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
O = I}, and O Sp(n) acts by
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 =
where Ψ+++ commutes with all three Ii, Ψ+−− commutes with I1 and anti-commutes
with the others two and etc. Explicitly,
= Ψ I1ΨI1 I2ΨI2 I3ΨI3,
= Ψ I1ΨI1 + I2ΨI2 + I3ΨI3,
= Ψ + I1ΨI1 I2ΨI2 + I3ΨI3,
= Ψ + I1ΨI1 + I2ΨI2 I3ΨI3.
The two Sp(n)Sp(1)-invariant components are given by
(2.7) Ψ[3] =
Ψ[−1] =
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
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