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Essential Parabolic Structures and Their Infinitesimal Automorphisms
Differential Geometry · Submitted 1 Nov 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.99 | 0.94 | 0.13 | asserted 1.00 |
| s1 | We show that a parabolic structure is inessential whenever the automorphism group acts properly on the base space. | result 1.00 | 0.89 | 0.30 | asserted 1.00 |
| s2 | As a corollary of the generalized Ferrand-Obata theorem proved by C. | result 0.67 | 0.26 | 0.06 | unclear 0.79 |
| s3 | 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. | result 0.64 | 0.60 | 0.31 | asserted 1.00 |
| s4 | 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. | contribution 0.96 | 0.90 | 0.12 | asserted 1.00 |
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The paper
Original 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.
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How the literature uses it
2 arXiv papers cite this in 9 sentences
Papers citing it · 5 sentences without a confident label
Contests it 1 paper
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We imagine that this was the intended definition in [ref] , since all of the results from that paper seem to work otherwise verbatim with this definition, but as written it was assumed that $λ_*^$ acts by scalar multiplication on each grading component $𝔤_i$, in which case it is not hard to see that $λ_*^$ must be a scalar multiple of the grading element.
Intrinsic holonomy and curved cosets of Cartan geometries · Inessential automorphism groups of parabolic geometries · contest 0.79
Compares with it 1 paper
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While Proposition [ref] does, more or less immediately, recover Theorem 1.2 from [ref] , and it does tell us that $ξ$ is $λ$-essential if $ω_ℯ^norm(ξ)∉(λ_*)$, the considerations above severely limit its ability to obtain more global results about $λ$-inessential infinitesimal automorphisms.
Intrinsic holonomy and curved cosets of Cartan geometries · Inessential automorphism groups of parabolic geometries · 0.86
What it cites · 18 sentences without a confident label
Builds on 2 papers
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Then, we apply the identity reviewed in Section [ref] to prove a generalization of results of [ref] , which give a “dictionary” relating essentiality of an infinitesimal automorphism near a singularity $x_0$ to properties of its holonomy $h^t$ at $x_0$, a one-parameter subgroup of $P$ which is determined up to conjugacy (cf. Definition [ref]; it should be emphasized that this notion of “holonomy” of the singularity of an infinitesimal automorphism is distinct from the holonomy of, e.g., a Cartan connection which is common in the literature).
Local Dynamics of Conformal Vector Fields · Organization of the text and summary of main results · 0.96
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The following identity was proven for parabolic geometries in [ref] and the proof carries over without substantive changes to Cartan geometries of general type (cf. also Lemma 1.5.12 in [ref] ):
math.DG/0508535 · Background results on infinitesimal automorphisms · 0.80
Uses the method or tool of 2 papers
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The assumption that the parabolic geometries are of rank one type is key for the argument in [ref] , because it allows the author to exploit the so-called “north-south dynamics” on the homogeneous model $G/P$.
math.DG/0608537 · Lichnérowicz theorem for rank one parabolic geometries · 0.80
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I am grateful to Charles Frances for discussions about the methods used in [ref] and [ref] , and to Felipe Leitner for useful comments on an earlier version of the text.
Local Dynamics of Conformal Vector Fields · Acknowledgements · 0.97
Cites as background 2 papers
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Let us begin by recalling the definitions of parabolic geometries and their Weyl structures (the latter, introduced by A. Čap and J. Slovák in [ref] , will be central to our notion of essential parabolic structures).
math.DG/0001166 · Background on parabolic geometries and their Weyl structures · 0.97
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We recall some general notions, mainly following the development of [ref] (cf. also 1.5 of [ref] ), in the setting of a general Cartan geometry $(→M,ω)$ of type $(G,P)$ (for now not assumed to be of parabolic type).
math.DG/0508535 · Background results on infinitesimal automorphisms · 1.00
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