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 · ¯ q,
where q =
(q1,q2,...,qn),
qo = (qo,qo,...,qo
1 2 n)

Hn
and ω, ωo Im H with
qo · ¯ q =

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 ¯ η is another form, with corresponding ¯ g [g] and almost complex structures
7
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