Talk:Section (fiber bundle)

Vector fields?

Suppose I have a particular vector field, ${\vec {v}}(x,y)$ over $R^{2}$ . Is that a section of the tangent bundle of $R^{2}$ ? —Ben FrantzDale 23:21, 28 April 2007 (UTC)[reply]

Yes. Vector fields are just sections of a vector bundle (usually the tangent bundle). -- Fropuff 01:32, 29 April 2007 (UTC)[reply]

Sweet. I'll add that to this page. Now sections make sense :-) —Ben FrantzDale 02:09, 29 April 2007 (UTC)[reply]

Regarding my edit: the set of all sections is not the bundle itself. Consider for example of a trivial line bundle over the real line. The space E is just R^2 and the base space is the x-axis. Sections of E are just continuous functions R → R represented as graphs. The space of all sections is just the space of all such functions, which is a different thing than the plane itself. -- Fropuff 02:28, 29 April 2007 (UTC)[reply]

Clarity of Obstructions in Global Section

I just edited the paragraph on global sections, explaining a little bit how global sections relate to obstructions which are represented by characteristic classes. It'd be nice if someone could review it, and if satisfactory remove the ^[why?] template.

—Pqnelson (talk) 20:35, 23 February 2012 (UTC)[reply]

That definitely helps, thank you. The language of "this leads to the study" is poor, because "this" hasn't been established. It's claiming but not describing a connection between the problem of obstructions and the motivation for sheaf cohomology. ᛭ LokiClock (talk) 06:57, 24 February 2012 (UTC)[reply]

Yes, I thought the same thing (it was like that when I got here and, being lazy, I left it). However, I tried clarifying that sheaf cohomology is a generalization of studying the local sections on a bundle, and characteristic classes are likewise generalizations of the notion of obstructions.

Additionally, I reworded the explanation of obstructions to make it clear that an obstruction obstructs extending a local section to a global section (which is why it is called an obstruction!). I think it works a little better now, although the sheaf cohomology stuff may need to be placed in a section "Generalizations".

—Pqnelson (talk) 15:38, 27 February 2012 (UTC)[reply]

Perhaps this sectioning will do. ᛭ LokiClock (talk) 21:15, 27 February 2012 (UTC)[reply]

I've tried editing the extension problem section, added the subsection on generalizations. I think I'm going to think for a few days before editing it any further...although I, personally, believe the section needs to be heavily re-worked. —Pqnelson (talk) 23:06, 27 February 2012 (UTC)[reply]