S^6 and the geometry of nearly Kähler 6-manifolds
aa r X i v : . [ m a t h . DG ] J u l S AND THE GEOMETRY OF NEARLY K ¨AHLER -MANIFOLDS ILKA AGRICOLA, ALEKSANDRA BOR ´OWKA, THOMAS FRIEDRICH
Abstract.
We review results on and around the almost complex structure on S , bothfrom a classical and a modern point of view. These notes have been prepared for the Work-shop “(Non)-existence of complex structures on S ” ( Erste Marburger ArbeitsgemeinschaftMathematik – MAM-1 ), held in Marburg in March 2017. Introduction
It is well known that the sphere S admits an almost Hermitian structure induced byoctonionic multiplication, and that this structure stems from the transitive action of thecompact exceptional Lie group G on it. In 1955, Fukami and Ishihara were presumablythe first authors to devote a separate paper to the detailed investigation of S and showedin particular that S is the naturally reductive space G / SU(3) [FI55]. In 1958, Calabistudied hypersurfaces in the space of imaginary octonions and proved that the inducedalmost complex structure is never integrable if the hypersurface is compact [Cal58]. In factthe almost Hermitian structure on the 6-sphere is a very special one: Already in [FI55], it isobserved that the Levi Civita derivative of J satisfies( ∗ ) ( ∇ gX J ) X = 0 for all vector fields X. Such manifolds are called nearly K¨ahler and they were investigated intensively by Gray ina series of papers [Gr66, Gra70, G76]. In particular, he showed in dimension 6 that theyare Einstein and their first Chern class vanishes. In fact, for many reasons dimension 6 isof particular interest for nearly K¨ahler geometry [Na10]. For a long time the only compactexamples of nearly K¨ahler manifolds were the four homogeneous examples: S , S × S , CP and the flag manifold F . The aim of this paper is to provide a concise review of propertiesof nearly K¨ahler manifolds in dimension 6 with special attention given to the sphere S .After some historical remarks, we start by recalling Calabi’s result about hypersurfaces inthe space of imaginary octonions R . Then we discuss the intrinsic torsion approach andnaturally reductive spaces and briefly recall Gray’s [G76] and Kirichenko’s [Ki77] results.Next we present the spinoral approach of R. Grunewald [Gru90] and a modern view onit. We finish by giving an overview of L. Foscolo and M. Haskins contribution [FH17].They discovered non-homogeneous cohomogeneity one nearly K¨ahler structures on S andconjectured that these are the only cohomogeneity one examples.2. Some historical comments
Clearing the facts around the almost complex structure on S took several independentsteps. In particular, it was not noticed immediately that (and how) it was related to thetransitive action of G . ontgomery and Samelson proved in 1943 that the only compact connected simple Liegroup which can be transitive on S n is SO(2 n + 1)— except for a a finite number of n ′ s [MS43, Thm II, p.462]. Their method was of topological nature and required the knowledgeof the homology rings of simple Lie groups, which was not yet available for the five exceptionalsimple Lie groups; hence they couldn’t give any further information on the exceptional cases.Six years later, Armand Borel proceeded by constructing the homogeneous spaces directly,which lead him to the result that the only sphere with a transitive group G acting that isnot orthogonal is S with G = G [Bo49, Thm III, p. 586]. This completed the classificationof transitive sphere actions and showed, in particular, that G is the only exceptional Liegroup with such an action.Meanwhile, Adrian Kirchhoff had noticed in 1947 that S carries an almost complex struc-ture induced from octonionic multiplication ([Ki47]; see also [Eh50]). In his main theorem,Kirchhoff related the existence of an almost complex structure on S n to the parallelism of S n +1 .In 1951, Ehresmann and Libermann [EL51] as well as Eckmann and Fr¨olicher [EF51]observed independently that this almost complex structure on S is not integrable — infact, their articles appeared in the same volume of the Comptes Rendus Hebdomadaires desS´eances de l’Acad´emie des Sciences, Paris. While Eckmann and Fr¨olicher were interested informulating the integrability condition and treated S merely as an example where it didn’thold, the aim of Ehresmann and Libermann was the local description of locally homogeneousalmost hermitian manifolds in terms of Cartan structural equations, and they found thatthe equations exhibited an exceptional structure for n = 6. They stated [EL51, p. 1282]:“La structure consid´er´ee est donc localement ´equivalente `a une structurepresque hermitienne sur S admettant G comme groupe d’automorphismes.Ce groupe ne peut laisser invariante sur S qu’une seule structure presquecomplexe. Celle-ci est donc isomorphe `a la structure presque complexe d´efinie`a l’aide des octaves de Cayley. Comme la deuxi`eme torsion dans les formules(5) n’est pas nulle, cette structure ne d´erive pas d’une structure complexe.” Hence, they seem to be the first authors to connect the transitive G -action on S toits octonionic almost complex structure. A detailed account of the results of [EF51] andfurther material was given by Fr¨olicher four years later [Fr55]; however, in his discussion ofhomogeneous almost complex manifolds, he doesn’t mention S . Remarkably, he describedalready (as did [FI55]) the characteristic connection of S and proved that its torsion is givenby the Nijenhuis tensor.In [FI55], all these thoughts on S are brought together for the first time, and the charac-teristic connection is proved to coincide with the canonical connection of the homogeneousspace G / SU(3).The first author to suggest the investigation of manifolds satisfying the abstract nearlyK¨ahler condition ( ∗ ) was Tachibana in [Ta59], who called such manifolds K -spaces andproved, amongst other things, that their Nijenhuis tensor is totally antisymmetric. No “The structure we considered is therefore locally equivalent to an almost hermitian structure on S admitting G as its group of automorphisms. This group can leave invariant only one almost complexstructure on S . It is therefore isomorphic to the almost complex structure defined with the help of Cayley’soctonions. Since the second torsion of the formulas (5) doesn’t vanish, this structure is not induced from acomplex structure.” (translated by the authors) xamples were discussed, although it is clear from the reference made to [FI55, Fr55] thatthe inspiration came from S . The paper [Ko60] continued the investigation of K -spaces.Inspired by the papers of Calabi [Cal58] and Koto [Ko60], Alfred Gray used in 1966 forthe first time the term nearly K¨ahler manifold [Gr66]. He writes in the introduction:“The manifolds we discuss include complex and almost K¨ahler manifolds; also S with the almost complex structure derived from the Cayley numbers fallsinto a class of manifolds which we call nearly K¨ahlerian.”This was the starting point of the career of S as a most remarkable nearly K¨ahler manifold.Surprisingly, most classes of almost Hermitian manifolds that were systemized later in theGray-Hervella classification [GH80] appear already in this paper.3. The almost complex structures induced from octonions
In this section we present an explicit approach for constructing the nearly K¨ahler structureon S . The construction goes back to Calabi [Cal58], who studied hypersurfaces in R witha complex structure induced from octonions.3.1. Seven-dimensional cross products.
Recall that the octonion algebra O is the unique8-dimensional composition algebra (or equivalently normed division algebra). It can bedefined from quaternions using the Cayley-Dickson construction O = H ⊕ J H , with thefollowing operations: q + J q = q − J q , ( q + J q )( q + J q ) = q q + q q + J ( q q + q q ) . Consequently, the octonions can be viewed as an 8 dimensional (non-associative) algebrawith basis 1 , e , . . . , e , where e i are imaginary units and the multiplication between themis defined above. Note that we can take e , e , e to be imaginary quaternions. Considerthe vector subspace of imaginary octonions Y := span { e , . . . , e } which, as a vector space,is isomorphic to R . The octonion multiplication induces (by restriction and projection) across product on Y via the formula A × B := 12 ( AB − BA ) . The group of automorphisms of Y is the exceptional group G . We strongly recommendthe article by Cristina Draper in this volume for a very thorough description of G and itsrelation to octonions, cross products, and spinors [Dr17]. The vector space Y together withthe cross product and the scalar product is sometimes called the Cayley space. The crossproduct has the following properties: • h A, ( B × C ) i = h ( A × B ) , C i =: ( ABC ) ( scalar triple product identity ), • A × ( A × B ) = −| A | B + h A, B i A ( Lagrange or Malcev identity ), • If C is orthogonal to A and B , then( A × B ) × C = A × ( B × C ) − h A, B i C .
However, there are some significant differences between dimensions 3 and 7 coming from thenon-associativity of octonions—for example, the Jacobi identity does not hold in dimension7. .2. Almost complex structure on hypersurfaces of the Cayley space.
The resultsdescribed in this section are mainly due to Calabi [Cal58]. Let S be a 6-dimensional orientedmanifold immersed into the Cayley space Y . The canonical orientation on Y induces anormal vector field on S , called N . Consider its second fundamental form and denote by K its shape operator. The eigenvalues of K are just the principal curvatures. We define J ∈ End ( T S ) by J ( X ) := N × X, X ∈ T S, and take g to be the metric on S induced by the pull back of the scalar product on Y . Lemma 3.1. J is an almost complex structure on S such that ( S, J, g ) is an almost Hermitianmanifold, and it satisfies the identity h ( ∇ gX J )( Y ) , Z i = h K ( X ) × Y, Z i . Proof.
First we need to show that J = − id . By the Malcev identity, N × ( N × X ) = −| N | X + h N, X i N, for any X ∈ ( T S ). As X is perpendicular to N , this proves the claim.Then, using the scalar triple product and Malcev identities we obtain h J ( X ) , J ( Y ) i = h N × X, N × Y i = h N × X ) × N, Y i = −h N × ( N × X ) , Y i = h X, Y i , showing the J -invariance of the metric. Finally, let us compute ( ∇ gX J )( Y ) for any tangentvectors X, Y :( ∇ gX J )( Y ) = ∇ gX ( J ( Y )) − J ( ∇ gX Y ) = ∇ gX ( N × Y ) − N × ∇ gX Y = ∇ gX N × Y + N × ∇ gX Y − N × ∇ gX Y = K ( X ) × Y .
This yields the claimed formula for ∇ gX J . (cid:3) Definition 3.1.
Let (
M, g, J ) be an almost Hermitian manifold. If ( ∇ gX J ) X = 0 and ∇ g J = 0, then ( M, g, J ) is called nearly K¨ahler manifold.The almost complex structure of a nearly K¨ahler manifold is never integrable, see [G76] or[AFS05]. In fact, an easy calculation shows that its Nijenhuis tensor is given by N ( X, Y ) =4( ∇ gX J ) J Y . A direct consequence of the last lemma is the following fact.
Proposition 3.1.
Let g be the standard metric on S and J the almost complex structuredefined by the cross product on the Cayley space. Then ( S , g, J ) is a nearly K¨ahler manifold. Proposition 3.2 (Calabi [Cal58]) . If ( S, I ) is a compact, oriented -manifold with an immer-sion into the Cayley space Y , the induced almost complex structure J on S is non-integrable. To prove this theorem Calabi studied the shape operator K of the hypersurface. He foundthat the integrability condition for J is that K is complex anti-linear, K ◦ J = − J ◦ K .
In any closed hypersurface of the Euclidian space, there exists an open subset on which thesecond fundamental form is positive or negative definite. But if the tangent vector X is aneigenvector of the shape operator with eigenvalue λ , then J ( X ) is again an eigenvector witheigenvalue − λ . This yields a contradiction. . S as naturally reductive space A homogeneous Riemannian space M = G/H is called reductive if there exists an Ad( H )-invariant subspace m such that g = h ⊕ m . Denote by h .. , .. i the inner product in m definingthe G -invariant metric. If it satisfies( ∗∗ ) h [ X, Y ] m , Z i = −h [ X, Z ] m , Y i , the homogeneous space is called naturally reductive. In this case the tensor T ( X, Y, Z ) := −h [ X, Y ] m , Z i is totally skew symmetric, i.e. T is a 3-form. The canonical connection ∇ c of G/H is theunique metric connection with skew symmetric torsion tensor T , ∇ c = ∇ g + 12 T .
The holonomy of ∇ c is contained in the isotropy group H , i.e. the canonical connectionis much more adapted to the space G/H then the Levi-Civita connection ∇ g . A naturallyreductive space with vanishing torsion T is a Riemannian symmetric space. Naturally re-ductive homogeneous spaces have the special property that the torsion and the curvatureare parallel with respect to the canonical connection, ∇ c R c = 0 , ∇ c T c = 0 . Observe that a homogenous Riemannian manifold can be naturally reductive in differentways, it depends on the choice of the subgroup G ⊂ Iso( M ) of the isometry group. However,this happens only for spheres or Lie groups, see the recent results of C. Olmos and S. Reggiani[OR12].Any nearly K¨ahler manifold admits a unique hermitian connection with skew symmetrictorsion, too. This connection has been introduced by A. Gray and is called the characteristicconnection of the nearly K¨ahler manifold, see [G76]. Let us explain the proof as well as theformula for the characteristic torsion. Proposition 4.1.
Let ( M, g, J ) be a nearly K¨ahler manifold. There exists a unique metricconnection preserving the almost complex structure and with skew symmetric torsion, andits torsion -form is given by the formula T c ( X, Y, Z ) = h ( ∇ gX J )( J Y ) , Z i . Proof.
Consider a metric connection ∇ = ∇ g + T with an arbitrary skew symmetric torsion.The condition ∇ J = yields the equation0 = h ( ∇ gX J )( Y ) , Z i + 12 T ( X, J Y, Z ) + 12 T ( X, Y, J Z ) . Symmetrizing the latter equation with respect to X and Y , we obtain0 = h ( ∇ gX J )( Y ) , Z i + h ( ∇ gY J )( X ) , Z i + 12 T ( X, J Y, Z ) + 12 T ( Y, J X, Z ) . In case the almost complex structure is nearly K¨ahler we obtain the condition T ( X, J Y, Z ) = − T ( Y, J X, Z ) , nd moreover T ( X, Y, J Z ) = T ( Y, J Z, X ) = − T ( Z, J Y, X ) = T ( X, J Y, Z ) . Inserting the latter formula into the first one, we finally obtain the formula for the charac-teristic torsion 0 = h ( ∇ gX J )( Y ) , Z i + T ( X, J Y, Z ) . (cid:3) If a nearly K¨ahler manifold is a reductive homogeneous space, the canonical connectionin the sense of reductive spaces coincides with the characteristic connection in the sense ofnearly K¨ahler manifolds. With respect to the full isometry group G = SO(7), the roundsphere S becomes a symmetric space, S = SO(7) / SO(6) , the canonical connection coincides with the Levi-Civita connection and is hence torsion free.On the other side, taking into account the almost complex structure J induced by the oc-tonions, S becomes a naturally reductive space (see [FI55], [R93]). Indeed, by construction J and g are invariant under the action of the automorphism group of octonions. In partic-ular, the exceptional group G ⊂ SO (7) preserves the metric as well as the almost complexstructure. Moreover, G acts transitively on S . The isotropy subgroup preserves the linearcomplex structure of the tangent space. Consequently, it is an 8-dimensional subgroup ofU(3) isomorphic to SU(3), see [Dr17, Prop. 5.2]. Proposition 4.2. S = G / SU(3) is naturally reductive, its canonical connection coincideswith the characteristic connection, and its torsion -form is given by the formula T c ( X, Y, Z ) = −h J ( X × Y ) , Z i = −h N, ( X × Y ) × Z i . Proof.
An explicit description of the reductive space S = G / SU(3) is given in [Dr17,Remarks 5.3, 6.4]. In particular, it is proved that the metric satisfies the condition ( ∗∗ )for being naturally reductive, and that the torsion of the canonical connection is given bythe formula stated in the proposition. We now prove that the same formula describes thecharacteristic connection. Consider the metric connection ∇ defined by the formula h∇ X Y, Z i := h∇ gX Y, Z i − h J ( X × Y ) , Z i . We compute the covariant derivative ∇ J : h ( ∇ X J ) Y, Z i = h ( ∇ gX J ) Y, Z i − h J ( X × Y ) , Z i + 12 h J ◦ J ( X × Y ) , Z i . Next we apply the formula for the covariant derivative ∇ g J . Then we obtain2 h ( ∇ X J ) Y, Z i = h X × Y, Z i − h J ( X × J Y ) , Z i . Since N and X, Y are orthogonal, the sume of the right side vanishes. Indeed, we have h J ( X × J Y ) , Z i ) = h N × ( X × ( N × Y )) , Z i = −h N × ( N × ( X × Y )) , Z i = h ( X × Y, Z i . The computation proves that the connection ∇ is a metric connection preserving the almostcomplex structure J . Moreover, its torsion T c ( X, Y, Z ) = −h J ( X × Y ) , Z ) = −h N, ( X × Y ) × Z i s skew symmetric. Consequently, ∇ is the canonical (characteristic) connection of thenaturally reductive and nearly K¨ahler space G / SU(3). (cid:3)
By a theorem of Butruille [B05], no other nearly K¨ahler structure on S can be homogeneous.In fact he proved that the only homogeneous nearly K¨ahler 6-manifolds are S , S × S , CP and the flag manifold F with their standard metrics.5. G -structures and connections with skew symmetric torsion The connection defined above can be described from the point of view of G -structures. Let G be a closed Lie subgroup of SO ( n ) and let so ( n ) = g ⊕ m be the corresponding orthogonaldecomposition of the Lie algebra so ( n ). Then a G-structure on Riemannian manifold M is a reduction R of its frame bundle, which is principal SO ( n )-bundle, to the subgroup G .As the Levi-Civita connection is a 1-form with values in so ( n ), using the decomposition so ( n ) = g ⊕ m , we get a direct sum decomposition of its restriction to R into a connection inprincipal G -bundle R and a term Γ corresponding to m . Γ is a 1-form on M with values inthe associated bundle R × G m and is called the intrinsic torsion. It measures the integrabilityof G -structure; the structure is integrable if and only if Γ = 0. At a fixed point, Γ is anelement of the G -representation R n ⊗ m . Moreover, one can show that in any case when G is the isotropy group of some tensor T the algebraic G -types of Γ correspond to algebraic G -types of ∇ g T (see [F03]; we also recommend [Agr06] as a suitable review on characteristicconnections). We are looking again for metric connections with skew torsion and preservingthe fixed G -structure. If it exists, it is called the characteristic torsion of the fixed G -typeand we denote by T c . However, not all G -structures admit such a connection. The questionwhether or not a certain G -type admits a characteristic connection can be decided usingrepresentation theory. Indeed, consider the G -morphismΘ : Λ ( R n ) −→ R n ⊗ m , Θ( T ) := n X i =1 e i ⊗ pr m ( e i T ) . Theorem 1 ([F03]) . A G -structure of a Riemannian manifold admits a characteristic con-nection if and only if the intrinsic torsion Γ belongs to the image of Θ . In this case thecharacteristic torsion T and the intrinsic torsion are related by the formula · Γ = − Θ( T ) . Gray-Hervella classification
Let (
M, g, J ) be a 6-dimensional almost Hermitian manifold. Then the corresponding U (3)-structure is given by the Lie algebra decomposition so (6) = u (3) ⊕ m . One candirectly compute the decomposition of R ⊗ m . The U(3)-representation R ⊗ m splits intofour irreducible representations, R ⊗ m = W ⊕ W ⊕ W ⊕ W . These are the basic classes of U (3)-structures in the Gray-Hervella classification. The man-ifolds of type W are exactly the nearly K¨ahler manifolds. On the other side, the U(3)-representation Λ ( R ) splits into three irreducible components,Λ ( R ) = W ⊕ W ⊕ W . he reader can find an explicit description of these decompositions in the paper [AFS05].Together, this allows us to describe more explicitly the U(3)-structures admitting a charac-teristic connection: Corollary 6.1 ([FI02]) . A U(3) -structure admits a characteristic connection if and only ifthe W -component of the intrinsic torsion vanishes. Let us finally summarize some results of Gray and Kirichenko on nearly K¨ahler manifoldsin dimension 6. A nearly K¨ahler manifold is said to be of constant type if there exists apositive constant α such that for all vector fields k ( ∇ gX J )( Y ) k = α [ k X k k Y k − g ( X, Y ) − g ( J X, Y ) ] . Theorem 2 ([G76]) . Let ( M, g, J ) be a 6–dimensional nearly K¨ahler manifold that is notK¨ahler. Then ii) M is of constant type, iii) g is an Einstein metric on M , iv) the first Chern class of M vanishes. Theorem 3 ([Ki77]) . The characteristic torsion of a nearly K¨ahler -manifold is parallelwith respect to the characteristic connection, ∇ c T c = 0 . Spinorial approach
There is another characterization of 6-dimensional nearly K¨ahler manifolds due to R.Grunewald, involving the existence of so called real Killing spinors ([Gru90], see also [BFGK91]).Let us first introduce basic facts and definitions. The real Clifford algebra in dimensions 6is isomorphic to End( R ). The spin representation is real, 8-dimensional and we denote itby ∆ := R . By fixing an orthonormal basis e , . . . , e of the Euclidean space R , one choicefor the real representation of the Clifford algebra on ∆ is e = + E + E − E − E , e = − E + E + E − E ,e = − E + E − E + E , e = − E − E − E − E ,e = − E − E + E + E , e = + E − E − E + E , where the matrices E ij denote the standard basis elements of the Lie algebra so (8), i. e.the endomorphisms mapping e i to e j , e j to − e i and everything else to zero. The spinrepresentation admits a Spin (6)-invariant complex structure J : ∆ → ∆ defined be theformula J := e · e · e · e · e · e . Indeed, J = − J anti-commutes with the Clifford multiplication X · φ by vectors X ∈ R and spinors φ ∈ ∆; this reflects the fact that Spin(6) is isomorphic to SU(4). Thecomplexification of ∆ splits, ∆ ⊗ R C = ∆ + ⊕ ∆ − , which is a consequence of the fact that J is a real structure making (∆ , J ) complex-(anti)-linearly isomorphic to either ∆ ± , via φ → φ ± i · J ( φ ). Furthermore, any real spinor0 = φ ∈ ∆ decomposes ∆ into three pieces,∆ = R φ ⊕ R J ( φ ) ⊕ { X · φ : X ∈ R } . n particular, J preserves the subspaces { X · φ : X ∈ R } ⊂ ∆, and the formula J φ ( X ) · φ := J ( X · φ )defines an orthogonal complex structure J φ on R that depends on the spinor φ . Moreover,the spinor determines a 3-form by means of ω φ ( X, Y, Z ) := − ( X · Y · Z · φ , φ )where the brackets indicate the inner product on ∆. The pair ( J φ , ω φ ) is an SU(3)-structureon R , and any such arises in this fashion from some real spinor. All this can be summarizedin the known fact that SU(3)-structures on R correspond one-to-one with real spinors oflength one ( mod Z ), SO(6) / SU(3) = P (∆) = RP . These formulas proves the following
Proposition 7.1.
Let M be a simply connected, -dimensional Riemannian spin manifold.Then the SU(3) -structures on M correspond to the real spinor fields of length one defined onM.
The different types of SU(3)-structures in the sense of Gray-Hervella can be characterizedby certain spinoral field equation for the defining spinor φ . The first result of this typehas been obtained by R. Grunewald in 1990. A spinor field φ defined on a Riemannianspin manifold is called a real Killing spinor if it satisfies the following first order differentialequation ∇ gX φ = λ · X · φ , λ = const ∈ R . If λ = 0, the spinor field is simply parallel. Real Killing spinors are the eigenspinors of theDirac operator realizing the lower bound of the Dirac spectrum given by Th. Friedrich in1980, see [F80]. Now we can formulate the mentioned result: Theorem 4. (see [Gru90] ) Let ( M, g ) be a -dimensional spin manifold admitting a non-trivial real Killing spinor. Then M is nearly K¨ahler. Conversely, any simply connectednearly K¨ahler -manifold admits non-trivial Killing spinor. We sketch the proof of the first statement. Suppose that φ is a Killing spinor, ∇ gX φ = X · φ .We differentiate the equation J φ ( X ) · φ = J ( X · φ ) : ∇ gY ( J φ ( X )) · φ + J φ ( X ) · ∇ gY φ = J ( ∇ gY X · φ ) + J ( X · ∇ gY φ ) = J φ ( ∇ gY X ) · φ + J ( X · ∇ gY φ ) . This formula yields the derivative ∇ g J φ :( ∇ gY J φ )( X ) · φ = J ( X · ∇ gY φ ) − J φ ( X ) · ∇ gY φ = J ( X · Y · φ ) − J φ ( X ) · ( Y · φ ) . In particular, for X = Y we obtain( ∇ gX J φ )( X ) · φ = −k X k J ( φ ) − J φ ( X ) · X · φ = −k X k J ( φ ) + X · J φ ( X ) · φ = −k X k J ( φ ) + X · J ( X · φ ) = −k X k J ( φ ) − X · X · J ( φ ) = 0 . Finally, the almost complex structure J φ is nearly K¨ahler. Remark 7.1.
The spinor field equations for all other types of SU(3)-structures have beendiscussed in the paper [ACFH15] . xample 7.1. The 6-dimensional sphere admits real Killing spinors. Indeed, fix a constantspinor in the Euclidean space R and restrict it to S . Then it becomes a real Killing spinoron the sphere. Moreover, this spinor defines its standard nearly K¨ahler structure describedbefore. 8. Non-homogeneous nearly K¨ahler manifolds
Although it had been widely believed that non-homogeneous nearly K¨ahler manifoldsshould exist, their explicit construction was an open problem for many years, in contrary totheir odd-dimensional siblings, nearly parallel G -manifolds, had been much less reluctantto provide inhomogeneous examples. On the path to a solution, several approaches hadbeen tried that provided new insights into the shape and properties of nearly K¨ahler man-ifolds, but had not brought the answer to the original problem. For example, nearly hypostructures allow the construction of compact nearly K¨ahler structures with conical singulari-ties [FIMU08], and infinitesimal deformations of nearly K¨ahler structures lead to interestingspectral problems on Laplacians [MS11]. Local homogeneous non-homogeneous examples ofnearly K¨ahler manifolds were described in [CV15].The main breakthrough was obtained very recently by Foscolo and Haskins [FH17], whichwe shall now shortly describe as it relates directly to our object of investigation, S : Theorem 5 (Foscolo, Haskins) . There exists a non-homogeneous nearly K¨ahler structure on S and on S × S . These are the first example of non-homogeneous compact nearly K¨ahler 6-manifolds.Recall that Butruille [B05] showed that the only homogeneous compact nearly K¨ahler 6-manifolds are S , S × S , CP and the flag manifold F . The examples of L. Foscolo and M.Haskins are based on weakening of the assumption of homogeneity: they are cohomogeneityone, i.e., they admit an isometric action of a compact Lie group such that generic orbits ofthe action are of codimension one. The Lie group considered in this case is SU(2) × SU(2)and the generic orbits are S × S which is motivated by results of Podesta and Spiro [PS12]characterizing all possible groups and orbits for cohomogeneity one nearly K¨ahler. In factL. Foscolo and M. Haskins state the following conjecture. Conjecture 1.
The only simply connected cohomogeneity one compact nearly K¨ahler man-ifolds in dimension 6 are the structures found in [FH17] on S and S × S .For proof of Theorem 5 they use another, equivalent (see for example [R93]) descriptionof nearly K¨ahler 6-manifolds. Proposition 8.1. A -dimensional manifold ( M, g, J ) is nearly K¨ahler if and only if thereexists a three holomorphic form ω ∈ Λ , and a constant a such that the following conditionshold d Ω = 12 a Re( ω ) , d Im( ω ) = a Ω ∧ Ω , where Ω = g ( J · , · ) is the K¨ahler form. This approach can be used to make explicit relation between nearly K¨ahler 6-manifoldsand manifolds with G holonomy which could have been suggested by the construction of thestructure on S from imaginary octonions. To see this, consider a 7-dimensional Riemanniancone C ( M ) over a smooth compact 6-manifold M and assume that the holonomy of C ( M ) s contained in G . Then, C ( M ) is equipped with a G structure, i.e., a 3-form ϕ and itsHodge dual ∗ ϕ with special properties. On the level 1 of the cone (which can be identifiedwith M ) ϕ and ∗ ϕ induce SU(3) structure ( ω, Ω) satisfying nearly K¨ahler conditions fromProposition 8.1.The main idea of the Foscolo and Haskings’s proof is to consider so-called nearly hypostructures which are the SU(2) structures induced on oriented hypersurfaces of nearly K¨ahler6-manifolds from SU(3)-structures. They describe the space of nearly hypo structures on S × S invariant under SU(2) × SU(2) action showing that it is a smooth connected 4-manifold. Away from singular orbits, cohomogeneity one nearly K¨ahler manifolds correspondto curves on this space satisfying some ODE equations. It turns out that there is a 2-parameter family of solutions of the ODE, and to finish the proof they found conditionsunder which the solutions extend to compact nearly K¨ahler 6-manifold. It is important tonote that this is closely related with desingularizations of Calabi-Yau conifold.
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Ilka AgricolaFachbereich Mathematik und InformatikPhilipps-Universit¨at MarburgHans-Meerwein-StrasseD-35032 Marburg, Germany [email protected]
Aleksandra Bor´owkaFaculty of Mathematics and Computer ScienceJagiellonian Universityul. Lojasiewicza 630-348 Krakow, Poland [email protected]
Thomas FriedrichInstitut f¨ur MathematikHumboldt-Universit¨at zu BerlinSitz: WBC AdlershofD-10099 Berlin, Germany [email protected]@mathematik.hu-berlin.de