Talk:Cartan connection

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The main ideas and definitions are explained, with plenty of motivation, but applications, examples, context and history need work (with the intro to be rewritten accordingly). Geometry guy 19:35, 12 April 2007 (UTC)

1 Initial comment on new article
2 Name
3 AFFINE connection
4 The article is getting too long
5 our discussion is getting too long
6 What's a Cartan connection
7 Todo
8 Or not todo, that is the question
9 What's a Cartan connection? An answer
10 Rewriting
11 Different viewpoints on Cartan connections
12 Progress report
- 12.1 More progress

[edit] Initial comment on new article

This centralises the material here - but still needs plenty of work.

Charles Matthews 17:50, 23 Dec 2003 (UTC)

[edit] Name

Tell me if I'm wrong, I think there is no such thing as Cartan connection, there is connection which is decribed by Cartan formalism or my method of Moving frames, it is ok to have such an article with a bit wrong name, but it is wrong to make it central for this metter...

Tosha 21:40, 25 May 2004 (UTC)

This is discussed (a bit) on this page at PlanetMath:

http://planetmath.org/encyclopedia/Connection.html .

I noticed, doing a Google search, that the most obvious hits for 'Cartan connection' seemed to be in mathematical physics.

Anyway, 'Cartan connection' can be called a genuine topic.

Charles Matthews 04:44, 26 May 2004 (UTC)

I found nice name connection form and already put something inside (clearly more work needed). I think to remove Almost formal introduction from here, infact although it is almost correct I realized that it does not exactly belongs here, and I'm not that strong in history to find what Cartan did and what he did not. I have a feeling that he only considered connection on the tangent bundle or asociated principle bundle???

While I was writing connection form I looked in Koboyashi, he makes clear difference between connection and covariant derivative, one for Principle bundles, an other for asociated vector bundles. Is it indeed standard???

Tosha 20:23, 9 Jun 2004 (UTC)

[edit] AFFINE connection

Reading the article at PlanetMath made me wonder once again. Why is it called an AFFINE connection? Phys 05:33, 14 Aug 2004 (UTC)

My conjecture is that originally the name was given depending on property of correspondent parallel translation, i.e. if p.t. gives linear map it is linear connection, if affine then it is affine one. now word affine is used wrongly... Tosha 11:08, 14 Aug 2004 (UTC)

This is now thoroughly explained at affine connection. Geometry guy 16:10, 6 April 2007 (UTC)

[edit] The article is getting too long

It does not seem that someone has an idea what should be inside here. Everyone writes something related but there is no structure.

I do not know phisics enough to do this article, but I would suggest to put all mathematics into connection form and leave here mostly phisics, I mean that in phisicas there are specific notation, plus it is probably more oriented to spin-bundles and to pseudo-Riemannian mnfls...??? Tosha 18:14, 7 Sep 2004 (UTC)

My idea would be the opposite - to put the vierbein material on its own page, where the computational side could be developed.

Charles Matthews 18:20, 7 Sep 2004 (UTC)

so, do you want everything from connection form to be here?

as you mentioned before, the term Cartan connection used mostly in math phisics... but anyway I agree, I only sugggest that one should think about structure of this article. Tosha

Well, too much material is a better problem than too little. The page was created out of several treatments 'in parallel'. There is some basic difficulty with Cartan's work, it seems. Just translating it into modern language loses some geometry, without creating a unique, best formulation ... Anyway in that situation WP has to allow multiple points of view, as a matter of basic policy; in that way it is a little different from a mathematics textbook. I think this is an interesting example of that principle at work. Some day, it all should be edited seriously. But actually I'm not too worried, yet. The basic article on Maurer-Cartan equations was only created today(!), so it isn't very surprising that the more serious aspects are only slowly appearing.

Charles Matthews 21:16, 7 Sep 2004 (UTC)

so, do you want everything from connection form to be here?

Actually, a connection form is something a little different than a Cartan connection. The definition in connection form is of an Ehresmann connection, which is of a very different character. Although there is a technique for taking a Cartan connection and producing an Ehresmann connection out of it, this is almost never done because (I suspect) there are representation-theoretic obstacles to doing it, and because all of the natural features of Cartan geometries are already present in the Cartan connection picture so nothing is to be gained.

I do plan to write a brief discussion of how one uses a coupling of the Cartan connection to produce: (1) the spin connection, (2) the local twistor connection, and (3) the tractor connection, as these seem to include the special cases of interest to Tosha and others. The "coupled Cartan connection" seems to be what most physicists mean when they say Cartan connection, but I should emphasize that these coupled connections are not the fundamental object Cartan introduced although he did obtain coupled Cartan connections by... er... coupling.

In pure mathematics, there are many other examples of purely Cartanian connections such as the Webster-Stanton connection in complex analysis. Cartan, the man himself, introduced this connection in the special case of domains in $\mathbb C^2$ and solved the equivalence problem under biholomorphism for these domains.

151.204.12.219 17:39, 8 Sep 2004 (UTC)

Well, it would be great to have good write-ups of this whole area - and thank you for your contributions. Shuffling things under the correct headings is really a secondary process.

Charles Matthews 18:12, 8 Sep 2004 (UTC)

sure, but there is yet an other thing, the article should be written the way it would be easy to edit, that means that it must be a structure this article does not have. It might well happen that it will be easier to rewite it all instead of editing... The first thing is to deside what is in there ands state it. I see now that many things from connection form and covariant derivative appear her, soon it will be all Diff.geometry, I do not see much sense in this. At least someone should answer "whom this article might help?", I can not see even an imaginary person.

This article might be for example about original Crtan's work (which by the way not only for $\mathbb C^2$ ) or its modern meaning in phisics (which I do not know much about) or something else but it should not be everything.

I understand that it should not be perfect, but it should get better at least, and also wikipedia is not only for editors it is mostly for readers Tosha

I see now that many things from connection form and covariant derivative appear her, soon it will be all Diff.geometry, I do not see much sense in this.

Well, that is the point that I tried to make before: that Cartan connections are rather more subtle than Ehresmann connections and covariant derivatives defined thereupon. For instance, for an Ehresmann connection to induce covariant derivatives, it generally needs to be a reduction of the linear frame bundle which is not the case for most structures of interest to physicists. (I cite for instance, the spin bundle and local twistor bundle to name a few. Strangely, the article up until now utterly failed to take such bundles into account.) Also, the covariant derivative for a Cartan connection is defined completely differently than for the standard (Ehresmann) connections.

Having at one point been a physicist myself, I am aware that there are various notions of a Cartan connection floating about. As we all know, there is the Levi-Civita connection of a pseudoriemannian geometry -- which is the normal Cartan connection for Euc(n)/SO(n), the Euclidean group of R^n modulo the rotation group. In four-dimensional Lorentzian geometry, the Levi-Civita connection decouples into the Weyl spin-connection on irreducible homogeneous SL(2,C)-bundles (Weyl spinors), which is in turn related to the Newman-Penrose formalism of general relativity (or Gerald-Held-Penrose). The local twistor connection is an SU(2,2) Cartan connection. The Dirac operator is a symmetry reduction of what I am calling the fundamental D-operator. Functional determinants of quantum field theory are given as conformally invariant (or logarithmically conformally invariant) operators on a representation of the conformal group (for which all invariant data are expressed in terms of the Cartan connection).

The point is that "The Cartan Connection" is a very general idea, and therefore deserves to be treated as such. What I am doing here attempts to take Cartan's definition of the connection and translate it into slightly more modern language. I will try to work my way from the general to the specific. For instance, most of what I have done can be interpreted in terms of a fixed gauge, although it is somewhat more difficult to prove gauge-invaraince from this point of view. (But once we have gauge-invariance in hand, we don't need to worry about it.) If you would prefer, go and read Cartan by all means. I especially recommend "The Theory of Spinors," ISBN 0-486-64070-1. Cartan "OEuvres Complètes" is also very good reading, but incredibly difficult (and I don't mean this condescendingly in any way ;-)

Also, I had no wish to encroach on anyone else's turf. That's why I chose to start writing in the "General Theory" section. If you have a "favourite" application of Cartan connections, by all means include it in the article. But perhaps it is also appropriate to link out to Newman-Penrose, (vierbeins) tetrads, Geroch-Held-Penrose, Twistor theory, etc, as they are all applications of Cartan connections and otherwise have little to do with the real meat of Cartan's work. (I'd happily link out to Webster-Stanton connection and local twistor connection. The article on twistor theory looks rather pathetic. I'll also update the page on the Dirac operator to include a brief discussion of how it arises as a symmetry reduction of the D operator.)

Another possible solution is to have Cartan_connection_(mathematics) and Cartan_connection_(physics) although I can't for the life of me see how Cartan_connection_(physics) can be systematically organized. I can find several examples which are unrelated other than bearing the name of Cartan, and of course being one of the Cartan_connection_(mathematics).

[edit] our discussion is getting too long

Ok summarizing all above:

Cartan connection for mathematics does not mean much, it is might be one of two things:

way to discribe connection (which is covered in connection form)
historic way of introducing connection by Cartan

(maybe some more?)

In both of these meaning there is no reason to include here everything which is connected to connection.

For math phisics it might be different, and I do not know much about this, but we can make this article entirely on math phisics

One thing which is almost desided is to put vierbein material on its own page

I do not have any preference on what to choose, but something (from above or else) must be chousen Tosha

Ok summarizing all above:

Cartan connection for mathematics does not mean much, it is might be one of two things:

way to discribe connection (which is covered in connection form)
historic way of introducing connection by Cartan

Wrong on both counts. A Cartan connection is what it is defined to be. This is different from the definition in connection form which defines something called an Ehresmann connection. Think: a bicycle and motorcycle are both called "bikes," they both have two wheels, two handles, and are a transportation device, but they ARE NOT THE SAME THING!

Sorry, but the Cartan connection is a very different beast altogether like I've already said. And no, it is not "historic." It's still used in complex analysis, conformal geometry, projective geometry, twistor theory, string theory, quantum field theory (via gauge theory), and relativity theory. Besides, you yourself said that there doesn't appear to be any one connection in physics bearing the name of Cartan. There's a good reason for this. There is nothing a priori physical about the Cartan connection. It is a differential-geometric construction, period. It's correspondence with physics comes largely through gauge theory, although I am sure I can come up with other examples if I think about it for long enough. (I seem to remember a book about Cartan connections and fluid dynamics. But maybe I just imagined that one. Oh, there's the old "Rolling without slipping or twisting" problem, too.)

In some special cases, such as the Levi-Civita connection, the associated linear connection is a reduction of either the Cartan connection or the associated Ehresmann connection. It doesn't make much difference. But there are cases (spinors, twistors, tractors, Yang-Mills fields, spannors, plyors, and homogeneous vector bundles) where it does. And yes, cases like these crop up surprisingly often. I cite the collected works of Elie Cartan as evidence.

So, making an arbitrary distinction like "These connections are for mathematicians" and "These connections are for physicists" is ludicrous, when the Cartan connection is a well-defined general construction which differs substantially from those connections you would relegate to the domain of mathematicians, and particularly when the very problems in which Cartan himself introduced the connection had little if any bearing on the physics of the era. Jholland 03:50, 9 Sep 2004 (UTC)

do not write too much please or I will start yet an other subsection

Maybe I do not know much, and maybe I'm all wrong. Tell me what is the difference between connection and Cartan connection, maybe this term has more meanings than I thought. I always thouhgt of book of Cartan "Riemannian geometry in an orthogonal frame" where he just gave a way t describe conection. plus this term might be used to do some direct generalizations.

In mathematics this term is not used, and that gave me idea that it is used in phisics. (again maybe I'm wrong then give me a ref)

It seems that you know some other meaning??? Now tell me what exactly do you mean by Cartan connection. As soon as you state it it will be clear where to go put it rigt in the beggining of the article and that is it Tosha

Tosha, I don't really agree with your approach on this. I have met the Cartan connection idea in trying to read Cartan (impossible, really) and also in Dieudonné's treatment, which is Bourbaki-like. At present we have a three-section article, like

A Introductory things you and I have written
B Physics-oriented material
C Recent additions.

I feel we could be patient and wait. If you don't agree, we could do this:

Make C the basic article
Make B a separate article
Copy A to this talk page, or archive it as a subpage here.

Charles Matthews 09:00, 9 Sep 2004 (UTC)

Ok, "Make B a separate article" I assume it is about vierbein? I think it would be good start. It was your idea and as far as I see noone is against it.

The problem with this article, as I see it, is that it does not answer the main question: "what is Cartan connection?", and I think we should agree on that before going further... Tosha

From the article:

A Cartan geometry consists of the following. A smooth manifold M of dimension n, a Lie group H of dimension r having Lie algebra $\mathfrak h$ , a principal H-bundle P on M, and Lie group G of dimension n+r with Lie algebra $\mathfrak g$ containing H as a subgroup. A Cartan connection is a $\mathfrak g$ -valued 1-form on P satisfying

w is a linear isomorphism of the tangent space of P.
$(R h) * w = A d (h - 1) w$ for all h in H.
$w (X +) = X$ for all X in $\mathfrak h$ .

This covers all the cases where the term "Cartan connection" may be applied. (Of course, a Cartan connection is simply part of the data for a Cartan geometry, just like a connection form is part of the data of a principal bundle with connection.) Jholland 19:35, 9 Sep 2004 (UTC)

Thank you very much,

That seems indeed different from what I thought, and it is different from the connection in Cartan's book, but is is ok. Let's put this def in the beggining, at least it will keep stupid guys like me from editing this article. Tosha

[edit] What's a Cartan connection

Well, I think you have just hit the nail on the head. The trouble is, how can I explain to non-experts what a Cartan connection is? That is, if you don't know everything about principal bundles and so on. It is a conundrum. This may be a bit helpful, and should perhaps be incorporated into a later version of the article (in a clearer way than I have already written) that Cartan was interested in several things:

Primarily generalizing existing geometrical theories: those of Klein, and those of Riemann. He sought a unifying framework in which a Kleinian geometry was the "flat" version of a suitably generalized Riemannian geometry. During the time Cartan lived, geometry really had two distinct schools: the Riemannians and the Kleinians. I believe that it was the Riemannians that ultimately prevailed due to the advent of relativity theory. (This has its drawbacks: today there are geometers who have never heard of Felix Klein.)
He sought techniques for applying Kleinian geometrical methods (in the sense of symmetry) to ostensibly non-geometrical problems.

The Cartan connection was a way of breaking the symmetry of Kleinian methods just enough that geometric methods (a la Frobenius theorem and integrability condition) could still be applied) but so that the symmetry conditions inherent in a Kleinian geometry were not a hinderance. In this sense, a Cartan connection is a perturbation of the Maurer-Cartan 1-form on a Kleinian geometry.

When I wrote the extension of the article, I didn't think much about organization because I didn't want to tread on any toes. But I think we may finally be getting somewhere.

There is still the problem of things getting too long. It is quite difficult to define a Cartan connection ad hoc. It can be defined in a manifestly gauge-invariant fashion (which I have done), or it can be done in the gauge-dependent version, using moving frames (which is the way Cartan did it). Each approach has its advantages and disadvantages. It's important to have both. It's also important to have examples (because, again, that's the way Cartan did it). I think the Cartan conformal connection should be discussed somewhere, though not necessarily in this article.

Jholland 20:36, 9 Sep 2004 (UTC)

Oh, you can assume principal bundles - why not? Anyway, thanks for all this exposition. We'll probably get round to incorporating some of this talk page material. Charles Matthews 21:05, 9 Sep 2004 (UTC)

I've now moved the central section to Cartan connection applications, and changed most of this page's redirect to that page, or moving frame. Charles Matthews 08:06, 10 Sep 2004 (UTC)

[edit] Todo

Todo/wish list:

Do a full, ugly, Cartan-style absolute parallelism based presentation of an affine connection.
Simplify some of the more gruesome bits of the affine connection and relate it to A general theory of frames and Identifying the tangent bundle.
Concordantly clean up A general theory... and Identifying... so that they actually make sense.
Give Cartan's own formulation of the connection in terms of an absolute parallelism. This is pretty frightful at first blush, but hopefully reasonably intuitive if we keep the example of the affine connection in mind.
Segue into General theory in formal terms, which is a cleaner and more geometrical way of organizing the absolute parallelism.
Briefly tie all of this in with Cartan's equivalence method.

Silly rabbit 15:10, 15 June 2006 (UTC)

[edit] Or not todo, that is the question

This article clearly needs a lot of work, everyone seems to be agreed on that, but it seems to me that it is in danger of going in the wrong direction. The problem with Cartan connections (the reason that they seem technical and/or abstract and/or incomprehensible) is that abstraction arrived too late on the scene, post general relativity, and so our whole conception of a Cartan connection (which has its roots in the 19th century study of surfaces in R³ by geometers such as Bianchi and Darboux) is somewhat screwed-up. Cartan connections are motivated by submanifolds: even in Cartan's famous 1923 paper, only the first half concerns the construction of the "normal Cartan connection" in conformal geometry - the rest is on submanifold geometry. Unfortunately by the time the idea of a connection was being abstracted, noone was interested in surfaces in R³, but only in more abstract objects like (pseudo-)Riemannian manifolds. Anyway, this talk page is already too long, so I'll leave the background here, and try to get to the point. Geometry guy 21:53, 12 February 2007 (UTC)

[edit] What's a Cartan connection? An answer

Because the development is screwed-up, even professional geometers should not be blamed for making assertions like:

a Cartan connection is not a connection;
a Cartan connection is a generalization of a connection.

(Where "connection" means either "Ehresmann connection" or "principal connection".)

These statements are at best misleading, at worst false (and perhaps even contradict each other). In fact this article already has the right idea (and it appears in the initial version of the article, but has not been elaborated since):

The first type of definition in this set-up is that a Cartan connection for H is a specific type of principal G-connection.

The point of view that a Cartan connection is a specialization of the notion of a G-connection (on a principal or associated bundle) is easy to motivate and easy to define. I shall try to be as brief as I can so as not to add further to this talk page, but I'm willing to elaborate these ideas into the article if they find favour.

[edit] Motivation

Consider a smooth surface S in Euclidean space. Near to any point, S can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space. The affine subspaces are "model" surfaces - they are the nicest examples, and are homogeneous under the Euclidean group of the plane - and every smooth surface has a unique model surface tangent to it at each point. Now consider a curve on S with endpoints x and y. It is intuitively clear that you can roll the model surface tangent to x along the curve, and hence identify it with the model surface tangent to y. This is a Cartan connection. Notice that it defines a notion of parallel transport along a curve, hence it is a connection in the usual sense (in fact a G-connection, where G is the Euclidean group of the plane). However, there is something special about it: the model surface tangent to x has a distinguished point in it (the point at which it is tangent to S) and this point always moves under parallel translation (unless the curve is trivial). This generic condition characterizes Cartan connections.

Unfortunately in modern differential geometry, the movement of the origin of the tangent plane in Euclidean geometry is ignored, leading to the notion of a linear connection (and in particular, the Levi-Civita connection). This trick works for the Euclidean group of the plane, but not for other groups. In the conformal geometry of surfaces in the 3-sphere, for example, the tangent plane is replaced by a tangent sphere, and it is not possible to separate the motion of the tangent point from the rest of the parallel transport in a natural way.

[edit] Definition

[edit] Heuristic formulation

Let G be a group with a subgroup H, and M a manifold of the same dimension as G/H. Then a Cartan connection on M is a G-connection which is generic with respect to a reduction to H.

[edit] Principal bundle formulation

A Cartan connection is given by a principal G-connection ω (a 1-form with values in the Lie algebra Lie(G)) on a principal G-bundle Q over M together with a principal H-subbundle P of Q such that the pullback of ω to P defines an isomorphism from each tangent space of P to Lie(G) - the Cartan condition.

Okay, this isn't hugely intuitive (essentially because principal bundles are not) and the pullback of ω to P is indeed an "absolute parallism", which is the modern approach to Cartan connections expressed already in this article.

But how does it relate to the motivation? Well, there is an associated bundle $Q\times_G G/H$ over M whose fibers are copies of the model space G/H. The Cartan condition says that the natural map from T_x M to the tangent space of the model (which is the fiber of $P\times_H \mathfrak g/\mathfrak h$ at x) is an isomorphism, so the model spaces are "tangent" to M in some sense. This isomorphism is called a "solder form" in physics. The meaning of the Cartan condition is that the marked point (the identity coset of the model fibers) always moves under parallel transport by the connection.

[edit] What's the point?

Abstract principal bundles on a manifold M have very little to do with the geometry of M (okay that is a rather bold statement). The point about Cartan connections is that they tie the geometry of the principal bundle to the geometry of M via the solder form. This is one reason why they are of interest to physicists who wish to unify gauge theory (the theory of principal connections) with gravity (the geometry of spacetime).

I have the impression that the contributors to this article are seeking to express such a point of view, so I hope these (too long) comments are helpful. Geometry guy 21:53, 12 February 2007 (UTC)

[edit] Comments

Well, I think its clear from earlier comments on this page that there is a great deal of confusion as to exactly what a Cartan connection is. I, for one, find your comments above enlightening and would welcome their incorporation into the article. I've read parts of Sharpe's Differential Geometry but someone missed the idea that a Cartan connection could be realized as a special type of principal G-connection rather than just a gadget on a principal H-bundle. -- Fropuff 04:28, 13 February 2007 (UTC)

This is written in very few places and those who know (e.g. IP address 151... above) dismiss it as "nothing to be gained" (incidently, there are no "representation theoretic obstables" to expressing a Cartan connection as a principal G-connection), whereas in fact it short-circuits the argument that Cartan connections induce covariant derivatives on bundles associated to representations of G.

I'm happy to incorporate these ideas into the article, but I think that even if I maintain all points of view (and I want to do this), an almost complete rewrite of the article is required. So please (anyone) let me know if you are watching this page, have objections, or are willing to help. I have read the history pages, and understand some of the ideas that need to go in. I will try and proceed step by step to get other input. Geometry guy 22:35, 14 February 2007 (UTC)

I'm sorry for the long delay. I've now edited affine connection on these lines (although there is still a bit more to do) and I hope to be returning to this soon, so let me know what you think. When I start, I may proceed quite quickly, since no comments have been received for more than a month ;) Geometry guy 23:10, 21 March 2007 (UTC)

[edit] Rewriting

I have now done the main part of rewriting the motivation and definition, including a relatively conservative rewrite of the introduction. Much more could be done: this is just a start. The later sections also need to be rewritten in the light of the changes made so far. Geometry guy 16:20, 7 April 2007 (UTC)

The later sections have now been rewritten in a fairly minimal way, essentially just for internal consistency of the notation and approach. They probably need a rethink, and some examples and applications are needed, but this requires some reflection and input from other editors! Geometry guy 19:13, 12 April 2007 (UTC)

[edit] Different viewpoints on Cartan connections

I'm not entirely satisfied with using the lifted principal connection to define the covariant derivative (or other things associated to a Cartan connection), although I have to admit that it does simplify things tremendously. The group G (and hence the principal bundle Q) isn't necessarily known ab initio, so it's not really part of the data defining the Cartan connection. One can associate a covariant derivative to any vector bundle V associated to a representation V of the group H as long as there is also a compatible $\mathfrak{g}$ action on V; i.e., V is a ( $\mathfrak{g}$ , H)-module. I suspect this is a significant, albeit slight, difference. Silly rabbit 17:23, 15 April 2007 (UTC)

I take your point and am well aware of this subtlety, so I'm glad you've added it to the article (and a reference to work of \v Cap and/or Gover was long-overdue!). On the other hand, it is a matter of convention/opinion whether G is part of the definition or not, and the motivation (either in terms of a Kleinian model, or parallel transport) essentially presupposes a choice of G. Although a ( $\mathfrak{g}$ , H)-module does not necessarily extend (in general) to a G-module for some compatible G, this is mainly an issue for infinite-dimensional V, and I've not seen an example in the Cartan connection literature where the distinction really matters. Geometry guy 18:07, 15 April 2007 (UTC)

Ahh... After looking over at affine connection, I see what you're up to. Silly rabbit 17:35, 15 April 2007 (UTC)

That's good! I took a lot more care with that article, and also I think had better material (your discussion of Cartan's point of view) to build on. For me, what is really missing from the discussion of the differential operators in this article, is some idea what they are for. Anyway, I'm really glad you are back working on this! Geometry guy 18:07, 15 April 2007 (UTC)

I'd urge against making the principal connection a systematic part of the treatment. Although I can definitely see your point of view, I am not of the mind that G is part of the data for a Cartan connection (so I dismiss the G-bundle as non-geometrical out of hand). If you think of Cartan developments as rolling a model space along a curve, then a consequence is that you aren't limited to the particular G/H since development is a local property. This likely has consequences for holonomy, as different choices of G will yield different holonomy groups.

Moreover, nitty-gritty calculations in a Cartan connection are still often done in a moving frame, and the notion of a Cartan connection (whether by an absolute parallelism or principal H-bundle) really is just syntactic sugar for the method of moving frames.

Furthermore, it's easier to lose track of the solder form. For example, Kobayashi-Nomizu take a rather interesting approach to affine connections by first introducing a generalized affine connection as a principal Aff(n) connection, and then splitting the pullback to the frame bundle into two components by the semidirect product. If the Rⁿ-valued component is the solder form, then the connection is affine. A similar approach should work in general, with $\mathfrak{g}/\mathfrak{h}$ replacing Rⁿ. Although in this case, a suitable interpretation of "solder form" must be explored. This seems rather more in the spirit of Cartan connections than sweeping the details under the rug by a reduction of the G-bundle in which the pullback satisfies the Cartan condition. One can think of (by analogy with the solder form for affine connections) the Cartan connection as supplying the needed machinery to identify (solder) copies of $\mathfrak{g}/\mathfrak{h}$ with the tangent space in an Ad(H)-equivariant manner. Silly rabbit 14:11, 21 April 2007 (UTC)

After having another look at the article, it shouldn't be too hard to work this last comment into the treatment. Silly rabbit 14:19, 21 April 2007 (UTC)

Ok, so I've done it for reductive geometries, although I admit without a lot of foreplay at present. Also I've sandboxed the affine connection material (to Cartan connection/Sandbox/Affine connection). This needs a complete rewrite. I've tried to bring the intro paragraphs more into line with the approach of the Motivation section. I'm a little worried about overplaying the parallel transport card, and not paying enough attention to absolute parallelisms and the equivalence problem, but the article is still too much in its infancy to worry that much. I like the parallel transport metaphor, since it is easy to grasp and adapt to different settings, so I'll exploit it for exposition whenever I can. Silly rabbit 17:13, 21 April 2007 (UTC)

I have to think about your comments and edits some more, but here is an initial response. The edits you have made point to several improvements that need to be made to the article, but I don't agree entirely with the direction. For instance, the motivating example of parallel transport, defined by rolling the tangent plane over a surface, is not linear, but affine (indeed the existence of a linear point of view depends essentially on the fact that affine space is reductive, as you have noticed); I think the linear point of view should be suppressed since it doesn't generalize. On the other hand, I agree entirely that the role and importance of the solder form needs to be brought out more clearly. This is well defined for any Cartan connection, by taking the H-bundle – g-form point of view, and passing to the quotient of g by h. I am not wedded to principal connections (I prefer to think of any bundle with structure group G) but they do provide a systematic viewpoint. I don't want to prioritize this, but neither do I think it should be marginalized.

Yes, clearly the geometry needs to be reductive in order to get a linear map. But organizing Cartan connections in general around the notion of development also doesn't seem quite right because ideas associated with development live more naturally on the prolonged bundle. (In particular, to do it this way you need a model space. See below.) Silly rabbit 00:01, 22 April 2007 (UTC)

This is precisely why the G-bundle point of view should be taken seriously! Geometry guy 20:27, 22 April 2007 (UTC)

In my view the motivation does involve the choice of a particular model (to roll a model surface along a curve in a manifold, you need to choose the model surface!), and even if development is local, to define it precisely, a model is needed. Finally, the idea that Cartan connections are syntactic sugar for the method of moving frames is but one point of view, and there are many others. I think it is easier for the modern reader (and perhaps more faithful to the 19th century motivation) to approach the subject from a modern point of view on connections and parallel transport, rather than via absolute parallelism and the method of moving frames. Indeed, an article on the method of moving frames has not yet been written; it is just a redirect. I'd be happy to work with you to rectify this! Geometry guy 21:47, 21 April 2007 (UTC)

I disagree. You can roll any model space at all. Of course, to actually define the rolling, you need the model space, but the connection doesn't care which one we choose. But I suppose its a matter of opinion whether the connection is the rolling map itself, or the "wheels" that do the rolling. Silly rabbit 00:01, 22 April 2007 (UTC)

Of course you can. But there is nearly always an obvious choice of model. I don't mind whether the connection refers to the rolling or the wheels, but wheels serve little purpose if they are not attached to a car! Geometry guy 20:27, 22 April 2007 (UTC)

PS. I'm not sure it is helpful to sandbox sections of an article. I'd rather keep the section in place until a rewrite is proposed; I think this is more inclusive and more transparent.

I don't plan on keeping it out for long, but I need to fiddle around a bit with it to bring it up to snuff. I'll restore the original version intact in the meantime. Silly rabbit 00:01, 22 April 2007 (UTC)

Thanks for putting it back. Geometry guy 20:27, 22 April 2007 (UTC)

[edit] Progress report

I have provided a bit more motivation for considering Cartan connections as "deformed" homogeneous spaces by a bit of hand-waving to do with the idea of tangent model-spaces. I have completely rewritten the section on Affine connections to bring it in line with the definition of a Cartan connection offered here. I have yet to merge in the material on prolongation and affine development, which are important topics generalizing to all Cartan connections (not just the reductive ones). Because of some of the logical difficulties in organizing a definition of Cartan connections around the idea of development per se, I plan to reorganize and rephrase some of the discussion of the "principal connection" approach to Cartan connections in order to capture both views, while maintaining a sharp logical distinction between them. Silly rabbit 13:20, 22 April 2007 (UTC)

I am watching your progress with interest. You are adding a lot of useful content to the article, and ideally I'd like to see your vision for the article fully developed, rather than criticise or edit partial versions. I am particularly interested in (but have mixed feelings about) the way you have approached affine connections, as there are now more technical details here than there are in some parts of the affine connections article: I take this partly as a friendly indication of changes/additions/clarifications you would like to see in that article, and I agree there is scope for improvements there.

However, as indicated already by previous comments, I don't agree entirely with the course you are taking, so I should attempt to summarize this now.

The main disagreement we have is whether "development" (and the closely related notion of parallel transport) is a primary or secondary concept for Cartan connections. I think it is primary. Kuiper's Theorem, for example, is a key application of Cartan connections to the geometry of conformally flat manifolds. I think as much weight should be attached to this as to the absolute parallelism point of view. In the past you have expressed an aversion to the Ehresmann approach to connections, yet at the same time you shared my point of view that parallel transport is one of several primary concepts in the theory of connections.

Many people seem to perceive a conflict between the Cartan and Ehresmann points of view on connections. I don't understand this. Inspired (no doubt) by his supervisor, Ehresmann provided an infinitesimal formulation of parallel transport via horizontal lifts. Cartan's geometrical theory embeds naturally into this analytical one as the parallel transport of geometrical (rather than arbitrary) objects. This intuition is easily lost in the absolute parallelism formulation to which you appear to be wedded.

The issue of whether G is part of the definition or not is a red herring. I don't know of any example where it is not entirely obvious what the model space is. The absolute parallelism point of view, avoiding the choice of G, loses classical intuition, came late in the day, and was not Cartan's point of view at all! Some people even call the associated G/H bundle the Cartan bundle!
The article is getting too long. It has nearly doubled recently, and there are still applications, examples, the equivalence method, and so on, to discuss.
I don't think it is a good idea to define Cartan connections (as absolute parallelisms) in the affine case first. The formal definition in an article like this needs to come as early as possible, with only motivation preceding it. The goal of the two-pronged (Klein and affine) motivation was not to define Cartan connections in a special case, but to relate the two possible definitions to ideas that might be familiar to the reader. In its technical details, the affine/reductive case is more complicated than the general case, and this is a major contribution to the increase in size of the article. The solder form and torsion are natural concepts (quotients), but the "rest" of the connection and its curvature is not, and discussing them adds unnatural complication.

The above comments partially express a point of view on Cartan connections, just as you have expressed yours. However, this is an encyclopedia, not a textbook on Cartan connections, and so we should attempt to express concisely both points of view and the interaction between them. Thanks for all your work on the article. I hope you will take my reservations in a constructive spirit! Geometry guy 20:19, 22 April 2007 (UTC)

A few replies:

I'll see what I can do to incorporate your vision of Cartan connections into the article. The way it reads now, I admit, it takes the absolute parallelism as primitive and uses that to derive the Ehresmann connection and transport. Nevertheless, the motivation places transport in the foreground (as indeed it should). It's difficult to square the two away with each other in a satisfying and even-handed way. Hopefully as my own thoughts on this article mature in the light of your comments, it will become clear how to do this.
Is G part of the construction? Maybe it is and maybe it isn't. Let's just agree to disagree about that. It certainly isn't required by the construction, and many current applications of Cartan connections make no reference to G at all (just g). But the point is a relatively minor one, so I'm willing to concede at the moment.

Agreed. Lets leave it as a point of view, rather than a concession. (I actually like both points of view and have even emphasised the (g,H) viewpoint in my own work.)

You're probably right that affine connections should be treated more briefly here, and the article affine connection should be expanded to include a discussion of them in a more explicitly Cartanian context. My interpretation of the old version of the article was that Cartan connections are a direct generalization of the modern affine connection qua principal connection with solder form, with which the reader is no doubt more familiar. I think this is misleading. For the purposes of this article, I think it's better to get an affine connection as a specialization of Cartan connections than to try to go the other way around. Historically, of course, this was not the case. But then again, affine connections weren't yet understood in the way we understand them now. Readers will have a lot of baggage involving principal connections, because their role has been dramatically overstated in the differential geometry literature to the point where it is now difficult to find any fundamental treatment of connections and the relationships between them. (At least in my point of view, but that's an essay unto itself.) Any discussion of affine connections here should clearly indicate the difference between the principal connection view and the Cartan connection view. Using principal connections from the beginning is counter-productive to this end.

Silly rabbit 13:59, 23 April 2007 (UTC)

[edit] More progress

I'm sure that we will succeed in the end. One way forward is to improve related articles. I found it impossible to make progress on this article until I got affine connection into better shape. Now maybe affine connection needs more work, as do method of moving frames, Cartan's equivalence method, projective connection and conformal connection. I will find it easier to see where to go with Cartan connection after a bit more work on these, but do not want to discourage you from continuing your programme of work here.

I have some sympathy with your final remarks, but yet again I have to remind myself that this is an encyclopedia, not a textbook. Alas, its role is to describe the field as it is, not as we would wish it to be. Nevertheless, there can be room for a little subterfuge. For this reason, I actually don't find the use of principal connections counterproductive, since it is a way of coaxing readers (with their baggage) out of a gauge-theoretic world into this wonderful geometrical one! Such a mission is not far beneath the surface in affine connection as I am sure you have noticed! So the previous version was not intended to be misleading, rather the opposite: proleading perhaps? :) Geometry guy 18:27, 23 April 2007 (UTC)

You are really making rapid progress here! I like very much that you have added the G/H Ehresmann connection definition, but (as I think you realize) it is not right yet, because it does not include the condition that the connection is a G-connection. Unfortunately, it isn't yet clear to me how to express this condition in a straightforward and direct way. Geometry guy 21:12, 25 April 2007 (UTC)

Indeed, there doesn't seem to be an (obvious) way to express it for the connection form. This afternoon, I dug up Ehresmann's paper on the subject, and as far as I can tell he uses the associated bundle to express the form and then descends it to the fibration. (Which is roughly what the article does now.) There may not even be a nice way to express it without essentially reverting to this approach. I'll have more to say on this later. Silly rabbit 21:27, 25 April 2007 (UTC)

I look forward to it. In practice "G-connection" usually has a clear meaning. For instance in the linear ("tractor") point of view on Cartan connections, the "G-connection" condition usually means that the connection is linear and preserves some natural tensors, such as an inner product. I'd like it if this article captures some of this intuitive feeling. Geometry guy 21:40, 25 April 2007 (UTC)