TITLE:
On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric
AUTHORS:
Yoshinobu Kamishima
KEYWORDS:
Conformal Structure, Quaternionic CR-Structure, G-Structure, Conformally Flat Structure, Weyl Tensor, Integrability, Uniformization, Transformation Groups
JOURNAL NAME:
Applied Mathematics,
Vol.9 No.2,
February
22,
2018
ABSTRACT: 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-CR structure (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
tensor vanishes).