Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then it is known that $G/H$ can be equipped with a unique differentiable structure such that $G\xrightarrow{\pi} G/H$ is a locally trivial fiber bundle. Then, the standard theory of principal bundles, connections, associated vector bundles and the associated covariant derivatives applies to $G\xrightarrow{\pi} G/H$, and I guess some simplifications appear. My question is:
The theory of principal bundles connections, associated vector bundles and the associated covariant derivatives applied to principal bundles of the form $G\xrightarrow{\pi} G/H$ goes under some special name? Could you give me a reference where this theory is developed? Is this related to the notion of "Cartan connection"?
Thanks.
Best Answer
Vector bundles associated to the principal bundle $G\to G/H$ can be equivalently characterized as homogeneous vector bundles (i.e. as vector bundles endowed with a lift of the action of $G$ on $G/H$ to an action by vector bundle homomorphisms.) There also is a notion of homogeneous connections and so on.
There is a relation between homogeneous spaces and Cartan connection, which is that the Maurer-Cartan form on $G$ defines a Cartan connection on $G\to G/H$ and this is the so-called homogeneous model of Cartan geometries of type $(G,H)$. This is related to the fact that the tangent bundle $T(G/H)$ always is an associated bundle to $G\to G/H$. Roughly, you take the Maurer-Cartan form as defining a geometric structure on $G/H$ whose automorphisms are exactly the left actions of elements of $G$ and then look for "curved analogs" of this situation (which are Cartan geometries).
An introduction to geometry of homogeneous spaces and the relation to Cartan geometries can be found in sections 1.4 and 1.5 of this book by J. Slovak and myself.