Talk:Cartan connection

Brevity, please

This article is somewhat schitzophrenic and the talk page has become a personal blog for some authors. Is it possible to archive the talk page and start a new one with only the main points from the earlier version collected into a short section? ...but wait... I cannot resist contributing to this mess.

If I understand it correctly, it makes no sense to make statements such as "Cartan connection is different from Ehresmann/principal connection", since principal connection is an Ehresmann connection with an extra structure and Cartan connection is a principal connection with an extra structure. Also "Cartan connection solders the geometry of the principal bundle into the geometry of base manifold" does not make much sense because the base manifold on its own has no geometry, unless it is given one by introducing some extra structure such as metric, a principal bundle structure with a Cartan connection etc. Lapasotka (talk) 22:29, 28 October 2010 (UTC)

The lead says that a Cartan connection is a principal connection with additional structure. I don't think the article says anywhere that a Cartan connection is not a principal connection or an Ehresmann connection, but rather that it is one or the other (depending on the focus) with some additional structure. Certainly this discussion page seems to have a few misunderstandings and apparent philosophical differences; though I don't really see any evidence that this is a problem with the article. The article certainly discusses various different points of view on the question of what is a Cartan connection—it should do this.

Regarding the intuitive idea of "soldering" expressed in the lead paragraph, I'm not quite sure how better to state it succinctly. Consider, for instance, an affine connection. As an abstract connection in a vector bundle, it is meaningless to talk about geodesics. But there is also a "soldering" of an affine connection "to the geometry" of M because the underlying vector bundle can be identified with the tangent bundle. So there are geodesics that live purely on M. Thus one is talking somehow about geometric objects "on M", rather than objects that live in some bundle "over M". Another example is the rolling spheres along curves in M. Abstractly one has a sphere bundle with Ehresmann connection, but that connection is soldered to M along a preferred section because the rolling sphere is tangent at the point of contact. Maybe a better wording would be something more along the lines of this:

It may also be regarded as a specialization of the general concept of a principal connection, together with a solder form that allows geometric structures in the principal bundle to be interpreted as geometric structures intrinsic to M itself.

I suppose one could still raise the objection that it isn't clear what this is about either. Any better ideas? Sławomir Biały (talk) 02:23, 29 October 2010 (UTC)

Archive created.67.198.37.16 (talk) 21:12, 2 May 2019 (UTC)

“Definition as an Ehresmann connection” is wrong?

The last formal definition in the article introduces the induced G-connection on the associated bundle E with typical fiber G/H and claims that this construction is equivalent to the Cartan connection on P, and links to the page explaining how to reconstruct a principal bundle from an associated bundle. But that page correctly emphasizes that to be able to reconstruct a principal bundle, the G-action on the typical fiber needs to be effective. However, the action on G/H is generally not effective, so you can’t recover the Cartan connection from E. In other words, the last definition works only for effective Klein geometries G/H. (By the way, the existence of a G-connection is irrelevant to the ability to do this, contrary to the language in the article). Blackmail1807 (talk) 16:55, 8 May 2024 (UTC)

Nomenclature: fiber versus fibre

Edit permalink/1232794883 changed fibre to fiber with a comment Fixed typo. My first thought was to revert the edit, as the spelling is correct either way, but when I saw that the article was using a mixture of both spellings, I though that it was better to seek a consensus on the spelling for this article and then edit for consistency.

Thoughts? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:19, 23 July 2024 (UTC)

Notes