A principal bundle with a Lie group $H$ consists of a manifold $P$ and a free proper smooth $H$-action $P\times H\to P$. There is a unique smooth manifold structure on the quotient space $M=P/H$ such that the canonical map $\pi :P\to M$ is smooth. $M$ is called a base manifold and $H\to P\to M$ stands for the bundle. The most fundamental examples of principal bundles are the homogeneous spaces $H\subset G\to G/H$, where $H$ is a closed subgroup of $G$. The pair $(𝔤,𝔥)$ is a Klein pair. A model geometry consists of a Klein pair $(𝔤,𝔥)$ and a Lie group $H$ with Lie algebra $𝔥$. In this paper, the author describes a Klein geometry as a principal bundle $H\to P\to M$ equipped with a $𝔤$-valued 1-form $\omega$ on $P$ having the properties (i) $\omega :TP\to 𝔤$ is an isomorphism on each fibre, (ii) $R_h^{*}\omega =\text{Ad}\left(h^{-1}\right)\omega$ for all $h\in H$, (iii) $\omega \left(v^{†}\right)$ for each $v\in 𝔥$, (iv)

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

@article{701688,
     title = {An introduction to Cartan Geometries},
     booktitle = {Proceedings of the 21st Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {2002},
     pages = {[61]-75},
     mrnumber = {MR1972425},
     zbl = {1028.53026},
     url = {http://dml.mathdoc.fr/item/701688}
}

Sharpe, Richard. An introduction to Cartan Geometries, dans Proceedings of the 21st Winter School "Geometry and Physics", GDML_Books,  (2002), pp. [61]-75. http://gdmltest.u-ga.fr/item/701688/