On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric

Abstract

A CR -structure on a 2 n +1-manifold gives a conformal class of Lorentz metrics on the Fefferman S 1 -bundle. This analogy is carried out to the quarternionic conformal 3- CR structure (a generalization of quaternionic CR - structure) on a 4 n + 3 -manifold M . This structure produces a conformal class [g] of a pseudo-Riemannian metric g of type (4n + 3,3) on M × S 3 . Let (PSp( n +1,1), S 4 n+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), S 4 n+3 ) if and only if ( M × S 3 ,[g]) is conformally flat ( i.e. the Weyl conformal curvature tensor vanishes).