A CR-structure on a 2n +1-manifold gives a conformal class of Lorentz
metrics on the Fefferman S1-bundle. This analogy is carried out to the quarternionic conformal 3-CRstructure (a generalization of quaternionic CR- structure) on a 4n + 3 -manifold M. This structure produces a conformal
class [g] of a pseudo-Riemannian metric g of type (4n + 3,3) on M × S3.
Let (PSp(n +1,1), S4n+3) be the geometric model obtained from the projective
boundary of the complete simply connected quaternionic hyperbolic
manifold. We shall prove that M is locally modeled on (PSp(n +1,1), S4n+3)
if and only if (M × S3 ,[g]) is conformally flat (i.e. the Weyl conformal curvature
Alekseevsky, D.V. and Kamishima, Y. (2008) Pseudo-Conformal Quaternionic CR Structure on -Dimensional Manifolds. Annali di Matematica Pura ed Applicata, 187, 487-529. https://doi.org/10.1007/s10231-007-0053-2