Associated Families of Immersions of Three Dimensional CR Manifolds in Euclidean Spaces

TL;DR

This paper classifies isometric immersions of three-dimensional CR manifolds into Euclidean spaces, revealing that associated families of such immersions are highly restricted and characterizing their structure.

Contribution

It introduces a natural generalization of associated families for CR manifolds and provides a complete classification of when such deformations exist.

Findings

Associated families are highly restrictive in this setting.

Complete classification of immersions with associated families.

Generalization of associated family concept to CR manifolds.

Abstract

We consider isometric immersions in arbitrary codimension of three-dimensional strongly pseudoconvex pseudo-hermitian CR manifolds into the Euclidean space $\mathbb{R}^n$ and generalize in a natural way the notion of associated family. We show that the existence of such deformations turns out to be very restrictive and we give a complete classification.