Talk:Principal bundle
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| Mathematics rating: | Start Class | Mid Priority | Field: Topology | ||||||||||||||||||||
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[edit] Transitive action
Why did you suppress the fact that G acts transitively on the fiber. This is contained implicitly, but should be mentioned explicitly, since it is essential. Hottiger 17:39, 11 April 2006 (UTC)
[edit] Measure
For at least some principle bundles, there is a natural measure on he space of connections, is there not? I beleive this is the case when the fibers are compact; less clear of the situation when they're not compact. Would like to see a proper definition. May attempt to do this myself, if/when get around to it. linas 15:51, 20 July 2006 (UTC)
[edit] An Algebraic Characterization of Principal Bundles
Both transitivity and freeness of action on the fibres can be directly characterized by defining the quotient on the product bundle, along with the right multiplication by G, by the properties p\(pg) = g and p(p\q) = q, where p\q denotes the quotient. This has the advantage of making everything else that follows down the line more transparent. For instance, the connection form simply becomes p\dp, the differential of the quotient by the second argument. In general, the differential of the quotient serves as another equivalent way to define the connection. For a section S, the operator S dS\() + dS S\() is none other than the horizontal lift operator; and the connection relativized to a section is just S\dS. The local decomposition, through a section S, of the principal bundle becomes p = S(pG) S(pG)\p, where pG denotes the projection of p onto the base space. -- Mark, 11 October 2006
- It took me a while to understand this, but it seems to be quite clever. Let me expand a bit: the idea is to consider the map from the fiber product
to M sending a pair (p,q) to the unique element g=p\q of G with pg=q. This is generalizes the difference map (x,y)→y-x from an affine space to the associated vector space of translations. I think that the connection form is here meant to be understood fiberwise, as the generalization of the Maurer-Cartan form. Thus p\dp identifies the vertical bundle of P with the trivial bundle whose fiber is the Lie algebra of G. Does anyone know a reference for this point of view? Geometry guy 16:15, 1 March 2007 (UTC)
[edit] Here's an odd suggestion
How about adding an executive summary in the introduction for those of us that are not math students? I realize that such a simplification will not be accurate, yet it remains common practice for other fields to include one, despite it being inaccurate. As this page stands, there's not even a starting point for understanding what this is about if you don't already know it, which is unfortunate when it is linked from non-math subjects. I'm not mathematically impaired, and can pick up the requisite knowledge if you provide a starting point, provided it is possible to recursively apply the process of following the links and absorbing their contents. A "gentle" introduction can ease this process by giving a context and some ideas of how to organize the knowledge. There are many pages that give a lot less context that still get a tag about insufficient context for those not familiar with the field. I haven't watchlisted this, so CCing my talkpage with any replies would be appreciated. Zuiram 04:25, 11 February 2007 (UTC)
- This is a fair point and thanks for making it. A lot of mathematics articles suffer from this problem at the moment, because there is so much work to do still. Partly because of this, editors tend to concentrate on "getting the math right" first. For principal bundles, it will be quite a challenge to make an accessible, yet concise, summary, but I hope someone takes it up. Geometry guy 21:14, 21 February 2007 (UTC)
