The connection as a choice of horizontal subspace?

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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 V_p consists of vectors in T_pP which are also tangent to the fiber at p, and the horizontal space H_p is a set of vectors such that V_p+H_p=T_pP.
What I don't understand is why finding V_p doesn't uniquely specify H_p. It should be possible to construct T_pP without defining a connection, right? If so, wouldn't H_p just be every element of T_pP that is not also in V_p? I don't see how we are free to make this partition ourselves. Where am I going wrong?

Thanks for reading!

quasar987

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.

...?

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!