An introduction to Cartan Geometries

Sharpe, Richard. "An introduction to Cartan Geometries." Proceedings of the 21st Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 2002. [61]-75. <http://eudml.org/doc/220395>.

@inProceedings{Sharpe2002,
abstract = {A principal bundle with a Lie group $H$ consists of a manifold $P$ and a free proper smooth $H$-action $P\times H\rightarrow P$. There is a unique smooth manifold structure on the quotient space $M=P/H$ such that the canonical map $\pi : P \rightarrow M$ is smooth. $M$ is called a base manifold and $H\rightarrow P\rightarrow M$ stands for the bundle. The most fundamental examples of principal bundles are the homogeneous spaces $H\subset G\rightarrow G/H$, where $H$ is a closed subgroup of $G$. The pair $(\mathfrak \{g\},\mathfrak \{h\})$ is a Klein pair. A model geometry consists of a Klein pair $(\mathfrak \{g\},\mathfrak \{h\})$ and a Lie group $H$ with Lie algebra $\mathfrak \{h\}$. In this paper, the author describes a Klein geometry as a principal bundle $H\rightarrow P\rightarrow M$ equipped with a $\mathfrak \{g\}$-valued 1-form $\omega $ on $P$ having the properties (i) $\omega : TP\rightarrow \mathfrak \{g\}$ is an isomorphism on each fibre, (ii) $R^*_h\omega = \text\{Ad\}(h^\{-1\})\omega $ for all $h\in H$, (iii) $\omega (v ^\{\dag \})$ for each $v\in \mathfrak \{h\}$, (iv)},
author = {Sharpe, Richard},
booktitle = {Proceedings of the 21st Winter School "Geometry and Physics"},
keywords = {Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)},
location = {Palermo},
pages = {[61]-75},
publisher = {Circolo Matematico di Palermo},
title = {An introduction to Cartan Geometries},
url = {http://eudml.org/doc/220395},
year = {2002},
}

TY - CLSWK
AU - Sharpe, Richard
TI - An introduction to Cartan Geometries
T2 - Proceedings of the 21st Winter School "Geometry and Physics"
PY - 2002
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [61]
EP - 75
AB - A principal bundle with a Lie group $H$ consists of a manifold $P$ and a free proper smooth $H$-action $P\times H\rightarrow P$. There is a unique smooth manifold structure on the quotient space $M=P/H$ such that the canonical map $\pi : P \rightarrow M$ is smooth. $M$ is called a base manifold and $H\rightarrow P\rightarrow M$ stands for the bundle. The most fundamental examples of principal bundles are the homogeneous spaces $H\subset G\rightarrow G/H$, where $H$ is a closed subgroup of $G$. The pair $(\mathfrak {g},\mathfrak {h})$ is a Klein pair. A model geometry consists of a Klein pair $(\mathfrak {g},\mathfrak {h})$ and a Lie group $H$ with Lie algebra $\mathfrak {h}$. In this paper, the author describes a Klein geometry as a principal bundle $H\rightarrow P\rightarrow M$ equipped with a $\mathfrak {g}$-valued 1-form $\omega $ on $P$ having the properties (i) $\omega : TP\rightarrow \mathfrak {g}$ is an isomorphism on each fibre, (ii) $R^*_h\omega = \text{Ad}(h^{-1})\omega $ for all $h\in H$, (iii) $\omega (v ^{\dag })$ for each $v\in \mathfrak {h}$, (iv)
KW - Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)
UR - http://eudml.org/doc/220395
ER -