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.
Questions tagged [cartan-geometry]
57 questions
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Cartan geometry on manifolds with boundary
I was reading Sharpe's text on Cartan geometry, and I started to wonder:
Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth manifold with boundary?
Is there any significant…
ಠ_ಠ
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Definition of parallel transport by M. Berry as a transport without twisting around normal vector
I have been trying to read Michael Berry's article named "The Quantum Phase, Five Years After" but I am stuck at the introduction by Berry's definition of parallel transport (found on the 2nd page here:…
TheQuantumMan
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5
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Existence of special transversal on foliation
This is a somewhat technical question about a line in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Program in the proof of the structure theorem, Theorem 8.3 in chapter 2. We have assumed that the leaf space of a…
subrosar
- 5,086
5
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Torsion Free Spin Connection
Ok I am not exactly sure how much of this common notation/terminology, and how much is unique to the book I'm reading, so bear with me for a moment here. First we have a vector bundle $E$ associated to the orthonormal frame bundle of some manifold…
Chris
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Finding the orbits of $SL(n+1)$ and a parabolic subgroup $P$ in a representation?
To provide some motivation, given a Cartan geometry modeled on the Klein geometry $G/P$ and a parallel section of a tractor bundle for the $G$-representation $V$, the space is partitioned into different geometries determined by the $G$- and…
ಠ_ಠ
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1-Forms on $SO(3)$ and $S^2$
The is a mistake somewhere in the following reasoning and I can't seem to detect which argument is wrong.
Consider the lie group $SO(3)$ with $e_{1},e_{2},e_{3}$ as left invariant vector fields. Each of which generates an $S^1$ action. Quotienting…
u184
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4
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Definition of $(\mathfrak{g}, H)$-modules used in Cap and Slovak's parabolic geometry text?
On page 68 of Cap and Slovak's parabolic geometries text, they mention $(\mathfrak{g}, H)$-modules for $\mathfrak{g}$ the Lie algebra of a Lie group $G$ and $H \hookrightarrow G$ a closed subgroup. Though they are interested in the case where $G$ is…
ಠ_ಠ
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1 answer
moving frame with maple
I have already ask this question on stackoverflow, but since it concerns as mathematics than computer science, I ask it here too.
I would like to make a classical computation using maple. I would like define an abstract moving frame (e_1,e_2),…
Paul
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3
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1 answer
Understanding Cartan bundle for parabolic geometry
Parabolic Geometries I by Cap and Slovak outlines an equivalence between regular normal Cartan geometries and regular infinitesimal flag structures associated to a $|k|$-graded parabolic geometry $(G,P)$. In the case of $|1|$-graded parabolic…
subrosar
- 5,086
3
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1 answer
Explicit formula for the principal connection 1-form induced by a Cartan connection
Let $P \subseteq G$ be a closed Lie subgroup. Suppose that a principal $P$-bundle $\mathcal{P} \to M$ is equipped with a Cartan connection $\omega: T\mathcal{P} \to \mathfrak{g}$. Then the extended principal $G$-bundle $\mathcal{P} \times_P G \to M$…
ಠ_ಠ
- 11,310
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How do I get the expression for the Killing form of $\mathfrak{u}(2)$?
I saw in the book of Mark Hamilton "Mathematical of gauge theory" in the pag. 126 that the killing form to lie algebra $\mathfrak{u}(2)$ is
$$B(X, Y) = 4 Tr(XY) - 2 Tr(X)Tr(Y).$$
I want prove this for this case (not the general case…
José Psicodélico
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3
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What is the general setting for the Cauchy-Riemann equations and the triholomorphic equations?
Let $U \subseteq \mathbb{R}^n$ be an open subset, let $M_n(\mathbb{R})$ be the algebra of real $n \times n$ matrices, and let $B \subseteq M_n(\mathbb{R})$ be a real subalgebra. Assume the coordinates on $\mathbb{R}^n$ are $x_1,\ldots,x_n$. Let…
Malkoun
- 5,594
3
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1 answer
Understanding an abstract notion of "frame" for Klein geometries
The Wikipedia page for "Moving frame" introduces the following abstract notion of a "frame" for a Klein geometry:
Formally, a frame on a homogeneous space $G/H$ consists of a point in the tautological bundle $G → G/H$.
Based on the previous…
Adam Williams
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3
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1 answer
Model for symplectic geometry
An almost symplectic structure on a smooth even dimension manifold $M$ can be viewed as a reduction of structure group $Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$ for the principal frame bundle $\mathcal{F}(M)$. If this structure group…
user564024
3
votes
1 answer
Method of moving frames for curves on $S^2$
I am rather confused about elements of the method of moving frames and how to apply the method of moving frames.
$S^2$ is a homegeneous space for the Lie group $SO(3)$.
We want to construct a lift $\tilde{\alpha}:U\subset\mathbb{R} \to SO(3)$ of a…
Andrew Whelan
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