Our first observation is that if if n 1 and the qc-Ricci tensor is trace-free
(qc-Einstein condition) then the qc-scalar curvature is constant (Theorem 4.9).
Studying conformal transformations of QC structures preserving the qc-Einstein
condition, we describe explicitly all global functions on the quaternionic Heisenberg
group sending conformally the standard flat QC structure to another qc-Einstein
structure. Our result here is the following Theorem.
Theorem 1.1. Let Θ =
Θ be a conformal transformation of the standard
Θ on the quaternionic Heisenberg group G(H). If Θ is also qc-Einstein,
then up to a left translation the function h is given by
h = c
1 + ν
ν2 (x2
where c and ν are positive constants. All functions h of this form have this property.
The crucial fact which allows the reduction of the Yamabe equation to the sys-
tem preserving the qc-Einstein condition is Proposition 8.2 which asserts that, under
some ”extra” conditions, QC structure with constant qc-scalar curvature obtained
by a conformal transformation of a qc-Einstein structure on compact manifold must
be again qc-Einstein. The prove of this relies on detailed analysis of the Bianchi
identities for the Biquard connection. Using the quaternionic Cayley transform
combined with Theorem 1.1 lead to our second main result.
Theorem 1.2. Let η = f ˜ η be a conformal transformation of the standard
qc-structure ˜ η on the quaternionic sphere S4n+3. Suppose η has constant qc-scalar
curvature. If the vertical space of η is integrable then up to a multiplicative constant
η is obtained from ˜ η by a conformal quaternionic contact automorphism. In the case
n 1 the same conclusion holds when the function f is the real part of an anti-CRF
The definition of conformal quaternionic contact automorpism can be found
in Definition 7.6. The solutions (conformal factors) we find agree with those con-
jectured in [GV1]. It might be possible to dispense of the ”extra” assumption in
Theorem 1.2. In a subsequent paper [IMV] we give such a proof for the seven
dimensional sphere.
Studying the geometry of the Biquard connection, our main geometrical tool
towards understanding the geometry of the Yamabe equation, we show that the
qc-Einstein condition is equivalent to the vanishing of the torsion of Biquard con-
nection. In our third main result we give a local characterization of such spaces as
3-Sasakian manifolds.
Theorem 1.3. Let (M
Q) be a QC manifold with positive qc scalar
curvature Scal 0, assumed to be constant if n = 1. The next conditions are
a) (M
Q) is a qc-Einstein manifold.
b) M is locally 3-Sasakian, i.e., locally there exists an SO(3)-matrix Ψ with smooth
entries, such that, the local QC structure (
Ψ · η, Q) is 3-Sasakian.
c) The torsion of the Biquard connection is identically zero.
In particular, a qc-Einstein manifold of positive qc scalar curvature, assumed in
addition to be constant if n = 1, is an Einstein manifold of positive Riemannian
scalar curvature.
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