8 2. QUATERNIONIC CONTACT STRUCTURES AND THE BIQUARD CONNECTION

¯

I s, s = 1, 2, 3, then ¯ η = 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 K¨ 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 (η , 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 =

−Ji−1G−1GJj

=

−(GJi)−1(GJj)

=

−Ni−1Nj,

(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 Ik 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).