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Title: An introduction to Cartan Geometries (English)
Author: Sharpe, Richard
Language: English
Journal: Proceedings of the 21st Winter School "Geometry and Physics"
Volume:
Issue: 2001
Year:
Pages: [61]-75
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Category: math
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Summary: 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 $(\frak g,\frak h)$ is a Klein pair. A model geometry consists of a Klein pair $(\frak g,\frak h)$ and a Lie group $H$ with Lie algebra $\frak h$. In this paper, the author describes a Klein geometry as a principal bundle $H\to P\to M$ equipped with a $\frak g$-valued 1-form $\omega$ on $P$ having the properties (i) $\omega: TP\to\frak g$ 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 $v\in\frak h$, (iv) (English)
MSC: 53B15
MSC: 53C05
MSC: 53C10
MSC: 53C30
MSC: 58A05
idZBL: Zbl 1028.53026
idMR: MR1972425
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Date available: 2009-07-13T21:46:51Z
Last updated: 2012-09-18
Stable URL: http://hdl.handle.net/10338.dmlcz/701688
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