Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On uniqueness of submaximally symmetric parabolic geometries

Dennis The

Abstract

Among (regular, normal) parabolic geometries of type $(G,P)$, there is a locally unique maximally symmetric structure and it has symmetry dimension $\dim(G)$. The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When $G$ is a complex or split-real simple Lie group of rank at least three or when $(G,P) = (G_2,P_2)$, we establish a local uniqueness result for submaximally symmetric structures of type $(G,P)$.

On uniqueness of submaximally symmetric parabolic geometries

Abstract

Among (regular, normal) parabolic geometries of type , there is a locally unique maximally symmetric structure and it has symmetry dimension . The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When is a complex or split-real simple Lie group of rank at least three or when , we establish a local uniqueness result for submaximally symmetric structures of type .

Paper Structure

This paper contains 16 sections, 11 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Parabolic geometries and algebraic models
  3. Parabolic geometries
  4. Symmetry and Tanaka prolongation
  5. Homogeneous parabolic geometries
  6. Algebraic models
  7. The canonical curved model and local uniqueness
  8. Kostant's theorem and the canonical curved model
  9. The $(G_2,P_2)$ case
  10. Preparation for the general case and the twistor simplification
  11. Proof of the main theorem
  12. Examples
  13. Projective structures
  14. 2nd order ODE systems
  15. Conformal structures
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $G$ be a complex or split-real simple Lie group, $P \subsetneq G$ a parabolic subgroup, and $G_0$ its Levi factor. Let $({\mathcal{G}} \to M, \omega)$ be a regular, normal parabolic geometry of type $(G,P,{\mathbb V})$, where ${\mathbb V} \subseteq H_2({\mathfrak p}_+,{\mathfrak g})^1$ is a $G_

Figures (1)

  • Figure 1: $G_2$ root diagram with grading associated to $P_2$

Theorems & Definitions (32)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Definition 2.1: Extrinsic Tanaka prolongation
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • ...and 22 more