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).
Highlights
A CR-structure on a 2n +1-manifold gives a conformal class of Lorentz metrics on the Fefferman S1 -bundle
This paper concerns a geometric structure on (4n + 3) -manifolds which is related with CR-structure and quaternionic CR-structure
Given a quaternionic CR-structure { } ωα α=1,2,3 on a 4n + 3 -manifold M, we have proved in [3] that the associated endomorphism Jα on the 4n-bundle D naturally extends to a complex structure Jα on kerωα
Summary
This paper concerns a geometric structure on (4n + 3) -manifolds which is related with CR-structure and quaternionic CR-structure (cf. [1] [2]). A hypercomplex 3 CR-structure on a (4n + 3) -manifold M consists of Let D be a 4n-dimensional subbundle endowed with a quaternionic structure Q on a (4n + 3) -manifold M. The conformal class [g] is an invariant for quaternionic 3 CR-structure. A quaternionic 3 CR-manifold M is spherical (i.e. locally ( ) modeled on PSp (n +1,1), S 4n+3 ) if and only if the pseudo-Riemannian. We have constructed a conformal invariant on (4n + 3) -dimensional pseudoconformal quaternionic CR manifolds in [3]. In general if S1 induces a lightlike vector field with respect to a Lorentz metric of a Lorentz manifold, S1 is said to be a lightlike group acting as Lorentz isometries.
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