|WikiProject Mathematics||(Rated B+ class, Mid-importance)|
This article needs help
I don't have the time right now, but at least the first section of this needs to be changed. For example:
"The torsion of a connection is a smooth vector field."
This is wrong. The torsion of a connection is a rank (1, 2) tensor.
"...whilst keeping the inner product of that vector constant along the curve."
The 'inner product of a vector' doesn't make sense.
"The resulting connection coefficients are called Christoffel symbols and can be calculated directly from the metric. For this reason, this type of connection is often called a metric connection."
This is a bit misleading. A metric connection is one for which the inner product of any two vectors in unchanged when those vectors are parallel transported together.
Shambolic Entity 00:17, 27 July 2006 (UTC)
In fact, the whole first section needs re-writing. There is incorrect information, as above, but also no informal discussion which would help beginners to come to grips with the subject. I'm sticking a `needs help' template on here in the meantime. Shambolic Entity 04:14, 5 November 2006 (UTC)
- I have removed the 'need help' sticker. Things of course can be improved, but the article is in much better shape.
Sebastien 00:21, 24 December 2006 (UTC)
Merge with covariant derivative?
Sebastien 20:10, 23 December 2006 (UTC)
- That would be a good idea, because this page is filled with garbage such as the idea that an affine connection is the covariant derivative, rather than it being possible to define them in terms of each other. The covariant derivative combines two different ways a function changes from point-to-point -- the normal coordinate derivative, and how the manifold "changes under the function". The last bit is the connection. The confusion probably comes about because applying the covariant derivative to an ostensibly constant function (such as basis vectors) will leave only the last term, the connection. Differential geometry is hard enough without conflating two concepts and lying to people about it. 188.8.131.52 18:00, 22 February 2007 (UTC)
- Unfortunately, the terminology in this subject is very confused because of its historical origins. For some, the covariant derivative is only defined on vector and tensor fields, for others, it is defined on sections of any vector bundle. A Covariant derivative is sometimes called a linear connection. Other times, linear connections refer to linear Ehresmann connections, or to the affine connections of this article. An affine connection is equivalent to a covariant derivatives on the tangent bundle, but they are not really the same thing. I do not support merging this article, but rewriting it to highlight the distinction. Geometry guy 17:20, 11 March 2007 (UTC)
What to do with this?
Once upon a time, this article had a purpose. If you look back through the history, you will find the following version of 5 July 2006.:
The outline for the article is clear: introduce affine connections via parallel transport and as Cartan connections for the affine group, and explain how this relates to the (more familiar) approach via linear connections (or covariant derivatives) on the tangent bundle. Such an article provides an excellent opportunity to explain, in true encyclopaedic spirit, why on earth a covariant derivative on the tangent bundle is called an affine connection. Unfortunately, the editor with this plan, Silly rabbit apparently left WP without completing a first draft. Subsequently, perhaps not surprisingly, the more familiar linear description of affine connections has come to dominate the article, leading to a large overlap with other articles on linear connections and covariant derivatives. The article is in much better shape now, with properly fleshed out definitions, and even an example, but it has lost its sense of direction in my opinion.
I believe this can be fixed quite easily by reordering the material, and relating principal and Cartan connections on the frame bundle using the solder form. I will probably start to work this out soon. Any comments? Geometry guy 17:47, 11 March 2007 (UTC)
I've done the (substantial!) ground work for this now. I think we should aim quite high with this article: it is potentially one of the more fundamental articles in the Category: Connection (mathematics) — a Cartan/geometrical alternative to the more Ricci/tensorial covariant derivative article — and I'd be very happy if we could get it to GA status. Let me know if you are watching and/or interested in this project. My next mission is to get the Cartan part into shape, and use this as a springboard to sort out (finally) the Cartan connection article. Geometry guy 21:38, 20 March 2007 (UTC)
Okay, I've now done the hard bit now, at least in outline (there is rather too much reliance on other articles in my approach). In general, I've tried very hard to be explanatory and comprehensive enough so that other editors can make this material more accessible (although a very general reader will rarely happen upon this article, so I would not like the mathematical content to be too diluted). I still have to rewrite the section on torsion and curvature (and geodesics) in terms of the various definitions. Then I will move onto Cartan connection. Geometry guy 22:30, 21 March 2007 (UTC)
- Well, I think you've done a great job with a difficult subject. This is looking pretty good now. Somehow it seems a shame though that the modern definition of an affine connection as a Cartan connection is buried so deep in the article. I suppose this is unavoidable with all the motivation and history one has to get through.
- Perhaps it would be useful to summarize somewhere the various ways in which an affine connection can be defined (as a linear connection, a principal connection, or a Cartan connection) and the relationship between them. The article doesn't quite make it to the point of explaining exactly how an affine connection in the sense of Cartan is equivalent to specifying a principal connection on the frame bundle.
- Also it seems a little strange to me to refer to the pair (P, η) as an affine connection. I tend to think of η as the connection and the pair (P, η) as an affine geometry on M. Not really a big deal of course, it just reads a little funny (to me at least).
- Again, nice work. This should give us a great launching point for the Cartan connection article. -- Fropuff 05:47, 22 March 2007 (UTC)
Thanks for these helpful comments. I agree that it is a shame the modern definition comes so late: originally I wanted to put the covariant derivative definition at the end, but I just couldn't figure out how to make the article work like that. I now see this article as having an educational flavour (hopefully in encyclopaedic rather than textbook spirit) to introduce readers who might already be familiar with covariant derivatives into the development of the notion and the rich world of Cartan geometry. We have another chance in the Cartan connection article to cut more quickly to the chase, and perhaps the existence of this article will help us to do that, as you suggest.
Your second idea resonates well with a thought that has been going through my mind from time to time: maybe we need an article like Connection (definitions) that summarizes all the definitions and the relations between them. It shouldn't be so hard to put together once everything else is sorted out (the main torment here for me is to rewrite Connection form as a bridge between mathematics and gauge theory). Also, after all our hard work, I think a rewrite of the main article Connection (mathematics) (or at least parts of it) is well overdue.
Your other suggestions are spot-on. I'll try to weave them into the article. If I don't succeed, hit that edit button! Geometry guy 15:00, 22 March 2007 (UTC)
- I think we could probably incorporate the various definitions of connections and their relationships into connection (mathematics) which is meant to be an overview of the subject. I agree that that article will be due for some work after the other connection pages are sorted out. -- Fropuff 17:54, 22 March 2007 (UTC)
I'm not sure about this: I suspect that such an article will be too technical and not sufficiently short to incorporate as a section in the main article, although the current last section of Connection (mathematics) could certainly be adapted to provide an overview. (Meanwhile, I've discovered that the whole concept of curvature is also in a rather parlous state right now, sigh!) Geometry guy 19:29, 22 March 2007 (UTC)
Affine connections and linear connections
I'm over in Cartan connection, trying to work in the prototypical example of affine-connections-as-Cartan-connections. The trouble is that there are two ways of conceptualizing an affine connection: one as a linear connection (covariant derivative) which comes directly from a Cartan connection on the bundle of linear frames, and the other as its "prolongation" to the bundle of affine frames. This article doesn't make a clear logical distinction between the two concepts: one may be called a "linear" connection since it comes from the bundle of linear frames, and the other is more properly an "affine" connection since it takes place on the bundle of affine frames. For the purposes of this article, it may be helpful to point out the logical difference, but I don't want to interfere with an already very good and quite mature article and insert some mention of this at an inopportune place. (So a modest request for Fropuff and Geometry Guy ;)
Anyway, this doesn't really solve my problem. In Cartan connections, the distinction is much more problematic. The "linear connections" of more conventional connection-theory are the Cartan connections modeled on Aff(n)/GL(n). So these have to be called affine connections in the Cartanian context. The prolongation of the affine Cartan connection defines what (in conventional terms) would then be called an affine connection. By introducing affine connections in the first place, I'm trying to motivate the abstract definition of a Cartan connection (via H-bundle + g-form). I'd like to be able to say in what sense the connection "is affine" without direct appeal to absolute parallelism (since that will come later in the article), and the only way I can presently think to do so is via infinitesimal displacements. Any thoughts on how to do this another way would be greatly appreciated. Silly rabbit 18:27, 21 April 2007 (UTC)
- The term "linear connection" has several problems.
- It is ambiguous (see linear connection).
- As a term for a connection on the tangent bundle it is a misnomer (see footnote 3).
- It does not generalize to arbitrary Cartan connections (the existence of a linear parallel transport makes essential use of the fact that affine space is reductive).
- An affine connection is a term for a number of equivalent things (as is, more generally, a Cartan connection) and different authors prioritize different meanings. Anyway, I need to think some more about your comments here and at Talk:Cartan connection before responding more fully. Geometry guy 20:55, 21 April 2007 (UTC)
I agree 100% with you about the ambiguity in the term "linear connection". I think you in part illustrate the point I'm trying to make. (BTW: Thanks for pointing out footnote 3. I had missed that the first time around.) Re-read the first paragraph in my post: Affine connections (as linear connections) arise as Cartan connections defined on the bundle P = linear frame bundle. Affine connections per se (which are prolongations of Cartan connections in the previous sense) can be defined as a principal connection on the bundle Q = affine frame bundle, along with a "Cartan condition" on the reduction to P. So really, whether or not you object to the term "linear connection", the article still talks about two different (but equivalent) notions of an affine connection. It would be nice to set these up in contrast to each other. But (as you point out) there doesn't seem to be a completely accepted way to do that, since the term linear connection is tainted by other uses. Silly rabbit 21:31, 21 April 2007 (UTC)
- I'll think some more. I made some initial comments at Cartan connection, as I'm sure you noticed! Geometry guy 21:52, 21 April 2007 (UTC)
I'm confused by the picture (reproduced at right). First, the picture doesn't look like a sphere; it sounds like it should. Assuming it is what the text says it is, wouldn't a geodesic from one point to another correspond to a streight line in the tangent space of either endpoint? I could see this not being the case on another surface, but on a sphere it seems pretty obvious. [I'm new to this stuff; feel free to correct my language if it sounds not-quite-right. :-)] —Ben FrantzDale (talk) 12:49, 22 October 2009 (UTC)
- It should indeed look like a sphere (the blue part); if you can make a better (shaded?) version that would be great! A geodesic for an affine connection is pretty much by definition a curve whose development is a straight line. On the sphere (with its usual affine connection) the geodesics are arcs of great circles (aka airplane routes). Other curves on the sphere have developments which are not straight lines. In this image it would be better if the pink curve looked less like an arc of a great circle. Maybe we could illustrate both a geodesic and a non-geodesic between these two points? Geometry guy 13:41, 24 October 2009 (UTC)
Without a metric?
I'm trying to understand manifolds that don't "come with" a metric. In particular, I am interested in SE(3), the set of rigid-body transformations in 3D. In SE(3), there is no "natural" coordinate-system-invariant metric. For example, there is no coordinate-system-invariant answer to the question "Which is more, a translation of 1mm or a rotation of 1 radian about the origin?". However, the matrix exponential provides a coordinate-system-invariant exponential map in which geodesics are screw motions. If I understand this page correctly, this means that the matrix exponential provides a natural coordinate-system-invariant affine connection.
- SE(3) is an example of a Lie group. Every Lie group has an exponential map from its Lie algebra (the tangent space to the identity). It also has two flat affine connections (with torsion) whose parallel sections are the left and right invariant vector fields respectively. Its Lie algebra may be identified with the left or right invariant vector fields by parallel transport with respect to the corresponding affine connection. Left and right invariant metrics on a Lie group correspond to inner products on the Lie algebra (as tangent space to the identity). If there is an inner product on the Lie algebra which is invariant under the adjoint action, then it extends to a bi-invariant metric on the Lie group. In the case of SE(3), the tangent space to the identity is an extension of the translations by the infinitesimal rotations about a point. This does not have an inner product invariant under the adjoint action because the translations do not have an invariant complement in the Lie algebra. (The infinitesimal rotations about a point form a complement, but it is not invariant, as it depends upon the point.) Geometry guy 17:52, 15 May 2010 (UTC)
I believe there is a mistake in the Maurer-Cartan equations as they are mentioned in the section https://en.wikipedia.org/wiki/Affine_connection#Affine_frames_and_the_flat_affine_connection. The minus signs should be plus signs. Starting from
and computing as indicated on the aforementioned article, if is found that
as well as
Using the linear independence of the functions yields the Maurer-Cartan equations but with plus signs between the terms instead of the minus signs on the article.