Principal bundle triviality, groups and connections

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In summary, if the principal connection is not trivial, then the principal bundle might not be trivial either.
  • #1
RockyMarciano
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The principal connection contrary to other connections like the affine connection has a tensorial character respect to the principal bundle, does thin mean that if the principal connection is not trivial it follows that the principal bundle isn't trivial either(unlike the case with affine connections and vector bundles)?

If this was the case when constructing a principal bundle starting by its fibre, would the structure of the group bundle(like being abelian or not) obstruct the possible base manifolds in the sense of not being compatible due to topological features like contractibility that automatically make a bundle trivial?
 
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  • #2
RockyMarciano said:
The principal connection contrary to other connections like the affine connection has a tensorial character respect to the principal bundle,
Do you mean a connection on a principal bundle? What do you mean by "tensorial character"?

does thin mean that if the principal connection is not trivial it follows that the principal bundle isn't trivial either(unlike the case with affine connections and vector bundles)?
What is a trivial connection?

If this was the case when constructing a principal bundle starting by its fibre, would the structure of the group bundle(like being abelian or not) obstruct the possible base manifolds in the sense of not being compatible due to topological features like contractibility that automatically make a bundle trivial?

"group bundle"? What is that? Not sure what you are asking here. Can you give an example?
 
  • #3
Thanks for answering.

lavinia said:
Do you mean a connection on a principal bundle?
Yes.
What do you mean by "tensorial character"?
That it doesn't depend on the choice of a frame because is defined on the fiber bundle rather than on the base manifold.
What is a trivial connection?
One with vanishing curvature form.
"group bundle"? What is that?
Sorry I meant to write the fiber, that in principal bundles is the Lie group.
Not sure what you are asking here.Can you give an example?
Ok. First, actually upon some further reading I realized the answer to the the first question is obviously negative, because the principal connections are constructed in such way that they cannot determine the triviality or nontriviality of the bundle in any direct way. Not to mention the fact that a principal bundle can have many different connections and curvature forms.
But maybe even if it is not the case that nontrivial connection can cause nontriviality of the principal bundle, maybe I can salvage the second question a bit.
I'll try and give some context. There is a theorem that says: "Given a fibre bundle with base space X and structure group G, if either X or G is contractible then the bundle is trivial."
And it ocurred to me that if the principal connection(aka Ehresman connection) generalizes the usual connection forms defined on the base manifold(and frame dependent) by being defined on the principal bundle itself, it might make sense that a situation could arise where the group structure in the fibre might restrict the possible base manifolds M in the principal G-bundle. Since a fiber bundle only demands the direct product locally, I think it make sense to consider also the discrete topology for the base manifolds. In this way when confronted with a nontrivial connection we could choose either to have a discrete manifold as base if we wanted to keep triviality at all costs or else a manifold with some topological feature like not being contractible that allowed the fiber bundle to be nontrivial in accordance with the topology of the group G.

I realize that in most situations(at least in physics) one uses a fiber bundle structure having in mind a specific base manifold(like a certain space or spacetime) so all this wouldn't apply in those cases. But I was thinking that sometimes one is only interested in certain local invariants like curvature that are realized in open set of a manifold and doesn't care about the manifold as a whole, and fiber bundles only care for the direct product of the fibre and the manifold locally.
Hope this is more readable and I have not ignored something important that makes my point useless.
 
  • #4
What you are saying is unclear to me.

Every group can be the fiber of a principal bundle over any space.
Every bundle over a discrete space is trivial.
A connection with zero curvatue 2-form is not trivial. It is called "flat".
 
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  • #5
lavinia said:
Every group can be the fiber of a principal bundle over any space.
Every bundle over a discrete space is trivial.
A connection with zero curvatue 2-form is not trivial. It is called "flat".
Agreed. The semantic thing with curvature is just that I've seen some physicists in the context of gauge group discussion referring to connections with curvature as non-trivial. But I don't think it is standard.

lavinia said:
What you are saying is unclear to me.
Ok. Thanks anyway. I'll try to make my questions clearer later here or in a new thread.
 
  • #6
On a product bundle "trivial connection " could just mean the projection onto the Lie algebra of the structure group. This is a flat connection but not all flat connections are trivial. As an exercise try constructing flat connections on the torus considered as an SO(2) bundle over the circle.
 
  • #7
For the tangent bundle a zero curvature tensor (of a Levi-Civita connection) does not imply that the bundle is trivial but it does imply that the structure group is finite. In fact,the structure group is equal to the holonomy group of the connection so if the structure group is not trivial niether is the connection. For instance the holonomy group of the flat Klein bottle is ##Z_2## and its tangent bundle is not trivial.
 
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What is a principal bundle?

A principal bundle is a mathematical structure that describes the relationship between a group and a space. It consists of a manifold (the base space) and a Lie group (the structure group) acting on it in a way that preserves its geometric structure. It is used to model symmetries and transformations in various fields such as physics and differential geometry.

What does it mean for a principal bundle to be trivial?

A principal bundle is said to be trivial if it is isomorphic to the product of its base space and structure group. In other words, the bundle can be "untwisted" or "unrolled" to a direct product of two spaces. This is often used as a simplifying assumption in calculations and proofs.

What is the role of groups in principal bundles?

Groups play a crucial role in principal bundles as they act on the base space to preserve its geometric structure. They also determine the type of bundle, as the structure group must be compatible with the base space for the bundle to exist. Different types of groups, such as Lie groups, can lead to different types of bundles with unique properties.

What are connections in principal bundles?

Connections in principal bundles are mathematical objects that describe how the structure group acts on the base space. They can be thought of as "glue" that connects the different fibers of the bundle. Connections are used to define parallel transport, which is important in differential geometry and gauge theories.

What are some applications of principal bundles?

Principal bundles have many applications in mathematics and physics. In mathematics, they are used in differential geometry to study curvature and in topology to classify manifolds. In physics, they are used to model symmetries and gauge theories, such as in the Standard Model of particle physics. They are also used in engineering and computer science for data analysis and machine learning.

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