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I don't have the time right now, but at least the first section of this needs to be changed. For example:. 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.
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. An affine connection is basically just another name for a Covariant derivative and the article on covariant derivative is good. So why not delete this page or re-direct to covariant derivative? Sebastien , 23 December UTC.
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 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 , 11 March UTC. I've done the substantial!
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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 , 20 March 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.
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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 , 21 March 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.
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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.
Talk:Affine connection/Archive 1
I'll try to weave them into the article. If I don't succeed, hit that edit button! Geometry guy , 22 March 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!
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. Anyway, this doesn't really solve my problem.
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In Cartan connections, the distinction is much more problematic. 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. 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 , 21 April UTC.
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. 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. This article is very well done. Lots of good explanation of concepts. 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.
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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.
What is the mathematical language to describe the fact that SE 3 doesn't "come with" a metric or an inner product? The minus signs should be plus signs.
Starting from. Following several discussions in past years , these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
This could be a central article in the connections category, so input is particularly welcome. It needs pictures where appropriate, and an elementary lead. Silly rabbit , 2 June UTC.