An introduction to Cartan Geometries

Sharpe, Richard

Proceedings of the 21st Winter School "Geometry and Physics", GDML_Books, (2002), p. [61]-75 / Harvested from

Résumé

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)

Détail

MR MR1972425 | Zbl 1028.53026

EUDML-ID : urn:eudml:doc:220395
Mots clés: