  R-circles in general three dimensional CR manifolds (of contact type) are the
analogues to traces of Lagrangian totally geodesic planes on the 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 conjugate CR diffeomorphism.
