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
tensor vanishes).
References
[1]
Biquard, O. (2001) Quaternionic Contact Structures, Quaternionic Structures in Mathematics and Physics, Rome, World Sci. 1999 Publishing, River Edge, NJ. 23-30.
[2]
Webster, S. (1977) On the Transformation Group of a Real Hypersurfaces. Transactions of the American Mathematical Society, 231, 179-190.
https://doi.org/10.1090/S0002-9947-1977-0481085-5
[3]
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
[4]
Kobayashi, S. (1970) Transformation Groups in Differential Geometry. Ergebnisse Math., 70.
[5]
Lee, J. (1986) The Fefferman Metric and Pseudo Hermitian Invariants. Transactions of the American Mathematical Society, 296, 411-429.
https://doi.org/10.1090/S0002-9947-1986-0837820-2
[6]
Fefferman, C. (1976) Monge-Ampère Equations, the Bergman Kernel, and Geometry of Pseudo-Convex Domains. Annals of Mathematics, 103, 395-416.
https://doi.org/10.2307/1970945
[7]
Kulkarni, R. (1978) On the Principle of Uniformization. Journal of Differential Geo-metry, 13, 109-138. https://doi.org/10.4310/jdg/1214434351
