8 2. QUATERNIONIC CONTACT STRUCTURES AND THE BIQUARD CONNECTION ¯ s , s = 1, 2, 3, then ¯ = μ Ψ η for some Ψ SO(3) and a positive function μ Typ- ical examples of manifolds with QC-structures are totally umbilical hypersurfaces in quaternionic ahler or hyperk¨ ahler manifold, see Proposition 6.12 for the latter. It is instructive to consider the case when there is a globally defined one-form η. The obstruction to the global existence of η is encoded in the first Pontrjagin class [AK]. Besides clarifying the notion of a QC-manifold, most of the time, for example when considering the Yamabe equation, we shall work with a QC-structure for which we have a fixed globally defined contact form. In this case, if we rotate the R3-valued contact form and the almost complex structures by the same rotation we obtain again a contact form, almost complex structures and a metric (the latter is unchanged) satisfying the above conditions. On the other hand, it is important to observe that given a contact form the almost complex structures and the horizontal metric are unique if they exist. Finally, if we are given the horizontal bundle and a metric on it, there exists at most one sphere of associated contact forms with a corresponding sphere Q of almost complex structures [Biq1]. These observations are summed-up in the next Lemma. Lemma 2.2. [Biq1] a) If (η, Is,g) and (η, Is,g ) are two QC structures on M, then Is = Is, s = 1, 2, 3 and g = g . b) If (η, g) and , g) are two QC structures on M with Ker(η) = Ker(η ) = H then Q = Q and η = Ψ η for some matrix Ψ SO(3) with smooth functions as entries. Proof. a) Let us fix a basis {e1, ..., e4n} of H . Suppose the tensors g, dη1|H, dη2|H,dη3|H, I1,I2,I3 ( tensors on H) are given in local coordinates, respectively, by the matrices G, N1,N2,N3,J1,J2,J3 GL(4n). From (2.1) it follows (2.2) GJs = 1 2 Ns, s = 1, 2, 3. Let (i, j, k) be any cyclic permutation of (1, 2, 3). Using (2.2) we compute that Jk = JiJj = −J −1 i G−1GJj = −(GJi)−1(GJj) = −N −1 i Nj, (2.3) which accomplishes the proof. b) The condition Ker(η) = Ker(η ) = H implies that (2.4) ηk = 3 l=1 Ψkl ηl, k = 1, 2, 3 for some matrix Ψ GL(3) with smooth functions Ψij as entries. Applying the exterior derivative in (2.4) we find (2.5) dηk = dΨkl ηl + Ψkl dηl, k = 1, 2, 3. Let the H tensors Ik and I k be defined as usual with (2.1) using respectively η and η . Restricting the equation (2.5) to H and using the metric tensor g on H we have (2.6) g(IkX, Y ) = Ψklg(IlX, Y ), X, Y H or the equivalent equations Ik = ΨklIl on H. It is easy to see that this is possible if and only if Ψ SO(3).
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