Talk:Cartan connection

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

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

1 Name
2 AFFINE connection
3 The article is getting too long
4 our discussion is getting too long
5 What's a Cartan connection
6 Todo
7 Or not todo, that is the question
8 What's a Cartan connection? An answer

[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)

[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)

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