I would like to learn more about Cartan Geometry ("les espaces généralisés de Cartan"). I ordered Rick Sharpe's book "Differential Geometry: Cartan's generalization…", which would take a long time to arrive though. In the mean time, can someone recommend possibly some online lecture notes, or some online papers containing an introduction to Cartan Geometry, with I hope several examples worked out?
I kind of get what it is. When you model it on Euclidean geometry, it yields Riemannian Geometry. When you model it on affine space, it yields a manifold with an affine connection. When you model it on G/H, it gives a kind of curved space, which looks infinitesimally like G/H. Ok, this is the rough idea, but I would like to learn a bit more. Does anyone know of a few online resources on the topic by any chance?
Edit 1: I thank everyone who replied. I have learned a lot from various people, and I thank you all.
Edit 2: Check out the very nice and short introduction to Cartan geometry by Derek Wise (it is very well written and concise):
https://arxiv.org/abs/gr-qc/0611154
A hilarious point in the explanation, is the image of a hamster rolling inside a sphere tangent to the manifold (followed by expressions such as "hamster configurations" etc). It was really funny, and it explained the idea very well. I just realized that it is a shortened version of Derek Wise's thesis, which Tobias Fritz had already suggested as a reference in the comments below (thank you!).
Edit 3: R. Sharpe's book has arrived. I find it interesting that R. Sharpe's motivation for writing the book was this question "why is Differential Geometry the study of a connection on a principal bundle?". He wrote that he kept bugging differential geometers with this question, and that attempting to answer this question eventually led him to write his book on Cartan geometry! I appreciate anecdotes like this one.
Best Answer
There is a series of four recorded lectures by Rod Gover introducing conformal geometry and tractor calculus. Tractor bundles are natural bundles equipped with canonical linear connections associated to $(\mathfrak{g}, H)$-modules. Tractor connections play the same role in general Cartan geometries that the Levi-Civita connection plays in Riemannian geometry; for general Cartan geometries the tangent bundle does not have a canonical linear connection.
There's also a set of introductory notes on conformal tractor calculus written by Rod Gover and Sean Curry.
If you have the book in your library, I would also suggest having a look at Cap & Slovak's Parabolic Geometries text. This is the modern bible on Cartan geometry, and parabolic geometries in particular. It is more terse than Sharpe, but also covers much more. Parabolic geometries are Cartan geometries modelled on $(\mathfrak{g}, P)$ where $\mathfrak{g}$ is semisimple and $P$ is a parabolic subgroup. Parabolic geometries include conformal, projective geometry, CR geometry, and many more geometries of interest.
In the parabolic setting representation-theoretic tools are often used to construct invariant differential operators. For instance, there are so-called BGG sequences of operators associated to irreducible representations, which in the flat case compute the same sheaf cohomology groups as the twisted de Rham sequence.