Talk:Projective connection

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Mathematics rating:	Start Class	Low Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Considerable expansion is needed, e.g., using the vector bundle point of view. Geometry guy 23:08, 14 April 2007 (UTC)

Hello. I was just wondering, what of the Thomas/Veblen approach to projective connections? —Preceding unsigned comment added by 81.152.93.24 (talk) 22:31, 23 January 2008 (UTC)

[edit] Cartan quotes

I have produced rough translations of the quotations. The first quote turned out not to support the first sentence of the section (the quote is not about geodesics!), so I restructured the section accordingly. Geometry guy 15:42, 4 April 2007 (UTC)

No, indeed not. Thanks for fixing it. I had meant to return to this at one point, but started working on other (non-wikipedia) things. I wanted to understand how Cartan thought about projective connections, since he thought of these connections as providing soldering data for projective coordinate systems at points connected by a curve. This is not quite the modern idea of a frame (which is, in classical terms, an infinitesimal coordinate system). At the time, I couldn't quite square the two conflicting notions of frame up with each other. So I put the quotes down to indicate Cartan's (perhaps dated) point of view to prompt further work, and then vanished from Wikipedia (no, I'm not back yet). The ghost of Silly rabbit 12:26, 13 April 2007 (UTC)

Nice to hear from you! As you probably have seen, the lead you took has inspired me to work on a number of articles, in particular affine connection, which was in danger of mutating into a carbon copy of covariant derivative and then promptly being deleted. I reworked it as a way to introduce readers (who might have met the covariant derivative point of view) to Cartan's approach. This provides a handy launch pad for the Cartan connection article (which still needs a fair amount of work).

It is a pity you are not back yet, as I could use your help with some of the other connection articles (such as this one!). My understanding of Cartan's approach is that the projective coordinate system attached to a point is the frame (infinitesimal coordinate system) and the connection allows you to solder the two systems at two infinitesimally close points. I don't see curves here, other than infinitesimal ones (tangent vectors). Anyway, from the little I know about Cartan's work, if he does start to use curves, then I would guess he was trying (alas probably failing) to make his ideas more accessible to the new generation of geometers who did not understand his infinitesimal coordinates, only tensor calculus and parallel transport. After all, Cartan was already effectively working with infinitesimal coordinates and Cartan connections in 1910! Anyway, if you've found (or written ;) any references on this IRL that you could point me too, I'd be very happy to know. So far the most helpful sources I found were the Encyclopaedia of Mathematics articles by Lumiste. Geometry guy 13:12, 13 April 2007 (UTC)