Diffeomorphisms preservingR-circles in three dimensional CR manifolds

Abstract

R-circles in (non-degenerate) three dimensional CR manifolds are the analogues to traces of Lagrangian totally geodesic planes on the three sphere viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain Legendrian curves on the manifold. We prove that a diffeomorphism between three dimensional CR manifolds which preserve circles is either a CR diffeomorphism or a conjugate CR diffeomorphism.