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

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