CHAPTER 2 Quaternionic contact structures and the Biquard connection The notion of Quaternionic Contact Structure has been introduced by O. Biquard in [Biq1] and [Biq2]. Namely, a quaternionic contact structure (QC structure for short) on a (4n+3)-dimensional smooth manifold M is a codimen- sion 3 distribution H, such that, at each point p M the nilpotent step two Lie algebra Hp (TpM/Hp) is isomorphic to the quaternionic Heisenberg alge- bra Hn Im H. The quaternionic Heisenberg algebra structure on Hn Im H is obtained by the identification of Hn Im H with the algebra of the left in- variant vector fields on the quaternionic Heisenberg group, see Section 5.2. In particular, the Lie bracket is given by the formula [(qo,ωo), (q, ω)] = 2 Im qo · ¯ where q = (q1,q2,...,qn), qo = (qo,qo,...,qo 1 2 n ) Hn and ω, ωo Im H with qo · ¯ = n α=1 qo α · qα, see Section 6.1.1 for notations concerning H. It is important to observe that if M has a quaternionic contact structure as above then the def- inition implies that the distribution H and its commutators generate the tangent space at every point. The following is another, more explicit, definition of a quaternionic contact structure. Definition 2.1. [Biq1] A quaternionic contact structure ( QC-structure) on a 4n + 3 dimensional manifold M, n 1, is the data of a codimension three distribution H on M equipped with a CSp(n)Sp(1) structure, i.e., we have: i) a fixed conformal class [g] of metrics on H ii) a 2-sphere bundle Q over M of almost complex structures, such that, locally we have Q = {aI1 + bI2 + cI3 : a2 + b2 + c2 = 1}, where the almost complex structures Is : H H, Is 2 = −1, s = 1, 2, 3, satisfy the commutation relations of the imaginary quaternions I1I2 = −I2I1 = I3 iii) H is locally the kernel of a 1-form η = (η1,η2,η3) with values in R3 and the following compatibility condition holds (2.1) 2g(IsX, Y ) = dηs(X, Y ), s = 1, 2, 3, X, Y H. A manifold M with a structure as above will be called also quaternionic con- tact manifold (QC manifold) and denoted by (M, [g], Q). The distribution H is frequently called the horizontal space of the QC-structure. With a slight abuse of notation we shall use the letter Q to also denote the rank-three bundle consisting of endomorphisms of H locally generated by three almost complex structures I1,I2,I3 on H. The meaning should be clear from the context. We note that if in some local chart ¯ is another form, with corresponding ¯ [g] and almost complex structures 7
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