All Title Author
Keywords Abstract

On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric

DOI: 10.4236/am.2018.92008, PP. 114-129

Keywords: Conformal Structure, Quaternionic CR-Structure, G-Structure, Conformally Flat Structure, Weyl Tensor, Integrability, Uniformization, Transformation Groups

Full-Text   Cite this paper   Add to My Lib


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).


[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.
[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.
[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.
[6]  Fefferman, C. (1976) Monge-Ampère Equations, the Bergman Kernel, and Geometry of Pseudo-Convex Domains. Annals of Mathematics, 103, 395-416.
[7]  Kulkarni, R. (1978) On the Principle of Uniformization. Journal of Differential Geo-metry, 13, 109-138.


comments powered by Disqus

Contact Us


微信:OALib Journal