An introduction to Cartan Geometries

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