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 V_x. think of a smooth manifold, where each point has a tangent space. "compatibility" doesn't refer to a single V_x 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 {V_x| x in U} looks just like a disjoint collection (topologically) in E. put another way, locally the way you attach V_x 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)