Talk:Vector bundle

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[edit] New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as vector bundles, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

This definition seems a bit weird. The definition should be of either smooth vector bundle or the general notion of vector bundle without restricting cases where the base space is a smooth manifold. The definition of smooth vector bundle includes restrictions on the transition functions too.

Why don't you try and rework it? Be bold! :) Dysprosia 12:05, 23 Oct 2003 (UTC)

The phrasing is too dense: hard to edit from here. In what sense is the fibre 'isomorphic' to V? What category is that in?

Charles Matthews 09:15, 29 Nov 2003 (UTC)

category of vector spaces.--MarSch 11:15, 29 January 2007 (UTC)

Also: what is a "smooth projection from a vector space to a manifold"? The term "smooth" only applies to maps between two smooth manifolds, and the term "projection" only applies to a map from a direct product to one of the factors. AxelBoldt 15:05, 4 Dec 2003 (UTC)

The bundle projection is a smooth map from the total space to the base space. For every point of the base space there exists a neighbourhood such that the bundle projection (restricted to the inverse image of this neighbourhood) is a projection. --MarSch 11:21, 29 January 2007 (UTC)

I'm likewise finding this rather vague. In this context, what's to 'attach' to a topological space, and what's 'compatible' as opposed to a vector space that is not? Sojourner001 21:35, 5 August 2006 (UTC)

pretty late but here is a reply. "attach" means that to each point x of X, we assign to it a vector space Vx. think of a smooth manifold, where each point has a tangent space. "compatibility" doesn't refer to a single Vx but all of the total space E. more precisely, it means that, given x, you can always take some small neighborhood of x such that, in their totality, the collection of vector spaces {Vx| x in U} looks just like a disjoint collection (topologically) in E. put another way, locally the way you attach Vx to x is the trivial one. ("trivial" here means there is no glueing involved and the topology is simply the product topology.) this may not be true if U is replaced by all of X. take again the example of smooth manifolds. depending on the manifold, the collection of tangent space may not have the disjoint topology when topologized in the natural way. Mct mht 06:43, 18 January 2007 (UTC)
I've tried to rewrite the introduction to clarify some of these points. Let me know what you think. Geometry guy 14:40, 11 February 2007 (UTC)

[edit] What else needs to be done?

I've made quite a few edits to this article today, partly to link up with the (tidied-up) pullback bundle and (new) bundle map articles, but also to clarify a few points, such as bundle isomorphism and trivialization. There is clearly still work to be done here though. For instance the section on "Operations on vector bundles" needs to be expanded, and this is probably next on my agenda, unless someone else does it before me! Anything else on your wishlist for this article? Geometry guy 14:48, 11 February 2007 (UTC)

Dual bundle should be mentioned somewhere. Tensor product bundle still needs to be written, but it wouldn't hurt to provide a redlink. At some point it would be nice to mention the connection between vector bundles and their associated principal (frame) bundles, but this isn't the most pressing issue. -- Fropuff 16:56, 11 February 2007 (UTC)
Dual bundles and tensor product bundles are mentioned very briefly in vector bundle. This is the section I think should be expanded. The relation with frame bundles is another issue, and I agree it should be covered somewhere. Geometry guy 00:02, 13 February 2007 (UTC)

The section on operations on vector bundles has now been expanded. Please leave your comments and suggestions here. Geometry guy 00:11, 1 March 2007 (UTC)


I am not sure the definition of vector bundle morphism is correct here. The one given in Milnor's characteristic classes is such that there exists a morphism if and only if the domain is the pullback of the range. The amazing power of vector bundles as a tool flows from their ability to be classified using homotopy theory, there is not a hint of that here. Finally, characteristic classes are inseparable from the theory of vector bundles and there needs to be some hint of that.

Topoman

You raise a valid point. The definition of vector bundle morphism given here is correct (from my point of view). But it's also in some sense the weakest definition availiable, and generally not a very well-behaved one. The image of a vector bundle may fail to be locally trivial, for instance. I've seen the "pullback of the image" version elsewhere in topology as well (e.g., in Steenrod's Topology of Fibre Bundles). In this case, the image of a vector bundle is a vector bundle. There appears to be a distinction between the category of differentiable vector bundles (including the smooth ones), and the topological vector bundles. In the former, one can at least impose some rank conditions on the morphisms under consideration, but in the latter that is no longer possible. Silly rabbit 11:29, 23 April 2007 (UTC)

[edit] Bundles vs. Sections

The article says "Tangent bundles are not, in general, trivial bundles: for example, the tangent bundle of the (two dimensional) sphere is not trivial by the Hairy ball theorem." But doesn't the hairy ball theorem talk about tangent-vector fields (sections of the tangent vector bundle), not tangent vector bundles themselves? I see how the hairy ball theorem applies to vector fields, but how does it imply that the tangent bundle of a 2-D sphere is nontrivial? —Ben FrantzDale 03:07, 29 April 2007 (UTC)

On a related note, when would you talk about a vector bundle without immediately starting to talk about a vector field? —Ben FrantzDale 03:07, 29 April 2007 (UTC)

The hairy ball theorem says every vector field on the 2-sphere vanishes somewhere. Even though it talks about vector fields, this does actually tell you something about the tangent bundle. You see, a trivial bundle always has nowhere vanishing sections (even constant sections), therefore the tangent bundle of the 2-sphere can't possibly be trivial (that would contradict the hairy ball theorem). -- Fropuff 06:20, 29 April 2007 (UTC)
OK. At Fiber_bundle#Trivial_bundle, I don't see mention of this fact and hairy ball theorem also does not mention trivial bundles. Is that fact mentioned anywhere on Wikipedia? Thanks. —Ben FrantzDale 06:29, 29 April 2007 (UTC)
Well it's mentioned in passing at tangent bundle, but only briefly. In fairness the article on sections only got started very recently. It is still in a very stubby state with much left unsaid. The article on the hairy ball theorem should really mention this fact somewhere too. If I find the time I'll try to incorporate it. -- Fropuff 16:28, 29 April 2007 (UTC)
Fairness is good. I am just using my learning process as an opportunity to find and fix holes in Wikipedia's coverage of the topic. Thanks for your help. —Ben FrantzDale 16:47, 29 April 2007 (UTC)

[edit] Metrics in vector bundles

Anyone know where I can link for a metric in a vector bundle? Metric tensor seems to be populated exclusively with metrics in the tangent (and other tensor) bundles. Silly rabbit 03:30, 22 May 2007 (UTC)

You, sir, are an idiot. Please look in the most obvious place: metric (vector bundle), an article I myself wrote over a month ago! Silly rabbit 16:48, 31 May 2007 (UTC)