Talk:Parallel transport

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[edit] Generality of this article?

There are various levels of generality at which one can discuss parallel transport:

  • for vector fields;
  • for sections of arbitrary vector bundles;
  • for sections of arbitrary geometric bundles (e.g. frame bundles and their associated bundles);
  • for sections of arbitrary fiber bundles (e.g. principal bundles and their associated bundles);

At what level should this article employ most of its energies? Geometry guy 16:08, 12 March 2007 (UTC)

Why not all of them? But if I had to pick one, I believe it should focus primarily on vector fields since that is likely to be the area of interest to most people looking in the general community. At least, it should start out with vector fields. The other levels of generality are also important, though, for the article holonomy (which also needs a lot of work). Silly rabbit 14:11, 13 April 2007 (UTC)

I tend to agree: work with vector fields, then explain that it all generalizes to each of the above levels (with fewer details in the more general cases). Geometry guy 14:41, 13 April 2007 (UTC)

[edit] Another question

In the earlier part of the 20th century, connections were defined in terms of parallel transport satisfying certain properties (see e.g. the Springer Encyclopedia articles on connections). Should we present this approach here? Geometry guy 22:21, 20 March 2007 (UTC)

I'm all for it, since PT really is the unifying theme behind all the connection concepts. So organizing Category:connection (mathematics) around this idea was one of my original goals in this bulk revision. That said, I don't think it can be done in a way that will be at all familiar to readers who have just recently encountered the covariant derivative (*) and are looking for some further reading on the subject. So, if it is to be done, it shouldn't be the primary approach: perhaps a section later on in the article like Defining the covariant derivative in terms of parallel transport.
(*) Ok, anywhere except Kobayashi-Nomizu.
Silly rabbit 14:20, 13 April 2007 (UTC)

I agree with your conclusion here, especially as this is an encyclopedia, not a new book on connections ;) ! Also, I would not say PT is the unifying theme, since there are several points of view, such as Cartan's infinitesimal one. And parallel transport is quite hard to use as a primitive definition (see again the Encyclopedia of Mathematics) because one has to capture the idea that it is not arbitrary, but only depends on 1-jets. This involves taking an awkward derivative at time zero. I feel some of the infinitesimal intuition gets lost when PT is overemphasised. I hope with both of us working on this we can achieve a good balance between the different points of view ;) Geometry guy 14:41, 13 April 2007 (UTC)

For future reference, some starting material for this point of view is buried in Talk:Connection (vector bundle). Silly rabbit 19:23, 13 April 2007 (UTC)

Yes, I put it there, to save it from being buried even deeper in the history archives. Geometry guy 19:28, 13 April 2007 (UTC)