Are vector bundles principal bundles?

In summary, The conversation discusses the differences between principal bundles and vector bundles, and the fact that not all vector bundles are principal bundles. The Möbius strip is an example of a vector bundle that is not a principal bundle, as it does not have a global section and its symmetry group does not match the structure group. Some sources use different terminology for principal bundles, but in most cases, principal bundles, principal fiber bundles, and principal G-bundles all refer to the same thing. The Möbius strip is an R-bundle over the circle, with a local R-action on its two trivializations. However, it does not have a consistent global G-action.
  • #1
mma
245
1
As far as I know, principal bundle is a fiber bundle with a fiber beeing a principal homogeneous space (or a topological group). According this definition vector bundle is a special principal bundle, because vector space with vector addition as group operation is a topological group.
But I feel that something is wrong with this, because there is a theorem that a principal bundle is trivial if and only if it has a global section (see C. Nash, S. Sen: Topology and Geometry for Physicists, p. 152). Möbius strip (regarded as vector bundle) has a global section (the identically 0 section) but it isn't trivial. Where is the error?
 
Physics news on Phys.org
  • #2
mma said:
Möbius strip (regarded as vector bundle) has a global section (the identically 0 section) but it isn't trivial. Where is the error?
Are you sure the Möbius strip is an R-bundle over the circle? How does R act on it?
 
  • #3
A key element of the definition of principal fibre bundles is that the transition mappings of a principal fibre bundle are left multiplications of elements from the group G. For a vector space viewed as an additive group, these are translations. But the transition mappings of a vector bundle are contained in the general linear group of the relevant dimension and in fact will *never* be nontrivial translations.

So, nontrivial vector bundles are not principal fibre bundles. Although, of course, every vector bundle induces a principal Gl(n,R)-fibre bundle consisting of its fibre-wise endomorphisms.
 
  • #4
Doodle Bob said:
A key element of the definition of principal fibre bundles is that the transition mappings of a principal fibre bundle are left multiplications of elements from the group G.

Yes, this is what I forgot. In other words, in the case of principal bundles, G (the symmetry group of the fiber) must be the structural group of the fiber bundle, and cannot be an arbitrary group. The Möbius vector bundle doesn't comply this. But for example the cylinder does. Thank you!
 
  • #5
I think that my trouble caused that some sources make a difference between principal bundles and principal G-bundles. (e.g. http://en.wikipedia.org/wiki/Principal_bundle" [Broken]), while others call principal bundle only principal G-bundles.
The equality of the symmetry group of the fiber and the structure group of the fiber bundle is required only in the case of principal G-bundles, but for the more general general principal bundles isn't. My book talks about principal bundles but it means principal G-bundles. The theorem cited in the first post is valid only for principal G-bundles.

So, I think that the correct answer to my question is that vector bundles are all principal bundles, but not all of them are principal G-bundles.
 
Last edited by a moderator:
  • #6
mma said:
My book talks about principal bundles but it means principal G-bundles. The theorem cited in the first post is valid only for principal G-bundles.

I would say that in the vast majority of the literature the following phrases mean the same thing: principal bundles, principal fibre bundles, principal G-bundles. So, it's best to assume that this is the case when reading a refereed paper or text.

Wikipedia is famous for not cohering very well with the current state of mathematical jargon. My favorite is "polychoron" which is supposedly the term for 4-dimensional polytopes. But a quick search of the literature (in MathSciNet) reveals that not a single refereed paper uses it.
 
  • #7
This reminds somewhat of the fact that parallelizability of a manifold is equivalent to the existence of a global section in the frame bundle over the manifold. The frame bundle is a principal bundle with group GL(n).

For a one-dimensional manifold the frame bundle and tangent bundle are the same.
 
  • #8
The cylinder and the Möbius strip are principal bundles with group [itex](\mathbf{R},+)[/itex]. They are also vector bundles. The cylinder and the Möbius strip with the zero section removed are principal bundles with group [itex](\mathbf{R}\backslash 0,\cdot)[/itex]. The cylinder without the zero section is the frame bundle of the circle.
 
  • #9
OrderOfThings said:
The cylinder and the Möbius strip are principal bundles with group [itex](\mathbf{R},+)[/itex].
Are you sure the Möbius strip is an R-bundle over the circle? How does R act on it?
 
  • #10
Hurkyl said:
Are you sure the Möbius strip is an R-bundle over the circle? How does R act on it?

Erh, well,... :rolleyes:
 
  • #11
Hurkyl said:
Are you sure the Möbius strip is an R-bundle over the circle? How does R act on it?

I'm not sure that I understand the concern here. Clearly, it's not the actual bounded Moebius strip that is being discussed but it's unbounded cousin, which can be defined piecewise over the circle by letting U and V be open sets on S^1 such that their intersection is two disconnected arcs, setting U'=UxR and V'=VxR, defining an equivalence relation on the disconnected union of U' and V' by: For u' in U' and v' in V', u'~v' iff u'=(x,t) and v'=(x,-t) (and otherwise an element is equivalent just to itself), then setting M=(U' union V')/~.

This both defines the unbounded Moebius band and gives it a vector bundle structure over S^1. We have a local R-actions on both trivializations with a certain amount of compatibility, which is all that we would want anyway. I don't think that there has to be a *global* G-action on a principal fibre bundle, only a very well-behaved local action. Similarly, there isn't necessarily a global Gl(V)-action on a vector bundle with fibres isomorphic to V.
 
  • #12
Doodle Bob said:
We have a local R-actions on both trivializations with a certain amount of compatibility,
The notion of a global G action on a bundle over a manifold M is equivalent to the notion of a local action by the trivial bundle GxM on that same bundle.

A G-bundle is not "a bundle upon which one can locally define an action of G, and it's not even "a bundle upon which one can globally define an action of G"... a G-bundle is "a bundle for which we have chosen a specific action of G"... and the Möbius strip does not have a global G action. Nor can you define a consistent 'local' G-action -- as you piece together the local G-actions, you will either have to run into a discontinuity someplace, or a fiber where the action is trivial (which violates the condition that G acts freely and transitively)


Now, what we do have is that Möbius strip can act on itself by addition. (For comparison, it doesn't have a multiplicative action on itself! The natural multiplication operation for sections of the Möbius strip actually has its values in the cylinder.
 
Last edited:
  • #13
Sorry, Hurkyl, I was getting confused over the terminology. When I say "R-bundle", I was thinking of R as a 1-diml. vector space and thus was trying to describe the Moebius strip as a *vector* bundle over S^1, which is what I did. This must be one of the reasons for the "principal" in the phrase "principal fibre bundle."
 
  • #14
The relevant part is being a G-bundle -- that terminology (like G-set or G-module) is used for a bundle together with an action by the group G. The "principal" part just adds a bit more about the behavior of the action.

Yes, I agree that the Möbius strip is, in fact, a vector bundle over R.
 
  • #15
Hurkyl said:
The relevant part is being a G-bundle -- that terminology (like G-set or G-module) is used for a bundle together with an action by the group G. The "principal" part just adds a bit more about the behavior of the action.

It's that damn R, which can mean anything from a vector space to a group to a field to just a simple 1-diml. manifold -- I've read many a paper referring to "S^2-bundles," which of course do not have a global S^2-action on them.
 

1. What is a vector bundle?

A vector bundle is a type of mathematical structure that combines the concepts of a topological space and a vector space. It is a collection of vector spaces attached to each point in a topological space in a consistent manner.

2. What is a principal bundle?

A principal bundle is a mathematical structure that is used to describe the relationship between a locally symmetric space and a group. It is a type of fiber bundle where the fibers are copies of a group and the base space is a locally symmetric space.

3. How are vector bundles and principal bundles related?

Vector bundles can be thought of as a special case of principal bundles, where the group is the general linear group of the vector space. In other words, vector bundles are a type of principal bundle with a specific choice of group.

4. What are the main differences between vector bundles and principal bundles?

The main difference between vector bundles and principal bundles lies in their group structure. In vector bundles, the group is always the general linear group of the vector space, while in principal bundles, the group can vary and is often chosen to describe the symmetries of the underlying space.

5. Are all vector bundles principal bundles?

No, not all vector bundles are principal bundles. As mentioned earlier, vector bundles are a special case of principal bundles where the group is the general linear group of the vector space. However, there are many other types of principal bundles that cannot be described as vector bundles.

Similar threads

  • Differential Geometry
Replies
6
Views
2K
  • Differential Geometry
Replies
9
Views
423
Replies
3
Views
2K
  • Differential Geometry
Replies
15
Views
3K
  • Differential Geometry
Replies
11
Views
3K
  • Differential Geometry
Replies
11
Views
3K
  • Special and General Relativity
Replies
29
Views
1K
Replies
6
Views
879
Replies
1
Views
3K
Replies
16
Views
3K
Back
Top