## The connection as a choice of horizontal subspace?

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?

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 Recognitions: Gold Member Homework Help Science Advisor 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.

 Quote by quasar987 (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!

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