Non-umbilical quaternionic contact hypersurfaces in hyper-Kähler manifolds
aa r X i v : . [ m a t h . DG ] S e p NON-UMBILICAL QUATERNIONIC CONTACT HYPERSURFACES INHYPER-K ¨AHLER MANIFOLDS
STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV
Abstract.
We show that any compact quaternionic contact (abbr. qc) hypersurfaces in a hyper-K¨ahlermanifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard3-Sasakian sphere. We also show that any nowhere umbilical qc hypersurface in a hyper-K¨ahler manifold isendowed with an involutive 7-dimensional distribution whose integral leafs are locally qc-conformal to thestandard 3-Sasakian sphere.
Contents
1. Introduction 12. Preliminaries 32.1. Quaternionic contact manifolds 32.2. Quaternionic contact hypersurfaces 43. The system of differential equations for the calibrating function 64. Compact qc-hypersurfaces 104.1. Proof of Theorem 1.1 105. Locally embedded qc-hypersurfaces 116. Appendix. 13References 141.
Introduction
Any real hypersurface in a complex manifold carries a natural CR structure which in the case of a strictlypositive Levi form endows the surface with a natural pseudo-Hermitian structure. The goal of this paper isto consider a hyper-K¨ahler manifold and describe the real hypersurfaces which carry a natural quaternioniccontact (qc) structure. The concept of a qc structure was originally introduced by O. Biquard [2] as amodel for the conformal boundary at infinity of the quaternionic hyperbolic space. According to a resultin [2, 4], every real analytic qc structure is the conformal infinity of a unique (asymptotically hyperbolic)quaternionic-K¨ahler metric defined in a neighborhood of the qc structure. Similar to the CR case thequestion of embedded quaternionic contact hypersurfaces is a natural one, but in contrast to the CR case itimposes a rather strong conditions on the hypersurface. The situation has the flavor of the K¨ahler versus thehyper-K¨ahler case. As well known any complex submanifold of a K¨ahler manifold is a K¨ahler manifold and aK¨ahler metric is locally given by a K¨ahler potential. In contrast, a hyper-complex manifold of a hyper-K¨ahlermanifold must be totally geodesic and (in general) there is no hyper-K¨ahler potential (the structure is rigid).This suggests that we can expect that there are few quaternionic contact hypersurfaces in a hyper-K¨ahlermanifold. Indeed, we showed in [9] that given a connected qc-hypersurface M in the flat quaternion space H n +1 , then, up to a quternionic affine transformation of H n +1 , M is contained in one of the following threehyperquadrics (the 3-Sasakain sphere, the hyperboloid and the quaternionic Heisenberg group):(1.1) ( i ) | q | + · · · + | q n | + | p | = 1 , ( ii ) | q | + · · · + | q n | −| p | = − , ( iii ) | q | + · · · + | q n | + R e ( p ) = 0 , Date : September 24, 2018.1991
Mathematics Subject Classification.
Key words and phrases. quaternionic contact, hypersurfaces, hyper-K¨ahler, quaternionic projective space, 3-Sasaki. where ( q , q , . . . q n , p ) denote the standard quaternionic coordinates of H n +1 . We recall that the above threeexamples are locally qc-conformal. Furthermore, it was shown [9] in the general hyper-K¨ahler case that theRiemannian curvature of the ambient space has to be degenerate along the normal to the qc-hypersurfacevector field.The notion of qc-hypersurface was first defined by Duchemin [5] in the general setting of quaternionicmanifold. A manifold K is called quaternionic if K is endowed with a 3-dimensional sub-bundle Q K ⊂ End ( T K ) locally generated by a pointwise quaternionic structure J , J , J together with a torsion freeconnection that preserves Q K .An embedding ι : M → K of a qc manifold M with a horizontal space H equipped with a quaternionstructure Q H , see Section 2.1 for precise definition, into a quaternionic manifold ( K, Q K ) is called a qcembedding if the differential ι ∗ intertwines Q K and Q H , i.e., if Q H = ι − ∗ Q K ι ∗ is satisfied at each point of M , where Q H denotes the point-wise quaternionic structure of the horizontaldistribution H ⊂ T M . In particular, the image ι ∗ ( H ) coincides with the maximal Q K − invariant subspaceof ι ∗ ( T M ) ⊂ T K . A real hypersurface M ⊂ K in a quaternionic manifold K is called a qc hypersurface ifthere exists a qc structure on M for which the inclusion map is a qc embedding. Notice that, if such a qcstructure exists, then it is unique, since the qc distribution H is the maximal Q K invariant subspace of T M .Duchemin [5]showed that a real analytic qc manifold can be realized as a qc-hypersurface in an appropriatequaternionic manifold.In this paper we consider qc-hypersurfaces in a hyper-K¨ahler manifold. Our main result in the case of acompact embedded qc-hypersurface is the following.
Theorem 1.1.
Let M be a compact qc-hypersurface of a hyper-K¨ahler manifold. If M is not a totallyumbilical hypersurface, then the qc-conformal class of the embedded qc structure contains a qc-Einsteinstructure of positive qc-scalar curvature which is locally qc-equivalent to the 3-Sasakian sphere. We note that the existence of a conformal factor leading to a qc-Einstein structure, called calibratedqc-structure, was established earlier by the authors, see [9, Theorem 1.2]. Thus, the main new result here isthe qc-conformal flatness of the calibrated qc-Einstein structure. In the connected simply-connected case theabove Theorem implies that the qc-conformal class of the embedded qc structure contains a qc structure qc-equivalent to the round 3-Sasakian sphere, see also Theorem 4.1. It is well known that any totally umbilicalhypersurface of a hyper-K¨ahler manifold is a qc-hypersurface whose qc structure is generated by its induced3-Sasakian metric. Furthermore, a 3-Sasakian space can be embedded as a totally umbilical qc-hypersurfacein a hyper-K¨ahler manifold, namely in its metric cone. The hyperquadric | q | + · · · + | q n | + 2 | p | = 1in H n +1 is an example of a compact qc-hypersurface which is not totally umbilical with respect to thestandard flat hyper-K¨ahler metric of H n +1 .The case of a local qc-embedding is considered in Section 5 where we prove results which in the sevendimensional case give the following theorem. Theorem 1.2.
A seven dimensional everywhere non-umbilical qc-hypersurface M embedded in a hyper-K¨ahler manifold is qc-conformal to a qc-Einstein structure which is locally qc-equivalent to the 3-Sasakiansphere, the quaternionic Heisenberg group or the hyperboloid. The proofs of the main results rely on the known and some new properties of the ”calibratng” qc-conformalfactor. More precisely, as shown in [9], given a qc-hypersurface M in a hyper-K¨ahler manifold K there isa positive function f on M called ”calibrating” function so that the qc structure on M obtained from theembedded one with f as a qc-conformal factor is qc-Einstein, see [9, Lemma 3.7]. Furthermore, if II is thesecond fundamental form of M , then the (0,2) tensor f II extends to a covariant constant along M , see [9,Theorem 3.1]. The new key points for the results of the current paper are certain identities for the secondand third order (horizontal) covariant derivative of the calibrating function f . Using the bracket generatingcondition and the relation between the Biquard and Levi-Civita connections these identities lead to a third ON-UMBILICAL QUATERNIONIC CONTACT HYPERSURFACES IN HYPER-K¨AHLER MANIFOLDS 3 order differential system on M well studied in the Riemannian case by several authors, see [17, 6, 18, 16].In the compact case, this system is known to have the remarkable property that it admits a non-constantsolution only on Riemannian manifolds which are locally isometric to the round sphere. Convention 1.3.
Throughout the paper, unless explicitly stated otherwise, we will use the following notation.a) All manifolds are assumed to be C ∞ and connected.b) The triple ( i, j, k ) denotes any positive permutation of (1 , , .c) s, t are any numbers from the set { , , } , s, t ∈ { , , } .d) For a given decomposition T M = V ⊕ H we denote by [ . ] V and [ . ] H the corresponding projections to V and H .e) A, B, C , etc. will denote sections of the tangent bundle of M , A, B, C ∈ T M .f )
X, Y, Z, U will denote horizontal vector fields,
X, Y, Z, U ∈ H . Acknowledgments.
S.I. and I.M. are partially supported by Contract DFNI I02/4/12.12.2014 andContract 195/2016 with the Sofia University ”St.Kl.Ohridski”. I.M. is supported by a SoMoPro II Fellowshipwhich is co-funded by the European Commission from “People” specific programme (Marie Curie Actions)within the EU Seventh Framework Programme on the basis of the grant agreement REA No. 291782. Itis further co-financed by the South-Moravian Region. DV was partially supported by Simons Foundationgrant Preliminaries
Quaternionic contact manifolds.
Here, we recall briefly the relevant facts and notation needed forthis paper and refer to [2], [7] and [13] for a more detailed exposition. A quaternionic contact (qc) manifoldis a (4 n + 3)-dimensional manifold M with a codimension three distribution H equipped with an Sp ( n ) Sp (1)structure locally defined by an R -valued 1-form η = ( η , η , η ). Thus, H = ∩ s =1 Ker η s carries a positivedefinite symmetric tensor g , called the horizontal metric, and a compatible rank-three bundle Q H consistingof endomorphisms of H locally generated by three orthogonal almost complex structures I s , satisfying theunit quaternion relations: (i) I I = − I I = I , I I I = − id | H ; (ii) g ( I s ., I s . ) = g ( ., . ); and (iii) thecompatibility conditions 2 g ( I s X, Y ) = dη s ( X, Y ), X, Y ∈ H hold true. Unlike the CR case, in the qc casethe horizontal space determines uniquely the qc-conformal class, cf. [9]. For this reason very often we willidentify the qc structure with the R -valued 1-form η while supressing the remaining data. We also notethat by virtue of its definition a quaternionic contact manifold is orientable.Two qc structures η and ¯ η on a manifold M are called qc-conformal to each other if ¯ η = µ Ψ η for apositive smooth function µ and an SO (3) matrix Ψ with smooth functions as entries. A diffeomorphism F between two qc manifolds M and ¯ M is called quaternionic contact conformal (qc-conformal) transformation if F ∗ ¯ η = µ Ψ η . The qc-conformal curvature tensor W qc , introduced in [11], is the obstruction for a qcstructure to be locally qc-conformally to the standard 3-Sasakian structure on the (4 n +3)-dimensional sphere[11, 13]. As already noted in the introduction the 3-Sasakain sphere, the hyperboloid and the quaternionicHeisenberg group are all locally qc-conformal to each other.As shown in [2], there is a ”canonical” connection associated to every qc manifold of dimension at leasteleven. In the seven dimensional case the existence of such a connection requires the qc structure to beintegrable [4]. The integrability condition is equivalent to the existence of Reeb vector fields [4], which(locally) generate the supplementary to H distribution V . The Reeb vector fields { ξ , ξ , ξ } are determinedby [2](2.1) η s ( ξ t ) = δ st , ( ξ s y dη s ) | H = 0 , ( ξ s y dη t ) | H = − ( ξ t y dη s ) | H , where y denotes the interior multiplication. Henceforth, by a qc structure in dimension 7 we shall meana qc structure satisfying (2.1) and refer to the ”canonical” connection as the Biquard connection . TheBiquard connection is the unique linear connection preserving the decomposition T M = H ⊕ V and the This article reflects only the author’s views and the EU is not liable for any use that may be made of the informationcontained therein.
STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV Sp ( n ) Sp (1) structure on H with torsion T determined by T ( X, Y ) = − [ X, Y ] | V while the endomorphisms T ( ξ s , . ) : H → H belong to the orthogonal complement ( sp ( n ) + sp (1)) ⊥ ⊂ GL (4 n, R ).The covariant derivatives with respect to the Biquard connection of the endomorphisms I s and the Reebvector fields are given by(2.2) ∇ I i = − α j ⊗ I k + α k ⊗ I j , ∇ ξ i = − α j ⊗ ξ k + α k ⊗ ξ j . The sp (1)-connection 1-forms α , α , α , defined by the above equations satisfy [2] α i ( X ) = dη k ( ξ j , X ) = − dη j ( ξ k , X ) , X ∈ H. Let R = [ ∇ , ∇ ] −∇ [ .,. ] be the curvature tensor of ∇ and R ( A, B, C , D ) = g ( R A,B C , D ) be the correspondingcurvature tensor of type (0,4). The qc Ricci tensor Ric , the qc-Ricci forms ρ s and the normalized qc scalarcurvature S are defined by Ric ( A, B ) = n X a =1 R ( e a , A, B, e a ) , nρ s ( A, B ) = n X a =1 R ( A, B, e a , I s e a ) , n ( n +2) S = Scal = n X a =1 Ric ( e a , e a ) , where e , . . . , e n of H is an g -orthonormal frame on H .We say that ( M, η ) is a qc-Einstein manifold if the restriction of the qc-Ricci tensor to the horizontalspace H is trace-free, i.e., Ric ( X, Y ) =
Scal n g ( X, Y ) = 2( n + 2) Sg ( X, Y ) , X, Y ∈ H. The qc-Einstein condition is equivalent to the vanishing of the torsion endomorphism of the Biquard con-nection, T ( ξ s , X ) = 0 [7]. It is also known [7, 8] that the qc-scalar curvature of a qc Einstein manifold isconstant and the vertical distribution is integrable.By [8, Theorem 5.1], see also [12] and [13, Theorem 4.4.4] for alternative proofs in the case Scal = 0, aqc-Einstein structure is characterised by either of the following equivalent conditionsi) locally, the given qc structure is defined by 1-form ( η , η , η ) such that for some constant S , we have(2.3) dη i = 2 ω i + Sη j ∧ η k ;ii) locally, the given qc structure is defined by a 1-form ( η , η , η ) such that the corresponding connection1-forms vanish on H and (cf. the proof of Lemma 4.18 of [7])(2.4) α s = − Sη s . The correspondin (pseudo) Riemannian geometry.
Let M be a qc-Einstein manifold. Note that, byapplying an appropriate qc homothetic transformation, we can aways reduce a general qc-Einstein structureto one whose normalized qc-scalar curvature S equals 0 , −
2. Consider the one-parameter family of(pseudo) Riemannian metrics h µ , µ = 0 on M by letting(2.5) h µ = g | H + µ ( η + η + η ) . Let ∇ µ be the Levi-Civita connection of h µ . Note that h µ is a positive-definite metric when µ > n,
3) when µ <
0. The difference L = ∇ µ − ∇ between the Levi-Cevita connection ∇ µ and theBiquard connection ∇ is given by [7, 8](2.6) L ( A, B ) ≡ ∇ µA B − ∇ A B = S A ] V × [ B ] V + X s =1 n − ω s ( A, B ) ξ s + µη s ( A ) I s B + µη s ( B ) I s A o , where . V × . V is the standard vector cross product on the 3-dimensional vertical space V .2.2. Quaternionic contact hypersurfaces.
In this section we summarize some results from [9] which arethe starting point of the subject of the current paper. For ease of reading we follow [9] closely.
ON-UMBILICAL QUATERNIONIC CONTACT HYPERSURFACES IN HYPER-K¨AHLER MANIFOLDS 5 qc-hypersurfaces.
Let K be a hyper-K¨ahler manifold with hyper-complex structure ( J , J , J ), quater-nionic bundle Q K , and hyper-K¨ahler metric G . In particular, the Levi-Civita connection D is a torsion freeconnection on K preserving Q K .For a real hypersurface M ⊂ K the maximal Q K -invariant subspace T M is denoted by H and refereed toas the horizontal distribtution. If ι : M → K is the natural inclusion map, then M is a qc-hypersurface if itis a qc manifold with respect to the induced quaternionic structure ι − ∗ ( Q K ) ι ∗ on H . In order to simplify thenotation we shell identify the corresponding points and tensor fields on M with their images through ι in K .An equivalent characterization of a qc-hypersurface M is that the restriction of the second fundamental formof M to the horizontal space H is a definite symmetric form, which is invariant with respect to the inducedquaternion structure, see [5, Proposition 2.1]. After choosing the unit normal vector N to M appropriately,we will assume that the second fundamental form of M , II ( A, B ) = − G ( D A N, B ) , A, B ∈ T M, is negative definite on the horizontal space H . The defining tensors of the embedded qc structure on M aregiven by(2.7) ˆ η s ( A ) = G ( J s N, A ) , ˆ ξ s = J s N + ˆ r s , ˆ ω s ( X, Y ) = − II ( I s X, Y ) , ˆ g ( X, Y ) = − ˆ ω s ( I s X, Y ) , where I s = J s | H and ˆ ξ s , are the Reeb vector fields corresponding to ˆ η s , see [9, Section 2.2].2.2.2. The calibrating function.
Let ˆ ω s be the fundamental 2-forms corresponding to ˆ η s , given by2ˆ ω s ( X, Y ) = d ˆ η s ( X, Y ) , X, Y ∈ H and ˆ ξ t y ˆ ω s = 0 , s, t = 1 , , . Following [9, Section 3.1], consider thecomplex 2-forms on M ,ˆ γ i = ˆ ω j + √− ω k , Γ i ( A, B ) = G ( J j A, B ) + √− G ( J k A, B ) . Using a type decomposition argument it was shown in [9, Section 3.1] that(2.8) Γ ns ≡ µ s ˆ γ ns mod { ˆ η , ˆ η , ˆ η } , for s = 1 , , µ s and, in fact, µ = µ = µ = µ for a positive (realvalued) function µ on M . The calibrating function of M was defined by f = µ n +2 . The calibrated qc structure.
The qc structure( η , η , η ) def = f (ˆ η , ˆ η , ˆ η )is called calibrated . As shown in [9], it satisfies the structure equations (2.3). In particular, it is a qc-Einsteinstructure. Moreover, by [9, Lemma 3.9] the horizontal metric g of the calibrated qc structure is related tothe second fundamental form of the qc-embedding by the formula(2.9) g ( A ′′ , B ′′ ) = − f II ( A, B ) − S X s =1 η s ( A ) η s ( B ) , A, B ∈ T M, where for A ∈ T M we let A ′′ = A − P s =1 η s ( A ) ξ s be the horizontal part of A . The corresponding Reebvector fields ξ s are given by(2.10) ξ s = J s (cid:16) f − N + r (cid:17) , where r ∈ H is determined by II ( r, X ) = f − df ( X ) , X ∈ H . In fact, we have [9, Lemma 3.8] r = − f − ∇ f, (2.11) df ( ξ s ) = 0 , s = 1 , , , (2.12)where ∇ f ∈ H denotes the horizontal gradient of f , df ( X ) = g ( ∇ f, X ).The calibrated transversal to M vector field is defined by(2.13) ξ = f − N + r. STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV
From (2.10) and (2.13) we have(2.14) ξ s = J s ξ. With the obvious identifications, the bundle
T K | M → M decomposes as a direct sum,(2.15) T K | M = H ⊕ V ⊕ R ξ, where V is the span of the Reeb vector fields ξ s of the calibrated qc structure on M . For v ∈ T p K we define(2.16) v ′ = v − λ ( v ) ξ ( p ) ∈ T p M = H p ⊕ V p , v ′′ = πv = v ′ − X s =1 η s ( v ′ ) ξ s ∈ H p , where λ = f G ( N, . ) so that v ′ is the projection of v on T p M = H p ⊕ V p parallel to the calibrated transversalfield ξ and π : T K | M → H is the projection on the horizontal space using the decomposition (2.15). Thus,for v ∈ T K | M we have(2.17) λ ( J s v ) = η s ( v ′ )and the decomposition(2.18) v = πv + X s =1 η s ( v ′ ) ξ s + λ ( v ) ξ ∈ H ⊕ V ⊕ R ξ. Following [9, (3.23)] consider the symmetric bilinear form W ∈ T ∗ K | M ⊗ T ∗ K | M , (2.19) W ( v, w ) def = − f II ( v ′ , w ′ ) + S λ ( v ) λ ( w ) = g ( πv, πw ) + S X s =1 η s ( v ′ ) η s ( w ′ ) + S λ ( v ) λ ( w ) . Clearly, W ( J s ., .J s . ) = W ( ., . ), s = 1 , , , and W as the unique J s -invariant extension of the symmetricbilinear form − f II on T M to a symmetric bilinear form on
T K | M . A very important property of thecalibrated qc structure is that W is constant along M with respect to the Levi-Civita connection D of thehyper-K¨ahler metric G , see [9, Theorem 3.1]), i.e., we have(2.20) D A W = 0 , A ∈ T M.
Finally, we record an important relation between the calibrating function and the parallel bilinear form,see [9, (2.16)](2.21) W ( N, A ) = − f II ( N ′ , A ) = f II ( r, A ) = − f g ( r, A ′′ ) = df ( A ′′ ) = df ( A ) . The system of differential equations for the calibrating function
We begin with a lemma relating the Levi-Civita connection D of the hyper-K¨ahler metric G to the Biquardconnection ∇ of the calibrated qc structure on M . Lemma 3.1.
For any A ∈ T M and X ∈ H we have:i) D A X = ∇ A X + P t =1 (cid:16) ( S/ η t ( A ) I t X − ω t ( πA, X ) ξ t (cid:17) − g ( πA, X ) ξ .ii) D A ξ = ( S/ A and D A ξ s = ( S/ J s A .Proof. First we shall prove the formula in part i) for a horizontal vector field A ,(3.1) D X Y = ∇ X Y − ω s ( X, Y ) ξ s − g ( X, Y ) ξ. We start with the computation of the horizontal part of D X Y ,(3.2) ∇ X Y = π ( D X Y ) , X, Y ∈ H, recalling that π is the projection on the horizontal space, see (2.18). From (2.19) and (2.20) we have0 = ( D X W )( Y, Z ) = X (cid:16) W ( Y, Z ) (cid:17) − W ( D X Y, Z ) − W ( Y, D X Z )= X (cid:16) g ( Y, Z ) (cid:17) − g (cid:16) π ( D X Y ) , Z (cid:17) − g (cid:16) Y, π ( D X Z ) (cid:17) . ON-UMBILICAL QUATERNIONIC CONTACT HYPERSURFACES IN HYPER-K¨AHLER MANIFOLDS 7
Letting F ( X, Y ) def = ∇ X Y − π ( D X Y ) , we compute0 = ( ∇ X g )( Y, Z ) = X (cid:16) g ( Y, Z ) (cid:17) − g (cid:16) π ( D X Y ) + F ( X, Y ) , Z (cid:17) − g (cid:16) Y, π ( D X Z ) + F ( X, Z ) (cid:17) = − g (cid:16) F ( X, Y ) , Z (cid:17) − g (cid:16) F ( X, Z ) , Y (cid:17) , while on the other hand0 = g (cid:16) π ( T ( X, Y )) , Z (cid:17) = g (cid:16) ∇ X Y − ∇ Y X − π ([ X, Y ]) , Z (cid:17) = g (cid:16) ∇ X Y − ∇ Y X − π ( D X Y − D Y X ) , Z (cid:17) = g (cid:16) F ( X, Y ) , Z (cid:17) − g (cid:16) F ( Y, X ) , Z (cid:17) . Thus, the tensor g (cid:16) F ( X, Y ) , Z (cid:17) is both symmetric in X, Y and skew-symmetric in
Y, Z which implies thatit vanishes.The remaining part of D X Y in the decomposition based on (2.18) can be computed easily as follows, λ ( D X Y ) = − f G ( D X N, Y ) = f II ( X, Y ) = − g ( X, Y )and η s (( D X Y ) ′ ) = − λ ( J s D X Y ) = − λ ( D X ( J s Y )) = g ( X, J s Y ) = − ω s ( X, Y ) . From the above the formula in part i) in the case when A is a horizontal vector field follows.Next we prove the formula(3.3) D X N = ∇ X ∇ f + Sf X − df ( J s X ) ξ s . In order to determine the horizontal part of D X N we recall (2.20) and then compute the (horizontal) Hessianof f as follows ∇ f ( X, Y ) = X ( df ( Y )) − df ( ∇ X Y ) = X ( W ( N, Y )) − df ( ∇ X Y )= W ( D X N, Y ) + W ( N, D X Y ) − df ( ∇ X Y ) = W ( D X N, Y ) + W ( N, D X Y − ∇ X Y )using (2.21) in the last equality. From (2.19) and (3.1) it follows ∇ f ( X, Y ) = g ( π ( D X N ) , Y ) − f Sg ( X, Y )noting that W ( N, ξ ) = f S . The vertical part of D X N is computed with the the help of (2.9) and (2.13) η s ( D X N ) = − f G (cid:16) N, J s ( D X N ) , (cid:17) = f G (cid:16) D X N, J s N (cid:17) = − f II ( X, J s N ) = − df ( J s X ) . The proof of formula (3.3) is complete.An immediate consequence of (2.13), (2.11), (3.1) and (3.3) is the following formula(3.4) D X ξ (2.10) = 12 SX.
At this point we can complete the proof of part i) . Since the calibrated qc structure is qc-Einstein and the1-forms η s satisfy the structure equations (2.3), we have ∇ ξ s X = [ ξ s , X ]. Therefore,(3.5) ∇ ξ s X = [ ξ, X ] = D ξ s X − D X ξ s = D ξ s X − J s ( D X ξ ) (3.4) = D ξ s X − S J s X. Finally, we compute D A X = D πA X + η s ( A ) D ξ s X (3.1) , (3.5) = ∇ πA X − ω s ( πA, X ) ξ s − g ( πA, X ) ξ + η s ( A ) (cid:16) ∇ ξ s X + S J s X (cid:17) = ∇ A X − ω s ( πA, X ) ξ s − g ( πA, X ) ξ + S η s ( A ) J s X. Turning to the proof of ii) , we have from (2.9) and (2.17) the formula G ( D ξ s N, A ) = − II ( ξ s , A ) = 12 f Sη s ( A ) = S G ( J s N, A ) , STEFAN IVANOV, IVAN MINCHEV, AND DIMITER VASSILEV hence(3.6) D ξ s N = 12 SJ s N = 12 Sξ s − SJ s ∇ r. From (2.12) and T ( ξ s , X ) = 0 it follows ∇ ξ s ∇ f = 0, hence (2.13), (2.11), (3.6) and (2.10) give D ξ s ξ = S ξ s , which together with (3.4) completes the proof of part ii ) after recalling (2.14). (cid:3) Corollary 3.2. M is a totally umbilical qc-hypersurface of a hyper-K¨ahler manifold iff the calibrating func-tion is locally constant.Proof. In view of (3.6) and (2.11) it follows the horizontal gradient of f vanishes ∇ f = 0, hence f is locallyconstant taking into account that the horizontal space is bracket generating. (cid:3) As customary, let W : M → End ( T K ) | M also denote the (1,1) tensor corresponding to the symmetricbilinear form W , i.e., G ( W u, v ) = W ( u, v ) for all u, v ∈ T K | M . Then W J s = J s W and, since both G and W are D -parallel along M , we also have(3.7) ( D A W )( u ) = 0 , A ∈ T M, u ∈ T K | M . An almost immediate corollary of the proof of Lemma 3.1 is the following formula for W in terms of thecalibrating function. Lemma 3.3.
For X ∈ H we have:i) W X = f ∇ X ∇ f + ( Sf / X + df ( X ) ∇ f − f P s =1 df ( I s X ) ξ s + f df ( X ) ξ ; ii) W ξ = ( Sf / ∇ f + ( Sf / ξ ; iii) W ξ s = ( Sf / I s ∇ f + ( Sf / ξ s , s = 1 , , .Proof. By definition (2.19), recall also (2.16), we have W ( X, u ) = − f II ( X, u ′ ) = f G ( D X N, u ′ ) = f G ( D X N, u ) − f G ( D X N, ξ ) λ ( u )= f G ( D X N, u ′ ) = f G ( D X N, u ) − f G ( D X N, ξ ) G ( N, u ) . Now, the formula of part i) follows by a direct substitution using (2.13), (2.11) and (3.3). Finally, part iii)follows from J s ξ = ξ s , see after equation (2.13).Part ii) is proved similarly with the help of (3.6) instead of (3.3). (cid:3) After the preceding technical lemmas we turn to the key result which gives a system of partial differentialequations for the calibrating function. With the help of (2.6) is then expressed in terms of Levi-Civitaconnection in the subsequent lemma.
Lemma 3.4.
The function φ def = f satisfies the following equations dφ ( ξ s ) = 0;(3.8) ∇ φ ( X, Y ) = ∇ φ ( I s X, I s Y );(3.9) ∇ φ ( X, Y, Z ) +
Sdφ ( X ) g ( Y, Z ) + S dφ ( Y ) g ( Z, X ) + S dφ ( Z ) g ( X, Y )(3.10) = S X s =1 h dφ ( I s Y ) ω s ( X, Z ) + dφ ( I s Z ) ω s ( X, Y ) i . Proof.
Since dφ = f df and ∇ φ = f ∇ f + df ⊗ df, (2.21) gives (3.8). Recalling the decomposition (2.18),see also (2.16), by Lemma 3.3 we have(3.11) g (cid:16) ( W X ) ′′ , Y (cid:17) = ∇ φ ( X, Y ) +
Sφg ( X, Y ) . From W J s = J s W and g ( I s X, I s Y ) = g ( X, Y ) for s = 1 , ,
3, (3.9) follows from0 = g (cid:16) ( W X ) ′′ , Y (cid:17) − g (cid:16) ( W I s X ) ′′ , I s Y (cid:17) = ∇ φ ( X, Y ) − ∇ φ ( I s X, I s Y ) . ON-UMBILICAL QUATERNIONIC CONTACT HYPERSURFACES IN HYPER-K¨AHLER MANIFOLDS 9
We turn to the proof of (3.10). A differentiation of (3.11) gives(3.12) ∇ φ ( X, Y, Z ) +
Sdφ ( X ) g ( Y, Z ) = g (cid:0) ∇ X ( W Y ) ′′ , Z (cid:1) − g ( π W ∇ X Y, Z ) . Taking into account (3.2), (2.18) and (3.7) we can rewrite the first term in the right-hand side of the aboveidentity as follows g (cid:0) ∇ X ( W Y ) ′′ , Z (cid:1) = g (cid:0) πD X ( W Y ) ′′ , Z (cid:1) = g ( πW D X Y, Z ) − η s (cid:0) ( W Y ) ′ (cid:1) g ( πD X ξ s , Z ) − λ ( W Y ) g ( πD X ξ, Z ) . Now we use Lemma 3.3 to compute η s (cid:0) ( W Y ) ′ (cid:1) = − f df ( I s Y ) and λ ( W Y ) = f df ( Y ) . A substitution of the last two equations in (3.12) gives ∇ φ ( X, Y, Z ) +
Sdφ ( X ) g ( Y, Z ) = g ( π W ( D X Y − ∇ X Y ) , Z ) + Sf ω s ( X, Z ) df ( I s Y ) − Sf g ( X, Z ) df ( Y )= S (cid:16) ω s ( X, Y ) dφ ( I s Z ) − g ( X, Y ) dφ ( Z ) + ω s ( X, Z ) dφ ( I s Y ) − g ( X, Z ) dφ ( Y ) (cid:17) using Lemma 3.1 and Lemma 3.3 in the last equality. The proof of Lemma 3.4 is complete. (cid:3) We continue with our main technical result, which allows the partial reduction to a Riemannian geometryproblem.
Proposition 3.5.
Let ( M, η, Q ) be a (4n+3)-dimensional qc-Einstein space with constant qc-scalar curvature S = 0 and φ be a smooth function which satisfies identities (3.8) , (3.9) and (3.10) . With respect to theLevi-Civita connection ∇ S of the (pseudo) Riemannian metric given by (2.5) for µ = S , the function φ satisfies the following identity (3.13) ( ∇ S ) φ ( A, B, C )+ Sdφ ( A ) h S ( B, C )+ S dφ ( B ) h S ( C, A )+ S dφ ( C ) h S ( A, B ) = 0 , A, B, C ∈ Γ( T M ) . Proof.
From (3.8), the properties of the Biquard connection, the Ricci identities, the vanishing of the torsionof the Biquard connection and the integrability of the vertical space we have the following equalities0 = ∇ φ ( X, ξ s ) = ∇ φ ( ξ s , X ) = ∇ φ ( ξ s , ξ t ) , (3.14) ∇ φ ( X, Y ) − ∇ φ ( Y, X ) = 2 X s =1 ω s ( X, Y ) dφ ( ξ s ) = 0 . (3.15)Next, using the equality (2.6) together with the Ricci identities for the Levi-Civita connection, (3.14) givesthe identities ( ∇ S ) φ ( Y, X ) = ( ∇ S ) φ ( X, Y ) = ∇ φ ( X, Y ) − dφ ( L ( X, Y )) = ∇ φ ( X, Y ) . (3.16) ( ∇ S ) φ ( X, ξ s ) = ( ∇ S ) φ ( ξ s , X ) = ∇ φ ( ξ s , X ) − dφ ( L ( ξ s , X ) = − Sdφ ( I s X );(3.17) ( ∇ S ) φ ( ξ s , ξ t ) = ( ∇ S ) φ ( ξ t , ξ s ) = ∇ φ ( ξ s , ξ t ) − dφ ( L ( ξ s , ξ t ) = 0 . (3.18)Now we turn to the computation of the third derivative. Using (3.16) and (3.17) we obtain the identities(3.19) ( ∇ S ) φ ( X, Y, Z ) = ∇ φ ( X, Y, Z ) − ( ∇ S ) φ ( L ( X, Y ) , Z ) − ( ∇ S ) φ ( Y, L ( X, Z ))= ∇ φ ( X, Y, Z ) + X s =1 h ω s ( X, Y )( ∇ S ) φ ( ξ s , Z ) + ω s ( X, Z )( ∇ S ) φ ( Y, ξ s ) i = ∇ φ ( X, Y, Z ) − S X s =1 h ω s ( X, Y ) dφ ( I s Z ) + ω s ( X, Z ) dφ ( I s Y ) i = Sdf ( X ) g ( Y, Z ) + S df ( Y ) g ( Z, X ) + S df ( Z ) g ( X, Y ) , where we used (3.10) in the last equality. Proceeding in the same fashion, we obtain(3.20) ( ∇ S ) φ ( ξ s , Y, Z ) = ∇ φ ( ξ s , Y, Z ) − ( ∇ S ) φ ( L ( ξ s , Y ) , Z ) − S ∇ S ) φ ( Y, L ( ξ s , Z ))= 0 − S ∇ S ) φ ( I s Y, Z ) − ( ∇ S ) φ ( Y, I s Z ) = − S ∇ φ ( I s Y, Z ) − S ∇ φ ( Y, I s Z ) = 0 , where we used (3.9) in the last equality. A similar computation shows(3.21) ( ∇ S ) φ ( Y, Z, ξ s ) = ( ∇ S ) φ ( Y, ξ s , Z )= ∇ ( ∇ S ) ( Y, ξ, Z ) − ( ∇ S ) φ ( L ( Y, ξ s ) , Z ) − S ( ∇ S ) φ ( ξ, L ( Y, Z ))= − S ∇ φ ( Y, I s Z ) − S ∇ S ) φ ( I s Y, Z ) = − S ∇ φ ( I s Y, Z ) − S ∇ φ ( Y, I s Z ) = 0 , where we used (3.9) in the last equality, and also( ∇ S ) φ ( Y, ξ s , ξ s ) = ∇ φ ( Y, ξ s , ξ s ) − ∇ S ) φ ( L ( Y, ξ s ) ξ s ) = − ∇ φ ( I s Y, ξ s ) = − Sdφ ( Y );(3.22) ( ∇ S ) φ ( Y, ξ s , ξ t ) = ∇ φ ( Y, ξ s , ξ t ) − ( ∇ S ) φ ( L ( Y, ξ s ) , ξ t ) − ( ∇ S ) φ ( ξ s , L ( Y, ξ t )) = 0 . (3.23)Finally, we calculate( ∇ S ) φ ( ξ s , ξ s , Y ) = ∇ φ ( ξ s , ξ s , Y ) − ( ∇ S ) φ (( L ( ξ s , ξ s ) , Y ) − ( ∇ S ) φ ( ξ s , L ( ξ s , Y )) = − S dφ ( Y );(3.24) ( ∇ S ) φ ( ξ s , ξ t , Y ) = ∇ φ ( ξ s , ξ t , Y ) − ( ∇ S ) φ ( L ( ξ s , ξ t ) , Y ) − ( ∇ S ) φ ( ξ s , L ( ξ t , Y ))(3.25) = − S dφ ( I s I t Y ) − S dφ ( I t I s Y ) = 0 . (3.26)Equations (3.16)-(3.25) show the validity of (3.13) for all A, B, C ∈ Γ( T M ). This completes the proof of theProposition. (cid:3) Compact qc-hypersurfaces
Proof of Theorem 1.1.
Proof.
We begin by showing that if a function φ satisfies (3.10), then h def = △ φ is necessarily an eigenfunctionfor the sub-Laplacian △ h = tr g ( ∇ h ). Indeed, see [18, (2.7)] for the analogous calculation in the Riemanniancase, taking a trace in (3.10) we obtain that X ( △ φ ) = − n + 1) Sdφ ( X ) which yields ∇ △ φ ( X, Y ) = − n + 1) S ∇ φ ( X, Y ) and △ h = − n + 1) Sh.
Since M is compact it follows S ≥ S = 0, then it follows φ = const , which contradicts our assumptionthat M is non-umbilic, see Corollary 3.2. Thus, we have S >
0. In fact, after a qc-homothety, we canassume that S = 2. Let h def = h S be the corresponding Riemannian metric on M . Now, in view of (3.13), byGallot-Obata-Tanno’s theorem [18, 6, 17] it follows that the Riemannian manifold ( M, h ) is isometric to theround sphere of radius 1. Therefore, the curvature tensor R h of the Levi-Civita connection ∇ h of h is givenby(4.1) R h ( A, B, C, D ) = h ( B, C ) h ( A, B ) − h ( B, D ) h ( A, C ) . The relation between the curvatures of the Levi-Civita connection and the Biquard connection for qc-Einsteinspaces with S = 2 (i.e., 3-Sasakian spaces) [7, Corollary 4.13] or [13, Theorem 4.4.3] together with (4.1) yields(4.2) R ( X, Y, Z, W ) = R h ( X, Y, Z, W )+ X s =1 h ω s ( Y, Z ) ω s ( X, W ) − ω s ( X, Z ) ω s ( Y, W ) − ω s ( X, Y ) ω s ( Z, W ) i = h ( Y, Z ) h ( X, W ) − h ( Y, W ) h ( X, Z )+ X s =1 h ω s ( Y, Z ) ω s ( X, W ) − ω s ( X, Z ) ω s ( Y, W ) − ω s ( X, Y ) ω s ( Z, W ) i . ON-UMBILICAL QUATERNIONIC CONTACT HYPERSURFACES IN HYPER-K¨AHLER MANIFOLDS 11
According to [11, Proposition 4.2], the qc conformal curvature tensor W qc can by expressed in terms ofthe curvature R of the Biquard connection, in general, on a qc-Einstein spaces with qc scalar curvature S by the formula(4.3) W qc ( X, Y, Z, W ) = R ( X, Y, Z, W ) + S n − g ( X, W ) g ( Y, Z ) + g ( X, Z ) g ( Y, W )+ X s =1 h − ω s ( X, W ) ω s ( Y, Z ) + ω s ( X, Z ) ω s ( Y, W ) + 2 ω s ( X, Y ) ω s ( Z, W ) io . Then, since in our case g ( X, Y ) = h ( X, Y ) and S = 2, (4.2) implies that W qc = 0 and therefore, ( M, η ) isqc-conformally flat (cf. [11, Theorem 1.3]). Now, the result follows by Theorem 6.1.Let us remark that the final step of the proof is similar to an argumentation that had been already usedbefore in the proof of [10, Theorem 1.3]. (cid:3)
In the case of a positive qc-scalar curvature of the calibrated qc structure we can substitute the compact-ness with completeness assumption of the Riemannian metric noting that the Gallot-Obata-Tanno’s theoremholds for a complete Riemannian manifold. In particular, the manifold is compact. In addition, the localqc-conformal maps considered in the proof of Theorem 1.1 define a global qc-conformality to the roundsphere, see Theorem 6.1. Therefore, we have
Theorem 4.1.
Let M be a simply connected qc hypersurface of a hyper-K¨ahler manifold which is not totallyumbilical. Suppose that the calibrated qc structure ( η , η , η ) on M has a positive qc-scalar curvature andthat it is complete with respect to the natural Riemannian metric h = g + η + η + η . Then the calibratedqc structure on M is qc-homothetic to the standard 3-Sasakian sphere. Locally embedded qc-hypersurfaces
In the non-compact case we show
Theorem 5.1.
Let M be a qc-hypersurface in a hyper-K¨ahler manifold such that all points of M are non-umbilic. Then there exists a 7 dimensional involutive distribution D on M such that the induced qc structureon each integral leaf of D is locally qc-conformal to the standard 7-dimensional 3-Sasakian sphere.Proof. We achieve Theorem 5.1 with a series of lemmas. We begin with the following
Lemma 5.2.
Let M be a qc Einstein space with local qc 1-forms η , η , η satisfying the structure equations (2.3) and let ξ , ξ , ξ be the corresponding Reeb vector fields. If there exists a function φ with a nowherevanishing horizontal gradient ∇ φ on M , satisfying (3.10) - (3.8) , then the 7-dimensional distribution D = span { ξ , ξ , ξ , ∇ φ, I ∇ φ, I ∇ φ, I ∇ φ } is integrable.Proof. Since η , η , η satisfy (2.3), the vertical distribution spand { ξ , ξ , ξ } is integrable and we have ∇ X ξ s = 0. Moreover,(5.1) [ ∇ φ, I i ∇ φ ] = ∇ ∇ φ ( I i ∇ φ ) − ∇ I i ∇ φ ( ∇ φ ) − T ( ∇ φ, I i ∇ φ )= − I i ∇ ∇ φ ( ∇ φ ) − ∇ I i ∇ φ ( ∇ φ ) − X t =1 ω t ( ∇ φ, I i ∇ φ ) ξ t (3.9) = − g ( ∇ φ, ∇ φ ) ξ i . We have also that T ( ξ s , X ) = 0, which leads to[ ξ s , ∇ φ ] = ∇ ξ s ∇ φ − ∇ ∇ φ ξ s − T ( ξ s , ∇ φ ) = ∇ φ ( ξ s , e a ) e a − ∇ ∇ φ ξ s (3.8) = ∇ ∇ φ ξ s ⊂ D. Similarly, [ ξ s , I t ∇ φ ] ⊂ D and thus the integrability of the distribution D is proved. (cid:3) We need the following
Lemma 5.3.
The qc-conformal curvature of a qc-Einstein space has the property W qc ( X, Y, Z, U ) = W qc ( Z, U, X, Y ) = W qc ( X, Y, I s Z, I s U ) = W qc ( I s X, I s Y, Z, U ) . Proof.
The first equality in the lemma is already known, see e.g. [8]. The second equality follows after asmall calculation using formula (4.3) combined with(5.2) ρ s = − Sω s , R ( X, Y, Z, W ) = R ( Z, W, X, Y )(cf. [8, (3.28)] and [11, Theorem 3.1]). (cid:3)
We proceed with
Lemma 5.4.
Let M be a 7-dimensional qc Einstein space with local qc 1-forms η , η , η , satisfying thestructure equations (2.3) , corresponding Reeb vector fields ξ , ξ , ξ and Biquard connection ∇ . If thereexists a function φ on M satisfying at each point: (i) ∇ φ = 0 , (ii) dφ ( ξ ) = dφ ( ξ ) = dφ ( ξ ) = 0 and (iii) ∇ φ ( X, Y ) = hg ( X, Y ) , for a smooth function h on M , then M is locally qc-conformally flat.Proof. Since we assume that the qc 1-forms η s satisfy (2.3), we have ∇ X ξ s = 0 and thus(5.3) ∇ φ ( ξ s , X ) = ∇ φ ( X, ξ s ) = X (cid:16) dφ ( ξ s ) (cid:17) = 0 . By differentiating (iii) we get(5.4) ∇ φ ( X, Y, Z ) = dh ( X ) g ( Y, Z ) . The Ricci identity for the Biquard connection ∇ implies that ∇ X,Y ∇ φ − ∇ Y,X ∇ φ = R ( X, Y ) ∇ φ − ∇ T ( X,Y ) ∇ φ = R ( X, Y ) ∇ φ − ω s ( X, Y ) ∇ ξ s ∇ φ (5.3) = R ( X, Y ) ∇ φ, which by means of (5.4) gives(5.5) R ( X, Y, Z, ∇ φ ) = −∇ φ ( X, Y, Z ) + ∇ φ ( Y, X, Z ) = − dh ( X ) g ( Y, Z ) + dh ( Y ) g ( X, Z ) . We take a trace in (5.5) to obtain(5.6)
Ric ( X, ∇ φ ) = − dh ( X ) . On the other hand, since M is qc Einstein, Ric ( X, Y ) = 6 Sg ( X, Y ), hence
Ric ( X, ∇ φ ) = 6 Sdφ ( X ).Therefore,(5.7) 2 Sdφ ( X ) + dh ( X ) = 0 . The qc-conformal curvature tensor is given by (4.3), which, by (5.5) and (5.7), implies that(5.8) W qc ( X, Y, Z, ∇ φ ) = R ( X, Y, Z, ∇ φ ) + 2 S (cid:16) − dφ ( X ) g ( Y, Z ) + dφ ( Y ) g ( X, Z ) (cid:17) − (cid:16) Sdφ ( X ) + dh ( X ) (cid:17) g ( Y, Z ) + (cid:16) Sdφ ( Y ) + dh ( Y ) (cid:17) g ( X, Z ) = 0 . Since the dimension of M is seven and since by assumption ∇ φ = 0 on M , the vector fields ∇ φ, I ∇ φ, I ∇ φ, I ∇ φ form an orthogonal frame of the 4-dimensional horizontal distribution H . Then,by (5.8) and Lemma 5.3, we have W qc ( X, Y, Z, I s ∇ φ ) = − W qc ( X, Y, I s Z, ∇ φ ) = 0 which implies that W qc ( X, Y, Z, W ) = 0, i.e. M is locally qc conformally flat. (cid:3) The next lemma together with [11, Theorem 3.1] completes the proof of Theorem 5.1.
Lemma 5.5.
Let M be a qc Einstein space, φ be the non-constant function satisfying (3.10) - (3.8) and D = span { ξ , ξ , ξ , ∇ φ, I ∇ φ, I ∇ φ, I ∇ φ } be the integrable distribution from Lemma 5.2. Then each integralmanifold ι : N → M of D caries an induced qc structure, defined locally by the 1-forms ι ∗ ( η ) , ι ∗ ( η ) , ι ∗ ( η ) ,which is qc conformally flat and qc-Einstein with qc-scalar curvature with the same sign as the qc-scalarcurvature of M . ON-UMBILICAL QUATERNIONIC CONTACT HYPERSURFACES IN HYPER-K¨AHLER MANIFOLDS 13
Proof.
Let j : N → M be any integral manifold of D . Then the pull-back 1-forms j ∗ ( η ) , j ∗ ( η ) , j ∗ ( η ) on N define a qc structure on N with Reeb vector fields ˜ ξ s = j − ∗ ( ξ s ). The horizontal distribution on N is then just˜ H = j − ∗ ( H ) and the corresponding quaternionic structure on it is given by the endomorphisms ˜ I s = j − ∗ I s j ∗ .Moreover, the pull-backs of the structure equations (2.3) remain satisfied on N and thus the induced qcstructure on N is again qc Einstein with the same qc scalar curvature as M . Let us denote the correspondingBiquard connection on N by ˜ ∇ and consider the function ˜ φ = j ∗ φ . Then, clearly, ˜ ∇ ( ˜ φ ) = j − ∗ ∇ φ and thus,for any s = 1 , , ∇ ˜ φ, ˜ I s ˜ ∇ ˜ φ ] = j − ∗ [ ∇ φ, I s ∇ φ ] (5.1) = j − ∗ (cid:16) − g ( ∇ φ, ∇ φ ) ξ s (cid:17) = − g ( ˜ ∇ ˜ φ, ˜ ∇ ˜ φ ) ˜ ξ s . Therefore, − g ( ˜ ∇ ˜ φ, ˜ ∇ ˜ φ ) ˜ ξ s = [ ˜ ∇ ˜ φ, ˜ I s ˜ ∇ ˜ φ ] = ˜ I s ( ˜ ∇ ˜ ∇ ˜ φ ˜ ∇ ˜ φ ) − ˜ ∇ ˜ I s ˜ ∇ ˜ φ ˜ ∇ ˜ φ − X t =1 ˜ ω t ( ˜ ∇ ˜ φ, ˜ I s ˜ ∇ ˜ φ ) ˜ ξ t , i.e. we have ˜ ∇ ˜ φ ( ˜ ∇ ˜ φ, ˜ I s X ) = − ˜ ∇ ˜ φ ( ˜ I s ˜ ∇ ˜ φ, X ) , X ∈ ˜ H. Since the four vector fields ˜ ∇ ˜ φ, ˜ I ˜ ∇ ˜ φ, ˜ I ˜ ∇ ˜ φ, ˜ I ˜ ∇ ˜ φ define a frame for the distribution ˜ H we obtain that˜ ∇ φ ( ˜ I s X, ˜ I s Y ) = ˜ ∇ φ ( X, Y )for any
X, Y ∈ ˜ H and s = 1 , ,
3. This implies that ˜ ∇ ˜ φ ( X, Y ) = h ˜ g ( X, Y ) and thus the function ˜ φ satisfiesthe assertions of Lemma 5.4. Therefore, the integral manifold N is locally qc-conformally flat. (cid:3)(cid:3) We finish the section with the prof of Theorem 1.2.
Proof.
The proof is similar to that of Theorem 1.1 noting that, here, the qc-conformal flatness follows fromLemma 5.4. However, the (constant) qc-scalar curvature is not necessarily positive. The proof is completetaking into account Theorem 6.1. (cid:3) Appendix.
In the course of the paper we used several times the fact that a qc-Einstein qc-conformally flat manifoldis locally qc-homothetic to one of the standard model qc-spaces (1.1). As indicated below, this fact has beenessentially proved before, but due to its independent interest we formulate it explicitly. Furthermore, weinclude an argument for global equivalence.
Theorem 6.1.
A qc-conformally flat qc-Einstein manifold M is locally qc-homothetic to one of the followingthree model spaces: the 3-Sasakian sphere S n +3 , the quaternionic Heisenberg group G ( H ) or the hyperboloid S n depending on the sign of the qc-scalar curvature, respectively. If in addition M is connected, simplyconnected with complete Biquard connection then we have a global qc-homothety with the model spaces (1.1) .Proof. By a qc-homothety, depending on the sign of the qc-scalar curvature, we can reduce the claim to oneof the cases S = 2, S = 0 or S = −
2. We recall that the model spaces (1.1) are qc-Einstein qc-conformallyflat manifolds with positive qc-scalar curvature S = 2 in the case i) of the 3-Sasakian sphere [7, 10], flat inthe case of the quaternionic Heisenberg group iii) [7], and negative qc-scalar curvature S = −
2, [9], for thehyperboloid ii) .One proof of the local equivalence goes as follows. Due to the local qc-conformality with the quaternionicHeisenberg group, with the help of [14, Theorem 6.2], see [7, Theorem 1.2] for the positive qc-scalar curvaturecase, we can determine the exact form of the conformal factor relating the invariant qc structure on theHeisenberg group to the image by a qc-conformal transformation of the given qc-Einstein structure. Theproof of the local equivalence statement in Theorem 1.1 follows, for more details see [7, Theorem 1.2] in thecase of positive qc-scalar curvature, the paragraph after [14, Lemma 8.6] in the zero qc-scalar curvature case,while the negative qc-scalar curvature case follows analogously. The global result in the case of a compactmanifold is achieved by a monodromy argument and Liouville’s theorem [14, Theorem 8.5], [3]. Below is an argument using that in our case Biquard’s connection is an affine connection with parallel torsion andparallel curvature, hence we can invoke the results in [15, Chapter VI].For a qc-Einstein manifold we have from [7, 8] T = U = 0, the qc-scalar curvature is constant, S = const and the vertical space is integrable. As a consequence, on a qc-Einstein manifold we have [7, 11, 13, 8] T ( X, Y ) = 2 X s =1 ω s ( X, Y ) ξ s ; T ( ξ i , ξ j ) = − Sξ k , (6.1) R ( ξ s , X, Y, Z ) = R ( ξ s , ξ t , X, Y ) = 0 , R ( A, B ) ξ = − S X s =1 ω s ( A, B ) ξ s × ξ. (6.2)Using (2.2), we obtain from (6.1) that the torsion of the Biquard connection is parallel, ∇ T = 0. Similarly,(6.2) implies that ∇ R ( ξ s , A, B, C ) = ∇ R ( A, B, C, ξ s ) = 0.For the horizontal part of R we apply the second condition of the qc-conformal flatness, W qc = 0. Asubstitution of (5.2) into (4.3) gives(6.3) R ( X, Y, Z, W ) = S h g ( Y, Z ) g ( X, W ) − g ( Y, W ) g ( X, Z ) i + S X s =1 h ω s ( Y, Z ) ω s ( X, W ) − ω s ( X, Z ) ω s ( Y, W ) − ω s ( X, Y ) ω s ( Z, W ) i . Hence, by (2.2), it follows that the horizontal curvature of the Biquard connection is parallel as well, i.e., wehave ∇ T = ∇ R = 0.Let F be a linear isomorphism between the tangent spaces T p ( M ) and T ′ p ( M ′ ) of a point p in M anda point p ′ in the model space (1.1) of same qc-scalar curvature, such that, F maps an orthonormal basis { e a , I e a , I e a , I e a } na =1 of the horizontal space at p to the an orthonormal basis { e ′ a , I ′ e ′ a , I ′ e ′ a , I ′ e a } na =1 ofthe horizontal space at p ′ and also sends the corresponding Reeb vector fields at p to those at p ′ . Thus, F preserves the horizontal and vertical spaces F ( H p ) = H ′ q , F ( V p ) = V ′ q , and the Sp ( n ) Sp (1)-structure, i.e.,it maps the tensors g p , ( I s ) | p , ( ξ s ) | p at the point p ∈ M into the tensors g ′ q , ( I ′ s ) | q , ( ξ ′ s ) | q . Taking into account S = S ′ , (6.1) together with (6.2), and (6.3) show that F maps the torsion T p and the curvature R p at p intothe torsion T ′ q and the curvature R ′ q at q ∈ M ′ , respectively.Now, we can apply the affine equivalence theorem [15, Theorem 7.4] to obtain an affine local isomorphismbetween M and the coresponding model space. Since the qc structure ( H ⊕ V, Q , g ) is parallel the affinelocal isomorphism is a qc-homothety.Finally, if in addition M is connected, simply connected with a complete Biquard connections then [15,Theorem 7.8] gives us a global qc-homothety to the corresponding model case. We note that the Biquardconnection in each of the model cases is complete since the 3-Sasakian spere is compact, the Biquard connec-tion on the qc Heisenberg group is an invariant connection of a homogeneous space, while the hyperboloidis Sp ( n, Sp (1) /Sp ( n ) Sp (1), see e.g. [1, Theorem 5.1], with the invariant Biquard connection determinedby (2.6). (cid:3) References [1] Alekseevsky, D., Kamishima, Y.,
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E-mail address : [email protected] (Ivan Minchev) University of Sofia, Faculty of Mathematics and Informatics, blvd. James Bourchier 5, 1164Sofia, Bulgaria; Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, 61137 Brno, CzechRepublic
E-mail address : [email protected] (Dimiter Vassilev) University of New Mexico, Albuquerque, NM 87131
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