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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.
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)
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)
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)
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)
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.
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)
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)
For a start, the definition given in the article is wrong! An additional condition, namely that the local trivialization p^(-1) (U) -> U X V is required to induce a k-linear transformation between vector spaces (for a field k) when restricted to each fibre, is required. The other parts of the definition do not necessarily imply this! Something has to be done...
Topology Expert ( talk) 11:59, 2 October 2008 (UTC)
In the first picture, the transversal must be an open interval to be locally homeomorphic to the real line. With a closed interval, homeomorphic to the extended real line. -- kmath ( talk) 14:06, 25 May 2009 (UTC)
Maybe I'm getting confused, but a finite dimensional real vector space must have continuum cardinality. What if some fibres have more guys? Then a finite dimensional structure won't exist... Money is tight ( talk) 10:22, 24 May 2010 (UTC)
3. For every x∈X there exists a structure of a topological vector space on the fiber π−1({x}) so that the following compatibility condition is satisfied: around any point in X there is an open neighborhood U, a topological vector space V and a homeomorphism such that for every x∈U
Why the term "rank" rather than "dimension"? The latter seems more familiar when talking about vector spaces.. is there a reason why the different term is used here? Cesiumfrog ( talk) 11:00, 17 October 2010 (UTC)
There are problems with the operations section. Firstly, the notation is never defined (it clearly means the fiber space at x, but that's only clear if you already know about vector bundles, in which case you don't need this article...). Secondly we are in a context where both x and X mean something. When you see Ex it is not clear (because of the sans serif font) whether the subscript is big or little X; this makes the section potentially very confusing. They should be in math tags as and look totally different. I will fix this myself later when I have some time. Tinfoilcat ( talk) 18:24, 2 February 2012 (UTC)
I myself was using this page as learning material on vector bundles so I'm not sure where to go in order to fix what's in the article, but:
In the definition of a real vector bundle, the function pi is defined from E to X, so that pi^-1 is from X to E. But later, phi is defined from U x R^k to pi^-1(U), which means that U is an element of X. But U is a neighborhood of points from X, which leads to an infinite number of sets of sets of sets ... being members of X.
Like I said I don't know what is supposed to be written, I just see a problem. — Preceding unsigned comment added by 74.243.202.152 ( talk) 20:58, 31 August 2013 (UTC)
There should be a section discussing algebraic vector bundles, including the construction of vector bundles from locally free -modules. This should include the (co)tangent bundles and (co)normal bundles with explicit examples.
A non-trivial example could be useful - perhaps the Möbius bundle ( https://math.uchicago.edu/~chonoles/expository-notes/promys/promys2011-vectorbundles.pdf). Someone familiar with that should add it, or maybe I'll get around to it (if I can understand it...) Tomtheebomb ( talk) 01:23, 11 October 2017 (UTC)
Some documents require that must be a continuous surjection but in this definition, it's fine for the first condition to require that is a continuous map rather than a continuous surjection. If is not surjective, then for some but the second condition requires that is a vector space, i.e. it must have at least an element. Hence the second condition implies that is a surjection.
The definition in Atiyah's (1994) lecture note about vector bundle and K-theory doesn't require that is a surjection. Duybao20 ( talk) 08:57, 24 September 2022 (UTC)
![]() | This article is rated B-class on Wikipedia's
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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.
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)
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)
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)
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)
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.
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)
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)
For a start, the definition given in the article is wrong! An additional condition, namely that the local trivialization p^(-1) (U) -> U X V is required to induce a k-linear transformation between vector spaces (for a field k) when restricted to each fibre, is required. The other parts of the definition do not necessarily imply this! Something has to be done...
Topology Expert ( talk) 11:59, 2 October 2008 (UTC)
In the first picture, the transversal must be an open interval to be locally homeomorphic to the real line. With a closed interval, homeomorphic to the extended real line. -- kmath ( talk) 14:06, 25 May 2009 (UTC)
Maybe I'm getting confused, but a finite dimensional real vector space must have continuum cardinality. What if some fibres have more guys? Then a finite dimensional structure won't exist... Money is tight ( talk) 10:22, 24 May 2010 (UTC)
3. For every x∈X there exists a structure of a topological vector space on the fiber π−1({x}) so that the following compatibility condition is satisfied: around any point in X there is an open neighborhood U, a topological vector space V and a homeomorphism such that for every x∈U
Why the term "rank" rather than "dimension"? The latter seems more familiar when talking about vector spaces.. is there a reason why the different term is used here? Cesiumfrog ( talk) 11:00, 17 October 2010 (UTC)
There are problems with the operations section. Firstly, the notation is never defined (it clearly means the fiber space at x, but that's only clear if you already know about vector bundles, in which case you don't need this article...). Secondly we are in a context where both x and X mean something. When you see Ex it is not clear (because of the sans serif font) whether the subscript is big or little X; this makes the section potentially very confusing. They should be in math tags as and look totally different. I will fix this myself later when I have some time. Tinfoilcat ( talk) 18:24, 2 February 2012 (UTC)
I myself was using this page as learning material on vector bundles so I'm not sure where to go in order to fix what's in the article, but:
In the definition of a real vector bundle, the function pi is defined from E to X, so that pi^-1 is from X to E. But later, phi is defined from U x R^k to pi^-1(U), which means that U is an element of X. But U is a neighborhood of points from X, which leads to an infinite number of sets of sets of sets ... being members of X.
Like I said I don't know what is supposed to be written, I just see a problem. — Preceding unsigned comment added by 74.243.202.152 ( talk) 20:58, 31 August 2013 (UTC)
There should be a section discussing algebraic vector bundles, including the construction of vector bundles from locally free -modules. This should include the (co)tangent bundles and (co)normal bundles with explicit examples.
A non-trivial example could be useful - perhaps the Möbius bundle ( https://math.uchicago.edu/~chonoles/expository-notes/promys/promys2011-vectorbundles.pdf). Someone familiar with that should add it, or maybe I'll get around to it (if I can understand it...) Tomtheebomb ( talk) 01:23, 11 October 2017 (UTC)
Some documents require that must be a continuous surjection but in this definition, it's fine for the first condition to require that is a continuous map rather than a continuous surjection. If is not surjective, then for some but the second condition requires that is a vector space, i.e. it must have at least an element. Hence the second condition implies that is a surjection.
The definition in Atiyah's (1994) lecture note about vector bundle and K-theory doesn't require that is a surjection. Duybao20 ( talk) 08:57, 24 September 2022 (UTC)