12 g (( ∇ X J )( JY ) , Z )+ 14 g (( ∇ Z J )( JY ) − ( ∇ Y J )( JZ ) − ( ∇ JZ J )( Y ) + ( ∇ JY J )( Z ) , X )It belongs to the distinguished 1-parameter family of Hermitian connections ∇ u , u ∈ R , defined by P. Gauduchon [52] :(9) g ( ∇ uX Y, Z ) = g ( ∇ X Y, Z ) + 12 g (( ∇ X J )( JY ) , Z )+ u g (( ∇ Z J )( JY ) − ( ∇ Y J )( JZ ) − ( ∇ JZ J )( Y ) + ( ∇ JY J )( Z ) , X )The Chern connection corresponds to u = 1, whereas for u = − N, g, J ) and δ Ω the co-differential of Ω withrespect to ∇ . Denote by ϕ and ψ the 2-forms on N defined by(10) ϕ ( X, Y ) =
T race ( Z → g (( ∇ X J )( JZ ) , ( ∇ Y J )( Z )))(11) ψ ( X, Y ) = ρ ∗ ( X, JY )where ρ ∗ is the ∗ -Ricci tensor of ( N, g, J ).The formula in the next lemma appears in [52] without proof and we refer thereader to [34] for its proof.
Lemma 4.
The first Chern form γ u of the Gauduchon connection ∇ u on an almostHermitian manifold ( N, g, J ) is given by πγ u = − ϕ − ψ + 2 udδ ΩLet (
M, g ) be an oriented Riemannian 4-manifold with twistor space Z . Denoteby D cn the Chern connection of the almost-Hermitian manifold ( Z , h t , J n ), n = 1 , γ t,n its first Chern form. In the case when the base manifold ( M, g ) isself-dual, an explicit formula for γ t, has been given by P. Gauduchon [51]. Foran arbitrary oriented Riemannian 4-manifold ( M, g ), the first Chern forms γ t,n , n = 1 ,
2, can be computed by means of the following formula.
Proposition 7. ([34])
The first Chern form γ t,n of the twistor space ( Z , h t , J n ) , n = 1 , , is given by πγ t,n ( E, F ) = [1 + ( − n +1 ][ g ( R ( σ ) , X ∧ Y ) + g ( A, σ × B )] where E, F ∈ T σ Z and X = π ∗ E , Y = π ∗ F , A = V E , B = V F . Now, we consider the problem of when the curvature tensor R cn of the Chernconnection D cn is of type (1 , R cn ( J n E, J n F ) G = R cn ( E, F ) G for all E, F, G ∈ T Z . Proposition 8. ([34]) ( i ) The curvature tensor R c is of type (1 ,
1) if and only ifthe base manifold (
M, g ) is self-dual.( ii ) The curvature tensor R c is of type (1 ,
1) if and only if the base manifold(
M, g ) is Einstein and self-dual.The next proposition solves the problem when of the Chern connections D c and D c of a twistor space have constant holomorphic sectional curvatures. Proposition 9. ([34]) ( i ) The Chern connection D c of the almost-Hermitian mani-fold ( Z , h t , J ) has constant holomorphic sectional curvature κ if and only if κ > M, g ) is of constant sectional curvature κ , and t = 1 /κ .( ii ) The holomorphic sectional curvature of the Chern connection D c of ( Z , h t , J )is never constant.8. Holomorphic curvatures of twistor spaces
Given an almost Hermitian manifold (
M, g, J ) one can define various types ofcurvatures related to the almost Hermitian structure ( g, J ). The most importantare the holomorphic sectional curvature [69] and the holomorphic, Hermitian, andorthogonal (totally real) bisectional curvatures [54], [17], [20]. These curvatureshave intensively been studied on K¨ahler manifolds and a lot of important resultshave been obtained. For example, the well-known uniformization theorem for com-plete K¨ahler manifolds of constant holomorphic sectional curvature states that anysuch manifold is either a complex projective space CP n with the Fubini-Study met-ric, a quotient of C n with the flat metric or a quotient of the unit ball in C n withthe hyperbolic metric [69]. Moreover, by the solution of the Frankel conjecturegiven by Mori [76] and by Siu and Yau [88], we know that the complex projectivespaces are the only compact complex manifolds admitting K¨ahler metrics of pos-itive holomorphic bisectional curvature. Note also that Mok [75] has proved theso-called generalized Frankel conjecture stating that any compact simply-connectedK¨ahler manifold with nonnegative holomorphic bisectional curvature is biholomor-phic to a compact Hermitian symmetric space. We refer the reader to [68], [27], [60]for analogous results under some weaker conditions on the holomorphic bisectionalcurvature. The case of negative holomorphic bisectional curvature is not so rigid.For example, recently To and Yeung [92] have constructed such K¨ahler metrics onany Kodaira surface.In the non-K¨ahler case the holomorphic curvatures mentioned above are not sowell studied. Complete results have been obtained only for complex dimension 2in which case it has been proved that every compact Hermitian surface of constantholomorphic or Hermitian sectional curvature is a complex space form [9]. In higherdimensions it is still an open question posed by Balas and Gauduchon [17, 18]whether there are compact non-K¨ahler Hermitian manifolds of non-zero constantholomorphic sectional curvature of the Chern connection. Holomorphic bisectional curvature.
The holomorphic bisectional curva-ture H t,n of the twistor space ( Z , h t , J n ) , n = 1 ,
2, of an oriented Riemannian4-manifold (
M, g ) can be computed by means of Proposition 1. For the sake ofsimplicity we give the respective formula only in the case when the base manifoldis self-dual and Einstein.
Proposition 10. ([3]) Let (
M, g ) be a self-dual Einstein manifold with scalarcurvature s and let E, F ∈ T σ Z be arbitrary h t -unit tangent vectors with X = π ∗ E , Y = π ∗ F , V = V E , W = V F . Then H t,n ( E, F ) = R ( X, K σ X, Y, K σ Y ) + t k V k k W k + 2 t ( s
24 ) ( k X k k Y k − g ( X, Y ) − g ( K σ X, Y ) )+ ( − n (2( ts
24 ) − ts
12 )( k X k k W k + k Y k k V k )+ (2( ts
24 ) (1 + ( − n ) − ts
12 )( g ( K σ X, Y ) g ( σ × V, W )+ ( − n g ( X, Y ) g ( V, W )) , (12)where K σ is the complex structure on T π ( σ ) M determined by σ .We next consider two particular cases of Proposition 10. Corollary 5.
Let ( M, g ) be a 4-manifold of constant sectional curvature and scalarcurvature s . Then H t,n ( E, F ) = s
12 ( g ( X, Y ) + g ( K σ X, Y ) ) + t k V k k W k + 2 t ( s
24 ) ( k X k k Y k − g ( X, Y ) − g ( K σ X, Y ))+ ( − n (2( ts
24 ) − ts
12 )( k X k k W k + k Y k k V k )+ (2( ts
24 ) (1 + ( − n ) − ts
12 )( g ( K σ X, Y ) g ( σ × V, W )+ ( − n g ( X, Y ) g ( V, W )) . (13) Corollary 6.
Let ( M, g ) be a self-dual Einstein manifold with sectional curvatureK and scalar curvature s , and let E ∈ T σ Z be arbitrary h t -unit tangent vector with X = π ∗ E and V = V E . The holomorphic sectional curvature of ( Z , h t , J n ) is givenby H t,n ( E ) = K ( X, K σ X ) k X k + t k V k +(2( st
24 ) (3( − n +1)+( − n +1 st
24 ) k X k k V k Using Proposition 1 and Corollary 6 we obtain the following.
Theorem 6. ([36]) ( i ) The almost Hermitian manifold ( Z , h t , J ) has constantholomorphic sectional curvature X if and only if the base manifold ( M, g ) hasconstant sectional curvature X = 1 /t .( ii ) The holomorphic sectional curvature of ( Z , h t , J ) is never constant. This together with Corollary 5 implies
Theorem 7. ([3]) The holomorphic bisectional curvature of the twistor space( Z , h t , J n ) , n = 1 , , of an oriented Riemannian 4-manifold ( M, g ) is never con-stant.In the next theorem, we consider the case when the base manifold (
M, g ) is a realspace form and determine all t > Z , h t , J n ) is strictly positive. In particular, it follows that the”squashed” metric on CP ([19], Example 9.83) is a non-K¨ahler Hermitian-Einsteinmetric of positive holomorphic bisectional curvature. This shows that a recentresult of Kalafat and Koca [66] in dimension four can not be extended to higherdimensions. Theorem 8. ([3]) Let (
M, g ) be an oriented Riemannian 4-manifold of constantsectional curvature.(i) The holomorphic bisectional curvature of ( Z , h t , J ) is positive if and only if0 < ts < M, g ) is a flat manifold, the holomorphic bisectional curvature of ( Z , h t , J n )is non-negative, n = 1 , Z , h , J ) of a 4-torus T with its standard flat metric. Then Z = T × S , h is theproduct metric and J is the complex structure defined by Blanchard [24]. So, theholomorphic bisectional curvature of ( T × S , h , J ) is non-negative. Note that J is not a product of complex structures on T and S .8.2. Orthogonal bisectional curvature.
The orthogonal (totally real) bisec-tional curvature B of an almost Hermitian manifold ( N, h, J ) is defined in [20]by B ( X, Y ) = h ( R ( X, JX ) Y, JY )for
X, Y ∈ T N such that X ⊥{ Y, JY } and || X || = || Y || = 1. It is well known [64]that the orthogonal bisectional curvature of a K¨ahler manifold of complex dimension ≥ Theorem 9. ([3]) Let (
M, g ) be a self-dual Einstein 4-manifold. Then its twistorspace (
Z, h t , J n ) has constant orthogonal bisectional curvature if and only if n = 1and ( M, g ) is of constant sectional curvature χ = 1 /t . Remark . Let (
M, g ) have a constant sectional curvature. Then the orthogonalbisectional curvature B t, of the twistor space ( Z , h t , J ) is strictly positive if andonly if 0 < ts < Hermitian bisectional curvature.
The Hermitian bisectional curvature H c of an almost Hermitian manifold ( N, h, J ) is defined as the holomorphic bisectionalcurvature of its Chern connection. As we have already noted, the curvature of thisconnection is directly related to the Chern classes of (
N, J ). In particular, if γ isthe first Chern form of ( N, h, J ), then for any X ∈ T N we have (14) γ ( X, JX ) = n X i =1 h ( H c ( X, JX ) E i , JE i ) , where ( E , . . . , E n , JE , . . . , JE n ) is a unitary frame.According to Theorem 7, the holomorphic bisectional curvature of the twistorspace of an oriented Riemannian 4-manifold is never constant. As for the Hermitianbisectional curvature, we have the following more general result which was pointedout to us by S. Kobayashi (private communication, April 2012). Theorem 10. ([3]) The Hermitian bisectional curvature of a Hermitian manifoldof complex dimension ≥ γ which impliesthat if the Hermitian bisectional curvature of a Hermitian manifold is a non-zeroconstant c , then the manifold is K¨ahler. Hence it is a complex space form and thewell-known formula for its curvature [69] implies that c = 0, a contradiction. Notealso that Theorem 10 gives a partial negative answer to the question of Balas andGauduchon [17, 18] mentioned at the beginning of this section. Remark. ([3]) Formula (14) for the first Chern form implies that if an almostHermitian manifold has non-zero constant Hermitian bisectional curvature, then itis an almost K¨ahler manifold, i.e. its K¨ahler 2-form is closed. Hence it is naturalto ask the following questions: • Are there compact non-K¨ahler and non-flat Hermitian manifolds of complexdimension ≥ • Are there compact non-K¨ahler almost K¨ahler manifolds of constant Her-mitian bisectional curvature?By a result of Vezzoni [96, Theorem 4.8], if (
N, h, J ) is an almost K¨ahler mani-fold whose holomorphic and Hermitian bisectional curvatures coincide, then it isa K¨ahler manifold. This result can be extended to a more general class of almostHermitian manifolds.
Theorem 11. ([3]) Let (
N, h, J ) be an almost Hermitian manifold such that(15) ( ∇ X J )( X ) = ε ( ∇ JX J )( JX ) , where ε = ±
1. Then its holomorphic and Hermitian bisectional curvatures coincideif and only if (
N, h, J ) is a K¨ahler manifold.
Remarks . ([3]) . According to the Gray-Hervella terminology [59] the almostHermitian manifolds satisfying (15) with ε = 1 are called G -spaces. This classcontains the Hermitian and nearly K¨ahler manifolds. The identity (15) with ε = − ∇ X J )( Y ) + ( ∇ JX J )( JY ) = 0). . The proof of Theorem 11 shows that the above mentioned result of Vezzoni for al-most K¨ahler manifolds holds true under the weaker condition that the holomorphicand Hermitian sectional curvatures coincide.Finally, we describe the twistor spaces whose holomorphic and Hermitian sec-tional curvatures coincide. Theorem 12. ([3]) Let (
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Email address : [email protected] Oleg Mushkarov, Institute of Mathematics and Informatics, Bulgarian Academy ofSciences, Acad. G.Bonchev Str. Bl.8, 1113 Sofia, Bulgaria,andSauth-West University, 2700 Blagoevgrad, Bulgaria
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