Essential Parabolic Structures and Their Infinitesimal Automorphisms

TL;DR

This paper generalizes the concepts of essential structures and automorphisms from conformal geometry to all parabolic geometries, establishing criteria for essentiality and extending classical theorems.

Contribution

It introduces a unified framework for essential parabolic structures using Weyl structures and generalizes key theorems like Ferrand-Obata and Lichnerowicz to broader geometries.

Findings

Proper automorphism group action implies inessentiality.

Generalization of the Ferrand-Obata theorem to parabolic geometries.

Characterization of essentiality via holonomy at singularities.

Abstract

Using the theory of Weyl structures, we give a natural generalization of the notion of essential conformal structures and conformal Killing fields to arbitrary parabolic geometries. We show that a parabolic structure is inessential whenever the automorphism group acts properly on the base space. As a corollary of the generalized Ferrand-Obata theorem proved by C. Frances, this proves a generalization of the "Lichnerowicz conjecture" for conformal Riemannian, strictly pseudo-convex CR, and quaternionic/octonionic contact manifolds in positive-definite signature. For an infinitesimal automorphism with a singularity, we give a generalization of the dictionary introduced by Frances for conformal Killing fields, which characterizes (local) essentiality via the so-called holonomy associated to a singularity of an infinitesimal automorphism.