aa r X i v : . [ m a t h . DG ] D ec ON THE STRUCTURE OF NEARLY PSEUDO-K ¨AHLER MANIFOLDS
LARS SCH ¨AFER
Abstract.
Firstly we give a condition to split off the K¨ahler factor from a nearly pseudo-K¨ahler manifold and apply this to get a structure result in dimension 8 . Secondly we extend theconstruction of nearly K¨ahler manifolds from twistor spaces to negatively curved quaternionicK¨ahler manifolds and para-quaternionic K¨ahler manifolds. The class of nearly pseudo-K¨ahlermanifolds obtained from this construction is characterized by a holonomic condition. The combi-nation of these results enables us to give a classification result in (real) dimension 10. Moreover,we show that a strict nearly pseudo-K¨ahler six-manifold is Einstein.
Introduction
Nearly K¨ahler geometry was introduced and studied in a series of papers by A. Gray in theseventies in the context of weak holonomy. To our best knowledge he only considers pseudo-Riemannian metrics in his paper on 3-symmetric spaces [13]. In the analysis of Killing spinors onpseudo-Riemannian manifolds [14] nearly pseudo-K¨ahler and nearly para-K¨ahler manifolds appearin a natural way. Levi-Civita flat nearly K¨ahler manifolds provide a special class of solutions ofthe topological-antitopological fusion equations on the tangent bundle [24, 25]. There is a strongsimilarity to special K¨ahler geometry. For these reasons we became interested in Levi-Civita flatnearly K¨ahler manifolds and were able to give a constructive classification [8, 9]. In particular itfollows, that non-K¨ahlerian examples only exist in pseudo-Riemannian geometry and that the realdimension is at least 12. In other words, nearly K¨ahler geometry in the pseudo-Riemannian worldcan be very different from the better-understood Riemannian world. In (real) dimension six nearlypseudo-K¨ahler manifolds satisfy an exterior system analogue to the Riemannian case. Details canbe found in [23]. This system is used there to study such structures on products G × G, where G is a simple three-dimensional Lie group.An interesting class of nearly K¨ahler manifolds M n +2 can be constructed from twistor spacesover positive quaternionic K¨ahler manifolds. This class is characterized [5, 19] by the reducibilityof the holonomy of the canonical connection ¯ ∇ to U ( n ) × U (1). We show in this article that thetwistor spaces over negative quaternionic K¨ahler manifolds and para-quaternionic K¨ahler manifoldscarry a nearly pseudo-K¨ahler structure and characterize the class of such examples by a holonomiccondition.Using this result, we classify nice and decomposable (cf. Definitions 2.7 and 2.10) nearly pseudo-K¨ahler manifolds in dimension ten. Theorem A.
Let ( M , J, g ) be a nice decomposable nearly K¨ahler manifold, then the universalcover of M is either the product of a pseudo-K¨ahler surface and a (strict) nearly pseudo-K¨ahlermanifold M or a twistor space of an eight-dimensional (para-)quaternionic K¨ahler manifold en-dowed with its canonical nearly pseudo-K¨ahler structure.A strict nearly pseudo-K¨ahler six-manifold M is shown to be Einstein in Theorem 2.11. Indimension eight simply connected strict nearly pseudo-K¨ahler manifolds are shown (Theorem 2.8)to be products Σ × M of a Riemannian surface Σ and a strict nearly pseudo-K¨ahler manifold M . Date : December 11, 2009.2000
Mathematics Subject Classification.
Key words and phrases.
Nearly K¨ahler manifold, twistor spaces, pseudo-Riemannian metrics.
In the first section of this paper we recall the definition of a nearly pseudo-K¨ahler manifoldand generalize some facts and curvature identities to arbitrary signature. In the second sectionwe give a general condition to split off the K¨ahler factor from a nearly pseudo-K¨ahler manifold,see Theorem 2.5. Using some linear algebra of three-forms this shows the splitting result for nice nearly pseudo-K¨ahler manifolds in dimension 8 . The argument also holds true for a Riemannianmetric and gives an alternative proof for the known result. If a nice nearly pseudo-K¨ahler ten-manifold is in addition decomposable , we find two cases: In the first we can split off the K¨ahlerfactor and in the second the holonomy of ¯ ∇ is reducible with a complex one dimensional factor.This is one motivation to study twistor spaces. Before doing this in section four we recall someinformation on pseudo-Riemannian submersions in the third section. In the pseudo-Riemanniansetting twistor spaces are a good source of examples, since quaternionic geometry is richer innegative scalar curvature than in positive (cf. Remark 4.7) and since we have the additional classof twistor spaces over para-quaternionic manifolds. In the last section we prove that a nearlypseudo-K¨ahler manifold M of twistorial type (cf. Definition 5.11) is obtained from the abovementioned construction on a twistor space. This is done as follows: We prove that M comesfrom a pseudo-Riemannian submersion π : M → N. Then we use the nearly K¨ahler data on M toendow N with the structure of a (para-)quaternionic manifold. The proof is finished by identifyingthe twistor space of N with M. The former proofs [5, 19] in the Riemannian case all use the inversetwistor construction of Penrose or LeBrun, which does not seem to be developed for the situationsoccurring in this text. As the reader might observe, the approach presented here holds also truefor Riemannian metrics.
Acknowledgments.
The author thanks Vicente Cort´es for discussions.1.
Nearly pseudo-K¨ahler manifolds
Definition 1.1.
An almost pseudo-hermitian manifold (
M, J, g ) is called nearly pseudo-K¨ahlermanifold if it holds ( ∇ X J ) X = 0 , ∀ X ∈ Γ( T M ) , where ∇ is the Levi-Civita connection of the (pseudo-)Riemannian metric g. A nearly pseudo-K¨ahler manifold is called strict if it holds ∇ X J = 0 for all X ∈ T M.
Curvature identities in the pseudo-Riemannian case.
The starting point of a series ofcurvature identities are R ( W, X, Y, Z ) − R ( W, X, JY, JZ ) = g (( ∇ W J ) X, ( ∇ Y J ) Z ) , (1.1) R ( W, X, W, Z ) + R ( W, JX, W, JZ ) (1.2) − R ( W, JW, X, JZ ) = 2 g (( ∇ W J ) X, ( ∇ W J ) Z ) ,R ( W, X, Y, Z ) = R ( JW, JX, JY, JZ ) , (1.3)which were already proven for pseudo-Riemannian metrics by Gray [12]. Let { e i } ni =1 be a localorthonormal frame field, then the Ricci- and the Ricci*-tensor are given by g ( Ric X, Y ) = n X i =1 ǫ i R ( X, e i , Y, e i ) , g ( Ric ∗ X, Y ) = 12 n X i =1 ǫ i R ( X, JY, e i , Je i )with ǫ i = g ( e i , e i ) = g ( Je i , Je i ) and X, Y ∈ T M.
The frame { e i } ni =1 is called adapted if it holds Je i = e i + n for i = 1 , . . . , n. Then it follows using an adapted frame from equations (1.2) and (1.3)that g ( rX, Y ) := g (( Ric − Ric ∗ ) X, Y ) = n X i =1 ǫ i g (( ∇ X J ) e i , ( ∇ Y J ) e i ) . (1.4)Using the right hand-side we see [ J, r ] = 0 . For the second derivative of the complex structure one has the identity2 g ( ∇ W,X ( J ) Y, Z ) = − σ X,Y,Z g (( ∇ W J ) X, ( ∇ Y J ) JZ ) , (1.5) N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 3 which was proven in [12] for Riemannian metrics and holds true in the pseudo-Riemannian setting,cf. [14] Proposition 7.1. This identity implies n X i =1 ǫ i ∇ e i ,e i ( J ) Y = − r ( JY ) . (1.6)1.2. The canonical connection and some of its properties.
An important property of nearlyK¨ahler geometry is the existence of a canonical Hermitian connection [10] (see [8] for pseudo-Riemannian metrics). This is the unique connection ¯ ∇ with skew-symmetric torsion, which par-allelizes the metric g and the almost complex structure J. Explicitely it is given by¯ ∇ X Y = ∇ X Y − J ( ∇ X J ) Y, for X, Y ∈ Γ( T M ) (1.7)and its torsion equals ¯ T ( X, Y ) = − J ( ∇ X J ) Y. Proposition 1.2.
Denote by ¯ ∇ the canonical connection of a nearly pseudo-K¨ahler manifold ( M, J, g ) . Then one has ¯ ∇ ( T ) = ¯ ∇ ( ∇ J ) = 0 . Proof.
The proof given in [5] essentially uses the explicit form (1.7) of the connection ¯ ∇ and theidentity (1.5). Therefore the proposition generalizes to the pseudo-Riemannian case. (cid:3) From Proposition 1.2 and the relation (1.7) of ∇ and ¯ ∇ one obtains the following identities forthe curvature tensor ¯ R of ¯ ∇ and the curvature tensor R of the Levi-Civita connection ∇ ¯ R ( W, X, Y, Z ) = R ( W, X, Y, Z ) − g (( ∇ W J ) X, ( ∇ Y J ) Z )+ 14 [ g (( ∇ W J ) Y, ( ∇ X J ) Z ) − g (( ∇ W J ) Z, ( ∇ X J ) Y )] (1.8)= 14 [3 R ( W, X, Y, Z ) + R ( W, X, JY, JZ )+ σ XY Z R ( W, X, JY, JZ )] , ¯ R ( W, JW, Y, JZ ) = 14 [5 R ( W, JW, Y, JZ ) − R ( W, Y, W, Z ) − R ( W, JY, W, JZ )] . (1.9)With the help of the equation (1.8) it follows¯ R ( W, X, Y, Z ) = ¯ R ( Y, Z, W, X ) = − ¯ R ( X, W, Y, Z ) = − ¯ R ( W, X, Z, Y ) . (1.10)Using ¯ ∇ J = 0 and ¯ ∇ g = 0 we obtain¯ R ( W, X, Y, Z ) = ¯ R ( W, X, JY, JZ ) (1.11)= ¯ R ( JW, JX, Y, Z ) = ¯ R ( JW, JX, JY, JZ ) . The general form of the first Bianchi identity (cf. chapter III of [16]) for a connection with torsionyields in the case of parallel torsion: σ XY Z ¯ R ( W, X, Y, Z ) = − σ XY Z g (( ∇ W J ) X, ( ∇ Y J ) Z ) . (1.12)In a similar way we get from the second Bianchi identity (cf. chapter III of [16]) for a connectionwith parallel torsion or from the second Bianchi identity for ∇− σ V W X ¯ ∇ V ( ¯ R )( W, X, Y, Z ) = σ V W X ¯ R (( ∇ V J ) JW, X, Y, Z ) . (1.13)From deriving equation (1.8) and the second Bianchi identity of ∇ one gets after a direct compu-tation σ V W X ∇ V ( ¯ R )( W, X, Y, Z ) = 12 g (( ∇ Y ) Z, σ
V W X ( ∇ X J )( ∇ V J ) JW ) , (1.14)which implies σ V W X ∇ V ( ¯ R )( W, X, Y, JY ) = 0 . (1.15) LARS SCH¨AFER
Proposition 1.3.
The tensor r on a nearly pseudo-K¨ahler manifold ( M, J, g ) is parallel withrespect to the canonical connection ¯ ∇ . Proof.
Deriving g ( rX, X ) = P ni =1 ǫ i g (( ∇ e i J ) X, ( ∇ e i J ) X ) one obtains g (( ∇ U r ) X, X ) = 2 n X i =1 ǫ i g (( ∇ U,e i J ) X, ( ∇ e i J ) X ) (1.5) = − n X i =1 ǫ i [ g (( ∇ U J ) e i , ( ∇ X J ) J ( ∇ e i J ) X )+ g (( ∇ U J )( ∇ e i J ) X, ( ∇ e i J ) JX )+ g (( ∇ U J ) X, ( ∇ ( ∇ ei J ) X J ) Je i )] . We observe that in the first two terms exchanging e i by Je i gives a minus sign. Hence taking anadapted orthogonal frame { e i } ni =1 yields: g (( ∇ U r ) X, X ) = n X i =1 ǫ i g (( ∇ U J ) X, ( ∇ Je i J )( ∇ e i J ) X ) = − n X i =1 ǫ i g (( ∇ U J ) X, ( ∇ e i J )( ∇ e i J ) JX )= n X i =1 ǫ i g (( ∇ e i J )( ∇ U J ) X, ( ∇ e i J ) JX ) = g ( r ( ∇ U J ) X, JX ) . Polarizing this expression shows using that r is g -symmetric the following identity g (( ∇ U r ) X, Y ) = 12 g ( r ( ∇ U J ) X, JY ) + 12 g ( JX, r ( ∇ U J ) Y ) . As the difference of ¯ ∇ and ∇ is − J ∇ J the last equation is exactly ¯ ∇ r = 0 . (cid:3) Theorem 1.4.
Let ( M, J, g ) be a nearly pseudo-K¨ahler manifold and let W, X be vector fieldson M then it holds n X i,j =1 ǫ i ǫ j g ( re i , e j ) [ R ( W, e i , X, e j ) − R ( W, e i , JX, Je j )] = 0 . (1.16) Proof.
Let { e i } ni =1 be an adapted orthogonal frame field. One observes n X i =1 ǫ i ¯ R ( W, X, e i , ( ∇ V J ) e i ) (1.11) = 12 n X i =1 ǫ i (cid:2) ¯ R ( W, X, e i , ( ∇ V J ) e i ) − ¯ R ( W, X, Je i , ( ∇ V J ) Je i ) (cid:3) = 0and one gets after derivation of the left hand-side n X i =1 ǫ i (cid:2) ∇ U (cid:0) ¯ R )( W, X, e i , ( ∇ V J ) e i (cid:1) + ¯ R (cid:0) W, X, e i , ∇ U,V ( J ) e i (cid:1)(cid:3) = 0 . (1.17)Taking the trace in U, V and applying the identity (1.6) yields on the second term n X i,k =1 ǫ i ǫ k (cid:2) ¯ R (cid:0) W, X, e i , ∇ e k ,e k ( J ) e i (cid:1)(cid:3) = n X i =1 ǫ i ¯ R ( W, X, e i , − r ( Je i ))= − n X i,j =1 ǫ i ǫ j g ( re i , e j ) ¯ R ( W, X, e i , Je j ) . From (1.9) we obtain4 ¯ R ( W, JW, e i , Je j ) = 5 R ( W, JW, e i , Je j ) − R ( W, e i , W, e j ) − R ( W, Je i , W, Je j ) . N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 5
The first Bianchi identity using an adapted frame implies for i, j ∈ { , . . . , n } : R ( W, JW, e i , Je j ) = − R ( JW, e i , W, Je j ) − R ( e i , W, JW, Je j )= − R ( W, Je j , JW, e i ) + R ( W, e i , JW, Je j )= R ( W, e j + n , JW, Je i + n ) + R ( W, e i , JW, Je j ) . In an adapted frame it is g ( re i + n , e j + n ) = g ( re i , e j ) with i, j ∈ { , . . . , n } . Therefore taking thetrace in an adapted frame and polarizing yields the claimed identity if we show the vanishing ofthe trace on the first term of (1.17). n X i,j =1 ǫ i ǫ j ∇ e j (cid:0) ¯ R )( W, JW, e i , ( ∇ e j J ) e i (cid:1) = n X i,j,k =1 ǫ i ǫ j ǫ k g (( ∇ e j J ) e i , e k ) ∇ e j (cid:0) ¯ R (cid:1) ( W, JW, e i , e k )= 13 n X i,j,k =1 σ ijk (cid:2) ǫ i ǫ j ǫ k g (( ∇ e j J ) e i , e k ) ∇ e j ( ¯ R )( W, JW, e i , e k ) (cid:3) . Since ǫ i ǫ j ǫ k g (( ∇ e j J ) e i , e k ) is constant under cyclic permutation of i, j, k, the last expression van-ishes as a consequence of the curvature identities (1.10) and the identity (1.15). Polarization in W finishes the proof. (cid:3) First structure results
Small dimensions.
For a nearly pseudo-K¨ahler manifold ∇ ω is a differential form of type(3 ,
0) + (0 , . In consequence real two- or four-dimensional nearly pseudo-K¨ahler manifolds areautomatically pseudo-K¨ahler. Six dimensional nearly pseudo-K¨ahler manifolds are either pseudo-K¨ahler manifolds or strict nearly pseudo-K¨ahler manifolds. In the strict case a nearly pseudo-K¨ahler manifold ( M , J, g ) is of constant type , i.e. it holds g (( ∇ X J ) Y, ( ∇ X J ) Y ) = α (cid:0) g ( X, X ) g ( Y, Y ) − g ( X, Y ) − g ( JX, Y ) (cid:1) . (2.1)The sign of the type constant α depends on the signature ( p, q ) with p + q = 6. In fact it issign( α ) = sign( p − q ) , see section 7 of [14].2.2. Linear algebra of three-forms.
In the following section we consider a (finite dimensional)pseudo-hermitian vector space (
V, J, h· , ·i ) . Let η ∈ Λ V ∗ be a three-form. We define the support of η by Σ η = span { X y Y y η | X, Y ∈ V } ⊂ V, (2.2)where we identified V and V ∗ by means of h· , ·i . The name support is motivated by the observation,that for a given η ∈ Λ V ∗ it already holds η ∈ Λ Σ ∗ η , compare Lemma 7 of [8].In the present paper we are essentially interested in the three-form g p (( ∇ X J ) p Y, Z ) for
X, Y, Z ∈ T p M on a nearly pseudo-K¨ahler manifold ( M, J, g ) . This three-form is a real form of type (3 ,
0) +(0 , . The type condition implies that Σ η is a J -invariant subspace. In particular it follows thatthe complex dimension of the support of a non-zero such form is at least three.In [8] the classification of Levi-Civita flat nearly K¨ahler manifolds was related to the existenceof real three-forms of type (3 ,
0) + (0 ,
3) with isotropic support, i.e. such that Σ η is an isotropicsubspace.We define the kernel of a three-form η ∈ Λ V ∗ by K = K η = ker( X X y η ) . Lemma 2.1.
One has K = Σ ⊥ η and Σ η = K ⊥ . Proof.
Suppose, that X is in K η , i.e. X y η = 0 . By definition Σ η is spanned by vectors U satisfying h U, ·i = η ( Y, Z, · ) for Y, Z ∈ V. This implies h U, X i = η ( Y, Z, X ) = 0 , since η is a three-form. If X is perpendicular to Σ η the claim follows from the last equation and η ∈ Λ Σ ∗ η . This means X isin K η if and only if X is perpendicular to Σ η . It follows Σ η = K ⊥ . (cid:3) Lemma 2.2.
Let ( V, J, h· , ·i ) be a pseudo-hermitian vector space with dim R ( V ) = 8 then a realthree-form η of type (3 ,
0) + (0 , and of non-vanishing length has a (complex) one dimensionalkernel K η , which admits an orthogonal complement ( K η ) ⊥ . Moreover one has Σ η = ( K η ) ⊥ . LARS SCH¨AFER
Proof.
Let us identify V and V , . Denote by { α i } i =1 a unitary basis of ( V , ) ∗ and define a(4 , v by v = α ∧ α ∧ α ∧ α . The mapΦ : V , → Λ ( V , ) ∗ , ζ ζ y v yields an isometry. Therefore the (3 , ρ = η + iJ ∗ η is given by ρ = Z y v for some Z ∈ V , and consequently it follows Z y ρ = 0 . As Φ is an isometry and ρ has non-zero length, we concludethat Z is not isotropic. Denote by L ⊂ K ρ the complex line spanned by Z and by L ⊥ ⊃ K ⊥ ρ itsorthogonal complement. It remains to prove L = K : On the one hand we have Σ ρ = K ⊥ ρ ⊂ L ⊥ and on the other hand from ρ = 0 we get dim C Σ ρ ≥ C L ⊥ . This shows Σ ρ = L ⊥ and K ρ = L . (cid:3) Remark . As the reader observes, if η has length zero, one can replace the orthogonal comple-ment by the null-space and obtain an analogous statement as in the last proposition. Lemma 2.4.
Let ( V, J, h· , ·i ) be a pseudo-hermitian vector space with dim R ( V ) = 10 then a realthree-form η of type (3 ,
0) + (0 , and of non-vanishing length is of the following possible types: (i) There exists an orthonormal real basis { f i } i =1 = { e , Je , . . . , e , Je } and real numbers α, β such that η ( e , e , e ) = α = 0; η ( e , e , e ) = β (2.3) and η ( f i , f j , f k ) = 0 for the cases which are not obtained from (2.3) by skew-symmetryand type relations. (ii) There exists an orthonormal real basis { f i } i =1 = { e , Je , . . . , e , Je } and real numbers α, β such that η ( e , e , e ) = α = 0; η ( e , e , e + e ) = β with h e , e i = −h e , e i (2.4) and η ( f i , f j , f k ) = 0 for the cases which are not obtained from (2.4) by skew-symmetryand type relations.Proof. Denote by { α i } i =1 a unitary basis of ( V , ) ∗ and define a (5 , v by v = α ∧ α ∧ α ∧ α ∧ α . The map Φ : Λ V , → Λ ( V , ) ∗ , ϕ ϕ y v yields an isometry. Therefore the (3 , ρ = η + iJ ∗ η is given by ρ = ϕ y v for some ϕ ∈ Λ V , . As Φ is an isometry and η has non-zero length, we conclude that ϕ is not isotropic. Define Z ∈ ( V , ) ∗ by Z = ϕ y ρ = ϕ y ( ϕ y ρ ) . From h Z, Z i = h ϕ, ϕ i h ρ, ρ i we obtain that Z is not isotropic.Choosing a unitary basis { √ ( e − iJe ) , √ ( e − iJe ) } of the plane ϕ and Z = α ′ √ ( e − iJe ) ∗ for a unit vector e ∈ V and α ′ ∈ R −{ } we consider B , := { √ ( e − iJe ) , √ ( e − iJe ) , √ ( e − iJe ) } . Claim:
For ζ, χ ∈ span C B , it follows ρ ( ζ, χ, · ) ∈ span C ( B , ) ∗ . Let us define the map ˜Φ : Λ (span C B , ) → ( V , ) ∗ , by linear extension of ζ ∧ χ ρ ( ζ, χ, · ) . We observe that Λ (span C B , ) is a vector space of complex dimension three, which implies thatdim C (im ˜Φ) ≤ . As ρ is a three-form one easily sees that the duals of e j − iJe j for j = 1 , , . By the bound on the dimension of im ˜Φ the components orthogonal to(span B , ) ∗ vanish. This proves the claim.Let W be the orthogonal complement of span C B , . Choose an orthogonal basis { e − iJe , e − iJe } of W. Using that ρ is skew-symmetric we conclude that ˜ Z = ρ ( e − iJe , e − iJe , · ) isperpendicular to the dual W ∗ of W and hence an element ˜ Z of (span C B , ) ∗ . If h ˜ Z, ˜ Z i 6 = 0we can adapt the basis of B , such that ˜ Z = β ′ √ ( e − iJe ) ∗ . If h ˜ Z, ˜ Z i = 0 we can achieve˜ Z = β ′ √ [( e − iJe ) + ( e − iJe )] with h e , e i = −h e , e i . Passing to the real basis yieldssome new constants α, β and the claim of the Lemma. (cid:3)
N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 7
K¨ahler factors and the structure in dimension 8.
The aim of this subsection is to splitoff the pseudo-K¨ahler factor of a nearly pseudo-K¨ahler manifold. This will be done by means ofthe kernel of ∇ J and allows to reduce the (real) dimension from eight to six.For p ∈ M we set K p = ker( X ∈ T p M
7→ ∇ X J ) . Theorem 2.5.
Let ( M, J, g ) be a nearly pseudo-K¨ahler manifold. Suppose, that the distribution K has constant dimension and admits an orthogonal complement, (i) then M is locally a pseudo-Riemannian product M = K × M of a pseudo-K¨ahler manifold K and a strict nearly pseudo-K¨ahler manifold M . (ii) if M is complete and simply connected then it is a pseudo-Riemannian product M = K × M of a pseudo-K¨ahler manifold K and a strict nearly pseudo-K¨ahler manifold M . Proof.
The distribution K is parallel for the canonical connection ¯ ∇ , since ∇ J is ¯ ∇ -parallel. Bythe formula (1.7) and the nearly K¨ahler condition it follows ¯ ∇ X K = ∇ X K for sections K in K and X in T M.
This implies that K is parallel for the Levi-Civita connection and in consequence itsorthogonal complement ( K ) ⊥ is Levi-Civita parallel. The proof of (i) finishes by the local versionof the theorem of de Rham and the proof of (ii) by the global version. (cid:3) Remark . There exist nearly pseudo-K¨ahler manifolds (
M, J, g ) without pseudo-K¨ahler deRham factor, such that K η = { } admits no orthogonal complement. In fact there are Levi-Civitaflat nearly pseudo-K¨ahler manifolds constructed in Theorem 1 and 3 of [8] such that the three-form η p ( X, Y, Z ) = g p ( J ( ∇ X J ) Y, Z ) , for p ∈ M, has a support Σ η ⊂ T p M which is a maximallyisotropic subspace (Here we identified T p M and T ∗ p M via the metric g. ). Obviously, J ( ∇ X J ) Y and J ( ∇ U J ) V are elements of the support of η for arbitrary X, Y, U, V ∈ T p M. It then follows0 = g ( J ( ∇ X J ) Y, J ( ∇ U J ) V ) = g ( J ( ∇ J ( ∇ X J ) Y J ) U, V ) for all V ∈ T p M. Hence it is Σ η ⊂ K η . Moreover for general reasons we have shown before Σ η = K ⊥ η which shows K η ∩ K ⊥ η = { } for theabove examples. From these examples we learn, that the Theorem 2.5 does not hold true, if thereis no orthogonal complement. Definition 2.7.
A nearly pseudo-K¨ahler manifold (
M, J, g ) is called nice if the three-form g (( ∇ · J ) · , · ) has non-zero length in each point p ∈ M. Theorem 2.8.
Let ( M , J, g ) be a complete simply connected eight-dimensional nice nearlypseudo-K¨ahler manifold. Then M = M × M where M is a two-dimensional K¨ahler manifoldand M is a six-dimensional strict nearly pseudo-K¨ahler manifold.Proof. Since (
M, J, g ) is a nice nearly pseudo-K¨ahler manifold we can use Lemma 2.2 to obtain anorthogonal splitting in the two-dimensional distribution K and its orthogonal complement, whichcoincides with Σ η . Therefore we are in the situation of Theorem 2.5 (ii). (cid:3)
Einstein condition versus reducible holonomy.Theorem 2.9.
Let ( M, J, g ) be a nearly pseudo-K¨ahler manifold. (i) Suppose that r has more than one eigenvalue, then the canonical Hermitian connectionhas reduced holonomy. (ii) If the tensor field r has exactly one eigenvalue then M is a pseudo-Riemannian Einsteinmanifold.Proof. (i) Let µ i for i = 1 , . . . , l be the eigenvalues of r. Then the decomposition in the accordingeigenbundles Eig( µ i ) is ¯ ∇ -parallel and hence its holonomy is reducible.(ii) From the identity of Theorem 1.4 and r = µ T M we obtain0 = n X i =1 ǫ i ( R ( W, e i , X, e i ) − R ( W, e i , JX, Je i )) = g (( Ric − Ric ∗ ) W, X ) , where we used the Bianchi identity and an adapted frame to obtain the last equality. This showscomparing with r = Ric − Ric ∗ that it holds Ric = µ. (cid:3) LARS SCH¨AFER
Let us recall, that in the pseudo-Riemannian setting the decomposition into the eigenbundlesis not automatically ensured to be an orthogonal direct decomposition. Therefore we introducethe following notion:
Definition 2.10.
A nearly pseudo-K¨ahler manifold (
M, J, g ) is called decomposable if the abovedecomposition into the eigenbundles of the tensor r is orthogonal. Theorem 2.11.
A strict nearly pseudo-K¨ahler six-manifold ( M , J, g ) of constant type α is apseudo-Riemannian Einstein manifold with Einstein constant α .Proof. In an adapted basis we obtain from the symmetries of ∇ Jg ( rX, X ) = 2 X i =1 ǫ i g (( ∇ X J ) e i , ( ∇ X J ) e i ) = − X i =1 ǫ i g (( ∇ X J ) e i , e i ) . This is exactly minus the trace of the operator ( ∇ X J ) which has a simple form in a cyclic frame.It follows after polarizing g ( rX, Y ) = 4 αg ( X, Y ) . From Theorem 2.9 we compute the Einsteinconstant 5 α where α is the type constant of the strict nearly pseudo-K¨ahler manifold M . (cid:3) Proposition 2.12.
Let ( M , J, g ) be a nice nearly pseudo-K¨ahler ten-manifold. (i) Then the tensor r in a frame of the first type in Lemma 2.4 is given by re = 4( α + β ) e ,re = 4 α e , re = 4 α e ,re = 4 β e , re = 4 β e ,r ( Je i ) = Jr ( e i ) , i = 1 , . . . , , where α, β are constants. (ii) For a frame of the second type in Lemma 2.4 the tensor r is given by r e e e = 4 α + β ǫ ǫ β ǫ ǫ α β ǫ ǫ α + β ǫ ǫ e e e re = 0 ,re = 4 β (2 ǫ ǫ − e ,r ( Je i ) = Jr ( e i ) , i = 1 , . . . , . The eigenvalues are {
0; 4 α ; 4 β (2 ǫ ǫ − α + 2 β ǫ ǫ ) } , where the eigenbundles aregiven as Ker( r ) = span { e , Je } , Eig( r, α ) = span {− e + e , e , − Je + Je , Je } , Eig( r, β (2 ǫ ǫ − { e , Je } , Eig( r, α + 2 β ǫ ǫ ))) = span { e + e , Je + Je } , where α, β are constants. For β = 0 the second case is not decomposable. (iii) Suppose β = 0 in the case (i) and (ii). Then it follows Eig( r, α ) = span { e , e , e , Je , Je , Je } , Ker( r ) = span { e , e , Je , Je } . Proof.
In an adapted basis we obtain from the symmetries of ∇ Jg ( rX, Y ) = 2 X i =1 ǫ i g (( ∇ X J ) e i , ( ∇ Y J ) e i ) = − X i =1 ǫ i g (( ∇ Y J )( ∇ X J ) e i , e i ) . This is exactly minus the trace of the operator ( ∇ Y J )( ∇ X J ) . Using the form of Lemma 2.4 onecan calculate r by hand or using computer algebra systems to obtain the claimed results. (cid:3) N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 9
Theorem 2.13.
Let ( M , J, g ) be a complete simply connected nice decomposable nearly pseudo-K¨ahler manifold of dimension ten. Then M is of one of the following types (i) the tensor r has a kernel and M = K × M is a product of a four-dimensional pseudo-K¨ahler manifold K and a strict nearly pseudo-K¨ahler six-manifold M . (ii) the tensor r has trivial kernel and r has eigenvalues α + β ) with multiplicity , α , β with multiplicity for some α, β = 0 , A nice nearly pseudo-K¨ahler manifold ( M , J, g ) is decomposable if the dimension of the kernelof r is not equal to two.Proof. Since we suppose, that ( M , J, g ) is a nice and decomposable nearly pseudo-K¨ahler man-ifold, Proposition 2.12 implies that one has the two different cases:(i) the distribution K , which is the tangent space of the K¨ahler factor has dimension four andadmits an orthogonal complement of dimension six. This is part (iii) of Proposition 2.12. Part (i)of the Theorem now follows from Theorem 2.5.(ii) the tensor r has trivial kernel and we are in the situation of Proposition 2.12 part (i) with α, β = 0 and part (ii) follows. (cid:3) Remark . Nearly pseudo-K¨ahler manifolds falling in the second case of the last theorem willbe shown to be related to twistor spaces in section 5.5.3.
Pseudo-Riemannian submersions
Let us consider the setting of a pseudo-Riemannian submersion π : ( M, g ) → ( N, h ) . Thetangent bundle
T M of M splits orthogonally into the direct sum T M = H ⊕ V . (3.1)Denote by ι H , ι V the canonical inclusions and by π H , π V the canonical projections. We recall thedefinition [3, 21] of the fundamental tensorial invariants A and T of the submersion πT ζ = π H ◦ ∇ π V ( ζ ) ◦ π V + π V ◦ ∇ π V ( ζ ) ◦ π H ,A ζ = π H ◦ ∇ π H ( ζ ) ◦ π V + π V ◦ ∇ π H ( ζ ) ◦ π H , where ζ is a vector field on M. The components of the Levi-Civita connection ∇ are given in the next proposition (compare [21],[3] 9.24 and 9.25). Proposition 3.1.
Let π : ( M, g ) → ( N, h ) be a pseudo-Riemannian submersion, denote by ∇ the Levi-Civita connection of g and define ∇ V := π V ◦ ∇ ◦ ι V . For vector fields
X, Y in H and U, V in V we have the following identities ∇ U V = ∇ V U V + T U V, (3.2) ∇ U X = T U X + π H ( ∇ U X ) , (3.3) ∇ X U = π V ( ∇ X U ) + A X U, (3.4) ∇ X Y = A X Y + π H ( ∇ X Y ) , (3.5) π V [ X, Y ] = 2 A X Y, (3.6) g ( A X Y, U ) = − g ( A X U, Y ) , or more generally A is alternating. (3.7)The canonical variation of the metric g for t ∈ R − { } is given by g t := g ( X, Y ) , for X, Y ∈ H ,tg ( V, W ) , for V, W ∈ V ,g ( V, X ) = 0 , for V ∈ V , X ∈ H . Lemma 3.2.
Denote by
X, Y vector fields in H and by U, V vector fields in V . (1) Let A t and T t be the tensorial invariants for g t and A and T those for g = g . Then itholds A tX Y = A X Y, A tX U = tA X U and (3.8) T tU V = tT U V, T tU X = T U X ; (3.9)(2) ∇ t V U V = ∇ V U V ;(3) π H ( ∇ tX Y ) = π H ( ∇ X Y ) and π V ( ∇ tX V ) = π V ( ∇ X V ); (3.10)(4) π H ( ∇ tV X − ∇ V X ) = ( t − A X V ; (3.11) ∇ tU V = ∇ U V + ( t − T U V. (3.12) Proof.
The first part can be found in Lemma 9.69 of [3]. On the right hand-side of the Koszulformulas one only needs the metric g t on V to determine ∇ t V . This shows ∇ t V = ∇ V . An analogousargument using the Koszul formulas shows π H ( ∇ tX Y ) = π H ( ∇ X Y ) and π V ( ∇ tX V ) = π V ( ∇ X V ) . The first part of the point (4) follows from the identities (3.6) and (3.7) and the Koszul formulas.The last equation follows from (1) and (2): ∇ tU V = ∇ t V U V + T tU V = ∇ V U V + tT U V = ∇ U V + ( t − T U V. (cid:3) Twistor spaces over quaternionic and para-quaternionic K¨ahler manifolds
In this section we consider pseudo-Riemannian submersions π : ( M, g ) → ( N, h ) endowed witha complex structure J on M which is compatible with the decomposition (3.1). Lemma 4.1.
Let π : ( M, g ) → ( N, h ) be a pseudo-Riemannian submersion endowed with acomplex structure J on M which is compatible with the decomposition (3.1) . Then ( M, g, J ) is apseudo-K¨ahler manifold if and only if the following equations are satisfied π H (( ∇ X J ) Y ) = π H (( ∇ V J ) X ) = 0 , (4.1)( ∇ V U J ) V = π V (( ∇ X J ) V ) = 0 , (4.2) A X ( JY ) − JA X Y = 0 , A X ( JV ) − JA X V = 0 , (4.3) T V ( JX ) − JT V X = 0 , T U ( JV ) − JT U V = 0 , (4.4) where X, Y are vector fields in H and U, V are vector fields in V . Proof.
Let
X, Y be vector fields in H and U, V be vector fields in V . Then it follows from comparingcomponents in
H ⊕ V ( ∇ X J ) Y = π H (( ∇ X J ) Y ) + π V ( ∇ X J ) Y ) = π H (( ∇ X J ) Y ) + ( A X ( JY ) − JA X Y ) , ( ∇ X J ) V = π H (( ∇ X J ) V ) + π V ( ∇ X J ) V ) = π V (( ∇ X J ) V ) + ( A X ( JV ) − JA X V ) , ( ∇ V J ) X = π H (( ∇ V J ) X ) + π V ( ∇ V J ) X ) = π H (( ∇ V J ) X ) + ( T V ( JX ) − JT V X ) , ( ∇ U J ) V = π H (( ∇ U J ) V ) + π V ( ∇ U J ) V ) = ( ∇ V U J ) V + ( T U ( JV ) − JT U V ) . (cid:3) Further we define a second complex structure byˆ J := ( J on H , − J on V . We observe that ˆˆ J = J. This construction was made in [19] for the Riemannian setting and imitatesthe construction on twistor spaces.
N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 11
Proposition 4.2.
Suppose, that the foliation induced by the pseudo-Riemannian submersion π is totally geodesic and that ( M, J, g ) is a pseudo-K¨ahler manifold and J is compatible withthe decomposition (3.1) , then the manifold ( M, ˆ g = g , ˆ J ) is a nearly pseudo-K¨ahler manifold.The distributions H and V are parallel with respect to the canonical Hermitian connection ¯ ∇ of ( M, ˆ g, ˆ J ) . In other words the nearly pseudo-K¨ahler manifold ( M, ˆ g, ˆ J ) has reducible holonomy.Proof. Let
U, V be vector fields in V and X, Y be vector fields in H : In the following ˆ ∇ is the Levi-Civita connection of ˆ g. Since the fibers are totally geodesic, i.e. T ≡ , we obtain from equation(3.2), that ˆ ∇ U V = ˆ ∇ V U V + ˆ T U V = ∇ V U V + T U V = ∇ U V, which yields ( ˆ ∇ U ˆ J ) V = − ( ∇ U J ) V = 0 . In the sequel we denote the O’Neill tensors of the pseudo-Riemannian foliations induced by V on( M, g ) and on ( M, ˆ g ) by A and ˆ A, respectively. From Lemma 3.2 it follows A X Y = ˆ A X Y andconsequently the same Lemma yields ∇ X Y = ˆ ∇ X Y. Since (
M, g ) is K¨ahler, Lemma 4.1 implies A ◦ J = J ◦ A and we compute( ˆ ∇ X ˆ J ) Y = ˆ ∇ X ( ˆ JY ) − ˆ J ˆ ∇ X Y (4.5)= π H [ ˆ ∇ X ( JY )] + π V [ ˆ ∇ X ( JY )] − ˆ J ( π H ( ˆ ∇ X Y ) + π V ( ˆ ∇ X Y ))= π H [ ˆ ∇ X ( JY ) − J ˆ ∇ X Y ] + π V [ ˆ ∇ X ( JY ) + J ˆ ∇ X Y ]= π H (( ˆ ∇ X J ) Y ) + ˆ A X ( JY ) + J ˆ A X Y (3.8) , (3.10) , (4.3) = π H (( ∇ X J ) Y ) + 2 A X ( JY ) (4.1) = 2 A X ( JY ) = 2 JA X Y. With the identity A X V = 2 ˆ A X V of Lemma 3.2 we get( ˆ ∇ X ˆ J ) V = ˆ ∇ X ( ˆ JV ) − ˆ J ˆ ∇ X V (4.6)= − π V ( ˆ ∇ X ( JV )) − π H ( ˆ ∇ X ( JV )) + Jπ V ( ˆ ∇ X V ) − Jπ H ( ˆ ∇ X V )= − π V (( ˆ ∇ X J ) V ) − ˆ A X JV − J ˆ A X V (3.8) , (3.10) , (4.3) = − π V (( ∇ X J ) V ) − JA X V = − A X JV.
The vanishing of the second fundamental form T, equation (3.9) and a second time A X V = 2 ˆ A X V show ( ˆ ∇ V ˆ J ) X = π V ( ˆ ∇ V ( JX )) + π V ( J ˆ ∇ V X ) + π H ( ˆ ∇ V ( JX ) − J ˆ ∇ V X ) (4.7) (3.11) = ˆ T V ( JX ) + J ( ˆ T V X ) + π H (( ∇ V J ) X ) + 12 ( JA X V − A JX V ) = JA X V, where we used A JX V = − JA X V which follows, since A X is alternating (compare equation (3.7))and commutes with J. The next Lemma finishes the proof. (cid:3)
Lemma 4.3. ( M, ˆ J, ˆ g ) is a nearly pseudo-K¨ahler manifold and ˆ J is compatiblewith the decomposition (3.1) , then the following statements are equivalent: (i) the splitting (3.1) is ¯ ∇ -parallel, (ii) the fundamental tensors ˆ A and ˆ T satisfy: ˆ T V X = 0 , ˆ J ˆ T V W = − ˆ T V ˆ JW ⇔ ˇ J ˆ T V W = ˆ T V ˇ JW for ˇ J = ˆˆ J, (4.8)ˆ A X V = 12 ˆ J ( ˆ ∇ X ˆ J ) V, ˆ A X Y = 12 π V (cid:16) ˆ J ( ˆ ∇ X ˆ J ) Y (cid:17) . (4.9) ( ˆ ∇ V ˆ J ) W = 0 then ¯ ∇ V W ∈ V for V, W ∈ V is equivalent to T V W = 0 . Moreover itis ( ˆ ∇ V V ˆ J ) W = 0 . Proof.
First we compute¯ ∇ V W = ˆ ∇ V W −
12 ˆ J ( ˆ ∇ V ˆ J ) W = ˆ ∇ V V W + ˆ T V W −
12 ˆ J ( ˆ ∇ V V ˆ J ) W −
12 ( ˆ J ˆ T V ( ˆ JW ) + ˆ T V W )= ˆ ∇ V V W −
12 ˆ J ( ˆ ∇ V V ˆ J ) W + 12 ( ˆ T V W − ˆ J ˆ T V ( ˆ JW )) . The first two terms lie in V , the second terms lie in H and therefore the expression is in V if and onlyif ˆ J ˆ T V ( ˆ JW ) = ˆ T V W ⇔ ˆ T V ( ˆ JW ) = − ˆ J ˆ T V W. From 0 = π H ( ¯ ∇ X V ) = π H ( ˆ ∇ X V − ˆ J ( ˆ ∇ X ˆ J ) V )one determines ˆ A X V = π H ( ˆ ∇ X V ) = 12 π H (cid:16) ˆ J ( ˆ ∇ X ˆ J ) V (cid:17) = 12 ˆ J ( ˆ ∇ X ˆ J ) V. The last equality follows from Lemma 5.3 (ii) part b). Conversely, if ˆ A X V is given by the lastformula one gets¯ ∇ X V = ˆ ∇ X V −
12 ˆ J ( ˆ ∇ X ˆ J ) V = ˆ ∇ X V − ˆ A X V = π V ( ˆ ∇ X V ) ∈ V . With the same identity we calculate π V ( ¯ ∇ V X ) = π V (cid:18) ˆ ∇ V X −
12 ˆ J ( ˆ ∇ V ˆ J ) X (cid:19) = π V ( ˆ ∇ V X + ˆ A X V ) = π V ( ˆ ∇ V X ) . This is in H if and only if ˆ T V X = π V ( ˆ ∇ V X ) = 0 . The last component, i.e. π V ( ¯ ∇ X Y ) = π V (cid:16) ˆ ∇ X Y − ˆ J ( ˆ ∇ X ˆ J ) Y (cid:17) is zero if and only if we have ˆ A X Y = π V ( ˆ ∇ X Y ) = π V (cid:16) ˆ J ( ˆ ∇ X ˆ J ) Y (cid:17) . ∇ V ˆ J ) W = 0 we calculate¯ ∇ V W = ˆ ∇ V W + 12 ( ˆ ∇ V ˆ J ) ˆ J W = ˆ ∇ V V W + ˆ T V W. This lies in V if and only if ˆ T V W = 0 . (cid:3) We apply Proposition 4.2 to twistor spaces and obtain.
Corollary 4.4.
The twistor space Z of a quaternionic K¨ahler manifold of dimension k withnegative scalar curvature admits a canonical nearly pseudo-K¨ahler structure of reducible holonomycontained in U (1) × U (2 k ) . Proof.
We remark that in negative scalar curvature the twistor space of a quaternionic K¨ahlermanifold is the total space of a pseudo-Riemannian submersion with totally geodesic fibers. Itadmits a compatible pseudo-K¨ahler structure of signature (2 , k ) , cf. Besse [3] 14.86 b). Theassumption of positive scalar curvature is often made to obtain a positive definite metric on Z . Here we focus on pseudo-Riemannian metrics and consequently on negative scalar curvature. (cid:3)
Proposition 4.5.
The twistor spaces Z of non-compact duals of Wolf spaces and of Alekseevskianspaces admit a nearly pseudo-K¨ahler structure.Proof. Non-compact duals of Wolf spaces are known [22] to be quaternionic K¨ahler manifolds ofnegative scalar curvature. The same holds for Alekseevskian spaces [1, 6]. (cid:3)
Studying the lists given in [1, 6, 22] we find examples of six-dimensional nearly pseudo-K¨ahlermanifolds.
Corollary 4.6.
The twistor spaces Z of ˜ H P = Sp(1 , / Sp(1)Sp(1) and SU (1 , /S ( U (1) U (2)) provide six-dimensional nearly pseudo-K¨ahler manifolds.Remark . The situation in negative scalar curvature is more flexible than in the positive case.This is illustrated by the following results in this area: In the main theorem of [17] it is shownthat the moduli space of complete quaternionic K¨ahler metrics on R n is infinite dimensional. Aconstruction of super-string theory, called the c-map [11], yields continuous families of negativelycurved quaternionic K¨ahler manifolds. These results show that Corollary 4.4 is a good source ofexamples.Another source of examples is given by twistor spaces over para-quaternionic K¨ahler mani-folds . Since these manifolds are less classical than quaternionic K¨ahler manifolds, we recall somedefinitions (cf. [2] and references therein). N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 13
Definition 4.8.
Let ( ǫ , ǫ , ǫ ) = ( − , ,
1) or some permutation thereof. An almost para-quaternionic structure on a differentiable manifold M k is a rank 3 sub-bundle Q ⊂ End ( T M ) , which is locally generated by three anti-commuting endomorphism-fields J , J , J = J J . Thesesatisfy J i = ǫ i Id for i = 1 , . . . , . Such a triple is called standard local basis of Q. A linear torsion-free connection preserving Q is called para-quaternionic connection. An almost para-quaternionicstructure is called a para-quaternionic structure if it admits a para-quaternionic connection. An almost para-quaternionic hermitian structure ( M, Q, g ) is a pseudo-Riemannian manifold endowedwith a para-quaternionic structure such that Q consists of skew-symmetric endomorphisms. For n > M k , Q, g ) is a para-quaternionic K¨ahler manifold if Q is preserved by the Levi-Civitaconnection of g. In dimension 4 a para-quaternionic K¨ahler manifold M is an anti-self-dual Einsteinmanifold.We use the same notions omitting the word ”para” for the quaternionic case. The conditionthat Q is preserved by the Levi-Civita connection is in a given standard local basis { J i } i =1 of Q equivalent to the equations ∇ X J i = − θ k ( X ) ǫ j J j + θ j ( X ) ǫ k J k , for X ∈ T M, (4.10)where i, j, k is a cyclic permutation of 1 , , { θ i } i =1 are local one-forms. In the context ofpara-quaternionic manifolds one can define twistor spaces for s = 1 , , − Z s := { A ∈ Q | A = sId, with A = 0 } . The case of interest in this text is Z = Z − , since this twistor space is a complex manifold,such that the conditions of Proposition 4.2 hold true (cf. [2]). Therefore we obtain the followingexamples of nearly pseudo-K¨ahler manifolds. Corollary 4.9.
The twistor space Z of a para-quaternionic K¨ahler manifold with non-zeroscalar curvature of dimension k admits a canonical nearly pseudo-K¨ahler structure of reducibleholonomy contained in U ( k, k ) × U (1) . Example 4.10.
The para-quaternions e H are the R -algebra generated by { , i, j, k } subject tothe relations i = − , j = k = 1 , ij = − ji = k. Like the quaternions, the para-quaternionsare a real Clifford algebra which in the convention of [18] is e H = C l , ∼ = C l , ∼ = R (2) . Onedefines the para-quaternionic projective space e H P n by the obvious equivalence relation on thepara-quaternionic right-module e H n +1 of ( n + 1)-tuples of para-quaternions. The manifold e H P n isa para-quaternionic K¨ahler manifold [4] in analogue to quaternionic projective space H P n . Thisyields examples of the type described in the last Corollary.5.
Reducible nearly pseudo-K¨ahler manifolds
In this section we study the case of a nearly pseudo-K¨ahler manifold ( M n , J, g ) , such that theholonomy of the canonical connection ¯ ∇ is reducible, in the sense that the tangent bundle T M admits a splitting
T M = H ⊕ V into two ¯ ∇ -parallel sub-bundles H , V , which are orthogonal and invariant with respect to thealmost complex structure J. General properties.
In this subsection we carefully check, generalizing [20] to pseudo-Riemannian foliations, the information which follows from the decomposition into the J -invariantsub-bundles. Lemma 5.1.
In the situation of this section and for a vector field X in H , a vector field Y in T M and vector fields
U, V in V it is ¯ R ( X, Y, U, V ) = g ([ ∇ U J, ∇ V J ] X, Y ) − g (( ∇ X J ) Y, ( ∇ U J ) V ) . (5.1) Proof.
Since H and V are ¯ ∇ -parallel it follows ¯ R ( Y, U, X, V ) = 0 and using equation (1.8) we get R ( Y, U, X, V ) = 12 g (( ∇ Y J ) U, ( ∇ X J ) V ) −
14 [ g (( ∇ Y J ) X, ( ∇ U J ) V ) − g (( ∇ Y J ) V, ( ∇ U J ) X )]= − g (( ∇ V J )( ∇ U J ) Y, X ) −
14 [ g (( ∇ Y J ) X, ( ∇ U J ) V ) − g (( ∇ U J )( ∇ V J ) Y, X )] . The first Bianchi identity yields R ( X, Y, U, V ) = − R ( Y, U, X, V ) − R ( U, X, Y, V )= − R ( Y, U, X, V ) + R ( X, U, Y, V )= 34 g ([( ∇ V J ) , ( ∇ U J )] Y, X ) + 12 g (( ∇ Y J ) X, ( ∇ U J ) V ) . Replacing the last expression into¯ R ( X, Y, U, V ) = R ( X, Y, U, V ) − g (( ∇ X J ) Y, ( ∇ U J ) V )+ 14 [ g (( ∇ X J ) U, ( ∇ Y J ) V ) − g (( ∇ X J ) V, ( ∇ Y J ) U )]proves the Lemma. (cid:3) Corollary 5.2.
For vector fields
X, Y in H and V, W in V one has (i) ( ∇ X J )( ∇ V J ) W = 0; ( ∇ V J )( ∇ X J ) Y = 0;(ii) ( ∇ X J )( ∇ Y J ) Z belongs to H for all Z ∈ Γ( H );(iii) ( ∇ V J )( ∇ W J ) X belongs to H ; and ( ∇ X J )( ∇ Y J ) V belongs to V . Proof. (i) follows from the fact, that ¯ R ( JX, JY, V, W ) = ¯ R ( X, Y, V, W ) and that the first term of equation(5.1) has the same symmetry with respect to J. This yields on the one hand g (( ∇ JX J ) JY, ( ∇ V J ) W ) = g (( ∇ X J ) Y, ( ∇ V J ) W )and on the other hand it is g (( ∇ JX J ) JY, ( ∇ V J ) W ) = − g (( ∇ X J ) Y, ( ∇ V J ) W ) . Consequently one has g (( ∇ X J ) Y, ( ∇ V J ) W ) = 0 . Exchanging H and V finishes part (i).(ii) From (i) one gets the vanishing of g (( ∇ V J )( ∇ Y J ) Z, X ) = g ( Z, ( ∇ Y J )( ∇ V J ) X ) = − g ( Z, ( ∇ Y J )( ∇ X J ) V ) = − g (( ∇ X J )( ∇ Y J ) Z, V ) . (iii) From (i) it follows 0 = ¯ R ( X, U, V, W ) = g ([ ∇ V J, ∇ W J ] X, U ) . This yields [ ∇ V J, ∇ W J ] X ∈ H and by [ ∇ V J, ∇ JW J ] JX = −{∇ V J, ∇ W J } X ∈ H we get the first part. The second part followsby replacing H and V . (cid:3) Co-dimension two.
Motivated by the above section on twistor spaces we suppose from nowon that the real dimension of V is two. Lemma 5.3.
Let dim R ( V ) = 2 . (i) Then the restriction of the metric g is either of signature (2 , or (0 , . (ii) a) T ( V, W ) = 0 for all
V, W ∈ V . b) T ( X, U ) ∈ H for all X ∈ H and U ∈ V . c) In dimension six it is T ( X, Y ) ∈ V for all X, Y ∈ H . d) Span { π V ( T ( X, Y )) | X, Y ∈ H} = V . N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 15
Proof.
Let V ∈ V with g ( V, V ) = 0 , then it is V = span { V, JV } . (i) It holds g ( JV, JV ) = g ( V, V ) =0 . This proves (i).(ii) In the following we denote by
X, Y (local) sections of H and by U, V (local) sections of V : The part a) follows from T ( V, V ) = T ( JV, JV ) = 0 and from the formula for the torsion T ( V, JV ) = − J ( ∇ V J ) JV = − ( ∇ V J ) V = 0 . Part b) follows from the fact that g (( ∇ U J ) Y, V ) and g (( ∇ U J ) Y, JV ) are three-forms: This implies,that one has g (( ∇ U J ) X, V ) = − g (( ∇ U J ) V, X ) a ) = 0 and g (( ∇ U J ) X, JV ) = − g (( ∇ U J ) V, JX ) a ) = 0 . Hence the projection of T ( X, U ) on V vanishes and part b) follows. Next we show part d). Thereexists a pair X, Y, such that π V ( T ( X, Y )) = 0 . Then it follows π V ( T ( X, JY )) = − Jπ V ( T ( X, Y )) =0 and part d) holds true. If there were no such pair X, Y then it follows − g ( T ( X, Y ) , V ) = g ( J ( ∇ X J ) Y, V ) = − g ( J ( ∇ V J ) Y, X ) = 0 for all
X, Y ∈ H , V ∈ V and consequently using a) andb) it follows ∇ V J = 0 . This contradicts the fact, that M is strict nearly K¨ahler. It remains part c).As T ( X, Y ) = 0 it follows that X and Y are linear independent. From the symmetries of T ( X, Y )one concludes, that span R { X, Y, JX, JY } ⊂ H has real dimension 4 and hence coincides with H . Using the symmetries of ∇ J one gets T ( X, Y ) , T ( X, JY ) , T ( JX, Y ) , T ( JX, JY ) ∈ V . This finishesthe proof. (cid:3)
Corollary 5.4.
Let dim R ( V ) = 2 . Then the foliation V has totally geodesic fibers and the O’Neilltensor is given by A X Y = π V ( J ( ∇ X J ) Y ) and A X V = J ( ∇ X J ) V. Moreover it is ∇ V J = 0 . Proof.
From Lemma 5.3 (ii) a) we obtain ( ∇ V J ) W = 0 with V, W ∈ Γ( V ) . By Lemma 4.3 part 2)it follows T V W = 0 and ∇ V J = 0 , since the decomposition H ⊕ V is ¯ ∇ parallel. Part 1) of Lemma4.3 finishes the proof. (cid:3) Proposition 5.5.
Let ( M, J, g ) be a nearly pseudo-K¨ahler manifold such that the property ofLemma 5.3 (ii) c) is satisfied and such that V has dimension 2, then ( M, ˇ J = ˆ J, ˇ g = g ) is apseudo-K¨ahler manifold . It is natural to suppose the property of Lemma 5.3 (ii) c), since this holds true in the cases of twistorial type which are studied in the next sections.
Proof.
By the last Corollary the data of the submersion is ˇ T = T ≡ , A X Y = ˇ A X Y = π V ( J ( ∇ X J ) Y ) and ˇ A X V = 2 A X V = J ( ∇ X J ) V. Since A anti-commutes with J it commuteswith ˇ J. This yields the conditions (4.3) and (4.4) of Lemma 4.1 on the triple ˇ A, ˇ T , ˇ J. Further itholds ∇ V J = 0 . From the reasoning of equation (4.5) we obtain π H (( ˇ ∇ X ˇ J ) Y ) = π H (( ∇ X J ) Y )which vanishes by the property of Lemma 5.3 (ii) c). By an analogous argument we get from equa-tion (4.6) the identity π V (( ˇ ∇ X ˇ J ) V ) = − π V (( ∇ X J ) V ) . This vanishes by Lemma 5.3 (ii) b). Fromequation (4.7) we derive − π H (( ∇ X J ) V ) n.K. = π H (( ∇ V J ) X ) = π H (( ˇ ∇ V ˇ J ) X ) + 2 π H ( JA X V ) . Thedefinition of A X V yields π H (( ˇ ∇ V ˇ J ) X ) = 0 . These are all the identities needed to apply Lemma4.1. (cid:3)
Proposition 5.6.
Let
X, Y be vector fields in H and V , V , V be vector fields in V . Supposethat it holds T ( V, W ) = 0 for all
V, W ∈ V then it is ¯ R (( ∇ X J ) JY, V , V , V ) = g ( JY, [ ∇ V J, [ ∇ V J, ∇ V J ]] X ) . (5.2) Moreover, one has ¯ ∇ U ¯ R ( V , V , V , V ) = 0 . Proof.
For V , V , V ∈ V and X ∈ H the second Bianchi identity gives − σ XY V ¯ ∇ X ( ¯ R )( Y, V , V , V ) = σ XY V ¯ R (( ∇ X J ) JY, V , V , V ) . As the decomposition
H ⊕ V is ¯ ∇ -parallel the terms on the left hand-side vanish due to thesymmetries (1.10) of the curvature tensor ¯ R. The right hand-side is determined with the helpof Lemma 5.1 and Corollary 5.2. If we apply ¯ ∇ to the formula (5.2) we obtain by ¯ ∇ ( ∇ J ) = 0the identity g (cid:0) ¯ ∇ U ( ¯ R )( V , V , V ) , ( ∇ X J ) Z (cid:1) = 0 with Z = JY.
This yields the proposition usingLemma 5.3 (ii) part d). (cid:3) Here we use ˇ · for the inverse construction of ˆ · . Six-dimensional nearly pseudo-K¨ahler manifolds.
Before analyzing the general casewe first focus on dimension six.
Lemma 5.7.
On a six-dimensional nearly pseudo-K¨ahler manifold ( M , J, g ) the integral man-ifolds of the foliation V have Gaussian curvature α and constant curvature κ = 4 α, where α isthe type constant. Recall that the sign of α is completely determined by the signature of the metric g, cf. section 2.1. Proof.
Let X and V be (local) vector fields of constant length in H and V , respectively. Then itfollows from equation (1.7) and the skew-symmetry of ∇ J : g ( ∇ V V, ∇ X X ) = g ( ¯ ∇ V V, ¯ ∇ X X ) = 0 . This identity yields R ( X, V, X, V ) = − g ( ∇ V ∇ X + ∇ [ X,V ] V, X ) NK = g ( ∇ X V, ∇ V X ) − g ( J ( ∇ [ X,V ] J ) V, X ) L. ( ii ) b ) = g ( ∇ X V, ∇ V X ) − g ( J ( ∇ X J ) V, π H ([ V, X ])) ( ∗ ) = g ( ∇ X V, ∇ V X ) − g ( J ( ∇ X J ) V, ∇ V X − J ( ∇ X J ) V )= g ( ¯ ∇ X V, ∇ V X ) + 14 g (( ∇ X J ) V, ( ∇ X J ) V ) L. ( ii ) b ) = g ( ¯ ∇ X V, ¯ ∇ V X ) + 14 g (( ∇ X J ) V, ( ∇ X J ) V )= 14 g (( ∇ X J ) V, ( ∇ X J ) V ) . At the equality ( ∗ ) we use Lemma 5.3 (ii) b) which implies, that ∇ V X = ¯ ∇ V X + J ( ∇ V J ) X ∈ H to show π H ([ V, X ]) = ∇ V X − π H ( ¯ ∇ X V ) − J ( ∇ X J ) V = ∇ V X − J ( ∇ X J ) V. Since M is strict, we obtain R ( X, V, X, V ) = αg ( X, X ) g ( V, V ) . In addition it holds
Ric = 5 αg which implies5 αg ( V, V ) = g ( Ric ( V ) , V ) = g ( V, V ) R ( JV, V, JV, V ) + X i =1 g ( e i , e i ) R ( e i , V, e i , V )= g ( V, V ) R ( JV, V, JV, V ) + α X i =1 g ( e i , e i ) g ( V, V ) , where { e i } i =1 is an orthogonal frame of H . This equation yields R ( JV, V, JV, V ) = 4 α and itfollows that the fibers have Gaussian curvature K = R ( JV, V, JV, V ) g ( V, V ) g ( JV, JV ) − g ( V, JV ) = 4 α and constant curvature κ = 4 α. (cid:3) Proposition 5.8.
The manifold ( M, J, g ) is the total space of a pseudo-Riemannian submersion π : ( M, g ) → ( N, h ) where ( N, h ) is an almost pseudo-hermitian manifold and the fibers are totallygeodesic hermitian symmetric spaces. In particular, the fibers are simply connected.Proof. The foliation which is induced by V is totally geodesic and each leaf is by Proposition 5.6a locally hermitian symmetric space of complex dimension one.It is shown in Lemma 5.7 that each leaf has constant curvature κ. In the case κ >
N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 17 trivial and that the foliation comes from a (smooth) submersion (cf. p. 90 of [26]). In the case κ <
M, J, − g ) is a nearly pseudo-K¨ahler manifold of constant type − α. Thesame argument shows that the fibers are simply connected. (cid:3)
Lemma 5.9.
Let ( M , g, J ) be a strict nearly pseudo-K¨ahler six-manifold of constant type α. For an arbitrary normalized local vector field V ∈ V , i.e. ǫ V = g ( V, V ) ∈ {± } , we considerthe endomorphisms ˜ J := J |H , ˜ J : H ∋ X ( ∇ V J ) X ∈ H and ˜ J = ˜ J ˜ J . Then the triple ( ˜ J , ˜ J , ˜ J ) defines an ǫ -quaternionic triple on H with ǫ = − and ǫ = ǫ = sign ( − αǫ V ) and itis π H [( ∇ χ ˜ J i ) Y ] = − θ k ( χ ) ǫ j ˜ J j Y + θ j ( χ ) ǫ k ˜ J k Y, for a cyclic permutation of i,j,k and with θ ( χ ) = sign ( α ) g ( JV, ¯ ∇ χ V ) , θ ( χ ) = − sign ( α ) p | α | g ( V, Jχ ) and θ ( χ ) = sign ( α ) p | α | g ( V, χ ) . The sub-bundle of endomorphisms spanned by ( ˜ J , ˜ J , ˜ J ) doesnot depend on the choice of V. Proof.
Let A ( X ) = ( ∇ V J ) X for a fixed V ∈ V with ǫ V = g ( V, V ) ∈ {± } and an arbitrary X ∈ H . Then it is g ( A ( X ) , X ) = − g ( A ( X ) , A ( X )) = − g (( ∇ V J ) X, ( ∇ V J ) X )= − αg ( V, V ) g ( X, X )and we find after polarizing the last expression in X the identity A ( X ) = − αǫ V X. Furthermore A is a skew-symmetric endomorphism field and in consequence trace-free. Thereforethe endomorphism field ˜ J = 1 p | α | ( ∇ V J )is a hermitian structure if αǫ V > αǫ V < . Next we set ˜ J = J |H and ˜ J := ˜ J ◦ ˜ J = − ˜ J ◦ ˜ J , which follows from ( ∇ J ) ◦ J = − J ◦ ( ∇ J )and observe ˜ J = ˜ J . Moreover these are (para-)hermitian structures, since J and ∇ V J are skew-symmetric w.r.t. the metric g. Hence the triple ( ˜ J , ˜ J , ˜ J ) is a (para-)quaternionic triple on H with ǫ = − ǫ = ǫ = sign( − αǫ V ) . In the following we suppose, that it holds( ∇ X ˜ J ) Y ∈ V , for X, Y ∈ H . (5.3)This identity yields π H (( ∇ X ˜ J ) Y ) = 0 and in consequence it is π H (( ∇ χ ˜ J ) Y ) = ǫ V g ( V, χ )( ∇ V ˜ J ) Y + ǫ V g ( JV, χ )( ∇ JV ˜ J ) Y = ǫ V g ( V, χ )( ∇ V ˜ J ) Y + ǫ V g ( V, Jχ ) J ( ∇ V ˜ J ) Y = ǫ V p | α | g ( V, χ ) ˜ J Y + ǫ V p | α | g ( V, Jχ ) ˜ J Y !! = − θ ( χ ) ǫ ˜ J Y + θ ( χ ) ǫ ˜ J Y, where we have to define θ ( χ ) = ǫ ǫ V p | α | g ( V, Jχ ) = − sign ( α ) p | α | g ( V, Jχ ) ,θ ( χ ) = − ǫ ǫ V p | α | g ( V, χ ) = sign ( α ) p | α | g ( V, χ ) . Further we compute using the relation (1.7) for ∇ and ¯ ∇ ( ∇ χ ˜ J ) Y = ¯ ∇ χ ( ˜ J Y ) − ˜ J ¯ ∇ χ Y + 12 p | α | [ J ( ∇ χ J ) , ( ∇ V J )] Y, for χ ∈ T M, Y ∈ H (5.4)and get using ¯ ∇ ( ∇ J ) = 0 π H [( ¯ ∇ χ ˜ J ) Y ] = 1 p | α | π H (cid:2) ( ¯ ∇ χ ( ∇ V J )) Y (cid:3) = − p | α | π H [( ∇ ¯ ∇ χ V J ) Y ]= − p | α | ǫ V g ( JV, ¯ ∇ χ V )( ∇ JV J ) Y = 1 p | α | ǫ V g ( JV, ¯ ∇ χ V ) J ( ∇ V J ) Y, Constant non-zero length suffices. where we recall that ¯ ∇ χ V ∈ V has no part parallel to V. Due to equation (5.3) and Lemma 5.3(ii) the last term of (5.4) lies in V if χ is in H and vanishes if χ is a multiple of JV.
For χ = V weget [ J ( ∇ V J ) , ( ∇ V J )] Y = 2 | α | ˜ J ˜ J Y = 2 ǫ | α | ˜ J Y. This shows π H [( ∇ χ ˜ J ) Y ] = ǫ p | α | ǫ V g ( V, χ ) ˜ J Y + ǫ V g ( JV, ¯ ∇ χ V ) ˜ J Y, = − θ ( χ ) ǫ ˜ J Y + θ ( χ ) ǫ ˜ J Y, if we set θ ( χ ) = − ǫ ǫ V g ( JV, ¯ ∇ χ V ) = sign ( α ) g ( JV, ¯ ∇ χ V ) . It remains to differentiate the third(para-)complex structure:( ∇ χ ˜ J ) Y = ¯ ∇ χ ( ˜ J Y ) − ˜ J ¯ ∇ χ Y + 12 p | α | [ J ( ∇ χ J ) , J ( ∇ V J )] Y. (5.5)Again one obtains using ¯ ∇ J = 0 and ¯ ∇ ( ∇ J ) = 0 : π H [( ¯ ∇ χ ˜ J ) Y ] = 1 p | α | π H [( ¯ ∇ χ ( J ∇ V J )) Y ] = − p | α | π H [ J ( ∇ ¯ ∇ χ V J ) Y ]= − p | α | ǫ V g ( JV, ¯ ∇ χ V ) J ( ∇ JV J ) Y = θ ( χ ) ǫ ˜ J Y. The last term of equation (5.5) lies (with the help of equation (5.3) and Lemma 5.3 (ii)) in V for χ ∈ H and vanishes if χ is a multiple of V. For χ = JV we compute [ J ( ∇ JV J ) , J ( ∇ V J )] Y =[( ∇ V J ) , J ( ∇ V J )] Y = 2 | α | ˜ J ˜ J Y = − ǫ | α | ˜ J Y. This yields π H [( ∇ χ ˜ J ) Y ] = − θ ( χ ) ǫ ˜ J Y + θ ( χ ) ǫ ˜ J Y. Given a second section U in V with g ( U, U ) = g ( V, V ) one has U = aV + bJV for real functions a, b with a + b = 1 . Using this one easily sees that the triple induced by V and the one by U (locally) spans the same sub-bundle Q of endomorphisms of H . (cid:3) Lemma 5.10.
Let ( M , g, J ) be a strict nearly pseudo-K¨ahler six-manifold of constant type α. Let s : U ⊂ N → M be a (local) section of π on some open set U. Define φ by φ = s ∗ ◦ π ∗ : H s ( n ) π ∗ → T n N s ∗ → s ∗ ( T n N ) ⊂ T s ( n ) M, for n ∈ N and set J i | n := π ∗ ◦ ˜ J i | s ( n ) ◦ ( π ∗|H ) − for i = 1 , . . . , , where ˜ J i are defined in Lemma 5.9. Then ( J , J , J ) defines a local ǫ -quaternionic basis preserved by the Levi-Civita connection ∇ N of N. Proof.
We choose U such that the section s is a diffeomorphism onto W = s ( U ) and a vector field V in V defined on a subset containing W. As π is a pseudo-Riemannian submersion we obtain from π ∗ ◦ s ∗ = that s is an isometry from U onto W. Therefore it holds s ∗ ( ∇ NX Y ) = π s ∗ T N [ ∇ s ∗ X s ∗ Y ]which yields ∇ NX Y = π ∗ ( ∇ s ∗ X s ∗ Y ) and( π ∗|H ) − ( ∇ NX Y ) = π H ( ∇ s ∗ X s ∗ Y ) . (5.6)For convenience let us identify U and W or in other words consider s as the inclusion W ⊂ M. Then the projection on s ∗ T N is φ = s ∗ π ∗ = π ∗ |H . Moreover we need the (tensorial) relation ∇ NX ( π ∗ Z ) − π ∗ π H ( ∇ MX Z ) = 0 or equivalently ∇ NX ˜ Z − π ∗ π H ( ∇ MX φ − ˜ Z ) = 0 , which can be directly checked for basic vector fields. Using this identity we get for i = 1 , . . . , ∇ NX ( J i Y ) = ∇ NX ( φ ˜ J i φ − Y ) = φ ∇ MX ( ˜ J i φ − Y ) = φ ( ∇ MX ˜ J i ) φ − Y + φ ˜ J i ∇ MX ( φ − Y )= φ ( ∇ MX ˜ J i ) φ − Y + φ ˜ J i φ − ∇ NX Y = φ ( ∇ MX ˜ J i ) φ − Y + J i ∇ NX Y, which reads ( ∇ NX J i ) Y = φ ( ∇ MX ˜ J i ) φ − Y. This finishes the proof, since the right hand-side iscompletely determined by Lemma 5.9. Therefore we have checked the condition (4.10), i.e. themanifold N is endowed with a parallel skew-symmetric (para-)quaternionic structure, see also [3]10.32 and 14.36. (cid:3) Local sections exist, since π is locally trivial [3] 9.3. N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 19
General dimension.
In the last section we have seen that in dimension six the tensor ∇ V J induces a (para-)complex structure on H . This motivates the following definition.
Definition 5.11.
The foliation induced by
T M = H ⊕ V is called of twistorial type if for all p ∈ M there exists a V ∈ V p such that the endomorphism ∇ V J : H p → H p is injective.Obviously, if ∇ V J defines a (para-)complex structure, then the foliation is of twistorial type. Proposition 5.12. (a)
If the metric induced on H is definite, then the foliation is of twistorial type. (b) If the foliation is of twistorial type, then for all p ∈ M and all = U ∈ V p the endomor-phism ∇ U J : H p → H p is injective. (c) It holds with A := ∇ V J for some vector field V in V of constant length and for vectorfields X ∈ H and χ ∈ T M ¯ ∇ χ ( A ) X = 0 . (5.7) Further it holds [ A , ( ∇ U J )] = 0 for all U ∈ V and ∇ U ( A ) X = 0 (5.8) for vector fields U in V . Proof.
Part (a) follows from ( ∇ V J ) X ∈ H for X ∈ H and V ∈ V , cf. Lemma 5.3 (i). For (b) weobserve, that if ∇ V J is injective so is ∇ JV J = − J ∇ V J. As V is of dimension two { V, JV } with V = 0 is an orthogonal basis. With a, b ∈ R it follows g (( a ∇ V J + b ∇ JV J ) X, ( a ∇ V J + b ∇ JV J ) X ) =( a + b ) g (( ∇ V J ) X, ( ∇ V J ) X ) , which yields, that ∇ aV + bJV J : H p → H p is injective since a = 0or b = 0 . It remains to prove part (c). We first observe, that, since V has constant length andsince ¯ ∇ is a metric connection and preserves V , it follows ¯ ∇ χ V = α ( χ ) JV for some one-form α. From ¯ ∇ ( ∇ J ) = 0 we obtain( ¯ ∇ χ A ) X = ( ¯ ∇ χ ( ∇ V J )) X = ( ∇ ¯ ∇ χ V J ) X = α ( χ )( ∇ JV J ) X = − α ( χ ) JAX and we compute using { A, J } = 0¯ ∇ χ ( A ) X = A ( ¯ ∇ χ A ) X + ( ¯ ∇ χ A ) AX = − α ( χ )[ A ( J ( AX )) + JA X ] = 0 . The expression [ A , ( ∇ U J )] = 0 is tensorial in U and vanishes for U = V. Therefore we onlyneed to compute [ A , ( ∇ JV J )] = − [ A , J ( ∇ V J )] = − J [ A , ( ∇ V J )] = 0 , where we used that A commutes with J. This implies ∇ U ( A ) X = ¯ ∇ U ( A ) X + 12 [ J ( ∇ U J ) , A ] X = −
12 [( ∇ JU J ) , A ] X = 0and proves part (c). (cid:3) In the following V is a local vector field of constant length ǫ V = g ( V, V ) ∈ {± } . We denote by Ω the curvature form of the connection induced by ¯ ∇ on the (complex) line bundle V , which is given by ¯ R ( X, Y ) V = Ω( X, Y ) JV, for
X, Y ∈ T M, V ∈ V . Proposition 5.13.
If the foliation is of twistorial type, (i) then the endomorphism A := ∇ V J |H satisfies A = κǫ V H for some real constant κ = 0 and Ω = − κ (2 ω V − ω H ) , where ω H ( X, Y ) = g ( X, JY ) is the restriction of the fundamental two-form ω to H ;(ii) for X, Y in H it is ( ∇ X J ) Y ∈ V . The proof of this proposition is divided in several steps.
Lemma 5.14. (i)
For
X, Y in H and V in V it is ¯ R ( X, Y, V, JV ) = − g (( ∇ V J ) X, JY ) . (ii) For a given X in H and V in V it follows ¯ R ( X, V, V, JV ) = 0 . Proof. (i) Since H is ¯ ∇ -parallel we obtain, that σ XY V ¯ R ( X, Y, V, JV ) = ¯ R ( X, Y, V, JV ) . This is theleft hand-side of the first Bianchi identity (1.12) . The right hand-side reads − σ XY V g (( ∇ X J ) Y, ( ∇ V J ) JV ) = − g (( ∇ V J ) X, ( ∇ Y J ) JV ) − g (( ∇ Y J ) V, ( ∇ X J ) JV )= − g (( ∇ V J ) X, JY ) . (ii) From the symmetries (1.10) of the curvature tensor ¯ R it follows ¯ R ( X, V, V, JV ) = ¯ R ( V, JV, X, V ) . This expression vanishes since H is ¯ ∇ -parallel. (cid:3) From the last lemma we derive the more explicit expression of the curvature form Ω :Ω = f ω V + ǫ V α, (5.9)where f is a smooth function, ω V is the restriction of the fundamental two-form ω = g ( · , J · ) to V and α ( X, Y ) = − g ( A X, JY ) . Lemma 5.15.
It holds with U ∈ V and X, Y ∈ H : dω V ( X, U, JU ) = 0 , (5.10) dα ( X, U, JU ) = 0 , (5.11) dω V ( U, X, Y ) = − g ( ∇ U J ) X, Y ) , (5.12) dα ( U, X, Y ) = 4 g ( A ( ∇ U J ) X, Y ) . (5.13) Proof.
For vector fields
A, B, C on M it is( ∇ A ω V )( B, C ) = Aω V ( B, C ) − ω V ( ∇ A B, C ) − ω V ( B, ∇ A C ) = Aω V ( B, C ) − ω V (cid:18) ¯ ∇ A B + 12 J ( ∇ A J ) B, C (cid:19) − ω V (cid:18) B, ¯ ∇ A C + 12 J ( ∇ A J ) C (cid:19) . If two of them are
X, Y ∈ H and one is U ∈ V we check using the definition of ω V , the informationof Lemma 5.3 and that the decomposition H ⊕ V is ¯ ∇ -parallel: ∇ U ω V ( X, Y ) = 0 , ∇ X ω V ( U, Y ) = − ω V ( U, ∇ X Y ) = − ω V ( U, J ( ∇ X J ) Y ) , ∇ Y ω V ( X, U ) = − ω V ( ∇ Y X, U ) = − ω V ( J ( ∇ Y J ) X, U ) . By the symmetries of ω ( J ∇ · J · , · ) we conclude dω V ( U, X, Y ) = − g (( ∇ U J ) X, Y ) . Next we suppose X ∈ H and U ∈ V and obtain with Lemma 5.3: ∇ X ω V ( U, JU ) = − ω V ( ¯ ∇ X U, JU ) − ω V ( U, ¯ ∇ X ( JU )) = 0 , ∇ U ω V ( X, JU ) = − ω V ( J ( ∇ U J ) X, JU ) − ω V ( X, J ( ∇ U J ) JU ) = 0 , ∇ JU ω V ( U, X ) = − ω V ( J ( ∇ JU J ) U, X ) − ω V ( U, J ( ∇ JU J ) X ) = 0 . This shows dω V ( X, U, JU ) = 0 . Let
X, Y ∈ H and U ∈ V . From α ( U, · ) = 0 we conclude( ∇ U α )( X, Y ) = − (cid:2) g ( ∇ U ( A ) X, Y ) + g ( A X, ( ∇ U J ) Y ) (cid:3) (5.8) = − g ( A X, ( ∇ U J ) Y ) , ( ∇ X α )( U, Y ) = − g ( A ( ∇ U J ) X, Y ) , ( ∇ Y α )( U, X ) = − g ( A ( ∇ U J ) Y, X ) , N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 21 which finishes the proof of dα ( U, X, Y ) = 4 g ( A ( ∇ U J ) X, Y ) , since [ A , ∇ U J ] = 0 for all U ∈ V . We now prove the last identity( ∇ U α )( JU, Y ) = − α ( ¯ ∇ U JU + 12 J ( ∇ U J ) JU, Y ) − α ( JU, ¯ ∇ U Y + 12 J ( ∇ U J ) Y ) = 0 , ( ∇ JU α )( Y, U ) = − α ( ¯ ∇ JU Y + 12 J ( ∇ JU J ) Y, U ) − α ( Y, ¯ ∇ JU U + 12 J ( ∇ JU J ) U ) = 0 , ( ∇ Y α )( U, JU ) = − α ( ¯ ∇ Y U + 12 J ( ∇ Y J ) U, JU ) − α ( U, ¯ ∇ Y JU + 12 J ( ∇ Y J ) JU ) = 0 , where we used α ( W, · ) = − α ( · , W ) = 0 for W ∈ V . This finally shows dα ( X, U, JU ) = 0 . (cid:3) Proof. (of the Proposition 5.13) (i) Let
X, Y be vector fields in H and V be a local vector field in V of constant length. Since Ω as a curvature form of a (complex) line bundle is closed, we obtainfrom equation (5.9) − ǫ V dα = f dω V + df ∧ ω V . The equations (5.10) and (5.11) imply df |H = 0 . This implies [
X, Y ] f = 0 and using that H is ¯ ∇ -parallel we obtain ( ¯ ∇ X Y ) f = 0 = ( ¯ ∇ Y X ) f whichyields finally 0 = T ¯ ∇ ( X, Y )( f ) = − [ J ( ∇ X J ) Y ]( f ) . By Lemma 5.3 (ii) d) the last equation shows df |V = 0 . Since M is connected, it follows f ≡ − κ for a constant κ. Again using d Ω( V, X, Y ) = 0 equation (5.12) and (5.13) yield for arbitrary
X, Yκg (( ∇ V J ) X, Y ) + 4 ǫ V g ( A ( ∇ V J ) X, Y ) = 0 . This implies ( ∇ V J )( κ H + 4 ǫ V A ) = 0 . Since the foliation is of twistorial type, it follows A = − ǫ V κ H = − ǫ V α H if we set 4 α = κ in analgogue to dimension six.(ii) Since Ω is closed, it follows from part (i) and dω V ( X, Y, Z ) = 0 for
X, Y, Z ∈ H that it is dω H ( X, Y, Z ) = 0 . Using dω H ( X, Y, Z ) = 3 g (( ∇ X J ) Y, Z ) yields part (ii). (cid:3)
Proposition 5.16.
Let ( M k +2 , g, J ) be a strict nearly pseudo-K¨ahler manifold of twistorialtype. Let s : U ⊂ N → M be a (local) section of π on some open set U. Define φ by φ = s ∗ ◦ π ∗ : H s ( n ) π ∗ → T n N s ∗ → s ∗ ( T n N ) ⊂ T s ( n ) M, for n ∈ N and set J i | n := π ∗ ◦ ˜ J i | s ( n ) ◦ ( π ∗|H ) − for i = 1 , . . . , , where ˜ J i are defined in Lemma 5.9. Then ( J , J , J ) defines a local ǫ -quaternionic basis preserved by the Levi-Civita connection ∇ N of N. Proof.
The proof of Proposition 5.9 only uses A = κǫ V and ( ∇ X J ) Y ∈ V for X, Y ∈ H . There-fore we can generalize it by means of Proposition 5.13 to strict nearly pseudo-K¨ahler manifolds oftwistorial type. (cid:3)
The twistor structure.
In this subsection we finally characterize the nearly pseudo-K¨ahlerstructures, which are related to the canonical nearly K¨ahler structure of twistor spaces.
Theorem 5.17. (i)
The manifold ( M, J = ˇ J, ˇ g = g ) is a twistor space of a quaternionic pseudo-K¨ahlermanifold, if it is ǫ V α > . (ii) The manifold ( M, J = ˇ J, ˇ g = g ) is a twistor space of a para-quaternionic K¨ahler manifold,if it is ǫ V α < . Proof.
Denote by π Z : Z → N the twistor space of the manifold N endowed with the parallel skew-symmetric (para-)quaternionic structure constructed from the foliation π : M → N of twistorialtype, cf. Proposition 5.9 for dimension six and Proposition 5.16 for general dimension. We observethat the restriction of J to H yields a (smooth) map ϕ : M → Z , m dπ m ◦ J m |H ◦ ( dπ m |H ) − =: j π ( m ) , which by construction satisfies π Z ◦ ϕ = π and as a consequence dπ Z ◦ dϕ = dπ. Since π and π Z are pseudo-Riemannian submersions, the last equation implies that dϕ induces an isometry of theaccording horizontal distributions and maps the vertical spaces into each other. Let us determine the differential of ϕ on V . Claim:
For V ∈ V one has dϕ ( V ) = 2 dπ ◦ ( ∇ V J ) ◦ ( dπ |H ) − ,dϕ ( JV ) = 2 dπ ◦ ( ∇ JV J ) ◦ ( dπ |H ) − = − dπ ◦ J ( ∇ V J ) ◦ ( dπ |H ) − . To prove the claim we consider a (local) vector field V ∈ V and a (local) integral curve γ of V onsome interval I ∋ γ (0) = m. Let X be a vector field in N. Denote by ˜ X the horizontal liftof X. The Lie transport of ˜ X along the vertical curve γ projects to X, i.e. it holds dπ γ ( t ) ( ˜ X ) = X for all t ∈ I and in consequence (cid:0) dπ γ ( t ) |H (cid:1) − X = ˜ X. In other words dπ commutes with this Lietransport, which implies dϕ ( V ) X = dπ (( L V J ) ˜ X ) , as one directly checks using basic vector fields. Therefore we need to determine the Lie-derivative L of J : π H (( L V J ) ˜ X ) = π H ([ V, J ˜ X ] − J [ V, ˜ X ])= π H (cid:16) ∇ V ( J ˜ X ) − ∇ J ˜ X V − J ∇ V ˜ X + J ∇ ˜ X V (cid:17) = π H (cid:18) ( ∇ V J ) ˜ X − J ( ∇ J ˜ X J ) V + 12 J ( J ( ∇ ˜ X J )) V (cid:19) = 2( ∇ V J ) ˜ X. This shows dϕ ( V ) = 2 dπ ◦ ( ∇ V J ) ◦ ( dπ |H ) − , which implies dϕ ( JV ) = 2 dπ ◦ ( ∇ JV J ) ◦ ( dπ |H ) − = − dπ ◦ J ( ∇ V J ) ◦ ( dπ |H ) − . Given a local section s : N → M and the associated adapted frameof the (para-)quaternionic structure it follows that ϕ ◦ s is J , dϕ ( V ) is related to J and dϕ ( JV )to − J which span the tangent space of the fiber F π ( m ) = S in ϕ ( m ) . The complex structure of Z maps J to J . Hence dϕ is complex linear for the opposite complex structure ˇ J on M. Furtherone sees in this local frame that ϕ maps horizontal part into horizontal part. Therefore ϕ is anisometry for the metric ˇ g = g , i.e. the parameter t = 2 in the canonical variation of the metric g. This means that ( M, ˇ J, ˇ g = g ) is isometrically biholomorph to Z . (cid:3) Combining Theorem 2.13 and Theorem 5.17 we obtain
Theorem A . References [1] D. Alekseevsky,
Classification of quaternionic spaces with transitive solvable group of motions,
Izv. Akad.Nauk SSSR Ser. Mat. (1975), no. 2, 315–362.[2] D. Alekseevsky, V. Cort´es, The twistor spaces of a para-quaternionic K¨ahler manifold,
Osaka J. Math. (2008), no. 1, 215–251.[3] A. Besse, Einstein manifolds,
Springer (1987).[4] N. Blaˇzi´c,
Paraquaternionic projective space and pseudo-Riemannian geometry , Publications de l’InstitutMath´ematique Nouvelle s´erie, tome (74) (1996), 101-107.[5] F. Belgun, A. Moroianu, Nearly K¨ahler 6-manifolds with reduced holonomy,
Ann. Global Anal. Geom. (2001), no. 4, 307–319.[6] V. Cort´es, Alekseevskian spaces,
Diff. Geom. Appl. (1996), no. 2, 129-168.[7] V. Cort´es, Th. Leistner, L. Sch¨afer, F. Schulte-Hengesbach, Half-flat Structures and Special Holonomy ,arXiv:0907.1222.[8] V. Cort´es, L. Sch¨afer,
Flat nearly K¨ahler manifolds , Ann. Global Anal. Geom. (2007), no. 4, 379-389.[9] V. Cort´es, L. Sch¨afer, Geometric Structures on Lie groups with flat bi-invariant metric , J. Lie Theory (2009), 423–437.[10] Th. Friedrich, S. Ivanov, Parallel spinors and connections with skew-symmetric torsion in string theory ,Asian J. Math. (2002), no. 2, 303-335.[11] S. Ferrara, S. Sabharwal, Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces ,Nucl. Phys. B (1990), 317-332.[12] A. Gray,
The structure of nearly K¨ahler manifolds , Math. Ann. (1976), no. 3, 233-248.[13] A. Gray,
Riemannian manifolds with geodesic symmetries of order
3, J. Diff. Geom. (1972), 343–369.[14] I. Kath, Killing Spinors on Pseudo-Riemannian Manifolds,
Habilitationsschrift an der Humboldt-Universit¨at zu Berlin (1999).
N THE STRUCTURE OF NEARLY PSEUDO-K¨AHLER MANIFOLDS 23 [15] S. Kobayashi,
On compact K¨ahler manifolds with positive definite Ricci tensor,
Ann. of Math. (1961),570–574.[16] S. Kobayashi, K. Nomizu, Foundations of differential geometry,
Vol. I/II Interscience Publishers JohnWiley & Sons (1969).[17] C. LeBrun,
On complete quaternionic-K¨ahler manifolds,
Duke Math. J. (1991), no.3, 723–743.[18] H. B. Lawson, M.-L.Michelson, Spin geometry , Princeton University Press, (1989).[19] P.-A. Nagy,
On nearly-K¨ahler geometry,
Ann. Global Anal. Geom. (2002), no. , 167–178.[20] P.-A. Nagy, Nearly K¨ahler geometry and Riemannian foliations,
Asian J. Math. (2002), no. , 481–504.[21] B. O’Neill, The fundamental equations of a submersion,
Mich. Math. J. (1966), 459–469.[22] J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces,
J. Math. Mech. (1965) 1033–1047.[23] L. Sch¨afer, F. Schulte-Hengesbach, Nearly pseudo-K¨ahler and nearly para-K¨ahler six-manifolds ,arXiv:0912.3271, to appear in Handbook of pseudo-Riemannian geometry and supersymmetry, ed. byV. Cort´es, IRMA Lect. Math. Theor. Phys. (2010).[24] L. Sch¨afer, tt ∗ -geometry on the tangent bundle of an almost complex manifold , J. Geom. Phys. (2007),no. 3, 999–1014.[25] L. Sch¨afer, Para- tt ∗ -bundles on the tangent bundle of an almost para-complex manifold , Ann. GlobalAnal. Geom. (2007), no. 2, 125–145.[26] R. W. Sharpe, Differential geometry, Cartan’s generalization of Klein’s Erlangen program , Springer(1997).
Lars Sch¨afer, Institut Differentialgeometrie, Leibniz Universit¨at Hannover, Welfengarten 1, D-30167 Hannover, Germany
E-mail address ::