Diffeomorphisms preserving R-circles in three dimensional CR manifolds

TL;DR

This paper characterizes diffeomorphisms that preserve R-circles in three-dimensional CR manifolds, showing they are either CR or conjugate CR diffeomorphisms, thus revealing their geometric invariance properties.

Contribution

It establishes a rigidity result for circle-preserving diffeomorphisms in three-dimensional CR manifolds, identifying them as CR or conjugate CR diffeomorphisms.

Findings

Diffeomorphisms preserving R-circles are either CR or conjugate CR diffeomorphisms.

R-circles are analogous to traces of Lagrangian totally geodesic planes in complex hyperbolic space.

The result characterizes the geometric invariance of R-circles under diffeomorphisms.

Abstract

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.