# Online References for Cartan Geometry

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.

• Bill Goldman has written some very nice papers explaining the theory of flat Cartan geometries: "Locally homogeneous geometric manifolds" and "Geometric structures on manifolds". My work on Cartan geometries consists in (1) look at the flat cases and then (2) argue that the curved cases are not much different. – Ben McKay Aug 6 '17 at 20:23
• Derek Wise's thesis at math.ucr.edu/home/baez/thesis_wise.pdf also contains a nice introduction. – Tobias Fritz Aug 6 '17 at 22:52
• I am currently reading a paper by Derek Wise: arxiv.org/abs/gr-qc/0611154 It has a very nice introduction to Cartan geometry with some nice figures. – Malkoun Aug 7 '17 at 9:30

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.

• It sounds interesting! Thank you, I will have a look. I think I have stumbled upon the expression "tractor calculus" or something close, but did not know that it was directly related to Cartan geometry. Thank you! – Malkoun Aug 6 '17 at 9:23
• Thank you. It looks like Gover and Curry applied Cartan geometry modeled on the conformal sphere to get various conformally invariant tensor quantities. It is very interesting, but do you know of some other references where they discuss more the Cartan geometry part please? – Malkoun Aug 6 '17 at 9:37
• So that's what "parabolic geometry" means! Ok, in some sense Cartan geometry modeled on a kind of generalized partial flag manifold (I am not sure if I am using the words right). Thank you again. I will accept your answer. – Malkoun Aug 6 '17 at 9:46
• Regarding BGG resolutions, Baston and Eastwood wrote a book entitled "The Penrose Transform", where they apply the BGG resolutions to Twistor theory (this is how I have first learned about them). BGG resolutions are more efficient than De Rham, being adapted to the given parabolic geometry. Parabolic geometries are broad enough to include the Cartan geometries that I am interested in at the moment. Thank you for this informative reply. – Malkoun Aug 6 '17 at 10:04
• This paper by D. Calderbank and T. Diemmer seems relevant too: arxiv.org/abs/math/0001158 – Malkoun Aug 6 '17 at 10:07

You can download some shorter text dealing with conformal geometries from Slovák's homepage One could download his book with Čap from the infamous Russian server (which appears to be down at the moment). I'm not sure whether the Sharpe's book is there as well.

I think a really good introductory text is the book Cartan for beginners by Ivey and Landsberg which doesn't really deal with Cartan geometries per se but rather teaches the Cartan method which, in a sense, is precisely the machinery that really makes the Cartan geometries work.

Tractor connections and tractor bundles are not really part of Cartan geometries but rather an independent (and in many cases equivalent) approach to study geometrical problems. In conformal geometry they were discovered by T. Y. Thomas in the mid twenties. See Thomas's structure bundle for conformal, projective and related structures. for details.

• Thank you so much. I am not completely ignorant of Cartan's methods, but I am always willing to learn more. There is a lot I don't understand deeply though. I have downloaded the text by Slovák, thanks. Now I have a lot to read. I really like that the Cartan point of view, particularly Cartan geometry, provides a general and unifying point of view for differential geometry (Riemannian, conformal, projective etc.). – Malkoun Aug 6 '17 at 12:45
• I guess the notorious difficulty of Cartan's papers has something to do with that. :) Sharpe's book explains this unifying point of view really well. It's point of view is that a Cartan geometry is to Klein geometry what Riemannian geometry is to Euclidean geometry. – Vít Tuček Aug 6 '17 at 12:50
• I agree that E. Cartan is difficult to read. – Malkoun Aug 6 '17 at 12:51
• I would say that tractor connections are part of Cartan geometry. You can define them as associated bundles to the Cartan principal bundle which come from $(\mathfrak{g}, H)$-modules. It's just a matter of perspective whether you want to build the Cartan principal bundle first and then take associated bundles, or construct tractor bundles and then take their frame bundles (for faithful reps). – ಠ_ಠ Aug 6 '17 at 19:22
• Well... it's a matter of taste. You are right of course, for any Cartan geometry you get plenty of tractor bundles and tractor connections. But my point is that you can work with tractor bundles and connections without ever mentioning Cartan connection. – Vít Tuček Aug 6 '17 at 19:25