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On CR manifolds of CR dimension 1

Boris Kruglikov

Abstract

We classify all maximal symmetry models of CR dimension 1, depending on their Bloom-Graham and Tanaka types, give coordinate realization to some of those models and prove a general extension principle.

On CR manifolds of CR dimension 1

Abstract

We classify all maximal symmetry models of CR dimension 1, depending on their Bloom-Graham and Tanaka types, give coordinate realization to some of those models and prove a general extension principle.
Paper Structure (7 sections, 11 theorems, 39 equations)

This paper contains 7 sections, 11 theorems, 39 equations.

Table of Contents

  1. Formulation of the Problem and the Results
  2. Bloom-Graham type, growth vector and Tanaka prolongation
  3. Prolongation rigidity and symmetry bound
  4. Growth vector $(2,1,\dots,1)$
  5. General rank 2 distributions: extensions, reductions and prolongations
  6. CR structures of CR dimension 1 with maximal symmetry dimension
  7. Examples of coordinate models

Key Result

Theorem 1

The symmetry dimension of the bracket-generating CR manifolds $(M,D,J)$ of CR dimension 1 does not exceed $m+2$, where $m=\dim M$, provided $m>3$. In fact, for every CR symbol, the most symmetric structures of this kind have symmetry dimension $m+r$, where $1\leq r\leq 2$.

Theorems & Definitions (22)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['Th1']}
  • proof : Proof of Corollary \ref{['Cor1']}
  • Proposition 2
  • ...and 12 more