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Intrinsic holonomy and curved cosets of Cartan geometries
Differential Geometry · Submitted 11 Sep 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We conclude the paper by using the machinery developed to generalize the de Rham decomposition theorem for Riemannian manifolds and give a potentially useful characterization of inessential automorphism groups for parabolic geometries.
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The abstract, sentence by sentence
We provide an intrinsic notion of curved cosets for arbitrary Cartan geometries, simplifying the existing construction of curved orbits for a given holonomy reduction. To do this, we define an intrinsic holonomy group, which is shown to coincide precisely with the standard definition of the holonomy group for Cartan geometries in terms of an associated principal connection. These curved cosets retain many characteristics of their homogeneous counterparts, and they behave well under the action of automorphisms. We conclude the paper by using the machinery developed to generalize the de Rham decomposition theorem for Riemannian manifolds and give a potentially useful characterization of inessential automorphism groups for parabolic geometries.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We provide an intrinsic notion of curved cosets for arbitrary Cartan geometries, simplifying the existing construction of curved orbits for a given holonomy reduction. | contribution 1.00 | 0.95 | 0.33 | asserted 1.00 |
| s1 | To do this, we define an intrinsic holonomy group, which is shown to coincide precisely with the standard definition of the holonomy group for Cartan geometries in terms of an associated principal connection. | method 0.67 | 0.91 | 0.11 | asserted 1.00 |
| s2 | These curved cosets retain many characteristics of their homogeneous counterparts, and they behave well under the action of automorphisms. | result 0.80 | 0.75 | 0.09 | asserted 0.99 |
| s3 | We conclude the paper by using the machinery developed to generalize the de Rham decomposition theorem for Riemannian manifolds and give a potentially useful characterization of inessential automorphism groups for parabolic geometries. | contribution 0.58 | 0.94 | 0.78 | hedged possibility 0.79 |
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The paper
Original abstract
We provide an intrinsic notion of curved cosets for arbitrary Cartan geometries, simplifying the existing construction of curved orbits for a given holonomy reduction. To do this, we define an intrinsic holonomy group, which is shown to coincide precisely with the standard definition of the holonomy group for Cartan geometries in terms of an associated principal connection. These curved cosets retain many characteristics of their homogeneous counterparts, and they behave well under the action of automorphisms. We conclude the paper by using the machinery developed to generalize the de Rham decomposition theorem for Riemannian manifolds and give a potentially useful characterization of inessential automorphism groups for parabolic geometries.
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How the literature uses it
1 arXiv paper cite this in 3 sentences
What it cites · 3 sentences without a confident label
Contests 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.
Essential Parabolic Structures and Their Infinitesimal Automorphisms · Inessential automorphism groups of parabolic geometries · contest 0.79
Compares with 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.
Essential Parabolic Structures and Their Infinitesimal Automorphisms · Inessential automorphism groups of parabolic geometries · 0.86
Cites as background 2 papers
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For parabolic geometries, the topic of tractor connections is covered extensively in [ref] , and their holonomy (defined in the usual way for vector bundle connections) has been thoroughly studied in several cases (see [ref] , [ref] , and [ref] ).
Projective Geometry I: Principles and Properties · Relation to tractor connections · 1.00
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For parabolic geometries, the topic of tractor connections is covered extensively in [ref] , and their holonomy (defined in the usual way for vector bundle connections) has been thoroughly studied in several cases (see [ref] , [ref] , and [ref] ).
Projective Geometry II: Cones and Complete Classifications · Relation to tractor connections · 1.00
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