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)

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

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

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)

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 template.

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

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)

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)

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

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)