Questions tagged [cartan-geometry]

Cartan geometry is the geometry of spaces that are locally (infinitesimally, tangentially) like coset spaces G/H, i.e. like Klein geometries. Intuitively, Cartan geometry studies the geometry of a manifold by ‘rolling without sliding’ the ‘model geometry’ G/H along it. Hence Cartan geometry may be thought of as the globalization of the program of Klein geometry initiated in the Erlangen program.

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What exactly is a Cartan radius vector (and its role in Poincaré gauge theories)

I am studying approaches to gravity where the Poincaré group is "gauged". The original motivation of this is to understand what is meant on the statement that "Teleparallel gravity is a gauge theory ...
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How to prove two curves in the frame bundle to project to the same curve on base manifold?

There is a problem about Cartan's development, arising from the paper 'Kinetic Brownian motion on Riemannian manifolds', Subsection 2.4.1. To be precise, let $(M,g)$ be a $d$-dimensional complete ...
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Termination of Cartan's equivalence method

The Cartan-Kuranishi theorem guarantees that a PDE or EDS can always be completed, by prolongation, to involution. My question is, and this is quite murky to gauge from the literature, whether Cartan'...
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Is the development map in Hyperbolic geometry related to development in Cartan geometry?

I am more familiar with Cartan geometry, and in this setting we have a notion of development of curves. As described in Cap & Slovak 1.5.17, on a Cartan geometry $(\mathcal{P} \to M, \omega)$ ...
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Universal covariant derivative decomposition in a Cartan geometry

I'm reading the book "Differential Geometry: Cartan's Generalisation of Klein's Erlangen Program" from Sharpe. Given a reductive model geometry, where $\mathfrak g=\mathfrak h\oplus \mathfrak p$ with $...
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Cartan's Structure Equations VS Cartan's Method of Equivalence

There have been a number of posts on related questions, such as: Geometric interpretation of Cartan's structure equations, What is the geometric significance of Cartan's structure equations? ...
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Online References for Cartan Geometry

I would like to learn more about Cartan Geometry ("les espaces généralisés de Cartan"). I ordered Rick Sharpe's book "Differential Geometry: Cartan's generalization...", which would take a long time ...
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What are the Cartan geometries modeled on $\mathbb{H}P^m$?

I am not an expert on Cartan Geometry (in fact, I have just read and understood the definition, at a basic level). I have the following questions: 1) Can someone please describe what are the Cartan ...
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The standard projective cotractor bundle and its cocycle of transition functions

Let $M$ be a smooth $n$-manifold. A projective structure on $M$ is a class $p$ of torsion-free connections on $TM$ which have the same geodesics as unparametrized curves. The bundle of densities of ...
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What is the curved version of the Tits fibration for $G_2$?

Let $\require{AMScd}$ \begin{CD} G_2/(P_1\cap P_2) @= G_2/(P_1\cap P_2)=:\mathbb{I}\\ @V \lambda V V @VV \pi V\\ \mathbb{Q}_5:=G_2/P_1 && G_2/P_2=:\mathbb{N}_5 \end{CD} be the Tits ...
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What is the space parametrising the curved sub-Cartan geometries of a flat Cartan geometry?

I'm basically wondering how to make "curved" the first column of the diagram $\require{AMScd}$ \begin{CD} P_1 @>\textrm{inclusion} >> G\\ @V \omega_0 V P_1\cap P_2 V @V\omega V P_2 V\\...
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Relationship between parabolic subgroups and parabolic subalgebras over non-algebraically-closed fields

Let $\mathbb{K}$ be an arbitrary field, and $G$ an algebraic group (or group variety?) over this field. A Borel subgroup of $G$ is a connected solvable subgroup variety $B$ of $G$ such that $G/B$ is ...
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Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5

Up to topology, the 5D homogeneous space $$ G_2/P $$ of the (real form of the) 14D exceptional Lie group $G_2$ is the 5D jet space $$ M:=J^1(2,1)=\{(x,y,u,p,q)\} $$ of scalar functions in two ...
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Intuition for the Cartan connection and “rolling without slipping” in Cartan geometry

Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$. The Cartan connection is supposed to formalize what it means to "roll ...