10 2. QUATERNIONIC CONTACT STRUCTURES AND THE BIQUARD CONNECTION 1-dimensional, i.e. the [3]-component of any symmetric endomorphism Ψ on H is proportional to the identity, Ψ3 = tr(Ψ) 4 Id|H. There exists a canonical connection compatible with a given quaternionic con- tact structure. This connection was discovered by O. Biquard [Biq1] when the dimension (4n + 3) 7 and by D. Duchemin [D] in the 7-dimensional case. The next result due to O. Biquard is crucial in the quaternionic contact geometry. Theorem 2.4. [Biq1] Let (M, g, Q) be a quaternionic contact manifold of di- mension 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 metric g ii) for X, Y H, one has T (X, Y ) = −[X, Y ]|V iii) preserves the Sp(n)Sp(1)-structure on H, i.e., ∇g = 0 and ∇Q Q iv) for ξ V , the endomorphism T (ξ, .)|H of H lies in (sp(n) sp(1))⊥ gl(4n) v) the connection on V is induced by the natural identification ϕ of V with the subspace sp(1) of the endomorphisms of H, i.e. ∇ϕ = 0. In (iv) the inner product on End(H) is given by (2.9) g(A, B) = tr(B∗A) = 4n a=1 g(A(ea),B(ea)), where A, B End(H), {e1, ..., e4n} is some g-orthonormal basis of H. We shall call the above connection the Biquard connection. Biquard [Biq1] also described the supplementary ”vertical” (sub-)space V explicitly, namely, locally V is generated by vector fields {ξ1,ξ2,ξ3}, such that (2.10) ηs(ξk) = δsk, (ξs dηs)|H = 0, (ξs dηk)|H = −(ξk dηs)|H, s, k {1, 2, 3}. The vector fields ξ1,ξ2,ξ3 are called Reeb vector fields or fundamental vector fields. If the dimension of M is seven, the conditions (2.10) do not always hold. Duchemin shows in [D] that if we assume, in addition, the existence of Reeb vec- tor fields as in (2.10), then Theorem 2.4 holds. Henceforth, by a QC structure in dimension 7 we shall always mean a QC structure satisfying (2.10). Notice that equations (2.10) are invariant under the natural SO(3) action. Using the Reeb vector fields we extend g to a metric on M by requiring (2.11) span{ξ1,ξ2,ξ3} = V H and g(ξs,ξk) = δsk. The extended metric does not depend on the action of SO(3) on V , but it changes in an obvious manner if η is multiplied by a conformal factor. Clearly, the Biquard connection preserves the extended metric on TM, ∇g = 0. We shall also extend the quternionic structure by setting Is|V = 0. Suppose the Reeb vector fields {ξ1,ξ2,ξ3} have been fixed. The restriction of the torsion of the Biquard connection to the vertical space V satisfies [Biq1] (2.12) T (ξi,ξj) = λξk [ξi,ξj]| H , where λ is a smooth function on M, which will be determined in Corollary 3.14. We recall that in the paper the indices follow the convention 1.4. Further properties of
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