The connection as a choice of horizontal subspace?

In summary, the conversation discusses the concept of a connection on a principle bundle and the partitioning of the tangent space into vertical and horizontal spaces. The speaker is stuck on why finding the vertical space does not uniquely determine the horizontal space. They also discuss how to construct the horizontal space and clarify misunderstandings about vector subspaces.
  • #1
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Hi,
I'm trying to understand the fiber bundle formulation of gauge theory at the moment, and I'm stuck on the connection. Every reference I've found introduces the idea of a connection on a principle bundle as a kind of partitioning of the tangent space at all points in the total space into a "vertical space" and a "horizontal space". The vertical space Vp consists of vectors in TpP which are also tangent to the fiber at p, and the horizontal space Hp is a set of vectors such that Vp+Hp=TpP.
What I don't understand is why finding Vp doesn't uniquely specify Hp. It should be possible to construct TpP without defining a connection, right? If so, wouldn't Hp just be every element of TpP that is not also in Vp? I don't see how we are free to make this partition ourselves. Where am I going wrong?

Thanks for reading!
 
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  • #2
Take R² for instance, and for simplicity, assume that V = {(0,y) | y in R}. Then you're saying "take H:= R² - V". But that's not a subspace! (Perhaps you overlooked the fact that H is supposed to be a vector subspace?)

On the other hand, H:={(x,0) | x in R} is a natural candidate... but there are (infinitely many) other choice as H:={ (x,ax) | x in R} for any a in R would do just as well.
 
  • #3
quasar987 said:
(Perhaps you overlooked the fact that H is supposed to be a vector subspace?)
Yeah, I did. I was also getting mixed up over how to take the direct sum of two vector spaces. But now I see how it all works. Cheers!
 

1. What is meant by "the connection as a choice of horizontal subspace"?

The connection as a choice of horizontal subspace refers to the mathematical concept of choosing a particular subspace, or subset, of a larger space to represent the horizontal directions in a given system. In other words, it is a way of selecting a specific set of directions to be considered as the "flat" or "horizontal" directions within a larger space.

2. What is the purpose of choosing a horizontal subspace in a connection?

The purpose of choosing a horizontal subspace in a connection is to define a specific set of directions that are considered to be "flat" or "horizontal" within a given system. This allows for a clearer understanding and analysis of the system, as well as the ability to perform calculations and make predictions based on the chosen subspace.

3. How is a horizontal subspace chosen in a connection?

The choice of a horizontal subspace in a connection is typically made by the scientist or mathematician studying the system. This choice is based on the specific needs and goals of the analysis, as well as the properties and characteristics of the system itself.

4. What are the advantages of using a horizontal subspace in a connection?

There are several advantages to using a horizontal subspace in a connection. Firstly, it allows for a better understanding and analysis of the system, as it provides a specific set of directions to focus on. Additionally, it allows for easier calculations and predictions to be made, as the chosen subspace is often simpler and more manageable than the entire system.

5. Are there any limitations to using a horizontal subspace in a connection?

While using a horizontal subspace in a connection can be beneficial, there are some limitations to consider. One limitation is that the choice of subspace may not accurately represent the entire system, leading to potential errors in analysis and predictions. Additionally, the complexity of the system may make it difficult to determine a suitable horizontal subspace, making the process more challenging.

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