STEFAN IVANOV, IVAN MINCHEV, and DIMITER VASSILEV 9

Besides the non-uniqueness due to the action of SO(3), the 1-form η can be

changed by a conformal factor, in the sense that if η is a form for which we can

find associated almost complex structures and metric g as above, then for any

Ψ ∈ SO(3) and a positive function μ, the form μ Ψ η also has an associated complex

structures and metric. In particular, when μ = 1 we obtain a whole unit sphere of

contact forms, and we shall denote, as already mentioned, by Q the corresponding

sphere bundle of associated triples of almost complex structures. With the above

consideration in mind we introduce the following notation.

Notation 2.3. We shall denote with (M, η) a QC-manifold with a fixed globally

defined contact form. (M, g, Q) will denote a QC-manifold with a fixed metric g

and a sphere bundle of almost complex structures Q. In this case we have in fact

a Sp(n)Sp(1) structure, i.e., we are working with a fixed metric on the horizontal

space. Correspondingly, we shall denote with η any (locally defined) associated

contact form.

We recall the definition of the Lie groups Sp(n), Sp(1) and Sp(n)Sp(1). Let

us identify Hn = R4n and let H acts on Hn by right multiplications, λ(q)(W ) =

W · q−1. This defines a homomorphism λ : {unit quaternions} −→ SO(4n) with

the convention that SO(4n) acts on R4n on the left. The image is the Lie group

Sp(1). Let λ(i) = I0,λ(j) = J0,λ(k) = K0. The Lie algebra of Sp(1) is

sp(1) = span{I0,J0,K0}.

The group Sp(n) is

Sp(n) = {O ∈ SO(4n) : OB = BO, B ∈ Sp(1)}

or Sp(n) = {O ∈ Mn(H) : O

¯t

O = I}, and O ∈ Sp(n) acts by

(q1,q2,...,qn)t

→ O

(q1,q2,...,qn)t.

Denote by Sp(n)Sp(1) the product of the two groups in SO(4n). Abstractly,

Sp(n)Sp(1) = (Sp(n) × Sp(1))/Z2. The Lie algebra of the group Sp(n)Sp(1) is

sp(n) ⊕ sp(1).

Any endomorphism Ψ of H can be decomposed with respect to the quaternionic

structure (Q,g) uniquely into four Sp(n)-invariant parts

Ψ =

Ψ+++

+

Ψ+−−

+

Ψ−+−

+

Ψ−−+,

where Ψ+++ commutes with all three Ii, Ψ+−− commutes with I1 and anti-commutes

with the others two and etc. Explicitly,

4Ψ+++

= Ψ − I1ΨI1 − I2ΨI2 − I3ΨI3,

4Ψ+−−

= Ψ − I1ΨI1 + I2ΨI2 + I3ΨI3,

4Ψ−+−

= Ψ + I1ΨI1 − I2ΨI2 + I3ΨI3,

4Ψ−−+

= Ψ + I1ΨI1 + I2ΨI2 − I3ΨI3.

The two Sp(n)Sp(1)-invariant components are given by

(2.7) Ψ[3] =

Ψ+++,

Ψ[−1] =

Ψ+−−

+

Ψ−+−

+

Ψ−−+.

Denoting the corresponding (0,2) tensor via g by the same letter one sees that

the Sp(n)Sp(1)-invariant components are the projections on the eigenspaces of the

Casimir operator

(2.8) † = I1 ⊗ I1 + I2 ⊗ I2 + I3 ⊗ I3

corresponding, respectively, to the eigenvalues 3 and −1, see [CSal]. If n = 1

then the space of symmetric endomorphisms commuting with all Ii,i = 1, 2, 3 is