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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2 votes
1 answer
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Characterization of reductive Klein geometries

In my struggle to understand Cartan/Klein geometries, I have the intuition that reductive Klein geometries are the link to connect the "classical" differential geometry approach with this &...
13 votes
2 answers
970 views

Moving frames method

I want to grasp the moving frames method but I find some obstacles. I don't know if this question is suitable for MO, if it is not the case please let me know and I will move it. I am aware there are ...
1 vote
3 answers
215 views

Equivalence problem of classifying heat equations

I have tried to search for references online but I am unable to do so. I am looking for references that uses Cartan's method of moving frames to classify heat equations. Also are there references that ...
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4 votes
0 answers
96 views

Cartan geometry: jet space perspective on the tractor bundle

Let $G$ a Lie group and $H\subset G$ a Lie subgroup. For simplicity we assume that the adjoint action of $H$ on $\mathfrak g/\mathfrak h$ is faithful. Let $M$ a differentiable manifold of the same ...
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5 votes
0 answers
146 views

Torsion-free Cartan connections

Let $M$ a differentiable manifold and $H\subset G$ a Lie group with a closed subgroup such that $G/H$ is connected. A $H\subset G$-Cartan connection on $M$ can be defined by A principal $G$-bundle on ...
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1 vote
2 answers
190 views

Understanding the space of structures

Apologies if this question is a little vague. I have seen written in a few places that the space of projective structures on a Riemann surface is an affine space modelled on the space of holomorphic ...
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3 votes
0 answers
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Restriction of complete 1-forms to closed submanifolds (Sharpe's book on Cartan geometries)

In his Book Differential Geometry: Cartan's generalization of Klein's Erlangen Program, Sharpe gives the following definition of a complete 1-form: Soon thereafter he gives the following example: I ...
4 votes
0 answers
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Eigenvalues and eigenvectors of the exceptional simple Lie group E6, E7, E8

What is the significance of the eigenvalues and eigenvectors of the exceptional simple Lie group root lattice to the Lie group or other mathematics branches? For example, E6, we have $$ \left( \begin{...
5 votes
1 answer
153 views

When are principal bundles supporting Cartan connections isomorphic?

Suppose I have two Cartan geometries $(\mathscr{G}_1,\omega_1)$ and $(\mathscr{G}_2,\omega_2)$ of type $(G,H)$ over the same manifold $M$. What conditions on $G$ and $H$ allow us to conclude that $\...
6 votes
1 answer
358 views

Name for a class of almost symplectic manifolds

A $2n$-dimensional manifold $M$ is said to be almost symplectic if it possesses a non-degenerate two-form $\omega \in \Omega^2(M)$. Equivalently, an almost symplectic structure is a $G$-subbundle $P \...
6 votes
1 answer
173 views

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 ...
5 votes
2 answers
307 views

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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7 votes
1 answer
285 views

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'...
3 votes
2 answers
386 views

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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2 votes
0 answers
135 views

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 $...
6 votes
1 answer
878 views

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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15 votes
2 answers
2k views

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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2 votes
0 answers
70 views

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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3 votes
1 answer
225 views

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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3 votes
1 answer
219 views

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 ...
2 votes
1 answer
198 views

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\\...
5 votes
2 answers
599 views

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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8 votes
2 answers
559 views

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 ...
14 votes
1 answer
1k views

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 ...
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