QUATERNIONIC CONTACT MANIFOLDS WITH A CLOSED FUNDAMENTAL 4-FORM

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1 Available at: IC/2008/090 United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS QUATERNIONIC CONTACT MANIFOLDS WITH A CLOSED FUNDAMENTAL 4-FORM Stefan Ivanov 1 University of Sofia, Faculty of Mathematics and Informatics, Blvd. James Bourchier 5, 1164, Sofia, Bulgaria and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and Dimiter Vassilev 2 Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, 87131, USA. Abstract We show that the torsion of the Biquard connection of a quaternionic contact manifold of dimension at least eleven vanishes if and only if the fundamental four form is closed. This condition characterizes quaternionic contact structures which are locally qc homothetic to 3- Sasakian structures. MIRAMARE TRIESTE December Senior Associate of ICTP. ivanovsp@fmi.uni-sofia.bg 2 vassilev@math.unm.edu

2 1. Introduction A quaternionic contact (qc) structure, introduced in [Biq1, Biq2, appears naturally as the conformal boundary at infinity of the quaternionic hyperbolic space. Such structures have been considered in connection with the quaternionic contact Yamabe problem, [W, IMV1, IMV2. A particular case of this problem amounts to finding the extremals and the best constant in the L 2 Folland-Stein Sobolev-type embedding, [Fo and [FSt, on the quaternionic Heisenberg group, see [GV1 and [IMV2. A qc structure on a real (4n+3)-dimensional manifold M is a codimension three distribution H, called the horizontal space, locally given as the kernel of a 1-form η = (η 1,η 2,η 3 ) with values in R 3, such that, the three 2-forms dη i H are the fundamental 2-forms of a quaternionic structure on H. The 1-form η is determined up to a conformal factor and the action of SO(3) on R 3. Therefore H is equipped with a conformal class [g of Riemannian metrics and a 2-sphere bundle of almost complex structures, the quaternionic bundle Q. The 2-sphere bundle of one forms determines uniquely the associated metric and a conformal change of the metric is equivalent to a conformal change of the one forms. To every metric in the fixed conformal class one can associate a linear connection preserving the qc structure, see [Biq1, which we shall call the Biquard connection. The qc scalar curvature of the Biquard connection is obtained by taking horizontal traces of the curvature tensor. The transformations preserving a given qc structure η, i.e. η = µψη for a positive smooth function µ and a SO(3) matrix Ψ with smooth functions as entries, are called quaternionic contact conformal (qc conformal) transformations. If the function µ is constant we have quaternionic contact homothetic (qc-homothetic) transformations. The Biquard connection is invariant under qc homothetic transformations. Examples of qc manifolds are given in [Biq1, Biq2, IMV1, D1. In particular, any totally umbilic hypersurface of a quaternionic Kähler or hyperkähler manifold carries such a structure. An extensiveley studied class of examples of quaternionic contact structures are provided by the 3-Sasakian manifolds. The latter can be defined as (4n + 3)-dimensional Riemannian manifolds whose Riemannian cone is a hyperkähler manifold. It was shown in [IMV1, Theorem 1.3 that the torsion endomorphism of the Biquard connection is the obstruction for a given qc-structure to be locally qc homothethic to a 3-Sasakian one provided the qc scalar curvature is not identically zero. Explicit examples of qc manifolds with non-zero torsion which are not obtained with a qc conformal transformation from a locally 3-Sasaki structure were recently given in [AFIV. The quaternionic Heisenberg group, the quaternionic sphere of dimension 4n + 3 with its standard 3- sasakian structure and the qc structures locally qc conformal to them are characterized in [IV by the vanishing of a tensor invariant, the qc-conformal curvature defined in terms of the curvature and torsion of the Biquard connection. Explicit examples of non-qc conformally flat qc manifolds are recently constructed in [AFIV. 2

3 In this article we consider the 4-form Ω defining the Sp(n)Sp(1) structure on the horizontal distribution. We shall call it the fundamental four-form. It is defined (globally) on the horizontal distribution H by (1.1) Ω = ω 1 ω 1 + ω 2 ω 2 + ω 3 ω 3, where ω s are the fundamental two-forms defined below in (2.2). The purpose of the paper is to show that when the dimensions of the manifold is grater than seven, the fundamnetal form is closed iff the qc structure is locally homothetic to a 3-sasakian one. We prove the following Theorem 1.1. The torsion of the Biquard connection on a qc manifold vanishes if and only if the fundamental 4-from Ω is closed, dω = 0, provided the dimension of the manifold is grater than seven. In particular, in such a case the qc scalar curvature is constant and the vertical distribution is integrable. Combining the last Theorem with Theorem 1.3 and Theorem 7.11 in [IMV1 we obtain Theorem 1.2. Let (M 4n+3,η, Q) be a 4n + 3-dimensional qc manifold. For n > 1 the following conditions are equivalent a) (M 4n+3,η, Q) has closed fundamental four form, dω = 0; b) (M 4n+3,g, Q) is qc-einstein manifold; c) Each Reeb vector field, defined in (2.1), preserves the horizontal metric and the quaternionic structure simultaneously. If in addition the qc scalar curvature is not identically zero, 0, then each of a), b) and c) is equivalent to the following condition d). d) M 4n+3 is locally qc homothetic to a 3-Sasakian manifold, i.e., locally, there exists a SO(3)- matrix Ψ with smooth entries depending on an auxiliary parameter, such that, the local qc structure ( 16n(n+2) Ψ η,q) is 3-Sasakian. As an application of Theorem 1.1 we give in the last section a proof of the equivalence of a) and d) in Theorem 1.2. Thus, when the dimension of the qc manifold is grater than seven, we establish in a slightly different manner Theorem 3.1 in [IMV1. Remark 1.3. When the dimension of the qc manifold is seven, if the torsion of the Biquard connection vanishes then the fundamental four form is closed. We do not know whether the converse holds or if there exists an example of a seven dimensional qc manifold with a closed fundamental four form and a non-vanishing torsion. Organization of the paper. The paper relies heavily on the notion of Biquard connection introduced in [Biq1 and the properties of its torsion and curvature discovered in [IMV1. In order to make the present paper self-contained, in Section 2 we give a review of the notion of a quaternionic contact structure and collect formulas and results from [Biq1 and [IMV1 that will be used. 3

4 Convention 1.4. a) We shall use X, Y, Z, U to denote horizontal vector fields, i.e. X, Y, Z, U H; b) {e 1,...,e 4n } denotes an orthonormal basis of the horizontal space H; c) The summation convention over repeated vectors from the basis {e 1,...,e 4n } is used. For example, the formula k = P(e b,e a,e a,e b ) means k = 4n a,b=1 P(e b,e a,e a,e b ). d) The triple (i, j, k) denotes any cyclic permutation of (1, 2, 3). e) s and t will be any numbers from the set {1, 2, 3}, s, t {1, 2, 3}. 2. Quaternionic contact manifolds In this section we will briefly review the basic notions of quaternionic contact geometry and recall some results from [Biq1 and [IMV1. For the purposes of this paper, a quaternionic contact (qc) manifold (M,g, Q) is a 4n+3 dimensional manifold M with a codimension three distribution H equipped with a metric g and an Sp(n)Sp(1) structure, i.e., we have i) a 2-sphere bundle Q over M of almost complex structures I s : H H, I 2 s = 1, satisfying the commutation relations of the imaginary quaternions I 1 I 2 = I 2 I 1 = I 3 and Q = {ai 1 + bi 2 + ci 3 : a 2 + b 2 + c 2 = 1}; ii) H is locally the kernel of a 1-form η = (η 1,η 2,η 3 ) with values in R 3 satisfying the compatibility condition 2g(I s X,Y ) = dη s (X,Y ). On a quaternionic contact manifold there exists a canonical connection defined in [Biq1 when the dimension (4n + 3) > 7, and in [D in the 7-dimensional case. Theorem 2.1. [Biq1 Let (M,g, Q) be a quaternionic contact manifold of dimension 4n + 3 > 7 and a fixed metric g on H in the conformal class [g. Then there exists a unique connection with torsion T on M 4n+3 and a unique supplementary subspace V to H in TM, such that: i) preserves the decomposition H V and the Sp(n)Sp(1)-structure on H, g = 0, Q Q; ii) for X,Y H, one has T(X,Y ) = [X,Y V ; iii) for ξ V, the endomorphism T(ξ,.) H of H lies in (sp(n) sp(1)) gl(4n); We shall call the above connection the Biquard connection. Biquard [Biq1 also described the supplementary subspace V. Locally, V is generated by vector fields {ξ 1,ξ 2,ξ 3 }, such that (2.1) η s (ξ k ) = δ sk, (ξ s dη s ) H = 0, (ξ s dη k ) H = (ξ k dη s ) H. The vector fields ξ 1,ξ 2,ξ 3 are called Reeb vector fields or fundamental vector fields. If the dimension of M is seven, the conditions (2.1) do not always hold. Duchemin shows in [D that if we assume, in addition, the existence of Reeb vector fields as in (2.1), then Theorem 2.1 holds. Henceforth, by a qc structure in dimension 7 we shall mean a qc structure satisfying (2.1). The fundamental 2-forms ω s of the quaternionic structure Q are defined by (2.2) 2ω s H = dη s H, ξ ω s = 0, ξ V. 4

5 Letting Ω = ω 1 ω 1 + ω 2 ω 2 + ω 3 ω 3 we see that Ω = 0 since preserves Q. The torsion endomorphism T ξ = T(ξ,.) : H H, ξ V, plays an important role in the qc geometry. Decomposing the endomorphism T ξ (sp(n) + sp(1)) into symmetric part T 0 ξ and skew-symmetric part b ξ, T ξ = T 0 ξ + b ξ Biquard shows in [Biq1 that T ξ is completely trace-free, tr T ξ = tr T ξ I = 0,I Q and describes the properties of the two components. Using the two Sp(n)Sp(1)-invariant trace-free symmetric 2-tensors T 0 and U on H defined in [IMV1 by T 0 (X,Y ) def = g((t 0 ξ 1 I 1 + T 0 ξ 2 I 2 + T 0 ξ 3 I 3 )X,Y ), U(X,Y ) def = g(i s b ξs X,Y ), the properties of T ξ outlined in [Biq1 give the following identities, cf. [IMV1, (2.3) (2.4) T 0 (X,Y ) + T 0 (I 1 X,I 1 Y ) + T 0 (I 2 X,I 2 Y ) + T 0 (I 3 X,I 3 Y ) = 0, U(X,Y ) U(I 1 X,I 1 Y ) U(I 2 X,I 2 Y ) U(I 3 X,I 3 Y ) = 0. If n = 1 then U vanishes identically, U = 0, and the torsion is a symmetric tensor, T ξ = Tξ 0. The covariant derivatives with respect to the Biquard connection of the almost complex structures and the vertical vectors are given by I i = α j I k + α k I j, ξ i = α j ξ k + α k ξ j, where, as shown in [Biq1, the sp(1)-connection 1-forms α s on H are (2.5) α i (X) = dη k (ξ j,x) = dη j (ξ k,x), X H, ξ i V. The sp(1)-connection 1-forms α s on the vertical space V are calculated in [IMV1 (2.6) ( α i (ξ s ) = dη s (ξ j,ξ k ) δ is 16n(n + 2) + 1 ) 2 ( dη 1(ξ 2,ξ 3 ) + dη 2 (ξ 3,ξ 1 ) + dη 3 (ξ 1,ξ 2 )), where s {1,2,3}. It turns out that the vanishing of the sp(1)-connection 1-forms on H is equivalent to the vanishing of the torsion endomorphism of the Biquard connection, see [IMV The Ricci two forms. Let R = [, [, be the curvature tensor of. We denote the curvature tensor of type (0,4) by the same letter, R(A,B,C,D) := g(r(a,b)c,d), A,B,C,D Γ(T M). The Ricci two forms and the scalar curvature of the Biquard connection, called qc-ricci forms and qc-sacalr curvature, respectively, are defined in [IMV1 by 4nρ s (X,Y ) = R(X,Y,e a,i s e a ), = R(e b,e a,e a,e b ). The sp(1)-part of R is determined by the Ricci 2-forms and the connection 1-forms by (2.7) R(A,B,ξ i,ξ j ) = 2ρ k (A,B) = (dα k + α i α j )(A,B), A,B Γ(TM). The horizontal part of the Ricci forms can be expressed in terms of the torsion of the Biquard connection [IMV1. We collect with slight modifications using the equality 4T 0 (ξ s,i s X,Y ) = T 0 (X,Y ) T 0 (I s X,I s Y ) the necessary facts from [IMV1, Theorem 1.3, Theorem 3.12, Corollary 3.14, Proposition 4.3 and Proposition 4.4, using the formulas in the form presented in [IV. 5

6 Theorem 2.2. [IMV1 On a (4n + 3)-dimensional QC manifold, n > 1 the next formulas hold ρ s (X,I s Y ) = 1 [ T 0 (X,Y ) + T 0 (I s X,I s Y ) 2U(X,Y ) g(x,y ), 2 8n(n + 2) T(ξ i,ξ j ) = 8n(n + 2) ξ k [ξ i,ξ j H, = 8n(n + 2)g(T(ξ 1,ξ 2 ),ξ 3 ) 1 T(ξ i,ξ j,x) = ρ k (I i X,ξ i ) = ρ k (I j X,ξ j ), ρ i (ξ i,ξ j ) + ρ k (ξ k,ξ j ) = 16n(n + 2) ξ j(); ρ i (X,ξ i ) = X() 32n(n + 2)) ( ρ i(ξ j,i k X) + ρ j (ξ k,i i X) + ρ k (ξ i,i j X)). For n = 1 the above formulas hold with U = 0. In particular, the vanishing of the horizontal trace-free part of the Ricci forms is equivalent to the vanishing of the torsion endomorphism of the Biquard connection. In this case the horizontal distribution is integrable, the qc scalar curvature is constant and if 0 then the qc-structure is 3-sasakian up to a multiplication by a constant and an SO(3)-matrix with smooth entries depending on an auxiliary parameters. For the last part of the above Theorem, we remind that a (4n+3)-dimensional Riemannian manifold (M,g) is called 3-Sasakian if the cone metric g c = t 2 g + dt 2 on C = M R + is a hyperkähler metric, namely, it has holonomy contained in Sp(n+1) [BGN. A 3-Sasakian manifold of dimension (4n+3) is Einstein with positive Riemannian scalar curvature (4n+2)(4n+3) [Kas and if complete it is compact with a finite fundamental group, due to Mayer s theorem (see [BG for a nice overview of 3-Sasakian spaces). 3. Local structure equations of qc manifolds First we derive the local structure equations of a qc structure in terms of the sp(1)-connection forms of the Biquard connection and the qc scalar curvature. Proposition 3.1. Let (M 4n+3,η, Q) be a (4n+3)- dimensional qc manifold with qc scalar curvature. Let s = 8n(n+2) has s = 2. The following equations hold be the normilized qc scalar curvature, so that a 3-Sasakian manifold (3.1) (3.2) (3.3) 2ω i = dη i + η j α k η k α j + sη j η k, dω i = ω j (α k + sη k ) ω k (α j + sη j ) ρ k η j + ρ j η k ds η j η k, dω = [2η i (ρ k ω j ρ j ω k ) + ds ω i η j η k, (ijk) where α s are the sp(1)-connection 1-forms of the Biquard connection, ρ s are the Ricci 2-forms and (ijk) is the cyclic sum of even permutations of {1,2,3}. In particular, the structure equations of a 3-Sasaki manifold have the form (3.4) dη i = 2ω i + 2η j η k. 6

7 Proof. From the definition (2.2) of the fundamental 2-forms ω s we have (see also [IMV1) (3.5) 2ω l = (dη l ) H = dη l 3 η s (ξ s dη l ) + s=1 1 s<t 3 dη l (ξ s,ξ t )η s η t, where is the interior multiplication. A straightforward calculation using (2.5) and (2.6) gives the equivalence of (3.5) and (3.1). Taking the exterior derivative of (3.1), followed by an application of (3.1) and (2.7) implies (3.2). The last formula, (3.3), follows from (3.2) and definition (1.1). For a 3-Sasaki manifold we have [IMV1 = 16n(n + 2),dη i (ξ j,ξ k ) = 2,α s = 2η s and the structure equations (3.1) become (3.4). Conversely, if (3.4) holds it is straightforward to check that the Kähler forms F i = t 2 (ω i + η j η k ) + tdt η i on the cone N = M R + are closed and therefore the cone metric g N = t 2 (g + 3 s=1 η s η s ) + dt dt is hyper Kähler due to Hitchin s theorem [Hit. 4. Proof of Theorem 1.1 We shall show that the condition dω = 0 is equivalent to the vanishing of the torsion of the Biquard connection. Indeed, if T 0 = U = 0, then Theorem 2.2 implies (4.1) ρ s (X,Y ) = sω s (X,Y ), ρ s (ξ t,x) = 0, ρ i (ξ i,ξ j ) + ρ k (ξ k,ξ j ) = 0, since is constant and the horizontal distribution is integrable. Using the just obtained identities in (3.3) gives dω = 0. The converse follows from the next Lemma 4.1. On a qc manifold of dimension (4n + 3) > 7 we have the identities U(X,Y ) = 1 [ (4.2) dω(ξ i,x,i k Y,e a,i j e a ) + dω(ξ i,i i X,I j Y,e a,i j e a ) 16 T 0 1 [ (4.3) (X,Y ) = dω(ξ i,x,i k Y,e a,i j e a ) dω(ξ i,i i X,I j Y,e a,i j e a ). 8(1 n) (ijk) Proof. The equation (3.3) together with the second equality in Theorem 2.2 yield (4.4) dω(ξ i,x,i k Y,e a,i j e a ) = 4(n 1)ρ 0 k (X,I ky ) + 2ρ 0 j (X,I jy ) 2ρ 0 j (I ix,i k Y ), where ρ 0 is the horizontal trace-free part of ρ given by (4.5) ρ 0 s (X,I sy ) = 1 [ T 0 (X,Y ) + T 0 (I s X,I s Y ) 2U(X,Y ). 2 A substitution of (4.5) in (4.4), combined with the properties of the torsion, (2.3) and (2.4) give [ (4.6) 2(n 1) T 0 (X,Y ) + T 0 (I s X,I s Y ) + 8nU(X,Y ) = dω(ξ i,x,i k Y,e a,i j e a ). Applying again (2.3) and (2.4) in (4.6) we see that U and T 0 satisfy (4.2) and (4.3), respectively. Clearly, Lemma 4.1 completes the proof of Theorem

8 4.1. Proof a) = d) in Theorem 1.2. The idea is the same as in the proof of Theorem 3.1 in [IMV1. Let dω = 0. Theorem 1.1 implies that the torsion of the Biquard connection vanishes, while Theorem 2.2 shows that the qc scalar curvature is constant and the vertical distribution is integrable. The qc structure η = 16n(n+2) η has normalized qc scalar curvature s = 2 and dω = 0 provided 0. For simplicity, we shall denote η with η and, in fact, drop the everywhere. In the first step of the proof we show that the Riemannian cone N = M R + with the metric g N = t 2 (g + 3 s=1 η s η s )+dt dt has holonomy contained in Sp(n+1). To this end we consider the following four form on N (4.7) F = F 1 F 1 + F 2 F 2 + F 3 F 3, where the two forms F s are defined by (4.8) F i = t 2 (ω i + η j η k ) + tdt η i. From (4.8) it follows df i = tdt (2ω i + 2η j η k dη i ) + t 2 d(ω i + η j η k ), hence df = t 4 (ijk)t 4[ d(ω i + η j η k ) (ω i + η j η k ) + t 3 dt (ijk) [(2ω i + 2η j η k dη i ) (ω i + η j η k ) d(ω i + η j η k ) η i. A short calculation using (3.1), (3.2) and (3.3) gives df = t 4 dω 2t 3 (ijk)dt (ρ i + 2ω i ) η j η k = 0, taking into account the first equality in Theorem 2.2, (4.1) and s = 2, which hold when dω = 0 by Theorem 1.1. Hence, df = 0 and the holonomy of the cone metric is contained in Sp(n+1)Sp(1) provided n > 1 [S, i.e. the cone is quaternionic Kähler manifold provided n > 1. Note that when n = 1 this conclusion can not be reached due to the 8-dimensional example constructed by S. Salamon [Sal. It is a classical result (see e.g [Bes and references therein) that a quaternionic Kähler manifolds of dimension bigger than four are Einstein. This fact implies that the cone N = M R + with the warped product metric g c must be Ricci flat (see e.g. [Bes, p.267) and therefore it is locally hyperkähler (see e.g. [Bes, p.397). This means that locally there exists a SO(3)-matrix Ψ with smooth entries, possibly depending on t, such that the triple of two forms ( F 1, F 2, F 3 ) = Ψ (F 1,F 2,F 3 ) t consists of closed 2-forms constitute the fundanental 2-forms of the local hypekähler structure. Consequently (M, Ψ η) is locally a 3-Sasakian manifold. Acknowledgments. This research was done during the visit of S. Ivanov to the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, as a Senior Associate. S.I. thanks ICTP for providing support and an excellent research environment. S.I. is partially supported by the Contract 154/2008 with the University of Sofia St.Kl.Ohridski. 8

9 References [AFIV L. de Andres, M. Fernandez, S. Ivanov, Joseba, L.Ugarte & D. Vassilev, Explicit non-flat quaternionic contact structures, in preparation. [Bes Besse, A., Einstein manifolds, Ergebnisse der Mathematik, vol. 3, Springer-Verlag, Berlin, [Biq1 O. Biquard, Métriques d Einstein asymptotiquement symétriques. Astérisque 265 (2000). [Biq2, Quaternionic contact structures. Quaternionic structures in mathematics and phisics (Rome, 1999), (electronic), Univ. Studi Roma La Sapienza, Roma, [BG Boyer, Ch. & Galicki, K., 3-Sasakian manifolds, Surveys in differential geometry: essays on Einstein manifolds, , Surv. Differ. Geom., VI, Int. Press, Boston, MA, [BGN Boyer, Ch., Galicki, K. & Mann, B., The geometry and topology of 3-Sasakian manifolds J. Reine Angew. Math., 455 (1994), [D D. Duchemin, Quaternionic contact structures in dimension 7. Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, [D1, Quaternionic contact hypersurfaces. math.dg/ [Fo G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Math., 13 (1975), [FSt G. B. Folland & E. M. Stein, Estimates for the b Complex and Analysis on the Heisenberg Group, Comm. Pure Appl. Math., 27 (1974), [GV1 Garofalo, N. & Vassilev, D., Symmetry properties of positive entire solutions of Yamabe type equations on groups of Heisenberg type, Duke Math J, 106 (2001), no. 3, [Hit N. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) [IMV1 Ivanov, S., Minchev, I., & Vassilev, Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem. preprint, math.dg/ [IMV2 Ivanov, S., Minchev, I., & Vassilev, Extremals for the Sobolev inequality on the seven dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem, J. Eur. Math. Soc., to appear, math.dg/ [IV S. Ivanov, & D. Vassilev, Conformal quaternionic contact curvature and the local sphere theorem, arxiv: , MPIM, Bonn, Preprint: MPIM [Kas Kashiwada, T., A note on Riemannian space with Sasakian 3-structure, Nat. Sci. Reps. Ochanomizu Univ., 22 (1971), 1 2. [S A. Swann, Quaternionic Khler geometry and the fundamental 4-form, Proceedings of the Workshop on Curvature Geometry (Lancaster, 1989), , ULDM Publ., Lancaster, [Sal S. Salamon, Almost parallel structures Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), , Contemp. Math., 288, Amer. Math. Soc., Providence, RI, [W Wang, W., The Yamabe problem on quaternionic contact manifolds, Ann. Mat. Pura Appl., 186 (2007), no. 2,